diff --git "a/random/random_our/pretrain_eval_data.jsonl" "b/random/random_our/pretrain_eval_data.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/random_our/pretrain_eval_data.jsonl"
@@ -0,0 +1,13660 @@
+{"state": {"context": ["V : Type u_1", "V₁ : Type u_2", "V₂ : Type u_3", "V₃ : Type u_4", "inst✝³ : SeminormedAddCommGroup V", "inst✝² : SeminormedAddCommGroup V₁", "inst✝¹ : SeminormedAddCommGroup V₂", "inst✝ : SeminormedAddCommGroup V₃", "f✝ g✝ f : NormedAddGroupHom V₁ V₂", "K : AddSubgroup V₂", "C C' : ℝ", "h : f.SurjectiveOnWith K C", "H : C ≤ C'", "g : V₁", "k_in : f g ∈ K", "hg : ‖g‖ ≤ C * ‖f g‖"], "goal": "‖g‖ ≤ C' * ‖f g‖"}}
+{"state": {"context": ["α : Type u", "Ring α", "a b : α", "u : αˣ"], "goal": "a - b /ₚ u = (a * ↑u - b) /ₚ u"}}
+{"state": {"context": ["α : Type u", "Mul α"], "goal": "WithOne.map (MulHom.id α) = MonoidHom.id (WithOne α)"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "σ : Type u_1", "a : R", "s : σ →₀ ℕ", "CommSemiring R", "CommSemiring S₁", "f : R →+* S₁", "g : σ → S₁"], "goal": "MvPolynomial.eval₂ f g ((MvPolynomial.monomial s) a) = f a * s.prod fun n e => g n ^ e"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "n : ℕ", "hzero : Tendsto (fun x_1 => -↑x⁻¹ * x_1) (𝓝 0) (𝓝 0)", "t : R", "ht : inverse (↑x + t) = inverse (1 + ↑x⁻¹ * t) * ↑x⁻¹", "ht' : inverse (1 - -↑x⁻¹ * t) = ∑ i ∈ range n, (-↑x⁻¹ * t) ^ i + (-↑x⁻¹ * t) ^ n * inverse (1 - -↑x⁻¹ * t)"], "goal": "inverse (↑x + t) = (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * inverse (↑x + t)"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "m1 m2 m : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "inst✝ : IsMarkovKernel κ", "p1 p2 : Set (Set Ω)", "h1 : m1 ≤ m", "h2 : m2 ≤ m", "hp1 : IsPiSystem p1", "hp2 : IsPiSystem p2", "hpm1 : m1 = generateFrom p1", "hpm2 : m2 = generateFrom p2", "hyp : IndepSets p1 p2 κ μ", "t1 t2 : Set Ω", "ht1 : t1 ∈ {s | MeasurableSet s}", "ht2 : t2 ∈ {s | MeasurableSet s}"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : MonoidalCategory C", "W X Y Z : C"], "goal": "(α_ W (X ⊗ Y) Z).inv ≫ ((α_ W X Y).inv ⊗ 𝟙 Z) ≫ (α_ (W ⊗ X) Y Z).hom = (𝟙 W ⊗ (α_ X Y Z).hom) ≫ (α_ W X (Y ⊗ Z)).inv"}}
+{"state": {"context": ["k : Type u_1", "E : Type u_2", "PE : Type u_3", "inst✝³ : Field k", "inst✝² : AddCommGroup E", "inst✝¹ : Module k E", "inst✝ : AddTorsor E PE", "f : k → E", "x y : k"], "goal": "slope (-f) x y = (-slope f) x y"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "n : ℕ", "b : Basis (Fin n) R M", "k : Fin (n + 1)", "l : Fin n", "h : k ≤ l.castSucc"], "goal": "b.flag k ≤ LinearMap.ker (b.coord l)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "Module R M", "Module R N", "Module R P", "Module R Q", "g : P ≃ₗ[R] Q", "f : N ≃ₗ[R] P"], "goal": "LinearEquiv.lTensor M (f ≪≫ₗ g) = LinearEquiv.lTensor M f ≪≫ₗ LinearEquiv.lTensor M g"}}
+{"state": {"context": [], "goal": "Fintype.card Bool = 2"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Sub α", "s t₁ t₂ : Finset α"], "goal": "s - (t₁ ∪ t₂) = s - t₁ ∪ (s - t₂)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "v : α → M"], "goal": "LinearMap.range (Finsupp.total α M R v) = Submodule.span R (Set.range v)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x y : X", "e : x = y"], "goal": "x ⤳ y"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "R : Type u_2", "inst✝¹ : Semiring k", "inst✝ : Monoid G", "a : G", "b : k"], "goal": "1 = single 1 1"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α", "f : ℕ → α → α", "a : α", "hf : Primrec₂ f"], "goal": "Primrec fun t => Nat.rec a f t"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N : Bimod Y Z"], "goal": "((α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) ((α_ M.X Y.X (TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)).hom ≫ M.X ◁ TensorBimod.actLeft { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) ((α_ M.X Y.X (TensorBimod.X { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N)).hom ≫ M.X ◁ TensorBimod.actLeft { X := Y.X, actLeft := Y.mul, one_actLeft := ⋯, left_assoc := ⋯, actRight := Y.mul, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } N) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ LeftUnitorBimod.hom N) (M.X ◁ LeftUnitorBimod.hom N) ⋯ ⋯) = coequalizer.π (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul) ▷ N.X ≫ coequalizer.desc M.actRight ⋯ ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one"}}
+{"state": {"context": ["α : Type u_2", "A : Set α", "S : Set (Set α)"], "goal": "Subtype.val ⁻¹' ⋃₀ S = ⋃₀ {x | ∃ B ∈ S, Subtype.val ⁻¹' B = x}"}}
+{"state": {"context": ["X Y : CommGrp", "f : X ⟶ Y"], "goal": "CommGrp.uliftFunctor.map f =\n  CommGrp.ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom))"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "N N' : LieSubmodule R L M", "x : M"], "goal": "x ∈ N ⊓ N' ↔ x ∈ N ∧ x ∈ N'"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "f g : α → E"], "goal": "MeasureTheory.eLpNormEssSup (f + g) μ ≤ MeasureTheory.eLpNormEssSup f μ + MeasureTheory.eLpNormEssSup g μ"}}
+{"state": {"context": ["a b : ℝ"], "goal": "∫ (x : ℝ) in a..b, Real.cos x ^ 2 = (Real.cos b * Real.sin b - Real.cos a * Real.sin a + b - a) / 2"}}
+{"state": {"context": ["α : Type u", "β✝ : Type u_1", "t : TopologicalSpace α", "B : Set (Set α)", "s✝ : Set α", "β : Type u_2", "inst✝ : TopologicalSpace β", "s : Set α", "f : α → β", "hf : Continuous f", "c : Set α", "c_count : c.Countable", "hc : s ⊆ closure c"], "goal": "IsSeparable (f '' s)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "p : ℝ≥0∞", "NormedAddCommGroup F", "MeasurableSpace α", "f : α → F"], "goal": "MeasureTheory.eLpNorm f p 0 = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "Preorder β", "Preorder γ", "f : α → β", "s : Set α", "a : α", "hf : IsMinOn f s a", "g : β → γ", "hg : Antitone g"], "goal": "IsMaxOn (g ∘ f) s a"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y Z : Cᴹᵒᵖ"], "goal": "(CategoryTheory.MonoidalCategory.associator X Y Z).hom.unmop =\n  (CategoryTheory.MonoidalCategory.associator Z.unmop Y.unmop X.unmop).inv"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝ : MeasurableSpace β", "μ✝ ν ν₁ ν₂ : Measure α", "s✝ t : Set α", "μ : Measure α", "h this : SigmaFinite μ", "s : ℕ → Set α := spanningSets μ", "hs_univ : ⋃ i, s i = univ", "hs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ"], "goal": "μ univ < ⊤"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "S : Set α", "hS : BooleanGenerators S", "T₁ T₂ : Set α", "hT₁ : T₁ ⊆ S", "hT₂ : T₂ ⊆ S", "X : Set α", "hX : X ⊆ S", "hX' : sSup T₁ ⊓ sSup T₂ = sSup X", "I : α", "hI : I ∈ X"], "goal": "I ∈ T₁ ∩ T₂"}}
+{"state": {"context": ["n : ℕ∞"], "goal": "1 ≤ ↑n ↔ 1 ≤ n"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "α : Type u'", "l : ℕ", "θ : L.BoundedFormula α l.succ", "v : α → M", "xs : Fin l → M"], "goal": "θ.all.Realize v xs ↔ ∀ (a : M), θ.Realize v (Fin.snoc xs a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : αᵒᵈ → βᵒᵈ", "a : WithTop α"], "goal": "WithBot.map f (WithTop.toDual a) = WithTop.map (⇑OrderDual.toDual ∘ f) a"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ", "hf : LSeriesSummable f s", "hg : LSeriesSummable g s"], "goal": "LSeriesSummable (f ⍟ g) s"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝⁴ : TopologicalSpace α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "inst✝¹ : RCLike 𝕜", "μ : Measure α", "inst✝ : IsFiniteMeasure μ", "f g : α →ᵇ 𝕜", "hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᶠ[ae μ] ⇑f"], "goal": "(fun a => ⟪↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a, ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a⟫_𝕜) =ᶠ[ae μ] fun a => (starRingEnd 𝕜) (f a) * g a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "𝔖 : Set (Set α)", "h : s ∈ 𝔖"], "goal": "⨅ i ∈ 𝔖, 𝓤 (α →ᵤ[𝔖] β) ≤ comap (fun x => ((⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖)) x.1, (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖)) x.2)) (𝓤 (↑s →ᵤ β))"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "∀ (d : D), CategoryTheory.IsFiltered (CategoryTheory.StructuredArrow d F)"], "goal": "F.Final"}}
+{"state": {"context": ["a : USize", "b : Nat"], "goal": "(a.modn b).toNat = a.toNat % b"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "x y : E"], "goal": "⟪x, y⟫_𝕜 = (↑‖x + y‖ ^ 2 - ↑‖x - y‖ ^ 2 + (↑‖x - RCLike.I • y‖ ^ 2 - ↑‖x + RCLike.I • y‖ ^ 2) * RCLike.I) / 4"}}
+{"state": {"context": ["K : Type v", "Field K", "p : K[X]"], "goal": "Polynomial.Splits (RingHom.id K) p ↔ Multiset.card p.roots = p.natDegree"}}
+{"state": {"context": [], "goal": "tan (π / 6) = 1 / √3"}}
+{"state": {"context": [], "goal": "Int.bitwise or = Int.lor"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "f : α → Option α", "a b : α", "l : List α", "h : f a = some b"], "goal": "(match f a with | some b => b :: l | none => lookmap.go f l (#[].push a)) = b :: l"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P", "h : ∡ p₁ p₂ p₃ = ↑(π / 2)"], "goal": "dist p₃ p₂ / (∡ p₃ p₁ p₂).sin = dist p₁ p₃"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹² : CommRing R", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R M", "I : Ideal R", "S : Type u_3", "inst✝⁹ : CommSemiring S", "inst✝⁸ : Algebra S R", "inst✝⁷ : Module S M", "inst✝⁶ : IsScalarTower S R M", "R' : Type u_4", "M' : Type u_5", "inst✝⁵ : CommRing R'", "inst✝⁴ : AddCommGroup M'", "inst✝³ : Module R' M'", "inst✝² : Algebra R R'", "inst✝¹ : Module R M'", "inst✝ : IsScalarTower R R' M'", "p : R[X]", "q : PolynomialModule R M", "r : R"], "goal": "(eval r ∘ₗ comp p) q = (eval (Polynomial.eval r p)) q"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "f : α → α", "s✝ : Set α", "μ : Measure α", "β : Type u_2", "m' : MeasurableSpace β", "μ' : Measure β", "s' : Set β", "g : α → β", "hg : MeasurePreserving g μ μ'", "hf : PreErgodic f μ", "f' : β → β", "h_comm : g ∘ f = f' ∘ g", "s : Set β", "hs₀ : MeasurableSet s", "hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s", "hs₂ : g ⁻¹' s =ᶠ[ae μ] ∅"], "goal": "s =ᶠ[ae μ'] ∅"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommRing R", "M : Submonoid R", "S : Type u_2", "inst✝² : CommRing S", "inst✝¹ : Algebra R S", "P : Type u_3", "inst✝ : CommRing P", "a c : R", "b d : ↥M"], "goal": "mk a b - mk c d = mk (↑d * a - ↑b * c) (b * d)"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "p : F[X]", "hFE : FiniteDimensional F E", "sp : Polynomial.IsSplittingField F E p", "hp : p.Separable", "K : Type u_3", "Field K", "Algebra F K", "Algebra K E", "IsScalarTower F K E", "x : E", "hx : x ∈ p.aroots E", "Fintype (K →ₐ[F] E)", "Fintype (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E)"], "goal": "Fintype.card (↥(IntermediateField.restrictScalars F K⟮x⟯) →ₐ[F] E) =\n  Fintype.card (K →ₐ[F] E) * FiniteDimensional.finrank K ↥K⟮x⟯"}}
+{"state": {"context": [], "goal": "Continuous Complex.re"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "𝕜 : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedSpace 𝕜 E", "G : Type u_4", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "x : E", "inst✝ : l.NeBot", "hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)", "hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2", "hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))", "this : (fun a => (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) = ((fun a => (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a => (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) + fun a => (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)"], "goal": "Tendsto (((fun a => (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a => (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) + fun a => (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)) (l.curry (𝓝 x)) (𝓝 0)"}}
+{"state": {"context": ["f g : ℝ → ℝ", "f' g' x y p : ℝ", "s : Set ℝ", "hf : HasDerivAt f f' x", "hg : HasDerivAt g g' x", "h : 0 < f x"], "goal": "HasDerivAt (fun x => f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * Real.log (f x)) x"}}
+{"state": {"context": ["α : Type u_1", "CommGroup α", "LinearOrder α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b : α"], "goal": "a / b ≤ b / a ↔ a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "s t : Set α", "x : α", "f : α ≃o β", "x✝ : β"], "goal": "x✝ ∈ ↑(upperClosure (⇑f '' s)) ↔ x✝ ∈ ↑((UpperSet.map f.symm).symm (upperClosure s))"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K L M : HomologicalComplex C c", "φ : K ⟶ L", "ψ : L ⟶ M", "i j✝ k✝ : ι", "inst✝ : K.HasHomology i", "A' A : C", "k : A ⟶ K.X i", "j : ι", "hj : c.next i = j", "hk : k ≫ K.d i j = 0", "α : A' ⟶ A"], "goal": "α ≫ K.liftCycles k j hj hk = K.liftCycles (α ≫ k) j hj ⋯"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "n : ℕ"], "goal": "Set.image f^[n] = (Set.image f)^[n]"}}
+{"state": {"context": ["P : TopCat → Prop", "self : CompHausLike.HasExplicitFiniteCoproducts P", "α : Type w", "Finite α", "X : α → CompHausLike P"], "goal": "CompHausLike.HasExplicitFiniteCoproduct X"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f : X ⟶ Y", "c : CategoryTheory.Limits.Fork f f", "h : CategoryTheory.Limits.IsLimit c"], "goal": "CategoryTheory.IsIso c.ι"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : CommMonoidWithZero α", "p : α", "hp✝ hp : Prime p", "a : α", "n : ℕ", "ih : p ∣ a ^ n → p ∣ a", "h : p ∣ a * a ^ n"], "goal": "p ∣ a"}}
+{"state": {"context": ["α : Type u", "lsize rsize : Nat", "self : Lean.Diff.Histogram.Entry α lsize rsize"], "goal": "self.rightCount = 0 ↔ self.rightIndex = none"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "inst✝³ : MeasurableSpace α", "μ : Measure α", "l : Filter ι", "inst✝² : NormedAddCommGroup E", "inst✝¹ : l.NeBot", "inst✝ : l.IsCountablyGenerated", "φ : ι → Set α", "hφ : AECover μ l φ", "f : α → E", "I : ℝ", "hfi : ∀ (i : ι), IntegrableOn f (φ i) μ", "hbounded : ∀ᶠ (i : ι) in l, (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I", "hfm : AEStronglyMeasurable f μ", "i : ι", "hi : (∫⁻ (a : α) in φ i, ↑‖f a‖₊ ∂μ).toReal ≤ I"], "goal": "∫⁻ (x : α) in φ i, ↑‖f x‖₊ ∂μ ≤ ENNReal.ofReal I"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : RCLike 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : TopologicalSpace E", "inst✝² : Module 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "A B : E →WOT[𝕜]F", "h : ∀ (x : E) (y : NormedSpace.Dual 𝕜 F), y (A x) = y (B x)"], "goal": "∀ (x : E), A x = B x"}}
+{"state": {"context": ["α : Type u_1", "G H : SimpleGraph α", "h : G ≤ H"], "goal": "G.cliqueSet ≤ H.cliqueSet"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁸ : Semiring R", "inst✝⁷ : Semiring R₂", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup M₂", "inst✝⁴ : Module R M", "inst✝³ : Module R₂ M₂", "τ₁₂ : R →+* R₂", "inst✝² : RingHomSurjective τ₁₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F τ₁₂ M M₂", "f : F", "p p' : Submodule R M"], "goal": "map f p ≤ map f p' ↔ p ≤ p' ⊔ ker f"}}
+{"state": {"context": ["α : Sort u", "p : α → Prop", "w : α", "h₁ : p w", "h₂ : ∀ (y : α), p y → y = w"], "goal": "∃! x, p x"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DistribLattice α", "inst✝ : BoundedOrder α", "a b x y z x' y' : α", "h : IsCompl x y", "h' : IsCompl x' y'"], "goal": "(x ⊔ x') ⊓ (y ⊓ y') = ⊥"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : TopologicalSpace 𝕜", "inst✝ : OrderTopology 𝕜", "p q : 𝕜", "hp : Fact (0 < p)", "n : ℕ", "hn : n > 0"], "goal": "Nat.card { u // addOrderOf u = n } = n.totient"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "W.HasLeftCalculusOfFractions", "X Y Z : C", "z₁ : W.LeftFraction X Y", "z₂ : W.LeftFraction Y Z"], "goal": "(CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.mk z₁).comp\n    (CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.mk z₂) =\n  z₁.comp z₂"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : EuclideanDomain R", "inst✝ : DecidableEq R", "s : R"], "goal": "(xgcd 0 s).2 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s : Finset α", "β' : Type u_4", "f : β ↪ γ", "g : α ↪ β", "f' : α ↪ β'", "g' : β' ↪ γ", "h_comm : ∀ (a : α), f (g a) = g' (f' a)"], "goal": "Finset.map f (Finset.map g s) = Finset.map g' (Finset.map f' s)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝ : PseudoEMetricSpace α", "f g : α → ℝ", "Kf Kg : ℝ≥0", "hf : LipschitzWith Kf f", "a : ℝ"], "goal": "LipschitzWith Kf fun x => max a (f x)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : Type u'", "inst✝ : Category.{v', u'} D", "W : MorphismProperty C", "F₁ F₂ : C ⥤ D", "e : F₁ ≅ F₂"], "goal": "(∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsIso (F₁.map f)) ↔ ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f → IsIso (F₂.map f)"}}
+{"state": {"context": ["α : Type u", "a b : α", "l l₁ l₂ : List α", "h : (a :: b :: l).split = (l₁, l₂)"], "goal": "l₁.length < (a :: b :: l).length ∧ l₂.length < (a :: b :: l).length"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "f : ℂ → E", "c : ℂ", "R : ℝ", "hf : CircleIntegrable f c R"], "goal": "CircleIntegrable (-f) c R"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "v : PicardLindelof E", "f₁ f₂ : v.FunSpace", "n : ℕ", "t : ↑(Set.Icc v.tMin v.tMax)"], "goal": "dist ((PicardLindelof.FunSpace.next^[n] f₁).toFun t) ((PicardLindelof.FunSpace.next^[n] f₂).toFun t) ≤\n  (↑v.L * |↑t - ↑v.t₀|) ^ n / ↑n ! * dist f₁ f₂"}}
+{"state": {"context": ["ι : Sort u_1", "f : ι → ℕ", "s : Set ℕ"], "goal": "⨅ i, ↑(f i) ≠ ⊤ ↔ Nonempty ι"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "CommRing R", "AddCommGroup M", "AddCommGroup N", "AddCommGroup P", "Module R M", "Module R N", "Module R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "Q : Type u_5", "AddCommGroup Q", "Module R Q", "hfg : Function.Exact ⇑f ⇑g", "hg : Function.Surjective ⇑g"], "goal": "Function.Exact ⇑(LinearMap.lTensor Q f) ⇑(LinearMap.lTensor Q g)"}}
+{"state": {"context": ["R : Type u_5", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "l : Multiset R", "x : M"], "goal": "l.sum • x = (Multiset.map (fun r => r • x) l).sum"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a : α", "a1 : 1 ≤ a", "m n : ℕ", "mn : m ≤ n"], "goal": "1 / a ^ n ≤ 1 / a ^ m"}}
+{"state": {"context": ["α : Type u", "m : ℕ", "n' : Type uₙ", "o' : Type uₒ", "NonUnitalNonAssocSemiring α", "Fintype n'", "A : Matrix (Fin m.succ) n' α", "B : Matrix n' o' α", "i : Fin m", "j : o'"], "goal": "(A * B) i.succ j = (Matrix.of (Matrix.vecTail (Matrix.of.symm A)) * B) i j"}}
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+{"state": {"context": ["α : Type u_1", "Fintype α", "CategoryTheory.SmallCategory α", "CategoryTheory.FinCategory α", "a : CategoryTheory.FinCategory.AsType α"], "goal": "(CategoryTheory.FinCategory.asTypeToObjAsType α).obj a = id a"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_3", "β : Type u_4", "Monoid α", "Monoid β", "FunLike F α β", "MonoidHomClass F α β", "s : Multiset α", "comm : {x | x ∈ s}.Pairwise Commute", "f : F"], "goal": "{x | x ∈ Multiset.map (⇑f) s}.Pairwise Commute"}}
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+{"state": {"context": ["α : Type u_1", "DivisionMonoid α", "a b : α", "h : 1 / a = 1 / b"], "goal": "a = b"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : Rel α β"], "goal": "Monotone r.core"}}
+{"state": {"context": ["x : SetTheory.PGame"], "goal": "x ≈ x"}}
+{"state": {"context": ["R : Type u_3", "OrderedSemiring R", "a : R", "m : ℕ", "ha : 1 ≤ a", "h : m ≠ 0"], "goal": "a ≤ a ^ m"}}
+{"state": {"context": ["K : Type u_1", "g : GenContFract K", "DivisionRing K"], "goal": "g.nums 0 = g.h"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "AddGroup G", "AddAction G α", "MeasurableSpace α", "ν : MeasureTheory.Measure α", "𝓕 : Set α", "h𝓕 : MeasureTheory.IsAddFundamentalDomain G 𝓕 ν", "μ : MeasureTheory.Measure (Quotient (AddAction.orbitRel G α))", "MeasureTheory.AddQuotientMeasureEqMeasurePreimage ν μ"], "goal": "MeasureTheory.MeasurePreserving (Quotient.mk (AddAction.orbitRel G α)) (ν.restrict 𝓕) μ"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "Z : Type u_6", "AddZeroClass Z", "f : C(X, Y)", "g : LocallyConstant Y Z"], "goal": "(LocallyConstant.comapAddMonoidHom f) g = LocallyConstant.comap f g"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "f f₁ f₂ : (i : ι) → Filter (α i)", "s : (i : ι) → Set (α i)", "p : (i : ι) → α i → Prop", "inst✝ : ∀ (i : ι), Nonempty (α i)"], "goal": "(Filter.coprodᵢ f).NeBot ↔ ∃ d, (f d).NeBot"}}
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+{"state": {"context": ["α : Type u_1", "Preorder α", "PredOrder α", "NoMinOrder α", "a b : α"], "goal": "Set.Icc a (Order.pred b) = Set.Ico a b"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "self : StrongFEPair E"], "goal": "self.g₀ = 0"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "l : Ordnode α", "x : α", "r : Ordnode α", "o₁ : WithBot α", "o₂ : WithTop α", "hl : Ordnode.Valid' o₁ l ↑x", "hr : Ordnode.Valid' (↑x) r o₂"], "goal": "Ordnode.BalancedSz l.size r.size → Ordnode.Valid' o₁ (l.glue r) o₂ ∧ (l.glue r).size = l.size + r.size"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "f g : PadicSeq p", "h : ∀ (k : ℕ), padicNorm p (↑f k) = padicNorm p (↑g k)", "hf : ¬f ≈ 0", "hg : ¬g ≈ 0"], "goal": "f.norm = g.norm"}}
+{"state": {"context": ["K : Type u_1", "Field K", "NumberField K", "A : Type u_2", "NormedField A", "IsAlgClosed A", "NormedAlgebra ℚ A", "B : ℝ"], "goal": "{x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}.Finite"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "e : E ≃ₛₗᵢ[σ₁₂] E₂", "x y : E"], "goal": "edist (e x) (e y) = edist x y"}}
+{"state": {"context": ["z w : ℂ"], "goal": "Complex.normSq (z / w) = Complex.normSq z / Complex.normSq w"}}
+{"state": {"context": ["N : Type u_1", "G : Type u_2", "Group N", "Group G", "φ : G →* MulAut N"], "goal": "SemidirectProduct.left 1 = 1"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "S : CategoryTheory.ShortComplex C", "s s' : S.Splitting", "h : s.s = s'.s"], "goal": "s = s'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Sort u_5", "inst✝ : CompleteLattice α", "B : Set ι", "Bcbl : B.Countable", "f : ι → α", "i₀ : ι", "h : f i₀ = ⊤", "Bnonempty : B.Nonempty"], "goal": "∃ x, ⨅ t ∈ B, f t = ⨅ i, f (x i)"}}
+{"state": {"context": ["δ : Type u_4", "π : δ → Type u_6", "(a : δ) → MeasurableSpace (π a)", "s : Set δ", "t : (i : δ) → Set (π i)", "hs : s.Countable", "ht : ∀ i ∈ s, MeasurableSet (t i)"], "goal": "MeasurableSet (s.pi t)"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "R : Type v", "inst✝ : CommRing R", "M N : Matrix n n R"], "goal": "∑ τ : Perm n, ∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, M (σ i) (τ i) * N (τ i) i = ∑ σ : Perm n, ∑ τ : Perm n, (∏ i : n, N (σ i) i) * ↑↑(sign τ) * ∏ j : n, M (τ j) (σ j)"}}
+{"state": {"context": ["n✝ n : ℕ"], "goal": "(1 - 1) * ∑ i ∈ range (log 1 n).succ, n / 1 ^ i.succ = n - (digits 1 n).sum"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedAddCommGroup F", "m : MeasurableSpace α", "μ : Measure α", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : SMulCommClass ℝ 𝕜 E", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace E"], "goal": "Continuous fun f => integralCLM f"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m✝ : MeasurableSpace α", "μ : Measure α", "T : Set α → F →L[ℝ] F'", "hT_empty : T ∅ = 0", "m : MeasurableSpace α", "x : F", "hs : MeasurableSet Set.univ", "hs_empty : Set.univ.Nonempty"], "goal": "∑ x_1 ∈ (piecewise Set.univ hs (const α x) (const α 0)).range, (T (↑(piecewise Set.univ hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = (T Set.univ) x"}}
+{"state": {"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "hf : Tendsto f atBot (𝓝 m)"], "goal": "ContinuousWithinAt f (Iic a) a"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α"], "goal": "(Finset.image Sym2.mk s.offDiag).card = s.card.choose 2"}}
+{"state": {"context": ["N : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace X", "x : X", "inst✝ : DecidableEq N", "i j : N", "f g : ↑(Ω^ N X x)"], "goal": "⟦transAt i f g⟧ = ⟦transAt j f g⟧"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ", "η : ProbabilityTheory.Kernel (α × β) γ", "ProbabilityTheory.IsSFiniteKernel η", "a : α", "s : Set (β × γ)"], "goal": "∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂κ a ≤ ((κ.compProd η) a) s"}}
+{"state": {"context": ["p n pow : ℕ", "hp : 1 < p", "hn : 0 < n", "h : p ^ pow ∣ n", "c : ℕ", "hc : n = p ^ pow * c"], "goal": "pow ≤ p.maxPowDiv n"}}
+{"state": {"context": ["A : Type u_1", "Add A", "Mul A", "Star A"], "goal": "⇑StarRingEquiv.refl = id"}}
+{"state": {"context": ["E : Type u", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "s : Set E", "a x y : E", "h : IsLocalMinOn f s a", "hy : y ∈ posTangentConeAt s a", "hy' : -y ∈ posTangentConeAt s a", "hf : ¬DifferentiableWithinAt ℝ f s a"], "goal": "0 y = 0"}}
+{"state": {"context": ["K : Type u", "V : Type v", "DivisionRing K", "AddCommGroup V", "Module K V", "FiniteDimensional K V", "S : Submodule K V", "h : FiniteDimensional.finrank K ↥S = FiniteDimensional.finrank K V"], "goal": "S = ⊤"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v} → Ordinal.{max u v}", "H : ∀ (i : ι), IsNormal (f i)"], "goal": "(∀ (a : Ordinal.{max u v}), derivFamily f a ∈ ⋂ i, fixedPoints (f i)) ∧ ∀ b ∈ ⋂ i, fixedPoints (f i), ∃ a, derivFamily f a = b"}}
+{"state": {"context": ["F : Type u_1", "α✝ : Type u_2", "β : Type u_3", "R : Type u_4", "α : Type u_5", "f : α → α", "n✝ : ℕ", "hf : Involutive f", "n : ℕ"], "goal": "f^[2 * n] = id"}}
+{"state": {"context": ["k m n : Nat"], "goal": "k ∣ m → k ∣ n → k ∣ m.gcd n"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "s t : Set α", "hs : IsClosed s", "ht : IsClosed t"], "goal": "EMetric.hausdorffEdist s t = 0 ↔ s = t"}}
+{"state": {"context": ["C : Type u₂", "CategoryTheory.Category.{v₂, u₂} C", "D : Type u₃", "CategoryTheory.Category.{v₃, u₃} D", "E : Type u₄", "CategoryTheory.Category.{v₄, u₄} E", "F : CategoryTheory.Functor (C × D) E"], "goal": "∀ {X Y : C} (f : X ⟶ Y) (Y_1 : D),\n  ((CategoryTheory.curry.obj F).map f).app Y_1 = F.map (f, CategoryTheory.CategoryStruct.id Y_1)"}}
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+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3"], "goal": "¬Nonempty (α → β) ↔ ¬(IsEmpty α ∨ Nonempty β)"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "I : Box ι"], "goal": "(volume ↑I).toReal = ∏ i : ι, (I.upper i - I.lower i)"}}
+{"state": {"context": ["n : ℕ"], "goal": "(Polynomial.hermite n).coeff n = 1"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : LinearOrderedAddCommGroup α"], "goal": "comap abs atTop = atBot ⊔ atTop"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "F : Type uF", "P : Type uP", "NormedAddCommGroup E", "NormedAddCommGroup E'", "NormedAddCommGroup F", "RCLike 𝕜", "NormedSpace 𝕜 E", "NormedSpace 𝕜 E'", "NormedSpace ℝ F", "NormedSpace 𝕜 F", "CompleteSpace F", "MeasurableSpace G", "NormedAddCommGroup G", "BorelSpace G", "NormedSpace 𝕜 G", "NormedAddCommGroup P", "NormedSpace 𝕜 P", "μ : MeasureTheory.Measure G", "μ.IsAddLeftInvariant", "μ.IsNegInvariant", "L : E' →L[𝕜] E →L[𝕜] F", "s : Set P", "n : ℕ∞", "v : P → G", "hv : ContDiffOn 𝕜 n v s", "f : G → E", "g : P → G → E'", "k : Set G", "hs : IsOpen s", "hk : IsCompact k", "hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0", "hf : MeasureTheory.LocallyIntegrable f μ", "hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ Set.univ)"], "goal": "ContDiffOn 𝕜 n (fun x => MeasureTheory.convolution (g x) f L μ (v x)) s"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddZeroClass M", "AddZeroClass N"], "goal": "(AddMonoid.Coprod.swap M N).comp AddMonoid.Coprod.inl = AddMonoid.Coprod.inr"}}
+{"state": {"context": ["n : ℕ"], "goal": "‖↑n‖₊ = ↑n"}}
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+{"state": {"context": ["𝕜 : Type u𝕜", "E : Type uE", "E' : Type uE'", "F : Type uF", "NormedAddCommGroup E", "NormedAddCommGroup E'", "NormedAddCommGroup F", "RCLike 𝕜", "NormedSpace 𝕜 E", "NormedSpace 𝕜 E'", "NormedSpace ℝ F", "NormedSpace 𝕜 F", "f₀ : 𝕜 → E", "g₀ : 𝕜 → E'", "L : E →L[𝕜] E' →L[𝕜] F", "μ : MeasureTheory.Measure 𝕜", "μ.IsAddLeftInvariant", "MeasureTheory.SFinite μ", "μ.IsNegInvariant", "hcf : HasCompactSupport f₀", "hf : ContDiff 𝕜 1 f₀", "hg : MeasureTheory.LocallyIntegrable g₀ μ", "x₀ : 𝕜"], "goal": "HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution (deriv f₀) g₀ L μ x₀) x₀"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝³ : Monoid M", "inst✝² : Monoid N", "inst✝¹ : Monoid P", "l✝ l₁ l₂ : List M", "a : M", "inst✝ : DecidableEq M", "l : List M", "ih : a ∈ l → (∀ x ∈ l, ∀ y ∈ l, x * y = y * x) → a * (l.erase a).prod = l.prod", "ha : a ∈ a :: l", "comm : ∀ x ∈ a :: l, ∀ y ∈ a :: l, x * y = y * x"], "goal": "a * ((a :: l).erase a).prod = (a :: l).prod"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y"], "goal": "DenseEmbedding ⇑h"}}
+{"state": {"context": ["R : Type u_1", "Rack R", "G : Type u_2", "Group G", "f : R →◃ Quandle.Conj G", "a b : Rack.PreEnvelGroup R"], "goal": "Rack.PreEnvelGroupRel' R a b → Rack.toEnvelGroup.mapAux f a = Rack.toEnvelGroup.mapAux f b"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝¹ : CancelCommMonoid α", "inst✝ : CancelCommMonoid β", "A : Set α", "B : Set β", "f : α → β", "m n : ℕ", "hmn : m ≤ n", "hf : IsMulFreimanIso n A B f", "s t : Multiset α", "hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A", "htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A", "hs : card s = m", "ht : card t = m", "a✝ : α", "ha✝ : a✝ ∈ A", "a : α", "ha : a ∈ s + replicate (n - m) a✝"], "goal": "a ∈ A"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α", "h : a ∈ l"], "goal": "[a] <+ l"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "P : CategoryTheory.Functor Cᵒᵖ (Type v₁)"], "goal": "(CategoryTheory.Presheaf.tautologicalCocone P).pt = P"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : Type u", "inst✝¹ : CommSemiring R", "inst✝ : Nontrivial R", "M : Ideal R", "hm : M.IsMaximal", "non_field : ¬IsField R"], "goal": "⊥ < M"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "r : α → α → Prop", "inst✝¹ : PartialOrder α", "inst✝ : Preorder β", "s : Set (Lex (α × β))", "hα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x => (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)", "hβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO", "f : ℕ → Lex (α × β)", "hf : ∀ (n : ℕ), f n ∈ s", "g : ℕ ↪o ℕ", "hg : Monotone fun n => (ofLex f (g n)).1", "hhg : ∀ (n : ℕ), (ofLex f (g 0)).1 ≤ (ofLex f (g n)).1"], "goal": "∃ m n, m < n ∧ (fun x x_1 => x ≤ x_1) (f m) (f n)"}}
+{"state": {"context": ["X : AlgebraicGeometry.LocallyRingedSpace", "r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))"], "goal": "X.toΓSpecMapBasicOpen r = X.toRingedSpace.basicOpen r"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F G : J ⥤ C", "α : F ≅ G"], "goal": "(CategoryTheory.Limits.Cones.postcomposeEquivalence α).functor = CategoryTheory.Limits.Cones.postcompose α.hom"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_7", "f : M [⋀^ι]→ₗ[R] N", "S : Type u_10", "Monoid S", "DistribMulAction S N", "SMulCommClass R S N", "c : S", "m : ι → M"], "goal": "(c • f) m = c • f m"}}
+{"state": {"context": ["α : Sort u_4"], "goal": "IsEmpty (PLift α) ↔ IsEmpty α"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "PredOrder α", "NoMinOrder α", "a : α"], "goal": "Order.pred a ⋖ a"}}
+{"state": {"context": [], "goal": "ContinuousOn Real.tan {x | Real.cos x ≠ 0}"}}
+{"state": {"context": ["K : Type v", "V : Type w", "inst✝³ : Field K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "f : End K V", "μ : K", "h : (minpoly K f).IsRoot μ", "p : K[X]", "hp : minpoly K f = (X - C μ) * p", "con : LinearMap.ker (f - (algebraMap K (End K V)) μ) = ⊥", "u : (V →ₗ[K] V)ˣ", "hu : ↑u = f - (algebraMap K (End K V)) μ", "p_ne_0 : p ≠ 0"], "goal": "False"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hs : Convex 𝕜 s", "hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)", "x✝ y✝ z✝ : 𝕜", "hx : x✝ ∈ s", "hz : z✝ ∈ s", "hxy : x✝ < y✝", "hyz : y✝ < z✝"], "goal": "-(((-f) z✝ - (-f) y✝) / (z✝ - y✝)) < -(((-f) y✝ - (-f) x✝) / (y✝ - x✝))"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s s₁ s₂ : Finset α", "a : α", "f g : α → β", "inst✝¹ : CommMonoid β", "inst✝ : DecidableEq α", "h : Disjoint s₁ s₂"], "goal": "∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommSemiring R", "CommSemiring S", "f : R →+* S", "x y : S", "p : R[X][Y]"], "goal": "Polynomial.eval₂ (Polynomial.eval₂RingHom f x) y p =\n  Polynomial.evalEval x y (Polynomial.map (Polynomial.mapRingHom f) p)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Group α", "AddMonoid β", "DistribMulAction α β", "x : α"], "goal": "∀ (a : β), (DistribMulAction.toAddEquiv β x) a = x • a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "P : CategoryTheory.MorphismProperty C", "CategoryTheory.Limits.HasPullbacks C", "P.RespectsIso", "hP₂ : ∀ (X Y S : C) (f : X ⟶ S) (g : Y ⟶ S), P g → P (CategoryTheory.Limits.pullback.fst f g)"], "goal": "P.StableUnderBaseChange"}}
+{"state": {"context": ["z : ℂ", "this : HasSum (fun x => ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2) ((NormedSpace.exp ℂ (z * I) + NormedSpace.exp ℂ (-z * I)) / 2)", "k : ℕ"], "goal": "HasSum (fun c => ((z * I) ^ (2 * (k, c).1 + ↑(k, c).2) / ↑(2 * (k, c).1 + ↑(k, c).2)! + (-z * I) ^ (2 * (k, c).1 + ↑(k, c).2) / ↑(2 * (k, c).1 + ↑(k, c).2)!) / 2) ((z * I) ^ (2 * k) / ↑(2 * k)!)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "H : J.OneHypercoverFamily", "self : H.IsGenerating", "X : C", "S : CategoryTheory.Sieve X", "hS : S ∈ J.sieves X"], "goal": "∃ E, ∃ (_ : H E), E.sieve₀ ≤ S"}}
+{"state": {"context": ["F : Type u", "Field F", "f : RatFunc F", "r : F"], "goal": "↑(r • f) = r • ↑f"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁹ : Category.{u_3, u_1} C", "inst✝⁸ : Preadditive C", "𝕜 : Type u_2", "inst✝⁷ : Field 𝕜", "inst✝⁶ : IsAlgClosed 𝕜", "inst✝⁵ : Linear 𝕜 C", "inst✝⁴ : HasKernels C", "X Y : C", "inst✝³ : FiniteDimensional 𝕜 (X ⟶ X)", "inst✝² : FiniteDimensional 𝕜 (X ⟶ Y)", "inst✝¹ : Simple X", "inst✝ : Simple Y"], "goal": "finrank 𝕜 (X ⟶ Y) = 0 ↔ ¬finrank 𝕜 (X ⟶ Y) = 1"}}
+{"state": {"context": ["R : Type u", "CommRing R", "P Q : Fin 3 → R", "hx : P x * Q z ^ 2 ≠ Q x * P z ^ 2"], "goal": "WeierstrassCurve.Jacobian.addZ P Q ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ✝ μ : Measure α", "inst✝ : SFinite μ", "s : Set α"], "goal": "μ.toFiniteAux s = ∑' (n : ℕ), (2 ^ (n + 1) * (sFiniteSeq μ n) Set.univ)⁻¹ * (sFiniteSeq μ n) s"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "‖y‖ / (o.oangle x (x + y)).sin = ‖x + y‖"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "-b - -(toIocDiv hp (-a) b • p + 1 • p) = p - (b - toIocDiv hp (-a) b • p)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "x : E", "s t : Set E", "h : HasFDerivWithinAt f f' t x", "hst : s ⊆ t"], "goal": "HasFDerivWithinAt f f' s x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "LinearOrder α", "Preorder β", "f : α → β", "hf : StrictMono f", "a b : α"], "goal": "f a ≤ f b ↔ a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "MulOneClass α", "Zero α", "Preorder α", "PosMulStrictMono α", "ha : 1 ≤ a", "hb : 1 < b", "a0 : 0 < a"], "goal": "1 < a * b"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "CommSemiring R", "CommSemiring S", "Semiring A", "Algebra R S", "Algebra R A", "Algebra S A", "IsScalarTower R S A", "hRS : Algebra.FiniteType R S", "hSA : Algebra.FiniteType S A"], "goal": "Algebra.FiniteType R A"}}
+{"state": {"context": ["p : ℕ", "a : ZMod p", "ha : a ^ (p - 1) = 1", "hd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1"], "goal": "Nat.Prime p"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Field R", "p q : R[X]", "ha : a ≠ 0", "x✝ : p ∣ C a * q", "r : R[X]", "hr : C a * q = p * r"], "goal": "q = p * (C a⁻¹ * r)"}}
+{"state": {"context": ["p : ℕ", "h : p ∈ primeFactorsList 0"], "goal": "p ≤ 0"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : AffineSubspace k P", "p : P", "hp : p ∈ s", "v : V"], "goal": "v ∈ s.direction ↔ ∃ p2 ∈ s, v = p -ᵥ p2"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "Fintype σ"], "goal": "MvPolynomial.psum σ R 0 = ↑(Fintype.card σ)"}}
+{"state": {"context": ["𝒪 : Type u", "K : Type v", "Γ : Type w", "inst✝⁵ : CommRing 𝒪", "inst✝⁴ : IsDomain 𝒪", "inst✝³ : Field K", "inst✝² : Algebra 𝒪 K", "inst✝¹ : LinearOrderedCommGroupWithZero Γ", "v : Valuation K Γ", "hh : v.Integers 𝒪", "inst✝ : ValuationRing 𝒪"], "goal": "IsFractionRing 𝒪 K ↔ (∀ (x : K), ∃ a, x = (algebraMap 𝒪 K) a ∨ x⁻¹ = (algebraMap 𝒪 K) a) ∧ Function.Injective ⇑(algebraMap 𝒪 K)"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "a b : E"], "goal": "nndist ‖a‖₊ ‖b‖₊ ≤ ‖a - b‖₊"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "K : ℝ≥0", "f : α → α", "hf : ContractingWith K f", "Nonempty α", "CompleteSpace α", "g : α → α", "hg : ContractingWith K g", "C : ℝ", "hfg : ∀ (z : α), dist (f z) (g z) ≤ C"], "goal": "dist (ContractingWith.fixedPoint f hf) (ContractingWith.fixedPoint g hg) ≤ C / (1 - ↑K)"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : HasFiniteCoproducts C", "X : Karoubi (ChainComplex C ℕ)", "n : ℕ"], "goal": "((𝟭 (Karoubi (ChainComplex C ℕ))).map X.decompId_i).f.f n ≫ X.p.f n ≫ ((Γ₀.splitting X.X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n])) = X.p.f n ≫ ((Γ₀.splitting X.X).cofan (op [n])).inj (Splitting.IndexSet.id (op [n]))"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝⁸ : CommRing P", "inst✝⁷ : Algebra R P", "loc : IsLocalization S P", "K : Type u_3", "K' : Type u_4", "inst✝⁶ : Field K", "inst✝⁵ : Field K'", "inst✝⁴ : Algebra R K", "inst✝³ : IsFractionRing R K", "inst✝² : Algebra R K'", "inst✝¹ : IsFractionRing R K'", "I J : FractionalIdeal R⁰ K", "h : K →ₐ[R] K'", "inst✝ : Nontrivial R", "hI : I ≠ 0", "y : K", "y_mem : y ∈ I", "y_not_mem : y ∉ ⊥"], "goal": "∃ x, x ≠ 0 ∧ (algebraMap R K) x ∈ I"}}
+{"state": {"context": ["R✝ : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "k : Type u_5", "S : Type u_6", "S₃ : Type u_7", "T : Type u_8", "M✝ : Type u_9", "M₁ : Type u_10", "M₂✝ : Type u_11", "M₃ : Type u_12", "N₁ : Type u_13", "N₂ : Type u_14", "N₃ : Type u_15", "ι : Type u_16", "R : outParam (Type u_17)", "M : outParam (Type u_18)", "M₂ : outParam (Type u_19)", "inst✝⁶ : Semiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : AddCommMonoid M₂", "inst✝³ : Module R M", "inst✝² : Module R M₂", "F : Type u_20", "inst✝¹ : FunLike F M M₂", "inst✝ : LinearMapClass F R M M₂", "f : F", "r : R", "x : M"], "goal": "f (r • x) = r • f x"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "S : Type u_2", "Semiring S", "R₂ : Type u_3", "Semiring R₂", "S₂ : Type u_4", "Semiring S₂", "M : Type u_5", "N : Type u_6", "P : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "Module R M", "Module S N", "Module R₂ P", "Module S₂ P", "SMulCommClass S₂ R₂ P", "ρ₁₂ : R →+* R₂", "σ₁₂ : S →+* S₂", "f g : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P", "h : f = g", "x : M", "y : N"], "goal": "(f x) y = (g x) y"}}
+{"state": {"context": ["z w : ℍ", "r R : ℝ", "a b c : ℍ"], "goal": "(dist ↑a ↑b * dist (↑c) ((starRingEnd ℂ) ↑b) + dist (↑a) ((starRingEnd ℂ) ↑b) * dist ↑b ↑c) / (2 * 2 * (√(a.im * b.im) * √(b.im * c.im))) = (dist ↑a ↑b * dist (↑c) ((starRingEnd ℂ) ↑b) + dist (↑a) ((starRingEnd ℂ) ↑b) * dist ↑b ↑c) / (2 * 2 * (√(a.im * c.im) * b.im))"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "M : ι → Type u_4", "N : Type u_5", "inst✝⁴ : DecidableEq ι", "inst✝³ : Ring R", "inst✝² : AddCommGroup N", "inst✝¹ : Module R N", "inst✝ : NoZeroSMulDivisors R N", "p : ι → Submodule R N", "hp : Independent p", "v : ι → N", "hv : ∀ (i : ι), v i ∈ p i", "hv' : ∀ (i : ι), v i ≠ 0", "x✝¹ : DecidableEq ι := Classical.decEq ι", "x✝ : DecidableEq R := Classical.decEq R", "l : ι →₀ R", "hl : (Finsupp.total ι N R v) l = 0", "a : Π₀ (i : ι), ↥(p i) := (mapRange.linearMap fun i => LinearMap.toSpanSingleton R ↥(p i) ⟨v i, ⋯⟩) l.toDFinsupp"], "goal": "l = 0"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "P : Type u_3", "Q : Type u_4", "inst✝⁷ : CommSemiring R", "inst✝⁶ : CommSemiring S", "inst✝⁵ : CommSemiring P", "inst✝⁴ : CommSemiring Q", "M : Submonoid R", "T : Submonoid P", "inst✝³ : Algebra R S", "inst✝² : Algebra P Q", "inst✝¹ : IsLocalization M S", "inst✝ : IsLocalization T Q", "g : R →+* P", "hT : Submonoid.map g M = T", "e : ↥(Ideal.map (algebraMap R S) (ker g)) ≃ₛₗ[id S] ↥(ker (IsLocalization.map Q g ⋯)) := LinearEquiv.ofEq (Ideal.map (algebraMap R S) (ker g)) (ker (IsLocalization.map Q g ⋯)) ⋯"], "goal": "IsLocalizedModule M (↑(LinearEquiv.restrictScalars R e) ∘ₗ Algebra.idealMap S (ker g))"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve R", "C : WeierstrassCurve.VariableChange R"], "goal": "(W.variableChange C).b₆ = ↑C.u⁻¹ ^ 6 * (W.b₆ + 2 * C.r * W.b₄ + C.r ^ 2 * W.b₂ + 4 * C.r ^ 3)"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "d : α"], "goal": "(Set.range fun n => l[n]?.getD d) = insert d {x | x ∈ l}"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Fintype ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x y✝ : ι → ℝ", "δ : ℝ", "hs : IsUpperSet s", "hx : x ∈ closure s", "hδ : 0 < δ", "y : ι → ℝ", "hy : y ∈ s", "hxy : dist x y < δ / 4", "z : ι → ℝ", "hz : z ∈ closedBall (x + const ι (3 / 4 * δ)) (δ / 4)", "i : ι"], "goal": "y i < z i"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : CancelCommMonoidWithZero α", "inst✝¹ : NormalizationMonoid α", "inst✝ : UniqueFactorizationMonoid α", "x : α", "n : ℕ", "ih : normalizedFactors (x ^ n) = n • normalizedFactors x", "h0 : ¬x = 0"], "goal": "normalizedFactors (x ^ (n + 1)) = (n + 1) • normalizedFactors x"}}
+{"state": {"context": ["X : Type u", "s t : Set X", "TopologicalSpace X", "h₁ : s ⊆ t", "h₂ : IsClosed t"], "goal": "closure s ⊆ t"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "K : Type u", "Field K", "Algebra F K", "Algebra E K", "IsScalarTower F E K", "S : Set K", "hprim : IntermediateField.adjoin F S = ⊤"], "goal": "IntermediateField.adjoin E S = ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r✝ R r r' : ℝ≥0", "hr : ↑r + ↑r' < p.radius"], "goal": "Summable ((fun s => ‖p (s.fst + s.snd.fst)‖₊ * r ^ s.snd.fst * r' ^ s.fst) ∘ ⇑changeOriginIndexEquiv.symm)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "AddCommMonoid M", "f : α → β"], "goal": "Finsupp.mapDomain f 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "Semiring R", "AddCommMonoid M", "Module R M", "p p' : Submodule R M", "h : Disjoint p p'", "x : ↥p'"], "goal": "↑x ∈ p ↔ x = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "CategoryTheory.HasShift C ℤ", "n : ℤ"], "goal": "∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y),\n  ((CategoryTheory.Pretriangulated.Triangle.shiftFunctor C n).map f).hom₁ = (CategoryTheory.shiftFunctor C n).map f.hom₁"}}
+{"state": {"context": ["C : Type uC", "inst✝¹ : Category.{uC', uC} C", "W : MorphismProperty C", "D : Type uD", "inst✝ : Category.{uD', uD} D", "G : C ⥤ D", "hG : W.IsInvertedBy G", "X Y : C", "f : X ⟶ Y"], "goal": "composePath (G.map f).toPath = G.map f"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x y : X", "γ : Path x y"], "goal": "γ.extend 0 = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "f✝ f₁ f₂ : Filter α", "g✝ g₁ g₂ : Set α → Filter β", "f : ι → Filter α", "g : Set α → Filter β", "hg : ∀ (s t : Set α), g (s ∩ t) = g s ⊓ g t", "hg' : g univ = ⊤"], "goal": "(iInf f).lift g = ⨅ i, (f i).lift g"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_7", "M₂ : Type u_8", "M₁' : Type u_9", "n : Type u_12", "m : Type u_13", "n' : Type u_14", "CommSemiring R", "AddCommMonoid M₁", "Module R M₁", "AddCommMonoid M₂", "Module R M₂", "DecidableEq n", "Fintype n", "DecidableEq m", "Fintype m", "b₁ : Basis n R M₁", "b₂ : Basis m R M₂", "AddCommMonoid M₁'", "Module R M₁'", "b₁' : Basis n' R M₁'", "Fintype n'", "DecidableEq n'", "B : M₁ →ₗ[R] M₂ →ₗ[R] R", "f : M₁' →ₗ[R] M₁"], "goal": "(LinearMap.toMatrix₂ b₁' b₂) (B ∘ₗ f) = ((LinearMap.toMatrix b₁' b₁) f)ᵀ * (LinearMap.toMatrix₂ b₁ b₂) B"}}
+{"state": {"context": ["L : Type u_1", "inst✝ : CompleteLattice L", "a b : L"], "goal": "(fun u => {x | x u}) (a ⊓ b) = (fun u => {x | x u}) a ⊓ (fun u => {x | x u}) b"}}
+{"state": {"context": ["G : Type u_3", "Mul G", "IsLeftCancelMul G", "a : G"], "goal": "Function.Injective fun x => a * x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "K : Type u_4", "inst✝³ : Semiring S", "inst✝² : CommSemiring R", "inst✝¹ : Semiring A", "inst✝ : Field K", "p q : R[X]", "r : R", "h : p.natDegree = q.natDegree", "n : ℕ"], "goal": "(r ^ (p.natDegree - (p + q).natDegree) • (p + q).scaleRoots r).coeff n = (p.scaleRoots r + q.scaleRoots r).coeff n"}}
+{"state": {"context": ["d : ℤ", "h₀ : 0 < d", "hd : ¬IsSquare d", "x y : ℤ", "h : x ^ 2 - d * y ^ 2 = 1", "hy : y ≠ 0"], "goal": "0 < d * y ^ 2"}}
+{"state": {"context": ["α : Type u", "m : ℕ", "x : α", "u : Fin m → α"], "goal": "Matrix.vecHead (Matrix.vecCons x u) = x"}}
+{"state": {"context": ["α : Type u", "s : Stream'.WSeq α", "n : ℕ", "T : (s.get? n).Terminates"], "goal": "s.destruct.Terminates"}}
+{"state": {"context": ["𝕜 : Type u", "n : ℕ", "Ei : Fin n.succ → Type wEi", "G : Type wG", "NontriviallyNormedField 𝕜", "(i : Fin n.succ) → NormedAddCommGroup (Ei i)", "(i : Fin n.succ) → NormedSpace 𝕜 (Ei i)", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ContinuousMultilinearMap 𝕜 Ei G", "x : Ei 0", "m : (i : Fin n) → Ei i.succ"], "goal": "(f.curryLeft x) m = f (Fin.cons x m)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "inst✝¹ : CompleteSpace F", "μ ν : Measure α", "s t N : Set α", "f : α → ℝ", "inst✝ : IsProbabilityMeasure μ", "hf : Integrable f μ", "hN : μ N = 0"], "goal": "∃ x ∉ N, f x ≤ ∫ (a : α), f a ∂μ"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Countable ι", "a : ι → ℂ", "p : ι → ℝ", "F : ℝ → ℂ", "s : ℂ", "hp : ∀ (i : ι), a i = 0 ∨ 0 < p i", "hs : 0 < s.re", "hF : ∀ t ∈ Ioi 0, HasSum (fun i => a i * ↑(rexp (-p i * t))) (F t)", "h_sum : Summable fun i => ‖a i‖ / p i ^ s.re"], "goal": "HasSum (fun i => Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type u_1", "L.Structure M", "L.Structure N", "f : L.Hom M N", "hf : Function.Surjective ⇑f"], "goal": "Function.Surjective (FirstOrder.Language.Substructure.map f)"}}
+{"state": {"context": ["ι : Sort u_1", "𝕜 : Type u_2", "E : Type u_3", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "s✝ t✝ u✝ : Set E", "x y : E", "s t u : Set E"], "goal": "convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 s (convexJoin 𝕜 t u)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "φ ψ : ℝ → ℝ", "s✝ : Set E", "a b m✝ : ℝ", "x y : E", "f✝ g : E → ℝ", "s : Set E", "f : E → ℝ", "m n : ℝ"], "goal": "StrongConcaveOn s 0 f ↔ ConcaveOn ℝ s f"}}
+{"state": {"context": ["ι : Type u_2", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Preadditive V", "c : ComplexShape ι", "C D : HomotopyCategory V c", "f : C ⟶ D"], "goal": "(HomotopyCategory.quotient V c).map (Quot.out f) = f"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f₂ : 𝕜 → F", "s₂ : Set 𝕜", "hs : UniqueDiffOn 𝕜 s₂"], "goal": "ContDiffOn 𝕜 ⊤ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ⊤ (derivWithin f₂ s₂) s₂"}}
+{"state": {"context": ["x : PartENat", "y : ℕ", "h : x ≤ ↑y"], "goal": "x.Dom"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "f : R[X]", "CommRing S", "i : R →+* S", "a : S", "h : Polynomial.eval₂ i a f = 0", "g : R[X]"], "goal": "(AdjoinRoot.lift i a h) ((AdjoinRoot.mk f) g) = Polynomial.eval₂ i a g"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "f : α → β", "hf : Function.Bijective f"], "goal": "Function.Bijective fun g => g ∘ f"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "P : Type u_10", "NormedField 𝕜", "SeminormedAddCommGroup V", "NormedSpace 𝕜 V", "PseudoMetricSpace P", "NormedAddTorsor V P", "s : AffineSubspace 𝕜 P", "Nonempty ↥s"], "goal": "s.subtypeₐᵢ.linear = s.direction.subtype"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N' : Type u_6", "AddCommGroup N'", "Module R N'", "ι : Type u_7", "g g₂ : M [⋀^ι]→ₗ[R] N'"], "goal": "↑(g - g₂) = ↑g - ↑g₂"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A : C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.Exponentiable A", "I : C", "t : CategoryTheory.Limits.IsInitial I"], "goal": "(CategoryTheory.zeroMul t).hom = CategoryTheory.Limits.prod.snd"}}
+{"state": {"context": ["n : ℕ+", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "ce : IsCyclotomicExtension {n} ℚ K", "hζ : IsPrimitiveRoot ζ ↑n", "this : NumberField K", "H₁ : (aeval (IsPrimitiveRoot.powerBasis ℚ hζ).gen) (X - 1) = (subOnePowerBasis ℚ hζ).gen", "H₂ : (aeval (subOnePowerBasis ℚ hζ).gen) (X + 1) = (IsPrimitiveRoot.powerBasis ℚ hζ).gen", "i : Fin (IsPrimitiveRoot.powerBasis ℚ hζ).dim", "j : Fin (subOnePowerBasis ℚ hζ).dim"], "goal": "minpoly ℚ (IsPrimitiveRoot.powerBasis ℚ hζ).gen = Polynomial.map (algebraMap ℤ ℚ) (minpoly ℤ (IsPrimitiveRoot.powerBasis ℚ hζ).gen)"}}
+{"state": {"context": ["G : Type u_7", "Group G", "H K : Subgroup G", "h : H ≤ K", "g : ↥(H.subgroupOf K)"], "goal": "↑((Subgroup.subgroupOfEquivOfLe h) g) = ↑↑g"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "p q : A → Prop", "inst✝²⁰ : Semifield R", "inst✝¹⁹ : StarRing R", "inst✝¹⁸ : MetricSpace R", "inst✝¹⁷ : TopologicalSemiring R", "inst✝¹⁶ : ContinuousStar R", "inst✝¹⁵ : Semifield S", "inst✝¹⁴ : StarRing S", "inst✝¹³ : MetricSpace S", "inst✝¹² : TopologicalSemiring S", "inst✝¹¹ : ContinuousStar S", "inst✝¹⁰ : TopologicalSpace A", "inst✝⁹ : Ring A", "inst✝⁸ : StarRing A", "inst✝⁷ : Algebra S A", "inst✝⁶ : ContinuousFunctionalCalculus S q", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra R A", "inst✝³ : IsScalarTower R S A", "inst✝² : StarModule R S", "inst✝¹ : ContinuousSMul R S", "inst✝ : CompleteSpace R", "f : C(S, R)", "halg : UniformEmbedding ⇑(algebraMap R S)", "h0 : p 0", "h : ∀ (a : A), p a ↔ q a ∧ SpectrumRestricts a ⇑f", "h_cpct : ∀ (a : A), q a → CompactSpace ↑(spectrum S a)", "a : A", "ha : p a"], "goal": "ClosedEmbedding ⇑(starAlgHom (cfcHom ⋯) ⋯)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "x : α"], "goal": "(InitialSeg.refl r) x = x"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "MulOneClass α", "Zero α", "Preorder α", "MulPosStrictMono α", "hb : 0 < b", "h : a < 1"], "goal": "a * b < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "MetricSpace β", "μ : MeasureTheory.Measure α", "s : Set α", "ε : ℝ", "f : ι → α → β", "g : α → β", "SemilatticeSup ι", "Nonempty ι", "Countable ι", "hε : 0 < ε", "hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)", "hg : MeasureTheory.StronglyMeasurable g", "hsm : MeasurableSet s", "hs : μ s ≠ ⊤", "hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (g x))", "n : ℕ"], "goal": "∃ j, μ (s ∩ MeasureTheory.Egorov.notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "σ : Type u_4", "inst✝³ : Primcodable α", "inst✝² : Primcodable β", "inst✝¹ : Primcodable γ", "inst✝ : Primcodable σ", "f : α → β → σ", "x✝¹ : α", "x✝ : ℕ"], "goal": "Option.map (f x✝¹) (decode x✝) = (decode x✝).bind (Option.some ∘ f x✝¹)"}}
+{"state": {"context": ["n d : ℤ"], "goal": "↑n / ↑d = Rat.divInt n d"}}
+{"state": {"context": ["a b x y u v : ℕ", "ha : ↑a = ↑x ^ 2 + ↑y ^ 2", "hb : ↑b = ↑u ^ 2 + ↑v ^ 2", "r s : ℤ", "h : ↑a * ↑b = r ^ 2 + s ^ 2"], "goal": "∃ r s, ↑a * ↑b = ↑r ^ 2 + ↑s ^ 2"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "n : Type w", "Nontrivial R", "NoZeroDivisors R", "B : LinearMap.BilinForm R M", "v : Basis n R M", "hO : B.iIsOrtho ⇑v"], "goal": "B.Nondegenerate ↔ ∀ (i : n), ¬B.IsOrtho (v i) (v i)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Monoid β", "a : α", "f : α → β"], "goal": "{a}.noncommProd f ⋯ = f a"}}
+{"state": {"context": ["N : ℕ", "χ : DirichletCharacter ℂ N", "f : ℕ → ℂ", "s : ℂ", "h : LSeriesSummable f s"], "goal": "LSeriesSummable ((fun n => χ ↑n) * f) s"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α", "n : ℤ"], "goal": "a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1"}}
+{"state": {"context": ["α : Sort u", "β : α → Sort v", "α' : Sort w", "DecidableEq α", "DecidableEq α'", "g : (a : α) → β a", "f : α' → α", "hf : Function.Injective f", "i : α'", "a : β (f i)"], "goal": "(fun j => Function.update g (f i) a (f j)) = Function.update (fun i => g (f i)) i a"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "T0Space X"], "goal": "Function.Injective 𝓝"}}
+{"state": {"context": ["n : ℕ", "inst✝ : NeZero n", "i : ZMod n"], "goal": "orderOf (r ↑i.val) = n / n.gcd i.val"}}
+{"state": {"context": ["K : Type u", "Field K", "x : RatFunc K"], "goal": "(-x).num * x.denom = -x.num * (-x).denom"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "R : α → β → Prop", "s : WSeq α", "t : WSeq β", "H : LiftRel R s t", "n : ℕ"], "goal": "Option (α × WSeq α) → Option (β × WSeq β) → Prop"}}
+{"state": {"context": ["m : Type u_3", "n : Type u_4", "α : Type u_5", "Fintype m", "Fintype n", "SeminormedAddCommGroup α", "A : Matrix m n α"], "goal": "‖A‖ = ↑(Finset.univ.sup fun i => Finset.univ.sup fun j => ‖A i j‖₊)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "M : AlgebraCat R"], "goal": "M.ofSelfIso.inv = 𝟙 M"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "x : α", "ι : Sort u_3", "δ : Type u_4", "CompleteLinearOrder δ", "f : ι → α → δ", "h : ∀ (i : ι), UpperSemicontinuousAt (f i) x"], "goal": "UpperSemicontinuousAt (fun x' => ⨅ i, f i x') x"}}
+{"state": {"context": ["e₁ : ONote", "n₁ : ℕ+", "o₁ e₂ : ONote", "n₂ : ℕ+", "o₂ : ONote", "h₁ : (e₁.oadd n₁ o₁).NF", "h : e₁ < e₂"], "goal": "e₁.oadd n₁ o₁ < e₂.oadd n₂ o₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "F : ℕ → α → β", "f : α → β", "bound : α → ℝ", "h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a", "h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a))"], "goal": "∀ᵐ (a : α) ∂μ, ENNReal.ofReal ‖f a‖ ≤ ENNReal.ofReal (bound a)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁶ : Semiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid M'", "inst✝² : Module R M'", "inst✝¹ : Finite ι", "inst✝ : DecidableEq ι", "e : M ≃ₗ[R] ι → R", "i j : ι"], "goal": "e ((ofEquivFun e) i) j = e (e.symm (update 0 i 1)) j"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "inst✝¹ : Semiring α", "inst✝ : Semiring β", "a b : α", "m n : ℕ"], "goal": "Even a ↔ 2 ∣ a"}}
+{"state": {"context": ["α : Type u", "a : α"], "goal": "↑(FreeRing.of a) = FreeCommRing.of a"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "n : ℕ", "x : α"], "goal": "(l ++ [x]).rdrop (n + 1) = l.rdrop n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l₁✝ l₂✝ : List α", "r : α → α → Prop", "a✝ b : α", "inst✝ : DecidableEq α", "l₁ : List α", "a : α", "l₂ : List α", "hl₁ : l₁.Nodup"], "goal": "l₁.diff (a :: l₂) = List.filter (fun x => decide (x ∉ a :: l₂)) l₁"}}
+{"state": {"context": ["S : Type u_1", "inst✝³ : AddCommGroupWithOne S", "C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasZeroMorphisms C", "inst✝ : HasShift C S", "n : S", "X✝ Y✝ Z✝ : DifferentialObject S C", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝"], "goal": "autoParam ((CategoryTheory.shiftFunctor C n).map (f.f ≫ g.f) = (CategoryTheory.shiftFunctor C n).map f.f ≫ (CategoryTheory.shiftFunctor C n).map g.f) _auto✝"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "PseudoEMetricSpace α", "OpensMeasurableSpace α", "e : ℕ → α", "N : ℕ", "x : α"], "goal": "↑(MeasureTheory.SimpleFunc.nearestPtInd e (N + 1)) x =\n  if ∀ k ≤ N, edist (e (N + 1)) x < edist (e k) x then N + 1 else ↑(MeasureTheory.SimpleFunc.nearestPtInd e N) x"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "𝕜 : Type u_4", "E : Type u_5", "inst✝³ : DecidableEq α", "inst✝² : CommMonoid α", "inst✝¹ : CommMonoid β", "inst✝ : DecidableEq β", "A : Finset α", "B : Finset β", "f : α → β", "hf : IsMulFreimanHom 2 (↑A) (↑B) f", "hf' : Set.BijOn f ↑A ↑B", "s : Finset β", "hsB : s ⊆ B", "hcard : s.card = mulRothNumber B", "hs : ThreeGPFree ↑s", "hsA : invFunOn f ↑A '' ↑s ⊆ ↑A", "hfsA : Set.SurjOn f ↑A ↑s"], "goal": "mulRothNumber B ≤ mulRothNumber A"}}
+{"state": {"context": ["P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme", "hP : AlgebraicGeometry.IsLocalAtTarget P", "X Y : AlgebraicGeometry.Scheme", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "H : ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i)"], "goal": "P f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "NormedAddCommGroup X", "NormedAddCommGroup Y", "NormedSpace ℝ X", "NormedSpace ℝ Y", "f g : X → Y", "x : X", "u : Set X", "hx : x ∈ u", "hu : IsOpen u", "hf : ConformalAt f x", "h : ∀ x ∈ u, g x = f x"], "goal": "ConformalAt g x"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : AffineSubspace k P"], "goal": "s.direction = vectorSpan k ↑s"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E"], "goal": "∀ (x x_1 : C × D × E) (f : x ⟶ x_1), (CategoryTheory.prod.inverseAssociator C D E).map f = ((f.1, f.2.1), f.2.2)"}}
+{"state": {"context": [], "goal": "LipschitzWith 1 fun x => max x 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type u_1", "inst✝² : OmegaCompletePartialOrder α", "inst✝¹ : OmegaCompletePartialOrder β", "inst✝ : OmegaCompletePartialOrder γ", "f : α →o β", "g : β →o γ", "x : β", "c : Chain α", "z : β"], "goal": "Function.const α x (ωSup c) ≤ z ↔ ∀ (i : ℕ), (Function.const α x ∘ ⇑c) i ≤ z"}}
+{"state": {"context": ["n : ℕ"], "goal": "(∑' (i : ℕ), if n ≤ i then 2⁻¹ ^ i else 0) = 2 * 2⁻¹ ^ n"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "hS : S.Exact", "hS' : S.ShortExact", "v : ι → ↑S.X₁", "β : Type u_4", "u : ι ⊕ β → ↑S.X₂", "huv : u ∘ Sum.inl = ⇑S.f ∘ v", "hv : ⊤ ≤ span R (range v)", "hw : ⊤ ≤ span R (range (⇑S.g ∘ u ∘ Sum.inr))", "m : ↑S.X₂", "a✝ : m ∈ ⊤", "cm : β →₀ R", "hm : (cm.sum fun i a => a • (⇑S.g ∘ u ∘ Sum.inr) i) = S.g m", "m' : ↑S.X₂ := cm.sum fun j a => a • u (Sum.inr j)"], "goal": "m ∈ span R (range u)"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{?u.14791, u_1} C", "P : Karoubi C"], "goal": "P.p = P.p ≫ P.p ≫ P.p"}}
+{"state": {"context": ["α : Type u_1", "G : SimpleGraph α", "H : SimpleGraph α", "Fintype α", "DecidableEq α", "DecidableRel G.Adj", "n : ℕ", "DecidableRel H.Adj", "h : G ≤ H"], "goal": "G.cliqueFinset n ⊆ H.cliqueFinset n"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_7", "AddMonoid M", "AddAction M α", "b : α"], "goal": "0 +ᵥ b = b"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedSemifield α", "a : α", "m n : ℤ", "h₀ : 0 < a", "h₁ : a ≠ 1"], "goal": "a ^ m = a ^ n ↔ m = n"}}
+{"state": {"context": ["α : Type u", "c : Cardinal.{u}", "hreg : c.IsRegular", "s : Set α"], "goal": "s ∈ Filter.cocardinal α hreg ↔ #↑sᶜ < c"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "x y : E"], "goal": "openSegment 𝕜 x y = (fun p => p.1 • x + p.2 • y) '' {p | 0 < p.1 ∧ 0 < p.2 ∧ p.1 + p.2 = 1}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "a a' : α", "v : Vector α n"], "goal": "a ∈ v.toList ↔ ∃ i, v.get i = a"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "ϕ : L →ᴸ L'", "L'' : FirstOrder.Language", "ψ : L'' →ᴸ L'", "L3 : FirstOrder.Language", "θ : L' →ᴸ L3"], "goal": "θ.comp (ϕ.sumElim ψ) = (θ.comp ϕ).sumElim (θ.comp ψ)"}}
+{"state": {"context": ["I : Type w₁", "C : I → Type u₁", "(i : I) → CategoryTheory.Category.{v₁, u₁} (C i)", "J : Type w₂", "K : Type w₃", "f : K → J", "g : J → I", "X : (i : K) × (fun i => C (g (f i))) i"], "goal": "(CategoryTheory.Sigma.mapComp C f g).inv.app X = CategoryTheory.CategoryStruct.id ⟨g (f X.fst), X.snd⟩"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "s : Set E", "x y : E"], "goal": "[x-[𝕜]y] ⊆ s ↔ ∀ (a b : 𝕜), 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s"}}
+{"state": {"context": ["K : Type u_2", "Field K", "NumberField K", "x : K"], "goal": "∏ w : NumberField.InfinitePlace K, w x ^ w.mult = ↑|(Algebra.norm ℚ) x|"}}
+{"state": {"context": ["m : Nat", "b : Int", "H : 0 < b"], "goal": "-[m+1] / b = -((↑m).div b + 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f g : C(α, β)", "h : ∀ (a : α), f a = g a"], "goal": "f = g"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty Cᵒᵖ", "X Y : Cᵒᵖ", "φ : W.LeftFraction X Y"], "goal": "φ.unop.s = φ.s.unop"}}
+{"state": {"context": ["F : Type u", "Field F"], "goal": "Function.Injective RatFunc.coeToLaurentSeries_fun"}}
+{"state": {"context": ["θ ψ : ℝ", "Hcos : cos θ = cos ψ", "Hsin : sin θ = sin ψ", "hc : ↑θ = -↑ψ"], "goal": "↑θ = ↑ψ"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{?u.26563, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : HasFiniteCoproducts C", "X : SimplicialObject C", "Δ Δ' : SimplexCategoryᵒᵖ", "θ : Δ ⟶ Δ'", "A : Splitting.IndexSet Δ"], "goal": "(image.ι (θ.unop ≫ A.e)).op ≫ (A.pull θ).e.op = A.e.op ≫ θ"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "s : Set V"], "goal": "SimpleGraph.induceHom SimpleGraph.Hom.id ⋯ = SimpleGraph.Hom.id"}}
+{"state": {"context": ["ι : Type u_1", "σ : Type u_2", "R : Type u_3", "A : Type u_4", "inst✝⁵ : CommSemiring A", "inst✝⁴ : DecidableEq ι", "inst✝³ : AddMonoid ι", "inst✝² : SetLike σ A", "inst✝¹ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝ : GradedRing 𝒜", "I✝ I J : Ideal A", "HI : ∃ S, I = span (Subtype.val '' S)", "HJ : ∃ S, J = span (Subtype.val '' S)"], "goal": "∃ S, I * J = span (Subtype.val '' S)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "f g : Π₀ (_a : α), ℕ"], "goal": "DFinsupp.toMultiset f ≤ DFinsupp.toMultiset g ↔ f ≤ g"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : ConcreteCategory C", "inst✝² : (forget C).Corepresentable", "J : Type w", "inst✝¹ : Category.{t, w} J", "G : J ⥤ C", "inst✝ : HasLimit G"], "goal": "Small.{v, max v w} ↑(G ⋙ forget C).sections"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "s : Set G"], "goal": "AddSubgroup.center G ≤ AddSubgroup.centralizer s"}}
+{"state": {"context": ["G : Type u_1", "Group G", "MeasurableSpace G", "TopologicalSpace G", "TopologicalGroup G", "BorelSpace G", "PolishSpace G", "Γ : Subgroup G", "Countable ↥Γ", "Γ.Normal", "T2Space (G ⧸ Γ)", "SecondCountableTopology (G ⧸ Γ)", "ν : MeasureTheory.Measure G", "MeasureTheory.SigmaFinite ν", "ν.IsHaarMeasure", "ν.IsMulRightInvariant", "K : TopologicalSpace.PositiveCompacts (G ⧸ Γ)", "𝓕 : Set G", "h𝓕 : MeasureTheory.IsFundamentalDomain (↥Γ.op) 𝓕 ν", "h𝓕_finite : ν 𝓕 ≠ ⊤"], "goal": "MeasureTheory.QuotientMeasureEqMeasurePreimage ν (ν (QuotientGroup.mk ⁻¹' ↑K ∩ 𝓕) • MeasureTheory.Measure.haarMeasure K)"}}
+{"state": {"context": ["α : Type u_1", "s s' : Multiset α", "hs : s = s'", "p p' : α → Prop", "DecidablePred p", "DecidablePred p'", "hp : ∀ x ∈ s, p x = p' x"], "goal": "Multiset.countP p s = Multiset.countP p' s'"}}
+{"state": {"context": ["m n : Nat"], "goal": "(n * m).gcd n = n"}}
+{"state": {"context": ["K : Type u", "V : Type v", "DivisionRing K", "AddCommGroup V", "Module K V", "FiniteDimensional K V", "f g : V →ₗ[K] V"], "goal": "f ∘ₗ g = LinearMap.id ↔ g ∘ₗ f = LinearMap.id"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "OrderedCommGroup α", "TopologicalSpace α", "TopologicalGroup α", "OrderClosedTopology α", "f g : ι → α", "hf : Multipliable f", "hg : Multipliable g", "h : f < g"], "goal": "∏' (n : ι), f n < ∏' (n : ι), g n"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t u : Finset α", "P : Finpartition s", "a : α"], "goal": "s.card % P.parts.card ≤ P.parts.card"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "β : Type u_7", "inst✝ : SeminormedAddCommGroup β", "T T' : Set α → β", "C C' : ℝ", "c : ℝ≥0∞", "hc_ne_top : c ≠ ⊤", "h : ∀ (s : Set α), MeasurableSet s → c • μ s = ⊤ → μ s = ⊤", "s : Set α", "hs : MeasurableSet s", "hμs : μ s < ⊤", "hcμs : c * μ s ≠ ⊤", "hT : FinMeasAdditive (c • μ) T ∧ ∀ (s : Set α), MeasurableSet s → c * μ s < ⊤ → ‖T s‖ ≤ C * (c.toReal * (μ s).toReal)"], "goal": "C * (c.toReal * (μ s).toReal) = c.toReal * C * (μ s).toReal"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "M : Matrix m n α", "i : m", "j : n", "c : m → α", "DecidableEq n", "j' : n"], "goal": "M.updateColumn j c i j' = if j' = j then c i else M i j'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f✝ : ℕ → α → β", "g✝ : α → β", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "f : ι → α → β", "hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ", "h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε", "g : ι → α → β := fun i => Exists.choose ⋯", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖Exists.choose ⋯ x‖₊}.indicator (Exists.choose ⋯)) p μ ≤ ENNReal.ofReal ε"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "b : β", "s : Set β", "h : b ∈ s"], "goal": "(fun x => b) ⁻¹' s = Set.univ"}}
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+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "F : J ⥤ C", "e : C ≌ D"], "goal": "(CategoryTheory.Limits.Cones.functorialityEquivalence F e).unitIso =\n  CategoryTheory.NatIso.ofComponents\n    (fun c => CategoryTheory.Limits.Cones.ext (e.unitIso.app ((𝟭 (CategoryTheory.Limits.Cone F)).obj c).1) ⋯) ⋯"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "G₁ G₂ : G.Finsubgraph"], "goal": "↑(G₁ \\ G₂) = ↑G₁ \\ ↑G₂"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : (o.oangle x y).sign = -1", "hx : ¬x = 0", "hy : ¬y = 0", "hxy : o.oangle x y = ↑(InnerProductGeometry.angle x y)"], "goal": "False"}}
+{"state": {"context": ["H : Type u", "H' : Type u_1", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝³ : TopologicalSpace H", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "G : StructureGroupoid H", "inst✝ : HasGroupoid M G", "s U : Opens M", "x : ↥U", "e : PartialHomeomorph M H := chartAt H ↑x"], "goal": "↑e.symm =ᶠ[𝓝 (↑e ↑x)] Subtype.val ∘ ↑(chartAt H x).symm"}}
+{"state": {"context": ["t : Type u → Type u → Type u", "Bitraversable t", "LawfulBitraversable t", "α β : Type u"], "goal": "Bitraversable.tfst pure = pure"}}
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+{"state": {"context": ["α : Type u_1", "p : ℂ × ℂ", "hp_fst : p.1 ≠ 0"], "goal": "IsOpen fun x => x.1 = 0 → False"}}
+{"state": {"context": ["V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "CategoryTheory.Limits.HasZeroObject V", "ι : Type u_1", "DecidableEq ι", "c : ComplexShape ι", "K : HomologicalComplex V c", "j : ι", "A : V", "f g : K ⟶ (HomologicalComplex.single V c j).obj A", "hfg : f.f j = g.f j"], "goal": "f = g"}}
+{"state": {"context": ["G : Type u_1", "inst✝⁹ : TopologicalSpace G", "inst✝⁸ : Group G", "inst✝⁷ : TopologicalGroup G", "inst✝⁶ : MeasurableSpace G", "inst✝⁵ : BorelSpace G", "inst✝⁴ : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝³ : IsProbabilityMeasure μ", "inst✝² : IsProbabilityMeasure μ'", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "this : CompactSpace G", "A : ∀ (s : Set G), μ' s = μ'.haarScalarFactor μ • μ s"], "goal": "μ' = μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "F' : Type u_14", "NormedAddCommGroup F'", "NormedSpace 𝕜 F'", "G' : Type u_15", "TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_16", "TopologicalSpace N'", "ChartedSpace G' N'", "x : M", "f : M → M'", "g : M → N'", "hf : SmoothAt I I' f x", "hg : SmoothAt I J' g x"], "goal": "SmoothAt I (I'.prod J') (fun x => (f x, g x)) x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "s : Set X", "t : Set Y", "x : X", "h : e.IsImage s t", "hx : x ∈ e.source"], "goal": "Filter.map (↑e) (𝓝[s] x) = 𝓝[t] ↑e x"}}
+{"state": {"context": ["A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "R : CategoryTheory.Functor B T", "L₁ L₂ L₃ : CategoryTheory.Functor A T", "l : L₁ ⟶ L₂", "l' : L₂ ⟶ L₃", "X : CategoryTheory.Comma L₃ R"], "goal": "((CategoryTheory.Comma.mapLeftComp R l l').hom.app X).right = CategoryTheory.CategoryStruct.id X.right"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "inst✝ : MeasurableSpace E", "m : MeasurableSpace Ω", "ℙ : Measure Ω", "μ : Measure E", "X : Ω → E", "h : ¬pdf X ℙ μ =ᶠ[ae μ] 0"], "goal": "AEMeasurable X ℙ"}}
+{"state": {"context": ["R : Type u_5", "σ : Type u_6", "CommRing R", "φ : MvPolynomial σ R", "n : ℕ", "hφ : φ.IsHomogeneous n"], "goal": "(-φ).IsHomogeneous n"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDomain R", "CharZero R", "p : R[X]", "t : R", "hpt : p.IsRoot t"], "goal": "Polynomial.rootMultiplicity t (Polynomial.derivative p) = Polynomial.rootMultiplicity t p - 1"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "y : R", "hI : I.IsRadical", "h : y ∉ I.toAddSubmonoid", "n : ℕ", "hx' : (fun x => y ^ x) n ∈ ↑I"], "goal": "(fun x => y ^ x) n ∈ ⊥"}}
+{"state": {"context": ["E'' : Type u_9", "NormedAddCommGroup E''", "c : E''"], "goal": "(fun _x => c) =o[Filter.atBot] id"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁰ : Group G", "inst✝⁹ : MeasurableSpace G", "inst✝⁸ : TopologicalSpace G", "inst✝⁷ : TopologicalGroup G", "inst✝⁶ : BorelSpace G", "inst✝⁵ : PolishSpace G", "ν : Measure G", "Γ : Subgroup G", "inst✝⁴ : Countable ↥Γ", "inst✝³ : T2Space (G ⧸ Γ)", "inst✝² : SecondCountableTopology (G ⧸ Γ)", "μ : Measure (G ⧸ Γ)", "inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ", "inst✝ : ν.IsMulLeftInvariant", "hasFun : HasFundamentalDomain (↥Γ.op) G ν", "g : G", "A : Set (G ⧸ Γ)", "hA : MeasurableSet A", "meas_π : Measurable QuotientGroup.mk", "𝓕 : Set G", "h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν", "h𝓕_translate_fundom : IsFundamentalDomain (↥Γ.op) (g • 𝓕) ν"], "goal": "μ ((fun x => g • x) ⁻¹' A) = μ A"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : Semiring R", "inst✝ : Semiring S", "I✝ I' : Ideal R", "J J' I : Ideal S", "h : I.IsPrime", "this : (I.prod ⊤).IsPrime := isPrime_ideal_prod_top"], "goal": "(map (↑RingEquiv.prodComm) (I.prod ⊤)).IsPrime"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : MeasurableSpace α", "μ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "inst✝ : TopologicalSpace δ", "f : α →ₘ[μ] β", "g : α →ₘ[μ] γ"], "goal": "f.pair g = mk (fun x => (↑f x, ↑g x)) ⋯"}}
+{"state": {"context": ["F : PFunctor.{u}", "n : ℕ", "x y : F.M"], "goal": "PFunctor.Approx.Agree (x.approx n) (y.approx (n + 1)) ↔ PFunctor.M.Agree' n x y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "UniformSpace β", "𝔖 : Set (Set α)", "ι : Type u_5", "Preorder ι", "IsDirected ι fun x x_1 => x ≤ x_1", "t : ι → Set α", "V : ι → Set (β × β)", "ht : ∀ (n : ι), t n ∈ 𝔖", "hmono : Monotone t", "hex : ∀ s ∈ 𝔖, ∃ n, s ⊆ t n", "hb : (𝓤 β).HasAntitoneBasis V"], "goal": "(𝓤 (α →ᵤ[𝔖] β)).HasAntitoneBasis fun n => UniformOnFun.gen 𝔖 (t n) (V n)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "D : Type u_2", "CategoryTheory.Category.{u_4, u_2} D", "F : CategoryTheory.Functor C D", "F.ReflectsEffectiveEpis", "X Y : C", "f : X ⟶ Y", "h : CategoryTheory.EffectiveEpi (F.map f)"], "goal": "CategoryTheory.EffectiveEpi f"}}
+{"state": {"context": ["a : Nat", "xs : List Nat", "m : a ∈ xs", "h : a ≠ 0"], "goal": "xs.nonzeroMinimum ≤ a"}}
+{"state": {"context": ["V : Type u_1", "α : Type u_2", "G : SimpleGraph V", "DecidableRel G.Adj", "Zero α", "One α"], "goal": "(SimpleGraph.adjMatrix α G).IsSymm"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "CategoryTheory.Bicategory.HasLeftKanExtension f g", "x : B", "h : c ⟶ x", "CategoryTheory.Bicategory.Lan.CommuteWith f g h"], "goal": "(CategoryTheory.Bicategory.Lan.CommuteWith.lanCompIsoWhisker f g h).inv.right =\n  (CategoryTheory.Bicategory.Lan.CommuteWith.isKan f g h).desc\n    (CategoryTheory.Bicategory.lanLeftExtension f (CategoryTheory.CategoryStruct.comp g h))"}}
+{"state": {"context": ["α : Type u", "Zero α", "LT α", "a : α"], "goal": "0 < ↑a ↔ 0 < a"}}
+{"state": {"context": ["α : Type u", "r : ℕ → ℕ → Prop", "n a : ℕ"], "goal": "r a 0 → (Chain r 0 (map succ (range n)) ↔ Chain' r (0 :: map succ (range n)))"}}
+{"state": {"context": ["a b c : PartENat", "hc : c ≠ ⊤"], "goal": "a + c = b + c ↔ a = b"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "LinearOrderedField α", "ConditionallyCompleteLinearOrderedField β", "Archimedean α", "a : α", "ha : 0 < a", "b : β", "hba : b < LinearOrderedField.inducedMap α β a * LinearOrderedField.inducedMap α β a"], "goal": "∃ c ∈ LinearOrderedField.cutMap β (a * a), b < c"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "FiniteDimensional 𝕜 E", "n : ℕ", "hn : FiniteDimensional.finrank 𝕜 E = n", "i : Fin n"], "goal": "T ((hT.eigenvectorBasis hn) i) = ↑(hT.eigenvalues hn i) • (hT.eigenvectorBasis hn) i"}}
+{"state": {"context": ["α : Type u", "a a₁ : α", "l₁ : List α", "a₀ : α", "l₀ : List α"], "goal": "Stream'.cycleG (a, a₁ :: l₁, a₀, l₀) = (a₁, l₁, a₀, l₀)"}}
+{"state": {"context": ["R : Type u_4", "NormedRing R", "CompleteSpace R", "x : R", "h : ‖x‖ < 1"], "goal": "x * ∑' (i : ℕ), x ^ i = ∑' (i : ℕ), x ^ (i + 1)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : AlgebraicGeometry.SheafedSpace C"], "goal": "(𝟙 X).base = 𝟙 ↑X.toPresheafedSpace"}}
+{"state": {"context": ["ι : Type u_1", "M : ι → Type u_2", "inst✝² : (i : ι) → Monoid (M i)", "N✝ : Type u_3", "inst✝¹ : Monoid N✝", "N : Type u_4", "inst✝ : Monoid N", "f : (i : ι) → M i →* N"], "goal": "⨆ i, Submonoid.closure (range fun b => (f ⟨i, b⟩.fst) ⟨i, b⟩.snd) = ⨆ i, MonoidHom.mrange (f i)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "a✝ a : α", "s : Finset α", "hs : a ∉ s", "ih : ∀ {𝒜 ℬ : Finset (Finset α)}, IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → 𝒜.card * ℬ.card ≤ 2 ^ s.card * (𝒜 ∩ ℬ).card", "𝒜 ℬ : Finset (Finset α)", "h𝒜 : IsLowerSet ↑𝒜", "hℬ : IsLowerSet ↑ℬ", "h𝒜s : ∀ t ∈ 𝒜, t ⊆ insert a s", "hℬs : ∀ t ∈ ℬ, t ⊆ insert a s"], "goal": "2 * ((Finset.memberSubfamily a 𝒜).card * (Finset.memberSubfamily a ℬ).card + (Finset.nonMemberSubfamily a 𝒜).card * (Finset.nonMemberSubfamily a ℬ).card) ≤ 2 * (2 ^ s.card * (𝒜 ∩ ℬ).card)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "IsTrans γ t", "f : r ≺i s", "g : s ≺i t", "a : α"], "goal": "(f.trans g).toRelEmbedding a = g.toRelEmbedding (f.toRelEmbedding a)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f g : E → F", "p pf pg : FormalMultilinearSeries 𝕜 E F", "x✝ : E", "r r' : ℝ≥0∞", "n m : ℕ", "x y : E", "hf : HasFiniteFPowerSeriesOnBall f p x n r", "h : y ∈ EMetric.ball x r"], "goal": "CPolynomialAt 𝕜 f y"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "s✝ s t : Multiset α", "_l₁ _l₂ : List α"], "goal": "_l₁ ≈ _l₂ ↔ ∀ (a : α), count a ⟦_l₁⟧ = count a ⟦_l₂⟧"}}
+{"state": {"context": ["n : ℕ", "R : Type u_1", "hpos : 0 < n", "Ring R", "Nontrivial R"], "goal": "0 < (Polynomial.cyclotomic n R).degree"}}
+{"state": {"context": ["n : Nat", "i : Fin n"], "goal": "i.castSucc.succ = i.succ.castSucc"}}
+{"state": {"context": ["R₂ : Type u_5", "CommSemiring R₂", "n : Type u_11", "Fintype n", "DecidableEq n", "B : LinearMap.BilinForm R₂ (n → R₂)", "i j : n"], "goal": "LinearMap.BilinForm.toMatrix' B i j =\n  (B ((LinearMap.stdBasis R₂ (fun x => R₂) i) 1)) ((LinearMap.stdBasis R₂ (fun x => R₂) j) 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "SMul α β", "Preorder α", "Preorder β", "Zero α", "PosSMulMono α β", "a : α", "s : Set β", "hs : BddAbove s", "ha : 0 ≤ a"], "goal": "BddAbove (a • s)"}}
+{"state": {"context": ["α : Type u", "S : Set (Set (Set α))"], "goal": "TopologicalSpace.generateFrom (⋃₀ S) = ⨅ s ∈ S, TopologicalSpace.generateFrom s"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_4", "TopologicalSpace X", "f g : ι → Set X", "hf : LocallyFinite f", "hg : ∀ (i : ι), g i ⊆ f i"], "goal": "LocallyFinite g"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a : α"], "goal": "1 < a⁻¹ ↔ 0 < a ∧ a < 1"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "inst✝⁸ : Category.{u_4, u_1} C₁", "inst✝⁷ : Category.{u_5, u_2} C₂", "W₁ : MorphismProperty C₁", "W₂ : MorphismProperty C₂", "Φ : LocalizerMorphism W₁ W₂", "inst✝⁶ : Φ.IsLocalizedEquivalence", "inst✝⁵ : W₁.IsMultiplicative", "inst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)", "inst✝³ : Φ.arrow.HasRightResolutions", "inst✝² : W₂.ContainsIdentities", "D : Type u_3", "inst✝¹ : Category.{u_6, u_3} D", "L : C₂ ⥤ D", "inst✝ : L.IsLocalization W₂", "X₂ : C₂", "X₃ : D", "y : L.obj X₂ ⟶ X₃", "w : TwoSquare Φ.functor (Φ.functor ⋙ L) L (𝟭 D) := TwoSquare.mk Φ.functor (Φ.functor ⋙ L) L (𝟭 D) (Φ.functor ⋙ L).rightUnitor.inv", "this : Nonempty (w.CostructuredArrowDownwards y)", "c : C₁", "g : X₂ ⟶ Φ.functor.obj c", "x : L.obj (Φ.functor.obj c) ⟶ X₃", "fac : L.map g ≫ 𝟙 (L.obj (Φ.functor.obj c)) ≫ x = y"], "goal": "∃ Y, Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac) ((fromRightResolution Φ L y).obj Y)"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "E : Type u_2", "LE E", "hm : m ≤ m0", "f₁ f₂ : α → E", "h12 : f₁ ≤ᵐ[μ.trim hm] f₂"], "goal": "f₁ ≤ᵐ[μ] f₂"}}
+{"state": {"context": ["n : ℕ", "q : ℝ", "hn' : 3 ≤ n", "hq' : 1 < q", "hn : 0 < n", "hq : 0 < q"], "goal": "eval q (cyclotomic n ℝ) < (q + 1) ^ φ n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝ : Semiring R", "p q r : R[X]", "hq : q.Monic", "hpq : p.degree < q.degree"], "goal": "(p + q).Monic"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "E : Type u_4", "F : Type u_6", "NormedAddCommGroup E", "NormedAddCommGroup F", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜₂", "NormedSpace 𝕜 E", "NormedSpace 𝕜₂ F", "σ₁₂ : 𝕜 →+* 𝕜₂", "Nontrivial E", "RingHomIsometric σ₁₂", "f : E →ₛₗᵢ[σ₁₂] F"], "goal": "‖f.toContinuousLinearMap‖ = 1"}}
+{"state": {"context": ["P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop", "h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P"], "goal": "∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y], sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] => P) f → sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] => P) (f ≫ e.hom)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "x✝ y : α", "inst✝³ : DecidableRel f.SameCycle", "inst✝² : DecidableRel g.SameCycle", "inst✝¹ : DecidableRel (f * g).SameCycle", "inst✝ : DecidableRel (g * f).SameCycle", "h : f.Disjoint g", "x : α"], "goal": "(f * g).cycleOf x = f.cycleOf x * g.cycleOf x"}}
+{"state": {"context": ["x : ℝ", "h : Irrational x", "m : ℕ", "hm : m ≠ 0"], "goal": "Irrational (x * ↑m)"}}
+{"state": {"context": ["C : Type v₁", "CategoryTheory.Category.{v₁, v₁} C", "D : Type u₂", "CategoryTheory.Category.{v₁, u₂} D", "F : CategoryTheory.Functor C D", "CategoryTheory.IsFilteredOrEmpty C"], "goal": "F.Final ↔\n  (∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)) ∧\n    ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),\n      ∃ c' t, CategoryTheory.CategoryStruct.comp s (F.map t) = CategoryTheory.CategoryStruct.comp s' (F.map t)"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "inst✝¹⁸ : Field F", "inst✝¹⁷ : Field K", "inst✝¹⁶ : Algebra F K", "K₁ : Type u_3", "K₂ : Type u_4", "K₃ : Type u_5", "inst✝¹⁵ : Field F", "inst✝¹⁴ : Field K₁", "inst✝¹³ : Field K₂", "inst✝¹² : Field K₃", "inst✝¹¹ : Algebra F K₁", "inst✝¹⁰ : Algebra F K₂", "inst✝⁹ : Algebra F K₃", "ϕ : K₁ →ₐ[F] K₂", "χ : K₁ ≃ₐ[F] K₂", "ψ : K₂ →ₐ[F] K₃", "ω : K₂ ≃ₐ[F] K₃", "E : Type u_6", "inst✝⁸ : Field E", "inst✝⁷ : Algebra F E", "inst✝⁶ : Algebra E K₁", "inst✝⁵ : Algebra E K₂", "inst✝⁴ : Algebra E K₃", "inst✝³ : IsScalarTower F E K₁", "inst✝² : IsScalarTower F E K₂", "inst✝¹ : IsScalarTower F E K₃", "inst✝ : Normal F E", "x✝ : E"], "goal": "(algebraMap E K₃) (((ψ.restrictNormal E).comp (ϕ.restrictNormal E)) x✝) = (algebraMap E K₃) (((ψ.comp ϕ).restrictNormal E) x✝)"}}
+{"state": {"context": ["M : Type u_1", "inst✝⁵ : Monoid M", "inst✝⁴ : MeasurableSpace M", "inst✝³ : MeasurableMul₂ M", "μ ν ρ : Measure M", "inst✝² : SFinite μ", "inst✝¹ : SFinite ν", "inst✝ : SFinite ρ"], "goal": "Measurable fun x => x.1 * x.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "b : β", "p : β → Prop", "s : (x : β) → x = b ∨ p x → Set α"], "goal": "⋃ x, ⋃ (h : x = b ∨ p x), s x h = s b ⋯ ∪ ⋃ x, ⋃ (h : p x), s x ⋯"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Add β", "F : AddHom (FreeAddMagma α) β"], "goal": "∀ (a : α), FreeAddMagma.lift.symm F a = (⇑F ∘ FreeAddMagma.of) a"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MeasurableSpace α", "f : α → ℝ", "g : ℝ → ℝ", "s : Set α", "μ : Measure α", "inst✝ : SFinite μ", "f_nn : 0 ≤ f", "f_mble : Measurable f", "g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t", "g_mble : Measurable g", "g_nn : ∀ t > 0, 0 ≤ g t", "g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t", "integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)"], "goal": "∫⁻ (y : ℝ), ∫⁻ (x : α), (Ioc 0 (f x)).indicator (fun t => ENNReal.ofReal (g t)) y ∂μ ∂volume = ∫⁻ (a : ℝ), (Ioi 0).indicator (fun t => μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a"}}
+{"state": {"context": ["α : Type u_1", "Zero α", "One α", "LE α", "ZeroLEOneClass α", "x : α", "hx : 0 ≤ x"], "goal": "⟨x, hx⟩ = 1 ↔ x = 1"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁷ : Ring R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : AddCommGroup M''", "inst✝³ : Module R M", "inst✝² : Module R M'", "inst✝¹ : Module R M''", "a b : R", "x✝ y : M", "inst✝ : Nontrivial R", "hv : LinearIndependent R v", "x : ι", "f : ι →₀ R", "h : x ∉ ↑f.support"], "goal": "(Finsupp.total ι M R v) f ≠ v x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "TopologicalSpace M", "AddMonoid M", "ContinuousAdd M", "f : ι → α → M", "x : Filter α", "a : ι → M", "l : List ι"], "goal": "(∀ i ∈ l, Filter.Tendsto (f i) x (𝓝 (a i))) →\n  Filter.Tendsto (fun b => (List.map (fun c => f c b) l).sum) x (𝓝 (List.map a l).sum)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "hg₁ : S₁.g = 0", "c₁ : CategoryTheory.Limits.CokernelCofork S₁.f", "hc₁ : CategoryTheory.Limits.IsColimit c₁", "hg₂ : S₂.g = 0", "c₂ : CategoryTheory.Limits.CokernelCofork S₂.f", "hc₂ : CategoryTheory.Limits.IsColimit c₂", "f : c₁.pt ⟶ c₂.pt", "comm :\n  CategoryTheory.CategoryStruct.comp φ.τ₂ (CategoryTheory.Limits.Cofork.π c₂) =\n    CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Cofork.π c₁) f"], "goal": "(CategoryTheory.ShortComplex.LeftHomologyMapData.ofIsColimitCokernelCofork φ hg₁ c₁ hc₁ hg₂ c₂ hc₂ f comm).φK = φ.τ₂"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "x y z : R", "h : IsCoprime (x + y * z) y"], "goal": "IsCoprime x y"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Group α", "MulAction α β", "s : Set β"], "goal": "⋃ g, g⁻¹ • s = ⋃ g, g • s"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C"], "goal": "(CategoryTheory.LaxBraidedFunctor.id C).ε = 𝟙 (𝟙_ C)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedAddCommGroup G", "𝕜 : Type u_5", "inst✝⁴ : NormedDivisionRing 𝕜", "inst✝³ : MulActionWithZero 𝕜 E", "inst✝² : Module 𝕜 F", "inst✝¹ : BoundedSMul 𝕜 E", "inst✝ : BoundedSMul 𝕜 F", "c : 𝕜", "f : α → F", "hc : c ≠ 0"], "goal": "↑‖c‖₊ * eLpNorm f p μ ≤ eLpNorm (c • f) p μ"}}
+{"state": {"context": ["R S Rₘ Sₘ : Type u", "inst✝¹⁵ : CommRing R", "inst✝¹⁴ : CommRing S", "inst✝¹³ : CommRing Rₘ", "inst✝¹² : CommRing Sₘ", "M : Submonoid R", "inst✝¹¹ : Algebra R S", "inst✝¹⁰ : Algebra R Sₘ", "inst✝⁹ : Algebra S Sₘ", "inst✝⁸ : Algebra R Rₘ", "inst✝⁷ : Algebra Rₘ Sₘ", "inst✝⁶ : IsScalarTower R Rₘ Sₘ", "inst✝⁵ : IsScalarTower R S Sₘ", "inst✝⁴ : IsLocalization M Rₘ", "inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ", "inst✝² : FormallySmooth R Sₘ", "Q : Type u", "inst✝¹ : CommRing Q", "inst✝ : Algebra Rₘ Q", "I : Ideal Q", "e : I ^ 2 = ⊥", "f✝ : Sₘ →ₐ[Rₘ] Q ⧸ I", "this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra", "this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl", "f : Sₘ →ₐ[Rₘ] Q := let __src := lift I ⋯ (AlgHom.restrictScalars R f✝); { toRingHom := __src.toRingHom, commutes' := ⋯ }"], "goal": "(Ideal.Quotient.mkₐ Rₘ I).comp f = f✝"}}
+{"state": {"context": ["m n : ℕ"], "goal": "↑(Nat.bitwise (fun x y => x && !!y) m n) = ↑(m &&& n)"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "SMul 𝕜 E", "p : Seminorm 𝕜 E", "c : ℝ≥0", "hc : 0 < c", "r : ℝ", "x : E"], "goal": "(c • p).closedBall x r = p.closedBall x (r / ↑c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : PartialEquiv α β", "s : Set β"], "goal": "e.target ∩ ↑e.symm ⁻¹' (↑e ⁻¹' s) = e.target ∩ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "Preorder α", "Preorder β", "a : α", "h : StrictAnti f"], "goal": "f '' Set.Ioi a ⊆ Set.Iio (f a)"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "f : V ≃ₗᵢ[ℝ] V", "hd : 0 < LinearMap.det ↑f.toLinearEquiv", "this : Nontrivial V", "x : V", "hx : x ≠ 0"], "goal": "f = o.rotation (o.oangle x (f x))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "D : Type w", "CategoryTheory.Category.{max v u, w} D", "CategoryTheory.ConcreteCategory D", "CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget D)", "∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D", "∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X), CategoryTheory.Limits.HasMultiequalizer (S.index P)", "(X : C) → CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)", "X : C", "P : CategoryTheory.Functor Cᵒᵖ D", "S T : J.Cover X", "x : CategoryTheory.Meq P S", "y : CategoryTheory.Meq P T"], "goal": "CategoryTheory.GrothendieckTopology.Plus.mk x = CategoryTheory.GrothendieckTopology.Plus.mk y ↔\n  ∃ W h1 h2, x.refine h1 = y.refine h2"}}
+{"state": {"context": ["a✝ b✝ : ℝ", "f f' : ℝ → ℝ", "s : Set ℝ", "hs : s.OrdConnected", "hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x", "a : ℝ", "ha : a ∈ s", "b : ℝ", "hb : b ∈ s", "m : ℝ", "hma : f' a < m", "hmb : m < f' b", "hab : a ≤ b"], "goal": "m ∈ f' '' s"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Ring R", "p✝ q✝ p q : R[X]"], "goal": "(p - q).natDegree ≤ max p.natDegree q.natDegree"}}
+{"state": {"context": ["c : Cardinal.{u_1}"], "goal": "c ^< ℵ₀ ≤ max c ℵ₀"}}
+{"state": {"context": ["z : ℝ"], "goal": "z + 2 * π = z + ↑1 * (2 * π)"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "inst✝⁶ : Field K", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝³ : H.IsCartanSubalgebra", "inst✝² : IsTriangularizable K (↥H) L", "inst✝¹ : IsKilling K L", "inst✝ : CharZero K", "x : ↥H", "hαx : 0 x = 0", "hx : x ∈ corootSpace 0"], "goal": "x = 0"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "n : ℕ", "n.AtLeastTwo"], "goal": "↑(OfNat.ofNat n) = OfNat.ofNat n"}}
+{"state": {"context": ["E : Type u_2", "SeminormedAddCommGroup E", "NormedSpace ℝ E", "δ : ℝ", "hδ : 0 < δ", "s : Set E", "x : E"], "goal": "EMetric.infEdist x (Metric.thickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "φ : PowerSeries R"], "goal": "φ.order.Dom ↔ φ ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁵ : TopologicalSpace α", "ι : Type u_5", "π : ι → Type u_6", "inst✝⁴ : (i : ι) → TopologicalSpace (π i)", "inst✝³ : TopologicalSpace β", "inst✝² : TopologicalSpace γ", "inst✝¹ : TopologicalSpace δ", "p : α → Prop", "f g : α → β", "inst✝ : (a : α) → Decidable (p a)", "hp : ∀ a ∈ frontier {x | p x}, f a = g a", "hf : ContinuousOn f (closure {x | p x})", "hg : ContinuousOn g (closure {x | ¬p x})"], "goal": "ContinuousOn (fun a => if p a then f a else g a) univ"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SMul α β", "f : Filter α", "g : Filter β"], "goal": "(f • g).NeBot ↔ f.NeBot ∧ g.NeBot"}}
+{"state": {"context": ["z : ℂ"], "goal": "Complex.normSq (-z) = Complex.normSq z"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l : α → β", "u : β → α", "Preorder α", "Preorder β", "gi : GaloisInsertion l u", "a b : β"], "goal": "u a ≤ u b ↔ a ≤ b"}}
+{"state": {"context": [], "goal": "Continuous ⇑NNReal.sqrt"}}
+{"state": {"context": ["α : Type u_1", "NormedRing α", "Nontrivial α", "x : αˣ"], "goal": "0 < ‖↑x‖₊"}}
+{"state": {"context": ["E : Type u", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℂ E", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℂ F", "inst✝¹ : Nontrivial E", "inst✝ : FiniteDimensional ℂ E", "f : E → F", "U : Set E", "hb : Bornology.IsBounded U", "hne : U.Nonempty", "hd : DiffContOnCl ℂ f U", "hc : IsCompact (closure U)", "w : E", "hle : IsMaxOn (norm ∘ f) (closure U) w", "hwU : w ∈ interior U", "this : interior U ≠ univ", "z : E", "hzU : z ∈ frontier (interior U)", "hzw : infDist w (interior U)ᶜ = dist w z"], "goal": "∃ z ∈ frontier U, IsMaxOn (norm ∘ f) (closure U) z"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : Infinite R", "σ : Type u_2", "n : ℕ", "f : Fin n → σ", "hf : Function.Injective f", "p : MvPolynomial (Fin n) R", "h : ∀ (x : σ → R), (eval x) ((rename f) p) = 0"], "goal": "(rename f) p = 0"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "CommRing R", "hp : Fact (Nat.Prime p)", "n : ℕ"], "goal": "MvPolynomial.constantCoeff (W_ R n) = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "n : ℕ", "p : ℕ → ℝ", "hp : ∀ (k : ℕ), 0 ≤ p k", "r a : ℝ", "hr : 0 ≤ r", "ha : 0 ≤ a", "k : ℕ", "c : Composition k", "hd : (2 ≤ k ∧ k < n + 1) ∧ 1 < c.length"], "goal": "2 ≤ c.length ∧ c.length < n + 1 ∧ ∀ (j : Fin c.length), c.blocksFun j < n"}}
+{"state": {"context": ["V : Type u_1", "R : Type u_2", "Fintype V", "G : SimpleGraph V", "DecidableRel G.Adj", "AddCommMonoidWithOne R", "i : V"], "goal": "↑(G.degree i) = ∑ j : V, if G.Adj i j then 1 else 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹⁴ : CommRing R", "inst✝¹³ : CommRing S", "inst✝¹² : Algebra R S", "P : Generators R S", "R' : Type u'", "S' : Type v'", "inst✝¹¹ : CommRing R'", "inst✝¹⁰ : CommRing S'", "inst✝⁹ : Algebra R' S'", "P' : Generators R' S'", "inst✝⁸ : Algebra R R'", "inst✝⁷ : Algebra S S'", "inst✝⁶ : Algebra R S'", "inst✝⁵ : IsScalarTower R R' S'", "inst✝⁴ : IsScalarTower R S S'", "T : Type uT", "inst✝³ : CommRing T", "inst✝² : Algebra R T", "inst✝¹ : Algebra S T", "inst✝ : IsScalarTower R S T", "f g : P.Hom P'", "x : P.Ring"], "goal": "(f.toAlgHom.toLinearMap - g.toAlgHom.toLinearMap) x ∈ Submodule.restrictScalars R P'.ker"}}
+{"state": {"context": ["n✝ b b' : ℕ", "c : ℤ", "h : ↑b ∣ ↑b' - c", "n : ℕ"], "goal": "b ∣ n ↔ ↑b ∣ ofDigits c (b'.digits n)"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f✝ f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x✝ : 𝕜", "s✝ t✝ : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "f : 𝕜 → F", "s t : Set 𝕜", "h : s ⊆ closure (s ∩ t)", "x : 𝕜", "H : (𝓝[s \\ {x}] x).NeBot", "H' : DifferentiableWithinAt 𝕜 f s x", "I : (𝓝[(s ∩ t) \\ {x}] x).NeBot", "this : Tendsto (slope f x) (𝓝[(s ∩ t) \\ {x}] x) (𝓝 (derivWithin f s x))"], "goal": "∀ᶠ (x_1 : 𝕜) in 𝓝[(s ∩ t) \\ {x}] x, slope f x x_1 ∈ ↑(Submodule.span 𝕜 (f '' t)).topologicalClosure"}}
+{"state": {"context": ["k : ℕ", "x : ℝ", "hk : k ≠ 0", "hx : x ∈ Icc 0 1"], "goal": "hurwitzZetaEven (↑x) (-(2 * ↑k)) = 0"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "CommGroup α", "e : α", "x : Finset α × Finset α"], "goal": "Finset.mulDysonETransform e (Finset.mulDysonETransform e x) = Finset.mulDysonETransform e x"}}
+{"state": {"context": ["M : Type u_5", "CommMonoid M", "ι : Type u_6", "s : Finset ι", "f g : ι → M", "h : ∀ i ∈ s, f i ~ᵤ g i"], "goal": "∏ i ∈ s, f i ~ᵤ ∏ i ∈ s, g i"}}
+{"state": {"context": ["G : Type w", "TopologicalSpace G", "μ : MeasureTheory.Content G", "R1Space G", "U : ℕ → Set G", "hU : ∀ (i : ℕ), IsOpen (U i)"], "goal": "μ.innerContent { carrier := ⋃ i, U i, is_open' := ⋯ } ≤ ∑' (i : ℕ), μ.innerContent { carrier := U i, is_open' := ⋯ }"}}
+{"state": {"context": ["α : Type u_1", "dec : DecidableEq α", "a b : α", "h : a ≠ b"], "goal": "(RegularExpression.char a).deriv b = 0"}}
+{"state": {"context": ["G₁ : Type u_2", "G₂ : Type u_3", "TopologicalSpace G₂", "T2Space G₂", "Neg G₁", "Neg G₂", "ContinuousNeg G₂"], "goal": "IsClosed {f | ∀ (x : G₁), f (-x) = -f x}"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι✝ : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t✝ : Set X", "ι : Type v", "hs : IsLindelof s", "t : ι → Set X", "htc : ∀ (i : ι), IsClosed (t i)", "hst : s ∩ ⋂ i, t i = ∅", "U : ι → Set X := tᶜ", "hUo : ∀ (i : ι), IsOpen (U i)", "hsU : s ⊆ ⋃ i, U i", "u : Set ι", "hucount : u.Countable", "husub : Disjoint (⋃ i ∈ u, U i)ᶜ s"], "goal": "s ∩ ⋂ i ∈ u, t i = ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹² : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝⁷ : NormedAddCommGroup G", "inst✝⁶ : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝⁵ : NormedAddCommGroup G'", "inst✝⁴ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "𝔸 : Type u_6", "𝔸' : Type u_7", "inst✝³ : NormedRing 𝔸", "inst✝² : NormedCommRing 𝔸'", "inst✝¹ : NormedAlgebra 𝕜 𝔸", "inst✝ : NormedAlgebra 𝕜 𝔸'", "a b : E → 𝔸", "a' b' : E →L[𝕜] 𝔸", "c d : E → 𝔸'", "c' d' : E →L[𝕜] 𝔸'", "hc : HasStrictFDerivAt c c' x", "hd : HasStrictFDerivAt d d' x", "z : E"], "goal": "(d x • c') z = (c'.smulRight (d x)) z"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "s : σ →₀ ℕ", "x : MvPolynomial σ R", "s' : σ →₀ ℕ"], "goal": "MvPolynomial.coeff s' (x.divMonomial s) = MvPolynomial.coeff (s + s') x"}}
+{"state": {"context": ["n k : ℕ", "h : n ∈ k.smoothNumbers"], "goal": "∃ s ∈ k.primesBelow.powerset, ∃ m, n = m ^ 2 * s.prod id"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "S : IntermediateField F E", "t : Finset E", "ht : Algebra.adjoin F ↑t = S.toSubalgebra"], "goal": "S.FG"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "NormalizedGCDMonoid α", "DecidableEq α", "a : α", "s : Multiset α"], "goal": "(Multiset.ndinsert a s).gcd = gcd a s.gcd"}}
+{"state": {"context": ["s : Set ℤ", "h : ¬BddAbove s"], "goal": "sSup s = 0"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "y : M", "h : x ∈ span R {y}"], "goal": "∃ a, a • y = 0"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝⁷ : CommRing R", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra R L", "inst✝⁴ : LieAlgebra.IsNilpotent R L", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : LieRingModule L M", "inst✝ : LieModule R L M"], "goal": "(weightSpace M 0).normalizer ≤ weightSpace M 0"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "M : Type u_3", "M₂ : Type u_4", "M₃ : Type u_5", "M₄ : Type u_6", "inst✝⁹ : CommRing R", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : AddCommGroup M₂", "inst✝⁶ : AddCommGroup M₃", "inst✝⁵ : AddCommGroup M₄", "inst✝⁴ : Module R M", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M₄", "inst✝ : Module.Flat R M₂", "rs : List R", "h : IsWeaklyRegular M rs"], "goal": "IsWeaklyRegular (M₂ ⊗[R] M) rs"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "I : CategoryTheory.Limits.MultispanIndex C", "K : CategoryTheory.Limits.Multicofork I", "desc : (E : CategoryTheory.Limits.Multicofork I) → K.pt ⟶ E.pt", "fac :\n  ∀ (E : CategoryTheory.Limits.Multicofork I) (i : I.R), CategoryTheory.CategoryStruct.comp (K.π i) (desc E) = E.π i", "uniq :\n  ∀ (E : CategoryTheory.Limits.Multicofork I) (m : K.pt ⟶ E.pt),\n    (∀ (i : I.R), CategoryTheory.CategoryStruct.comp (K.π i) m = E.π i) → m = desc E", "E : CategoryTheory.Limits.Multicofork I"], "goal": "(CategoryTheory.Limits.Multicofork.IsColimit.mk K desc fac uniq).desc E = desc E"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "s✝ s : Multiset α", "f : α → β", "x : { x // x ∈ s }"], "goal": "map (fun x => (f ∘ val) x) (s.attach.erase x) = map f (s.erase ↑x)"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "P : Cᵒᵖ ⥤ Type (max v u)", "X : C", "R : Presieve X", "S : Sieve X", "z₁ z₂ : FirstObj P R", "h : ∀ (Y : C) (f : Y ⟶ X) (hf : R f), Pi.π (fun f => P.obj (op f.fst)) ⟨Y, ⟨f, hf⟩⟩ z₁ = Pi.π (fun f => P.obj (op f.fst)) ⟨Y, ⟨f, hf⟩⟩ z₂"], "goal": "∀ (j : Discrete ((Y : C) × { f // R f })), limit.π (Discrete.functor fun f => P.obj (op f.fst)) j z₁ = limit.π (Discrete.functor fun f => P.obj (op f.fst)) j z₂"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s t : Set α", "hμst : μ (symmDiff s t) ≠ ⊤"], "goal": "μ s = ⊤ ↔ μ t = ⊤"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "V W X Y Z : Mon_ C", "M : Bimod V W", "N : Bimod W X", "P : Bimod X Y", "Q : Bimod Y Z"], "goal": "coequalizer.desc ((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N P) ((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)) ⋯) ⋯ ▷ Q.X ≫ ((PreservesCoequalizer.iso (tensorRight Q.X) (M.actRight ▷ TensorBimod.X N P) ((α_ M.X W.X (TensorBimod.X N P)).hom ≫ M.X ◁ TensorBimod.actLeft N P)).inv ≫ coequalizer.desc ((α_ M.X (TensorBimod.X N P) Q.X).hom ≫ M.X ◁ coequalizer.π (TensorBimod.actRight N P ▷ Q.X) ((α_ (TensorBimod.X N P) Y.X Q.X).hom ≫ TensorBimod.X N P ◁ Q.actLeft) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q) ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q)) ⋯) ≫ colimMap (parallelPairHom (M.actRight ▷ TensorBimod.X (N.tensorBimod P) Q) ((α_ M.X W.X (TensorBimod.X (N.tensorBimod P) Q)).hom ≫ M.X ◁ TensorBimod.actLeft (N.tensorBimod P) Q) (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q)) ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q)) ((M.X ⊗ W.X) ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ⋯) (M.X ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight Q.X) (N.actRight ▷ P.X) ((α_ N.X X.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ N.X P.X Q.X).hom ≫ N.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q)) ⋯) ⋯) ⋯ ⋯) = ((PreservesCoequalizer.iso (tensorRight Q.X) (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) X.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ (TensorBimod.X M N) P.X Q.X).hom ≫ TensorBimod.X M N ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X Y.X Q.X).hom ≫ P.X ◁ Q.actLeft) ≫ coequalizer.π (TensorBimod.actRight M N ▷ TensorBimod.X P Q) ((α_ (TensorBimod.X M N) X.X (TensorBimod.X P Q)).hom ≫ TensorBimod.X M N ◁ TensorBimod.actLeft P Q)) ⋯) ≫ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight (TensorBimod.X P Q)) (M.actRight ▷ N.X) ((α_ M.X W.X N.X).hom ≫ M.X ◁ N.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X (TensorBimod.X P Q)).hom ≫ M.X ◁ coequalizer.π (N.actRight ▷ TensorBimod.X P Q) ((α_ N.X X.X (TensorBimod.X P Q)).hom ≫ N.X ◁ TensorBimod.actLeft P Q) ≫ coequalizer.π (M.actRight ▷ TensorBimod.X N (P.tensorBimod Q)) ((α_ M.X W.X (TensorBimod.X N (P.tensorBimod Q))).hom ≫ M.X ◁ TensorBimod.actLeft N (P.tensorBimod Q))) ⋯) ⋯"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Group α", "e : α", "x : Finset α × Finset α"], "goal": "x.2.card + (e⁻¹ • x.2).card = 2 * x.2.card"}}
+{"state": {"context": ["g : ℝ → ℝ", "g' u : ℝ", "L' : Filter ℝ", "h : HasGradientAtFilter g g' u L'"], "goal": "HasDerivAtFilter g g' u L'"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "G : Type u_3", "G₀ : Type u_4", "R : Type u_5", "K : Type u_6", "inst✝ : Monoid M", "x : Mˣ", "u : (↥(Submonoid.centralizer {↑x}))ˣ"], "goal": "(Units.map (Submonoid.centralizer {↑x}).subtype) u * x = x * (Units.map (Submonoid.centralizer {↑x}).subtype) u"}}
+{"state": {"context": ["R : Type u_2", "M₁ : Type u_3", "M₂ : Type u_4", "P : Type u_7", "CommSemiring R", "AddCommMonoid M₁", "AddCommMonoid M₂", "AddCommMonoid P", "Module R M₁", "Module R M₂", "Module R P", "PartialOrder P", "CovariantClass P P (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Q₁ : QuadraticMap R M₁ P", "Q₂ : QuadraticMap R M₂ P", "h₁ : Q₁.PosDef", "h₂ : Q₂.PosDef"], "goal": "(Q₁.prod Q₂).PosDef"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : SeminormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "x : F", "h : ‖x‖ = ‖0‖"], "goal": "SameRay ℝ x 0 ↔ x = 0"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁶ : CommSemiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid N", "inst✝² : Module R N", "inst✝¹ : AddCommMonoid P", "inst✝ : Module R P", "f : N →ₗ[R] P", "Q : QuadraticMap R M N", "B : BilinMap R M N", "h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y", "x y : M"], "goal": "(fun x => f (Q x)) (x + y) = (fun x => f (Q x)) x + (fun x => f (Q x)) y + ((LinearMap.compr₂ B f) x) y"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SMul α β", "s₁ s₂ : Set α", "t₁ t₂ : Set β"], "goal": "(s₁ ∩ s₂) • (t₁ ∪ t₂) ⊆ s₁ • t₁ ∪ s₂ • t₂"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "AddCommMonoid M", "f : α → M", "s : Set α", "DecidablePred fun x => x ∈ s", "hf : (Function.support f).Finite"], "goal": "∑ᶠ (i : α) (_ : i ∈ s), f i = ∑ i ∈ Finset.filter (fun x => x ∈ s) hf.toFinset, f i"}}
+{"state": {"context": ["α : Type u", "l : List α", "a : α"], "goal": "{x | x ∈ a :: l} = insert a {x | x ∈ l}"}}
+{"state": {"context": ["α : Type u_1", "s t : Set α", "hs : s.Finite := by toFinite_tac", "ht : t.Finite := by toFinite_tac"], "goal": "s.ncard < t.ncard ↔ (s \\ t).ncard < (t \\ s).ncard"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f₁ : α → Option β", "f₂ : β → Option α", "h : ∀ (a : α) (b : β), a ∈ f₂ b ↔ b ∈ f₁ a", "x : α"], "goal": "{ toFun := f₁, invFun := f₂, inv := h } x = f₁ x"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "P' : Type u_3", "CommRing P'", "Algebra R P'", "I : FractionalIdeal S P", "g : P →ₐ[R] P'", "y : P'"], "goal": "y ∈ FractionalIdeal.map g I ↔ ∃ x ∈ I, g x = y"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝¹ : Add R", "inst✝ : Mul R", "c : RingCon R"], "goal": "ringConGen ⇑c ≤ c"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "CategoryTheory.Category.{u_5, u_1} C₁", "CategoryTheory.Category.{u_6, u_2} C₂", "W₁ : CategoryTheory.MorphismProperty C₁", "W₂ : CategoryTheory.MorphismProperty C₂", "Φ : CategoryTheory.LocalizerMorphism W₁ W₂", "X₂ : C₂", "W₁.IsMultiplicative", "L L' : Φ.LeftResolution X₂", "φ₁ φ₂ : L ⟶ L'", "h : φ₁.f = φ₂.f"], "goal": "φ₁ = φ₂"}}
+{"state": {"context": ["α : Type u_1", "a : α", "as : List α", "i : Fin (as ++ [a]).length", "h : ¬↑i < as.length"], "goal": "(as ++ [a]).get i = a"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommRing R", "S : Type u_2", "inst✝⁸ : CommRing S", "inst✝⁷ : Algebra R S", "inst✝⁶ : IsDomain R", "inst✝⁵ : IsDomain S", "inst✝⁴ : IsDedekindDomain R", "inst✝³ : IsDedekindDomain S", "inst✝² : Module.Finite R S", "inst✝¹ : Module.Free R S", "I J : Ideal S", "inst✝ : Nontrivial R", "h✝ : Nontrivial S", "P : Ideal R", "hP : P.IsMaximal", "hP0 : ¬P = ⊥", "P' : Submonoid S := Algebra.algebraMapSubmonoid S P.primeCompl", "this✝⁴ : Algebra (Localization.AtPrime P) (Localization P') := localizationAlgebra P.primeCompl S", "this✝³ : IsScalarTower R (Localization.AtPrime P) (Localization P')", "h : P' ≤ S⁰", "this✝² : IsDomain (Localization P')", "this✝¹ : IsDedekindDomain (Localization P')", "this✝ : DecidableEq (Ideal (Localization P')) := Classical.decEq (Ideal (Localization P'))", "this : IsPrincipalIdealRing (Localization P')"], "goal": "map (algebraMap R (Localization.AtPrime P)) (spanNorm R (I * J)) = map (algebraMap R (Localization.AtPrime P)) (spanNorm R I * spanNorm R J)"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I✝ J : Box ι", "i✝ : ι", "x✝ : ℝ", "I : Box ι", "i : ι", "x : ℝ"], "goal": "(split I i x).iUnion = ↑I"}}
+{"state": {"context": ["p : ℕ → Prop", "hf : (setOf p).Finite", "n : ℕ", "hn : hf.toFinset.card ≤ n"], "goal": "Nat.nth p n = 0"}}
+{"state": {"context": ["F : Type u_1", "CommSemiring F", "q : ℕ", "f : Polynomial F", "hf : Polynomial.HasSeparableContraction q f"], "goal": "∃ m, hf.degree * q ^ m = f.natDegree"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "s : ι → Set (Set Ω)", "s' : Set (Set Ω)", "_mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "hyp : ∀ (n : ι), IndepSets (s n) s' κ μ", "t1 t2 : Set Ω", "ht1 : ∃ i, t1 ∈ s i", "ht2 : t2 ∈ s'"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "inst✝¹⁹ : Field F", "inst✝¹⁸ : Field K", "inst✝¹⁷ : Algebra F K", "K₁ : Type u_3", "K₂ : Type u_4", "K₃ : Type u_5", "inst✝¹⁶ : Field F", "inst✝¹⁵ : Field K₁", "inst✝¹⁴ : Field K₂", "inst✝¹³ : Field K₃", "inst✝¹² : Algebra F K₁", "inst✝¹¹ : Algebra F K₂", "inst✝¹⁰ : Algebra F K₃", "ϕ : K₁ →ₐ[F] K₂", "χ : K₁ ≃ₐ[F] K₂", "ψ : K₂ →ₐ[F] K₃", "ω : K₂ ≃ₐ[F] K₃", "E✝ : Type u_6", "inst✝⁹ : Field E✝", "inst✝⁸ : Algebra F E✝", "inst✝⁷ : Algebra E✝ K₁", "inst✝⁶ : Algebra E✝ K₂", "inst✝⁵ : Algebra E✝ K₃", "inst✝⁴ : IsScalarTower F E✝ K₁", "inst✝³ : IsScalarTower F E✝ K₂", "inst✝² : IsScalarTower F E✝ K₃", "inst✝¹ : Algebra F K", "E : IntermediateField F K", "inst✝ : Normal F ↥E", "f : ↥E →ₐ[F] K", "this : Algebra ↥E ↥E := Algebra.id ↥E", "g : ↥E ≃ₐ[F] ↥E := f.restrictNormal' ↥E"], "goal": "f.fieldRange = E"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "s : Setoid α", "t : Setoid β", "f : α → β → γ", "h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), Setoid.r a₁ b₁ → Setoid.r a₂ b₂ → f a₁ a₂ = f b₁ b₂", "a : α", "b : β"], "goal": "⟦a⟧.liftOn₂' ⟦b⟧ f h = f a b"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "NormedAddCommGroup F", "NormedSpace ℝ F", "NormedAddCommGroup F'", "NormedSpace ℝ F'", "NormedAddCommGroup G", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "T : Set α → F →L[ℝ] F'", "h_add : MeasureTheory.FinMeasAdditive μ T", "f : MeasureTheory.SimpleFunc α G", "hf : MeasureTheory.Integrable (↑f) μ", "g : G → F", "hg : g 0 = 0"], "goal": "MeasureTheory.SimpleFunc.setToSimpleFunc T (MeasureTheory.SimpleFunc.map g f) = ∑ x ∈ f.range, (T (↑f ⁻¹' {x})) (g x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "β : Type u_5", "OrderedSemiring 𝕜", "TopologicalSpace β", "LinearOrderedCancelAddCommMonoid β", "OrderTopology β", "Module 𝕜 β", "OrderedSMul 𝕜 β", "r s : β"], "goal": "StrictConvex 𝕜 (Set.Icc r s)"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "H : Type u_2", "TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "γ : ℝ → M", "v : (x : M) → TangentSpace I x", "s : Set ℝ", "t₀ : ℝ", "hγ : IsIntegralCurveOn γ v s", "ht : t₀ ∈ s"], "goal": "ContinuousAt γ t₀"}}
+{"state": {"context": ["S : Type u_3", "AddSemigroup S", "a b c : S", "hab : AddCommute a b", "hac : AddCommute a c"], "goal": "AddCommute a (b + c)"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p✝ q✝ : ℝ≥0", "p q : ℝ≥0∞", "hpq : p ≤ q", "hp : 0 < p", "hq : q < ⊤"], "goal": "p = 0 ∧ q = 0 ∨ p = 0 ∧ q = ⊤ ∨ p = 0 ∧ 0 < q.toReal ∨ p = ⊤ ∧ q = ⊤ ∨ 0 < p.toReal ∧ q = ⊤ ∨ 0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝¹ : TopologicalSpace α", "ι : Type u_5", "π : ι → Type u_6", "inst✝ : (i : ι) → TopologicalSpace (π i)", "f : β → α", "a : α", "s : Set α", "l : Filter β", "h : ∀ (i : Set α), a ∈ i ∧ IsOpen i → (∀ᶠ (x : β) in l, f x ∈ i) ∧ ∀ᶠ (x : β) in l, f x ∈ s"], "goal": "∀ᶠ (i : β) in l, f i ∈ s"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V"], "goal": "∀ {X Y : (Finset V)ᵒᵖ} (f : X ⟶ Y) (C : G.ComponentCompl ↑(Opposite.unop X)),\n  G.componentComplFunctor.map f C = SimpleGraph.ComponentCompl.hom ⋯ C"}}
+{"state": {"context": ["b p : ℕ", "hb : 0 < b", "hp : 1 ≤ p"], "goal": "b ^ (2 * p) - 1 - (b ^ 2 - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)"}}
+{"state": {"context": ["E : ℕ → Type u_1", "x : (n : ℕ) → E n", "n : ℕ"], "goal": "x ∈ PiNat.cylinder x n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "inst✝² : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "s✝ : Set α", "a✝ : α", "inst✝ : MeasurableSingletonClass α", "a : α", "s : Set α"], "goal": "s ∈ ae (dirac a) ↔ s ∈ pure a"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n : ℕ", "χ : DirichletCharacter R n", "m d : ℕ", "hm : n ∣ m", "hd : m ∣ d"], "goal": "(changeLevel ⋯) χ = (changeLevel hd) ((changeLevel hm) χ)"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s : Set P", "hs : Cospherical s", "p : Fin 3 → P", "hps : Set.range p ⊆ s", "hpi : Function.Injective p"], "goal": "¬Collinear ℝ (Set.range p)"}}
+{"state": {"context": ["n m a b : ℕ", "han : a < n.succ", "hbn : b < n.succ"], "goal": "↑a < ↑b ↔ a < b"}}
+{"state": {"context": ["R : Type x", "NonUnitalNonAssocRing R", "x y : R"], "goal": "Commute x y ↔ ⁅x, y⁆ = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_4", "ι : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "TopologicalSpace M", "AddCommMonoid N", "Module R N", "TopologicalSpace N", "A : Type u_7", "Semiring A", "SMul R A", "Module A M", "Module A N", "IsScalarTower R A M", "IsScalarTower R A N", "f : M [⋀^ι]→L[A] N"], "goal": "⇑(ContinuousAlternatingMap.restrictScalars R f) = ⇑f"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝⁷ : Field K", "inst✝⁶ : CharZero K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : IsKilling K L", "inst✝² : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝¹ : H.IsCartanSubalgebra", "inst✝ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "n : ℤ"], "goal": "rootSpace H (n • ⇑α + ⇑β) ≠ ⊥ ↔ n ∈ Finset.Icc (-↑(chainBotCoeff (⇑α) β)) ↑(chainTopCoeff (⇑α) β)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p✝ : Perm α", "x✝ : α", "p : Perm α", "x y : α", "hy✝ : y ∈ p.toList x", "hy : p.SameCycle x y ∧ x ∈ p.support", "k : ℕ", "hk : k < (p.cycleOf x).support.card", "hk' : (p.toList x).nthLe k ⋯ = y"], "goal": "(p.cycleOf x).IsCycle"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "AddZeroClass M", "a : α", "b : M"], "goal": "(Finsupp.singleAddHom a) b = Finsupp.single a b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l₁ l₂ : List α", "f : α → β", "H : l₁ ⊆ l₂"], "goal": "List.map f l₁ ⊆ List.map f l₂"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "x : α"], "goal": "(FreeAddMonoid.countP p) (FreeAddMonoid.of x) = if p x = (true = true) then 1 else 0"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "R₃ : Type u_3", "R₄ : Type u_4", "E : Type u_5", "E₂ : Type u_6", "E₃ : Type u_7", "E₄ : Type u_8", "F : Type u_9", "𝓕 : Type u_10", "inst✝³³ : Semiring R", "inst✝³² : Semiring R₂", "inst✝³¹ : Semiring R₃", "inst✝³⁰ : Semiring R₄", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "σ₁₃ : R →+* R₃", "σ₃₁ : R₃ →+* R", "σ₁₄ : R →+* R₄", "σ₄₁ : R₄ →+* R", "σ₂₃ : R₂ →+* R₃", "σ₃₂ : R₃ →+* R₂", "σ₂₄ : R₂ →+* R₄", "σ₄₂ : R₄ →+* R₂", "σ₃₄ : R₃ →+* R₄", "σ₄₃ : R₄ →+* R₃", "inst✝²⁹ : RingHomInvPair σ₁₂ σ₂₁", "inst✝²⁸ : RingHomInvPair σ₂₁ σ₁₂", "inst✝²⁷ : RingHomInvPair σ₁₃ σ₃₁", "inst✝²⁶ : RingHomInvPair σ₃₁ σ₁₃", "inst✝²⁵ : RingHomInvPair σ₂₃ σ₃₂", "inst✝²⁴ : RingHomInvPair σ₃₂ σ₂₃", "inst✝²³ : RingHomInvPair σ₁₄ σ₄₁", "inst✝²² : RingHomInvPair σ₄₁ σ₁₄", "inst✝²¹ : RingHomInvPair σ₂₄ σ₄₂", "inst✝²⁰ : RingHomInvPair σ₄₂ σ₂₄", "inst✝¹⁹ : RingHomInvPair σ₃₄ σ₄₃", "inst✝¹⁸ : RingHomInvPair σ₄₃ σ₃₄", "inst✝¹⁷ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "inst✝¹⁶ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄", "inst✝¹⁵ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄", "inst✝¹⁴ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄", "inst✝¹³ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁", "inst✝¹² : RingHomCompTriple σ₄₂ σ₂₁ σ₄₁", "inst✝¹¹ : RingHomCompTriple σ₄₃ σ₃₂ σ₄₂", "inst✝¹⁰ : RingHomCompTriple σ₄₃ σ₃₁ σ₄₁", "inst✝⁹ : SeminormedAddCommGroup E", "inst✝⁸ : SeminormedAddCommGroup E₂", "inst✝⁷ : SeminormedAddCommGroup E₃", "inst✝⁶ : SeminormedAddCommGroup E₄", "inst✝⁵ : Module R E", "inst✝⁴ : Module R₂ E₂", "inst✝³ : Module R₃ E₃", "inst✝² : Module R₄ E₄", "inst✝¹ : NormedAddCommGroup F", "inst✝ : Module R F", "e✝ e : E ≃ₛₗᵢ[σ₁₂] E₂"], "goal": "range ⇑e = univ"}}
+{"state": {"context": ["α : Type u_1", "Append α", "a b : Part α", "ma mb : α", "ha : ma ∈ a", "hb : mb ∈ b"], "goal": "ma ++ mb ∈ a ++ b"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "PartialOrder α", "Preorder β", "s : Set (Lex (α × β))", "hαβ : s.IsPWO"], "goal": "((fun x => (ofLex x).1) '' s).IsPWO"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "self : StarSubalgebra R A", "a : A"], "goal": "a ∈ self.carrier → star a ∈ self.carrier"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F✝ : Type u_3", "𝕜 : Type u_4", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : SMulCommClass ℝ 𝕜 E", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace ℝ F✝", "inst✝⁴ : CompleteSpace F✝", "G : Type u_5", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace ℝ G", "f✝ g : α → E", "m : MeasurableSpace α", "μ✝ : Measure α", "X : Type u_6", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X", "μ : Measure α", "f : ℕ → α → ℝ", "F : α → ℝ", "hf : ∀ (n : ℕ), Integrable (f n) μ", "hF : Integrable F μ", "h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x", "h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))", "f' : ℕ → α → ℝ := fun n x => f n x - f 0 x", "hf'_nonneg : ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, 0 ≤ f' i a", "hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ", "hF_ge : 0 ≤ᶠ[ae μ] fun x => (F - f 0) x", "h_cont : ContinuousAt ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ)"], "goal": "Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal (f' n a) ∂μ) atTop (𝓝 (∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ))"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "g : G", "K : Compacts G", "eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) ⋯ K) - f K", "this : Continuous eval"], "goal": "prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "M : Type u_5", "M' : Type u_6", "N : Type u_7", "P : Type u_8", "G : Type u_9", "H : Type u_10", "R : Type u_11", "S : Type u_12", "inst✝¹ : AddCommMonoid M", "v v₁ v₂ : α →₀ M", "inst✝ : DecidableEq β", "f : α → β", "s : α →��� M", "hf : Set.InjOn f ↑s.support", "x : β", "hx : x ∈ image f s.support"], "goal": "x ∈ (mapDomain f s).support"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "h : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))", "n : ℕ+"], "goal": "IsUnit ↑↑n"}}
+{"state": {"context": ["α : Type u", "l l' : List α", "h : l ~r l'"], "goal": "l.Nodup ↔ l'.Nodup"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_4", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "f : α → β → γ", "hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂ → t (f a b₁) (f a b₂)", "hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂ → t (f a₁ b) (f a₂ b)", "a : α", "b : β"], "goal": "Quot.map₂ f hr hs (Quot.mk r a) (Quot.mk s b) = Quot.mk t (f a b)"}}
+{"state": {"context": ["p' : ℕ", "k : ℤ", "h : ω ^ 2 ^ p' * (ω ^ 2 ^ p' + ωb ^ 2 ^ p') = ω ^ 2 ^ p' * ↑((2 ^ (p' + 2) - 1) * k)", "t : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)"], "goal": "ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝¹ : Semiring R", "inst✝ : DecidableEq R", "n : ℕ"], "goal": "degreeLT R n ≤ Submodule.span R ↑(image (fun n => X ^ n) (range n))"}}
+{"state": {"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α", "f : α → Set α", "hf : ∀ (x : α), f x ∈ 𝓝 x"], "goal": "∃ s, s.Countable ∧ ⋃ x ∈ s, f x = univ"}}
+{"state": {"context": ["R : Type u", "a : R", "Ring R"], "goal": "Polynomial.C (-a) = -Polynomial.C a"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "m' _mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "inst✝ : IsMarkovKernel κ", "s t : Set Ω", "a✝ : s ∈ {s | MeasurableSet s}", "ht : t = ∅ ∨ t = Set.univ", "a : α"], "goal": "(κ a) (s ∩ t) = (κ a) s * (κ a) t"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "Module 𝕜 E", "x : E", "s : Set E", "hx : x ∈ s"], "goal": "StarConvex 𝕜 x s ↔ ∀ ⦃y : E⦄, y ∈ s → openSegment 𝕜 x y ⊆ s"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "Semiring k", "Add G", "a₁ a₂ : G", "b₁ b₂ : k", "ha : AddCommute a₁ a₂", "hb : Commute b₁ b₂"], "goal": "Commute (AddMonoidAlgebra.single a₁ b₁) (AddMonoidAlgebra.single a₂ b₂)"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "NonUnitalSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A"], "goal": "algebraMap R (Unitization R A) = Unitization.inlRingHom R A"}}
+{"state": {"context": ["X : Type u", "Lattice X", "JordanHolderLattice X", "s₁ s₂ : CompositionSeries X", "h : s₁.Equivalent s₂"], "goal": "s₁.length = s₂.length"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "M : Type u_3", "N : Type u_4", "inst✝⁹ : CommRing K", "inst✝⁸ : CommRing L", "inst✝⁷ : CommRing M", "inst✝⁶ : CommRing N", "i : K →+* L", "j : K →+* M", "k : K →+* N", "f : L →+* M", "g : L →+* N", "p : ℕ", "inst✝⁵ : ExpChar K p", "inst✝⁴ : ExpChar L p", "inst✝³ : ExpChar M p", "inst✝² : ExpChar N p", "inst✝¹ : PerfectRing M p", "inst✝ : IsPRadical i p", "x : K"], "goal": "(RingEquiv.refl M).symm (j x) = j x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommMonoid α", "TopologicalSpace α", "s : Finset β", "f : β → α"], "goal": "Multipliable (f ∘ Subtype.val)"}}
+{"state": {"context": ["μ : MeasureTheory.Measure ℝ", "MeasureTheory.IsProbabilityMeasure μ", "x : ℝ"], "goal": "ENNReal.ofReal (↑(ProbabilityTheory.cdf μ) x) = μ (Set.Iic x)"}}
+{"state": {"context": ["A : Type u_1", "M : Type u_3", "AddMonoid A", "Monoid M", "ψ : A →+ Additive M"], "goal": "⇑(AddChar.toAddMonoidHomEquiv.symm ψ) = ⇑Additive.toMul ∘ ⇑ψ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "T₁ T₂ : CategoryTheory.Monad C", "h : T₁ ≅ T₂"], "goal": "(CategoryTheory.Monad.algebraEquivOfIsoMonads h).unitIso =\n  CategoryTheory.Monad.algebraFunctorOfMonadHomId.symm ≪≫\n    CategoryTheory.Monad.algebraFunctorOfMonadHomEq ⋯ ≪≫ CategoryTheory.Monad.algebraFunctorOfMonadHomComp h.hom h.inv"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedSemifield α", "a b c d e : α", "m n : ℤ", "h₀ : 0 < a", "h₁ : a ≠ 1", "H : 1 < a"], "goal": "Injective fun x => a ^ x"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "f : α → Real.Angle", "s : Set α", "hf : ContinuousOn f s", "hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ ↑π"], "goal": "ContinuousOn (Real.Angle.sign ∘ f) s"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p q : R[X]"], "goal": "Polynomial.derivative (p.comp q) = Polynomial.derivative q * (Polynomial.derivative p).comp q"}}
+{"state": {"context": ["E : Type u_9", "self : SeminormedGroup E", "x y : E"], "goal": "dist x y = ‖x / y‖"}}
+{"state": {"context": ["α : Type u_3", "E : Type u_6", "SeminormedGroup E", "f : α → E", "a : α → ℝ", "t₀ : Filter α", "h : ∀ᶠ (n : α) in t₀, ‖f n‖ ≤ a n", "h' : Filter.Tendsto a t₀ (𝓝 0)"], "goal": "Filter.Tendsto f t₀ (𝓝 1)"}}
+{"state": {"context": [], "goal": "ClosedEmbedding ⇑natUnionInftyEmbedding"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder α", "TopologicalSpace α", "OrderTopology α", "LinearOrder β", "TopologicalSpace β", "OrderTopology β", "f : α → β", "s : Set α", "a : α", "h_mono : MonotoneOn f s", "hs : s ∈ 𝓝[≥] a", "hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioo (f a) b"], "goal": "ContinuousWithinAt f (Set.Ici a) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "ι : Sort u_5", "f : ι → α → ℝ≥0∞"], "goal": "⨆ i, ∫⁻ (a : α), f i a ∂μ ≤ ∫⁻ (a : α), (⨆ i, f i) a ∂μ"}}
+{"state": {"context": ["n : ℕ", "NeZero n"], "goal": "Function.RightInverse ZMod.val Nat.cast"}}
+{"state": {"context": ["k : ℤ", "hk : 3 ≤ k", "z : ℍ", "hk' : 2 < ↑k", "b : Fin 2 → ℤ"], "goal": "Complex.abs (↑(b 0) * ↑z + ↑(b 1)) ^ (-k) ≤ r z ^ (-↑k) * ‖b‖ ^ (-↑k)"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Field L'", "inst✝¹ : Algebra K L", "inst✝ : Algebra K L'", "S F E : IntermediateField K L"], "goal": "(∀ (x : L), x ∈ F.toSubalgebra ↔ x ∈ E.toSubalgebra) ↔ ∀ (x : L), x ∈ ↑F ↔ x ∈ ↑E"}}
+{"state": {"context": ["α : Type u_3", "SemilatticeInf α", "Nonempty α", "p : α → Prop", "h : ∃ᶠ (x : α) in Filter.atBot, p x", "a : α"], "goal": "∃ b ≤ a, p b"}}
+{"state": {"context": ["R : Type u_1", "NonAssocSemiring R", "Nontrivial R", "p : ℕ", "hp : Nat.Prime p"], "goal": "CharP R p ↔ ↑p = 0"}}
+{"state": {"context": ["X : Type w", "TopologicalSpace X", "UCompactlyGeneratedSpace X", "Y : Type u_1", "TopologicalSpace Y", "f : X → Y", "h : ∀ (S : CompHaus) (g : C(↑S.toTop, X)), Continuous (f ∘ ⇑g)"], "goal": "Continuous f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Ring α", "na nb nc : ℤ", "da db dc k : ℕ", "inv✝¹ : Invertible ↑da", "inv✝ : Invertible ↑db", "h₁ : na * nb = ↑k * nc", "h₂ : da * db = k * dc", "this✝ : Invertible ↑(da * db)", "this : Invertible ↑dc"], "goal": "↑na * ⅟↑da * (↑nb * ⅟↑db) = ↑nc * ⅟↑dc"}}
+{"state": {"context": ["α : Type u", "Preorder α", "s : Set α"], "goal": "¬BddAbove s ↔ ∀ (x : α), ∃ y ∈ s, ¬y ≤ x"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "inst✝¹ : Ring R", "inst✝ : Nontrivial R"], "goal": "-1 = 1 ↔ ringChar R = 2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder β", "l : Filter α", "f : α → β"], "goal": "f ≤ᶠ[l] f"}}
+{"state": {"context": ["R : Type uR", "ι : Type uι", "M₁ : ι → Type v₁", "M₂ : Type v₂", "Semiring R", "(i : ι) → AddCommMonoid (M₁ i)", "AddCommMonoid M₂", "(i : ι) → Module R (M₁ i)", "Module R M₂", "f : MultilinearMap R M₁ M₂", "P : ι → Prop", "DecidablePred P", "x : (i : { a // P a }) → M₁ ↑i", "z : (i : { a // ¬P a }) → M₁ ↑i"], "goal": "(f.domDomRestrict P z) x = f fun j => if h : P j then x ⟨j, h⟩ else z ⟨j, h⟩"}}
+{"state": {"context": ["q : ℚ", "x y : ℝ", "m : ℤ", "h : Irrational (↑↑m + x)"], "goal": "Irrational x"}}
+{"state": {"context": ["G : Type u_1", "Group G", "ι : Type u_2", "H : ι → Subgroup G", "g : ι → G", "s : Finset ι", "hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ"], "goal": "∃ k ∈ s, (H k).FiniteIndex"}}
+{"state": {"context": ["α : Sort u_4", "r : α → α → Prop", "a bot : α", "ha : Acc r a", "C : α → Prop", "ih : ∀ (b : α), b ≠ bot → C b → ∃ c, r c b ∧ C c"], "goal": "C a → C bot"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "a b c : R", "h : ∀ (s : R), discrim a b c ≠ s ^ 2", "x : R"], "goal": "a * x * x + b * x + c ≠ 0"}}
+{"state": {"context": ["p i : ℕ", "hp : Nat.Prime p"], "goal": "(ArithmeticFunction.sigma 0) (p ^ i) = i + 1"}}
+{"state": {"context": ["m : Type u → Type u", "inst✝¹ : _root_.Monad m", "inst✝ : LawfulMonad m", "X✝ Y✝ Z✝ : KleisliCat m", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝", "t : { obj := fun X => X, map := fun {X Y} f => f }.obj X✝"], "goal": "{ obj := fun X => X, map := fun {X Y} f => f }.map (f ≫ g) t = joinM (g <$> f t)"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "s : Set R", "t : NonUnitalSubsemiring R", "h₁ : s ⊆ ↑t", "h₂ : t ≤ NonUnitalSubsemiring.closure s"], "goal": "NonUnitalSubsemiring.closure s = t"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_3", "LinearOrder ι", "LocallyFiniteOrderBot ι", "IsWellOrder ι fun x x_1 => x < x_1", "Fintype ι", "FiniteDimensional 𝕜 E", "h : FiniteDimensional.finrank 𝕜 E = Fintype.card ι", "f : ι → E", "hf : Pairwise fun i j => ⟪f i, f j⟫_𝕜 = 0", "i : ι", "hi : f i ≠ 0"], "goal": "(gramSchmidtOrthonormalBasis h f) i = (↑‖f i‖)⁻¹ • f i"}}
+{"state": {"context": ["α : Type u_1"], "goal": "∀ {x : Setoid α} {r : α → α → Prop} (H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂) (a : Quotient x),\n  (Quotient.out'RelEmbedding H) a = a.out'"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : AddCommMonoid N", "inst✝ : Module R N", "w : Set M", "x : ↥(span R w)"], "goal": "(Finsupp.total (↑w) M R Subtype.val) ⋯.choose = ↑x"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "Small.{u_2, u_1} α", "x y : α"], "goal": "(equivShrink α) (x * y) = (equivShrink α) x * (equivShrink α) y"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimits C", "X Y : AlgebraicGeometry.PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y"], "goal": "Y.presheaf.stalkSpecializes ⋯ ≫ AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f x =\n  AlgebraicGeometry.PresheafedSpace.Hom.stalkMap f y ≫ X.presheaf.stalkSpecializes h"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommSemiring R", "M : Type u_2", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : Module R M", "N : Type u_3", "inst✝⁶ : AddCommMonoid N", "inst✝⁵ : Module R N", "ι : Type u_4", "inst✝⁴ : DecidableEq ι", "S : Type u_5", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "inst✝¹ : Module S M", "inst✝ : IsScalarTower R S M", "i : ι", "m : M"], "goal": "(finsuppScalarRight R M ι).symm (Finsupp.single i m) = m ⊗ₜ[R] Finsupp.single i 1"}}
+{"state": {"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "c : ℤ → ℂ := fun n => -I * cexp (2 * ↑π * I * ↑a * ↑n) / 2", "hc : ∀ (n : ℤ), ‖c n‖ = 1 / 2", "hF : ∀ (t : ℝ), 0 < t → HasSum (fun n => c n * ↑n * ↑(rexp (-π * ↑n ^ 2 * t))) (↑(sinKernel (↑a) t) / 2)", "h_sum : Summable fun i => ‖c i‖ / |↑i| ^ s.re"], "goal": "HasSum (fun n => (s + 1).Gammaℝ * -I * ↑n.sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (completedSinZeta (↑a) s)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "A : Type u_2", "CommRing A", "Algebra R A", "M : Type u_3", "AddCommGroup M", "Module A M", "Module R M", "D1 D2 : Derivation R A M", "a : A"], "goal": "(D1 - D2) a = D1 a - D2 a"}}
+{"state": {"context": ["α : Type u", "l : List α", "n m : ℕ", "hml : m < l.length"], "goal": "(drop (n % l.length) l ++ take (n % l.length) l)[m]? = l[(m + n) % l.length]?"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "Bornology α", "Bornology β", "Bornology γ", "Bornology δ", "f : LocallyBoundedMap γ δ", "g : LocallyBoundedMap β γ", "h : LocallyBoundedMap α β"], "goal": "(f.comp g).comp h = f.comp (g.comp h)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.IsFilteredOrEmpty C", "j₁ j₂ : C", "f g h : j₁ ⟶ j₂"], "goal": "CategoryTheory.CategoryStruct.comp f (CategoryTheory.IsFiltered.coeq₃Hom f g h) =\n  CategoryTheory.CategoryStruct.comp h (CategoryTheory.IsFiltered.coeq₃Hom f g h)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "s₁ s₂ : AList β", "h : s₁.entries ⊆ s₂.entries", "k : α", "hk : k ∈ s₁.keys", "this✝ : DecidableEq α := Classical.decEq α", "this : lookup k s₂ = some ((lookup k s₁).get ⋯)"], "goal": "k ∈ s₂.keys"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P✝ : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "P : Cᵒᵖ ⥤ D", "x✝ : Cᵒᵖ"], "goal": "colimMap (𝟙 (J.diagram P (unop x✝))) = 𝟙 ({ obj := fun X => colimit (J.diagram P (unop X)), map := fun {X Y} f => colimMap (J.diagramPullback P f.unop) ≫ colimit.pre (J.diagram P (unop Y)) (J.pullback f.unop).op, map_id := ⋯, map_comp := ⋯ }.obj x✝)"}}
+{"state": {"context": ["α : Type u", "Preorder α"], "goal": "LeftOrdContinuous id"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "MulZeroOneClass α", "MulZeroOneClass β", "f : α →*₀ β", "h1 : (↑f).toFun 1 = 1", "hmul : ∀ (x y : α), (↑f).toFun (x * y) = (↑f).toFun x * (↑f).toFun y"], "goal": "{ toZeroHom := ↑f, map_one' := h1, map_mul' := hmul } = f"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "N : Type u_5", "AddGroup N", "f : G →+ N", "H K : AddSubgroup G"], "goal": "AddSubgroup.map f H ≤ AddSubgroup.map f K ↔ H ≤ K ⊔ f.ker"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "D : Type u_2", "CategoryTheory.Category.{u_5, u_2} D", "E : Type u_3", "CategoryTheory.Category.{u_6, u_3} E", "G : C ⥤ D", "G' : D ⥤ E", "J : CategoryTheory.GrothendieckTopology C", "K : CategoryTheory.GrothendieckTopology D", "L : CategoryTheory.GrothendieckTopology E", "G.IsCocontinuous J K", "G'.IsCocontinuous K L"], "goal": "(G ⋙ G').IsCocontinuous J L"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m✝ : MeasurableSpace α", "μ✝ ν : Measure α", "ι : Type u_5", "m : MeasurableSpace α", "s : Finset ι", "f : α → ℝ≥0∞", "μ : ι → Measure α"], "goal": "∫⁻ (a : α), f a ∂∑ i ∈ s, μ i = ∑ i ∈ s, ∫⁻ (a : α), f a ∂μ i"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : UniformSpace α", "inst✝² : UniformSpace β", "inst✝¹ : UniformSpace γ", "l : Filter β", "inst✝ : l.NeBot", "f g : β → α", "a b : α", "ha : Tendsto f l (𝓝 a)", "hb : Tendsto g l (𝓝 b)", "h : Tendsto (fun x => (f x, g x)) l (𝓤 α)"], "goal": "Inseparable a b"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "f g : Equiv.Perm α", "h : f.support ⊆ g.support", "h' : ∀ x ∈ g.support, f x = g x"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "a : Part α", "Decidable a.Dom"], "goal": "a.toOption = none ↔ ¬a.Dom"}}
+{"state": {"context": ["α : Type u_1", "AddCommSemigroup α", "PartialOrder α", "ExistsAddOfLE α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Sub α", "OrderedSub α", "a b c : α", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "h₁ : b ≤ a", "h₂ : c ≤ a"], "goal": "a - b < a - c ↔ c < b"}}
+{"state": {"context": ["α : Type u_1", "x y : List α", "h : x <+: y", "hl : x.length < y.length"], "goal": "x ++ [y.get ⟨x.length, hl⟩] <+: y"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s : Sphere P", "p : P", "v : V"], "goal": "s.secondInter p v ∈ s ↔ p ∈ s"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "⊤.toSubmonoid = ⊤"}}
+{"state": {"context": [], "goal": "⇑NNRat.coeHom = NNRat.cast"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "a : ℤ", "ha : ↑a ≠ 0", "x y : ZMod p", "hx : x ≠ 0", "hxy : x ^ 2 - ↑a * y ^ 2 = 0"], "goal": "legendreSym p a = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x : α", "inst✝ : AddCommGroup α", "h : Periodic f c", "a : α"], "goal": "Periodic (fun x => f (x - a)) c"}}
+{"state": {"context": ["n p : ℕ", "hp : Prime p", "h_two : p ≠ 2", "h_three : p ≠ 3"], "goal": "5 ≤ p"}}
+{"state": {"context": ["L : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "S✝ : L.Substructure M", "s : Set M", "S : L.Substructure M := { carrier := range (Term.realize Subtype.val), fun_mem := ⋯ }"], "goal": "↑((closure L).toFun s) = range (Term.realize Subtype.val)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝ : Semiring R", "p q r : R[X]"], "goal": "comp 0 p = 0"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "Group α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α"], "goal": "mabs (mabs a) = mabs a"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝⁴ : CommMonoid M", "inst✝³ : CommMonoid N", "inst✝² : DivisionCommMonoid G", "k l : ℕ", "ζ : R", "inst✝¹ : CommRing R", "inst✝ : IsDomain R"], "goal": "1 ∈ primitiveRoots 1 R ∧ ∀ x ∈ primitiveRoots 1 R, x = 1"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q r : ℚ", "hq : ¬q = 0", "hr : ¬r = 0", "this : ↑p ≠ 0"], "goal": "padicNorm p (q * r) = padicNorm p q * padicNorm p r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "TopologicalSpace α", "TopologicalSpace β", "TopologicalSpace γ", "Sub γ", "ContinuousSub γ", "f g : C(β, γ)", "h : C(α, β)"], "goal": "(f - g).comp h = f.comp h - g.comp h"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "LocallyFiniteOrderBot α", "s : Set α", "hs : s.Infinite", "a : α"], "goal": "∃ b ∈ s, a < b"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : T1Space X", "x y : X", "s : Set X", "hxy : x ≠ y"], "goal": "𝓝[insert y s] x = 𝓝[s] x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommSemiring R", "inst✝¹ : Semiring S", "inst✝ : Algebra R S", "M N : Submodule R S", "hc : ∀ (m : ↥M) (n : ↥N), Commute ↑m ↑n", "n : ↥N", "m : ↥M"], "goal": "(N.mulMap M) (n ⊗ₜ[R] m) = (M.mulMap N ∘ₗ ↑(TensorProduct.comm R ↥N ↥M)) (n ⊗ₜ[R] m)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z"], "goal": "∀ (X_1 : CategoryTheory.Limits.WalkingSpan),\n  (CategoryTheory.Limits.spanOp f g).hom.app X_1 =\n    (Option.rec (CategoryTheory.Iso.refl (Opposite.op Z))\n        (fun val =>\n          CategoryTheory.Limits.WalkingPair.rec (CategoryTheory.Iso.refl (Opposite.op X))\n            (CategoryTheory.Iso.refl (Opposite.op Y)) val)\n        X_1).hom"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P✝ : Type u_3", "inst✝¹ : CommSemiring P✝", "P : Ideal R", "hp : P.IsPrime", "inst✝ : IsLocalization.AtPrime S P", "hze : (algebraMap R S) 0 = (algebraMap R S) 1", "t : ↥P.primeCompl", "ht : ↑t * 0 = ↑t * 1"], "goal": "False"}}
+{"state": {"context": ["α : Type u_2", "SubtractionMonoid α"], "goal": "Set.univ - Set.univ = Set.univ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ε : Type u_5", "ζ : Type u_6"], "goal": "@iterate = @iterateTR"}}
+{"state": {"context": ["s : ℝ", "hs : 0 < s"], "goal": "Gamma s * Gamma (s + 1 / 2) = Gamma s * Gamma (s + 1 / 2) * 2 ^ (2 * s - 1) / √π * (2 ^ (2 * s - 1))⁻¹ * √π"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Algebra F E", "K : Type w", "inst✝¹ : Field K", "inst✝ : Algebra F K", "L : IntermediateField F E", "h : IsPurelyInseparable (↥L) E", "x : E", "hx : x ∈ separableClosure F E"], "goal": "x ∈ L"}}
+{"state": {"context": ["α : Type u", "PartialOrder α", "OrderTop α", "a : α"], "goal": "¬a < ⊤ ↔ a = ⊤"}}
+{"state": {"context": ["n : ℕ", "a : Fin n"], "goal": "a ∈ List.finRange n"}}
+{"state": {"context": ["n : Type u_1", "α : Type u_2", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "inst✝ : CommRing α", "A B : Matrix n n α"], "goal": "(X • A.map ⇑C + B.map ⇑C).det.coeff 0 = B.det"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "e : E ≃ₛₗᵢ[σ₁₂] E₂", "c : R", "x : E"], "goal": "e (c • x) = σ₁₂ c • e x"}}
+{"state": {"context": ["R : Type u", "inst✝³ : CommRing R", "n G : Type v", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "α β : Type v", "inst✝ : DecidableEq α", "M✝ M : Matrix n n R"], "goal": "(1 + X • M.map ⇑C).det.coeff 1 = M.trace"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V", "h : ⟪x, -y⟫_ℝ = 0", "h0 : x ≠ 0 ∨ -y ≠ 0"], "goal": "angle x (x - y) = Real.arcsin (‖y‖ / ‖x - y‖)"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "CommRing A", "Field K", "CommRing B", "Field L", "Algebra A K", "Algebra B L", "Algebra A B", "Algebra K L", "Algebra A L", "IsScalarTower A K L", "IsScalarTower A B L", "IsDomain A", "IsFractionRing A K", "IsFractionRing B L", "IsDomain B", "NoZeroSMulDivisors A B", "I : Submodule B (FractionRing B)"], "goal": "Submodule.map (FractionRing.algEquiv B L) (Submodule.traceDual A (FractionRing A) I) =\n  Submodule.traceDual A K (Submodule.map (FractionRing.algEquiv B L) I)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "inst✝ : DenselyOrdered α"], "goal": "Ioi b ⊆ Ici a ↔ a ≤ b"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "P : Type u_3", "inst✝³ : SeminormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : PseudoMetricSpace P", "inst✝ : NormedAddTorsor E P", "s t : Set E", "hs : Convex ℝ s", "δ : ℝ", "hδ : δ ≤ 0"], "goal": "Convex ℝ (Metric.cthickening δ s)"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "Preorder α", "NoMaxOrder α"], "goal": "Set.Bounded (fun x x_1 => x ≤ x_1) s ↔ Set.Bounded (fun x x_1 => x < x_1) s"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R", "x : R", "s : Set R"], "goal": "x ∈ NonUnitalSubring.closure s ↔ ∀ (S : NonUnitalSubring R), s ⊆ ↑S → x ∈ S"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "G : C", "hG : IsCoseparator G", "X Y : C", "f g : X ⟶ Y", "hfg : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h", "H : C", "hH : H ∈ {G}", "h : Y ⟶ H"], "goal": "f ≫ h = g ≫ h"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "p : R[X]", "hp : p ≠ 0", "f : R →+* S", "z : S", "hz : Polynomial.eval₂ f z p = 0", "inj : ∀ (x : R), f x = 0 → x = 0"], "goal": "0 < p.natDegree"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "LinearOrderedField 𝕜", "DecidableEq α", "A : Finset α", "P : Finpartition A", "G : SimpleGraph α", "DecidableRel G.Adj", "ε δ : 𝕜", "a b : α"], "goal": "(SimpleGraph.regularityReduced P G ε δ).Adj a b =\n  (G.Adj a b ∧ ∃ U ∈ P.parts, ∃ V ∈ P.parts, a ∈ U ∧ b ∈ V ∧ U ≠ V ∧ G.IsUniform ε U V ∧ δ ≤ ↑(G.edgeDensity U V))"}}
+{"state": {"context": ["x✝¹ y✝ a : ℝ", "ha : 0 < a", "x : ℝ", "hex : rexp (1 / a) ≤ x", "y : ℝ", "x✝ : rexp (1 / a) ≤ y", "hxy : x ≤ y", "x_pos : 0 < x", "y_pos : 0 < y", "x_nonneg : 0 ≤ x", "y_nonneg : 0 ≤ y"], "goal": "x ^ a ∈ {x | rexp 1 ≤ x}"}}
+{"state": {"context": ["x₁ y₁ x₂ y₂ : SetTheory.PGame", "hx : x₁ ≈ x₂", "hy : y₁ ≈ y₂"], "goal": "x₁.LF y₁ ↔ x₂.LF y₂"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type w", "inst✝² : MetricSpace X", "inst✝¹ : MetricSpace Y", "Φ✝ : Z → X", "Ψ✝ : Z → Y", "ε✝ : ℝ", "inst✝ : Nonempty Z", "Φ : Z → X", "Ψ : Z → Y", "ε : ℝ", "H : ∀ (p q : Z), |dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)| ≤ 2 * ε", "x : X", "y : Y", "z : X", "p : Z"], "goal": "dist x z ≤ (⨅ p, dist z (Φ p) + dist y (Ψ p)) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))"}}
+{"state": {"context": ["z : ℤ"], "goal": "IsSquare ↑z ↔ IsSquare z"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "k : ℕ", "ζ : Mˣ"], "goal": "IsPrimitiveRoot (↑ζ) k ↔ IsPrimitiveRoot ζ k"}}
+{"state": {"context": ["n n' m : Nat", "i : Fin n'", "h : m + n' = m + n"], "goal": "Fin.cast h (Fin.natAdd m i) = Fin.natAdd m (Fin.cast ⋯ i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ring α", "AddCommGroup β", "Module α β", "NoZeroSMulDivisors α β", "PartialOrder α", "PartialOrder β"], "goal": "SMulPosReflectLE α β ↔ SMulPosReflectLT α β"}}
+{"state": {"context": ["k : Type u", "Field k", "R : Type u_1", "CommRing R", "IsAlgClosed k", "Algebra R k", "hinj : Function.Injective ⇑(algebraMap R k)", "p : R[X]", "hp : p.degree ≠ 0"], "goal": "∃ x, (Polynomial.aeval x) p = 0"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "S S' S'' : D", "Y Y' Y'' : C", "T T' : C ⥤ D", "A B : StructuredArrow S T", "f : A ⟶ B", "this : (Functor.fromPUnit S).map f.left ≫ B.hom = A.hom ≫ T.map f.right"], "goal": "A.hom ≫ T.map f.right = B.hom"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α"], "goal": "Finset.powersetCard s.card s = {s}"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "inst✝⁴ : DivisionRing K", "inst✝³ : AddCommGroup V", "inst✝² : AddCommGroup V'", "inst✝¹ : Module K V", "inst✝ : Module K V'", "v : ι → V", "s t : Set V", "y z✝ : V", "a : K", "z : V", "hz : z ∈ span K s", "h : a • y + z ∉ span K s", "a0 : a ≠ 0"], "goal": "y = a⁻¹ • (a • y + z) + -a⁻¹ • z"}}
+{"state": {"context": ["R : Type u_1", "N : Type u_9", "N₂ : Type u_10", "CommSemiring R", "AddCommMonoid N", "AddCommMonoid N₂", "Module R N", "Module R N₂", "qₗ : Submodule R N₂", "fₗ : N →ₗ[R] N₂", "c : R"], "goal": "Submodule.comap fₗ qₗ ≤ Submodule.comap (c • fₗ) qₗ"}}
+{"state": {"context": ["p : ℕ → Prop", "k a : ℕ", "h : a < nth p (k + 1)", "ha : p a", "hf : (setOf p).Finite"], "goal": "a ≤ nth p k"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "a : R", "a0 : a ≠ 0"], "goal": "(C a * T 0).degree = 0"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G", "Finite ↥H"], "goal": "Nat.card ↥H ≤ 1 ↔ H = ⊥"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "x y : α", "h : x < y", "z : α"], "goal": "x < z ∨ z < y"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "r : R"], "goal": "(MvPolynomial.C r).IsSymmetric"}}
+{"state": {"context": ["a b c : Ordinal.{u_1}", "a1 : 1 < a"], "goal": "a ^ b ≤ a ^ c ↔ b ≤ c"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "ι : Type u_4", "v : ι → E", "hz : ∀ (i : ι), v i ≠ 0", "ho : Pairwise fun i j => ⟪v i, v j⟫_𝕜 = 0"], "goal": "LinearIndependent 𝕜 v"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : (o.oangle x y).sign = -1"], "goal": "o.oangle x y = -↑(InnerProductGeometry.angle x y)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y : C", "a b c : X ⟶ Y"], "goal": "a - b + c = a - (b - c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq β", "f : α → β", "s : Finset α", "p : β → Prop", "DecidablePred p"], "goal": "Finset.filter p (Finset.image f s) = Finset.image f (Finset.filter (fun a => p (f a)) s)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "κ : β → Type u_9", "f : (i : β) → κ i → α"], "goal": "⨆ i, ⨆ j, f i j = ⨆ x, f x.fst x.snd"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "p : α → β → Bool", "l : AssocList α β"], "goal": "findEntryP? p l = List.find? (fun x => match x with | (a, b) => p a b) l.toList"}}
+{"state": {"context": ["x : ℝ", "n : ℕ∞", "hx : x ≠ 0"], "goal": "ContDiffAt ℝ n (fun x => √x) x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝² : PseudoMetricSpace α", "inst✝¹ : PseudoMetricSpace β", "inst✝ : PseudoMetricSpace γ", "K : ℝ≥0", "s : Set α", "f : α → β", "hf : LipschitzOnWith K f s", "u : ℕ → α", "hu : CauchySeq u", "h'u : range u ⊆ s", "b : ℕ → ℝ", "b_nonneg : ∀ (n : ℕ), 0 ≤ b n", "hb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ b N", "blim : Tendsto b atTop (𝓝 0)"], "goal": "Tendsto (fun n => ↑K * b n) atTop (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : DecidableEq α", "A : Finset α", "P : Finpartition A", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε δ : 𝕜", "u v s t : Finset α", "hP : P.IsEquipartition", "hε : 0 ≤ ε"], "goal": "∑ UV ∈ P.sparsePairs G ε, ↑(G.interedges UV.1 UV.2).card ≤ ∑ UV ∈ P.sparsePairs G ε, ε * (↑UV.1.card * ↑UV.2.card)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s t u : Set α", "inst✝² : MeasurableSpace α", "μ : Measure ℝ", "inst✝¹ : IsFiniteMeasureOnCompacts μ", "inst✝ : NoAtoms μ", "b : ℝ"], "goal": "Tendsto (fun δ => μ (Icc (b - δ) (b + δ))) (𝓝[<] 0) (𝓝 0) ∧ Tendsto (fun δ => μ (Icc (b - δ) (b + δ))) (𝓝[≥] 0) (𝓝 0)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq β", "inst✝ : DivisionMonoid α", "s t : Finset α"], "goal": "IsUnit ↑s ↔ IsUnit s"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "v : VitaliFamily μ", "E : Type u_2", "NormedAddCommGroup E", "SecondCountableTopology α", "BorelSpace α", "MeasureTheory.IsLocallyFiniteMeasure μ", "f : α → E", "hf : MeasureTheory.LocallyIntegrable f μ"], "goal": "∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "DecidableRel r"], "goal": "List.mergeSort r [] = []"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasPullbacks C", "B : C", "X Y : CategoryTheory.Subobject B"], "goal": "X.Factors (X ⊓ Y).arrow"}}
+{"state": {"context": ["α : Type u", "Add α", "a b : WithBot α"], "goal": "a + b ≠ ⊥ ↔ a ≠ ⊥ ∧ b ≠ ⊥"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "SeminormedRing 𝕜", "AddCommGroup E", "Module 𝕜 E", "s t : Set E", "hs : Balanced 𝕜 s", "ht : Balanced 𝕜 t"], "goal": "Balanced 𝕜 (s - t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝³ : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "p : ℝ≥0∞", "μ : Measure α", "G : Type u_7", "inst✝ : NormedLatticeAddCommGroup G", "hp : Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤", "g : { g // 0 ≤ g }", "this✝¹ : MeasurableSpace G := borel G", "this✝ : BorelSpace G", "hg_memℒp : Memℒp (↑↑↑g) p μ", "zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}", "this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})", "g_meas : Measurable ↑↑↑g", "x : ℕ → α →ₛ G := fun n => SimpleFunc.approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n", "hx_nonneg : ∀ (n : ℕ), 0 ≤ x n", "hx_memℒp : ∀ (n : ℕ), Memℒp (↑(x n)) p μ", "h_toLp : ∀ (n : ℕ), ↑↑(Memℒp.toLp ↑(x n) ⋯) =ᶠ[ae μ] ↑(x n)", "hx_nonneg_Lp : ∀ (n : ℕ), 0 ≤ toLp (x n) ⋯", "hx_tendsto : Tendsto (fun n => eLpNorm (↑(x n) - ↑↑↑g) p μ) atTop (𝓝 0)", "n : ℕ"], "goal": "↑(x n) =ᶠ[ae μ] ↑↑↑(toLp (x n) ⋯)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f : Filter α", "m : α → β", "m' : β → γ", "s : Set α", "t : Set β", "hf : Injective m", "h : m '' s ∈ map m f"], "goal": "s ∈ f"}}
+{"state": {"context": ["R : Type u", "Semiring R", "S : Type u_1", "SMulZeroClass S R", "a : S", "b : R[ℕ]"], "goal": "{ toFinsupp := a • b } = a • { toFinsupp := b }"}}
+{"state": {"context": ["n : Nat"], "goal": "Int.negSucc n < 0"}}
+{"state": {"context": ["m₂ m₁ : List Char", "s : Substring"], "goal": "s.Valid →\n  s.toString.data = m₁.reverse ++ m₂ →\n    ∀ (n : Nat), s.prevn n { byteIdx := String.utf8Len m₁ } = { byteIdx := String.utf8Len (List.drop n m₁) }"}}
+{"state": {"context": ["E : Type u_3", "NormedAddCommGroup E", "NormedSpace ℝ E", "a b : ℝ", "f : ℝ → E", "μ : MeasureTheory.Measure ℝ", "h : a ≤ b"], "goal": "∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in Set.Ioc a b, f x ∂μ"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "V2 : Type u_4", "P2 : Type u_5", "V3 : Type u_6", "P3 : Type u_7", "V4 : Type u_8", "P4 : Type u_9", "inst✝¹² : Ring k", "inst✝¹¹ : AddCommGroup V1", "inst✝¹⁰ : Module k V1", "inst✝⁹ : AffineSpace V1 P1", "inst✝⁸ : AddCommGroup V2", "inst✝⁷ : Module k V2", "inst✝⁶ : AffineSpace V2 P2", "inst✝⁵ : AddCommGroup V3", "inst✝⁴ : Module k V3", "inst✝³ : AffineSpace V3 P3", "inst✝² : AddCommGroup V4", "inst✝¹ : Module k V4", "inst✝ : AffineSpace V4 P4", "f : V1 →ᵃ[k] V2"], "goal": "⇑f.linear = ⇑f - fun x => f 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y z : E"], "goal": "⟪x - y, z⟫_𝕜 = ⟪x, z⟫_𝕜 - ⟪y, z⟫_𝕜"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_4, u_1} C", "inst✝³ : Category.{u_3, u_2} D", "inst✝² : Preadditive C", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "inst✝ : W.HasLeftCalculusOfFractions", "X Y Z : C", "f₁ f₂ : L.obj X ⟶ L.obj Y", "g : L.obj Y ⟶ L.obj Z", "α : W.LeftFraction₂ X Y", "h₁ : f₁ = α.fst.map L ⋯", "h₂ : f₂ = α.snd.map L ⋯", "β : W.LeftFraction Y Z", "hβ : g = β.map L ⋯", "γ : W.LeftFraction α.Y' β.Y'", "hγ : β.f ≫ γ.s = α.s ≫ γ.f"], "goal": "(LeftFraction.mk ((α.f + α.f') ≫ γ.f) (β.s ≫ γ.s) ⋯).map L ⋯ = (LeftFraction.mk (α.f ≫ γ.f + α.f' ≫ γ.f) (β.s ≫ γ.s) ⋯).map L ⋯"}}
+{"state": {"context": ["ι✝ : Type u_1", "α : Type u_2", "M : Matroid α", "F X Y : Set α", "e : α", "ι : Sort u_3", "I J B : Set α", "f x y : α", "hI : M.Indep I", "hJI : J ⊆ I", "hJ : M.Indep J"], "goal": "M.closure J ∩ I ⊆ J"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "s t u : Multiset α", "a b : α"], "goal": "0 ∪ s = s"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "DivisionRing k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : AffineSubspace k P", "p₁ p₂ p₃ : P", "hp₁p₂ : p₁ ≠ p₂", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "hp₃ : p₃ ∉ s"], "goal": "AffineIndependent k ![p₁, p₂, p₃]"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "X Y W : C", "f : Y ⟶ X", "CategoryTheory.EffectiveEpi f", "e : Y ⟶ W", "h :\n  ∀ {Z : C} (g₁ g₂ : Z ⟶ Y),\n    CategoryTheory.CategoryStruct.comp g₁ f = CategoryTheory.CategoryStruct.comp g₂ f →\n      CategoryTheory.CategoryStruct.comp g₁ e = CategoryTheory.CategoryStruct.comp g₂ e"], "goal": "CategoryTheory.CategoryStruct.comp f (CategoryTheory.EffectiveEpi.desc f e ⋯) = e"}}
+{"state": {"context": ["X Y : Scheme", "U : X.Opens", "hU : IsAffineOpen U", "f✝ : ↑Γ(X, U)", "V : X.Opens", "hV : IsAffineOpen V", "x : ↑↑X.toPresheafedSpace", "hx : x ∈ U ⊓ V", "f : ↑Γ(X, U)", "hf₁ : X.basicOpen f ≤ V", "hf₂ : ↑⟨x, ⋯⟩ ∈ X.basicOpen f", "g : ↑Γ(X, V)", "hg₁ : X.basicOpen g ≤ X.basicOpen f", "hg₂ : ↑⟨x, hf₂⟩ ∈ X.basicOpen g", "f' : ↑Γ(X, U)", "hf' : X.basicOpen f' = X.toRingedSpace.basicOpen g"], "goal": "∃ f g, X.basicOpen f = X.basicOpen g ∧ x ∈ X.basicOpen f"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E", "a b : E", "r : ℝ", "h : b ∈ Metric.ball a r"], "goal": "‖b‖ < ‖a‖ + r"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝⁶ : EMetricSpace X", "inst✝⁵ : EMetricSpace Y", "C K r : ℝ≥0", "f : X → Y", "s✝ t : Set X", "𝕜 : Type u_4", "E : Type u_5", "F : Type u_6", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "e : E ≃L[𝕜] F", "s : Set E"], "goal": "dimH s ≤ dimH (⇑e '' s)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "ι : Type u_2", "v : ι → R", "inst✝ : Nontrivial R"], "goal": "nodal ∅ v ≠ 0"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝¹ : NormedGroup E", "inst✝ : NormedGroup F", "a b : E", "h : ‖a / b‖ ≤ 0"], "goal": "a = b"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_4", "s : Set ι", "LinearOrder ι", "f : ι → Set α", "h : ¬s.PairwiseDisjoint f"], "goal": "∃ i ∈ s, ∃ j ∈ s, i < j ∧ ∃ x, x ∈ f i ∩ f j"}}
+{"state": {"context": ["P : ℕ → Prop", "inst✝ : DecidablePred P", "n : ℕ"], "goal": "↑n < find P ↔ ∀ m ≤ n, ¬P m"}}
+{"state": {"context": ["k : Type u_1", "V₁ : Type u_2", "P₁ : Type u_3", "V₂ : Type u_4", "P₂ : Type u_5", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AffineSpace V₂ P₂", "f : P₁ →ᵃ[k] P₂", "s : AffineSubspace k P₂"], "goal": "↑(AffineSubspace.comap f s) = ⇑f ⁻¹' ↑s"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "C : ℝ", "hs : (μ.restrict s) univ < ⊤", "hC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C"], "goal": "‖∫ (x : X) in s, f x ∂μ‖ ≤ C * ((μ.restrict s) univ).toReal"}}
+{"state": {"context": ["E : Type u_3", "SeminormedAddGroup E", "a b : E"], "goal": "dist (a + b) b = ‖a‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : UniformSpace α", "inst✝¹ : Group α", "inst✝ : UniformGroup α", "ι : Sort u_3", "p : ι → Prop", "U : ι → Set α", "h : (𝓝 1).HasBasis p U"], "goal": "(Filter.comap (fun x => x.1 / x.2) (𝓝 1)).HasBasis p fun i => {x | x.1 / x.2 ∈ U i}"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "𝕜 : Type u_5", "CommRing 𝕜", "PartialOrder 𝕜", "StarRing 𝕜", "StarOrderedRing 𝕜", "Fintype m", "Fintype n", "DecidableEq n", "A : Matrix m m 𝕜", "B : Matrix m n 𝕜", "D : Matrix n n 𝕜", "x : m → 𝕜", "y : n → 𝕜", "Invertible D", "hD : D.IsHermitian"], "goal": "Matrix.dotProduct (Matrix.vecMul (star (Sum.elim x y)) (Matrix.fromBlocks A B B.conjTranspose D)) (Sum.elim x y) =\n  Matrix.dotProduct (Matrix.vecMul (star ((D⁻¹ * B.conjTranspose).mulVec x + y)) D)\n      ((D⁻¹ * B.conjTranspose).mulVec x + y) +\n    Matrix.dotProduct (Matrix.vecMul (star x) (A - B * D⁻¹ * B.conjTranspose)) x"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "c : ComplexShape ι", "C : HomologicalComplex V c", "CategoryTheory.Limits.HasKernels V", "i j j' : ι", "r : c.Rel i j", "r' : c.Rel i j'"], "goal": "CategoryTheory.Limits.kernelSubobject (C.d i j) = CategoryTheory.Limits.kernelSubobject (C.d i j')"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "MeasurableSpace δ", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "f : α → γ", "g : β → γ", "ρ : MeasureTheory.Measure δ", "h : δ → γ", "h₁ : ProbabilityTheory.IdentDistrib f g μ ν", "h₂ : ProbabilityTheory.IdentDistrib g h ν ρ"], "goal": "ProbabilityTheory.IdentDistrib f h μ ρ"}}
+{"state": {"context": [], "goal": "\"\".data.head! = default"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "hC : CategoryTheory.Pretriangulated C", "T : CategoryTheory.Pretriangulated.Triangle C", "hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles", "X : C", "f : X ⟶ T.obj₃", "hf : f ≫ T.mor₃ = 0"], "goal": "∃ g, f = g ≫ T.mor₂"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_2", "GroupWithZero M", "Ring R", "MulSemiringAction M R", "a : M", "ha : a ≠ 0", "S T : Subring R"], "goal": "a • S ≤ a • T ↔ S ≤ T"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "x y : V"], "goal": "‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ InnerProductGeometry.angle x y = π / 2"}}
+{"state": {"context": ["α : Fin 2 → Type u_8", "(i : Fin 2) → MeasurableSpace (α i)"], "goal": "⇑(MeasurableEquiv.piFinTwo α) = fun f => (f 0, f 1)"}}
+{"state": {"context": ["n i j : ℕ"], "goal": "(n &&& 2 ^ i).testBit j = ((n.testBit i).toNat * 2 ^ i).testBit j"}}
+{"state": {"context": [], "goal": "⊥ = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : T2Space X", "inst✝ : CompactSpace X", "h : TotallyDisconnectedSpace X", "x : X", "x✝¹ : x ∈ univ", "y : X", "x✝ : y ∈ univ"], "goal": "(¬∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ univ ⊆ u ∪ v ∧ Disjoint u v) → x = y"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "ι' : Type x'", "inst✝⁴ : Semiring R", "φ : ι → Type u_1", "inst✝³ : (i : ι) → AddCommMonoid (φ i)", "inst✝² : (i : ι) → Module R (φ i)", "I : Set ι", "p q : (i : ι) → Submodule R (φ i)", "x : (i : ι) → φ i", "inst✝¹ : DecidableEq ι", "inst✝ : Finite ι", "val✝ : Fintype ι", "i : ι"], "goal": "map (single i) (p i) ≤ pi Set.univ p"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a : α"], "goal": "z + 1 ≤ ⌈a⌉ ↔ ↑z < a"}}
+{"state": {"context": ["o : Ordinal.{u_1}", "ho : o.card.ord = o", "ho' : ω ≤ o", "a : Ordinal.{u_1}", "ha : (aleph' a).ord = o"], "goal": "ω ≤ a"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X : BialgebraCat R"], "goal": "(CategoryTheory.Iso.refl X).toBialgEquiv = BialgEquiv.refl R ↑X.toBundled"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommRing R", "Ring A", "Algebra R A", "Nontrivial R", "n : ℤ"], "goal": "IsAlgebraic R ↑n"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝⁵ : CommRing P", "inst✝⁴ : Algebra R P", "loc : IsLocalization S P", "P' : Type u_3", "inst✝³ : CommRing P'", "inst✝² : Algebra R P'", "loc' : IsLocalization S P'", "P'' : Type u_4", "inst✝¹ : CommRing P''", "inst✝ : Algebra R P''", "loc'' : IsLocalization S P''", "I J : FractionalIdeal S P", "g : P →ₐ[R] P'", "s✝ : Set P", "x✝ : ∃ a ∈ S, ∀ b ∈ s✝, IsInteger R (a • b)", "a : R", "a_mem : a ∈ S", "h : ∀ b ∈ s✝, IsInteger R (a • b)", "b : P", "hb : b ∈ span R s✝", "s : R", "x : P", "hx : IsInteger R (a • x)"], "goal": "IsInteger R (s • a • x)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "K : LieSubalgebra R L"], "goal": "K.incl.range = K"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "ι₂ : Sort u_6", "s : ι → Set α", "t : ι₂ → Set α", "h : ∀ (i : ι), ∃ j, s i ⊆ t j"], "goal": "⋃ i, s i ⊆ ⋃ i, t i"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "v : VitaliFamily μ", "SecondCountableTopology α", "BorelSpace α", "MeasureTheory.IsLocallyFiniteMeasure μ", "ρ : MeasureTheory.Measure α", "MeasureTheory.IsLocallyFiniteMeasure ρ"], "goal": "∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (ρ.rnDeriv μ x))"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "OrderBot α", "DecidableEq α", "a : α", "parts : Finset α", "sup_indep : parts.SupIndep id", "sup_parts : parts.sup id = a"], "goal": "(Finpartition.ofErase parts sup_indep sup_parts).parts = parts.erase ⊥"}}
+{"state": {"context": ["α : Type u", "c : Cardinal.{u}", "l : Filter α", "CardinalInterFilter l c", "hc : ℵ₀ < c"], "goal": "CountableInterFilter l"}}
+{"state": {"context": ["G : Type u_2", "α : Type u_3", "Inv G", "MeasurableSpace G", "MeasurableInv G", "m : MeasurableSpace α", "f : α → G", "μ : MeasureTheory.Measure α", "hf : AEMeasurable f μ"], "goal": "AEMeasurable (fun x => (f x)⁻¹) μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type uG", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : E → F", "x : E", "n : ℕ∞", "e : F ≃L[𝕜] G"], "goal": "ContDiffAt 𝕜 n (⇑e ∘ f) x ↔ ContDiffAt 𝕜 n f x"}}
+{"state": {"context": ["z : ℂ", "i j : Fin 2"], "goal": "(leftMulMatrix basisOneI) z i j = !![z.re, -z.im; z.im, z.re] i j"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "f : α → β → γ", "s s' : Set α", "t t' : Set β"], "goal": "Set.image2 f (s ∩ s') (t ∪ t') ⊆ Set.image2 f s t ∪ Set.image2 f s' t'"}}
+{"state": {"context": ["X : Type u_1", "R : Type u_2", "Zero X", "TopologicalSpace X", "TopologicalSpace R", "CommSemiring R", "TopologicalSemiring R", "StarRing R", "ContinuousStar R", "f : ContinuousMapZero X R"], "goal": "ContinuousMapZero.toContinuousMapHom f = ↑f"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : α → F", "g : α → G", "c : ℝ≥0", "h : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ c * ‖g x‖₊", "p : ℝ", "hp : 0 < p"], "goal": "∫⁻ (a : α), ↑(‖f a‖₊ ^ p) ∂μ ≤ ∫⁻ (a : α), ↑(c ^ p * ‖g a‖₊ ^ p) ∂μ"}}
+{"state": {"context": ["I : Type u", "inst✝⁶ : AddMonoid I", "C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "inst✝⁴ : MonoidalCategory C", "X₁ X₂ X₃ X₄ : GradedObject I C", "inst✝³ : X₃.HasTensor X₄", "inst✝² : X₂.HasTensor (tensorObj X₃ X₄)", "inst✝¹ : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))", "j : I", "A : C", "f g : tensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) j ⟶ A", "inst✝ : X₂.HasGoodTensorTensor₂₃ X₃ X₄", "H : X₁.HasTensor₄ObjExt X₂ X₃ X₄", "h : ∀ (i₁ i₂ i₃ i₄ : I) (h : i₁ + i₂ + i₃ + i₄ = j), ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ f = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ g", "i₁ i₂₃₄ : I", "h' : i₁ + i₂₃₄ = j"], "goal": "ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ f = ιTensorObj X₁ (tensorObj X₂ (tensorObj X₃ X₄)) i₁ i₂₃₄ j h' ≫ g"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a : α", "s : Finset α"], "goal": "(insert a s).val = Multiset.ndinsert a s.val"}}
+{"state": {"context": ["x : EReal"], "goal": "⊥ + x = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ✝ : Measure α", "inst✝ : NormedAddCommGroup β", "μ : Measure α", "p : ℝ≥0∞", "f✝ : ℕ → α → β", "g✝ : α → β", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "f : ℕ → α → β", "g : α → β", "haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ", "hg' : Memℒp g p μ", "hui : UnifIntegrable f p μ", "hut : UnifTight f p μ", "hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))", "hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)", "hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯", "hui' : UnifIntegrable (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ", "hut' : UnifTight (fun n => AEStronglyMeasurable.mk (f n) ⋯) p μ", "hg : StronglyMeasurable (AEStronglyMeasurable.mk g ⋯)", "hgg' : g =ᶠ[ae μ] AEStronglyMeasurable.mk g ⋯"], "goal": "Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)"}}
+{"state": {"context": ["x y : ZFSet", "h : x.Nonempty"], "goal": "y ∈ ⋂₀ x ↔ ∀ z ∈ x, y ∈ z"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β", "r : ℝ", "hr : 0 ≤ r", "H1 : ∀ x ∈ s, ∃ y ∈ t, dist x y ≤ r", "H2 : ∀ x ∈ t, ∃ y ∈ s, dist x y ≤ r"], "goal": "∀ x ∈ t, infDist x s ≤ r"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "w x y z : α", "inst✝ : GeneralizedBooleanAlgebra α", "h : x < z \\ y", "hyz : y ≤ z", "h' : z \\ y ⊔ y ≤ x ⊔ y"], "goal": "(z \\ y) \\ y ≤ x"}}
+{"state": {"context": ["z x✝ y✝ : ℝ", "n : ℕ", "hn : n ≠ 0", "x y : ℝ", "h : x < y", "hy : 0 < y", "hn' : 0 < ↑n", "q : ℚ", "hqy : ↑q < y ^ (↑n)⁻¹", "this : 0 ≤ max 0 x ^ (↑n)⁻¹", "hq : 0 < ↑q", "hxq : (max 0 x ^ (↑n)⁻¹) ^ ↑n < ↑q ^ ↑n"], "goal": "∃ q, 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "StrictOrderedCommRing R", "AddCommGroup V", "Module R V", "AddTorsor V P", "s : AffineSubspace R P", "x y : P", "h : s.SOppSide x y"], "goal": "y ∉ s"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "x y z : α", "DecompositionMonoid α", "H1 : IsRelPrime x y", "H2 : x ∣ y * z"], "goal": "x ∣ z"}}
+{"state": {"context": ["σ : Type u_1", "inst✝¹ : Fintype σ", "R : Type u_2", "inst✝ : CommRing R", "k : ℕ", "h : 0 < k", "hesymm : ↑k * esymm σ R k = (-1) ^ (k + 1) * (∑ x ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2 + ∑ x ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2)", "sub_both_sides✝ : ↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2 = (-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2", "sub_both_sides : (fun x => (-1) ^ (k + 1) * x) (↑k * esymm σ R k - (-1) ^ (k + 1) * ∑ a ∈ filter (fun a => 0 < a.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2) = (fun x => (-1) ^ (k + 1) * x) ((-1) ^ (k + 1) * ∑ a ∈ filter (fun x => ¬0 < x.1) (filter (fun a => a.1 < k) (antidiagonal k)), (-1) ^ a.1 * esymm σ R a.1 * psum σ R a.2)"], "goal": "psum σ R k = (-1) ^ (k + 1) * ↑k * esymm σ R k - ∑ x ∈ filter (fun a => a.1 < k ∧ 0 < a.1) (antidiagonal k), (-1) ^ x.1 * esymm σ R x.1 * psum σ R x.2"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "sz : ℕ", "m : Ordnode α", "y : α", "r : Ordnode α"], "goal": "l.rotateL x (Ordnode.node sz m y r) = if m.size < Ordnode.ratio * r.size then l.node3L x m y r else l.node4L x m y r"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimits C", "X : TopCat", "F : TopCat.Presheaf C X", "U : TopologicalSpace.Opens ↑X", "x y : ↑X", "h : x ⤳ y", "hy : y ∈ U"], "goal": "F.germ ⟨y, hy⟩ ≫ F.stalkSpecializes h = F.germ ⟨x, ⋯⟩"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "K✝ : Type u_4", "A : Type u_5", "A' : Type u_6", "A'' : Type u_7", "V : Type u", "V' : Type u_8", "x : ι → A", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing A", "inst✝⁸ : CommRing A'", "inst✝⁷ : CommRing A''", "inst✝⁶ : Algebra R A", "inst✝⁵ : Algebra R A'", "inst✝⁴ : Algebra R A''", "a b : R", "K : Type u_9", "inst✝³ : CommRing K", "inst✝² : Algebra R K", "inst✝¹ : Algebra K A", "inst✝ : IsScalarTower R K A", "hinj : Injective ⇑(algebraMap R K)", "ai : AlgebraicIndependent K x"], "goal": "AlgebraicIndependent R x"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "o : Type u_4", "α : Type u_12", "AddZeroClass α", "M : Matrix (m × o) (n × o) α", "k : o"], "goal": "(Matrix.blockDiagAddMonoidHom m n o α) M k = M.blockDiag k"}}
+{"state": {"context": ["M : Type u_1", "inst✝¹ : Mul M", "M' : Type u_2", "inst✝ : Mul M'"], "goal": "commProb (M × M') = commProb M * commProb M'"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "𝒜 : Finset (Finset α)", "a : α"], "goal": "Finset.image (insert a) (Finset.memberSubfamily a 𝒜) = Finset.filter (fun x => a ∈ x) 𝒜"}}
+{"state": {"context": ["c : ℝ"], "goal": "∫ (x : ℝ) in Iic c, rexp x = rexp c"}}
+{"state": {"context": ["x✝ y✝ z x y : ℝ", "hx : x < 0"], "goal": "|x ^ y| ≤ |x| ^ y"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalSemiring R"], "goal": "PreQuasiregular.val 1 = 0"}}
+{"state": {"context": ["R' : Type u_10", "M'' : Type u_11", "N'' : Type u_13", "CommSemiring R'", "AddCommMonoid M''", "AddCommMonoid N''", "Module R' M''", "Module R' N''", "n : ℕ", "f g : M'' [⋀^Fin n.succ]→ₗ[R'] N''"], "goal": "(f + g).curryLeft = f.curryLeft + g.curryLeft"}}
+{"state": {"context": ["m : Type u_1", "n✝ : Type u_2", "inst✝⁴ : DecidableEq n✝", "inst✝³ : Fintype n✝", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "R : Type v", "inst✝ : CommRing R", "n : ℕ", "A : Matrix (Fin n.succ) (Fin n.succ) R", "i : Fin n.succ", "this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det", "j : Fin n.succ", "x✝ : j ∈ univ"], "goal": "(-1) ^ ↑j * A.submatrix (⇑i.cycleRange⁻¹) id 0 j * ((A.submatrix (⇑i.cycleRange⁻¹) id).submatrix Fin.succ j.succAbove).det = (-1) ^ ↑j * (A i j * (A.submatrix i.succAbove j.succAbove).det)"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "P : Sort u_5", "x y : α", "h₁ : x < y → P", "h₂ : x = y → P", "h₃ : y < x → P", "p : P", "hlt : ∀ (h : x < y), h₁ h = p", "heq : ∀ (h : x = y), h₂ h = p", "hgt : ∀ (h : y < x), h₃ h = p"], "goal": "ltByCases x y h₁ h₂ h₃ = p"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f g : A"], "goal": "basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f"}}
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+{"state": {"context": ["p : ℕ → Prop", "inst✝ : DecidablePred p", "n : ℕ"], "goal": "count p n = (filter p (range n)).card"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞"], "goal": "essSup f μ = 0 ↔ f =ᵐ[μ] 0"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "F : CategoryTheory.PrelaxFunctor B C", "a b : B", "f g : a ⟶ b", "η : f ⟶ g", "CategoryTheory.IsIso η"], "goal": "CategoryTheory.CategoryStruct.comp (F.map₂ (CategoryTheory.inv η)) (F.map₂ η) =\n  CategoryTheory.CategoryStruct.id (F.map g)"}}
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+{"state": {"context": ["xs : Lean.Omega.IntList"], "goal": "xs + [] = xs"}}
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+{"state": {"context": ["α : Type u"], "goal": "Stream'.Seq.nil.enum = Stream'.Seq.nil"}}
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+{"state": {"context": ["J : Type u", "CategoryTheory.Category.{v, u} J"], "goal": "CategoryTheory.Limits.HasColimitsOfShape J AddCommGrp"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R"], "goal": "algebraMap R (MvPolynomial σ R) = MvPolynomial.C"}}
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+{"state": {"context": ["a b c : Prop"], "goal": "(a ∧ b ↔ a ∧ c) ↔ a → (b ↔ c)"}}
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+{"state": {"context": ["α : Type u_1", "NormedDivisionRing α", "a : α", "ha : a ≠ 0"], "goal": "Filter.map (fun x => x * a) (Bornology.cobounded α) = Bornology.cobounded α"}}
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+{"state": {"context": ["R : Type u", "S : Type v", "A' : Type u_1", "CommSemiring R", "CommSemiring A'", "CommSemiring S", "Algebra S R", "Algebra S A'", "g : R →ₐ[S] A'", "y : A'", "x : R"], "goal": "(Polynomial.aevalTower g y) ((algebraMap R R[X]) x) = g x"}}
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+{"state": {"context": ["α : Type u_1", "GeneralizedBooleanAlgebra α", "a b : α"], "goal": "Booleanisation.lift a ≤ Booleanisation.lift b ↔ a ≤ b"}}
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+{"state": {"context": ["R : Type u", "n : ℕ", "Semiring R", "p : R[X]"], "goal": "p.natDegree ≤ n ↔ ∀ (N : ℕ), n < N → p.coeff N = 0"}}
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+{"state": {"context": ["a b : Int"], "goal": "0 ≤ a + b ↔ -b ≤ a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "DecidableRel fun x x_1 => x < x_1", "a : α"], "goal": "[a].minimum = ↑a"}}
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+{"state": {"context": ["l : Type u_1", "R : Type u_2", "inst✝² : DecidableEq l", "inst✝¹ : CommRing R", "inst✝ : Fintype l"], "goal": "J l R * -J l R = 1"}}
+{"state": {"context": ["α : Type u_1", "Nonempty α", "s : Set (Set α)", "h : ⋃₀ s = Set.univ"], "goal": "s.Nonempty"}}
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+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X", "hs : IsClosed s"], "goal": "IsLocallyClosed s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : Rel α β"], "goal": "Rel.comp Eq r = r"}}
+{"state": {"context": ["R : Type u", "CommRing R", "S : Type v", "CommRing S", "Algebra R S", "P : Algebra.PreSubmersivePresentation R S", "Fintype P.rels", "DecidableEq P.rels"], "goal": "P.jacobian = (algebraMap P.Ring S) P.jacobiMatrix.det"}}
+{"state": {"context": ["α : Type u_1", "l : List α"], "goal": "ofFn (get ⟨l, ⋯⟩) = ⟨l, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "Preorder α", "s : Set α"], "goal": "BddAbove (⇑OrderDual.ofDual ⁻¹' s) ↔ BddBelow s"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝¹ : (i : α) → NormedAddCommGroup (E i)", "inst✝ : Nonempty α", "f : ↥(lp E ⊤)"], "goal": "IsLUB (Set.range fun i => ‖↑f i‖) ‖f‖"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁴ : Field F", "inst✝³ : Field K", "inst✝² : Field L", "inst✝¹ : Algebra F K", "inst✝ : Algebra F L", "ne : Nonempty (K →ₐ[F] L)", "h : Normal F L", "φ : K →ₐ[F] L", "x : K"], "goal": "Splits (algebraMap F L) (minpoly F (φ x))"}}
+{"state": {"context": ["α : Type u_1", "Semifield α", "b d : α", "a c : α", "hb : b ≠ 0", "hd : d ≠ 0"], "goal": "a / b + c / d = (a * d + b * c) / (b * d)"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "ι✝ : Sort u_4", "f✝ f₁ f₂ : Filter α", "g✝ g₁ g₂ : Set α → Filter β✝", "ι : Sort u_6", "p : ι → Prop", "s✝ : ι → Set α", "f : Filter α", "hf : f.HasBasis p s✝", "β : ι → Type u_5", "pg : (i : ι) → β i → Prop", "sg : (i : ι) → β i → Set γ", "g : Set α → Filter γ", "hg : ∀ (i : ι), (g (s✝ i)).HasBasis (pg i) (sg i)", "gm : Monotone g", "s : Set γ"], "goal": "s ∈ f.lift g ↔ ∃ i, p i ∧ ∃ x, pg i x ∧ sg i x ⊆ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a : α", "b₁ b₂ : β", "OrderedRing α", "OrderedAddCommGroup β", "Module α β", "PosSMulStrictMono α β", "PosSMulReflectLT α β", "ha : a < 0"], "goal": "a • b₁ < a • b₂ ↔ b₂ < b₁"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "c x : P", "R : ℝ", "hR : R ≠ 0"], "goal": "c = EuclideanGeometry.inversion c R x ↔ x = c"}}
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+{"state": {"context": ["T : ℝ", "hT : T ≠ 0", "z✝¹ z✝ : ℝ", "h : expMapCircle (2 * π / T * z✝¹) = expMapCircle (2 * π / T * z✝)", "m : ℤ", "hm : 2 * π * z✝ = 2 * π * z✝¹ + ↑m * (2 * π) * T"], "goal": "↑m * T = -z✝¹ + z✝"}}
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+{"state": {"context": ["x : ℝ"], "goal": "HasDerivAt Real.exp (Real.exp x) x"}}
+{"state": {"context": ["a : Ordinal.{u_1}"], "goal": "ω ≤ (ord ∘ aleph) a"}}
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+{"state": {"context": ["α : Type u", "Monoid α", "a b : α"], "goal": "∀ {x : Invertible a}, a * (⅟a * b) = b"}}
+{"state": {"context": ["α : Type u", "Semiring α", "Preorder α", "a : α", "MulPosMono α", "ha : 0 ≤ a"], "goal": "Monotone fun x => x * a"}}
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+{"state": {"context": ["R : Type u", "CommSemiring R"], "goal": "Polynomial.Separable 1"}}
+{"state": {"context": ["α : Type u_2", "Fintype α", "Encodable α", "x : α"], "goal": "x ∈ Encodable.sortedUniv α"}}
+{"state": {"context": ["x✝ y x : ℝ"], "goal": "1 < rexp x ↔ 0 < x"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "self : AList β"], "goal": "self.entries.NodupKeys"}}
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+{"state": {"context": ["α : Type u_2", "Ring α", "a b : α", "ha : Odd a", "hb : Even b"], "goal": "Odd (a - b)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "t : Set (ProjectiveSpectrum 𝒜)"], "goal": "vanishingIdeal (closure t) = vanishingIdeal t"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : SeminormedCommGroup E", "ε δ : ℝ", "s t : Set E", "x✝ y : E", "hs : IsCompact s", "hδ : 0 ≤ δ", "x : E"], "goal": "x ∈ s * closedBall 1 δ ↔ x ∈ ⋃ x ∈ s, closedBall x δ"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : LinearOrderedAddCommGroup G", "inst✝ : Archimedean G", "H : AddSubgroup G", "a : G", "h₀ : 0 < a", "hd : Disjoint (↑H) (Ioo 0 a)", "hbot : H ≠ ⊥"], "goal": "∃ b, H = closure {b}"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z", "hyz' : 0 < z - y", "hxz' : 0 < z - x"], "goal": "(f z - f x) / (z - x) ≤ (f z - f y) / (z - y)"}}
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+{"state": {"context": ["n : ℕ", "α : TypeVec.{u_1} n"], "goal": "TypeVec.comp TypeVec.prod.fst TypeVec.prod.diag = TypeVec.id"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : Rel α β", "s : Set β"], "goal": "s ∩ r.codom ⊆ r.image (r.preimage s)"}}
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+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a : α", "𝒜 : Finset (Finset α)"], "goal": "(𝓓 a 𝒜).shatterer ⊆ 𝒜.shatterer"}}
+{"state": {"context": ["α : Sort u_4", "β : Sort u_5"], "goal": "IsEmpty (α ⊕' β) ↔ IsEmpty α ∧ IsEmpty β"}}
+{"state": {"context": ["R : Type u", "CommRing R", "p : R[X]", "p0 : p ≠ 0", "a : R"], "goal": "¬(Polynomial.X - Polynomial.C a) ^ (Polynomial.rootMultiplicity a p + 1) ∣ p"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : Fin 3 → F", "hP : W.Equation P", "hQ : W.Equation Q", "hPz : P z ≠ 0", "hQz : Q z ≠ 0", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3"], "goal": "W.dblXYZ P = W.dblZ P • ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), 1]"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_4", "Semiring R", "AddCommMonoid M", "Module R M", "s : Set M"], "goal": "(Submodule.span R s).toAddSubmonoid = AddSubmonoid.closure (Set.univ • s)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "M : Type u_3", "AddCommMonoid M", "TopologicalSpace M", "v : MeasureTheory.VectorMeasure α M", "N : Type u_4", "AddCommMonoid N", "TopologicalSpace N", "f : M →+ N", "hf : Continuous ⇑f", "s : Set α"], "goal": "↑(v.mapRange f hf) s = f (↑v s)"}}
+{"state": {"context": ["z τ J J' : ℂ"], "goal": "cexp (↑π * I * (z + 1) ^ 2 * τ) * (cexp (-↑π * I * (τ + 2 * (z * τ))) * (J' - 2 * ↑π * I * J) + (z * (cexp (-↑π * I * (τ + 2 * (z * τ))) * J) + cexp (-↑π * I * (τ + 2 * (z * τ))) * J) * (2 * ↑π * I)) = cexp (↑π * I * z ^ 2 * τ) * (J' + z * J * (2 * ↑π * I))"}}
+{"state": {"context": ["F : Type u → Type u", "q : QPF F", "α : Type u", "g₁ g₂ : Fix F → α", "h : ∀ (x : F (Fix F)), g₁ <$> x = g₂ <$> x → g₁ (mk x) = g₂ (mk x)", "x✝ : Fix F", "a : (P F).A", "f : (P F).B a → WType (P F).B", "ih : ∀ (a : (P F).B a), g₁ (Quot.mk Setoid.r (f a)) = g₂ (Quot.mk Setoid.r (f a))"], "goal": "g₁ <$> abs ⟨a, fun x => ⟦f x⟧⟩ = g₂ <$> abs ⟨a, fun x => ⟦f x⟧⟩"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "UniqueFactorizationMonoid α", "a p : α", "ha0 : a ≠ 0", "hp : Irreducible p"], "goal": "Associates.mk p ∈ (Associates.mk a).factors ↔ p ∣ a"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "a r : ℝ"], "goal": "volume (Metric.ball a r) = ofReal (2 * r)"}}
+{"state": {"context": ["P : TopCat → Prop", "α : Type w", "Finite α", "X : α → CompHausLike P", "CompHausLike.HasExplicitFiniteCoproduct X", "R : ↑(CompHausLike.finiteCoproduct X).toTop"], "goal": "∃ a r, R = (CompHausLike.finiteCoproduct.ι X a) r"}}
+{"state": {"context": ["p✝ : ℕ", "inst✝ : Fact (Nat.Prime p✝)", "n✝¹ p : ℕ", "h_prime : Fact (Nat.Prime p)", "n✝ n : ℕ", "h : n ≠ 0", "g_poly : (ZMod p)[X] := X ^ p ^ n - X", "hp : 1 < p"], "goal": "FiniteDimensional.finrank (ZMod p) (GaloisField p n) = n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : FirstCountableTopology X", "inst✝ : T1Space X", "x : X", "h : {x}.Subsingleton"], "goal": "IsClosed {x}"}}
+{"state": {"context": ["α : Type u", "β : Type v", "δ : Type x", "Preorder β", "f : α → β", "s : Set α", "a : α", "t : Set δ", "g : δ → α", "b : δ", "hf : IsMinOn f s a", "hg : Set.MapsTo g t s", "ha : g b = a"], "goal": "IsMinOn (f ∘ g) t b"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : ℝ → E", "x : ℝ", "n : ℕ", "s : Set ℝ", "hs : UniqueDiffOn ℝ s", "hf : ContDiffOn ℝ (↑n) f s", "i : ℕ", "hi : i ∈ Finset.range (n + 1)"], "goal": "ContinuousOn (iteratedDerivWithin i f s) s"}}
+{"state": {"context": ["x✝ y✝ x : ℝ", "hx : x < 0", "y : ℝ", "hy : y < 0", "hxy : x < y"], "goal": "log y < log x"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M"], "goal": "FirstOrder.Language.JointEmbedding (L.age M)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν✝ ν μ : Measure α", "inst✝¹ : SigmaFinite ν", "inst✝ : SigmaFinite μ", "r : ℝ≥0"], "goal": "(r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "f g₁ g₂ : R", "U : Opens ��(PrimeSpectrum.Top R)", "hu₁ : ∀ x ∈ U, g₁ ∈ x.asIdeal.primeCompl", "hu₂ : ∀ x ∈ U, g₂ ∈ x.asIdeal.primeCompl"], "goal": "f * g₁ * g₂ = f * (g₁ * g₂)"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "C₃ : Type u_3", "D : Type u_4", "I₁ : Type u_5", "I₂ : Type u_6", "I₃ : Type u_7", "J : Type u_8", "CategoryTheory.Category.{u_10, u_1} C₁", "CategoryTheory.Category.{u_12, u_2} C₂", "CategoryTheory.Category.{u_11, u_3} C₃", "CategoryTheory.Category.{u_9, u_4} D", "Zero I₂", "DecidableEq I₂", "CategoryTheory.Limits.HasInitial C₂", "F₁ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁)", "F₂ : CategoryTheory.Functor C₂ (CategoryTheory.Functor C₃ C₃)", "G : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₃ D)", "associator : CategoryTheory.bifunctorComp₁₂ F₁ G ≅ CategoryTheory.bifunctorComp₂₃ G F₂", "X₂ : C₂", "e₁ : F₁.flip.obj X₂ ≅ CategoryTheory.Functor.id C₁", "e₂ : F₂.obj X₂ ≅ CategoryTheory.Functor.id C₃", "(X₁ : C₁) → CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C₂) (F₁.obj X₁)", "(X₃ : C₃) → CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C₂) (F₂.flip.obj X₃)", "r : I₁ × I₂ × I₃ → J", "π : I₁ × I₃ → J", "τ : CategoryTheory.GradedObject.TriangleIndexData r π", "X₁ : CategoryTheory.GradedObject I₁ C₁", "X₃ : CategoryTheory.GradedObject I₃ C₃", "(((CategoryTheory.GradedObject.mapBifunctor F₁ I₁ I₂).obj X₁).obj\n        ((CategoryTheory.GradedObject.single₀ I₂).obj X₂)).HasMap\n    τ.p₁₂", "(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj\n            (CategoryTheory.GradedObject.mapBifunctorMapObj F₁ τ.p₁₂ X₁\n              ((CategoryTheory.GradedObject.single₀ I₂).obj X₂))).obj\n        X₃).HasMap\n    π", "(((CategoryTheory.GradedObject.mapBifunctor F₂ I₂ I₃).obj ((CategoryTheory.GradedObject.single₀ I₂).obj X₂)).obj\n        X₃).HasMap\n    τ.p₂₃", "(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj X₁).obj\n        (CategoryTheory.GradedObject.mapBifunctorMapObj F₂ τ.p₂₃ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂)\n          X₃)).HasMap\n    π", "CategoryTheory.GradedObject.HasGoodTrifunctor₁₂Obj F₁ G τ.ρ₁₂ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃", "CategoryTheory.GradedObject.HasGoodTrifunctor₂₃Obj G F₂ τ.ρ₂₃ X₁ ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃", "(((CategoryTheory.GradedObject.mapBifunctor G I₁ I₃).obj X₁).obj X₃).HasMap π", "triangle :\n  ∀ (X₁ : C₁) (X₃ : C₃),\n    CategoryTheory.CategoryStruct.comp (((associator.hom.app X₁).app X₂).app X₃) ((G.obj X₁).map (e₂.hom.app X₃)) =\n      (G.map (e₁.hom.app X₁)).app X₃"], "goal": "CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.GradedObject.mapBifunctorAssociator associator τ.ρ₁₂ τ.ρ₂₃ X₁\n        ((CategoryTheory.GradedObject.single₀ I₂).obj X₂) X₃).hom\n    (CategoryTheory.GradedObject.mapBifunctorMapMap G π (CategoryTheory.CategoryStruct.id X₁)\n      (CategoryTheory.GradedObject.mapBifunctorLeftUnitor F₂ X₂ e₂ τ.p₂₃ ⋯ X₃).hom) =\n  CategoryTheory.GradedObject.mapBifunctorMapMap G π\n    (CategoryTheory.GradedObject.mapBifunctorRightUnitor F₁ X₂ e₁ τ.p₁₂ ⋯ X₁).hom (CategoryTheory.CategoryStruct.id X₃)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsJacobson R", "P : Ideal R[X]", "hP : P.IsMaximal", "inst✝ : Nontrivial R", "hP' : ∀ (x : R), C x ∈ P → x = 0", "P' : Ideal R := comap C P", "hP'_prime : P'.IsPrime", "m : R[X]", "hmem_P : m ∈ P", "hm : ⟨m, hmem_P⟩ ≠ 0", "hm' : m ≠ 0", "φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯", "a : R ⧸ P' := (Polynomial.map (Quotient.mk P') m).leadingCoeff", "M : Submonoid (R ⧸ P') := Submonoid.powers a", "hp0 : a ≠ 0", "hM : 0 ∉ M"], "goal": "⊥.IsMaximal"}}
+{"state": {"context": ["f g : CircleDeg1Lift", "hf : Continuous ⇑f", "z : ℝ", "hz : ∀ (x : ℝ), f x < x + z"], "goal": "τ f < z"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "R✝ : Type u_4", "S : Type u_5", "π : ι → Type u_6", "R : Type u_7", "inst✝¹ : StrictOrderedSemiring R", "inst✝ : Archimedean R", "s : ℕ → Set α", "ω : α"], "goal": "Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop ↔ Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) atTop atTop"}}
+{"state": {"context": ["R : Type u_1", "AddCommMonoid R", "StarAddMonoid R", "x : R"], "goal": "IsSelfAdjoint (star x + x)"}}
+{"state": {"context": ["x y : ℕ∞"], "goal": "↑x ≤ ↑y ↔ x ≤ y"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "H : Type u_3", "TopologicalSpace H", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "CommMonoid G", "TopologicalSpace G", "ChartedSpace H G", "SmoothMul I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "x₀ : M", "f : ι → M → G", "n : ℕ∞", "lf : LocallyFinite fun i => Function.mulSupport (f i)", "h : ∀ (i : ι), ContMDiffAt I' I n (f i) x₀"], "goal": "ContMDiffAt I' I n (fun x => ∏ᶠ (i : ι), f i x) x₀"}}
+{"state": {"context": ["a b : ℝ", "f : C(↑(Set.Icc a b), ℝ)", "ε : ℝ", "pos : 0 < ε"], "goal": "∃ p, ‖p.toContinuousMapOn (Set.Icc a b) - f‖ < ε"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve R", "n : ℤ"], "goal": "(W.Φ n).coeff (n.natAbs ^ 2) = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "AddCommMonoid M", "e : α ≃ β", "f : β → M"], "goal": "∑ᶠ (i : α), f (e i) = ∑ᶠ (i' : β), f i'"}}
+{"state": {"context": ["Γ✝ : Type u_1", "inst✝¹⁴ : PartialOrder Γ✝", "R✝ : Type u_2", "V✝ : Type u_3", "W✝ : Type u_4", "inst✝¹³ : CommRing R✝", "inst✝¹² : AddCommGroup V✝", "inst✝¹¹ : Module R✝ V✝", "inst✝¹⁰ : AddCommGroup W✝", "inst✝⁹ : Module R✝ W✝", "Γ : Type u_5", "Γ' : Type u_6", "inst✝⁸ : OrderedCancelAddCommMonoid Γ", "inst✝⁷ : OrderedCancelAddCommMonoid Γ'", "R : Type u_7", "inst✝⁶ : CommRing R", "U : Type u_8", "V : Type u_9", "W : Type u_10", "inst✝⁵ : AddCommGroup U", "inst✝⁴ : Module R U", "inst✝³ : AddCommGroup V", "inst✝² : Module R V", "inst✝¹ : AddCommGroup W", "inst✝ : Module R W", "A : HVertexOperator Γ R V W", "B : HVertexOperator Γ' R U V", "u : U", "g' : Γ'", "hg' : ¬A (((of R).symm (B u)).coeff g') = 0", "hAB : ((of R).symm (B u)).coeff g' = 0"], "goal": "A 0 = 0"}}
+{"state": {"context": ["p : ℝ[X]", "hp' : derivative p ≠ 0", "hp : p ≠ 0", "x : ℝ", "hx : p.IsRoot x", "y : ℝ", "hy : p.IsRoot y", "hxy : x < y", "hxy' : ∀ z ∈ p.roots.toFinset, z ∉ Set.Ioo x y", "z : ℝ", "hz1 : z ∈ Set.Ioo x y", "hz2 : deriv (fun x => eval x p) z = 0"], "goal": "z ∈ (derivative p).roots.toFinset"}}
+{"state": {"context": ["α : Type u", "self : StrictOrderedSemiring α", "a b c : α"], "goal": "a < b → 0 < c → a * c < b * c"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "E' : Type u_3", "F : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹³ : RCLike 𝕜", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : InnerProductSpace 𝕜 E", "inst✝¹⁰ : CompleteSpace E", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : InnerProductSpace 𝕜 E'", "inst✝⁷ : CompleteSpace E'", "inst✝⁶ : NormedSpace ℝ E'", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedAddCommGroup G'", "inst✝¹ : NormedSpace ℝ G'", "inst✝ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "hm : m ≤ m0", "f : ↥(Lp E' 2 μ)", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤"], "goal": "∀ (c : E'), ∫ (x : α) in s, ⟪c, (↑↑↑((condexpL2 E' 𝕜 hm) f) - ↑↑f) x⟫_𝕜 ∂μ = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "α : X ≅ Y"], "goal": "α.symm.inv = α.hom"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "E : Type u_3", "AddCommGroup E", "Module 𝕜 E", "f : E → F", "x : E"], "goal": "HasLineDerivAt 𝕜 f 0 x 0"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "NormedAddCommGroup E", "𝕜 : Type u_3", "RCLike 𝕜", "NormedSpace 𝕜 E", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "g' : E → E →L[𝕜] G", "x : E", "l.NeBot", "s : Set E", "hs : IsOpen s", "hf' : TendstoLocallyUniformlyOn (fderiv 𝕜 ∘ f) g' l s", "hf : ∀ (n : ι), DifferentiableOn 𝕜 (f n) s", "hfg : ∀ x ∈ s, Filter.Tendsto (fun n => f n x) l (𝓝 (g x))", "hx : x ∈ s"], "goal": "HasFDerivAt g (g' x) x"}}
+{"state": {"context": ["K : Type u", "inst✝ : Field K", "a : K", "H : Irreducible (X ^ 0 - C a)"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type x", "UniformSpace β", "F : ι → α → β", "f : α → β", "s : Set α", "p : Filter ι", "p.NeBot", "hF : UniformCauchySeqOn F p s", "hF' : ∀ x ∈ s, Filter.Tendsto (fun n => F n x) p (𝓝 (f x))"], "goal": "TendstoUniformlyOn F f p s"}}
+{"state": {"context": ["m k : ℕ+", "h₀ : ↑m % ↑k + ↑k * (↑m / ↑k) = ↑m := Nat.mod_add_div ↑m ↑k", "this : ¬(↑m % ↑k = 0 ∧ ↑m / ↑k = 0)"], "goal": "↑(m.mod k) + ↑k * m.div k = ↑m"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f g : ℝ × ℝ → E", "f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E", "a b : �� × ℝ", "hle : a ≤ b", "s : Set (ℝ × ℝ)", "hs : s.Countable", "Hcf : ContinuousOn f (Set.Icc a b)", "Hcg : ContinuousOn g (Set.Icc a b)", "Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \\ s, HasFDerivAt f (f' x) x", "Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \\ s, HasFDerivAt g (g' x) x", "Hi : IntegrableOn (fun x => (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume", "e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm"], "goal": "∀ (x y : ℝ × ℝ), e x ≤ e y ↔ x ≤ y"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasFiniteCoproducts C", "K : ChainComplex C ℕ", "n : ℕ"], "goal": "(AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K).f.f n =\n  (AlgebraicTopology.DoldKan.Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X : Cᵒᵖ"], "goal": "(CategoryTheory.MonoidalCategory.leftUnitor X).unop =\n  (CategoryTheory.MonoidalCategory.leftUnitor (Opposite.unop X)).symm"}}
+{"state": {"context": ["α : Type u_3", "CommMonoid α", "s : Multiset α", "DecidableEq α", "a : α"], "goal": "a ^ Multiset.count a s = (Multiset.filter (Eq a) s).prod"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_6, u_1} C", "inst✝² : Category.{u_4, u_2} D", "L : C ⥤ D", "H : Type u_3", "inst✝¹ : Category.{u_5, u_3} H", "inst✝ : ∀ (F : C ⥤ H), L.HasLeftKanExtension F", "G : D ⥤ H"], "goal": "L.lanUnit.app (L ⋙ G) ≫ whiskerLeft L ((L.lanAdjunction H).counit.app G) = 𝟙 (L ⋙ G)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "ha : a ≠ 0"], "goal": "(C b).degree < (C a * X).degree"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "H : C ⥤ D", "F G : J ⥤ C", "α : F ≅ G", "c : CategoryTheory.Limits.Cocone G"], "goal": "H.mapCoconePrecomposeEquivalenceFunctor.hom.hom = 𝟙 (H.obj c.pt)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν✝ μ ν : Measure α", "inst✝ : SigmaFinite μ", "n : ℕ"], "goal": "μ (spanningSets μ n) ≠ ⊤"}}
+{"state": {"context": ["α : Type u_2", "CancelCommMonoidWithZero α", "GCDMonoid α", "a : α", "b : α", "ha : Squarefree a"], "goal": "Squarefree (gcd a b)"}}
+{"state": {"context": ["V : Type u_1", "G G' : SimpleGraph V", "M M' : G.Subgraph", "v w : V", "h : M.IsMatching", "hv : v ∈ M.verts", "hvw : M.Adj v w"], "goal": "Exists.choose ⋯ = w"}}
+{"state": {"context": ["l : Filter ℂ", "hl : IsExpCmpFilter l", "n : ℕ"], "goal": "(fun z => Real.exp z.re) = fun z => Real.exp z.re ^ 1"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹² : CommRing R", "inst✝¹¹ : LieRing L", "inst✝¹⁰ : LieAlgebra R L", "inst✝⁹ : AddCommGroup M", "inst✝⁸ : Module R M", "inst✝⁷ : LieRingModule L M", "inst✝⁶ : LieModule R L M", "k : ℕ", "N : LieSubmodule R L M", "M₂ : Type w₁", "inst✝⁵ : AddCommGroup M₂", "inst✝⁴ : Module R M₂", "inst✝³ : LieRingModule L M₂", "inst✝² : LieModule R L M₂", "inst✝¹ : Nontrivial M", "inst✝ : IsNilpotent R L M"], "goal": "(match nilpotencyLength R L M with | 0 => ⊥ | k.succ => lowerCentralSeries R L M k) ≠ ⊥"}}
+{"state": {"context": ["K : Type v", "L : Type w", "Field K", "Field L", "i : K →+* L", "x : K", "h : K[X]", "h_splits : Polynomial.Splits i h", "h_roots : (Polynomial.map i h).roots = {i x}"], "goal": "h = Polynomial.C h.leadingCoeff * (Polynomial.X - Polynomial.C x)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "U : (TopologicalSpace.Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ"], "goal": "(AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred 1"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝¹ : CompleteLattice β", "inst✝ : DecidableEq α", "x : α", "t : Finset α", "f : α → β", "s : β", "hx : x ∉ t"], "goal": "s ⊔ ⨆ x_1 ∈ t, update f x s x_1 = s ⊔ ⨆ i ∈ t, f i"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "K : Type u_4", "inst✝³ : Semiring S", "inst✝² : CommSemiring R", "inst✝¹ : Semiring A", "inst✝ : Field K", "p : S[X]", "f : S →+* R", "r : R", "s : S", "hr : eval₂ f r p = 0"], "goal": "eval₂ f (f s * r) (p.scaleRoots s) = 0"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "Module R M", "Module R N", "Module R P", "f : N ≃ₗ[R] P", "m : M", "n : N"], "goal": "(LinearEquiv.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m"}}
+{"state": {"context": ["T : ℝ", "hT : Fact (0 < T)", "f : ↥(Lp ℂ 2 haarAddCircle)"], "goal": "HasSum (fun i => ↑(fourierBasis.repr f) i • (fun i => fourierBasis.repr.symm (lp.single 2 i 1)) i) f"}}
+{"state": {"context": ["b x y : ℝ", "hb : 1 < b", "hy : 0 < y"], "goal": "x < logb b y ↔ b ^ x < y"}}
+{"state": {"context": ["θ : Angle", "hpi : ¬θ = ↑π"], "goal": "(2 • θ).sign = θ.sign ↔ |θ.toReal| < π / 2"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "a b c : α", "ab : a ≠ b", "ac : a ≠ c", "bc : b ≠ c"], "goal": "(swap a b * swap a c).support = {a, b, c}"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "NonUnitalSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "s t : Set A", "h : s ⊆ t"], "goal": "NonUnitalSubalgebra.centralizer R t ≤ NonUnitalSubalgebra.centralizer R s"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Prop", "Fintype { x // p x }", "Fintype { x // q x }", "Fintype { x // p x ∨ q x }"], "goal": "Fintype.card { x // p x ∨ q x } ≤ Fintype.card { x // p x } + Fintype.card { x // q x }"}}
+{"state": {"context": ["F : Type u → Type v", "Applicative F", "G : Type u → Type w", "Applicative G", "η : ApplicativeTransformation F G", "α β : Type u", "x : F (α → β)", "y : F α"], "goal": "(fun {α} => η.app α) (Seq.seq x fun x => y) = Seq.seq ((fun {α} => η.app α) x) fun x => (fun {α} => η.app α) y"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "x y : R", "A : Type v", "inst✝⁶ : CommSemiring R", "inst✝⁵ : Semiring A", "inst✝⁴ : Algebra R A", "ι : Type u_3", "M : Type u_4", "inst✝³ : Fintype ι", "inst✝² : DecidableEq ι", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "b : Basis ι R M", "f : M →ₗ[R] M"], "goal": "IsNilpotent ((toMatrix b b) f) ↔ IsNilpotent f"}}
+{"state": {"context": ["x : Lean.Omega.IntList", "m : Nat"], "goal": "List.length (x.bmod m) ≤ List.length x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type x", "k : Type y", "A : Type z", "a b : R", "m n : ℕ", "inst✝ : CommRing R", "f : R[X]", "x y : R"], "goal": "(f.sum fun e a => (RingHom.id R) a * (x + y) ^ e) = f.sum fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "N : Type u_2", "CommMonoid N", "P : Type u_3", "CommMonoid P", "f : S.LocalizationMap N", "g : M →* P", "T : Submonoid P", "hy : ∀ (y : ↥S), g ↑y ∈ T", "Q : Type u_4", "CommMonoid Q", "k : T.LocalizationMap Q", "x : M", "y : ↥S"], "goal": "(f.map hy k) (f.mk' x y) = k.mk' (g x) ⟨g ↑y, ⋯⟩"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "S : Type u_1", "Semiring S", "Algebra R S", "ι₁ : Type v₁", "ι₂ : Type v₂", "DecidableEq ι₁", "DecidableEq ι₂", "M₁ : ι₁ → Type w₁", "M₂ : ι₂ → Type w₂", "(i₁ : ι₁) → AddCommMonoid (M₁ i₁)", "(i₂ : ι₂) → AddCommMonoid (M₂ i₂)", "(i₁ : ι₁) → Module R (M₁ i₁)", "(i₂ : ι₂) → Module R (M₂ i₂)", "(i₁ : ι₁) → Module S (M₁ i₁)", "∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)", "i₁ : ι₁", "m₁ : M₁ i₁", "i₂ : ι₂", "m₂ : M₂ i₂"], "goal": "(TensorProduct.directSum R S M₁ M₂) ((DirectSum.lof S ι₁ M₁ i₁) m₁ ⊗ₜ[R] (DirectSum.lof R ι₂ M₂ i₂) m₂) =\n  (DirectSum.lof S (ι₁ × ι₂) (fun i => TensorProduct R (M₁ i.1) (M₂ i.2)) (i₁, i₂)) (m₁ ⊗ₜ[R] m₂)"}}
+{"state": {"context": ["α✝ : Type u", "β✝ : Type u_1", "γ : Type u_2", "r✝ : α✝ → α✝ → Prop", "s✝ : β✝ → β✝ → Prop", "t : γ → γ → Prop", "α β : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "inst✝¹ : IsWellOrder α r", "inst✝ : IsWellOrder β s", "f : r ≼i s", "a : α", "x✝ : ↑{b | s b (f a)}", "y : β", "h : y ∈ {b | s b (f a)}"], "goal": "∃ a_1, (RelEmbedding.codRestrict {b | s b (f a)} ((Subrel.relEmbedding r {b | r b a}).trans f.toRelEmbedding) ⋯) a_1 = ⟨y, h⟩"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeInf α", "OrderTop α", "l : List α"], "goal": "(↑l).inf = List.foldr (fun x x_1 => x ⊓ x_1) ⊤ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "inst✝¹ : ConditionallyCompleteLattice α", "s t : Set α", "a b : α", "inst✝ : SemilatticeSup β", "f g : β → α", "hf : Monotone f", "hg : Antitone g", "h : f ≤ g"], "goal": "⨆ n, f n ∈ ⋂ n, Icc (f n) (g n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : MeasurableSpace α", "μ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "inst✝ : TopologicalSpace δ", "f : α → β", "hf : AEStronglyMeasurable f μ"], "goal": "↑(mk f hf) =ᶠ[ae μ] f"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "A : Type u_3", "T : Type u_4", "B : Type u_5", "ι : Type u_6", "inst✝¹¹ : SemilatticeSup B", "inst✝¹⁰ : OrderBot B", "inst✝⁹ : SemilatticeInf T", "inst✝⁸ : OrderTop T", "inst✝⁷ : Semiring R", "inst✝⁶ : AddMonoid A", "inst✝⁵ : AddMonoid B", "inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝² : AddMonoid T", "inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "degb : A → B", "degt : A → T", "degb0 : degb 0 ≤ 0", "degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b", "n : ℕ", "f : R[A]"], "goal": "(List.map (fun f => f.support.sup degb) (List.replicate n f)).sum = (List.replicate n (f.support.sup degb)).sum"}}
+{"state": {"context": ["a b : Cardinal.{u}", "n m : ℕ"], "goal": "IsUnit a ↔ a = 1"}}
+{"state": {"context": ["α : Type u", "R : α → α → Prop"], "goal": "Equivalence R → Equivalence (Stream'.WSeq.LiftRel R)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "σ : Type u_4", "inst✝⁵ : Semiring A", "inst✝⁴ : DecidableEq ι", "inst✝³ : CanonicallyOrderedAddCommMonoid ι", "inst✝² : SetLike σ A", "inst✝¹ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝ : GradedRing 𝒜", "a : ↥(𝒜 0)"], "goal": "↑((projZeroRingHom' 𝒜) ↑a) = ↑a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedRing 𝕜", "AddCommGroup E", "Module 𝕜 E", "s : Set E"], "goal": "(convexHull 𝕜) s ⊆ ↑(affineSpan 𝕜 s)"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "o.oangle (x + y) y = ↑(Real.arcsin (‖x‖ / ‖x + y‖))"}}
+{"state": {"context": ["n k : Nat"], "goal": "n < k + n → 0 < k"}}
+{"state": {"context": ["G : Type u_1", "Group G", "s : Set G", "N : Subgroup G", "N.Normal"], "goal": "s ⊆ ↑N ↔ Subgroup.normalClosure s ≤ N"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : TopologicalSpace α", "inst✝³ : OpensMeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : TopologicalSpace β", "inst✝ : BorelSpace β", "f : α → β", "s : Set α", "μ : Measure α", "hf : ContinuousOn f s", "hs : MeasurableSet s", "a✝ : Nontrivial α", "inhabited_h : Inhabited α", "this : (s.piecewise f fun x => f default) =ᶠ[ae (μ.restrict s)] f"], "goal": "Measurable (s.piecewise f fun x => f default)"}}
+{"state": {"context": ["α✝ β✝ : Type u", "c : Cardinal.{?u.227041}", "α : Type u", "β : Type v", "f : α → β", "s : Set β", "h : Injective f"], "goal": "Injective fun x => f ↑x"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "p q : R[X]", "hq : q.Monic"], "goal": "p %ₘ q = 0 ↔ (Ideal.Quotient.mk (Ideal.span {q})) p = 0"}}
+{"state": {"context": ["G : Type w", "inst✝ : TopologicalSpace G", "μ : Content G", "K₁ K₂ : Compacts G", "h : ↑K₁ ⊆ ↑K₂"], "goal": "(fun s => ↑(μ.toFun s)) K₁ ≤ (fun s => ↑(μ.toFun s)) K₂"}}
+{"state": {"context": ["o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}", "ho : o ≠ 0", "H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)", "a : Ordinal.{max v u}", "b : Ordinal.{max u v}", "h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi b < o.nfpBFamily f a", "this : Nonempty (Quotient.out o).α"], "goal": "∀ (i : (Quotient.out o).α), IsNormal (o.familyOfBFamily f i)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Lattice α", "Lattice β", "BoundedOrder α", "BoundedOrder β", "f : BoundedLatticeHom α β"], "goal": "⇑f.toSupBotHom = ⇑f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "Z : Type u_5", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "e : PartialHomeomorph X Y", "f' : Y ≃ₜ Z"], "goal": "↑(e.transHomeomorph f').symm = ↑e.symm ∘ ⇑f'.symm"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "s : Set (Set α)", "hs : s.Countable", "h : ∀ t ∈ s, MeasurableSet t"], "goal": "MeasurableSet (⋃₀ s)"}}
+{"state": {"context": ["R : Type u", "x : R"], "goal": "Tropical.trop ↑x ≠ 0"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "N : Type u_2", "CommMonoid N", "f : S.LocalizationMap N", "x₁ x₂ : M", "y : ↥S"], "goal": "f.mk' x₂ y * f.toMap x₁ = f.mk' (x₁ * x₂) y"}}
+{"state": {"context": ["p q x y : ℝ", "r : ℚ", "m : ℤ", "n : ℕ", "C : ℝ", "hC : ∃ᶠ (n : ℕ) in atTop, ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p"], "goal": "∃ C, ∃ (_ : 0 < C), ∃ᶠ (n : ℕ) in atTop, 1 ≤ n ∧ ∃ m, x ≠ ↑m / ↑n ∧ |x - ↑m / ↑n| < C / ↑n ^ p"}}
+{"state": {"context": ["α : Type u", "l' : List α", "l : List α", "h : l' ∈ l.cyclicPermutations"], "goal": "l'.length = l.length"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "a : α", "s : Multiset α", "h : a ∈ Multiset.filter p s"], "goal": "p a"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "DecidableEq α", "DecidableEq β", "SemilatticeInf α", "SemilatticeInf β", "FunLike F α β", "InfHomClass F α β", "f : F", "hf : Function.Injective ⇑f", "s t : Finset α"], "goal": "Finset.map { toFun := ⇑f, inj' := hf } (s ⊼ t) =\n  Finset.map { toFun := ⇑f, inj' := hf } s ⊼ Finset.map { toFun := ⇑f, inj' := hf } t"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D", "W X X' Y Y' Z : C", "f : W ⟶ X", "g : X ⟶ Y", "h : Y ⟶ Z", "f' : W ⟶ X'", "g' : X' ⟶ Y'", "h' : Y' ⟶ Z"], "goal": "CategoryTheory.CategoryStruct.comp f\n      (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.comp h (e.unit.app Z))) =\n    CategoryTheory.CategoryStruct.comp f'\n      (CategoryTheory.CategoryStruct.comp g' (CategoryTheory.CategoryStruct.comp h' (e.unit.app Z))) ↔\n  CategoryTheory.CategoryStruct.comp f (CategoryTheory.CategoryStruct.comp g h) =\n    CategoryTheory.CategoryStruct.comp f' (CategoryTheory.CategoryStruct.comp g' h')"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "β : Type u_5", "MeasurableSpace β", "μb : MeasureTheory.Measure β", "g : β →ₘ[μb] E", "hg : g ∈ MeasureTheory.Lp E p μb", "f : α → β", "hf : MeasureTheory.MeasurePreserving f μ μb"], "goal": "g.compMeasurePreserving f hf ∈ MeasureTheory.Lp E p μ"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝⁷ : Semiring R", "inst✝⁶ : StarMul R", "inst✝⁵ : TrivialStar R", "inst✝⁴ : AddCommGroup A", "inst✝³ : Module R A", "inst✝² : StarAddMonoid A", "inst✝¹ : StarModule R A", "inst✝ : Invertible 2", "x y : A"], "goal": "↑((fun x => ⟨⅟2 • (x - star x), ⋯⟩) (x + y)) = ↑((fun x => ⟨⅟2 • (x - star x), ⋯⟩) x + (fun x => ⟨⅟2 • (x - star x), ⋯⟩) y)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "f g : PowerSeries R"], "goal": "(f + g).derivativeFun = f.derivativeFun + g.derivativeFun"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop"], "goal": "d * (n / d) < n ↔ ¬d ∣ n"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L : Filter 𝕜", "a : 𝕜", "H : DifferentiableAt 𝕜 (fun x => f (-x)) a", "x✝ : 𝕜"], "goal": "f x✝ = ((fun x => f (-x)) ∘ Neg.neg) x✝"}}
+{"state": {"context": ["α : Type u_1", "V : Type u_2", "P : Type u_3", "W : Type u_4", "Q : Type u_5", "inst✝⁵ : SeminormedAddCommGroup V", "inst✝⁴ : PseudoMetricSpace P", "inst✝³ : NormedAddTorsor V P", "inst✝² : NormedAddCommGroup W", "inst✝¹ : MetricSpace Q", "inst✝ : NormedAddTorsor W Q", "v : V", "x : P"], "goal": "dist x (v +ᵥ x) = ‖v‖"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "a✝ a b c : α", "h : 0 < c"], "goal": "(fun x => x * c) ⁻¹' Icc a b = Icc (a / c) (b / c)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "n : ℤ", "h₁ : U R (n + 1) = X * U R n + T R (n + 1)", "h₂ : U R (n + 1 + 1) = X * U R (n + 1) + T R (n + 1 + 1)"], "goal": "T R (n + 2) = X * T R (n + 1) - (1 - X ^ 2) * U R n"}}
+{"state": {"context": ["ι : Type u_1", "inst✝⁴ : DecidableEq ι", "A : ι → Type u_2", "inst✝³ : (i : ι) → AddCommMonoid (A i)", "inst✝² : AddMonoid ι", "inst✝¹ : GSemiring A", "inst✝ : (i : ι) → (x : A i) → Decidable (x ≠ 0)", "a a' : ⨁ (i : ι), A i"], "goal": "(DFinsupp.sum a fun a b => ((toAddMonoid fun x => ((compHom (of A (a + x))).comp (gMulHom A)).flip).flip b) a') = DFinsupp.sum a fun i ai => DFinsupp.sum a' fun j aj => (of A (i + j)) (GradedMonoid.GMul.mul ai aj)"}}
+{"state": {"context": ["n k m : ℕ"], "goal": "n + k - (m + k) + (m + k - (n + k)) = n - m + (m + k - (n + k))"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M"], "goal": "⨆ i, LinearMap.range (ι Q) ^ i = ⊤"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P", "h : ∡ p₁ p₂ p₃ = ↑π"], "goal": "p₁ ≠ p₃"}}
+{"state": {"context": ["α : Type u_1", "Zero α", "a : αᵐᵒᵖ"], "goal": "MulOpposite.unop a = 0 ↔ a = 0"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "f : α → β", "hf : Function.Bijective f"], "goal": "Function.Injective f"}}
+{"state": {"context": ["M : Type u_3", "Monoid M", "Preorder M", "CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : M", "ha : 1 < a", "k : ℕ", "hk : k ≠ 0"], "goal": "1 < a ^ k"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type u_3", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "f : E → F", "s : Set E", "x v : E", "f' : F", "h : HasDerivWithinAt (fun t => f (x + t • v)) f' ((fun t => x + t • v) ⁻¹' s) 0", "c : 𝕜", "g : 𝕜 → 𝕜 := fun t => c • t", "s' : Set 𝕜 := (fun t => x + t • v) ⁻¹' s", "A : HasDerivAt g c 0", "B : HasDerivWithinAt (fun t => f (x + t • v)) f' s' (g 0)", "Z : HasDerivWithinAt ((fun t => f (x + t • v)) ∘ g) (c • f') (g ⁻¹' s') 0"], "goal": "HasDerivWithinAt (fun t => f (x + t • c • v)) (c • f') ((fun t => x + t • c • v) ⁻¹' s) 0"}}
+{"state": {"context": ["n : Type u_1", "α : Type u_2", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "inst✝ : CommRing α", "A B : Matrix n n α"], "goal": "∀ x ∈ Finset.univ, (sign x • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (x i) i).coeff 0 = sign x • ∏ i : n, B (x i) i"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "U : TopCat", "X : PresheafedSpace C", "f : U ⟶ ↑X", "h : OpenEmbedding ⇑f", "x : ↑U", "V : (OpenNhds (f x))ᵒᵖ"], "goal": "colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj X.presheaf) V ≫ (X.restrictStalkIso h x).inv = colimit.ι (((whiskeringLeft (OpenNhds (f x))ᵒᵖ (Opens ↑↑X)ᵒᵖ C).obj (OpenNhds.inclusion (f x)).op).obj X.presheaf) V ≫ Hom.stalkMap (X.ofRestrict h) x"}}
+{"state": {"context": ["M : Type u_4", "Monoid M", "f : ℕ → M", "n : ℕ"], "goal": "(List.map f (List.range n.succ)).prod = f 0 * (List.map (fun i => f i.succ) (List.range n)).prod"}}
+{"state": {"context": ["L R : Type v", "fst snd : L → R", "X Y Z : CategoryTheory.Limits.WalkingMultispan fst snd", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "CategoryTheory.Limits.WalkingMultispan.Hom.comp f g = CategoryTheory.CategoryStruct.comp f g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "l : List α", "a m : α", "h : 0 < l.length"], "goal": "maximum_of_length_pos h ∈ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : α → β", "hf : Memℒp f p μ", "hmeas : StronglyMeasurable f", "ε : ℝ", "hε : 0 < ε", "M : ℝ", "hM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε", "x : α"], "goal": "{x | max M 1 ≤ ↑‖f x‖₊}.indicator (fun a => ‖f a‖) x ≤ {x | M ≤ ↑‖f x‖₊}.indicator (fun a => ‖f a‖) x"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "CommMonoidWithZero M", "S : Finset α", "p : M", "pp : Prime p", "g : α → M", "hS : ∀ a ∈ S, ¬p ∣ g a"], "goal": "¬p ∣ S.prod g"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "f g : CategoryTheory.ComposableArrows C 4", "app₀ : f.obj' 0 ⋯ ≅ g.obj' 0 ⋯", "app₁ : f.obj' 1 ⋯ ≅ g.obj' 1 ⋯", "app₂ : f.obj' 2 ⋯ ≅ g.obj' 2 ⋯", "app₃ : f.obj' 3 ⋯ ≅ g.obj' 3 ⋯", "app₄ : f.obj' 4 ⋯ ≅ g.obj' 4 ⋯", "w₀ :\n  CategoryTheory.CategoryStruct.comp (f.map' 0 1 ⋯ ⋯) app₁.hom =\n    CategoryTheory.CategoryStruct.comp app₀.hom (g.map' 0 1 ⋯ ⋯)", "w₁ :\n  CategoryTheory.CategoryStruct.comp (f.map' 1 2 ⋯ ⋯) app₂.hom =\n    CategoryTheory.CategoryStruct.comp app₁.hom (g.map' 1 2 ⋯ ⋯)", "w₂ :\n  CategoryTheory.CategoryStruct.comp (f.map' 2 3 ⋯ ⋯) app₃.hom =\n    CategoryTheory.CategoryStruct.comp app₂.hom (g.map' 2 3 ⋯ ⋯)", "w₃ :\n  CategoryTheory.CategoryStruct.comp (f.map' 3 4 ⋯ ⋯) app₄.hom =\n    CategoryTheory.CategoryStruct.comp app₃.hom (g.map' 3 4 ⋯ ⋯)"], "goal": "(CategoryTheory.ComposableArrows.isoMk₄ app₀ app₁ app₂ app₃ app₄ w₀ w₁ w₂ w₃).hom =\n  CategoryTheory.ComposableArrows.homMk₄ app₀.hom app₁.hom app₂.hom app₃.hom app₄.hom w₀ w₁ w₂ w₃"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "MulZeroClass α"], "goal": "⊤ * ⊤ = ⊤"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : Group α", "s t : Finset α", "a b : α"], "goal": "preimage 1 (fun x => a⁻¹ * x) ⋯ = {a}"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "s t : Finset α"], "goal": "Codisjoint s t ↔ ∀ ⦃a : α⦄, a ∉ s → a ∈ t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝¹ : CommMonoid α", "inst✝ : TopologicalSpace α", "f g : β → α", "a a₁ a₂ : α", "s : Finset β", "hf : mulSupport f ⊆ ↑s"], "goal": "∏' (b : β), f b = ∏ b ∈ s, f b"}}
+{"state": {"context": ["R : Type u", "A : Type v", "B : Type w", "CommSemiring R", "Semiring A", "Semiring B", "Algebra R A", "Algebra R B", "CoalgebraStruct R A", "CoalgebraStruct R B", "e : A ≃ₐc[R] B"], "goal": "e.invFun = ⇑e.symm"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "p : ι → P", "i₀ : ι", "v : V"], "goal": "v ∈ (fun x => x -ᵥ p i₀) '' (p '' (univ \\ {i₀})) ↔ v ∈ range fun i => p ↑i -ᵥ p i₀"}}
+{"state": {"context": ["M : Type u_1", "AddMonoid M"], "goal": "StrictMono AddSubgroup.ofAddUnits"}}
+{"state": {"context": ["n : ℕ"], "goal": "Odd n ↔ n % 2 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "g : β → ℝ≥0∞", "f : α →ₛ β"], "goal": "∑ x ∈ Finset.image g f.range, x * μ (↑(map g f) ⁻¹' {x}) = ∑ x ∈ f.range, g x * μ (↑f ⁻¹' {x})"}}
+{"state": {"context": ["n a : ℕ", "l : List ℕ", "hl : (a :: l).IsZeckendorfRep", "hn : ∀ a_1 ∈ (a :: l ++ [0]).head?, a_1 < n"], "goal": "(map fib (a :: l)).sum < fib n"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "Module R (Additive ℤˣ)", "s : ℤˣ"], "goal": "s ^ 1 = s"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "c₁ : ℝ", "hc₁_mem : c₁ ∈ Set.Ioo 0 1", "hc₁ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)", "c₂ : ℝ", "hc₂_mem : c₂ > 0", "hc₂ : ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u", "c₃ : ℝ", "hc₃_mem : c₃ ∈ Set.Ioo 0 1", "hc₃ : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n", "hc₁_pos : 0 < c₁", "hc₃' : 0 < 1 - c₃", "n : ℕ", "hn₁ : ∀ (i : α), c₁ * ↑n ≤ ↑(r i n)", "hn₂ : ∀ u ∈ Set.Icc (c₁ * ↑n) ↑n, c₂ * g ↑n ≤ g u", "hn₃ : ∀ (i : α), ↑(r i n) ≤ c₃ * ↑n", "hrpos : ∀ (i : α), 0 < r i n", "hr_lt_n : ∀ (i : α), r i n < n", "hn_pos : 0 < n", "i : α", "hrpos_i : 0 < r i n"], "goal": "min (c₂ * (1 - c₃)) ((1 - c₃) * c₂ / c₁ ^ (p a b + 1)) * g ↑n ≤ sumTransform (p a b) g (r i n) n"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : PointClass F", "a✝¹ : Fin 3 → F", "hP : W.NonsingularLift (Quot.mk Setoid.r a✝¹)", "a✝ : Fin 3 → F", "hQ : W.NonsingularLift (Quot.mk Setoid.r a✝)"], "goal": "W.NonsingularLift (W.addMap (Quot.mk Setoid.r a✝¹) (Quot.mk Setoid.r a✝))"}}
+{"state": {"context": ["T : Type u₁", "inst✝ : Category.{v₁, u₁} T", "X✝ X Y Z : T", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "autoParam (∀ (X_1 Y_1 : Under Z) (f_1 : X_1 ⟶ Y_1), (map (f ≫ g)).map f_1 = eqToHom ⋯ ≫ (map g ⋙ map f).map f_1 ≫ eqToHom ⋯) _auto✝"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "inst✝² : Semiring α", "inst✝¹ : LinearOrder α", "a b c d : α", "inst✝ : PosMulStrictMono α", "h : 0 < c"], "goal": "0 = c * 0"}}
+{"state": {"context": ["s : Set Ordinal.{u}", "a : Ordinal.{u}", "f : Ordinal.{u} → Ordinal.{u}"], "goal": "IsNormal f ↔ StrictMono f ∧ Continuous f"}}
+{"state": {"context": ["a b c : Cardinal.{u_1}"], "goal": "a ^< min b c = min (a ^< b) (a ^< c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "a b : Ordinal.{u_4}", "h : a < b"], "goal": "a / b = 0"}}
+{"state": {"context": ["M : Type u_2", "Monoid M", "n : ℕ"], "goal": "1 ^ n = 1"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "ι : Type u_6", "Fintype n", "NonUnitalNonAssocSemiring α", "M : Matrix m n α", "v : n → α"], "goal": "Matrix.row ι (M.mulVec v) = (M * Matrix.col ι v).transpose"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "AddGroup α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "a b c : α"], "goal": "a ⊔ b - c = (a - c) ⊔ (b - c)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : Infinite R", "σ : Type u_2", "p q : MvPolynomial σ R", "h : ∀ (x : σ → R), (eval x) p = (eval x) q"], "goal": "p = q"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "U : X.Opens"], "goal": "X.zeroLocus ∅ = Set.univ"}}
+{"state": {"context": ["b c : Ordinal.{u_1}"], "goal": "0 ^ (b + c) = 0 ^ b * 0 ^ c"}}
+{"state": {"context": ["γ : Type w", "SemilatticeInf γ", "a : γ", "b : γ", "s : Set γ", "hs : IsGLB s b"], "goal": "IsGLB (insert a s) (a ⊓ b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "q : Semiquot α", "f : α → Semiquot β", "b : β"], "goal": "b ∈ q.bind f ↔ ∃ a ∈ q, b ∈ f a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "F : Type u_1", "CommRing R", "CommRing S", "FunLike F R S", "rc : RingHomClass F R S", "f : F", "I J : Ideal R"], "goal": "Ideal.map f (I * J) = Ideal.map f I * Ideal.map f J"}}
+{"state": {"context": ["R : Type u", "S : Type v", "A : Type w", "B : Type u₁", "C : Type u_1", "inst✝⁶ : CommSemiring A", "inst✝⁵ : CommSemiring B", "inst✝⁴ : Semiring C", "inst✝³ : Algebra A B", "inst✝² : Algebra B C", "inst✝¹ : Algebra A C", "inst✝ : IsScalarTower A B C", "x : Finset C", "hx : Algebra.adjoin A ↑x = ⊤", "y : Finset C", "hy : span B ↑y = ⊤", "f : C → C → B", "hf : ∀ (x : C), ∑ i ∈ y, f x i • i = x", "s : Finset B := image₂ f (x ∪ y * y) y", "hxy : ∀ xi ∈ x, xi ∈ span ↥(Algebra.adjoin A ↑s) ↑(insert 1 y)"], "goal": "∃ B₀, B₀.FG ∧ ⊤.FG"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "L : CategoryTheory.Functor C D", "R : CategoryTheory.Functor D C", "h : L ⊣ R", "X : C"], "goal": "(CategoryTheory.Comonad.comparisonForget h).hom.app X =\n  CategoryTheory.CategoryStruct.id (((CategoryTheory.Comonad.comparison h).comp h.toComonad.forget).obj X)"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "S : NonUnitalSubalgebra R A", "x : ↥S"], "goal": "↑x = 0 ↔ x = 0"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "RCLike 𝕜", "E : Type u_3", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "b : HilbertBasis ι 𝕜 E", "x y : E"], "goal": "HasSum (fun i => ⟪x, (fun i => b.repr.symm (lp.single 2 i 1)) i⟫_𝕜 * ⟪(fun i => b.repr.symm (lp.single 2 i 1)) i, y⟫_𝕜)\n  ⟪x, y⟫_𝕜"}}
+{"state": {"context": ["α β : Type u_2", "m : α → β", "f : Filter α"], "goal": "m <$> f = Filter.map m f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "f : Filter α", "α₁ : Type u_5", "α₂ : Type u_6", "β₁ : Type u_7", "β₂ : Type u_8", "f₁ : Filter α₁", "f₂ : Filter α₂", "g₁ : Set α₁ → Filter β₁", "g₂ : Set α₂ → Filter β₂", "hg₁ : Monotone g₁", "hg₂ : Monotone g₂"], "goal": "f₁.lift g₁ ×ˢ f₂.lift g₂ = f₁.lift fun s => f₂.lift fun t => g₁ s ×ˢ g₂ t"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "φ ψ : ℝ → ℝ", "s : Set E", "a b m : ℝ", "x y : E", "f g : E → ℝ", "hf : UniformConcaveOn s φ f", "hg : UniformConvexOn s ψ g"], "goal": "UniformConcaveOn s (φ + ψ) (f - g)"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝¹ : (i : α) → NormedAddCommGroup (E i)", "ι : Type u_3", "l : Filter ι", "inst✝ : l.NeBot", "_i : Fact (1 ≤ p)", "F : ℕ → ↥(lp E p)", "hF : CauchySeq F", "f : ↥(lp E p)", "hf : Tendsto (id fun i => ↑(F i)) atTop (𝓝 ↑f)", "ε : ℝ", "hε : 0 < ε", "hε' : {p_1 | ‖p_1.1 - p_1.2‖ < ε} ∈ uniformity ↥(lp E p)", "n : ℕ", "hn : ∀ᶠ (l : ℕ) in atTop, ‖(fun f => F n - f) (F l)‖ < ε", "a : α"], "goal": "Tendsto (fun i => id (fun i => ↑((fun f => F n - f) (F i))) i a) atTop (𝓝 (↑(F n - f) a))"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "A : Type u_3", "S✝ : Type u_4", "inst✝¹ : Group G", "inst✝ : AddGroup A", "s : Set G", "ι : Sort u_5", "S : ι → Subgroup G", "C : G → Prop", "x : G", "hx : x ∈ closure (⋃ i, ↑(S i))", "mem : ∀ (i : ι), ∀ x ∈ S i, C x", "one : C 1", "mul : ∀ (x y : G), C x → C y → C (x * y)"], "goal": "C x"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : OrderedAddCommMonoid M", "f : ℕ → M", "u : ℕ → ℕ", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u", "n : ℕ", "this : ∀ k ∈ range n, (u (k + 2) - u (k + 1)) • f (u (k + 1)) ≤ C • (u (k + 1) - u k) • f (u (k + 1))"], "goal": "∑ k ∈ range n, (u (k + 2) - u (k + 1)) • f (u (k + 1)) ≤ C • ∑ k ∈ range n, (u (k + 1) - u k) • f (u (k + 1))"}}
+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝ : Monoid M", "a : M", "s : Set M", "p : (m : M) → m ∈ closure s → Prop", "one : p 1 ⋯", "mul_left : ∀ (x : M) (hx : x ∈ s) (y : M) (hy : y ∈ closure s), p y hy → p (x * y) ⋯", "x : M", "h✝ : x ∈ closure s", "h : x ∈ mrange (FreeMonoid.lift Subtype.val)"], "goal": "p x h✝"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "AddCommMonoid A", "Module R A", "Coalgebra R A", "f g : A →ₗc[R] R"], "goal": "f = g"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X✝ Y✝ Z✝ X Y Z : C", "f : Y ≅ X", "g g' : Y ⟶ Z"], "goal": "f.inv ≫ g = f.inv ≫ g' ↔ g = g'"}}
+{"state": {"context": ["b : Ordinal.{0}", "e : ONote", "n : ℕ+", "a o : ONote", "h₁ : (e.oadd n a).NFBelow b", "h₂ : o.NFBelow b", "h' : (a + o).NFBelow b"], "goal": "(e.oadd n a + o).NFBelow b"}}
+{"state": {"context": ["α : Type u_1", "m : MeasureTheory.OuterMeasure α", "S : Set (Set α)", "hS : S.Countable"], "goal": "m (⋃₀ S) = 0 ↔ ∀ s ∈ S, m s = 0"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "μ : M → N → N", "r : N → N → Prop", "h : Covariant M N μ r"], "goal": "Covariant M N μ (flip r)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "SeminormedAddCommGroup β", "b : β"], "goal": "‖BoundedContinuousFunction.const α b‖ ≤ ‖b‖"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "CategoryTheory.BraidedCategory D", "F G : CategoryTheory.BraidedFunctor C D", "i : F.toMonoidalFunctor ≅ G.toMonoidalFunctor"], "goal": "(CategoryTheory.BraidedFunctor.mkIso i).hom = i.hom"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_7", "M : Type u_8", "M₂ : Type u_10", "Semiring R", "Semiring S", "AddCommMonoid M", "AddCommMonoid M₂", "module_M : Module R M", "module_S_M₂ : Module S M₂", "σ : R →+* S", "σ' : S →+* R", "re₁ : RingHomInvPair σ σ'", "re₂ : RingHomInvPair σ' σ", "e : M ≃ₛₗ[σ] M₂", "c : M₂"], "goal": "e (e.symm c) = c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a✝ a₁ a₂ : α", "b b₁✝ b₂✝ : β", "γ : Type u_3", "inst✝⁸ : Zero α", "inst✝⁷ : Preorder α", "inst✝⁶ : Zero β", "inst✝⁵ : Preorder β", "inst✝⁴ : Zero γ", "inst✝³ : Preorder γ", "inst✝² : SMul α β", "inst✝¹ : SMul α γ", "f : β → γ", "inst✝ : PosSMulMono α γ", "a : α", "ha : 0 ≤ a", "b₁ b₂ : β", "hf : ∀ {b₁ b₂ : β}, True", "smul : α → β → True", "hb : f b₁ ≤ f b₂"], "goal": "a • f b₁ ≤ a • f b₂"}}
+{"state": {"context": ["M : Type u_11", "N : Type u_12", "α : Type u_13", "Monoid M", "Monoid N", "MulAction N α", "f : M →* N", "h : MulAction.IsPretransitive M α"], "goal": "MulAction.IsPretransitive N α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β"], "goal": "MeasurableSpace.generateFrom (Set.image2 (fun x x_1 => x ×ˢ x_1) {s | MeasurableSet s} {t | MeasurableSet t}) =\n  Prod.instMeasurableSpace"}}
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+{"state": {"context": ["σ : Type u_1", "R : Type u_3", "CommSemiring R", "i : σ", "n : ℕ"], "goal": "(MvPolynomial.X i ^ n).IsHomogeneous n"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "D H : Subgroup G", "inst✝ : D.FiniteIndex", "hD_le_H : D ≤ H", "t : Set ↥H", "ht : t ∈ leftTransversals ↑(D.subgroupOf H) ∧ 1 ∈ t", "hf : t.Finite"], "goal": "∃ t, ↑t ∈ leftTransversals ↑(D.subgroupOf H) ∧ ⋃ g ∈ t, ↑g • ↑D = ↑H"}}
+{"state": {"context": ["α : Type u_1", "GeneralizedBooleanAlgebra α", "DecidableRel Disjoint", "DecidableRel fun x x_1 => x ≤ x_1", "s : Finset α", "u v a : α", "DecidableEq α", "h : a ∈ UV.compression u v s", "ha : a ∉ s"], "goal": "Disjoint v a"}}
+{"state": {"context": [], "goal": "↑0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : α → β", "hf : Memℒp f ⊤ μ", "hmeas : StronglyMeasurable f", "hbdd : eLpNormEssSup f μ < ⊤"], "goal": "eLpNormEssSup f (μ.restrict {x | (eLpNorm f ⊤ μ + 1).toReal ≤ ↑‖f x‖₊}) = 0"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "hf : f.NeBot"], "goal": "f ≠ ⊥"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "v w : F", "h : ‖v‖ = ‖w‖", "R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ", "h₁ : R (v - w) = -(v - w)"], "goal": "R v + R v = w + w"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "P : Type u_3", "CommSemiring P", "IsLocalization M S", "g : R →+* P", "hg : ∀ (y : ↥M), IsUnit (g ↑y)", "x : R"], "goal": "(IsLocalization.lift hg) ((algebraMap R S) x) = g x"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Add α", "s t : Finset α", "hs : s.Nonempty", "ht : t.Nonempty"], "goal": "0 < s.addEnergy t"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "TopologicalAddGroup G", "K₀ : TopologicalSpace.PositiveCompacts G", "U : Set G", "hU : (interior U).Nonempty", "K₁ K₂ : TopologicalSpace.Compacts G", "h : ↑K₁ ⊆ K₂.carrier"], "goal": "MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₁ ≤ MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : Finite α", "this : Fintype α"], "goal": "1 < Nat.card α ↔ Nontrivial α"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "H : Type u_3", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "CategoryTheory.Category.{u_6, u_3} H", "L : CategoryTheory.Functor C D", "F : CategoryTheory.Functor C H", "E : L.RightExtension F", "h : E.IsPointwiseRightKanExtension", "G : L.RightExtension F", "f₁ f₂ : G ⟶ E"], "goal": "f₁ = f₂"}}
+{"state": {"context": ["n : ℕ"], "goal": "powHalf (n + 1) + powHalf (n + 1) ≈ powHalf n"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : MeasurableSpace α", "f : α → α", "s : Set α", "μ : Measure α", "hf : Conservative f μ", "hs : NullMeasurableSet s μ", "h0 : μ s ≠ 0", "t : ℕ → Set α := fun n => s ∩ f^[n] ⁻¹' s", "H : ¬∃ᶠ (m : ℕ) in atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0", "N : ℕ", "hN : μ (t N) ≠ 0", "hmax : ∀ n > N, μ (t n) = 0", "htm : ∀ {n : ℕ}, NullMeasurableSet (t n) μ", "T : Set α := t N \\ ⋃ n, ⋃ (_ : n > N), t n", "hT : T = t N \\ ⋃ n, ⋃ (_ : n > N), t n"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : Ring R", "E : Type u_2", "inst✝⁸ : AddCommGroup E", "inst✝⁷ : Module R E", "F : Type u_3", "inst✝⁶ : AddCommGroup F", "inst✝⁵ : Module R F", "G : Type u_4", "inst✝⁴ : AddCommGroup G", "inst✝³ : Module R G", "M : Type u_5", "inst✝² : Monoid M", "inst✝¹ : DistribMulAction M F", "inst✝ : SMulCommClass R M F", "y✝ : M", "f g : E →ₗ.[R] F", "h : f.graph ≤ g.graph", "x : E", "hx : x ∈ f.domain", "y : E", "hy : y ∈ g.domain", "hxy : ↑⟨x, hx⟩ = ↑⟨y, hy⟩"], "goal": "↑f ⟨x, hx⟩ = ↑g ⟨y, hy⟩"}}
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+{"state": {"context": ["R : Type u_2", "Ring R"], "goal": "IsSimpleModule R R ↔ Nontrivial R ∧ ∀ (x : R), x ≠ 0 → IsUnit x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "f g : Filter β", "u : β → α", "h : f ≤ g"], "goal": "Filter.IsBoundedUnder r g u → Filter.IsBoundedUnder r f u"}}
+{"state": {"context": ["N : ℕ", "inst✝ : NeZero N", "χ : DirichletCharacter ℂ N", "k j : ZMod N"], "goal": "toCircle (-(j * k)) • χ j = χ j * (stdAddChar.mulShift (-k)) j"}}
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+{"state": {"context": ["ζ : ℂ", "n : ℕ", "h : ζ ^ n = 1", "hn : n ≠ 0"], "goal": "‖ζ‖₊ = 1"}}
+{"state": {"context": ["ι : Type u_1", "inst✝⁴ : DecidableEq ι", "A : ι → Type u_2", "R : Type u_3", "inst✝³ : (i : ι) → AddCommMonoid (A i)", "inst✝² : AddMonoid ι", "inst✝¹ : GSemiring A", "inst✝ : Semiring R", "f : { f // f GradedMonoid.GOne.one = 1 ∧ ∀ {i j : ι} (ai : A i) (aj : A j), f (GradedMonoid.GMul.mul ai aj) = f ai * f aj }", "xi : ι", "xv : A xi"], "goal": "↑((fun F => ⟨fun {i} => (↑F).comp (of A i), ⋯⟩) ((fun f => toSemiring (fun x => ↑f) ⋯ ⋯) f)) xv = ↑f xv"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "V₁ : Type u_3", "V₁' : Type u_4", "V₂ : Type u_5", "V₃ : Type u_6", "V₄ : Type u_7", "P₁ : Type u_8", "P₁' : Type u_9", "P : Type u_10", "P₂ : Type u_11", "P₃ : Type u_12", "P₄ : Type u_13", "inst✝²⁵ : NormedField 𝕜", "inst✝²⁴ : SeminormedAddCommGroup V", "inst✝²³ : NormedSpace 𝕜 V", "inst✝²² : PseudoMetricSpace P", "inst✝²¹ : NormedAddTorsor V P", "inst✝²⁰ : SeminormedAddCommGroup V₁", "inst✝¹⁹ : NormedSpace 𝕜 V₁", "inst✝¹⁸ : PseudoMetricSpace P₁", "inst✝¹⁷ : NormedAddTorsor V₁ P₁", "inst✝¹⁶ : SeminormedAddCommGroup V₁'", "inst✝¹⁵ : NormedSpace 𝕜 V₁'", "inst✝¹⁴ : MetricSpace P₁'", "inst✝¹³ : NormedAddTorsor V₁' P₁'", "inst✝¹² : SeminormedAddCommGroup V₂", "inst✝¹¹ : NormedSpace 𝕜 V₂", "inst✝¹⁰ : PseudoMetricSpace P₂", "inst✝⁹ : NormedAddTorsor V₂ P₂", "inst✝⁸ : SeminormedAddCommGroup V₃", "inst✝⁷ : NormedSpace 𝕜 V₃", "inst✝⁶ : PseudoMetricSpace P₃", "inst✝⁵ : NormedAddTorsor V₃ P₃", "inst✝⁴ : SeminormedAddCommGroup V₄", "inst✝³ : NormedSpace 𝕜 V₄", "inst✝² : PseudoMetricSpace P₄", "inst✝¹ : NormedAddTorsor V₄ P₄", "e : P ≃ᵃⁱ[𝕜] P₂", "α : Type u_14", "inst✝ : TopologicalSpace α", "f : P → P₂", "hf : Isometry f", "p : P", "g : V → V₂", "hg : ∀ (v : V), g v = f (v +ᵥ p) -ᵥ f p"], "goal": "g = ⇑(vaddConst 𝕜 (f p)).symm ∘ f ∘ ⇑(vaddConst 𝕜 p)"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝⁶ : TopologicalSpace Ω", "inst✝⁵ : MeasurableSpace Ω", "inst✝⁴ : OpensMeasurableSpace Ω", "ι : Type u_2", "L : Filter ι", "inst✝³ : L.IsCountablyGenerated", "inst✝² : TopologicalSpace Ω", "inst✝¹ : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "c : ℝ≥0", "E : Set Ω", "E_mble : MeasurableSet E", "fs : ι → Ω →ᵇ ℝ≥0", "fs_bdd : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c", "fs_lim : ∀ᵐ (ω : Ω) ∂μ, Tendsto (fun i => (fs i) ω) L (𝓝 (E.indicator (fun x => 1) ω))"], "goal": "Tendsto (fun n => ∫⁻ (ω : Ω), ↑((fs n) ω) ∂μ) L (𝓝 (μ E))"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "NormedAddCommGroup E", "𝕜 : Type u_3", "RCLike 𝕜", "NormedSpace 𝕜 E", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "l.NeBot", "hf' : TendstoUniformly f' g' l", "hf : ∀ (n : ι) (x : E), HasFDerivAt (f n) (f' n x) x", "hfg : ∀ (x : E), Filter.Tendsto (fun n => f n x) l (𝓝 (g x))", "x : E"], "goal": "HasFDerivAt g (g' x) x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝³ : MeasurableSpace X", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "f g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "inst✝ : CompleteSpace E", "p : ℝ≥0∞", "hs : MeasurableSet s", "ht : MeasurableSet t", "hμt : μ t ≠ ⊤", "e : E"], "goal": "∫ (x : X) in s, t.indicator (fun x => e) x ∂μ = (μ (t ∩ s)).toReal • e"}}
+{"state": {"context": ["a b : Prop"], "goal": "¬a → ¬b ↔ b → a"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "β : Type u_2", "Ring β", "abv : β → α", "IsAbsoluteValue abv", "f g : CauSeq β abv"], "goal": "CauSeq.Completion.mk f + CauSeq.Completion.mk g = CauSeq.Completion.mk (f + g)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "c : α", "AddZeroClass α", "Neg β", "h : Function.Antiperiodic f c"], "goal": "f c = -f 0"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s✝ : β✝ → β✝ → Prop", "t : γ → γ → Prop", "β : Type u_4", "s : β → β → Prop", "inst✝ : IsTrans β s"], "goal": "WellFounded s ↔ ∀ (b : β), WellFounded (Subrel s {b' | s b' b})"}}
+{"state": {"context": [], "goal": "⋂ r, Iic ↑r = ∅"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "ε : Type u_6", "inst✝ : Monoid α", "f g : Filter α", "s✝ : Set α", "a : α", "m n : ℕ", "hf : 1 ≤ f", "s : Set α"], "goal": "(∃ t₁ ∈ f, t₁ * univ ⊆ s) → s = univ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "A : Type u₂", "CategoryTheory.Category.{v₂, u₂} A", "B : Type u₃", "CategoryTheory.Category.{v₃, u₃} B", "J : CategoryTheory.GrothendieckTopology C", "F G : CategoryTheory.Functor A B", "H : CategoryTheory.Functor A B", "η : F ⟶ G", "γ : G ⟶ H", "J.HasSheafCompose F", "J.HasSheafCompose G", "J.HasSheafCompose H"], "goal": "CategoryTheory.sheafCompose_map J (CategoryTheory.CategoryStruct.comp η γ) =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.sheafCompose_map J η) (CategoryTheory.sheafCompose_map J γ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "I : Type u_3", "f : I → Type u_4", "i' : I", "inst : ∀ (i : I), Nonempty (f i)", "inst✝ : Nontrivial (f i')", "this : DecidableEq ((i : I) → f i) := Classical.decEq ((i : I) → f i)"], "goal": "Nontrivial ((i : I) → f i)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "s : Set X", "t : Set Y", "h : Set.MapsTo f s t", "hc : Continuous f", "ht : IsClosed t"], "goal": "Set.MapsTo f (closure s) t"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "CommRing R", "AddCommGroup M", "Module R M", "I : Ideal R", "N N' : Submodule R M", "hN' : N'.FG", "hIJ : I ≤ ⊥.jacobson", "hNN : N' ≤ N ⊔ I • N'"], "goal": "N' ≤ N"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ : NumberField K", "inst✝ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "u : (𝓞 K)ˣ", "x✝ x : 𝓞 K", "h : λ ∣ -x - 1"], "goal": "λ ^ 4 ∣ x ^ 3 + 1"}}
+{"state": {"context": ["Γ : Type u_1", "Inhabited Γ", "p : Turing.ListBlank Γ → Prop", "q : Turing.ListBlank Γ", "h : ∀ (a : List Γ), p (Turing.ListBlank.mk a)"], "goal": "p q"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V", "hx : x ≠ 0", "hy : y ≠ 0", "hcos : Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1", "n : ℤ", "hn : angle x y + angle x (x - y) + angle y (y - x) = ↑n * π", "h0 : 0 ≤ angle x y + angle x (x - y) + angle y (y - x)"], "goal": "angle x y + angle x (x - y) + angle y (y - x) = π"}}
+{"state": {"context": ["α : Type u_2", "Fintype α"], "goal": "StrictMono Finset.dens"}}
+{"state": {"context": ["E : Type u", "NormedAddCommGroup E", "NormedSpace ℂ E", "F : Type v", "NormedAddCommGroup F", "NormedSpace ℂ F", "f : E → F", "s : Set E", "hd : DifferentiableOn ℂ f s"], "goal": "IsOpen {z | s ∈ 𝓝 z ∧ IsMaxOn (norm ∘ f) s z}"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "SMul 𝕜 E", "p : Seminorm 𝕜 E", "c : ℝ≥0", "hc : 0 < c", "r : ℝ", "x : E"], "goal": "(c • p).ball x r = p.ball x (r / ↑c)"}}
+{"state": {"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "inst✝⁴ : ConcreteCategory C", "inst✝³ : HasForget₂ C Ab", "inst✝² : Abelian C", "inst✝¹ : (forget₂ C Ab).Additive", "inst✝ : (forget₂ C Ab).PreservesHomology", "D : SnakeInput C", "x₃ : (forget C).obj D.L₀.X₃", "x₂ : (forget C).obj D.L₁.X₂", "x₁ : (forget C).obj D.L₂.X₁", "h₂ : D.L₁.g x₂ = D.v₀₁.τ₃ x₃", "h₁ : D.L₂.f x₁ = D.v₁₂.τ₂ x₂", "this✝ : PreservesFiniteLimits (forget₂ C Ab)", "this : PreservesFiniteLimits (forget C)", "eq : (forget C).map D.δ ((forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂)) = (forget C).map D.v₂₃.τ₁ ((forget C).map D.φ₁ (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂))", "eq₁ : (forget C).map (pullback.fst D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₂", "eq₂ : (forget C).map (pullback.snd D.L₁.g D.v₀₁.τ₃) (Concrete.pullbackMk D.L₁.g D.v₀₁.τ₃ x₂ x₃ h₂) = x₃"], "goal": "D.δ x₃ = D.v₂₃.τ₁ x₁"}}
+{"state": {"context": ["G : Type u_2", "Group G", "S : Set (Subgroup Gᵐᵒᵖ)"], "goal": "(sInf S).unop = sInf (Subgroup.op ⁻¹' S)"}}
+{"state": {"context": ["E : Type u_1", "f f' : ℝ → E", "a : ℝ", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "hderiv : ∀ x ∈ Set.Ioi a, HasDerivAt f (f' x) x", "f'int : MeasureTheory.IntegrableOn f' (Set.Ioi a) MeasureTheory.volume"], "goal": "Filter.Tendsto f Filter.atTop (𝓝 (limUnder Filter.atTop f))"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : Nontrivial R", "inst✝² : LinearOrderedCommRing S", "n : Type u_3", "inst✝¹ : Fintype n", "inst✝ : DecidableEq n", "ι : Type u_4", "s : Finset ι", "A : ι → Matrix n n R", "abv : AbsoluteValue R S", "x : S", "hx : ∀ (k : ι) (i j : n), abv (A k i j) ≤ x", "i j : n"], "goal": "abv ((∑ k ∈ s, A k) i j) = abv (∑ k ∈ s, A k i j)"}}
+{"state": {"context": ["J : Type u", "CategoryTheory.Category.{u_1, u} J", "F : J ⥤ Type v", "CategoryTheory.IsCofilteredOrEmpty J", "h : ∀ (j : J), ∃ i f, (Set.range (F.map f)).Finite"], "goal": "F.IsMittagLeffler"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "CategoryTheory.HasShift C ℤ"], "goal": "∀ {X Y : CategoryTheory.Pretriangulated.Triangle C} (f : X ⟶ Y),\n  ((CategoryTheory.Pretriangulated.rotate C).map f).hom₃ = (CategoryTheory.shiftFunctor C 1).map f.hom₁"}}
+{"state": {"context": ["F : Type u_6", "α : Type u_7", "β : Type u_8", "LE α", "LE β", "EquivLike F α β", "self : OrderIsoClass F α β", "f : F", "a b : α"], "goal": "f a ≤ f b ↔ a ≤ b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "m : M"], "goal": "(ExteriorAlgebra.GradedAlgebra.ι R M) m * (ExteriorAlgebra.GradedAlgebra.ι R M) m = 0"}}
+{"state": {"context": ["a b : ℕ"], "goal": "a ^ 2 - b ^ 2 = (a + b) * (a - b)"}}
+{"state": {"context": ["ι : Type u_6", "x : ℕ → ι", "p : ι → Prop", "l : Filter ι", "h_tendsto : Filter.Tendsto x Filter.atTop l", "h : ∃ᶠ (n : ℕ) in Filter.atTop, p (x n)"], "goal": "∃ ns, Filter.Tendsto (fun n => x (ns n)) Filter.atTop l ∧ ∀ (n : ℕ), p (x (ns n))"}}
+{"state": {"context": ["K : Type u", "inst✝³ : Field K", "V : Type v", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)", "j : ↑(Basis.ofVectorSpaceIndex K V)"], "goal": "∑ x : ↑(Basis.ofVectorSpaceIndex K V), ((Basis.ofVectorSpace K V).dualBasis j) ((Basis.ofVectorSpace K V) x) ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).dualBasis j"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "v w : Fin 3 → R"], "goal": "w ⬝ᵥ (crossProduct v) w = 0"}}
+{"state": {"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1"], "goal": "(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1"}}
+{"state": {"context": ["α : Type u_2", "Zero α"], "goal": "Set.Nonempty 0"}}
+{"state": {"context": ["m : ℤ", "n : ℕ", "ihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)"], "goal": "0 ≤ ∏ k ∈ Finset.range (n + 1 + (n + 1)), (m - ↑k)"}}
+{"state": {"context": ["α : Type u_1", "M : α → Type u_2", "u : Ultrafilter α", "L : FirstOrder.Language", "(a : α) → L.Structure (M a)", "∀ (a : α), Nonempty (M a)", "β : Type u_3", "n : ℕ", "φ : L.BoundedFormula β n", "x : β → (a : α) → M a", "v : Fin n → (a : α) → M a"], "goal": "(φ.Realize (fun i => Quotient.mk' (x i)) fun i => Quotient.mk' (v i)) ↔\n  ∀ᶠ (a : α) in ↑u, φ.Realize (fun i => x i a) fun i => v i a"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "inst✝ : HasOrthogonalProjection s.direction", "ps : Set P", "hnps : ps.Nonempty", "p : P", "hps : ps ⊆ ↑s", "hp : p ∉ s", "this : Nonempty ↥s", "cc : P", "cr : ℝ", "hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }", "hcc : cc ∈ s", "hcr : ps ⊆ Metric.sphere cc cr", "x : ℝ := dist cc ↑((orthogonalProjection s) p)", "y : ℝ := dist p ↑((orthogonalProjection s) p)", "hy0 : y ≠ 0", "ycc₂ : ℝ := (x * x + y * y - cr * cr) / (2 * y)", "cc₂ : P := (ycc₂ / y) • (p -ᵥ ↑((orthogonalProjection s) p)) +ᵥ cc", "cr₂ : ℝ := √(cr * cr + ycc₂ * ycc₂)"], "goal": "∃! cs₂, cs₂.center ∈ affineSpan ℝ (insert p ↑s) ∧ insert p ps ⊆ Metric.sphere cs₂.center cs₂.radius"}}
+{"state": {"context": ["A : Type u_1", "M : Type u_2", "M₂ : Type u_3", "Ring A", "AddCommGroup M", "AddCommGroup M₂", "Module A M", "Module A M₂", "TopologicalSpace M", "TopologicalSpace M₂", "R : Type u_4", "Ring R", "Module R M", "Module R M₂", "LinearMap.CompatibleSMul M M₂ R A", "f : M →L[A] M₂"], "goal": "↑(ContinuousLinearMap.restrictScalars R f) = ↑R ↑f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : AddGroup α", "a b : α", "n : ℕ", "x✝ : α"], "goal": "(Equiv.addRight a ^ n) x✝ = (Equiv.addRight (n • a)) x✝"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E", "a : E"], "goal": "Inseparable a 1 ↔ ‖a‖ = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hs : Convex 𝕜 s", "hf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f z - f y) / (z - y) < (f y - f x) / (y - x)"], "goal": "StrictConcaveOn 𝕜 s f"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "RCLike 𝕜", "m : MeasurableSpace α", "f : α → 𝕜", "μ : MeasureTheory.Measure α", "hf : AEMeasurable f μ"], "goal": "AEMeasurable (fun x => RCLike.im (f x)) μ"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsArtinianRing R", "Jac : Ideal R := ⊥.jacobson", "f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := ⋯ }", "n : ℕ", "hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m", "J : Ideal R := annihilator (Jac ^ n)"], "goal": "J = ⊤"}}
+{"state": {"context": ["α : Type u_1", "n : Nat", "a : α", "p : α → Bool"], "goal": "List.takeWhile p (List.replicate n a) = if p a = true then List.replicate n a else []"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "IsLowerModularLattice α", "a b : α"], "goal": "a ⋖ a ⊔ b → a ⊓ b ⋖ b"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "ι : Type u_4", "DecidableEq ι", "Fintype ι"], "goal": "(Pi.basisFun R ι).det = Matrix.detRowAlternating"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "E : Type u₃", "inst✝ : Category.{v₃, u₃} E", "F G H I : C ⥤ D", "α β : 𝟭 C ⟶ 𝟭 C", "X : C"], "goal": "(α ≫ β).app X = (β ≫ α).app X"}}
+{"state": {"context": ["R : Type u", "CommRing R", "n : ℕ", "M : Matrix (Fin n) (Fin n) R"], "goal": "Polynomial.eval 0 (Polynomial.derivative (1 + Polynomial.X • M.map ⇑Polynomial.C).det) = M.trace"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "x : α"], "goal": "|x - ↑(round x)| = min (Int.fract x) (1 - Int.fract x)"}}
+{"state": {"context": ["n : ℕ", "G : Type u_6", "AddGroup G", "h : (Nat.card G).Coprime n", "g : G"], "goal": "(nsmulCoprime h) (-g) = -(nsmulCoprime h) g"}}
+{"state": {"context": ["α : Type u", "s : Stream' α"], "goal": "Stream'.pure id ⊛ s = s"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "B' : Type u_3", "inst✝² : CommRing A", "inst✝¹ : Ring B", "inst✝ : Algebra A B", "x : B", "a : A[X]", "hx : IsIntegral A x", "hamonic : a.Monic", "hdvd : DvdNotUnit a (minpoly A x)", "ha : (Polynomial.aeval x) a = 0"], "goal": "a.natDegree < (minpoly A x).natDegree"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁸ : NontriviallyNormedField 𝕜", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : SMulCommClass ℝ 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : CompleteSpace F", "G : Type u_5", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "H : Type u_6", "β : Type u_7", "γ : Type u_8", "inst✝ : NormedAddCommGroup H", "m m0 : MeasurableSpace β", "μ : Measure β", "hm : m ≤ m0", "f : β →ₛ F", "hf_int : Integrable (↑f) μ", "hf : StronglyMeasurable ↑f", "hf_int_m : Integrable (↑f) (μ.trim hm)"], "goal": "∑ x ∈ f.range, (μ (↑f ⁻¹' {x})).toReal • x = ∑ x ∈ f.range, ((μ.trim hm) (↑f ⁻¹' {x})).toReal • x"}}
+{"state": {"context": ["α : Type u'"], "goal": "(FirstOrder.Language.constantsOn α).card = Cardinal.mk α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "X Y : Pointed", "f : X ⟶ Y", "a : X.X", "ha : a ≠ X.point"], "goal": "((pointedToPartialFun ⋙ partialFunToPointed).map f ≫ ((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) Y).hom).toFun (some ⟨a, ha⟩) = (((fun X => Pointed.Iso.mk (Equiv.optionSubtypeNe X.point) ⋯) X).hom ≫ (𝟭 Pointed).map f).toFun (some ⟨a, ha⟩)"}}
+{"state": {"context": ["n : ℕ", "α : TypeVec.{u_1} (n + 1)", "p : α.Arrow (TypeVec.repeat (n + 1) Prop)"], "goal": "TypeVec.lastFun (TypeVec.ofSubtype p) = id"}}
+{"state": {"context": ["α : Type u_2", "l : Filter α"], "goal": "l ≤ Filter.cofinite ↔ ∀ (x : α), {x}ᶜ ∈ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "hs : s.Subsingleton", "f : α → β"], "goal": "Set.InjOn f s"}}
+{"state": {"context": ["G : Type u_3", "AddSemigroup G", "a b : G", "h : AddCommute a b"], "goal": "Function.Commute (fun x => a + x) fun x => b + x"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "a✝ : α", "n : ℕ", "a : α", "x✝ : ℕ"], "goal": "x✝ ∈ Nat.cast ⁻¹' Iio a ↔ x✝ ∈ Iio ⌈a⌉₊"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteBooleanAlgebra α", "f : Filter β", "u : β → α"], "goal": "(Filter.liminf u f)ᶜ = Filter.limsup (compl ∘ u) f"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f : X ⟶ Y", "Z : C", "g : Y ⟶ Z", "W : C", "h : Z ⟶ W", "CategoryTheory.Limits.HasImage (g ≫ h)", "CategoryTheory.Limits.HasImage (f ≫ g ≫ h)", "CategoryTheory.Limits.HasImage h", "CategoryTheory.Limits.HasImage ((f ≫ g) ≫ h)"], "goal": "CategoryTheory.Limits.image.preComp f (g ≫ h) ≫ CategoryTheory.Limits.image.preComp g h =\n  CategoryTheory.Limits.image.eqToHom ⋯ ≫ CategoryTheory.Limits.image.preComp (f ≫ g) h"}}
+{"state": {"context": ["M : Type u", "inst✝⁵ : Monoid M", "G : Type u", "inst✝⁴ : Group G", "F : Type v", "inst✝³ : Field F", "inst✝² : MulSemiringAction M F", "inst✝¹ : MulSemiringAction G F", "m : M", "inst✝ : Fintype G", "x : F", "f g : Polynomial ↥(subfield G F)", "hf : f.Monic", "hg : g.Monic", "hfg : f * g = minpoly G F x"], "goal": "f = 1 ∨ g = 1"}}
+{"state": {"context": ["𝕜 : Type u_3", "Field 𝕜", "r : ℕ", "IH :\n  ∀ (M : Matrix (Fin r) (Fin r) 𝕜),\n    ∃ L₀ L₀' D₀,\n      (List.map Matrix.TransvectionStruct.toMatrix L₀).prod * M *\n          (List.map Matrix.TransvectionStruct.toMatrix L₀').prod =\n        Matrix.diagonal D₀", "M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜"], "goal": "∃ L L' D,\n  (List.map Matrix.TransvectionStruct.toMatrix L).prod * M * (List.map Matrix.TransvectionStruct.toMatrix L').prod =\n    Matrix.diagonal D"}}
+{"state": {"context": ["n m : Nat"], "goal": "n ^ (m + 1) = n ^ m * n"}}
+{"state": {"context": ["K : Type u", "V : Type v", "DivisionRing K", "AddCommGroup V", "Module K V", "f : Module.End K V", "k : ℕ", "h : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)", "m : ℕ"], "goal": "LinearMap.ker (f ^ k) = LinearMap.ker (f ^ (k + m))"}}
+{"state": {"context": ["a t : ℝ", "ht : 0 < t"], "goal": "HasSum (fun n => if n = 0 then 0 else cexp (2 * ↑π * I * ↑a * ↑n) * ↑(rexp (-π * ↑n ^ 2 * t))) (↑(cosKernel (↑a) t) - 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ε : Type u_5", "ζ : Type u_6", "r : α → α → Prop", "a b c d : α", "h : ReflTransGen r a b"], "goal": "a = b ∨ ∃ c, r a c ∧ ReflTransGen r c b"}}
+{"state": {"context": ["A B : Grp", "f : A ⟶ B", "a : ↑A", "this : f a ∈ {b | h b = g b}"], "goal": "g (f a) = h (f a)"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_5", "Semiring R", "SeminormedAddCommGroup E", "Module R E", "e e' : E ≃ₗᵢ[R] E"], "goal": "⇑(e * e') = ⇑e ∘ ⇑e'"}}
+{"state": {"context": ["A : Type u₁", "B : Type u₂", "C : Type u₃", "D : Type u₄", "inst✝³ : Category.{v₁, u₁} A", "inst✝² : Category.{v₂, u₂} B", "inst✝¹ : Category.{v₃, u₃} C", "inst✝ : Category.{v₄, u₄} D", "G : A ⥤ C", "H : B ⥤ D", "L₁ : A ⥤ B", "R₁ : B ⥤ A", "L₂ : A ⥤ B", "R₂ : B ⥤ A", "L₃ : C ⥤ D", "R₃ : D ⥤ C", "adj₁ : L₁ ⊣ R₁", "adj₂ : L₂ ⊣ R₂", "adj₃ : L₃ ⊣ R₃", "α : L₂ ⟶ L₁", "β : G ⋙ L₃ ⟶ L₂ ⋙ H", "b : B", "vcomp : (mateEquiv adj₁ adj₃) (leftAdjointSquare.vcomp (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) β) = rightAdjointSquare.vcomp ((mateEquiv adj₁ adj₂) (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv)) ((mateEquiv adj₂ adj₃) β)"], "goal": "((mateEquiv adj₁ adj₃) (leftAdjointConjugateSquare.vcomp α β)).app b = (rightAdjointConjugateSquare.vcomp ((conjugateEquiv adj₁ adj₂) α) ((mateEquiv adj₂ adj₃) β)).app b"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "n✝² n✝¹ n : ℕ", "h : n ≠ 0", "n✝ : ℕ", "h_prime : Fact (Nat.Prime (n✝ + 1))", "g_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X", "hp : 1 < n✝ + 1", "aux : g_poly ≠ 0", "key : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n", "nat_degree_eq : g_poly.natDegree = (n✝ + 1) ^ n", "x : GaloisField (n✝ + 1) n", "hx : x ∈ Subring.closure (Set.range ⇑(algebraMap (ZMod (n✝ + 1)) (GaloisField (n✝ + 1) n)) ∪ (X ^ (n✝ + 1) ^ n - X).rootSet (X ^ (n✝ + 1) ^ n - X).SplittingField)"], "goal": "(aeval 1) g_poly = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "x : α", "s t : Set α", "r : ℝ", "f : ℕ → α", "hf : CauchySeq f"], "goal": "Cauchy (map f Filter.cofinite)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_4", "Zero M", "s : Set α", "f : α → M"], "goal": "s.indicator f = f ↔ Function.support f ⊆ s"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "_mΩ : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "β : ι → Type u_10", "m : (i : ι) → MeasurableSpace (β i)", "f : (i : ι) → Ω → β i", "S T : Finset ι", "hST : Disjoint S T", "hf_Indep : ProbabilityTheory.iIndepFun m f μ", "hf_meas : ∀ (i : ι), Measurable (f i)"], "goal": "ProbabilityTheory.IndepFun (fun a i => f (↑i) a) (fun a i => f (↑i) a) μ"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "Fact (FiniteDimensional.finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "φ : E ≃ₗᵢ[ℝ] E", "hφ : 0 < LinearMap.det ↑φ.toLinearEquiv", "x : E"], "goal": "φ (o.rightAngleRotation x) = o.rightAngleRotation (φ x)"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Group G", "inst✝¹ : MulAction G α", "inst✝ : MulAction G β", "H : Subgroup G", "x : Quotient G α", "a b : ↑x.orbit", "c : α", "h : ⟦a⟧ = ⟦b⟧"], "goal": "↑b ∈ MulAction.orbit ↥H ↑a"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : MeasurableSpace E", "inst✝⁴ : BorelSpace E", "inst✝³ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝² : μ.IsAddHaarMeasure", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "s : Set E", "g : ℝ → F", "a : ℝ"], "goal": "∫ (x : ℝ), g (a⁻¹ * x) = |a| • ∫ (y : ℝ), g y"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "NontriviallyNormedField R", "NormedAddCommGroup F", "NormedSpace R F", "TopologicalSpace B", "ι : Type u_5", "Z : VectorBundleCore R B F ι", "i : ι", "b : B", "hb : b ∈ Z.baseSet i", "v : F"], "goal": "(Z.localTriv i).symm b v = (Z.coordChange i (Z.indexAt b) b) v"}}
+{"state": {"context": ["α : Type u_1", "Inhabited α", "a : α"], "goal": "a ∈ Semiquot.univ"}}
+{"state": {"context": ["X₁ : Type u_1", "X₂ : Type u_2", "Y₁ : Sort u_3", "Y₂ : Sort u_4", "Z₁ : Type u_5", "Z₂ : Type u_6", "TopologicalSpace X₁", "TopologicalSpace X₂", "TopologicalSpace Z₁", "TopologicalSpace Z₂", "f₁ : X₁ → Y₁", "g₁ : Z₁ → Y₁", "f₂ : X₂ → Y₂", "g₂ : Z₂ → Y₂", "mapX : X₁ → X₂", "contX : Continuous mapX", "mapY : Y₁ → Y₂", "mapZ : Z₁ → Z₂", "contZ : Continuous mapZ", "commX : f₂ ∘ mapX = mapY ∘ f₁", "commZ : g₂ ∘ mapZ = mapY ∘ g₁"], "goal": "Continuous (Function.mapPullback mapX mapY mapZ commX commZ)"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : CompleteLattice α", "inst✝ : IsAtomistic α", "x : α"], "goal": "x ∈ {a | IsAtom a} ↔ x ∈ {a | IsAtom a ∧ a ≤ ⊤}"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ"], "goal": "volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) ≠ ⊤"}}
+{"state": {"context": ["F : Type v'", "R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝¹⁷ : CommSemiring R", "inst✝¹⁶ : StarRing R", "inst✝¹⁵ : NonUnitalSemiring A", "inst✝¹⁴ : StarRing A", "inst✝¹³ : Module R A", "inst✝¹² : IsScalarTower R A A", "inst✝¹¹ : SMulCommClass R A A", "inst✝¹⁰ : StarModule R A", "inst✝⁹ : NonUnitalSemiring B", "inst✝⁸ : StarRing B", "inst✝⁷ : Module R B", "inst✝⁶ : IsScalarTower R B B", "inst✝⁵ : SMulCommClass R B B", "inst✝⁴ : StarModule R B", "inst✝³ : FunLike F A B", "inst✝² : NonUnitalAlgHomClass F R A B", "inst✝¹ : NonUnitalStarAlgHomClass F R A B", "S : NonUnitalStarSubalgebra R A", "ι : Type u_1", "inst✝ : Nonempty ι", "K : ι → NonUnitalStarSubalgebra R A", "dir : Directed (fun x x_1 => x ≤ x_1) K", "f : (i : ι) → ↥(K i) →⋆ₙₐ[R] B", "hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)", "i : ι", "x : ↥(iSup K)", "hx : ↑x ∈ K i"], "goal": "Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑(iSup K) ⋯ x = (f i) ⟨↑x, hx⟩"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R"], "goal": "W.Φ 1 = X"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "A : Type u_2", "inst✝¹ : AddGroup A", "H K : Subgroup G", "inst✝ : Finite ↥H", "h : Nat.card ↥H = Nat.card G", "this : Nonempty ↥H"], "goal": "H = ⊤"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "Field α", "CharZero α", "s : Finset ι", "f : ι → ℚ"], "goal": "↑(∏ i ∈ s, f i) = ∏ i ∈ s, ↑(f i)"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P n", "p p₁ p₂ : P", "r : ℝ", "hp₁ : ∃ r, ∃ p0 ∈ affineSpan ℝ (Set.range s.points), p₁ = r • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ p0", "hp₂ : ∃ r, ∃ p0 ∈ affineSpan ℝ (Set.range s.points), p₂ = r • (p -ᵥ ↑((orthogonalProjection (affineSpan ℝ (Set.range s.points))) p)) +ᵥ p0", "h₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r", "h₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ = r", "span_s : AffineSubspace ℝ P := affineSpan ℝ (Set.range s.points)", "h₁' : ↑(s.orthogonalProjectionSpan p₁) = s.circumcenter", "h₂' : ↑(s.orthogonalProjectionSpan p₂) = s.circumcenter"], "goal": "p₁ = p₂ ∨ p₁ = (reflection (affineSpan ℝ (Set.range s.points))) p₂"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : NormedAddCommGroup F'", "inst✝³ : NormedSpace ℝ F'", "inst✝² : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedField 𝕜", "inst✝ : NormedSpace 𝕜 E", "μ' : Measure α", "T : Set α → E →L[ℝ] F", "h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0", "h_add : FinMeasAdditive μ T", "hμ : μ ≪ μ'", "f : ↥(simpleFunc E 1 μ)", "f' : ↥(simpleFunc E 1 μ')", "h : ↑↑↑f =ᶠ[ae μ] ↑↑↑f'", "goal' : ↑↑↑f' =ᶠ[ae μ'] ↑(toSimpleFunc f')"], "goal": "↑↑↑f' =ᶠ[ae μ] ↑(toSimpleFunc f')"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "μ : Measure ℝ", "l : Filter ι", "inst✝³ : l.IsCountablyGenerated", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "a✝ b : ι → ℝ", "f : ℝ → E", "a : ℝ", "hfi : IntegrableOn f (Ioi a) μ", "hb : Tendsto b l atTop", "φ : ι → Set ℝ := fun i => Iic (b i)", "hφ : AECover (μ.restrict (Ioi a)) l φ"], "goal": "(fun i => ∫ (x : ℝ) in φ i, f x ∂μ.restrict (Ioi a)) =ᶠ[l] fun i => ∫ (x : ℝ) in a..b i, f x ∂μ"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "a x y : G₀"], "goal": "SemiconjBy a x⁻¹ y⁻¹ ↔ SemiconjBy a x y"}}
+{"state": {"context": ["n : ℕ", "f : Mathlib.Vector ℕ n → ℕ", "hf : Nat.Primrec' f"], "goal": "Nat.Primrec' fun v => (Nat.unpair (f v)).1"}}
+{"state": {"context": ["C : Type u", "inst✝ : Groupoid C", "S✝ S T : Subgroupoid C", "h : S ≤ T", "s t : ↑S.objs", "x✝¹ x✝ : s ⟶ t", "f : ↑s ⟶ ↑t", "hf : f ∈ S.arrows ↑s ↑t", "g : ↑s ⟶ ↑t", "hg : g ∈ S.arrows ↑s ↑t"], "goal": "f = g → f = g"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDedekindDomain R", "S : Type u_2", "CommRing S", "Algebra R S", "Module.Free R S", "Module.Finite R S", "p : Ideal R", "hp0 : p ≠ ⊥", "p.IsPrime", "Sₚ : Type u_3", "CommRing Sₚ", "Algebra S Sₚ", "IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ", "Algebra R Sₚ", "IsScalarTower R S Sₚ", "IsDedekindDomain Sₚ", "IsDomain S"], "goal": "IsPrincipalIdealRing Sₚ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "S : CategoryTheory.ShortComplex C"], "goal": "(CategoryTheory.ShortComplex.LeftHomologyData.ofAbelian S).i = CategoryTheory.Limits.kernel.ι S.g"}}
+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "γ₂ : Type u_4", "δ : Type u_5", "ι : Sort y", "s t u : Set α✝", "inst✝¹³ : TopologicalSpace α✝", "inst✝¹² : MeasurableSpace α✝", "inst✝¹¹ : BorelSpace α✝", "inst✝¹⁰ : TopologicalSpace β��", "inst✝⁹ : MeasurableSpace β✝", "inst✝⁸ : BorelSpace β✝", "inst✝⁷ : TopologicalSpace γ", "inst✝⁶ : MeasurableSpace γ", "inst✝⁵ : BorelSpace γ", "inst✝⁴ : MeasurableSpace δ", "α : Type u_6", "β : Type u_7", "inst✝³ : MeasurableSpace α", "inst✝² : TopologicalSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : TopologicalSpace β", "hβ : BorelSpace β", "e : α → β", "h'e : MeasurableEmbedding e", "h''e : Inducing e"], "goal": "inst✝³ = borel α"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X : Mon_ C"], "goal": "((Mon_.EquivLaxMonoidalFunctorPUnit.counitIso C).hom.app X).hom = 𝟙 X.X"}}
+{"state": {"context": ["ι : Type u_6", "κ : Type u_7", "α : Type u_8", "Fintype ι", "Fintype κ", "CommMonoid α", "DecidableEq κ", "g : ι → κ", "f : ι → α"], "goal": "∏ j : κ, ∏ i : { i // g i = j }, f ↑i = ∏ i : ι, f i"}}
+{"state": {"context": ["V : Type u_1", "inst✝² : DecidableEq V", "G : SimpleGraph V", "s t : V", "W : Type u_2", "G' : SimpleGraph W", "f : G ≃g G'", "inst✝¹ : Fintype ↑G.edgeSet", "inst✝ : Fintype ↑G'.edgeSet"], "goal": "{ x // x ∈ G.edgeFinset } ≃ { x // x ∈ G'.edgeFinset }"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "f : α → β", "a : α"], "goal": "⋯ = ⋯"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M"], "goal": "⇑DomMulAct.mkHomeomorph = ⇑DomMulAct.mk"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoMetricSpace α", "c : α", "f : β → α", "l : Filter β"], "goal": "Filter.Tendsto (fun x => dist (f x) c) l Filter.atTop ↔ Filter.Tendsto f l (Bornology.cobounded α)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M", "p : Submodule R M", "M₂ : Type u_3", "AddCommGroup M₂", "Module R M₂", "f g : M ⧸ p →ₗ[R] M₂", "h : ∀ (x : M), f (Submodule.Quotient.mk x) = g (Submodule.Quotient.mk x)"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedAddCommGroup α", "OrderedAddCommGroup β", "s : Set ι", "f : ι → α", "g : ι → β", "hfg : AntivaryOn f g s", "n : ℕ"], "goal": "AntivaryOn (n • f) g s"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : Category.{?u.8253, u_1} V", "inst✝ : Abelian V", "X Y Z : Vᵒᵖ", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0"], "goal": "f ≫ factorThruImage g = 0"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_3", "CommSemiring R", "φ ψ : MvPolynomial σ R", "m n : ℕ", "hφ : φ.IsHomogeneous m", "hψ : ψ.IsHomogeneous n"], "goal": "(φ * ψ).IsHomogeneous (m + n)"}}
+{"state": {"context": ["G : Type u_3", "DivisionCommMonoid G", "k : ℕ", "ζ : G", "h : IsPrimitiveRoot ζ k"], "goal": "IsPrimitiveRoot ζ⁻¹ k"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "inst✝ : CompleteLattice α", "f : ι → α", "s : Set α", "hne : s.Nonempty", "hsc : SupClosed s", "t : Finset α", "ht₁ : ↑t ⊆ s", "ht₂ : sSup s = t.sup id", "h : t = ∅"], "goal": "sSup s ∈ s"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{?u.96949, u_1} C", "inst✝² : Category.{?u.96953, u_2} D", "inst✝¹ : Preadditive C", "inst✝ : Preadditive D", "S S₁ S₂ : ShortComplex C", "s : S₁.Splitting", "e : S₁ ≅ S₂", "eq : e.inv.τ₂ ≫ (s.r ≫ S₁.f + S₁.g ≫ s.s) ≫ e.hom.τ₂ = e.inv.τ₂ ≫ 𝟙 S₁.X₂ ≫ e.hom.τ₂"], "goal": "(e.inv.τ₂ ≫ s.r ≫ e.hom.τ₁) ≫ S₂.f + S₂.g ≫ e.inv.τ₃ ≫ s.s ≫ e.hom.τ₂ = 𝟙 S₂.X₂"}}
+{"state": {"context": ["X : Type u_2", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "t : Set Y", "ht : IsClosed t", "s : Set X"], "goal": "IsClosed {f | Set.MapsTo (⇑f) s t}"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹¹ : AddCommGroup E", "inst✝¹⁰ : TopologicalSpace E", "inst✝⁹ : TopologicalAddGroup E", "inst✝⁸ : T2Space E", "inst✝⁷ : Module ℝ E", "inst✝⁶ : ContinuousSMul ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "F : Type u_2", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "G : Type u_3", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "inst✝ : FiniteDimensional ℝ G", "f g : F → G", "n : ℕ∞", "hf : ContDiff ℝ n f", "hg : ContDiff ℝ n g", "h : ∀ (x : F), f x ≠ g x"], "goal": "ContDiff ℝ n fun y => toEuclidean (f y)"}}
+{"state": {"context": ["F : Type u", "E : Type v", "Field F", "Field E", "Algebra F E", "q n : ℕ", "hF : ExpChar F q", "ι : Type u_1", "v : ι → E", "hsep : ∀ (i : ι), IsSeparable F (v i)", "h : LinearIndependent F v"], "goal": "LinearIndependent F fun x => v x ^ q ^ n"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "OrderedAddCommMonoid β", "l : Filter α", "f g : α → β", "hf : ∀ (x : α), 0 ≤ f x", "hg : Filter.Tendsto g l Filter.atTop"], "goal": "Filter.Tendsto (fun x => f x + g x) l Filter.atTop"}}
+{"state": {"context": ["f : ℕ → ℂ", "s a : ℂ", "hf : LSeriesHasSum f s a"], "goal": "LSeriesHasSum (-f) s (-a)"}}
+{"state": {"context": ["ι : Type u_1", "M : ι → Type u_2", "inst✝³ : (i : ι) → Monoid (M i)", "N : Type u_3", "inst✝² : Monoid N", "inst✝¹ : (i : ι) → DecidableEq (M i)", "inst✝ : DecidableEq ι", "i : ι", "m : M i", "w : Word M"], "goal": "(equivPair i) ((equivPair i).symm (let __src := (equivPair i) w; { head := m * ((equivPair i) w).head, tail := __src.tail, fstIdx_ne := ⋯ })) = { head := m * ((equivPair i) w).head, tail := ((equivPair i) w).tail, fstIdx_ne := ⋯ }"}}
+{"state": {"context": ["x y : SetTheory.PGame", "ih :\n  ∀ (a : Surreal.Multiplication.Args),\n    Surreal.Multiplication.ArgsRel a (Surreal.Multiplication.Args.P1 x y) → Surreal.Multiplication.P124 a"], "goal": "Surreal.Multiplication.IH1 x y"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "f : MvPowerSeries σ R", "n : σ →₀ ℕ", "a : R"], "goal": "(MvPowerSeries.coeff R n) (a • f) = a * (MvPowerSeries.coeff R n) f"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "OrderTop α", "LocallyFiniteOrder α", "a b : α"], "goal": "Finset.Ioc ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ioc a b)"}}
+{"state": {"context": ["α : Type u_1", "δ : Type u_4", "TopologicalSpace α", "MeasurableSpace α", "BorelSpace α", "MeasurableSpace δ", "ConditionallyCompleteLinearOrder α", "OrderTopology α", "SecondCountableTopology α", "ι : Sort u_5", "Countable ι", "f : ι → δ → α", "hf : ∀ (i : ι), Measurable (f i)"], "goal": "Measurable fun b => ⨅ i, f i b"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "k : ℕ", "hi : (X ^ k).content = 1"], "goal": "(X ^ (k + 1)).content = 1"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y z : V", "hx : x ≠ 0", "hy : y ≠ 0", "hz : z ≠ 0"], "goal": "o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = ↑π"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝¹⁰ : ConditionallyCompleteLinearOrder α", "inst✝⁹ : TopologicalSpace α", "inst✝⁸ : OrderTopology α", "inst✝⁷ : ConditionallyCompleteLinearOrder β", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : OrderTopology β", "inst✝⁴ : Nonempty γ", "inst✝³ : DenselyOrdered α", "a b : α", "δ : Type u_1", "inst✝² : LinearOrder δ", "inst✝¹ : TopologicalSpace δ", "inst✝ : OrderClosedTopology δ", "f : α → δ", "hf_c : Continuous f", "hf_i : Injective f"], "goal": "StrictMono f ∨ StrictAnti f"}}
+{"state": {"context": ["α✝ : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝⁷ : ConditionallyCompleteLinearOrder α", "inst✝⁶ : TopologicalSpace α", "inst✝⁵ : OrderTopology α", "inst✝⁴ : TopologicalSpace β", "inst✝³ : TopologicalSpace γ", "inst✝² : DenselyOrdered α", "inst✝¹ : ConditionallyCompleteLinearOrder β", "inst✝ : OrderTopology β", "f : α → β", "a b c : α", "h : ContinuousOn f (Icc a b)", "hc : c ∈ Icc a b"], "goal": "f c ≤ sSup (f '' Icc a b)"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "f : CauSeq ℚ_[p] ⇑padicNormE"], "goal": "∃ q, ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (q - ↑f i) < ε"}}
+{"state": {"context": ["J : Type v'", "inst✝³ : Category.{u', v'} J", "C : Type u", "inst✝² : Category.{v, u} C", "D : Type u''", "inst✝¹ : Category.{v'', u''} D", "X Y : C", "inst✝ : FinitaryPreExtensive C", "n : ℕ", "F : Discrete (Fin n) ⥤ C", "c : Cocone F", "hc : IsColimit c", "f : Fin n → C := F.obj ∘ Discrete.mk"], "goal": "IsUniversalColimit c"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "f : α → ℝ≥0∞"], "goal": "⨍⁻ (x : α), f x ∂0 = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : ClosedIicTopology α", "inst✝ : TopologicalSpace β", "a b c : α", "f : α → β", "H : a < b"], "goal": "Ioo a b ∈ 𝓝[<] b"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f g : α → E", "hf : MeasureTheory.Memℒp f p μ", "hg : MeasureTheory.Memℒp g p μ"], "goal": "MeasureTheory.Memℒp.toLp (f - g) ⋯ = MeasureTheory.Memℒp.toLp f hf - MeasureTheory.Memℒp.toLp g hg"}}
+{"state": {"context": ["x y z : ℤ", "h : PythagoreanTriple x y z", "hc : x.gcd y = 1", "hyo : y % 2 = 1", "hzpos : 0 < z", "h0 : ¬x = 0", "v : ℚ := ↑x / ↑z", "w : ℚ := ↑y / ↑z", "hq : v ^ 2 + w ^ 2 = 1", "hvz : v ≠ 0", "hw1 : w ≠ -1", "hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0", "hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}", "q : ℚ := (circleEquivGen hQ).symm ⟨(v, w), hp⟩", "ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)", "m : ℤ := ↑q.den", "n : ℤ := q.num"], "goal": "h.IsPrimitiveClassified"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "AddGroup α"], "goal": "0⁻ = 0"}}
+{"state": {"context": ["m : Type u → Type u", "inst✝¹ : _root_.Monad m", "inst✝ : LawfulMonad m", "X✝ Y✝ Z✝ : KleisliCat m", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝", "t : { obj := fun X => X, map := fun {X Y} f => f }.obj X✝"], "goal": "(f ≫ g) t = f t >>= g"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y : Cᵒᵖ", "f : CategoryTheory.coyoneda.obj X ⟶ CategoryTheory.coyoneda.obj Y"], "goal": "CategoryTheory.Coyoneda.fullyFaithful.preimage f =\n  Quiver.Hom.op (f.app (Opposite.unop X) (CategoryTheory.CategoryStruct.id (Opposite.unop X)))"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : InnerProductSpace ℝ V", "inst✝³ : MetricSpace P", "inst✝² : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "ps : Set P", "h : ps ⊆ ↑s", "inst✝¹ : Nonempty ↥s", "n : ℕ", "inst✝ : FiniteDimensional ℝ ↥s.direction", "hd : finrank ℝ ↥s.direction = n", "c : P", "hc : c ∈ s", "r : ℝ", "hcr : ∀ p ∈ ps, dist p c = r"], "goal": "∀ (sx : Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumradius = r"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "F' : Type u_14", "NormedAddCommGroup F'", "NormedSpace 𝕜 F'", "G' : Type u_15", "TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_16", "TopologicalSpace N'", "ChartedSpace G' N'", "s : Set M", "f : M → M'", "g : M → N'", "hf : SmoothOn I I' f s", "hg : SmoothOn I J' g s"], "goal": "SmoothOn I (I'.prod J') (fun x => (f x, g x)) s"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "DecidableRel fun x x_1 => x ∣ x_1", "a : α", "ha : ¬IsUnit a", "ha0 : a ≠ 0"], "goal": "multiplicity a a = 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "h : Function.Bijective f"], "goal": "Function.Bijective (List.map f)"}}
+{"state": {"context": ["R : Type u_1", "p m n : ℕ", "inst✝² : CommSemiring R", "inst✝¹ : ExpChar R p", "inst✝ : PerfectRing R p"], "goal": "⇑(iterateFrobeniusEquiv R p n) = ⇑((RingAut.toPerm R) (frobeniusEquiv R p ^ n))"}}
+{"state": {"context": ["α✝ : Type u", "β : Type u_1", "γ : Type u_2", "r✝ : α✝ → α✝ → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "a✝ : Ordinal.{u_3}", "α : Type u_3", "r : α → α → Prop", "wo : IsWellOrder α r", "a x : α", "x✝ : ∀ (y : α), r y x → Acc r y", "IH : ∀ (y : α), r y x → Acc (fun x x_1 => x < x_1) (typein r y)", "b : α", "h : typein r b < typein r x"], "goal": "Acc (fun x x_1 => x < x_1) (typein r b)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "Zero M", "a a' : α", "f : α →₀ M", "h : a' ≠ a"], "goal": "(Finsupp.erase a f) a' = f a'"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "Archimedean α", "ε : α", "x : α", "ε0 : 0 < ε"], "goal": "∃ q, |x - ↑q| < ε"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α"], "goal": "{I | M.Indep I} = ↑(lowerClosure {B | M.Base B})"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "α : Type u_1", "Applicative m", "LawfulApplicative m", "x : m α"], "goal": "SatisfiesM (fun x => True) x"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n m : ℕ", "ω : Ω", "ℱ : Filtration ℕ m0", "M : ℕ", "hNM : N ≤ M", "h : upperCrossingTime a b f N (n + 1) ω < N", "this : lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω"], "goal": "upperCrossingTime a b f M (n + 1) ω = upperCrossingTime a b f N (n + 1) ω ∧ lowerCrossingTime a b f M n ω = lowerCrossingTime a b f N n ω"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemifield α", "inst✝ : FloorSemiring α", "a : α", "n : ℕ", "ha : 0 ≤ a", "hn : n > 0"], "goal": "⌊a / ↑n⌋₊ = ⌊a⌋₊ / n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddGroup α", "AddAction α β", "g : α", "x y : β", "h : g +ᵥ x = g +ᵥ y"], "goal": "x = y"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommSemiring R", "inst✝¹ : Semiring S", "inst✝ : Algebra R S", "M N : Submodule R S", "κ : Type u_1", "ι : Type u_2", "m : Basis κ R ↥M", "n : Basis ι R ↥N", "i0 : (κ × ι →₀ R) ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := (finsuppTensorFinsupp' R κ ι).symm", "i1 : ↥M ⊗[R] ↥N ≃ₗ[R] (κ →₀ R) ⊗[R] (ι →₀ R) := TensorProduct.congr m.repr n.repr", "i : (κ × ι →₀ R) →ₗ[R] S := M.mulMap N ∘ₗ ↑(i0 ≪≫ₗ i1.symm)", "this : i = Finsupp.total (κ × ι) S R fun i => ↑(m i.1) * ↑(n i.2)", "H : Function.Injective ⇑(M.mulMap N)"], "goal": "M.LinearDisjoint N"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝⁴ : Preorder ι", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : CompleteSpace E", "f✝ g : ι → Ω → E", "ℱ : Filtration ι m0", "inst✝ : IsFiniteMeasure μ", "f : ι → Ω → ℝ", "hadp : Adapted ℱ f", "hint : ∀ (i : ι), Integrable (f i) μ", "hf : ∀ (i j : ι), i ≤ j → 0 ≤ᶠ[ae μ] μ[f j - f i|↑ℱ i]", "i j : ι", "hij : i ≤ j"], "goal": "f i ≤ᶠ[ae μ] μ[f j|↑ℱ i]"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "SuccOrder α", "a b : α", "ha : ¬IsMax a"], "goal": "b < Order.succ a ↔ b ≤ a"}}
+{"state": {"context": ["M : Type u", "G : Type v", "X : Type w", "inst✝³ : Group G", "inst✝² : PseudoMetricSpace G", "inst✝¹ : IsometricSMul G G", "inst✝ : IsometricSMul Gᵐᵒᵖ G", "a b c : G"], "goal": "nndist (a / b) (a / c) = nndist b c"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹³ : CommRing R", "inst✝¹² : IsDedekindDomain R", "S : Type u_2", "inst✝¹¹ : CommRing S", "inst✝¹⁰ : Algebra R S", "inst✝⁹ : Module.Free R S", "inst✝⁸ : Module.Finite R S", "p : Ideal R", "hp0 : p ≠ ⊥", "inst✝⁷ : p.IsPrime", "Sₚ : Type u_3", "inst✝⁶ : CommRing Sₚ", "inst✝⁵ : Algebra S Sₚ", "inst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ", "inst✝³ : Algebra R Sₚ", "inst✝² : IsScalarTower R S Sₚ", "inst✝¹ : IsDedekindDomain Sₚ", "inst✝ : IsDomain S", "P : Ideal Sₚ", "hP : P.IsPrime", "hP0 : P ≠ ⊥", "non_zero_div : Algebra.algebraMapSubmonoid S p.primeCompl ≤ S⁰", "this✝¹ : Algebra (Localization.AtPrime p) Sₚ := localizationAlgebra p.primeCompl S", "this✝ : IsScalarTower R (Localization.AtPrime p) Sₚ", "pid : IsPrincipalIdealRing (Localization.AtPrime p)", "p' : Ideal (Localization.AtPrime p)", "hpu : ∀ (y : Ideal (Localization.AtPrime p)), (fun P => P ≠ ⊥ ∧ P.IsPrime) y → y = p'", "hp'0 : p' ≠ ⊥", "hp'p : p'.IsPrime", "this : LocalRing.maximalIdeal (Localization.AtPrime p) ≠ ⊥"], "goal": "Ideal.map (algebraMap R Sₚ) p ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "x : 𝕜", "ι : Type u_1", "Fintype ι", "E' : ι → Type u_2", "(i : ι) → NormedAddCommGroup (E' i)", "(i : ι) → NormedSpace 𝕜 (E' i)", "φ : 𝕜 → (i : ι) → E' i", "φ' : (i : ι) → E' i"], "goal": "HasDerivAt φ φ' x ↔ ∀ (i : ι), HasDerivAt (fun x => φ x i) (φ' i) x"}}
+{"state": {"context": ["R : Type u", "inst✝⁷ : Ring R", "J : Type v", "inst✝⁶ : Category.{t, v} J", "F : J ⥤ ModuleCat R", "ι : Type v", "dec_ι : DecidableEq ι", "inst✝⁵ : Preorder ι", "G : ι → Type v", "inst✝⁴ : (i : ι) → AddCommGroup (G i)", "inst✝³ : (i : ι) → Module R (G i)", "f : (i j : ι) → i ≤ j → G i →ₗ[R] G j", "inst✝² : DirectedSystem G fun i j h => ⇑(f i j h)", "inst✝¹ : DecidableEq ι", "inst✝ : IsDirected ι fun x x_1 => x ≤ x_1", "s : Cocone (directLimitDiagram G f)", "m : (directLimitCocone G f).pt ⟶ s.pt", "h : ∀ (j : ι), (directLimitCocone G f).ι.app j ≫ m = s.ι.app j"], "goal": "m = (fun s => DirectLimit.lift R ι G f s.ι.app ⋯) s"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β", "fin : hausdorffEdist u t ≠ ⊤", "I : hausdorffDist u s ≤ hausdorffDist u t + hausdorffDist t s"], "goal": "hausdorffDist s u ≤ hausdorffDist s t + hausdorffDist t u"}}
+{"state": {"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "z w : K"], "goal": "z < w ↔ re z < re w ∧ im z = im w"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "M : Type u_2", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "p : Submodule R (CliffordAlgebra Q)", "n : ℕ"], "goal": "Submodule.map reverse (p ^ n) = Submodule.map reverse p ^ n"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "β : Type u_2", "MeasurableSpace β", "μ : MeasureTheory.FiniteMeasure α", "ν : MeasureTheory.FiniteMeasure β", "s : Set α", "t : Set β"], "goal": "(μ.prod ν) (s ×ˢ t) = μ s * ν t"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "G : Type u_4", "OrderedRing 𝕜", "AddCommGroup E", "AddCommGroup G", "Module 𝕜 E", "AddTorsor G E", "VAddCommClass G E E", "a : G", "b c : E"], "goal": "a +ᵥ [b-[𝕜]c] = [a +ᵥ b-[𝕜]a +ᵥ c]"}}
+{"state": {"context": ["α : Type u", "β : Type v", "s : Set α", "t : Set β", "h : (s ×ˢ t).Finite"], "goal": "t.Nonempty → s.Finite"}}
+{"state": {"context": ["γ : Type u", "x : γ", "f : Listβ γ (Listα.cons x) → WType (Listβ γ)"], "goal": "ofList γ (toList γ (mk (Listα.cons x) f)) = mk (Listα.cons x) f"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "a✝ a₁ a₂ b✝ c : G", "inst✝¹ : Group G", "inst✝ : Group H", "f : G → H", "hf : IsGroupHom f", "a b : G", "h : f a = f b"], "goal": "f (a * b⁻¹) = 1"}}
+{"state": {"context": ["n : Nat", "i : Fin (n + 1)"], "goal": "↑(i + 1) = if i = Fin.last n then 0 else ↑i + 1"}}
+{"state": {"context": ["α : Type u_6", "β : Type u_7", "PseudoMetricSpace α", "PseudoMetricSpace β", "f : α → β", "h : ∃ r, r ≠ 0 ∧ ∀ (x y : α), dist (f x) (f y) = ↑r * dist x y"], "goal": "⇑(Dilation.mkOfDistEq f h) = f"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "s : S₁.Splitting", "e : S₁ ≅ S₂"], "goal": "(s.ofIso e).r = CategoryTheory.CategoryStruct.comp e.inv.τ₂ (CategoryTheory.CategoryStruct.comp s.r e.hom.τ₁)"}}
+{"state": {"context": ["α : Type u", "s₁ s₂ : Stream' α"], "goal": "(∀ (n : ℕ), Stream'.take n s₁ = Stream'.take n s₂) → s₁ = s₂"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "LocallyCompactSpace X", "s : Set X", "hs : IsClosed s"], "goal": "LocallyCompactSpace ↑s"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "α : Type v", "DecidableEq l", "DecidableEq m", "A : Matrix m n α", "i : m", "r : n → α", "e : l ≃ m", "f : o ≃ n"], "goal": "(A.updateRow i r).submatrix ⇑e ⇑f = (A.submatrix ⇑e ⇑f).updateRow (e.symm i) fun i => r (f i)"}}
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+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "Semiring S", "f : R →+* S"], "goal": "Function.Injective ⇑f.kerLift"}}
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+{"state": {"context": ["β : Type u_2", "AddCommMonoid β", "f : Fin 0 → β"], "goal": "∑ i : Fin 0, f i = 0"}}
+{"state": {"context": ["α : Type u", "n : ℕ", "a : α", "l : List α"], "goal": "List.sublistsLen (n + 1) (a :: l) = List.sublistsLen (n + 1) l ++ List.map (List.cons a) (List.sublistsLen n l)"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "a b : α"], "goal": "Associates.mk a ∣ Associates.mk b ↔ a ∣ b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹² : CommRing R", "inst✝¹¹ : Ring S", "inst✝¹⁰ : Algebra R S", "K : Type u_4", "L : Type u_5", "F : Type u_6", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : Field F", "inst✝⁶ : Algebra K L", "inst✝⁵ : Algebra K F", "ι : Type w", "E : Type u_7", "inst✝⁴ : Field E", "inst✝³ : Algebra K E", "inst✝² : FiniteDimensional K L", "inst✝¹ : Algebra.IsSeparable K L", "inst✝ : IsAlgClosed E", "x : L", "hx : IsIntegral K x"], "goal": "(∏ σ : ↥K⟮x⟯ →ₐ[K] E, σ (adjoin.powerBasis hx).gen) ^ finrank (↥K⟮x⟯) L = ∏ σ : L →ₐ[K] E, σ x"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.SigmaFinite μ", "x : α"], "goal": "x ∈ MeasureTheory.spanningSets μ (MeasureTheory.spanningSetsIndex μ x)"}}
+{"state": {"context": ["X : Type u_1", "α : Type u_2", "TopologicalSpace X", "BaireSpace X", "t : Set α", "s : Set X", "hs : IsGδ s", "hd : Dense s", "ht : t.Countable", "f : α → Set X", "hc : ∀ i ∈ t, IsClosed (f i)", "hU : s ⊆ ⋃ i ∈ t, f i"], "goal": "Dense (⋃ i ∈ t, interior (f i))"}}
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+{"state": {"context": ["a b c : Prop", "h : Or a b", "left : a → c", "right : b → c"], "goal": "c"}}
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+{"state": {"context": ["x✝ y z x : ℝ", "hx : x ∈ interior (Ici 0)"], "goal": "0 < deriv cosh x"}}
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+{"state": {"context": ["n : ℕ", "p : Fin (n + 1)", "e : Equiv.Perm (Fin n)"], "goal": "(Equiv.Perm.decomposeFin.symm (p, e)) 0 = p"}}
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+{"state": {"context": ["P Q : Prop"], "goal": "Xor' P Q ↔ ¬(P ↔ Q)"}}
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+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "AddMonoid α", "l : List (Matrix m n α)"], "goal": "l.sumᵀ = (List.map Matrix.transpose l).sum"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "E : Type u_4", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝³ : NormedAddCommGroup E", "κ : Kernel α β", "inst✝² : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝¹ : IsSFiniteKernel η", "a : α", "δ : Type u_5", "inst✝ : TopologicalSpace δ", "f : β × γ → δ", "hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)"], "goal": "∀ᵐ (x : β) ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x))"}}
+{"state": {"context": ["α : Type u_6", "Zero α", "One α", "n : ℕ"], "goal": "(Matrix.circulant fun i => if ↑i = 0 then 1 else 0) = 1"}}
+{"state": {"context": ["α✝ : Type u_1", "r✝ : α✝ → α✝ → Prop", "α : Type u", "β : Type v", "r : α → α → Prop", "s : β → β → Prop", "inst✝ : IsRefl β s", "f : r ≃r s"], "goal": "Cardinal.lift.{max u v, u} (cof r) ≤ Cardinal.lift.{max u v, v} (cof s)"}}
+{"state": {"context": ["θ : Angle"], "goal": "0 < θ.cos ↔ |θ.toReal| < π / 2"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "A : Type u_4", "B : Type u_5", "C : Type u_6", "inst✝⁴ : AddCommMonoid A", "inst✝³ : AddCommMonoid B", "inst✝² : AddCommMonoid C", "t : ι → A → C", "h0 : ∀ (i : ι), t i 0 = 0", "h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y", "s : Finset α", "f✝ : α → ι →₀ A", "i : ι", "g✝ : ι →₀ A", "k : ι → A → γ → B", "x : γ", "β : Type u_7", "M : Type u_8", "M' : Type u_9", "N : Type u_10", "P : Type u_11", "G : Type u_12", "H : Type u_13", "R : Type u_14", "S : Type u_15", "inst✝¹ : Zero M", "inst✝ : CommMonoid N", "v : α →₀ M", "f : α ↪ β", "g : β → M → N"], "goal": "∏ x ∈ v.support, g (f x) ((embDomain f v) (f x)) = ∏ a ∈ v.support, g (f a) (v a)"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝ : PseudoEMetricSpace α", "δ ε : ℝ", "s t : Set α", "x : α", "hst : Disjoint s t", "hs : IsCompact s", "ht : IsClosed t", "r : ℝ≥0", "hr : 0 < r", "h : ∀ x ∈ s, ∀ y ∈ t, ↑r < edist x y", "z : α", "hzs : ∃ z_1 ∈ s, edist z z_1 < ENNReal.ofReal (↑r / 2)", "hzt : ∃ z_1 ∈ t, edist z z_1 < ENNReal.ofReal (↑r / 2)"], "goal": "z ∈ ⊥"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A : CategoryTheory.Functor Cᵒᵖ (Type v)", "P : CategoryTheory.Limits.IndObjectPresentation A", "X : P.I"], "goal": "(P.toCostructuredArrow.obj X).hom = P.cocone.ι.app X"}}
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+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : α → β → β", "op : α → α → α", "inst✝³ : Monoid β", "inst✝² : Monoid γ", "inst✝¹ : FunLike F β γ", "inst✝ : DecidableEq α", "f : α → β", "sl : List α", "sl' : sl.Nodup", "tl : List α", "tl' : tl.Nodup", "h : sl.Disjoint tl", "comm : {x | x ∈ sl.toFinset ∪ tl.toFinset}.Pairwise fun a b => Commute (f a) (f b)"], "goal": "(sl.toFinset ∪ tl.toFinset).noncommProd f comm = sl.toFinset.noncommProd f ⋯ * tl.toFinset.noncommProd f ⋯"}}
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+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "d₁ d₂ : G.Dart"], "goal": "d₁ = d₂ ↔ d₁.toProd = d₂.toProd"}}
+{"state": {"context": ["α : Type u_1", "l : List (α → Prop)", "H : ∀ (a : α), (map (fun p => p a) l).TFAE"], "goal": "(map (fun p => ∃ a, p a) l).TFAE"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n✝ : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "n : ℕ"], "goal": "(X ^ n).roots = n • {0}"}}
+{"state": {"context": ["ι : Type u_5", "Fintype ι"], "goal": "dimH Set.univ = ↑(Fintype.card ι)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "A : Type u₃", "CategoryTheory.Category.{v₃, u₃} A", "J : CategoryTheory.GrothendieckTopology C", "K : CategoryTheory.GrothendieckTopology D", "L : CategoryTheory.GrothendieckTopology A", "F : C ⥤ D", "hF : CategoryTheory.CoverPreserving J K F", "G : D ⥤ A", "hG : CategoryTheory.CoverPreserving K L G"], "goal": "CategoryTheory.CoverPreserving J L (F ⋙ G)"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : NonAssocRing R", "inst✝² : Pow R ℕ", "inst✝¹ : BinomialRing R", "inst✝ : NatPowAssoc R", "r : R", "k : ℕ"], "goal": "(descPochhammer ℤ (k + 1)).smeval (r + 1) = (k + 1).factorial • choose r k + (descPochhammer ℤ (k + 1)).smeval r"}}
+{"state": {"context": ["e : ℝ", "d n : ℕ", "h : NormNum.IsInt e (Int.negOfNat n)", "w : Nat.blt 1 n = true"], "goal": "1 < ↑n"}}
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+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "R : Type u_4", "inst✝² : NonUnitalNonAssocSemiring R", "inst✝¹ : SetLike M R", "inst✝ : MulMemClass M R", "S : M", "b : R", "hb : b ∈ AddSubmonoid.closure ↑S"], "goal": "∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * b ∈ AddSubmonoid.closure ↑S"}}
+{"state": {"context": ["α : Type u_2", "MulOneClass α"], "goal": "⇑Set.singletonMonoidHom = singleton"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : NormedAddCommGroup F'", "inst✝³ : NormedSpace ℝ F'", "inst✝² : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedField 𝕜", "inst✝ : NormedSpace 𝕜 E", "T : Set α → E →L[ℝ] F", "h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0", "h_add : FinMeasAdditive μ T", "f g : ↥(simpleFunc E 1 μ)"], "goal": "setToL1S T (f - g) = setToL1S T f - setToL1S T g"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_7", "ι' : Type u_8", "S : Type u_12", "Semiring S", "Module S N", "SMulCommClass R S N", "σ : ι ≃ ι'", "f : M [⋀^ι]→ₗ[R] N"], "goal": "(AlternatingMap.domDomCongrₗ S σ) f = AlternatingMap.domDomCongr σ f"}}
+{"state": {"context": [], "goal": "‖Complex.reCLM‖₊ = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "β : Type u_4", "inst✝⁴ : OrderedRing 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : AddCommGroup F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "s t : Set E", "f : E →ᵃ[𝕜] F", "hs : Convex 𝕜 s", "x : E", "hx : x ∈ s"], "goal": "StarConvex 𝕜 (f x) (⇑f '' s)"}}
+{"state": {"context": ["α : Type u_1", "V : Type u_2", "P : Type u_3", "W : Type u_4", "Q : Type u_5", "inst✝⁹ : SeminormedAddCommGroup V", "inst✝⁸ : PseudoMetricSpace P", "inst✝⁷ : NormedAddTorsor V P", "inst✝⁶ : NormedAddCommGroup W", "inst✝⁵ : MetricSpace Q", "inst✝⁴ : NormedAddTorsor W Q", "𝕜 : Type u_6", "inst✝³ : NormedField 𝕜", "inst✝² : NormedSpace 𝕜 V", "inst✝¹ : NormedSpace 𝕜 W", "inst✝ : Invertible 2", "p₁ p₂ p₃ p₄ : P"], "goal": "‖midpoint 𝕜 (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)‖ ≤ (‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖) / ‖2‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder β", "f : α → β", "l : List α", "a m : α"], "goal": "a ∈ l → m ∈ List.argmax f l → f a ≤ f m"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Group α", "e : α", "x : Finset α × Finset α"], "goal": "op e • x.1 * e⁻¹ • x.2 ⊆ x.1 * x.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "f g : α → E", "s t : Set α", "μ ν : Measure α"], "goal": "IntegrableOn f ∅ μ"}}
+{"state": {"context": ["G : Type u_1", "Group G", "N : Type u_5", "Group N", "f : G →* N", "K : Subgroup G", "y : N"], "goal": "y ∈ Subgroup.map f K ↔ ∃ x ∈ K, f x = y"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "UniqueFactorizationMonoid α", "a : Associates α"], "goal": "(∃ s, a.factors = ↑s) ↔ a ≠ 0"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "Z X : C", "h : Z ⟶ (shiftFunctor C 1).obj X", "Y' : C", "f' : (shiftFunctor C 1).obj X ⟶ Y'", "g' : Y' ⟶ (shiftFunctor C 1).obj Z", "mem : Triangle.mk h f' g' ∈ distinguishedTriangles", "T' : Triangle C := (Triangle.mk h f' g').invRotate.invRotate"], "goal": "∃ Y f g, Triangle.mk f g h ∈ distinguishedTriangles"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t s : Set α"], "goal": "(μ.restrict s) univ = μ s"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "m n : σ →₀ ℕ", "a b : R"], "goal": "(MvPowerSeries.monomial R m) a * (MvPowerSeries.monomial R n) b = (MvPowerSeries.monomial R (m + n)) (a * b)"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "inst✝⁴ : HasShift C ℤ", "inst✝³ : HasZeroObject C", "inst✝² : Preadditive C", "inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝ : Pretriangulated C", "T₁ T₂ : Triangle Cᵒᵖ", "hT₁ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₁) ∈ Pretriangulated.distinguishedTriangles", "hT₂ : Opposite.unop ((triangleOpEquivalence C).inverse.obj T₂) ∈ Pretriangulated.distinguishedTriangles", "a : T₁.obj₁ ⟶ T₂.obj₁", "b : T₁.obj₂ ⟶ T₂.obj₂", "comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁", "c : Opposite.unop T₂.obj₃ ⟶ Opposite.unop T₁.obj₃", "hc₁ : T₂.mor₂.unop ≫ b.unop = c ≫ T₁.mor₂.unop", "hc₂ : (shiftFunctor C 1).map T₂.mor₃.unop ≫ (shiftFunctor C 1).map c = ((opShiftFunctorEquivalence C 1).unitIso.hom.app T₂.obj₁).unop ≫ a.unop ≫ ((opShiftFunctorEquivalence C 1).unitIso.inv.app T₁.obj₁).unop ≫ (shiftFunctor C 1).map T₁.mor₃.unop"], "goal": "∃ c, T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ (shiftFunctor Cᵒᵖ 1).map a = c ≫ T₂.mor₃"}}
+{"state": {"context": [], "goal": "bernoulli' 4 = -1 / 30"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "inst✝¹ : TopologicalSpace X", "C₀ C₁ C₂ : Set X", "h₀ : C₀ ⊆ C₁ ∪ C₂", "h₁ : IsClosed C₁", "h₂ : IsClosed C₂", "f₁ : LocallyConstant (↑C₁) Z", "f₂ : LocallyConstant (↑C₂) Z", "inst✝ : DecidablePred fun x => x ∈ C₁", "hf : ∀ (x : X) (hx : x ∈ C₁ ∩ C₂), f₁ ⟨x, ⋯⟩ = f₂ ⟨x, ⋯⟩", "x : ↑C₀", "hx : ↑x ∈ C₁"], "goal": "(piecewise' h₀ h₁ h₂ f₁ f₂ hf) x = f₁ ⟨↑x, hx⟩"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "(o.oangle (x + y) y).tan * ‖y‖ = ‖x‖"}}
+{"state": {"context": ["m n : ℕ", "h : m ≤ n"], "goal": "m.gcd (n - m) = m.gcd n"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ✝ μ' : Measure α", "ν✝ ν' : Measure β", "τ✝ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "f : α × β → γ", "μ : Measure α", "ν : Measure β", "τ : Measure γ", "hf : Measurable f", "inst✝ : SFinite ν", "h2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving (fun y => f (x, y)) ν τ", "s : Set γ", "hs : MeasurableSet s", "h2s : τ s = 0"], "goal": "∫⁻ (x : α), ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p : (α × β) × γ"], "goal": "(Equiv.prodAssoc α β γ) p = (p.1.1, p.1.2, p.2)"}}
+{"state": {"context": ["β : Type w", "C : Type u", "CategoryTheory.Category.{v, u} C", "Unique β", "f : β → C", "s : CategoryTheory.Limits.Cocone (CategoryTheory.Discrete.functor f)"], "goal": "(CategoryTheory.Limits.colimitCoconeOfUnique f).isColimit.desc s = s.ι.app default"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : AddCommMonoid N", "inst✝ : Module R N", "w : Set M", "x : ↥(span R w)"], "goal": "(Finsupp.total (↑w) M R Subtype.val) (repr R w x) = ↑x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "TopologicalSpace β", "Group α", "MulAction α β", "ContinuousInv α", "ContinuousSMul α β", "s : Set α", "t : Set β", "ht : IsClosed t", "hs : IsCompact s"], "goal": "IsClosed (s • t)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹⁵ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹⁴ : TopologicalSpace M", "inst✝¹³ : ChartedSpace H M", "inst✝¹² : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹¹ : NormedAddCommGroup E'", "inst✝¹⁰ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁹ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁸ : TopologicalSpace M'", "inst✝⁷ : ChartedSpace H' M'", "inst✝⁶ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝⁵ : NormedAddCommGroup E''", "inst✝⁴ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝³ : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝² : TopologicalSpace M''", "inst✝¹ : ChartedSpace H'' M''", "inst✝ : SmoothManifoldWithCorners I'' M''", "s✝ : Set M", "x✝ : M", "s : Set (M × M')", "x : M × M'", "hxs : UniqueMDiffWithinAt (I.prod I') s x"], "goal": "mfderiv (I.prod I') I Prod.fst x = ContinuousLinearMap.fst 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2)"}}
+{"state": {"context": ["𝕂 : Type u_1", "inst✝⁵ : RCLike 𝕂", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace 𝕂 E", "F : Type u_3", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕂 F", "inst✝ : CompleteSpace E", "f : E → F", "f' : E ≃L[𝕂] F", "a : E", "n : ℕ∞", "hf : ContDiffAt 𝕂 n f a", "hf' : HasFDerivAt f (↑f') a", "hn : 1 ≤ n", "this : hf.localInverse hf' hn (f a) = a"], "goal": "ContDiffAt 𝕂 n (↑(toPartialHomeomorph f hf hf' hn)) (↑(toPartialHomeomorph f hf hf' hn).symm (f a))"}}
+{"state": {"context": ["q r : ℚ", "s : ℚ := q.num * r.num /. (↑q.den * ↑r.den)", "hs : ↑q.den * ↑r.den ≠ 0", "c : ℤ", "c_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num", "c_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den", "a✝ : ¬c = 0", "h : q * r = s"], "goal": "↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num = (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : Mod α", "a b : Part α", "ma mb : α", "ha : ma ∈ a", "hb : mb ∈ b"], "goal": "∃ a_1 ∈ a, ∃ a ∈ b, a_1 % a = ma % mb"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "κ : Type u_6", "R✝ : Type u_7", "R₂ : Type u_8", "M✝ : Type u_9", "M₂ : Type u_10", "M : Type u_11", "inst✝⁷ : TopologicalSpace M", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : T2Space M", "R : Type u_12", "inst✝⁴ : DivisionRing R", "inst✝³ : Module R M", "inst✝² : ContinuousConstSMul R M", "inst✝¹ : Group α", "inst✝ : MulAction α β", "f : β → M", "g : Quotient (orbitRel α β) → R", "x : Quotient (orbitRel α β)"], "goal": "∀ (a : β), automorphize (g ∘ Quotient.mk' • f) (Quotient.mk'' a) = (g • automorphize f) (Quotient.mk'' a)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C", "X Y : C", "f : X ⟶ Y"], "goal": "(CategoryTheory.kernelOpUnop f).hom =\n  (CategoryTheory.Limits.kernel.lift f.op (CategoryTheory.Limits.cokernel.π f).op ⋯).unop"}}
+{"state": {"context": ["R : Type u", "Ring R", "p q : R[X]"], "goal": "(p - q).natDegree = (q - p).natDegree"}}
+{"state": {"context": ["R : Type u_2", "Ring R", "S : Subring Rᵐᵒᵖ"], "goal": "↑S.unop = MulOpposite.op ⁻¹' ↑S"}}
+{"state": {"context": [], "goal": "∀ {x : Sort u_1} {cmp : x → x → Ordering} [inst : Batteries.TransCmp cmp] {x_1 y : x},\n  cmp x_1 y = Ordering.gt → cmp y x_1 ≠ Ordering.gt"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "inst✝² : DivisionRing K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "n : ℕ", "b : Basis (Fin n) K V", "i : Fin n"], "goal": "b i ∉ b.flag i.castSucc"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_4", "One M", "s : Set α", "f : β → α", "g : α → M", "x : β"], "goal": "(f ⁻¹' s).mulIndicator (g ∘ f) x = s.mulIndicator g (f x)"}}
+{"state": {"context": ["a : ℕ", "IH : ∀ m < a, ∀ {b : ℕ} {flip : Bool} {ha0 : m > 0}, b % 2 = 1 → b > 1 → fastJacobiSymAux m b flip ha0 = if flip = true then -J(↑m | b) else J(↑m | b)", "b : ℕ", "flip : Bool", "ha0 : a > 0", "hb2 : b % 2 = 1", "hb1 : b > 1", "ha4 : ¬a % 4 = 0", "ha2 : ¬a % 2 = 0", "ha1 : ¬a = 1", "hba : ¬b % a = 0"], "goal": "(if xor (decide (a % 4 = 3 ∧ b % 4 = 3)) flip = true then -J(↑b | a) else J(↑b | a)) = if flip = true then -if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a) else if a % 4 = 3 ∧ b % 4 = 3 then -J(↑b | a) else J(↑b | a)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "R : α → β → Prop", "s1 : WSeq α", "s2 : WSeq β", "h✝ : LiftRel R s1 s2", "s : WSeq β", "t : WSeq α", "h : LiftRel R t s"], "goal": "Computation.LiftRel (LiftRelO R (LiftRel R)) t.destruct s.destruct"}}
+{"state": {"context": ["G : Type u", "Group G", "TopologicalSpace G", "hmul : Filter.Tendsto (Function.uncurry fun x x_1 => x * x_1) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1)", "hinv : Filter.Tendsto (fun x => x⁻¹) (𝓝 1) (𝓝 1)", "hleft : ∀ (x₀ : G), 𝓝 x₀ = Filter.map (fun x => x₀ * x) (𝓝 1)", "hconj : ∀ (x₀ : G), Filter.Tendsto (fun x => x₀ * x * x₀⁻¹) (𝓝 1) (𝓝 1)"], "goal": "TopologicalGroup G"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "s : NonUnitalSubsemiring R", "x : R"], "goal": "x ∈ s.toAddSubmonoid ↔ x ∈ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : Option α ≃ Option β", "x : α", "h : e (some x) = none"], "goal": "some (e.removeNone_aux x) = e none"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "Ring R", "Invertible 2", "AddCommGroup V", "Module R V", "AddTorsor V P", "x y z : P"], "goal": "midpoint R x y = z ↔ (AffineEquiv.pointReflection R z) x = y"}}
+{"state": {"context": ["α : Type u", "s : Computation α", "a : α"], "goal": "a ∈ s → s.Promises a"}}
+{"state": {"context": [], "goal": "Path.Homotopy.transAssocReparamAux 1 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "f : α → β → δ", "l : Batteries.AssocList α β"], "goal": "(Batteries.AssocList.mapVal f l).toList =\n  List.map\n    (fun x =>\n      match x with\n      | (a, b) => (a, f a b))\n    l.toList"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "Disjoint s t ↔ ∀ ⦃a : α⦄, a ∈ t → a ∉ s"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Lattice α", "Lattice β", "f : LatticeHom α β"], "goal": "f.comp (LatticeHom.id α) = f"}}
+{"state": {"context": ["α : Type u_3", "DecidableEq α", "AddCommMonoid α", "m : Multiset α", "s : Finset α", "hs : m.toFinset ⊆ s"], "goal": "m.sum = ∑ i ∈ s, Multiset.count i m • i"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ"], "goal": "-cos b - -cos a = cos a - cos b"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "tensorObj : Mon_ C → Mon_ C → Mon_ C := fun M N => { X := M.X ⊗ N.X, one := (λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one), mul := tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul), one_mul := ⋯, mul_one := ⋯, mul_assoc := ⋯ }", "X₁ Y₁ X₂ Y₂ : Mon_ C", "f : X₁ ⟶ Y₁", "g : X₂ ⟶ Y₂"], "goal": "(tensor_μ C (X₁.X, X₂.X) (X₁.X, X₂.X) ≫ (X₁.mul ⊗ X₂.mul)) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ g.hom) ⊗ f.hom ⊗ g.hom) ≫ tensor_μ C (Y₁.X, Y₂.X) (Y₁.X, Y₂.X) ≫ (Y₁.mul ⊗ Y₂.mul)"}}
+{"state": {"context": ["ι : Type (max u_2 u_3)", "f : ι → Ordinal.{max u_2 u_3}", "c : Cardinal.{max u_2 u_3}", "hc : c.IsRegular", "hι : #ι < c"], "goal": "(∀ (i : ι), f i < c.ord) → Ordinal.sup f < c.ord"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : Seminorm 𝕜 E", "r : ℝ"], "goal": "Balanced 𝕜 (p.closedBall 0 r)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁸ : TopologicalSpace α", "inst✝⁷ : NormalSpace α", "inst✝⁶ : R1Space α", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : BorelSpace α", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "μ : Measure α", "p : ℝ≥0∞", "inst✝² : NormedSpace ℝ E", "inst✝¹ : WeaklyLocallyCompactSpace α", "inst✝ : μ.Regular", "f : α → E", "ε : ℝ", "hε : 0 < ε", "hf : Memℒp f (ENNReal.ofReal 1) μ"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal 1) μ"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.HasShift C ℤ", "T₁ T₂ : (CategoryTheory.Pretriangulated.Triangle C)ᵒᵖ", "φ : T₁ ⟶ T₂"], "goal": "((CategoryTheory.Pretriangulated.TriangleOpEquivalence.functor C).map φ).hom₃ = φ.unop.hom₁.op"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "inst✝⁵ : Ring R", "inst✝⁴ : Ring S", "M : Type u_4", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "m : Submodule R M", "N : Type u_5", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "A B : Submodule R M", "hAB : A ≤ B", "f : Submodule R ↥B ≃o ↑(Set.Iic B) := B.mapIic", "hf : f = B.mapIic"], "goal": "IsCoatom ⟨A, hAB⟩ ↔ IsCoatom (B.mapIic (Submodule.comap B.subtype A))"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "CommRing R", "AddCommGroup M₁", "AddCommGroup M₂", "Module R M₁", "Module R M₂", "Q₁ : QuadraticForm R M₁", "Q₂ : QuadraticForm R M₂"], "goal": "(CliffordAlgebra.toProd Q₁ Q₂).comp (CliffordAlgebra.ofProd Q₁ Q₂) =\n  AlgHom.id R (CliffordAlgebra (QuadraticMap.prod Q₁ Q₂))"}}
+{"state": {"context": ["F✝ : Type u_1", "X : Type u", "Y : Type v", "Z : Type w", "Z' : Type x", "ι : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "f₀ f₁ f₂ : C(X, Y)", "F : f₀.Homotopy f₁", "G : f₁.Homotopy f₂", "t : ↑I", "snd✝ : X"], "goal": "(if h : ↑(σ (t, snd✝).1, (t, snd✝).2).1 ≤ 1 / 2 then F (⟨2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1, ⋯⟩, (σ (t, snd✝).1, (t, snd✝).2).2) else G (⟨2 * ↑(σ (t, snd✝).1, (t, snd✝).2).1 - 1, ⋯⟩, (σ (t, snd✝).1, (t, snd✝).2).2)) = if h : ↑(t, snd✝).1 ≤ 1 / 2 then G.symm (⟨2 * ↑(t, snd✝).1, ⋯⟩, (t, snd✝).2) else F.symm (⟨2 * ↑(t, snd✝).1 - 1, ⋯⟩, (t, snd✝).2)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "P : Polynomial R"], "goal": "(KaehlerDifferential.D R (Polynomial R)) P =\n  Polynomial.derivative P • (KaehlerDifferential.D R (Polynomial R)) Polynomial.X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "σ : Type u_5", "inst✝⁴ : Primcodable α", "inst✝³ : Primcodable β", "inst✝² : Primcodable γ", "inst✝¹ : Primcodable δ", "inst✝ : Primcodable σ", "f : α → β", "g : α → ℕ × β → β", "hf : Primrec f", "hg : Primrec₂ g", "n : ℕ"], "goal": "Nat.unpaired (fun z n => Nat.casesOn n 0 fun y => Nat.unpaired (fun z n => Nat.rec (encode (Option.map f (decode z))) (fun y IH => encode (Option.map (fun p => g p.1 p.2) (decode (Nat.pair (Nat.unpair (Nat.pair z (Nat.pair y IH))).1 (Nat.pair (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).1 (Nat.unpair (Nat.unpair (Nat.pair z (Nat.pair y IH))).2).2.pred))))) n) (Nat.pair (Nat.unpair (Nat.pair z y)).2 (Nat.unpair (Nat.unpair (Nat.pair z y)).1).2)) (Nat.pair (id n) (encode (decode (Nat.unpair n).1))) = Nat.unpaired (fun m n => encode ((decode m).bind fun a => Option.map (fun n => Nat.rec (f a) (fun n IH => g a (n, IH)) n) (decode n))) n"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "a b✝ c✝ d✝ x y z : P", "r R : ℝ", "b c d : P", "hb : b ≠ d", "hc : c ≠ d"], "goal": "dist d c * dist b d ≤ dist d b * dist c d + dist b c * dist d d"}}
+{"state": {"context": ["R : Type u", "inst✝³ : CommRing R", "n G : Type v", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "α β : Type v", "inst✝ : DecidableEq α", "M : Matrix n n R", "c : n ≃ n", "hc : c ∈ univ.erase (Equiv.refl n)"], "goal": "(∏ i : n, M.charmatrix (c i) i).natDegree < Fintype.card n - 1"}}
+{"state": {"context": ["X Y : TopCat", "f g : C(↑X, ↑Y)", "H : f.Homotopy g", "x₀ x₁ : ↑X", "p : fromTop x₀ ⟶ fromTop x₁", "p' : Path (fromTop x₀).as (fromTop x₁).as"], "goal": "∀ (t : ↑I), g (p' t) = H.uliftMap (((Path.refl { down := 1 }).prod p') t)"}}
+{"state": {"context": ["o o' : Ordinal.{max u v}", "f : (a : Ordinal.{max u v}) → a < o → Ordinal.{max u v w}", "hf : ∀ {i j : Ordinal.{max u v}} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj", "g : (a : Ordinal.{max u v}) → a < o' → Ordinal.{max u v}", "hg : o'.blsub g = o"], "goal": "(o'.bsup fun a ha => f (g a ha) ⋯) = o.bsup f"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "n : ℕ", "Group G", "A : Rep k G", "f : (Fin n → G) → CoeSort.coe A", "x : Fin (n + 1) → G"], "goal": "((Rep.diagonalHomEquiv n A).symm f).hom (Finsupp.single x 1) = (A.ρ (x 0)) (f fun i => (x i.castSucc)⁻¹ * x i.succ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "o o' : Ordinal.{max u v}", "f : (a : Ordinal.{max u v}) → a < o → Ordinal.{max u v w}", "hf : ∀ {i j : Ordinal.{max u v}} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj", "g : (a : Ordinal.{max u v}) → a < o' → Ordinal.{max u v}", "hg : o'.blsub g = o", "i : Ordinal.{max u v}", "hi : i < o'"], "goal": "f (g i hi) ⋯ ≤ o.bsup f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "A B C : Set E", "hAC : IsExtreme 𝕜 A C", "hBA : B ⊆ A", "hCB : C ⊆ B"], "goal": "IsExtreme 𝕜 B C"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "MulOneClass M", "MulOneClass N", "s : Submonoid M", "t : Submonoid N", "u : Submonoid (M × N)"], "goal": "u ≤ s.prod t ↔ Submonoid.map (MonoidHom.fst M N) u ≤ s ∧ Submonoid.map (MonoidHom.snd M N) u ≤ t"}}
+{"state": {"context": ["a✝ b c a : Ordinal.{u}", "IH : ∀ k < a, k ♯ 1 = succ k", "i : Ordinal.{u}", "hi : i < a"], "goal": "i ♯ 1 < succ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "LinearOrderedAddCommGroup α", "OrderTopology α", "l : Filter β", "f g : β → α", "C : α", "hf : Filter.Tendsto f l (𝓝 C)", "hg : Filter.Tendsto g l Filter.atTop"], "goal": "Filter.Tendsto (fun x => f x + g x) l Filter.atTop"}}
+{"state": {"context": ["a : ℤ", "R : Type u_1", "inst✝ : CommSemiring R", "χ : R →* ℤ", "hp : ∀ (p : ℕ) (pp : Nat.Prime p), p ≠ 2 → legendreSym p a = χ ↑p", "b : ℕ", "hb : Odd b"], "goal": "List.pmap (fun p pp => legendreSym p a) b.primeFactorsList ⋯ = List.pmap (fun a x => (⇑χ ∘ Nat.cast) a) b.primeFactorsList ⋯"}}
+{"state": {"context": ["n : ℕ"], "goal": "(Fin.last n).succAbove = Fin.castSucc"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.exp x - Real.sinh x = Real.cosh x"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "LocalRing R", "x : R"], "goal": "(LocalRing.residue R) x ≠ 0 ↔ IsUnit x"}}
+{"state": {"context": ["n m : ℕ"], "goal": "m ∈ n.smoothNumbers ↔ ∀ (p : ℕ), Prime p → p ∣ m → p < n"}}
+{"state": {"context": ["B : Type u_1", "F : Type u_2", "inst✝¹ : TopologicalSpace B", "inst✝ : TopologicalSpace F"], "goal": "topologicalSpace B F = induced (Prod.fst ∘ ⇑(TotalSpace.toProd B F)) inst✝¹ ⊓ induced (Prod.snd ∘ ⇑(TotalSpace.toProd B F)) inst✝"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "Fintype m", "Fintype n", "NonUnitalNonAssocSemiring α", "u v : m → α", "x y : n → α"], "goal": "Sum.elim u x ⬝ᵥ Sum.elim v y = u ⬝ᵥ v + x ⬝ᵥ y"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝³ : Preorder ι", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f✝ g✝ : ι → Ω → E", "ℱ : Filtration ι m0", "f g : ι → Ω → ℝ", "hf : Submartingale f ℱ μ", "hg : Submartingale g ℱ μ", "i j : ι", "hij : i ≤ j"], "goal": "(f ⊔ g) i ≤ᶠ[ae μ] μ[(f ⊔ g) j|↑ℱ i]"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "f : ↥(lp E p)"], "goal": "f = 0 ↔ ↑f = 0"}}
+{"state": {"context": ["x y : ℝ"], "goal": "√x ≤ y ↔ 0 ≤ y ∧ x ≤ y ^ 2"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "φ : MvPolynomial σ R"], "goal": "↑φ = fun n => MvPolynomial.coeff n φ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : DecidableEq α", "inst✝ : GeneralizedBooleanAlgebra α", "s : Finset α"], "goal": "s.card ^ 2 ≤ (s \\\\ s).card ^ 2"}}
+{"state": {"context": ["R : Type uR", "A : Type uA", "B : Type uB", "C : Type uC", "D : Type uD", "CommSemiring R", "Semiring A", "Algebra R A", "Semiring B", "Algebra R B", "Semiring C", "Algebra R C", "Semiring D", "Algebra R D", "f : A →ₐ[R] B", "g : C →ₐ[R] D"], "goal": "(Algebra.TensorProduct.map f g).range =\n  (Algebra.TensorProduct.includeLeft.comp f).range ⊔ (Algebra.TensorProduct.includeRight.comp g).range"}}
+{"state": {"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α", "ι : Type u_1", "I : Set ι", "s : ι → Set α", "H : ∀ i ∈ I, IsOpen (s i)", "T : Set ↑I", "hTc : T.Countable", "hU : ⋃ i ∈ T, s ↑i = ⋃ i, s ↑i"], "goal": "∃ s_1, (Subtype.val '' s_1).Countable ∧ ⋃ y ∈ s_1, s ↑y = ⋃ i ∈ I, s i"}}
+{"state": {"context": ["x✝ x : ℝ", "h1 : 0 < x", "h2 : x < π / 2", "U : Set ℝ := Ico 0 (π / 2)", "intU : interior U = Ioo 0 (π / 2)", "half_pi_pos : 0 < π / 2"], "goal": "x < tan x"}}
+{"state": {"context": ["α : Type u_2", "M : Matroid α", "X I : Set α"], "goal": "M.Basis I (I ∪ X) ↔ M.Indep I ∧ X ⊆ M.closure I"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : Set α", "s.OrdConnected", "x y : ↑s"], "goal": "Subtype.val '' Set.Ico x y = Set.Ico ↑x ↑y"}}
+{"state": {"context": ["f : ℕ → NNReal", "hf : Summable f", "k : ℕ"], "goal": "Summable fun i => f (i + k)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "R : Type u_13", "Norm E", "SeminormedRing R", "f : α → E", "l : Filter α", "g : α → R", "c : R", "hc : IsUnit c", "h : f =o[l] g"], "goal": "f =o[l] fun x => c * g x"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)", "x : K", "h : IsIntegral ℤ x", "B : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ", "hint : IsIntegral ℤ B.gen", "this : FiniteDimensional ℚ K := finiteDimensional {p ^ k} ℚ K", "u : ℤˣ", "n : ℕ", "hun : Algebra.discr ℚ ⇑(IsPrimitiveRoot.subOnePowerBasis ℚ hζ).basis = ↑↑u * ↑↑p ^ n", "H : u.inv • (↑↑u * ↑↑p ^ n) • x ∈ adjoin ℤ {B.gen}"], "goal": "x ∈ adjoin ℤ {ζ}"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "inst✝² : Semiring α", "inst✝¹ : LinearOrder α", "a b c d : α", "inst✝ : PosMulStrictMono α", "h : ∀ (n : ℕ), 0 ≤ a * b ^ n", "this : 0 ≤ a"], "goal": "a = 0 ∨ 0 < a ∧ 0 ≤ b"}}
+{"state": {"context": ["R : Type u", "Ring R", "Nontrivial R", "a : R"], "goal": "(Polynomial.X - Polynomial.C a).degree = 1"}}
+{"state": {"context": ["G : Type u_3", "α : Type u_7", "Group G", "MulAction G α", "g : G", "a b : α"], "goal": "a = g⁻¹ • b ↔ g • a = b"}}
+{"state": {"context": ["R : Type u_1", "σ : Type u_2", "inst✝ : CommRing R", "r : R", "I : Ideal R", "f : MvPolynomial σ R"], "goal": "∀ (a : R), eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) ((Ideal.Quotient.mk (Ideal.map C I)) (C a))) = (Ideal.Quotient.mk (Ideal.map C I)) (C a)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedField α", "a✝ b c d : α", "n : ℤ", "a : α"], "goal": "a - a / 2 = a / 2"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "a b : V", "s : Set G.Subgraph"], "goal": "(sSup s).Adj a b ↔ ∃ G_1 ∈ s, G_1.Adj a b"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ", "n : ℕ", "hn : n ≠ 0"], "goal": "∑ i ∈ n.divisorsAntidiagonal, f i.1 * g i.2 / ↑n ^ s = ∑ p ∈ n.divisorsAntidiagonal, term f s p.1 * term g s p.2"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "x y : 𝕎 R", "h : ∀ (n : ℕ), x.coeff n = y.coeff n"], "goal": "x = y"}}
+{"state": {"context": ["ι : Sort uι", "R : Type u", "inst✝² : CommSemiring R", "A : Type v", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "S T : Set A", "M N P Q : Submodule R A", "m n : A"], "goal": "1 * M = M"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z", "hxy' : 0 < y - x", "hxz' : 0 < z - x"], "goal": "(f y - f x) / (y - x) ≤ (f z - f x) / (z - x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕜₃ : Type u_3", "E : Type u_4", "Eₗ : Type u_5", "F : Type u_6", "Fₗ : Type u_7", "G : Type u_8", "Gₗ : Type u_9", "𝓕 : Type u_10", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : NormedAddCommGroup F", "inst✝¹⁰ : NormedAddCommGroup G", "inst✝⁹ : NormedAddCommGroup Fₗ", "inst✝⁸ : NontriviallyNormedField 𝕜", "inst✝⁷ : NontriviallyNormedField 𝕜₂", "inst✝⁶ : NontriviallyNormedField 𝕜₃", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedSpace 𝕜₂ F", "inst✝³ : NormedSpace 𝕜₃ G", "inst✝² : NormedSpace 𝕜 Fₗ", "c : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f g : E →SL[σ₁₂] F", "x y z : E", "inst✝¹ : CompleteSpace F", "e : E →L[𝕜] Fₗ", "h_dense : DenseRange ⇑e", "N : ℝ≥0", "h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖", "inst✝ : RingHomIsometric σ₁₂"], "goal": "Continuous fun x => ‖(f.extend e h_dense ⋯) x‖"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "Group G", "H : Subgroup G", "hH : IsPGroup p ↥H", "K : Type u_2", "Group K", "ϕ : K →* G", "hϕ : Function.Injective ⇑ϕ"], "goal": "IsPGroup p ↥(Subgroup.comap ϕ H)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "A : Type u_3", "T : Type u_4", "B : Type u_5", "ι : Type u_6", "inst✝¹¹ : SemilatticeSup B", "inst✝¹⁰ : OrderBot B", "inst✝⁹ : SemilatticeInf T", "inst✝⁸ : OrderTop T", "inst✝⁷ : Semiring R", "inst✝⁶ : AddMonoid A", "inst✝⁵ : AddMonoid B", "inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝² : AddMonoid T", "inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "degb : A → B", "degt : A → T", "degt0 : 0 ≤ degt 0", "degtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)", "n : ℕ", "f : R[A]"], "goal": "n • f.support.inf degt ≤ (f ^ n).support.inf degt"}}
+{"state": {"context": ["α : Type u_2", "x y : α"], "goal": "Equiv.Perm.SameCycle 1 x y ↔ x = y"}}
+{"state": {"context": ["α : Type u_2", "InvolutiveInv α", "s : Set α"], "goal": "Inv.inv '' s = s⁻¹"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_8", "M₂ : Type u_10", "Semiring R", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂", "to_fun : M → M₂", "inv_fun : M₂ → M", "map_add : ∀ (x y : M), to_fun (x + y) = to_fun x + to_fun y", "map_smul :\n  ∀ (m : R) (x : M),\n    { toFun := to_fun, map_add' := map_add }.toFun (m • x) =\n      (RingHom.id R) m • { toFun := to_fun, map_add' := map_add }.toFun x", "left_inv : Function.LeftInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun", "right_inv : Function.RightInverse inv_fun { toFun := to_fun, map_add' := map_add, map_smul' := map_smul }.toFun"], "goal": "⇑{ toFun := to_fun, map_add' := map_add, map_smul' := map_smul, invFun := inv_fun, left_inv := left_inv,\n        right_inv := right_inv }.symm =\n  inv_fun"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasZeroMorphisms C", "S S₁ S₂ S₃ S₄ : ShortComplex C", "φ : S₁ ⟶ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "γ : HomologyMapData φ h₁ h₂"], "goal": "homologyMap' φ h₁ h₂ = (h₁.iso.hom ≫ (homologyMap' (opMap φ) h₂.op h₁.op).unop) ≫ h₂.iso.inv"}}
+{"state": {"context": ["α : Type u", "inst✝² : Group α", "inst✝¹ : LE α", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c : α"], "goal": "a⁻¹ * a ≤ 1 * a ↔ 1 ≤ a"}}
+{"state": {"context": ["n : Nat", "a b : Fin n"], "goal": "↑(a - b) = n + ↑a - ↑b ↔ a < b"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝ : PseudoEMetricSpace α", "δ✝ : ℝ", "s : Set α", "x : α", "δ : ℝ", "E : Set α"], "goal": "Eᶜᶜ ⊆ (thickening δ (thickening δ E)ᶜ)ᶜ"}}
+{"state": {"context": ["R : Type u_1", "MonoidWithZero R", "Nontrivial R"], "goal": "¬Squarefree 0"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R", "f : R[X]"], "goal": "f ∈ Ideal.map Polynomial.C I → (Polynomial.eval₂RingHom (Polynomial.C.comp (Ideal.Quotient.mk I)) Polynomial.X) f = 0"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Monoid α", "s t : Finset α", "hst : s ⊆ t", "n : ℕ"], "goal": "s ^ n ⊆ t ^ n"}}
+{"state": {"context": ["n✝ p n : ℕ"], "goal": "(p - 1) * ∑ i ∈ range (log p n).succ, n / p ^ i.succ = n - (p.digits n).sum"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "α : Type u_2", "inst✝ : Mul α", "J : Subgroup G", "g : G", "H K : Subgroup G", "a x : G"], "goal": "x ∈ ⋃ h, (↑h * a) • ↑K ↔ x ∈ doset a ↑H ↑K"}}
+{"state": {"context": ["x : ℝ", "hx : x ≠ 0", "r : ℂ", "hr : r ≠ -1"], "goal": "HasDerivAt (fun y => ↑y ^ (r + 1) / (r + 1)) (↑x ^ r) x"}}
+{"state": {"context": ["c : Code", "h : c.Ok", "v✝ v : List ℕ"], "goal": "Turing.eval step (Cfg.ret Cont.halt v) = pure (Cfg.halt v)"}}
+{"state": {"context": ["n : ℕ"], "goal": "(Fin.cycleRange 2).IsThreeCycle"}}
+{"state": {"context": ["Λ : ℂ → ℂ", "hf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s", "L : ℂ", "h_lim : Filter.Tendsto (fun s => s * Λ s) (𝓝[≠] 0) (𝓝 L)", "s : ℂ", "hs' : s ≠ 1"], "goal": "DifferentiableAt ℂ (Function.update (fun s => Λ s / s.Gammaℝ) 0 (L / 2)) s"}}
+{"state": {"context": ["α : Type u_3", "TopologicalSpace α", "s t : Set α"], "goal": "s ⊆ exterior t ↔ ∀ (U : Set α), IsOpen U → t ⊆ U → s ⊆ U"}}
+{"state": {"context": ["n x y : ℤ", "h : n = x ^ 2 + y ^ 2", "hc : IsCoprime x y", "u v : ℤ", "huv : u * x + v * n = 1", "H : u * y * (u * y) - -1 = n * (-v ^ 2 * n + u ^ 2 + 2 * v)"], "goal": "IsSquare (-1)"}}
+{"state": {"context": ["F : Type u", "K : Type v", "L : Type w", "inst✝⁷ : Field K", "inst✝⁶ : Field L", "inst✝⁵ : Field F", "inst✝⁴ : Algebra K L", "inst✝³ : Algebra F K", "inst✝² : Algebra F L", "inst✝¹ : IsScalarTower F K L", "f : K[X]", "inst✝ : IsSplittingField K L f", "h : Splits (RingHom.id K) f"], "goal": "↑(Multiset.map (⇑(algebraMap K L)) f.roots).toFinset ⊆ ↑⊥"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedSemifield α", "a b c d e : α", "m n : ℤ", "ha : 0 < a"], "goal": "a⁻¹ ≤ b ↔ 1 ≤ b * a"}}
+{"state": {"context": ["α : Type u_1", "self : Filter α"], "goal": "Set.univ ∈ self.sets"}}
+{"state": {"context": ["N : Type u_2", "r : N → N → Prop", "AddGroup N"], "goal": "Covariant N N (Function.swap fun x x_1 => x + x_1) r ↔ Contravariant N N (Function.swap fun x x_1 => x + x_1) r"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "x y : M", "c : ↥S", "h : f.toMap (↑c * x) = f.toMap (↑c * y)"], "goal": "f.toMap x = f.toMap y"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "S₁ : Type u_3", "R₂ : Type u_4", "S₂ : Type u_5", "M : Type u_6", "M₁ : Type u_7", "M₂ : Type u_8", "M₁' : Type u_9", "M₂' : Type u_10", "N₂ : Type u_11", "n : Type u_12", "m : Type u_13", "n' : Type u_14", "m' : Type u_15", "ι : Type u_16", "inst✝¹⁸ : CommSemiring R", "inst✝¹⁷ : AddCommMonoid M₁", "inst✝¹⁶ : Module R M₁", "inst✝¹⁵ : AddCommMonoid M₂", "inst✝¹⁴ : Module R M₂", "inst✝¹³ : AddCommMonoid N₂", "inst✝¹² : Module R N₂", "inst✝¹¹ : DecidableEq n", "inst✝¹⁰ : Fintype n", "inst✝⁹ : DecidableEq m", "inst✝⁸ : Fintype m", "b₁✝ : Basis n R M₁", "b₂✝ : Basis m R M₂", "b₁ : Basis n R M₁", "b₂ : Basis m R M₂", "inst✝⁷ : AddCommMonoid M₁'", "inst✝⁶ : Module R M₁'", "inst✝⁵ : AddCommMonoid M₂'", "inst✝⁴ : Module R M₂'", "b₁' : Basis n' R M₁'", "b₂' : Basis m' R M₂'", "inst✝³ : Fintype n'", "inst✝² : Fintype m'", "inst✝¹ : DecidableEq n'", "inst✝ : DecidableEq m'", "B : M₁ →ₗ[R] M₂ →ₗ[R] R", "f : M₁' →ₗ[R] M₁"], "goal": "((toMatrix b₁' b₁) f)ᵀ * (toMatrix₂ b₁ b₂) B * (toMatrix b₂ b₂) id = ((toMatrix b₁' b₁) f)ᵀ * (toMatrix₂ b₁ b₂) B"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "e₀ : E", "f₀ : F"], "goal": "HasFDerivAt (fun e => (e, f₀)) (ContinuousLinearMap.inl 𝕜 E F) e₀"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_7", "Semiring R", "f : α → R", "r : R", "h : (Function.support f).Finite"], "goal": "(∑ᶠ (a : α), f a) * r = ∑ᶠ (a : α), f a * r"}}
+{"state": {"context": ["z : ℤ", "hz : z ≠ 0"], "goal": "|z.sign| = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "f : Filter α", "g : Filter β"], "goal": "Filter.map₂ Prod.mk f g = f ×ˢ g"}}
+{"state": {"context": [], "goal": "USize.toNat 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoid α", "s : Submonoid α", "a₁ b₁ : α", "a₂ b₂ : ↥s"], "goal": "Localization.mk a₁ a₂ = Localization.mk b₁ b₂ ↔ ↑b₂ * a₁ = ↑a₂ * b₁"}}
+{"state": {"context": ["α : Type u_1", "t t₁ t₂ : TopologicalSpace α", "s : Set α", "inst✝ : TopologicalSpace α"], "goal": "DiscreteTopology α ↔ ∀ (x : α), 𝓝[≠] x = ⊥"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "c c' : Cardinal.{u_3}", "h : c.ord = c'.ord"], "goal": "c = c'"}}
+{"state": {"context": ["E : Type u_3", "SeminormedAddGroup E", "ι : Sort u_6", "p : ι → Prop", "s : ι → Set ℝ", "h : Filter.atTop.HasBasis p s"], "goal": "(Bornology.cobounded E).HasBasis p fun i => norm ⁻¹' s i"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "t : Batteries.RBNode α"], "goal": "Batteries.RBNode.Ordered cmp t.setBlack ↔ Batteries.RBNode.Ordered cmp t"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "R : Type u_2", "inst✝⁶ : Semiring k", "inst✝⁵ : DistribSMul R k", "inst✝⁴ : Mul G", "A : Type u₃", "inst✝³ : NonUnitalNonAssocSemiring A", "inst✝² : Module k A", "inst✝¹ : IsScalarTower k A A", "inst✝ : SMulCommClass k A A", "f : G →ₙ* A", "a₁ a₂ : MonoidAlgebra k G", "g : G → k → A := fun m t => t • f m", "h₁ : ∀ (m : G), g m 0 = 0", "h₂ : ∀ (m : G) (t₁ t₂ : k), g m (t₁ + t₂) = g m t₁ + g m t₂"], "goal": "(sum (sum a₁ fun a₁ b₁ => sum a₂ fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)) fun m t => t • f m) = sum a₁ fun x' v' => sum a₂ fun x v => (v' * v) • f (x' * x)"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "l : BoxIntegral.IntegrationParams", "I : BoxIntegral.Box ι", "π₀ : BoxIntegral.Prepartition I"], "goal": "(l.toFilteriUnion I π₀).HasBasis (fun r => ∀ (c : ℝ≥0), l.RCond (r c)) fun r =>\n  {π | ∃ c, l.MemBaseSet I c (r c) π ∧ π.iUnion = π₀.iUnion}"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝¹ : CommSemiring R", "p q : MvPolynomial σ R", "inst✝ : Nontrivial R"], "goal": "(X n).vars = {n}"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "CommSemiring A", "Algebra R A", "S : Subalgebra R A", "x : ↥S", "p : R[X]"], "goal": "(Polynomial.aeval ↑x) p = ↑((Polynomial.aeval x) p)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "R : Type u_5", "inst✝¹² : MeasurableSpace X", "inst✝¹¹ : TopologicalSpace X", "inst✝¹⁰ : MeasurableSpace Y", "inst✝⁹ : TopologicalSpace Y", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "f g : X → E", "μ : Measure X", "inst✝⁶ : BorelSpace X", "inst✝⁵ : ConditionallyCompleteLinearOrder X", "inst✝⁴ : ConditionallyCompleteLinearOrder E", "inst✝³ : OrderTopology X", "inst✝² : OrderTopology E", "inst✝¹ : SecondCountableTopology E", "inst✝ : IsFiniteMeasureOnCompacts μ", "hs : IsCompact ∅", "hmono : MonotoneOn f ∅"], "goal": "IntegrableOn f ∅ μ"}}
+{"state": {"context": ["α : Type u_2", "BooleanAlgebra α", "s : Finset α", "a : α"], "goal": "a ∈ s → aᶜ ∈ s.compls"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "hs : s.Finite := by toFinite_tac"], "goal": "s.ncard ≤ 1 ↔ ∀ a ∈ s, ∀ b ∈ s, a = b"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x✝ y x : V"], "goal": "arccos (⟪x, 0⟫_ℝ / (‖x‖ * ‖0‖)) = π / 2"}}
+{"state": {"context": ["Ω : Type u_1", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n : ℕ", "ω : Ω"], "goal": "0 ≤ MeasureTheory.upcrossingStrat a b f N n ω"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.cos x = -1 ↔ ∃ k, Real.pi + ↑k * (2 * Real.pi) = x"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "x : E"], "goal": "Filter.Tendsto (fun a => ‖a‖) (𝓝 x) (𝓝 ‖x‖)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_5", "M₂ : Type u_7", "CommRing R", "AddCommGroup M", "Module R M", "AddCommGroup M₂", "Module R M₂", "B : M →ₗ[R] M →ₗ[R] M₂", "f : Module.End R M"], "goal": "B.IsSkewAdjoint f ↔ (-B).IsAdjointPair B f f"}}
+{"state": {"context": ["G : Type u_1", "MeasurableSpace G", "Group G", "MeasurableMul G", "g : G"], "goal": "(MeasurableEquiv.mulRight g).toEquiv = Equiv.mulRight g"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I : Set α", "M.FiniteRk", "hI : M.Indep I"], "goal": "I.Finite"}}
+{"state": {"context": ["A : Type u_3", "Ring A", "Algebra ℝ A", "a : A", "t : ℝ≥0", "ht : spectralRadius ℝ a ≤ ↑t"], "goal": "SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I : Ideal R", "hqf : IsField (R ⧸ I)"], "goal": "I.IsMaximal"}}
+{"state": {"context": ["Ω : Type u_1", "Ω' : Type u_2", "α : Type u_3", "m : MeasurableSpace Ω", "m' : MeasurableSpace Ω'", "μ : Measure Ω", "s t : Set Ω", "hms : MeasurableSet s", "hmt : MeasurableSet t", "hcs : μ s ≠ ⊤", "u : Set Ω", "a✝ : MeasurableSet u", "hst : μ (s ∩ t) = 0"], "goal": "((μ s)⁻¹ * μ (s ∩ t))⁻¹ * ((μ s)⁻¹ * μ (s ∩ (t ∩ u))) = (μ (s ∩ t))⁻¹ * μ (s ∩ t ∩ u)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "LieAlgebra.IsNilpotent R L"], "goal": "∃ k, ∀ (x : L), (LieAlgebra.ad R L) x ^ k = 0"}}
+{"state": {"context": ["M : Type u_3", "TopologicalSpace M", "Monoid M", "ContinuousMul M", "s : Submonoid M"], "goal": "IsClosed ↑s.topologicalClosure"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝³ : DivisionRing k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "p₁ p₂ p₃ p₄ p₅ : P", "h₁ : p₁ ∈ affineSpan k {p₄, p₅}", "h₂ : p₂ ∈ affineSpan k {p₄, p₅}", "h₃ : p₃ ∈ affineSpan k {p₄, p₅}"], "goal": "Collinear k {p₁, p₂, p₃}"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "f : S₁ ⟶ S₂", "CategoryTheory.IsIso f.τ₁", "CategoryTheory.IsIso f.τ₂", "CategoryTheory.IsIso f.τ₃"], "goal": "CategoryTheory.IsIso f"}}
+{"state": {"context": ["R : Type u", "n : ℕ", "Semiring R", "p : R[X]", "pn : p.degree ≤ ↑n", "p1 : p.coeff n ≠ 0"], "goal": "p.degree = ↑n"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "q : ℚ", "v_eq_q : v = ↑q", "n : ℕ"], "goal": "Option.map (Pair.map Rat.cast) (Option.map (fun p => { a := 1, b := ↑p.b }) (IntFractPair.stream q (n + 1))) = Option.map (fun p => { a := 1, b := ↑p.b }) (IntFractPair.stream v (n + 1))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2"], "goal": "∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace β}, Measurable Prod.snd"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "p : ℝ", "hp : 1 ≤ p", "h₁ : 0 < p", "h₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i : ι, |x i| ^ p", "eq_norm : ∀ (x : ι → ℝ), ‖x‖ = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)", "this : Fact (1 ≤ ENNReal.ofReal p)", "eq_zero : ∀ (x : ι → ℝ), (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p) = 0 ↔ x = 0", "nm_zero : (∑ x : ι, |0 x| ^ p) ^ (1 / p) = 0", "nm_neg : ∀ (x : ι → ℝ), (∑ x_1 : ι, |(-x) x_1| ^ p) ^ (1 / p) = (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)", "nm_add : ∀ (x y : ι → ℝ), (∑ x_1 : ι, |(x + y) x_1| ^ p) ^ (1 / p) ≤ (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p) + (∑ x : ι, |y x| ^ p) ^ (1 / p)", "nm_smul : ∀ (r : ℝ) (x : ι → ℝ), (∑ x_1 : ι, |(r • x) x_1| ^ p) ^ (1 / p) ≤ |r| * (∑ x_1 : ι, |x x_1| ^ p) ^ (1 / p)"], "goal": "volume {x | ∑ i : ι, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (↑(card ι) / p + 1))"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "E : Type u_5", "NormedAddCommGroup E", "p : ℝ≥0∞", "s : Set α", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "c : E"], "goal": "MeasureTheory.Integrable (↑↑(MeasureTheory.indicatorConstLp p hs hμs c)) μ"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝³ : DivisionRing k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "s t : Finset V", "hs : AffineIndependent k Subtype.val", "hst : ↑s ⊆ ↑(affineSpan k ↑t)", "hs' : s.Nonempty", "ht' : t.Nonempty", "this✝ : Nonempty { x // x ∈ s }", "this : Nonempty ↑↑t", "direction_le : (affineSpan k ↑s).direction ≤ (affineSpan k ↑(affineSpan k ↑t)).direction"], "goal": "s.card ≤ t.card"}}
+{"state": {"context": ["n : Nat", "i : Fin n"], "goal": "i.castSucc < i.succ"}}
+{"state": {"context": ["K : Type u_1", "RCLike K"], "goal": "RCLike.normSq 0 = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : α → β → Prop", "C : Computation α → Computation β → Prop", "ca : Computation α", "cb : Computation β"], "goal": "Computation.LiftRelAux R C (Sum.inr ca) (Sum.inr cb) = C ca cb"}}
+{"state": {"context": ["Ω : Type u_1", "Nonempty Ω", "m0 : MeasurableSpace Ω", "μ : MeasureTheory.FiniteMeasure Ω", "s : Set Ω"], "goal": "μ s = μ.mass * μ.normalize s"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "a₁ a₂ b₁ b₂ : α", "ha : a₁ ∈ Set.Icc a₂ b₂", "hb : b₁ ∈ Set.Icc a₂ b₂"], "goal": "Set.uIcc a₁ b₁ ⊆ Set.Icc a₂ b₂"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β✝ : Type u_3", "inst✝¹ : LinearOrderedSemifield α", "a b c✝ d e : α", "m n : ℤ", "β : Type u_4", "inst✝ : Preorder β", "f : β → α", "hf : StrictMono f", "c : α", "hc : 0 < c"], "goal": "StrictMono fun x => f x / c"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "M : Type u_3", "a✝ b✝ : R", "s : S", "inst✝¹ : Monoid R", "inst✝ : MulAction R M", "a b : M", "ab : a = b"], "goal": "a = b"}}
+{"state": {"context": ["p : Prop"], "goal": "∀ {x : Decidable p}, decide p = false → p = False"}}
+{"state": {"context": ["z : ℂ", "hz : 0 < z.re", "hz₀ : z ≠ 0"], "goal": "Tendsto (fun x => ‖x.1 ^ x.2‖) (𝓝 (0, z)) (𝓝 0)"}}
+{"state": {"context": ["f : ℝ → ℝ", "x : ℝ", "n : ℕ", "hf_conv : ConvexOn ℝ (Ioi 0) f", "hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y", "hn : 2 ≤ n", "hx : 0 < x"], "goal": "f ↑n + x * log (↑n - 1) ≤ f (↑n + x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "f : ℕ → α", "m n : ℕ", "h : m ≤ n"], "goal": "dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))"}}
+{"state": {"context": ["α : Sort u", "r : α → α → Prop", "self : Equivalence r", "x : α"], "goal": "r x x"}}
+{"state": {"context": ["F : Type u", "K✝¹ : Type v", "L : Type w", "inst✝³ : Field K✝¹", "inst✝² : Field L", "inst✝¹ : Field F", "n✝ : ℕ", "K✝ : Type u", "inst✝ : Field K✝", "n : ℕ", "ih : (fun n => ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤) n", "K : Type u", "x✝ : Field K", "f : K[X]", "hfn : f.natDegree = n.succ", "hndf : f.natDegree ≠ 0", "hfn0 : f ≠ 0", "hmf0 : map (algebraMap K (SplittingFieldAux n.succ f)) f ≠ 0"], "goal": "Algebra.adjoin K (f.rootSet (SplittingFieldAux n.succ f)) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : α → β", "hp_top : p ≠ ⊤", "ε : ℝ", "hε : 0 < ε", "M : ℝ", "hf : ∀ (x : α), ‖f x‖ < M", "hM : 0 < M", "s : Set α", "hs : MeasurableSet s", "hμ : μ s ≤ ENNReal.ofReal ((ε / M) ^ p.toReal)"], "goal": "eLpNorm (s.indicator f) p μ ≤ ENNReal.ofReal ε"}}
+{"state": {"context": ["α : Type u_1", "V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P", "l : Filter α", "f g : α → P", "x y : P", "hf : Filter.Tendsto f l (𝓝 x)", "hg : Filter.Tendsto g l (𝓝 y)"], "goal": "Filter.Tendsto (f -ᵥ g) l (𝓝 (x -ᵥ y))"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "f : M ≃ₗ[R] M"], "goal": "↑(LinearEquiv.det f)⁻¹ = LinearMap.det ↑f.symm"}}
+{"state": {"context": ["f✝ f g : ℝ → ℝ", "hf : GrowsPolynomially f", "hg : GrowsPolynomially g", "this : (fun x => f x / g x) = fun x => f x * (g x)⁻¹"], "goal": "GrowsPolynomially fun x => f x / g x"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁶ : Category.{v₁, u₁} C", "A : Type u₂", "inst✝⁵ : Category.{v₂, u₂} A", "A' : Type u₂", "inst✝⁴ : Category.{max v₁ u₁, u₂} A'", "B : Type u₃", "inst✝³ : Category.{v₃, u₃} B", "J : GrothendieckTopology C", "U : C", "R : Presieve U", "P : Cᵒᵖ ⥤ A", "P' : Cᵒᵖ ⥤ A'", "s : A' ⥤ Type (max v₁ u₁)", "inst✝² : HasLimits A'", "inst✝¹ : PreservesLimits s", "inst✝ : s.ReflectsIsomorphisms", "this✝ : HasLimitsOfSize.{v₁, max v₁ u₁, max u₁ v₁, u₂} A'", "this : PreservesLimitsOfSize.{v₁, max v₁ u₁, max u₁ v₁, max u₁ v₁, u₂, max (u₁ + 1) (v₁ + 1)} s"], "goal": "IsSheaf J P' ↔ IsSheaf J (P' ⋙ s)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "S : Type v", "CommRing S", "I : Ideal R", "J : Ideal S", "f : R ≃+* S", "hIJ : J = Ideal.map (↑f) I", "a : S ⧸ J"], "goal": "(I.quotientEquiv J f hIJ).symm a = (Ideal.quotientMap I ↑f.symm ⋯) a"}}
+{"state": {"context": ["R : Type u₁", "L : Type u₂", "CommRing R", "LieRing L", "LieAlgebra R L", "A : Type u₃", "Ring A", "Algebra R A", "f : L →ₗ⁅R⁆ A"], "goal": "⇑((UniversalEnvelopingAlgebra.lift R) f) ∘ ⇑(UniversalEnvelopingAlgebra.ι R) = ⇑f"}}
+{"state": {"context": ["α : Type u_2", "s : Set α", "Preorder α", "InfSet α", "Inhabited ↑s"], "goal": "sInf = fun t =>\n  if ht : t.Nonempty ∧ BddBelow t ∧ sInf (Subtype.val '' t) ∈ s then ⟨sInf (Subtype.val '' t), ⋯⟩ else default"}}
+{"state": {"context": ["a : Bool", "x : ℕ"], "goal": "Nat.bit a x % 2 = 1 ↔ a = true"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "𝕜 : Type u_3", "inst✝⁴ : LinearOrderedField 𝕜", "G H : SimpleGraph α", "ε δ : 𝕜", "n : ℕ", "s : Finset α", "inst✝³ : Fintype α", "inst✝² : DecidableRel G.Adj", "inst✝¹ : DecidableRel H.Adj", "inst✝ : DecidableEq α", "tris : Finset (Finset α)", "htris : tris ⊆ G.cliqueFinset 3", "pd : (↑tris).Pairwise fun x y => (↑x ∩ ↑y).Subsingleton", "hHG : H ≤ G", "hH : H.CliqueFree 3", "hG : (G.edgeFinset \\ H.edgeFinset).card < tris.attach.card", "fx fy : ⦃t : Finset α⦄ → t ∈ tris → α", "hfx : ∀ ⦃t : Finset α⦄ (ht : t ∈ tris), fx ht ∈ t", "hfy : ∀ ⦃t : Finset α⦄ (ht : t ∈ tris), fy ht ∈ t", "hfne : ∀ ⦃t : Finset α⦄ (ht : t ∈ tris), fx ht ≠ fy ht", "fmem : ∀ ⦃t : Finset α⦄ (ht : t ∈ tris), s(fx ht, fy ht) ∈ G.edgeFinset \\ H.edgeFinset", "f : { x // x ∈ tris } → Sym2 α := fun t => s(fx ⋯, fy ⋯)", "hf : ∀ x ∈ tris.attach, f x ∈ G.edgeFinset \\ H.edgeFinset", "t₁ : Finset α", "ht₁ : t₁ ∈ tris", "t₂ : Finset α", "ht₂ : t₂ ∈ tris", "tne : ⟨t₁, ht₁⟩ ≠ ⟨t₂, ht₂⟩", "t : fx ⋯ = fx ⋯ ∧ fy ⋯ = fy ⋯ ∨ fx ⋯ = fy ⋯ ∧ fy ⋯ = fx ⋯", "i : (↑t₁ ∩ ↑t₂).Subsingleton"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "l : List α"], "goal": "[] <:+ l"}}
+{"state": {"context": ["α : Type u_5", "LinearOrderedSemiring α", "a b c : α", "ha : 0 ≤ a", "hb : b < 0", "hc : 0 ≤ c"], "goal": "a ≤ b * c ↔ a = 0 ∧ c = 0"}}
+{"state": {"context": ["K : Type u", "Field K", "A B : ValuationSubring K"], "goal": "B.principalUnitGroup ≤ A.principalUnitGroup ↔ A ≤ B"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f g : α → ℝ"], "goal": "∫ (x : α), rexp (g x) ∂μ.tilted f = (∫ (x : α), rexp ((f + g) x) ∂μ) / ∫ (x : α), rexp (f x) ∂μ"}}
+{"state": {"context": ["x y : ℂ"], "goal": "sin (x - y) = sin x * cos y - cos x * sin y"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : GlueData C", "inst✝ : HasLimits C", "i j : D.J", "U : Opens ↑↑(D.U i)"], "goal": "↑((Opens.map (D.ι j).base).obj (⋯.functor.obj U)) = ↑((opensFunctor (D.f j i)).obj ((Opens.map (D.t j i).base).obj ((Opens.map (D.f i j).base).obj U)))"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "D : Type u'", "inst✝³ : Category.{v', u'} D", "J : Type u₁", "inst✝² : Category.{v₁, u₁} J", "K : Type u₂", "inst✝¹ : Category.{v₂, u₂} K", "inst✝ : HasLimitsOfShape J C", "i j : K", "F : J ⥤ K ⥤ C", "f : i ⟶ j"], "goal": "(limitObjIsoLimitCompEvaluation F i).inv ≫ (limit F).map f = limMap (whiskerLeft F ((evaluation K C).map f)) ≫ (limitObjIsoLimitCompEvaluation F j).inv"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : MeasurableSpace α'", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace β'", "inst✝³ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : SFinite ν", "f : α → β → E", "hf : StronglyMeasurable (uncurry f)", "hE : CompleteSpace E", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "this : SeparableSpace ↑(range (uncurry f) ∪ {0})", "s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯", "s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) ⋯", "f' : ℕ → α → E := fun n => {x | Integrable (f x) ν}.indicator fun x => SimpleFunc.integral ν (s' n x)"], "goal": "StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "n : ℕ", "Aₙ : ℚ", "nth_num_eq : (of v).nums n = ↑Aₙ"], "goal": "∃ q, (of v).convs n = ↑q"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeSup α", "c : ClosureOperator α", "x y : α"], "goal": "c (x ⊔ c y) = c (x ⊔ y)"}}
+{"state": {"context": ["f : StieltjesFunction", "s : Set ℝ", "t : ℕ → Set ℝ", "ht : s ⊆ ⋃ i, t i", "ε : ℝ≥0", "ε0 : 0 < ε", "h : ∑' (i : ℕ), f.length (t i) < ⊤", "ε' : ℕ → ℝ≥0", "ε'0 : ∀ (i : ℕ), 0 < ε' i", "hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε", "g : ℕ → Set ℝ", "hg : ∀ (i : ℕ), t i ⊆ g i ∧ MeasurableSet (g i) ∧ f.outer (g i) ≤ f.length (t i) + ↑(ε' i)"], "goal": "f.outer (iUnion g) ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))"}}
+{"state": {"context": ["A B : AddCommGrp", "f : A ⟶ B", "CategoryTheory.Epi f"], "goal": "AddMonoidHom.range f = ⊤"}}
+{"state": {"context": ["V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "X₀ X₁ X₂ : V", "d₀ : X₁ ⟶ X₀", "d₁ : X₂ ⟶ X₁", "s : d₁ ≫ d₀ = 0", "succ : (S : CategoryTheory.ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0"], "goal": "(ChainComplex.mk X₀ X₁ X₂ d₀ d₁ s succ).X 0 = X₀"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : MetricSpace X", "inst✝ : CompleteSpace X", "x : X", "ε : ℝ", "ε_pos : 0 < ε", "ϕ : X → ℝ", "cont : Continuous ϕ", "nonneg : ∀ (y : X), 0 ≤ ϕ y", "H : ¬∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'"], "goal": "False"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "s t : Set K"], "goal": "Subfield.closure (s ∪ t) = Subfield.closure s ⊔ Subfield.closure t"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "Z X Y P : C", "f : Z ⟶ X", "g : Z ⟶ Y", "inl : X ⟶ P", "inr : Y ⟶ P", "h : CategoryTheory.IsPushout f g inl inr", "Z' X' Y' P' : C", "f' : Z' ⟶ X'", "g' : Z' ⟶ Y'", "inl' : X' ⟶ P'", "inr' : Y' ⟶ P'", "e₁ : Z ≅ Z'", "e₂ : X ≅ X'", "e₃ : Y ≅ Y'", "e₄ : P ≅ P'", "commf : f ≫ e₂.hom = e₁.hom ≫ f'", "commg : g ≫ e₃.hom = e₁.hom ≫ g'", "comminl : inl ≫ e₄.hom = e₂.hom ≫ inl'", "comminr : inr ≫ e₄.hom = e₃.hom ≫ inr'"], "goal": "CategoryTheory.IsPushout f' g' inl' inr'"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "x : M", "n : ℕ∞"], "goal": "ContMDiffOn I I n (↑(chartAt H x).symm) (chartAt H x).target"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "w x y z : α", "inst✝ : GeneralizedBooleanAlgebra α"], "goal": "(x ⊓ y ⊔ x \\ y) ⊓ y \\ x = x ⊓ y ⊓ y \\ x ⊔ x \\ y ⊓ y \\ x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e e' : PartialEquiv α β", "h : e ≈ e'"], "goal": "Set.EqOn (↑e.symm) (↑e'.symm) e.target"}}
+{"state": {"context": ["α : Type u", "m' : Type uₘ", "n' : Type uₙ", "o' : Type uₒ", "A : Matrix m' n' α", "row : Fin 0 → m'", "col : o' → n'"], "goal": "A.submatrix row col = ![]"}}
+{"state": {"context": ["R✝ : Type u_1", "A✝ : Type u_2", "inst✝¹¹ : CommSemiring R✝", "inst✝¹⁰ : NonUnitalRing A✝", "inst✝⁹ : Module R✝ A✝", "inst✝⁸ : IsScalarTower R✝ A✝ A✝", "inst✝⁷ : SMulCommClass R✝ A✝ A✝", "R : Type u_3", "S : Type u_4", "A : Type u_5", "inst✝⁶ : Semifield R", "inst✝⁵ : Semifield S", "inst✝⁴ : Ring A", "inst✝³ : Algebra R S", "inst✝² : Algebra R A", "inst✝¹ : Algebra S A", "a✝ : A", "f✝ : S → R", "inst✝ : IsScalarTower R S A", "h✝ : SpectrumRestricts a✝ f✝", "a b : A", "f : S → R", "ha : SpectrumRestricts a f", "h : spectrum S a = spectrum S b"], "goal": "Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S b)"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s : UpperSet α"], "goal": "s.compl.compl = s"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{?u.26563, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : HasFiniteCoproducts C", "X : SimplicialObject C"], "goal": "∀ (n : ℕ), (Γ₀.splitting K[X]).φ { app := fun Δ => (Γ₀.splitting K[X]).desc Δ fun A => PInfty.f (unop A.fst).len ≫ X.map A.e.op, naturality := ⋯ } n = (Γ₀.splitting K[X]).φ (((N₁ ⋙ Γ₂).obj X).p ≫ { app := fun Δ => (Γ₀.splitting K[X]).desc Δ fun A => PInfty.f (unop A.fst).len ≫ X.map A.e.op, naturality := ⋯ } ≫ ((toKaroubi (SimplicialObject C)).obj X).p) n"}}
+{"state": {"context": ["H : Type u_5", "NormedAddCommGroup H", "InnerProductSpace ℝ H", "s t : Set H"], "goal": "(s ∪ t).innerDualCone = s.innerDualCone ⊓ t.innerDualCone"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s : Set α", "a : α", "hr : Symmetric r"], "goal": "(insert a s).Pairwise r ↔ s.Pairwise r ∧ ∀ b ∈ s, a ≠ b → r a b"}}
+{"state": {"context": ["n✝ n : ℕ"], "goal": "∃ y ∈ {p | Prime p}, n < y"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : GCDMonoid α", "a b c : α", "hab : IsUnit (gcd b a)", "k : ℕ", "h✝ : Nontrivial α", "ha : ¬a = 0", "hb : ¬b = 0", "hk : k > 0", "hc✝ : c ∣ a * b", "d₁ d₂ : α", "hd₁ : d₁ ∣ a", "hd₂ : d₂ ∣ b", "hc : c = d₂ * d₁", "h0₁ : d₁ ^ k ≠ 0", "a' : α", "ha' : a = d₁ ^ k * a'", "h0₂ : d₂ ^ k ≠ 0", "b' : α", "h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k", "hb' : b = d₂ ^ k * b'", "h' : a' * b' = 1"], "goal": "d₁ ^ k * ↑(Units.mkOfMulEqOne a' b' h') = a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "x y z : α", "δ ε ε₁ ε₂ : ℝ", "s✝ : Set α", "F : ι → β → α", "f : β → α", "p : Filter ι", "s : Set β"], "goal": "TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, dist (f x) (F n x) < ε"}}
+{"state": {"context": ["α : Type u_1", "Γ₀ : Type u_2", "LinearOrderedCommGroupWithZero Γ₀", "l : Filter α", "f : α → Γ₀"], "goal": "Filter.Tendsto f l (𝓝 0) ↔ ∀ (γ₀ : Γ₀), γ₀ ≠ 0 → ∀ᶠ (x : α) in l, f x < γ₀"}}
+{"state": {"context": ["X : Type u_2", "Y : Type u_3", "Z : Type u_4", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "LocallyCompactSpace Y", "f : C(X, C(Y, Z))"], "goal": "Continuous (Function.uncurry fun x y => (f x) y)"}}
+{"state": {"context": ["A : Type u", "inst✝¹ : AddCommGroup A", "inst✝ : DivisibleBy A ℤ", "m : ℤ", "g : ↥(Submodule.span ℤ {m}) →ₗ[ℤ] A", "h0 : m ≠ 0", "gₘ : A := g ⟨m, ⋯⟩", "n : ℤ", "hn : n • m ∈ Submodule.span ℤ {m}"], "goal": "(LinearMap.toSpanSingleton ℤ A (DivisibleBy.div gₘ m)) (n • m) = g ⟨n • m, hn⟩"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "m : ℤ"], "goal": "padicNorm p ↑m = 1 ↔ ¬↑p ∣ m"}}
+{"state": {"context": ["R : Type u_1", "K : Type v", "L : Type w", "CommRing R", "Field K", "Field L", "Algebra R K", "Algebra R L", "f : R[X]", "h : Polynomial.Splits (algebraMap R K) f", "e : K →ₐ[R] L"], "goal": "Polynomial.Splits (algebraMap R L) f"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "Fu Gu : Type u", "NormedAddCommGroup Fu", "NormedSpace 𝕜 Fu", "NormedAddCommGroup Gu", "NormedSpace 𝕜 Gu", "g : Fu → Gu", "f : E → Fu", "n : ℕ", "s : Set E", "t : Set Fu", "x : E", "hg : ContDiffOn 𝕜 (↑n) g t", "hf : ContDiffOn 𝕜 (↑n) f s", "ht : UniqueDiffOn 𝕜 t", "hs : UniqueDiffOn 𝕜 s", "hst : Set.MapsTo f s t", "hx : x ∈ s", "C D : ℝ", "hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C", "hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i"], "goal": "‖iteratedFDerivWithin 𝕜 n (g ∘ f) s x‖ ≤ ↑n ! * C * D ^ n"}}
+{"state": {"context": ["n : ℕ"], "goal": "∃ k, Xor' (n = 2 * k) (n = 2 * k + 1)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "a : α"], "goal": "𝓝[Set.univ] a = 𝓝 a"}}
+{"state": {"context": ["x✝ x : ℝ", "h1 : 0 < x", "h2 : x < π / 2", "U : Set ℝ := Ico 0 (π / 2)", "intU : interior U = Ioo 0 (π / 2)", "half_pi_pos : 0 < π / 2", "cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y", "sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y", "tan_cts_U : ContinuousOn tan U", "tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U", "deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y", "mono : StrictMonoOn (fun y => tan y - y) (Ico 0 (π / 2))", "zero_in_U : 0 ∈ U"], "goal": "x < tan x"}}
+{"state": {"context": ["α : Sort u_1", "p : α → Prop", "P Q : (x : α) → p x → Prop"], "goal": "(∀ (x : α) (h : p x), P x h ∧ Q x h) ↔ (∀ (x : α) (h : p x), P x h) ∧ ∀ (x : α) (h : p x), Q x h"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "A : Type u_3", "CommSemiring R", "Semiring A", "Algebra R A", "AddCommMonoid M", "Module R M", "p : Submodule R M", "a : A", "m : M", "hm : m ∈ p"], "goal": "a ⊗ₜ[R] m ∈ Submodule.baseChange A p"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁵ : MeasurableSpace X", "μ : Measure X", "𝕜 : Type u_5", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : ℝ≥0∞", "f : X → 𝕜", "hf : Integrable f μ"], "goal": "∫ (a : X), ↑(re (f a)) + I • ↑(im (f a)) ∂μ = ∫ (x : X), f x ∂μ"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "β : Sort u_1", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F G : C ⥤ D", "e : F ≅ G", "hobj : ∀ (X : C), F.obj X = G.obj X", "happ : ∀ (X : C), e.hom.app X = eqToHom ⋯", "X Y : C", "f : X ⟶ Y"], "goal": "F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯"}}
+{"state": {"context": ["p n : ℕ", "hp : Prime p", "h : p ∣ n", "h1 : φ p * φ (p * n) = φ p * (φ n * p)"], "goal": "φ (p * n) = p * φ n"}}
+{"state": {"context": ["α : Type u_1", "CoheytingAlgebra α", "a : α"], "goal": "∂ (￢a) ≤ ∂ a"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "S : Set (L.Substructure M)", "x : M"], "goal": "x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "β : Type u_4", "inst✝⁵ : TopologicalSpace γ", "inst✝⁴ : PolishSpace γ", "inst✝³ : TopologicalSpace β", "inst✝² : T2Space β", "inst✝¹ : MeasurableSpace β", "inst✝ : OpensMeasurableSpace β", "f : γ → β", "f_cont : Continuous f", "f_inj : Injective f", "this : UpgradedPolishSpace γ := upgradePolishSpace γ"], "goal": "MeasurableSet (range f)"}}
+{"state": {"context": ["n : ℕ", "hn : Even n", "x : ℤ", "hx : x ∈ nonneg ℤ", "this : x = x.natAbs • 1 ^ n"], "goal": "x ∈ closure (range fun x => x ^ n)"}}
+{"state": {"context": ["R : Type u", "M : Type v", "N : Type w", "Ring R", "AddCommGroup M", "Module R M", "AddCommGroup N", "Module R N", "h : Cardinal.lift.{w, v} (Module.rank R M) ≤ Cardinal.lift.{v, w} (Module.rank R N)", "h' : Module.rank R N < ℵ₀"], "goal": "FiniteDimensional.finrank R M ≤ FiniteDimensional.finrank R N"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : α → β → Prop"], "goal": "(List.Forall₂ R ⇒ List.Forall₂ R ⇒ List.Forall₂ R) (fun x x_1 => x ++ x_1) fun x x_1 => x ++ x_1"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ"], "goal": "(∫ (x : ℝ) in a..b, sin x ^ (n + 2)) * (↑n + 2) = sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b + (↑n + 1) * ∫ (x : ℝ) in a..b, sin x ^ n"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "n : ℕ", "a : α"], "goal": "⌊↑n + a⌋ = ↑n + ⌊a⌋"}}
+{"state": {"context": ["α : Type u_2", "BooleanAlgebra α", "a b : α"], "goal": "a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "e₁ : α ↪ β", "e₂ : γ ↪ δ"], "goal": "⇑(e₁.sumMap e₂) = Sum.map ⇑e₁ ⇑e₂"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "S : D", "Y' : C", "T : CategoryTheory.Functor C D", "f : CategoryTheory.StructuredArrow S T", "g : f.right ⟶ Y'"], "goal": "(f.homMk' g).left = CategoryTheory.CategoryStruct.id f.left"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "PredOrder α", "a : α"], "goal": "¬Order.IsPredLimit a ↔ ∃ b, ¬IsMin b ∧ Order.pred b = a"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "self : AlgebraicGeometry.IsAffine X"], "goal": "CategoryTheory.IsIso (AlgebraicGeometry.ΓSpec.adjunction.unit.app X)"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "a : ℕ"], "goal": "legendreSym p ↑a = -1 ↔ ¬IsSquare ↑a"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "κ✝ η✝ : Kernel α γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : Kernel α γ", "inst✝¹ : IsFiniteKernel κ", "inst✝ : IsFiniteKernel η", "a : α"], "goal": "(κ.singularPart η) a = 0 ↔ ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = 0"}}
+{"state": {"context": ["b n : ℕ", "hb : 1 < b", "hn : 2 ≤ n"], "goal": "(n + b - 1) / b < n"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "f : Equiv.Perm α", "x : α"], "goal": "orderOf (f.cycleOf x) ∣ orderOf f"}}
+{"state": {"context": ["n : Type u_1", "𝕜 : Type u_2", "inst✝² : RCLike 𝕜", "inst✝¹ : Fintype n", "inst✝ : DecidableEq n", "A : Matrix n n 𝕜", "hA : A.IsHermitian", "r : ℝ"], "goal": "↑hA.eigenvectorUnitary * ((diagonal fun x => ↑r) * star ↑hA.eigenvectorUnitary) = ↑hA.eigenvectorUnitary * ((algebraMap ℝ (Matrix n n 𝕜)) r * star ↑hA.eigenvectorUnitary)"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "a b c d : Associates α", "h₁ : a ≤ b", "h₂ : c ≤ d"], "goal": "a * c ≤ b * d"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "δ : Type u_3", "ε : Type u_4", "f : α → β → γ", "p : γ → Prop", "a : α", "l₁ : List α", "b : β", "l₂ : List β", "h : (a :: l₁).length = (b :: l₂).length"], "goal": "Forall p (zipWith f (a :: l₁) (b :: l₂)) ↔ Forall₂ (fun x y => p (f x y)) (a :: l₁) (b :: l₂)"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "Field R", "AddCommGroup V", "TopologicalSpace R", "TopologicalSpace V", "TopologicalRing R", "Module R V", "SeparatingDual R V", "x : V", "hx : x ≠ 0"], "goal": "∃ f, f x = 1"}}
+{"state": {"context": ["α : Type u_1", "m n : ℕ", "f : Vector3 α 0 → Prop", "x✝ : Exists f", "v : Vector3 α 0", "fv : f v"], "goal": "f []"}}
+{"state": {"context": ["x y : ℝ"], "goal": "Real.toEReal ⁻¹' Ioo ↑x ↑y = Ioo x y"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "s : Set (Sym2 V)"], "goal": "G.deleteEdges s = G ↔ Disjoint G.edgeSet s"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "R S T U : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "L : Bimod T U"], "goal": "TensorBimod.actLeft (P.tensorBimod Q) L ≫ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ⋯ = R.X ◁ coequalizer.desc ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫ coequalizer.desc ((α_ P.X Q.X L.X).hom ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫ coequalizer.π (P.actRight ▷ TensorBimod.X Q L) ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L)) ⋯) ⋯ ≫ TensorBimod.actLeft P (Q.tensorBimod L)"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V"], "goal": "(o.oangle (x + y) y).sign = (o.oangle x y).sign"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p1 p2 p3 : P", "h2 : p2 ≠ p1", "h3 : p3 ≠ p1"], "goal": "EuclideanGeometry.angle p1 p2 p3 + EuclideanGeometry.angle p2 p3 p1 + EuclideanGeometry.angle p3 p1 p2 = π"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁸ : _root_.RCLike 𝕜", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : InnerProductSpace 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : InnerProductSpace ℝ F", "ι : Type u_4", "ι' : Type u_5", "ι'' : Type u_6", "E' : Type u_7", "inst✝³ : NormedAddCommGroup E'", "inst✝² : InnerProductSpace 𝕜 E'", "E'' : Type u_8", "inst✝¹ : NormedAddCommGroup E''", "inst✝ : InnerProductSpace 𝕜 E''", "v : Basis ι 𝕜 E", "hv : Orthonormal 𝕜 ⇑v", "i : ι"], "goal": "(hv.equiv hv (Equiv.refl ι)) (v i) = (LinearIsometryEquiv.refl 𝕜 E) (v i)"}}
+{"state": {"context": [], "goal": "Set.range ENNReal.toEReal = Set.Ici 0"}}
+{"state": {"context": ["α : Type u_2", "MulZeroClass α", "g : Filter α", "hg : g.NeBot"], "goal": "0 ≤ 0 * g"}}
+{"state": {"context": ["ι : Type u₁", "k : Type u₂", "V : Type u₃", "P : Type u₄", "AddCommGroup V", "AffineSpace V P", "CommRing k", "Module k V", "DecidableEq ι", "Fintype ι", "b b₂ : AffineBasis ι k P", "x : P"], "goal": "b.coords x ᵥ* (b.toMatrix ⇑b₂)⁻¹ = b₂.coords x"}}
+{"state": {"context": ["R : Type u_1", "Monoid R", "S : Submonoid R", "OreLocalization.OreSet S"], "goal": "1 = 1 /ₒ 1"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y Z : C", "f : Y ⟶ X", "CategoryTheory.Limits.HasPullbacks C", "g : Z ⟶ X"], "goal": "CategoryTheory.Presieve.pullbackArrows f (CategoryTheory.Presieve.singleton g) =\n  CategoryTheory.Presieve.singleton (CategoryTheory.Limits.pullback.snd g f)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "Z : D", "F : CategoryTheory.Functor C D", "hZ : CategoryTheory.Limits.IsTerminal Z", "X Y : CategoryTheory.WithTerminal C", "f : X ⟶ Y"], "goal": "(CategoryTheory.WithTerminal.liftToTerminal F hZ).map f =\n  match X, Y, f with\n  | CategoryTheory.WithTerminal.of x, CategoryTheory.WithTerminal.of y, f => F.map (CategoryTheory.WithTerminal.down f)\n  | CategoryTheory.WithTerminal.of x, CategoryTheory.WithTerminal.star, x_1 => hZ.from (F.obj x)\n  | CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star, x => CategoryTheory.CategoryStruct.id Z"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "R : α → β → Prop", "h : Relator.LeftTotal R"], "goal": "((R ⇒ fun x x_1 => x → x_1) ⇒ fun x x_1 => x → x_1) (fun p => ∃ i, p i) fun q => ∃ i, q i"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.IsFiltered C", "J : Type w", "CategoryTheory.SmallCategory J", "CategoryTheory.FinCategory J", "F : CategoryTheory.Functor J C"], "goal": "Nonempty (CategoryTheory.Limits.Cocone F)"}}
+{"state": {"context": ["m n a✝ b✝ c d : ℕ", "co : n.Coprime m", "a b z : ℕ", "hzan : z ≡ a [MOD n]", "hzbm : z ≡ b [MOD m]"], "goal": "z ≡ ↑(chineseRemainder co a b) [MOD n * m]"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Preadditive V", "c : ComplexShape ι", "C D : HomologicalComplex V c", "f g : C ⟶ D", "i : ι"], "goal": "(f + g).f i = f.f i + g.f i"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type v", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "CompleteSpace 𝕜", "r : ℝ", "rpos : 0 < r", "c : E", "h : IsCompact (Metric.closedBall c r)"], "goal": "FiniteDimensional 𝕜 E"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X Y Z : AlgebraCat R"], "goal": "(CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).inv =\n  (α_ ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj X)\n      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Y)\n      ((CategoryTheory.forget₂ (AlgebraCat R) (ModuleCat R)).obj Z)).inv"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "MeasurableSpace α", "NormedAddCommGroup E", "f : α → E", "μ : MeasureTheory.Measure α", "l l' : Filter α", "hl : MeasureTheory.IntegrableAtFilter f l μ"], "goal": "MeasureTheory.IntegrableAtFilter f (l ⊓ l') μ"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "f : G →g G'", "f' : G' →g G''", "u✝ v✝ u' v' : V", "p✝ : G.Walk u✝ v✝", "u v : V", "p : G.Walk u v", "H : SimpleGraph V", "hp : ∀ e ∈ p.edges, e ∈ H.edgeSet"], "goal": "∀ e ∈ p.reverse.edges, e ∈ H.edgeSet"}}
+{"state": {"context": ["F : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "Monoid M₁", "Monoid M₂", "FunLike F M₁ M₂", "MonoidHomClass F M₁ M₂", "f : F", "hf : Function.Surjective ⇑f"], "goal": "Monoid.exponent M₂ ∣ Monoid.exponent M₁"}}
+{"state": {"context": ["R✝ : Type u_1", "A : Type u_2", "B : Type u_3", "M✝ : Type u_4", "N : Type u_5", "inst✝⁸ : Semiring R✝", "inst✝⁷ : AddCommMonoid M✝", "inst✝⁶ : Module R✝ M✝", "inst✝⁵ : AddCommMonoid N", "inst✝⁴ : Module R✝ N", "R : Type u_6", "M : Type u_7", "inst✝³ : CommSemiring R", "inst✝² : AddCommMonoid M", "inst✝¹ : Module R M", "inst✝ : Finite R M", "f : End R M", "h : ∀ (m : M), ∃ n, (f ^ n) m = 0", "S : Finset M", "hS : span R ↑S = ⊤"], "goal": "IsNilpotent f"}}
+{"state": {"context": ["n m l : ℕ", "hnm : n ≤ m", "hml : m ≤ l"], "goal": "range' m (l - m) = range' (n + 1 * (m - n)) (l - m)"}}
+{"state": {"context": [], "goal": "∀ {X Y : CondensedSet} (f : X ⟶ Y), condensedSetToTopCat.map f = CondensedSet.toTopCatMap f"}}
+{"state": {"context": ["R : Type u", "A : Type v", "inst✝⁴ : CommRing R", "inst✝³ : Ring A", "inst✝² : Algebra R A", "L : Type v", "inst✝¹ : LieRing L", "inst✝ : LieAlgebra R L", "K : LieSubalgebra R L", "x : ↥K", "n : ℕ", "hn : (LieAlgebra.ad R L) ↑x ^ n = 0"], "goal": "(LieAlgebra.ad R ↥K) x ^ n = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "B : LinearMap.BilinForm R M", "M' : Type u_9", "AddCommMonoid M'", "Module R M'", "B' : LinearMap.BilinForm R M'", "f : M →ₗ[R] M'", "g : M' →ₗ[R] M", "M'' : Type u_11", "AddCommMonoid M''", "Module R M''", "B'' : LinearMap.BilinForm R M''", "f' : M' →ₗ[R] M''", "g' : M'' →ₗ[R] M'", "h : B.IsAdjointPair B' f g", "h' : B'.IsAdjointPair B'' f' g'"], "goal": "B.IsAdjointPair B'' (f' ∘ₗ f) (g ∘ₗ g')"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_4", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_6", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "i j : ↑(atlas H M)", "x : M"], "goal": "(tangentBundleCore I M).coordChange i j x =\n  fderivWithin 𝕜 (↑((↑j).extend I) ∘ ↑((↑i).extend I).symm) (Set.range ↑I) (↑((↑i).extend I) x)"}}
+{"state": {"context": ["R : Type u_1", "NonAssocRing R", "Pow R ℕ", "BinomialRing R", "NatPowAssoc R", "n : ℕ"], "goal": "Ring.multichoose (-↑n) (n + 1) = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "s : Set E", "hf : AnalyticOn 𝕜 f s"], "goal": "ContinuousOn f s"}}
+{"state": {"context": ["G : Type w", "inst✝² : TopologicalSpace G", "μ : Content G", "inst✝¹ : R1Space G", "S : MeasurableSpace G", "inst✝ : BorelSpace G", "U : Set G", "hU : U ∈ {s | IsOpen s}", "U' : Opens G", "this : Nonempty { L // ↑L ⊆ ↑U' ∩ U }"], "goal": "∀ (i : { i // ↑i ⊆ ↑U' ∩ U }), ↑(μ.toFun ↑i) + μ.outerMeasure (↑U' \\ U) ≤ μ.outerMeasure ↑U'"}}
+{"state": {"context": ["𝓕 : Type u_1", "α : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : SeminormedGroup E", "inst✝ : SeminormedGroup F", "s : Set E", "a✝ a₁ a₂ b✝ b₁ b₂ : E", "r r₁ r₂ : ℝ", "a b c : E"], "goal": "dist a (b / c) = dist (a * c) b"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasZeroMorphisms C", "S S₁ S₂ S₃ S₄ : ShortComplex C", "φ : S₁ ⟶ S₂", "h₁✝ : S₁.HomologyData", "h₂✝ : S₂.HomologyData", "e : S₁ ≅ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData"], "goal": "homologyMap' e.inv h₂ h₁ ≫ homologyMap' e.hom h₁ h₂ = 𝟙 h₂.left.H"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type uX", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f f₁ : E → F", "g : F → G", "x✝ x₀ : E", "c : F", "m✝ n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "H : ContinuousOn f s", "x : E", "hx : x ∈ s", "m : ℕ", "hm : ↑m ≤ 0", "this : ↑m = 0"], "goal": "∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn 0 f p u"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "I : (FractionalIdeal R⁰ (FractionRing R))ˣ", "I' : Ideal R", "hI : ↑I = ↑I'"], "goal": "(∃ x y, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' = Ideal.span {y} * ⊤) ↔ ∃ x, x ≠ 0 ∧ I' = Ideal.span {x}"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "s : Multiset α"], "goal": "a ∈ b ::ₘ s ↔ a = b ∨ a ∈ s"}}
+{"state": {"context": ["M₀ : Type u_1", "G₀ : Type u_2", "inst✝ : MulZeroClass M₀", "s : Set M₀"], "goal": "0 ∈ s.centralizer"}}
+{"state": {"context": ["C✝ : Type u", "𝒞 : Category.{v, u} C✝", "inst✝² : MonoidalCategory C✝", "C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : MonoidalCategory C", "U V W X✝ Y✝ Z✝ X Y Z X' : C", "f : X ⟶ X'"], "goal": "(f ⊗ 𝟙 (Y ⊗ Z)) ≫ (α_ X' Y Z).inv = (α_ X Y Z).inv ≫ ((f ⊗ 𝟙 Y) ⊗ 𝟙 Z)"}}
+{"state": {"context": ["α : Type u_1", "AddCommMonoid α", "n : ℕ", "f : Fin n → α", "i : ℕ"], "goal": "(List.take i (List.ofFn f)).sum = ∑ j ∈ Finset.filter (fun j => ↑j < i) Finset.univ, f j"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : E", "i : ℕ", "f g : E → F", "hf : ContDiff 𝕜 (↑i) f", "hg : ContDiff 𝕜 (↑i) g"], "goal": "iteratedFDeriv 𝕜 i (f + g) x = iteratedFDeriv 𝕜 i f x + iteratedFDeriv 𝕜 i g x"}}
+{"state": {"context": ["C : Type u", "inst✝⁷ : Category.{v, u} C", "inst✝⁶ : Preadditive C", "R : Type u_1", "inst✝⁵ : Ring R", "inst✝⁴ : Linear R C", "F G K L : CochainComplex C ℤ", "n m : ℤ", "D : Type u_2", "inst✝³ : Category.{u_3, u_2} D", "inst✝² : Preadditive D", "z z' : Cochain K L n", "f : K ⟶ L", "Φ✝ : C ⥤ D", "inst✝¹ : Φ✝.Additive", "n₁ n₂ n₁₂ : ℤ", "z₁ : Cochain F G n₁", "z₂ : Cochain G K n₂", "h : n₁ + n₂ = n₁₂", "Φ : C ⥤ D", "inst✝ : Φ.Additive", "p q : ℤ", "hpq : p + n₁₂ = q"], "goal": "Φ.map ((z₁.comp z₂ h).v p q hpq) = ((z₁.map Φ).comp (z₂.map Φ) h).v p q hpq"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f✝ fa : α → α", "fb : β → β", "x✝ y : α", "m n : ℕ", "r : α → α → Prop", "f : α → α", "x : α", "hx : x ∈ periodicPts f"], "goal": "(∀ n < minimalPeriod f x, r (f^[n] x) (f^[n + 1] x)) ↔ ∀ (n : ℕ), r (f^[n] x) (f^[n + 1] x)"}}
+{"state": {"context": ["ι : Type v", "Preorder ι", "G : ι → Type w", "(i : ι) → CommRing (G i)", "f : (i j : ι) → i ≤ j → G i →+* G j", "Nonempty ι", "IsDirected ι fun x x_1 => x ≤ x_1"], "goal": "Ring.DirectLimit.map (fun i => RingHom.id (G i)) ⋯ = RingHom.id (Ring.DirectLimit G fun x x_1 h => ⇑(f x x_1 h))"}}
+{"state": {"context": ["n : Nat"], "goal": "(Int.negSucc n).natAbs = n.succ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "hc : 0 ≤ c", "hfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖", "hgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖", "x : α", "hx : ‖f x‖ ≤ c * ‖g x‖", "hx' : ‖g x‖ ≤ c' * ‖k x‖"], "goal": "‖f x‖ ≤ c * c' * ‖k x‖"}}
+{"state": {"context": ["a b : Bool"], "goal": "(a || (a || b)) = (a || b)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "f g : α → α", "h : Function.Commute f g", "hf : StrictMono f", "hg : Monotone g", "x : α", "n : ℕ", "hn : 0 < n"], "goal": "f^[n] x ≤ g^[n] x ↔ f x ≤ g x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedCommGroup α", "LinearOrderedCommGroup β", "s : Set ι", "f : ι → α", "g₁ g₂ : ι → β", "h₁ : MonovaryOn f g₁ s", "h₂ : MonovaryOn f g₂ s"], "goal": "MonovaryOn f (g₁ * g₂) s"}}
+{"state": {"context": ["m0 n0 : ℤ", "hm : (m0 * 2 + 1) % 2 = 1", "hn : (n0 * 2 + 1) % 2 = 1", "h : (m0 * 2 + 1).gcd (n0 * 2 + 1) = 1", "h1 : (m0 * 2 + 1) ^ 2 + (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1)", "h2 : (m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2 = 2 * (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0))", "h3 : ((m0 * 2 + 1) ^ 2 - (n0 * 2 + 1) ^ 2) / 2 % 2 = 0", "h20 : 2 ≠ 0", "h4 : ¬(2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).gcd (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : p ∣ (2 * (m0 ^ 2 - n0 ^ 2 + m0 - n0)).natAbs", "hp2 : p ∣ (2 * (m0 ^ 2 + n0 ^ 2 + m0 + n0) + 1).natAbs"], "goal": "p ∣ (m0 * 2 + 1).gcd (n0 * 2 + 1)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "x : E", "s : Set E", "h : 𝓝[s \\ {x}] x = ⊥"], "goal": "HasFDerivWithinAt f f' s x"}}
+{"state": {"context": ["m n k : Nat", "h : m < n * k"], "goal": "m / n < k"}}
+{"state": {"context": ["α : Type u", "Inv α", "a : α"], "goal": "↑a⁻¹ = (↑a)⁻¹"}}
+{"state": {"context": ["K : Type u_1", "Field K", "f : K[X]", "a : K"], "goal": "f /ₘ (Polynomial.X - Polynomial.C a) +\n    (Polynomial.X - Polynomial.C a) * Polynomial.derivative (f /ₘ (Polynomial.X - Polynomial.C a)) =\n  Polynomial.derivative f"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝⁵ : CommSemiring R", "inst✝⁴ : StarRing R", "inst✝³ : Semiring A", "inst✝² : StarMul A", "inst✝¹ : Algebra R A", "inst✝ : StarModule R A", "r : R", "hr : IsSelfAdjoint r"], "goal": "IsSelfAdjoint ((algebraMap R A) r)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "m0 : MeasurableSpace α", "μ : Measure α", "v : VitaliFamily μ", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : SecondCountableTopology α", "inst✝² : BorelSpace α", "inst✝¹ : IsLocallyFiniteMeasure μ", "ρ : Measure α", "inst✝ : IsLocallyFiniteMeasure ρ", "s : Set α", "hs : MeasurableSet s", "this : IsLocallyFiniteMeasure (μ.restrict s)"], "goal": "∀ (a : α), Tendsto (fun a => (μ.restrict s) a / μ a) (v.filterAt a) (𝓝 ((μ.restrict s).rnDeriv μ a)) → (μ.restrict s).rnDeriv μ a = s.indicator 1 a → Tendsto (fun a => μ (s ∩ a) / μ a) (v.filterAt a) (𝓝 (s.indicator 1 a))"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "BoundedOrder α", "x y : α", "h : IsCompl x y"], "goal": "x ⊓ y = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "hκ : IsSFiniteKernel κ"], "goal": "(Kernel.sum fun n => Kernel.sum fun m => κ.seq n ⊗ₖ η.seq m) = Kernel.sum fun n => κ ⊗ₖ η.seq n"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "Group G", "MulAction G α", "x : MulAction.orbitRel.Quotient G α", "φ : MulAction.orbitRel.Quotient G α → α", "hφ : Function.RightInverse φ Quotient.mk'"], "goal": "x.orbit = MulAction.orbit G (φ x)"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "t : CategoryTheory.Bicategory.LeftExtension f g"], "goal": "t.whiskerOfCompIdIsoSelf.inv.right = (CategoryTheory.Bicategory.rightUnitor t.extension).inv"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "c : ℝ≥0∞", "f : α → ℝ≥0∞"], "goal": "∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "h : s.Nonempty"], "goal": "⊤.image s = Set.univ"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_3", "F : Type u_5", "RCLike 𝕜", "NormedAddCommGroup F", "CompleteSpace F", "u : α → ℝ", "NormedSpace 𝕜 F", "g g' : α → 𝕜 → F", "hu : Summable u", "hg : ∀ (n : α) (y : 𝕜), HasDerivAt (g n) (g' n y) y", "hg' : ∀ (n : α) (y : 𝕜), ‖g' n y‖ ≤ u n"], "goal": "Differentiable 𝕜 fun z => ∑' (n : α), g n z"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : CanonicallyLinearOrderedAddCommMonoid α", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a b c d : α", "h : b ≤ a"], "goal": "a - min a b = a - b"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "l : List (Sigma β)", "a : α", "b : β a"], "goal": "List.dlookup a (⟨a, b⟩ :: l) = some b"}}
+{"state": {"context": ["C : Type u", "X Y X✝ Y✝ Z✝ : F C"], "goal": "∀ (n : NormalMonoidalObject C), inclusionObj n ◁ (α_ X✝ Y✝ Z✝).inv ≫ (normalizeIsoApp' C ((X✝ ⊗ Y✝) ⊗ Z✝) n).hom = (normalizeIsoApp' C (X✝ ⊗ Y✝ ⊗ Z✝) n).hom ≫ inclusion.map (eqToHom ⋯)"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R"], "goal": "⇑MvPolynomial.constantCoeff = MvPolynomial.coeff 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "s : Set E", "a : E", "n : ℕ∞", "h : ContDiffWithinAt ℝ n f s a"], "goal": "ContDiffWithinAt ℝ n (fun x => Real.arsinh (f x)) s a"}}
+{"state": {"context": ["α : Type u_1", "p : α → Bool", "xs : List α", "h : ∃ x, x ∈ xs ∧ p x = true"], "goal": "List.findIdx p xs < xs.length"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝¹ : SeminormedRing α", "inst✝ : NormOneClass α", "l : List α"], "goal": "(map norm l).prod = (fun a => ↑a) (map nnnorm l).prod"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Lattice α", "Lattice β", "f : LatticeHom α β"], "goal": "⇑f.withBot = Option.map ⇑f"}}
+{"state": {"context": ["a : ℤ"], "goal": "|a| ≤ 1 ↔ a = 0 ∨ a = 1 ∨ a = -1"}}
+{"state": {"context": ["ε : Type u_1", "p : ε → Prop", "DecidablePred p", "ep : Equiv.Perm { a // p a }", "en : Equiv.Perm { a // ¬p a }"], "goal": "Equiv.symm (ep.subtypeCongr en) = Equiv.Perm.subtypeCongr (Equiv.symm ep) (Equiv.symm en)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "p : ℕ", "P : Sylow p G", "hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P", "Fact (Nat.Prime p)", "Finite (Sylow p G)", "(↑P).FiniteIndex", "Q : Subgroup G", "hQ : IsPGroup p ↥Q"], "goal": "Disjoint (MonoidHom.transferSylow P hP).ker Q"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "W : Type u_3", "P : Type u_4", "Ring R", "AddCommGroup V", "Module R V", "TopologicalSpace P", "AddTorsor V P", "AddCommGroup W", "Module R W", "TopologicalSpace W", "TopologicalAddGroup W", "f g : P →ᴬ[R] W"], "goal": "⇑(f + g) = ⇑f + ⇑g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : MeasurableSpace α", "r : α → β → Prop", "f : α →ₛ β", "h : ∀ (b : β), MeasurableSet {a | r a b}"], "goal": "MeasurableSet {a | r a (↑f a)}"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MetricSpace α", "β : Type u", "inst✝ : Nonempty β", "p : TauPackage β α", "N : ℕ", "hN : IsEmpty (SatelliteConfig α N p.τ)", "i : Ordinal.{u}", "IH : ∀ k < i, k < p.lastStep → p.color k < N", "hi : i < p.lastStep", "A : Set ℕ := ⋃ j, ⋃ (_ : (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {p.color ↑j}", "color_i : p.color i = sInf (univ \\ A)", "N_mem : N ∈ univ \\ A", "Inf_eq_N : sInf (univ \\ A) = N", "this : ∀ k < N, ∃ j < i, (closedBall (p.c (p.index j)) (p.r (p.index j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color j"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "inst✝⁸ : NontriviallyNormedField R", "inst✝⁷ : (x : B) → AddCommMonoid (E x)", "inst✝⁶ : (x : B) → Module R (E x)", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace R F", "inst✝³ : TopologicalSpace B", "inst✝² : TopologicalSpace (TotalSpace F E)", "inst✝¹ : (x : B) → TopologicalSpace (E x)", "inst✝ : FiberBundle F E", "ι : Type u_5", "Z : VectorBundleCore R B F ι", "b✝ : B", "a : F", "i j : ι", "b : B", "hb : b ∈ Z.baseSet i", "v : F"], "goal": "(Z.localTriv i).symm b v = (Z.coordChange i (Z.indexAt b) b) v"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "E : Type u_3", "Fintype ι", "Fintype ι'", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "b : Basis ι ℝ E", "e : ι ≃ ι'"], "goal": "(b.reindex e).addHaar = b.addHaar"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "G : SimpleGraph V", "a b c u v✝ w✝ : V", "e : Sym2 V", "s : Set (Sym2 V)", "v w : V"], "goal": "(fromEdgeSet Set.univ).Adj v w ↔ ⊤.Adj v w"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedLinearOrderedField 𝕜", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : StrictConvexSpace 𝕜 E", "x : E", "r : ℝ"], "goal": "StrictConvex 𝕜 (closedBall x r)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "TopologicalSpace β"], "goal": "∀ {x : MeasurableSpace α} {x_1 : MeasurableSpace γ} {f : α → β} {g : γ → α},\n  MeasureTheory.StronglyMeasurable f → Measurable g → MeasureTheory.StronglyMeasurable (f ∘ g)"}}
+{"state": {"context": ["X Y : Compactum", "f : X ⟶ Y"], "goal": "∀ (x : X.A) (g : Ultrafilter X.A), ↑g ≤ 𝓝 x → Tendsto f.f (↑g) (𝓝 (f.f x))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "hC : CategoryTheory.Pretriangulated C", "T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C", "hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles", "hT₂ : T₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles", "a : T₁.obj₁ ⟶ T₂.obj₁", "b : T₁.obj₂ ⟶ T₂.obj₂", "comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁"], "goal": "(CategoryTheory.Pretriangulated.completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ a b comm).hom₂ = b"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "a a₁ a₂ : α", "b b₁ b₂ : β", "x y : α × β", "h₁ : a₁ < a₂", "h₂ : b₁ ≤ b₂"], "goal": "(a₁, b₁) < (a₂, b₂)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "l : Filter α", "f g : α → β", "h : f =ᶠ[l] g"], "goal": "EventuallyConst f l ↔ EventuallyConst g l"}}
+{"state": {"context": ["α : Type u_1", "NormedDivisionRing α", "a b : α"], "goal": "‖a * b‖₊ = ‖a‖₊ * ‖b‖₊"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f✝ f : Perm α", "x : α"], "goal": "f.cycleOf x ∈ f.cycleFactorsFinset ↔ x ∈ f.support"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l✝¹ l✝ : List α", "x : α", "l : List α", "y : α", "hp : l.reverse = l → l.Palindrome", "hr : (x :: (l ++ [y])).reverse = x :: (l ++ [y])"], "goal": "(x :: (l ++ [y])).Palindrome"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X : CategoryTheory.Center C"], "goal": "(ρ_ X).inv.f = (ρ_ X.fst).inv"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "c : R", "f : RingSeminorm R", "hf1 : f 1 ≤ 1", "hc : f c ≠ 0", "hpm : IsPowMul ⇑f"], "goal": "∃ a, ∀ (b : ℕ), a ≤ b → seminormFromConst_seq c f 1 b = 1"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "Ring A", "StarRing A", "TopologicalSpace A", "Algebra R A", "self : UniqueContinuousFunctionalCalculus R A", "s : Set R", "CompactSpace ↑s", "φ ψ : C(↑s, R) →⋆ₐ[R] A", "hφ : Continuous ⇑φ", "hψ : Continuous ⇑ψ", "h : φ (ContinuousMap.restrict s (ContinuousMap.id R)) = ψ (ContinuousMap.restrict s (ContinuousMap.id R))"], "goal": "φ = ψ"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "t : Cofork f g"], "goal": "f ≫ t.π = g ≫ t.π"}}
+{"state": {"context": ["n : Type u_1", "p : Type u_2", "R : Type u₂", "𝕜 : Type u_3", "inst✝³ : Field 𝕜", "inst✝² : DecidableEq n", "inst✝¹ : DecidableEq p", "inst✝ : CommRing R", "r : ℕ", "M✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜", "IH : ∀ (M : Matrix (Fin r) (Fin r) 𝕜), ∃ L₀ L₀' D₀, (List.map toMatrix L₀).prod * M * (List.map toMatrix L₀').prod = diagonal D₀", "M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜", "L₁ L₁' : List (TransvectionStruct (Fin r ⊕ Unit) 𝕜)", "hM : ((List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod).IsTwoBlockDiagonal", "M' : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜 := (List.map toMatrix L₁).prod * M * (List.map toMatrix L₁').prod", "M'' : Matrix (Fin r) (Fin r) 𝕜 := M'.toBlocks₁₁", "L₀ L₀' : List (TransvectionStruct (Fin r) 𝕜)", "D₀ : Fin r → 𝕜", "h₀ : (List.map toMatrix L₀).prod * M'' * (List.map toMatrix L₀').prod = diagonal D₀", "c : 𝕜 := M' (inr ()) (inr ())", "this : M' = fromBlocks M'' 0 0 (diagonal fun x => c)"], "goal": "(List.map (toMatrix ∘ sumInl Unit) L₀).prod * M' * (List.map (toMatrix ∘ sumInl Unit) L₀').prod = diagonal (Sum.elim D₀ fun x => c)"}}
+{"state": {"context": ["P : TopCat → Prop", "α : Type w", "Finite α", "X : α → CompHausLike P", "CompHausLike.HasExplicitFiniteCoproduct X", "a : α"], "goal": "OpenEmbedding ⇑(CompHausLike.finiteCoproduct.ι X a)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_7", "v : ι → M"], "goal": "0 v = 0"}}
+{"state": {"context": ["α : Type u", "s : Set α", "DecidablePred fun x => x ∈ s", "a : α", "H : a ∉ s", "b : ↑s"], "goal": "(Equiv.Set.insert H).symm (Sum.inl b) = ⟨↑b, ⋯⟩"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n✝ : ℕ", "inst✝ : Semiring R", "p q : R[X]", "n i : ℕ", "toFinsupp✝ : R[ℕ]"], "goal": "(erase n { toFinsupp := toFinsupp✝ }).coeff i = if i = n then 0 else { toFinsupp := toFinsupp✝ }.coeff i"}}
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+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "G' : G.Subgraph", "G'' : G'.coe.Subgraph"], "goal": "(SimpleGraph.Subgraph.coeSubgraph G'').verts = Subtype.val '' G''.verts"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R"], "goal": "W.Φ 0 = 1"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "OrderedCommGroup α", "TopologicalSpace α", "TopologicalGroup α", "OrderClosedTopology α", "f g : ι → α", "a₁ a₂ : α", "hf : HasProd f a₁", "hg : HasProd g a₂", "h : f < g"], "goal": "a₁ < a₂"}}
+{"state": {"context": ["α : Sort u", "r : α → α → Prop", "wf : WellFounded r", "a : α"], "goal": "Acc r a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "hp : p.Monic", "r : R", "ha : (X + C r).natDegree = 0"], "goal": "False"}}
+{"state": {"context": ["V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P"], "goal": "UniformContinuous fun x => x.1 +ᵥ x.2"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "Group G", "A : Rep k G", "f : ↥(groupCohomology.twoCocycles A)", "g : G"], "goal": "(A.ρ g) (↑f (g⁻¹, g)) - ↑f (g, g⁻¹) = ↑f (1, 1) - ↑f (g, 1)"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : E → F", "hf_temperate : Function.HasTemperateGrowth f", "n : ℕ"], "goal": "∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "AddCommGroup α", "UniformSpace α", "UniformAddGroup α", "f : β → α", "CompleteSpace α", "i : γ → β", "hf : Summable f", "hi : Function.Injective i"], "goal": "Summable (f ∘ i)"}}
+{"state": {"context": ["R : Type u", "A : Type v", "B : Type w", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "NonUnitalNonAssocSemiring B", "Module R B", "IsScalarTower R B B", "SMulCommClass R B B", "f : A →ₙₐ[R] B"], "goal": "NonUnitalSubalgebra.comap f ⊤ = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → ℚ → ℝ", "hf : IsRatCondKernelCDFAux f κ ν", "a : α"], "goal": "∀ᵐ (t : β) ∂ν a, ∀ (q : ℚ), BddBelow (range fun r => f (a, t) ↑r)"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N₁ N₂ : Bimod Y Z", "f : N₁ ⟶ N₂"], "goal": "((α_ M.X Y.X N₁.X).hom ≫ M.X ◁ N₁.actLeft) ≫ M.X ◁ f.hom = (M.X ⊗ Y.X) ◁ f.hom ≫ (α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "N : Type u_2", "AddGroup N", "TopologicalSpace N", "H : OpenAddSubgroup N", "f : G →+ N", "hf : Continuous ⇑f", "x : G"], "goal": "x ∈ OpenAddSubgroup.comap f hf H ↔ f x ∈ H"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s✝ : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "ι : Type u_3", "f : ι → MvPolynomial σ R", "a : ι", "s : Finset ι", "has : a ∉ s", "hind : (s.sum f).totalDegree ≤ s.sup fun i => (f i).totalDegree"], "goal": "((Finset.cons a s has).sum f).totalDegree ≤ (Finset.cons a s has).sup fun i => (f i).totalDegree"}}
+{"state": {"context": ["L : FirstOrder.Language", "T : L.Theory", "α : Type w", "self : T.CompleteType α"], "goal": "(↑self).IsMaximal"}}
+{"state": {"context": ["b✝ : ℂ", "V : Type u_1", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : FiniteDimensional ℝ V", "inst✝¹ : MeasurableSpace V", "inst✝ : BorelSpace V", "b : ℝ", "hb : 0 < b"], "goal": "↑((π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)) = (↑π / ↑b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "X : C"], "goal": "(α_ X (𝟙_ C) (𝟙_ C)).inv ≫ (β_ (𝟙_ C) X).inv ▷ 𝟙_ C ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ 𝟙_ C ◁ (ρ_ X).hom =\n  X ◁ (ρ_ (𝟙_ C)).hom ≫ (β_ (𝟙_ C) X).inv"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "ι : Type u_2", "inst✝ : IsFiniteMeasure μ", "g : α → ℝ", "hint : Integrable g μ", "ℱ : ι → MeasurableSpace α", "hℱ : ∀ (i : ι), ℱ i ≤ m0", "A : MeasurableSpace α := m0", "hmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}", "hg : Memℒp g 1 μ", "ε : ℝ", "hε : 0 < ε", "hne : ¬eLpNorm g 1 μ = 0", "δ : ℝ", "hδ : 0 < δ", "h : ∀ (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator g) 1 μ ≤ ENNReal.ofReal ε", "C : ℝ≥0 := ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal", "hC : C = ⟨δ, ⋯⟩⁻¹ * (eLpNorm g 1 μ).toNNReal", "hCpos : 0 < C", "this : ∀ (n : ι), μ {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} ≤ ENNReal.ofReal δ", "n : ι", "hmeasℱ : MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊}"], "goal": "eLpNorm (μ[{x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g|ℱ n]) 1 μ ≤ eLpNorm ({x | C ≤ ‖(μ[g|ℱ n]) x‖₊}.indicator g) 1 μ"}}
+{"state": {"context": ["R : Type u", "inst✝ : AddGroupWithOne R"], "goal": "↑(-↑0) = -↑↑0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁹ : MeasurableSpace X", "μ : Measure X", "𝕜 : Type u_5", "inst✝⁸ : RCLike 𝕜", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "p : ℝ≥0∞", "inst✝³ : NormedSpace ℝ F", "inst✝² : CompleteSpace F", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "L : E →L[𝕜] F", "φ : X → E", "φ_int : Integrable φ μ"], "goal": "∀ ⦃f g : X → E⦄, Disjoint (support f) (support g) → Integrable f μ → Integrable g μ → (fun φ => ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)) f → (fun φ => ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)) g → (fun φ => ∫ (x : X), L (φ x) ∂μ = L (∫ (x : X), φ x ∂μ)) (f + g)"}}
+{"state": {"context": ["ι : Type u_1", "γ : Type u_4", "π : ι → Type u_5", "rι : ι → ι → Prop", "rπ : (i : ι) → π i → π i → Prop", "f : γ → ι", "g : (i : ι) → γ → π i", "s : Set γ", "hι : s.WellFoundedOn (rι on f)", "hπ : ∀ (i : ι), (s ∩ f ⁻¹' {i}).WellFoundedOn (rπ i on g i)"], "goal": "s.WellFoundedOn (Sigma.Lex rι rπ on fun c => ⟨f c, g (f c) c⟩)"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : Monoid G", "a b x✝ y : G", "n m : ℕ", "x : G"], "goal": "x ^ n * 1 = 1 ↔ x ^ n = 1"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f✝ g : Perm α", "x✝ y : α", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f : Perm α", "hf : f.IsCycle", "x : α", "hx : f x ≠ x ∧ ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y"], "goal": "Perm.sign f = Perm.sign (swap x (f x) * (swap x (f x) * f))"}}
+{"state": {"context": ["ι : Type u_1", "η : ι → Type u_4", "N : Type u_5", "R : Type u_6", "Monoid R", "AddMonoid N", "DistribMulAction R N", "r : R", "f : (i : ι) × η i →₀ N"], "goal": "sigmaFinsuppEquivDFinsupp (r • f) = r • sigmaFinsuppEquivDFinsupp f"}}
+{"state": {"context": ["K : Type u", "V V₁ : Type v", "V' V'₁ : Type v'", "V'' : Type v''", "inst✝⁷ : Ring K", "inst✝⁶ : AddCommGroup V", "inst✝⁵ : Module K V", "inst✝⁴ : AddCommGroup V₁", "inst✝³ : Module K V₁", "inst✝² : AddCommGroup V'", "inst✝¹ : Module K V'", "inst✝ : Nontrivial K"], "goal": "rank 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "G' : Type u_7", "G'' : Type u_8", "NormedLatticeAddCommGroup G''", "NormedSpace ℝ G''", "CompleteSpace G''", "NormedLatticeAddCommGroup G'", "NormedSpace ℝ G'", "T : Set α → G' →L[ℝ] G''", "C : ℝ", "hT : MeasureTheory.DominatedFinMeasAdditive μ T C", "hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x", "f g : α → G'", "hf : MeasureTheory.Integrable f μ", "hg : MeasureTheory.Integrable g μ", "hfg : f ≤ᵐ[μ] g"], "goal": "MeasureTheory.setToFun μ T hT f ≤ MeasureTheory.setToFun μ T hT g"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{u_2, u_1} C", "P : Karoubi (Karoubi C)"], "goal": "P.p.f ≫ P.p.f = P.p.f"}}
+{"state": {"context": ["α : Type u_1", "NonAssocSemiring α", "n : ℕ", "n.AtLeastTwo", "x : α"], "goal": "Commute (OfNat.ofNat n) x"}}
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+{"state": {"context": ["α : Type u_1", "PseudoMetricSpace α", "p : α → Prop", "x y : Subtype p"], "goal": "nndist x y = nndist ↑x ↑y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : ℕ × α", "n : ℕ", "l : List α", "h : x ∈ enumFrom n l"], "goal": "x.1 < n + l.length"}}
+{"state": {"context": ["F : Type u_1", "Field F", "x y : F", "hxy : x ≠ y"], "goal": "(Lagrange.basisDivisor x y).degree = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β"], "goal": "∀ (l : List α), ContinuousAt length l"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "P : Ideal R", "hP : P.IsPrime", "p q : R[X]", "hfl : (p * q).leadingCoeff ∉ P", "hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P", "hfd0✝ : 0 < (p * q).degree", "h0 : (p * q).coeff 0 ∉ P ^ 2", "hu : (p * q).IsPrimitive", "hf0 : p * q ≠ 0", "hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree", "hfd0 : 0 < (p * q).natDegree"], "goal": "IsUnit p ∨ IsUnit q"}}
+{"state": {"context": ["p : EReal × EReal", "h₁ : p.1 ≠ 0 ∨ p.2 ≠ ⊥", "h₂ : p.1 ≠ 0 ∨ p.2 ≠ ⊤", "h₃ : p.1 ≠ ⊥ ∨ p.2 ≠ 0", "h₄ : p.1 ≠ ⊤ ∨ p.2 ≠ 0"], "goal": "ContinuousAt (fun p => p.1 * p.2) p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : Fintype α", "inst✝¹ : DecidableEq β", "f : α ↪ β", "b✝ : ↑(Set.range ⇑f)", "inst✝ : Nonempty α", "b : β", "h : b ∈ Set.range ⇑f"], "goal": "f ((Set.range ⇑f).restrict (invFun ⇑f) ⟨b, h⟩) = f (f.invOfMemRange ⟨b, h⟩)"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁷ : Group G", "inst✝⁶ : Group G'", "inst✝⁵ : Group G''", "A : Type u_4", "inst✝⁴ : AddGroup A", "M : Type u_5", "S : Type u_6", "inst✝³ : DivInvMonoid M", "inst✝² : SetLike S M", "hSM : SubgroupClass S M", "H✝ K✝ : S", "inst✝¹ : SetLike S G", "inst✝ : SubgroupClass S G", "H K L : S", "h : ↑H ⊆ ↑K ∪ ↑L"], "goal": "H ≤ K ∨ H ≤ L"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝² : DistribLattice α", "inst✝¹ : OrderBot α", "s✝ : Finset ι", "f✝ : ι → α", "inst✝ : DecidableEq ι", "s : Finset ι'", "g : ι' → Finset ι", "f : ι → α", "hs : (↑s).PairwiseDisjoint fun i => (g i).sup f", "hg : ∀ i' ∈ s, (↑(g i')).PairwiseDisjoint f"], "goal": "(↑(s.sup g)).PairwiseDisjoint f"}}
+{"state": {"context": [], "goal": "RingHom.StableUnderBaseChange @RingHom.Finite"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "u v w : V", "p : G.Walk u v", "q : G.Walk u w"], "goal": "(p.reverseAux q).length = p.length + q.length"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "η : ProbabilityTheory.Kernel (α × β) γ", "ProbabilityTheory.IsSFiniteKernel η", "s : Set (β × γ)", "hs : MeasurableSet s", "a : α"], "goal": "Measurable fun b => (η (a, b)) (Prod.mk b ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α"], "goal": "l.concat a ≠ []"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "β : Type u_2", "Preorder β", "f : α → β", "x : α", "s : Set α", "h : UpperSemicontinuousAt f x"], "goal": "UpperSemicontinuousWithinAt f s x"}}
+{"state": {"context": ["R : Type u_1", "b c d : R", "Semiring R"], "goal": "{ a := 0, b := b, c := c, d := d }.toPoly.degree ≤ 2"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : ℝ → F", "K : Set F", "r x : ℝ", "hr : 0 < r", "ε : ℝ", "L₁ L₂ : F", "h₁ : x ∈ A f L₁ r ε", "h₂ : x ∈ A f L₂ r ε"], "goal": "‖(r / 2) • (L₁ - L₂)‖ ≤ r / 2 * (4 * ε)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "A : Type u_4", "inst✝² : DecidableEq ι", "inst✝¹ : AddMonoid ι", "inst✝ : Semiring M", "to_hom : M[ι] →+ ⨁ (x : ι), M := { toFun := toDirectSum, map_zero' := ⋯, map_add' := ⋯ }", "x✝ : NonUnitalNonAssocSemiring (ι →₀ M) := nonUnitalNonAssocSemiring", "xi : ι", "xv : M", "yi : ι", "yv : M"], "goal": "(⇑to_hom ∘ ⇑(AddMonoidHom.mul (Finsupp.single xi xv))) (Finsupp.single yi yv) = ((AddMonoidHom.mul.comp to_hom) (Finsupp.single xi xv)) (to_hom (Finsupp.single yi yv))"}}
+{"state": {"context": ["X Y : CompactlyGenerated", "f : ↑X.toTop ≃ₜ ↑Y.toTop"], "goal": "(CompactlyGenerated.isoOfHomeo f).hom = { toFun := ⇑f, continuous_toFun := ⋯ }"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_5", "Semiring R", "SeminormedAddCommGroup E", "Module R E", "f : E →ₗᵢ[R] E", "n : ℕ"], "goal": "⇑(f ^ n) = (⇑f)^[n]"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ✝ : MeasurableSpace Ω", "μ✝ : Measure Ω", "f : Ω → ℝ≥0∞", "X Y : Ω → ℝ", "Mf Mg mΩ : MeasurableSpace Ω", "μ : Measure Ω", "hMf : Mf ≤ mΩ", "hMg : Mg ≤ mΩ", "h_ind : Indep Mf Mg μ", "h_meas_f : Measurable f"], "goal": "∀ {g : Ω → ℝ≥0∞}, Measurable g → ∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ"}}
+{"state": {"context": ["X : Type u", "inst✝² : TopologicalSpace X", "inst✝¹ : CompactSpace X", "inst✝ : T2Space X", "h : Projective X", "U : Set X", "hU : IsOpen U", "Z₁ : Set (X × Bool) := Uᶜ ×ˢ {true}", "Z₂ : Set (X × Bool) := closure U ×ˢ {false}", "Z : Set (X × Bool) := Z₁ ∪ Z₂", "hZ₁₂ : Disjoint Z₁ Z₂", "hZ₁ : IsClosed Z₁", "hZ₂ : IsClosed Z₂", "hZ : IsClosed Z", "f : ↑Z → X := Prod.fst ∘ Subtype.val", "f_cont : Continuous f", "f_sur : Surjective f", "this : CompactSpace ↑Z", "g : X → ↑Z", "hg : Continuous g", "g_sec : f ∘ g = id", "φ : X → X × Bool := Subtype.val ∘ g", "hφ : Continuous φ"], "goal": "IsOpen (closure U)"}}
+{"state": {"context": ["m✝ n a b c d m : ℕ", "h : ↑(m * n) ∣ ↑b - ↑a"], "goal": "↑n ∣ ↑b - ↑a"}}
+{"state": {"context": ["α : Type u_1", "f g h✝ : Perm α", "h : f.Disjoint g", "x : α"], "goal": "f⁻¹ x = x ∨ g x = x"}}
+{"state": {"context": ["p q : Prop", "dpq : Decidable (p ∨ q)", "dp : Decidable p", "dq : Decidable q"], "goal": "decide (p ∨ q) = (decide p || decide q)"}}
+{"state": {"context": ["n : Type v", "α : Type w", "DecidableEq n", "Fintype n", "CommRing α", "A : Matrix n n α", "β : Type u_1", "s : Finset β", "f : β → n → α"], "goal": "∑ x ∈ s, A.cramer (f x) = A.cramer (∑ x ∈ s, f x)"}}
+{"state": {"context": ["m n : ℕ"], "goal": "0 < ack (m + 1) (n + 1)"}}
+{"state": {"context": ["G : Type u_3", "Semigroup G", "a b c : G", "h : SemiconjBy a b c"], "goal": "Function.Semiconj (fun x => a * x) (fun x => b * x) fun x => c * x"}}
+{"state": {"context": ["f : ℝ → ℝ", "x a c d : ℝ", "l : Filter ℝ", "hl : l ≤ 𝓝[≠] x", "hf : Tendsto (fun y => (f y - d) / (y - x)) l (𝓝 a)", "h' : Tendsto (fun y => y + c * (y - x) ^ 2) l l", "L : Tendsto (fun y => (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)", "Z : Tendsto (fun x_1 => ((fun y => (f y - d) / (y - x)) ∘ fun y => y + c * (y - x) ^ 2) x_1 * ((x_1 + c * (x_1 - x) ^ 2 - x) / (x_1 - x))) l (𝓝 a)"], "goal": "Tendsto (fun y => (f (y + c * (y - x) ^ 2) - d) / (y - x)) l (𝓝 a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "X✝ Y✝ Z✝ : Pointed", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝"], "goal": "{ obj := fun X => { x // x ≠ X.point }, map := fun {X Y} f => PFun.toSubtype (fun x => x ≠ Y.point) f.toFun ∘ Subtype.val }.map (f ≫ g) = { obj := fun X => { x // x ≠ X.point }, map := fun {X Y} f => PFun.toSubtype (fun x => x ≠ Y.point) f.toFun ∘ Subtype.val }.map f ≫ { obj := fun X => { x // x ≠ X.point }, map := fun {X Y} f => PFun.toSubtype (fun x => x ≠ Y.point) f.toFun ∘ Subtype.val }.map g"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "n : ℕ∞"], "goal": "ContDiff 𝕜 n f ↔ HasFTaylorSeriesUpTo n f (ftaylorSeries 𝕜 f)"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "inst✝¹ : HasBinaryBiproducts C", "W X Y Z : C", "f : W ⟶ Y", "g : X ⟶ Z", "inst✝ : IsIso (biprod.map f g)", "t : f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = biprod.inl ≫ biprod.fst"], "goal": "f ≫ biprod.inl ≫ inv (biprod.map f g) ≫ biprod.fst = 𝟙 W"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : MeasurableSpace α", "μ ν : Measure α", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : IsFiniteMeasure ν", "d : Set α → ℝ := fun s => ↑(μ s).toNNReal - ↑(ν s).toNNReal", "c : Set ℝ := d '' {s | MeasurableSet s}", "γ : ℝ := sSup c", "hμ : ∀ (s : Set α), μ s ≠ ⊤", "hν : ∀ (s : Set α), ν s ≠ ⊤", "to_nnreal_μ : ∀ (s : Set α), ↑(μ s).toNNReal = μ s", "to_nnreal_ν : ∀ (s : Set α), ↑(ν s).toNNReal = ν s", "d_split : ∀ (s t : Set α), MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)", "d_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n)))", "d_Inter : ∀ (s : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (s n)) → (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n)))", "bdd_c : BddAbove c", "c_nonempty : c.Nonempty", "d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ", "this : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s", "e : ℕ → Set α", "he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)", "he₁ : ∀ (n : ℕ), MeasurableSet (e n)", "he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)", "f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e", "hf : ∀ (n m : ℕ), MeasurableSet (f n m)", "f_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c", "f_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)", "le_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)", "s : Set α := ⋃ m, ⋂ n, f m n", "γ_le_d_s : γ ≤ d s"], "goal": "∃ s, MeasurableSet s ∧ (∀ (t : Set α), MeasurableSet t → t ⊆ s → ν t ≤ μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → μ t ≤ ν t"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁶ : CanonicallyOrderedAddCommMonoid A", "inst✝⁵ : Sub A", "inst✝⁴ : OrderedSub A", "inst✝³ : ContravariantClass A A (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝² : HasAntidiagonal A", "n m : A", "inst✝¹ : DecidablePred fun x => x = m", "inst✝ : Decidable (m ≤ n)", "this : (fun x => x.2 = m) ∘ Prod.swap = fun x => x.1 = m"], "goal": "map { toFun := Prod.swap, inj' := ⋯ } (filter ((fun x => x.2 = m) ∘ ⇑{ toFun := Prod.swap, inj' := ⋯ }) (antidiagonal n)) = if m ≤ n then {(n - m, m)} else ∅"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y", "s : Set X"], "goal": "⇑h '' interior s = interior (⇑h '' s)"}}
+{"state": {"context": ["z τ : ℂ"], "goal": "jacobiTheta₂' (z + τ) τ = cexp (-↑π * I * (τ + 2 * z)) * (jacobiTheta₂' z τ - 2 * ↑π * I * jacobiTheta₂ z τ)"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "N : Type u_5", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "toFun : M → N", "toFun_smul : ∀ (a : R) (x : M), toFun (a • x) = (a * a) • toFun x", "polar_add_left :\n  ∀ (x x' y : M), QuadraticMap.polar toFun (x + x') y = QuadraticMap.polar toFun x y + QuadraticMap.polar toFun x' y", "polar_smul_left : ∀ (a : R) (x y : M), QuadraticMap.polar toFun (a • x) y = a • QuadraticMap.polar toFun x y"], "goal": "∀ (a : M), (QuadraticMap.ofPolar toFun toFun_smul polar_add_left polar_smul_left) a = toFun a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : ProbabilityTheory.Kernel α (β × ℝ)", "ν : ProbabilityTheory.Kernel α β", "f : α × β → StieltjesFunction", "ProbabilityTheory.IsFiniteKernel κ", "hf : ProbabilityTheory.IsCondKernelCDF f κ ν", "a : α", "s : Set (β × ℝ)", "hs : MeasurableSet s"], "goal": "∫⁻ (b : β), ((ProbabilityTheory.IsCondKernelCDF.toKernel f hf) (a, b)) {y | (b, y) ∈ s} ∂ν a = (κ a) s"}}
+{"state": {"context": ["X : Type u_1", "F : Type u_4", "MeasurableSpace X", "𝕜 : Type u_5", "NormedField 𝕜", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "p : ℝ≥0∞", "μ : MeasureTheory.Measure X", "c : 𝕜", "f : ↥(MeasureTheory.Lp F p μ)", "s : Set X"], "goal": "MeasureTheory.Memℒp.toLp ↑↑(c • f) ⋯ = c • MeasureTheory.Memℒp.toLp ↑↑f ⋯"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : C ⥤ₑ D"], "goal": "(CategoryTheory.LeftExactFunctor.ofExact C D).obj F = { obj := F.obj, property := ⋯ }"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Type u_2", "CommRing S", "f : R →+* S", "n ν : ℕ"], "goal": "Polynomial.map f (bernsteinPolynomial R n ν) = bernsteinPolynomial S n ν"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝³ : Field K", "inst✝² : LieRing L", "inst✝¹ : LieAlgebra K L", "inst✝ : Module.Finite K L", "hLK : ↑(finrank K L) ≤ #K", "U : LieSubalgebra K L", "x : L", "hxU : x ∈ U", "y : L", "hyU : y ∈ U", "Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩", "Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩", "hUle : U ≤ ↑Ex", "hmin : ∀ E ≤ Ex, Ex ≤ E", "E : LieSubmodule K (↥U) L := let __src := engel K x; { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }", "hx₀ : x ≠ 0", "Q : Type u_2 := L ⧸ E", "r : ℕ := finrank K ↥↑E", "hr : r < finrank K L", "x' : ↥U := ⟨x, hxU⟩", "y' : ↥U := ⟨y, hyU⟩", "u : ↥U := y' - x'", "χ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K (↥↑E) x' u", "ψ : K[X][X] := LieAlgebra.engel_isBot_of_isMin.lieCharpoly K Q x' u", "i : ℕ", "hi : i < r", "hi0 : i ≠ 0"], "goal": "χ.coeff i = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "d : D", "X : (CategoryTheory.CostructuredArrow F d)ᵒᵖ"], "goal": "(CategoryTheory.CostructuredArrow.toStructuredArrow F d).obj X =\n  CategoryTheory.StructuredArrow.mk (Opposite.unop X).hom.op"}}
+{"state": {"context": [], "goal": "⋂ r, Set.Iic ↑r = ∅"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra R L", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : LieRingModule L M", "inst✝ : LieModule R L M", "N✝ N' : LieSubmodule R L M", "I J : LieIdeal R L", "ι : Sort u_1", "N : ι → LieSubmodule R L M", "C : (x : M) → x ∈ ⨆ i, N i → Prop", "hN : ∀ (i : ι) (x : M) (hx : x ∈ N i), C x ⋯", "h0 : C 0 ⋯", "hadd : ∀ (x y : M) (hx : x ∈ ⨆ i, N i) (hy : y ∈ ⨆ i, N i), C x hx → C y hy → C (x + y) ⋯", "x : M", "hx : x ∈ ⨆ i, N i"], "goal": "C x hx"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "f : G →* G'"], "goal": "Nat.card ↥f.range = Nat.card ↑(Set.range ⇑f)"}}
+{"state": {"context": ["F : Type u_1", "Field F", "ι : Type u_2", "DecidableEq ι", "s : Finset ι", "v : ι → F", "i : ι", "hvs : Set.InjOn v ↑s", "hi : i ∈ s"], "goal": "(Lagrange.basis s v i).natDegree = s.card - 1"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "n : ℕ", "ifp_succ_n : IntFractPair K"], "goal": "IntFractPair.stream v (n + 1) = some ifp_succ_n ↔ ∃ ifp_n, IntFractPair.stream v n = some ifp_n ∧ ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "α : Type u_2", "inst✝ : Mul α", "J : Subgroup G", "g : G", "H K : Subgroup G", "a x : G"], "goal": "(∃ i, a⁻¹ * (↑i)⁻¹ * x ∈ ↑K) ↔ ∃ x_1 ∈ ↑H, ∃ y ∈ ↑K, x = x_1 * a * y"}}
+{"state": {"context": ["𝕜 : Type u_3", "G : Type u_4", "RCLike 𝕜", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : 𝕜 → G", "hf : Differentiable 𝕜 f", "hf' : ∀ (x : 𝕜), deriv f x = 0", "x y : 𝕜"], "goal": "f x = f y"}}
+{"state": {"context": ["α : Type u_2", "self : NormedOrderedAddGroup α", "x y : α"], "goal": "dist x y = ‖x - y‖"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "IsWellOrder α fun x x_1 => x < x_1", "a b : α", "h : a < wellOrderSucc b"], "goal": "a ≤ b"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M"], "goal": "0 = ⟨0, ⋯⟩"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommMonoid α", "TopologicalSpace α", "T2Space α", "ContinuousMul α", "DecidableEq β", "f : β → α", "b : β", "hf : Multipliable (Function.update f b 1)"], "goal": "∏' (x : β), f x = f b * ∏' (x : β), if x = b then 1 else f x"}}
+{"state": {"context": ["x✝ y z : ℝ", "n : ℕ", "x : ℝ"], "goal": "x ^ 2 = x ^ 2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝³ : LinearOrderedRing α", "inst✝² : FloorRing α", "inst✝¹ : TopologicalSpace α", "inst✝ : OrderClosedTopology α", "n : ℤ"], "goal": "pure (IntCast.intCast (n + 1)) ≤ 𝓝[≥] ↑(n + 1)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "inst✝ : TopologicalSpace γ", "s : Opens α"], "goal": "CompleteLattice.IsCompactElement s ↔ IsCompact ↑s"}}
+{"state": {"context": ["R : Type u_2", "Add R", "Mul R", "r : R → R → Prop"], "goal": "ringConGen r = sInf {s | ∀ (x y : R), r x y → s x y}"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "y : Y", "f : X → Z"], "goal": "(Inducing fun x => (f x, y)) ↔ Inducing f"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "D : Type w", "CategoryTheory.Category.{max v u, w} D", "CategoryTheory.ConcreteCategory D", "CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget D)", "∀ (P : CategoryTheory.Functor Cᵒᵖ D) (X : C) (S : J.Cover X), CategoryTheory.Limits.HasMultiequalizer (S.index P)", "∀ (X : C), CategoryTheory.Limits.HasColimitsOfShape (J.Cover X)ᵒᵖ D", "(X : C) → CategoryTheory.Limits.PreservesColimitsOfShape (J.Cover X)ᵒᵖ (CategoryTheory.forget D)", "(CategoryTheory.forget D).ReflectsIsomorphisms"], "goal": "∀ {X Y : CategoryTheory.Functor Cᵒᵖ D} (η : X ⟶ Y), ((CategoryTheory.plusPlusSheaf J D).map η).val = J.sheafifyMap η"}}
+{"state": {"context": ["J : Type w", "C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f : J → (X ⟶ Y)", "CategoryTheory.Limits.HasWideCoequalizer f", "Nonempty J", "W : C", "k : Y ⟶ W", "h : ∀ (j₁ j₂ : J), CategoryTheory.CategoryStruct.comp (f j₁) k = CategoryTheory.CategoryStruct.comp (f j₂) k"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.wideCoequalizer.π f)\n    (CategoryTheory.Limits.wideCoequalizer.desc k h) =\n  k"}}
+{"state": {"context": ["R : CommRingCat", "f : ↑Γ(Spec R, ⊤)"], "goal": "(Spec R).basicOpen f = PrimeSpectrum.basicOpen ((Scheme.ΓSpecIso R).hom f)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "n : ℕ", "ζ : R", "hζ : IsPrimitiveRoot ζ n", "α a : R", "hn : 0 < n", "e : α ^ n = a"], "goal": "Polynomial.X ^ n - Polynomial.C a = ∏ i ∈ Finset.range n, (Polynomial.X - Polynomial.C (ζ ^ i * α))"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝⁵ : Category.{max v u, w} D", "inst✝⁴ : ConcreteCategory D", "inst✝³ : PreservesLimits (forget D)", "inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)", "P : Cᵒᵖ ⥤ D", "hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y", "X : C", "S : J.Cover X", "s : Meq (J.plusObj P) S", "inj : ∀ (X : C), Function.Injective ⇑((J.toPlus P).app (op X))", "T : (I : S.Arrow) → J.Cover I.Y", "t : (I : S.Arrow) → Meq P (T I)", "ht : ∀ (I : S.Arrow), ↑s I = mk (t I)", "B : J.Cover X := S.bind T", "Z : B.Arrow → C", "e1 : (I : B.Arrow) → I.Y ⟶ Z I", "e2 : (I : B.Arrow) → Z I ⟶ X", "he2 : ∀ (I : B.Arrow), S.sieve.arrows (e2 I)", "h✝ : ∀ (I : B.Arrow), (fun Y f h => ((fun Y f hf => (T { Y := Y, f := f, hf := hf }).sieve) Y f h).arrows) (Z I) (e2 I) ⋯ (e1 I)", "a✝ : ∀ (I : B.Arrow), e1 I ≫ e2 I = I.f", "w : Meq P B := meqOfSep P hsep X S s T t ht"], "goal": "Meq.mk S (mk w) = s"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : TopologicalSpace α", "inst✝¹ : Preorder α", "inst✝ : ∀ (x : α), (𝓝[<] x).NeBot", "s : Set α", "hs : IsAntichain (fun x x_1 => x ≤ x_1) s", "x : α", "hx : x ∈ interior s", "this : ∀ᶠ (y : α) in 𝓝 x, y ∈ s"], "goal": "False"}}
+{"state": {"context": ["G : Type w", "TopologicalSpace G", "InvolutiveNeg G", "ContinuousNeg G", "s : Set G", "hs : IsClosed s"], "goal": "IsClosed (-s)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J K : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "CategoryTheory.Category.{v₂, u₁} K", "F : J ⥤ C", "G : K ⥤ C", "h : F.cones ≅ G.cones", "CategoryTheory.Limits.HasLimit F"], "goal": "CategoryTheory.Limits.HasLimit G"}}
+{"state": {"context": ["b e : String.Pos", "s : Substring"], "goal": "s.Valid → { str := s.toString, startPos := b, stopPos := e }.Valid → (s.extract b e).toString = s.toString.extract b e"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝¹ : TopologicalSpace α", "s t u v : Set α", "inst✝ : LocallyConnectedSpace α", "F : Set α", "x : α", "hF : IsOpen F", "y : α", "hy : y ∈ connectedComponentIn F x"], "goal": "connectedComponentIn F y ∈ 𝓝 y"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "hi : Irreducible p", "x : R", "hx : p.IsRoot x", "g : R[X]", "hg : p = (X - C x) * g", "this : IsUnit (X - C x) ∨ IsUnit g", "h : IsUnit (X - C x)", "h₁ : (X - C x).degree = 1"], "goal": "p.degree = 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "PseudoMetricSpace β", "Zero β", "f : BoundedContinuousFunction α β"], "goal": "(∀ (x : α), f x = 0) ↔ f = 0"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "s t : Set α", "h : s ⊆ t"], "goal": "s ≤ᶠ[l] t"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "C : β → Type u_7", "(b : β) → SemilatticeInf (C b)", "(b : β) → OrderTop (C b)", "s : Finset α", "f : α → (b : β) → C b", "b : β"], "goal": "s.inf f b = s.inf fun a => f a b"}}
+{"state": {"context": ["f : ℂ ≃ₗᵢ[ℝ] ℂ", "h : f 1 = 1"], "goal": "f = LinearIsometryEquiv.refl ℝ ℂ ∨ f = Complex.conjLIE"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "φ ψ : ℝ → ℝ", "s : Set E", "f g : E → ℝ", "hf : UniformConvexOn s φ f", "hg : UniformConvexOn s ψ g"], "goal": "UniformConvexOn s (φ + ψ) (f + g)"}}
+{"state": {"context": ["R : Type u", "Ring R", "I : Ideal R"], "goal": "I.jacobson = I ↔ ∃ M, (∀ J ∈ M, J.IsMaximal ∨ J = ⊤) ∧ I = sInf M"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommGroup α", "UniformSpace α", "UniformGroup α", "f : β → α"], "goal": "(CauchySeq fun s => ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Zero β", "SMulZeroClass α β", "DecidableEq β", "t : Finset β", "a : α", "h : 0 ∈ t"], "goal": "0 ∈ a • t"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedCoheytingAlgebra α", "a b : α"], "goal": "a ∆ b ∆ (a ⊓ b) = a ⊔ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddCommMonoid α", "TopologicalSpace α", "f : β → α", "IsEmpty β"], "goal": "HasSum f 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "ε a₁ a₂ b₁ b₂ : α"], "goal": "|a₁ - b₁| < ε → |a₂ - b₂| < ε → |a₁ ⊔ a₂ - b₁ ⊔ b₂| < ε"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : PartialOrder α", "inst✝¹ : PredOrder α", "a b : α", "C : α → Sort u_2", "inst✝ : WellFoundedGT α", "H_pred : (a : α) → ¬IsMin a → C a → C (pred a)", "H_lim : (a : α) → IsPredLimit a → ((b : α) → b > a → C b) → C a", "ha : ¬IsMin a", "h : ¬IsPredLimit (pred a)"], "goal": "(let x := Classical.indefiniteDescription (fun x => ¬IsMin x ∧ pred x = pred a) ⋯; ⋯ ▸ H_pred ↑x ⋯ ((fun y x => ⋯.fix (fun a IH => if h : IsPredLimit a then H_lim a h IH else let x := Classical.indefiniteDescription (fun x => ¬IsMin x ∧ pred x = a) ⋯; ⋯ ▸ H_pred ↑x ⋯ (IH ↑x ⋯)) y) ↑x ⋯)) = H_pred a ha (PredOrder.limitRecOn a H_pred H_lim)"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.Scheme", "f : X ⟶ Y", "h : AlgebraicGeometry.IsOpenImmersion f", "CategoryTheory.Epi f.val.base"], "goal": "CategoryTheory.IsIso f"}}
+{"state": {"context": ["m n✝ n : ℕ", "inst✝ : NeZero n", "x : ZMod n"], "goal": "(0 ≤ if x.val ≤ n / 2 then ↑x.val else ↑x.val - ↑n) ↔ x.val ≤ n / 2"}}
+{"state": {"context": ["R : Type u_1", "AddMonoid R", "S : AddSubmonoid R", "AddOreLocalization.AddOreSet S", "X : Type u_3", "AddAction R X", "C : Sort u_2", "P : X → ↥S → X → ↥S → C", "hP :\n  ∀ (r₁ : X) (t₁ : R) (s₁ : ↥S) (ht₁ : t₁ + ↑s₁ ∈ S) (r₂ : X) (t₂ : R) (s₂ : ↥S) (ht₂ : t₂ + ↑s₂ ∈ S),\n    P r₁ s₁ r₂ s₂ = P (t₁ +ᵥ r₁) ⟨t₁ + ↑s₁, ht₁⟩ (t₂ +ᵥ r₂) ⟨t₂ + ↑s₂, ht₂⟩", "r₁ : X", "s₁ : ↥S", "r₂ : X", "s₂ : ↥S"], "goal": "AddOreLocalization.lift₂Expand P hP (r₁ -ₒ s₁) (r₂ -ₒ s₂) = P r₁ s₁ r₂ s₂"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s✝ : Set E", "i j : ι", "c : R", "t✝ : Finset ι", "w : ι → R", "z : ι → E", "s t : Set E"], "goal": "(convexHull R) (s - t) = (convexHull R) s - (convexHull R) t"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a", "i j n : ℕ", "ipos : 0 < i", "hin : i ≤ n", "j4n : j ≤ 4 * n", "h : xn a1 j ≡ xn a1 i [MOD xn a1 n]", "i2n : i ≤ 2 * n", "j2n : 2 * n < j"], "goal": "4 * n - j + ?m.167070 i2n j2n ≤ 2 * n + ?m.167070 i2n j2n"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a b : List α"], "goal": "a.toFinset = b.toFinset ↔ ∀ (x : α), x ∈ a ↔ x ∈ b"}}
+{"state": {"context": ["z : ℂ"], "goal": "|z.im| = Complex.abs z ↔ z.re = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "x y : α × β"], "goal": "Prod.Lex r s x y ↔ r x.1 y.1 ∨ x.1 = y.1 ∧ s x.2 y.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι' : Sort u_5", "la : Filter α", "lb : Filter β", "pb : ι' → Prop", "sb : ι' → Set β", "f : α → β", "H : Filter.Tendsto f la lb", "hlb : lb.HasBasis pb sb", "i : ι'"], "goal": "pb i → ∀ᶠ (x : α) in la, f x ∈ sb i"}}
+{"state": {"context": ["α : Type u_1", "CanonicallyLinearOrderedAddCommMonoid α", "Sub α", "OrderedSub α", "a b : α", "ha : AddLECancellable a"], "goal": "a - b < a ↔ 0 < a ∧ 0 < b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹⁹ : CommSemiring R", "inst✝¹⁸ : AddCommMonoid M", "inst✝¹⁷ : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝¹⁶ : CommRing R₁", "inst✝¹⁵ : AddCommGroup M₁", "inst✝¹⁴ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝¹³ : Field K", "inst✝¹² : AddCommGroup V", "inst✝¹¹ : Module K V", "M'✝ : Type u_7", "M'' : Type u_8", "inst✝¹⁰ : AddCommMonoid M'✝", "inst✝⁹ : AddCommMonoid M''", "inst✝⁸ : Module R M'✝", "inst✝⁷ : Module R M''", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M' : Type u_9", "inst✝⁶ : AddCommMonoid M'", "inst✝⁵ : Module R M'", "inst✝⁴ : FiniteDimensional K V", "ι✝ : Type u_10", "inst✝³ : DecidableEq ι✝", "inst✝² : Finite ι✝", "B : BilinForm K V", "hB : B.Nondegenerate", "ι : Type u_11", "inst✝¹ : Finite ι", "inst✝ : DecidableEq ι", "b : Basis ι K V", "i j : ι"], "goal": "(B ((B.dualBasis hB (B.flip.dualBasis ⋯ b)) i)) ((B.flip.dualBasis ⋯ b) j) = (B (b i)) ((B.flip.dualBasis ⋯ b) j)"}}
+{"state": {"context": ["p q : ℝ", "hpq : p.IsConjExponent q"], "goal": "p.toNNReal.IsConjExponent q.toNNReal"}}
+{"state": {"context": ["G₀ : Type u_1", "α : Type u_2", "GroupWithZero G₀", "Bornology G₀", "MulAction G₀ α", "t : Set α", "ι : Type u_3", "I : Set ι", "hI : I.Finite", "s : ι → Set α"], "goal": "Absorbs G₀ (⋂ i ∈ I, s i) t ↔ ∀ i ∈ I, Absorbs G₀ (s i) t"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s s₁ s₂ t✝ t₁ t₂ u : Set α", "p : Prop", "inst✝ : Decidable p", "t : p → Set α", "x : α"], "goal": "(x ∈ if h : p then t h else ∅) ↔ ∃ (h : p), x ∈ t h"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "κ : ProbabilityTheory.Kernel α α", "μ : MeasureTheory.Measure α", "hκ : κ.Invariant μ"], "goal": "κ ∘ₖ ProbabilityTheory.Kernel.const α μ = ProbabilityTheory.Kernel.const α μ"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "A : Type u₂", "inst✝² : Category.{v₂, u₂} A", "D : Type u_1", "inst✝¹ : Category.{u_2, u_1} D", "inst✝ : HasWeakSheafify J D", "P Q : Cᵒᵖ ⥤ D", "η γ : sheafify J P ⟶ Q", "hQ : Presheaf.IsSheaf J Q", "h : toSheafify J P ≫ η = toSheafify J P ≫ γ"], "goal": "sheafifyLift J (toSheafify J P ≫ η) hQ = γ"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t u : Set Ω", "hs : s.Finite", "hs' : s.Nonempty", "ht : s ⊆ t", "this : IsProbabilityMeasure (condCount s)"], "goal": "¬(condCount s) t < 1"}}
+{"state": {"context": ["R : Type r", "CommRing R", "W : WeierstrassCurve R", "m : ℕ"], "goal": "W.ΨSq (2 * (↑m + 3)) =\n  (W.preΨ' (m + 2) ^ 2 * W.preΨ' (m + 3) * W.preΨ' (m + 5) - W.preΨ' (m + 1) * W.preΨ' (m + 3) * W.preΨ' (m + 4) ^ 2) ^\n      2 *\n    W.Ψ₂Sq"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "h : X ≃ₜ Y", "f : Y → Z"], "goal": "Continuous (f ∘ ⇑h) ↔ Continuous f"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : TopologicalSpace X", "β : Type u_2", "inst✝ : TopologicalSpace β", "A : Set X", "f : X → β", "hf1 : Continuous f", "hf2 : Function.Injective f", "x : β", "hx : ∃ x_1, AccPt x_1 (𝓟 A) ∧ f x_1 = x"], "goal": "x ∈ derivedSet (f '' A)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : CompleteSpace E", "f : ℕ → α → E", "p : ℝ", "hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ", "hp1 : 1 ≤ p", "B : ℕ → ℝ≥0∞", "hB : ∑' (i : ℕ), B i ≠ ⊤", "h_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm' (f n - f m) p μ < B N", "h_summable : ∀ᵐ (x : α) ∂μ, Summable fun i => f (i + 1) x - f i x", "h : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => ∑ i ∈ Finset.range n, (f (i + 1) x - f i x)) atTop (𝓝 l)"], "goal": "∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) atTop (𝓝 l)"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : GroupWithZero G₀", "a b c d : G₀", "m n : ℕ"], "goal": "a / b = 0 ↔ a = 0 ∨ b = 0"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Preadditive C", "CategoryTheory.Preadditive D", "S : CategoryTheory.ShortComplex C", "s : S.Splitting", "F : CategoryTheory.Functor C D", "F.Additive"], "goal": "(s.map F).r = F.map s.r"}}
+{"state": {"context": ["α : Type u", "Group α", "i : ℕ", "a b : α"], "goal": "(a * b * a⁻¹) ^ i = a * b ^ i * a⁻¹"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrderedRing α", "LinearOrderedAddCommGroup β", "Module α β", "PosSMulStrictMono α β", "a : α", "b : β"], "goal": "a • b ≤ 0 ↔ (a < 0 → 0 ≤ b) ∧ (0 < b → a ≤ 0)"}}
+{"state": {"context": ["X : ℕ → Type u", "inst✝ : (n : ℕ) → MetricSpace (X n)", "f : (n : ℕ) → X n → X (n + 1)", "I : ∀ (n : ℕ), Isometry (f n)", "n : ℕ", "h : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I", "x✝ : TopologicalSpace ((n : ℕ) × X n) := UniformSpace.toTopologicalSpace", "x : X n"], "goal": "dist ⟨n.succ, f n x⟩ ⟨n, x⟩ = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "F.Final", "CategoryTheory.IsFiltered C"], "goal": "CategoryTheory.IsFiltered D"}}
+{"state": {"context": ["n : Type u_4", "α : Type u_5", "ι : Type u_7", "Fintype n", "Unique ι", "SeminormedAddCommGroup α", "v : n → α"], "goal": "‖Matrix.col ι v‖₊ = ‖(WithLp.equiv 2 (n → α)).symm v‖₊"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "l : Type u_3", "m : Type u_4", "n : Type u_5", "DecidableEq n", "Fintype n", "Fintype m", "DecidableEq m", "M : Matrix l m R", "N : Matrix m n R", "x : n → R"], "goal": "(Matrix.toLin' (M * N)) x = (Matrix.toLin' M) ((Matrix.toLin' N) x)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "inst✝¹ : CompleteSpace F", "μ ν : Measure α", "s t N : Set α", "f : α → ℝ≥0∞", "inst✝ : IsFiniteMeasure μ", "hμ : μ ≠ 0", "hf : AEMeasurable f μ", "hN : μ N = 0", "this : 0 < μ ({x | f x ≤ ⨍⁻ (a : α), f a ∂μ} \\ N)"], "goal": "∃ x ∉ N, f x ≤ ⨍⁻ (a : α), f a ∂μ"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f g : 𝕜 → F", "f' g' : F", "x : 𝕜", "hf : HasStrictDerivAt f f' x", "hg : HasStrictDerivAt g g' x"], "goal": "HasStrictDerivAt (fun x => f x - g x) (f' - g') x"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "Group G", "MulAction G α", "FaithfulSMul G α", "g h : G", "ne_one : g ≠ 1", "disjoint : Disjoint (MulAction.fixedBy α g)ᶜ (h • (MulAction.fixedBy α g)ᶜ)"], "goal": "¬Commute g h"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "W : Type u_3", "Q : Type u_4", "P : Type u_5", "inst✝⁸ : NormedField 𝕜", "inst✝⁷ : SeminormedAddCommGroup V", "inst✝⁶ : SeminormedAddCommGroup W", "inst✝⁵ : NormedSpace 𝕜 V", "inst✝⁴ : NormedSpace 𝕜 W", "inst✝³ : MetricSpace P", "inst✝² : PseudoMetricSpace Q", "inst✝¹ : NormedAddTorsor V P", "inst✝ : NormedAddTorsor W Q", "φ : P →ᵃⁱ[𝕜] Q", "s : Set P", "hs : s.Nonempty", "this : Nonempty ↑s"], "goal": "intrinsicClosure 𝕜 (⇑φ '' s) = ⇑φ '' intrinsicClosure 𝕜 s"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Relation.Comp List.Perm List.Perm = List.Perm"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : AddCommSemigroup α", "inst✝⁴ : PartialOrder α", "inst✝³ : ExistsAddOfLE α", "inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a b c d : α", "hb : AddLECancellable b", "hba : b ≤ a"], "goal": "a < b + c → a - b < c"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "v : α → M", "l : α →₀ R"], "goal": "(Finsupp.total α M R v) l = l.sum fun i a => a • v i"}}
+{"state": {"context": ["z τ : ℂ", "hτ : 0 < τ.im", "T : ℝ", "hT : 0 < T", "hτ' : T < τ.im", "S : ℝ", "hz : |z.im| < S", "V : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}", "hVo : IsOpen V", "hVmem : (z, τ) ∈ V", "hVp : IsPreconnected V", "f : ℤ → ℂ × ℂ → ℂ := fun n p => jacobiTheta₂_term n p.1 p.2", "f' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p => jacobiTheta₂_term_fderiv n p.1 p.2", "hf : ∀ (n : ℤ), ∀ p ∈ V, HasFDerivAt (f n) (f' n p) p", "u : ℤ → ℝ := fun n => 3 * π * ↑|n| ^ 2 * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))", "hu : ∀ (n : ℤ), ∀ x ∈ V, ‖f' n x‖ ≤ u n", "hu_sum : Summable u", "hf_sum : Summable fun n => f n (z, τ)"], "goal": "HasFDerivAt (fun p => jacobiTheta₂ p.1 p.2) (jacobiTheta₂_fderiv z τ) (z, τ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β", "g : β → γ", "h : Function.Injective (g ∘ f)"], "goal": "Set.InjOn g (Set.range f)"}}
+{"state": {"context": ["R : Type u", "inst✝¹⁰ : CommRing R", "S✝ : Submonoid R", "M : Type w", "inst✝⁹ : AddCommMonoid M", "inst✝⁸ : Module R M", "t : Finset R", "ht : Ideal.span ↑t = ⊤", "Mₚ : { x // x ∈ t } → Type u_1", "inst✝⁷ : (g : { x // x ∈ t }) → AddCommMonoid (Mₚ g)", "inst✝⁶ : (g : { x // x ∈ t }) → Module R (Mₚ g)", "Rₚ : { x // x ∈ t } → Type u", "inst✝⁵ : (g : { x // x ∈ t }) → CommRing (Rₚ g)", "inst✝⁴ : (g : { x // x ∈ t }) → Algebra R (Rₚ g)", "inst✝³ : ∀ (g : { x // x ∈ t }), IsLocalization.Away (↑g) (Rₚ g)", "inst✝² : (g : { x // x ∈ t }) → Module (Rₚ g) (Mₚ g)", "inst✝¹ : ∀ (g : { x // x ∈ t }), IsScalarTower R (Rₚ g) (Mₚ g)", "f : (g : { x // x ∈ t }) → M →ₗ[R] Mₚ g", "inst✝ : ∀ (g : { x // x ∈ t }), IsLocalizedModule (Submonoid.powers ↑g) (f g)", "H : ∀ (g : { x // x ∈ t }), Finite (Rₚ g) (Mₚ g)", "s₁ : (g : { x // x ∈ t }) → Finset (Mₚ g)", "s₂ : ∀ (g : { x // x ∈ t }), Submodule.span (Rₚ g) ↑(s₁ g) = ⊤", "sf : { x // x ∈ t } → Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers ↑x) (f x) (s₁ x)", "x : M", "r : ↑↑t", "S : Submonoid R := Submonoid.powers ↑r", "n₁ : ℕ", "hn₁ : (f r) (↑⟨(fun x => ↑r ^ x) n₁, ⋯⟩ • x) ∈ Submodule.span R ↑(s₁ r)"], "goal": "∃ n, ↑r ^ n • x ∈ ⨆ x, Submodule.span R ↑(sf x)"}}
+{"state": {"context": ["𝕜 : Type u_4", "NormedDivisionRing 𝕜", "CompleteSpace 𝕜", "r : 𝕜", "hr : ‖r‖ < 1"], "goal": "HasSum (fun n => ↑n * r ^ n) (r / (1 - r) ^ 2)"}}
+{"state": {"context": ["X : Type u", "x : X", "s : Set X", "TopologicalSpace X"], "goal": "x ∈ closure s ↔ (Filter.comap Subtype.val (𝓝 x)).NeBot"}}
+{"state": {"context": ["G : Type u", "inst✝ : AddCommGroup G", "hG : AddGroup.FG G", "n : ℕ", "ι : Type", "fι : Fintype ι", "p : ι → ℤ", "hp : ∀ (i : ι), Irreducible (p i)", "e : ι → ℕ", "f : G ≃ₗ[ℤ] (Fin n →₀ ℤ) × ⨁ (i : ι), ℤ ⧸ Submodule.span ℤ {p i ^ e i}"], "goal": "G ≃+ (Fin n →₀ ℤ) × ⨁ (i : ι), ZMod ((fun i => (p i).natAbs) i ^ e i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrderedField α", "inst✝¹ : Ring β", "abv : β → α", "inst✝ : IsAbsoluteValue abv", "f : CauSeq β abv", "hf : f.LimZero"], "goal": "(-1 * f).LimZero"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "h e f : L", "m : M", "μ : R", "t : IsSl2Triple h e f", "P : t.HasPrimitiveVectorWith m μ", "inst✝³ : IsNoetherian R M", "inst✝² : NoZeroSMulDivisors R M", "inst✝¹ : IsDomain R", "inst✝ : CharZero R", "hs : (range fun n => μ - 2 * ↑n).Infinite", "contra : ∀ (n : ℕ), ((toEnd R L M) f ^ n) m ≠ 0"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "f : MeasureTheory.SimpleFunc α β"], "goal": "f.bind (MeasureTheory.SimpleFunc.const α) = f"}}
+{"state": {"context": ["α : Type u_1", "P : Set α → Prop", "m : (s : Set α) → P s → ℝ≥0∞", "P0 : P ∅", "m0 : m ∅ P0 = 0", "PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)", "mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯", "msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯", "m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂", "s : Set α", "hs : (inducedOuterMeasure m P0 m0) s ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0", "h : (inducedOuterMeasure m P0 m0) s < (inducedOuterMeasure m P0 m0) s + ε"], "goal": "∃ t, P t ∧ s ⊆ t ∧ (inducedOuterMeasure m P0 m0) t ≤ (inducedOuterMeasure m P0 m0) s + ε"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "m : Type u_3", "n : Type u_4", "inst✝³ : DecidableEq m", "inst✝² : DecidableEq n", "inst✝¹ : Fintype m", "inst✝ : Fintype n", "M₁₁ : Matrix m m R", "M₁₂ : Matrix m n R", "M₂₁ : Matrix n m R", "M₂₂ M : Matrix n n R", "i j : n"], "goal": "(fromBlocks M₁₁ 0 M₂₁ M₂₂).charpoly = M₁₁.charpoly * M₂₂.charpoly"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "R : Type u_3", "CommSemiring R", "υ : Type u_6", "f : σ → MvPolynomial τ R", "g : τ → υ", "φ : MvPolynomial σ R"], "goal": "(MvPolynomial.rename g) ((MvPolynomial.bind₁ f) φ) = (MvPolynomial.bind₁ fun i => (MvPolynomial.rename g) (f i)) φ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁴ : CommMonoid α", "inst✝³ : TopologicalSpace α", "f✝ g✝ : β → α", "a a₁ a₂ : α", "inst✝² : T2Space α", "α' : Type u_5", "inst✝¹ : CommMonoid α'", "inst✝ : TopologicalSpace α'", "e : α' → α", "hes : Surjective e", "h1 : e 1 = 1", "f : β → α", "g : γ → α'", "h : ∀ {a : α'}, HasProd f (e a) ↔ HasProd g a", "hg : ¬Multipliable g", "hf : ¬Multipliable f"], "goal": "∏' (b : β), f b = e (∏' (c : γ), g c)"}}
+{"state": {"context": ["α : Type u_2", "BooleanAlgebra α", "s : Finset α"], "goal": "s.compls.compls = s"}}
+{"state": {"context": ["X : RingedSpace", "U : Opens ↑↑X.toPresheafedSpace", "f g : ↑(X.presheaf.obj (op U))"], "goal": "Subtype.val '' {x | IsUnit ((X.presheaf.germ x) (f * g))} = (fun a => ↑a) '' ({x | IsUnit ((X.presheaf.germ x) f)} ∩ {x | IsUnit ((X.presheaf.germ x) g)})"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "inst✝⁶ : CommRing A", "S : Submonoid A", "hS : S ≤ A⁰", "inst✝⁵ : Field K", "inst✝⁴ : Algebra A K", "inst✝³ : IsFractionRing A K", "B : Type u_3", "inst✝² : CommRing B", "inst✝¹ : Algebra A B", "inst✝ : IsLocalization S B", "x : K"], "goal": "x ∈ (mapToFractionRing K S B hS).range ↔ ∃ a s, ∃ (_ : s ∈ S), x = (algebraMap A K) a * ((algebraMap A K) s)⁻¹"}}
+{"state": {"context": ["a b : ℝ", "ha : a ≤ 0", "z : ℂ", "hz : |z.im| ≤ b", "hb : b ≤ π / 2"], "goal": "abs (cexp (↑a * (cexp z + cexp (-z)))) ≤ rexp (a * Real.cos b * rexp |z.re|)"}}
+{"state": {"context": ["α : Type u_1", "DivisionCommMonoid α", "a b c : α"], "goal": "a * (b / c) = b * (a / c)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "R : Type u_3", "M : Type u_5", "TopologicalSpace M", "SMul R M", "ContinuousConstSMul R M", "c : R", "f : C(α, M)", "a : α"], "goal": "(c • f) a = c • f a"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "S : Subalgebra R A", "L : List A", "h : ∀ x ∈ L, x ∈ S"], "goal": "L.prod ∈ S"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : β"], "goal": "(Sum.inr x).swap = Sum.inl x"}}
+{"state": {"context": ["x : ℝ", "hx : |x| ≤ 1"], "goal": "|rexp x - 1 - x| ≤ |x| ^ 2"}}
+{"state": {"context": ["p q : ℚ≥0", "n₁ n₂ d₁ d₂ d : ℕ", "h₁ : d₁ ≠ 0", "h₂ : d₂ ≠ 0"], "goal": "↑n₁ * ↑d₂ = ↑n₂ * ↑d₁ ↔ n₁ * d₂ = n₂ * d₁"}}
+{"state": {"context": ["α : Type u", "Group α", "LT α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α"], "goal": "c < a * b⁻¹ ↔ c * b < a"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "x : α"], "goal": "IsClosed (connectedComponent x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁷ : MeasurableSpace α", "K : Type u_5", "inst✝⁶ : SemilatticeSup β", "inst✝⁵ : OrderBot β", "inst✝⁴ : Zero β", "inst✝³ : TopologicalSpace β", "inst✝² : OrderClosedTopology β", "inst✝¹ : MeasurableSpace β", "inst✝ : OpensMeasurableSpace β", "i : ℕ → β", "f : α → β", "n : ℕ", "a : α", "hf : Measurable f", "k : ℕ"], "goal": "MeasurableSet {a | i k ≤ f a}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "α : Type u_4", "β : Type u_5", "ι : Type u_6", "inst✝⁶ : OrderedSemiring 𝕜", "inst✝⁵ : AddCommMonoid E", "inst✝⁴ : AddCommMonoid F", "inst✝³ : OrderedAddCommMonoid α", "inst✝² : OrderedAddCommMonoid β", "inst✝¹ : Module 𝕜 E", "inst✝ : SMul 𝕜 β", "s : Set E", "f : E → β", "hf : ConvexOn 𝕜 s f", "c x : E", "hx : x ∈ (fun z => c + z) ⁻¹' s", "y : E", "hy : y ∈ (fun z => c + z) ⁻¹' s", "a b : 𝕜", "ha : 0 ≤ a", "hb : 0 ≤ b", "hab : a + b = 1"], "goal": "f (c + (a • x + b • y)) = f (a • (c + x) + b • (c + y))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : Option α ≃ Option β", "h : e none = none"], "goal": "e.symm none = none"}}
+{"state": {"context": ["R : Type u", "x : Tropical R"], "goal": "Tropical.trop (Tropical.untrop x) = x"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "Zero M", "DecidableEq α", "DecidableEq M", "l : AList fun _x => M"], "goal": "l.lookupFinsupp.support = (List.filter (fun x => decide (x.snd ≠ 0)) l.entries).keys.toFinset"}}
+{"state": {"context": ["e₁ : ONote", "n₁ : ℕ+", "a₁ e₂ : ONote", "n₂ : ℕ+", "a₂ : ONote", "b : Ordinal.{0}", "h₁ : (e₁.oadd n₁ a₁).NFBelow b", "h₂ : (e₂.oadd n₂ a₂).NF"], "goal": "(e₁.oadd n₁ a₁ - e₂.oadd n₂ a₂).NFBelow b"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "MulOneClass α", "Zero α", "Preorder α", "PosMulMono α", "ha : a ≤ 1", "hb : b ≤ 1", "a0 : 0 ≤ a"], "goal": "a * b ≤ 1"}}
+{"state": {"context": ["G : Type u", "Group G", "a : G ⧸ (MonoidHom.id G).ker"], "goal": "QuotientGroup.quotientBot a = (QuotientGroup.kerLift (MonoidHom.id G)) a"}}
+{"state": {"context": ["α : Type u_1", "LE α", "S : Set (LowerSet α)"], "goal": "(sInf S).compl = ⨅ s ∈ S, s.compl"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "M' : Type u_3", "F : Type u_4", "G : Type u_5", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "N N₁ N₂ P P₁ P₂ : Submodule R M", "I : Ideal R"], "goal": "colon I ⊤ = I"}}
+{"state": {"context": ["x y : PSet", "h : x.Equiv y"], "goal": "ZFSet.mk x = ZFSet.mk y"}}
+{"state": {"context": ["R : Type u_1", "n : Type u_3", "DecidableEq n", "Semiring R", "Fintype n", "a : R", "i : n"], "goal": "∑ x : n, Finsupp.single x (if i = x then a else 0) = Finsupp.single i a"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "S : Set L", "h₁ : ∀ {a b : L}, a ∈ S → b ∈ S → a + b ∈ S", "h₂ : S 0", "h₃ :\n  ∀ (c : R) {x : L},\n    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier →\n      c • x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂ }.carrier", "h₄ :\n  ∀ {x y : L},\n    x ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →\n      y ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier →\n        ⁅x, y⁆ ∈ { carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃ }.carrier"], "goal": "↑{ carrier := S, add_mem' := h₁, zero_mem' := h₂, smul_mem' := h₃, lie_mem' := h₄ } = S"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "I : BoxIntegral.Box ι", "l : BoxIntegral.IntegrationParams", "π₁ π₂ : BoxIntegral.Prepartition I", "h : π₁.iUnion = π₂.iUnion"], "goal": "l.toFilteriUnion I π₁ = l.toFilteriUnion I π₂"}}
+{"state": {"context": ["α : Type u_1"], "goal": "(PartialEquiv.refl α).target = Set.univ"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : GeneralizedHeytingAlgebra α", "a b c d : α"], "goal": "c ⊓ (b ⊓ a) ≤ c"}}
+{"state": {"context": ["n : Type u_1", "p : Type u_2", "R : Type u₂", "𝕜 : Type u_3", "inst✝⁴ : Field 𝕜", "inst✝³ : DecidableEq n", "inst✝² : DecidableEq p", "inst✝¹ : CommRing R", "i j : n", "inst✝ : Fintype n", "a b : n", "ha : a ≠ i", "c : R", "M : Matrix n n R"], "goal": "(transvection i j c * M) a b = M a b"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "x y : α"], "goal": "Prime (x * y) ↔ Prime x ∧ IsUnit y ∨ IsUnit x ∧ Prime y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝³ : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "p : ℝ≥0∞", "μ : Measure α", "inst✝ : Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "f : ↥(Lp E p μ)"], "goal": "∃ x, (∀ (n : ℕ), x n ∈ Set.range Subtype.val) ∧ Tendsto x atTop (𝓝 f)"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "U : TopologicalSpace.Opens α"], "goal": "↑U = Set.univ ↔ U = ⊤"}}
+{"state": {"context": ["m : Type → Type", "ω σ : Type"], "goal": "∀ {α : Type} {p : α → Prop} {x : StateRefT' ω σ m α} [inst : Monad m],\n  SatisfiesM p x ↔ ∀ (s : ST.Ref ω σ), SatisfiesM p (x s)"}}
+{"state": {"context": ["z : ℂ"], "goal": "z.arg = π ↔ z.re < 0 ∧ z.im = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "MeasurableSpace α", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "MeasurableSpace β", "e : α → β", "ν : MeasureTheory.Measure β", "h₁ : MeasureTheory.MeasurePreserving e μ ν", "h₂ : MeasurableEmbedding e", "f : β → E", "s : Set β"], "goal": "MeasureTheory.IntegrableOn (f ∘ e) (e ⁻¹' s) μ ↔ MeasureTheory.IntegrableOn f s ν"}}
+{"state": {"context": ["α : Type u", "inst✝⁵ : Primcodable α", "inst✝⁴ : Inhabited α", "β : Type v", "inst✝³ : Primcodable β", "inst✝² : Inhabited β", "γ : Type w", "inst✝¹ : Primcodable γ", "inst✝ : Inhabited γ", "p : α → Prop", "q : β → Prop"], "goal": "Setoid.r (toNat p) (toNat q) ↔ ManyOneEquiv p q"}}
+{"state": {"context": ["R : Type uR", "inst✝³ : Semiring R", "S : Type uS", "inst✝² : CommSemiring S", "T : Type uT", "A : Type uA", "inst✝¹ : Semiring A", "inst✝ : Algebra S A", "r✝ r : R → R → Prop", "x y : R", "w : r x y"], "goal": "(mkRingHom r) x = (mkRingHom r) y"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddGroup E", "SMul 𝕜 E"], "goal": "∀ (a : Seminorm 𝕜 E) (a_1 : E), (Seminorm.coeFnAddMonoidHom 𝕜 E) a a_1 = a a_1"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹⁰ : Category.{u_4, u_1} C", "inst✝⁹ : Category.{u_3, u_2} D", "G : C ⥤ D", "A : Type w", "inst✝⁸ : Category.{w', w} A", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "inst✝⁷ : G.IsCocontinuous J K", "inst✝⁶ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F", "inst✝⁵ : G.IsCocontinuous J K", "inst✝⁴ : G.IsContinuous J K", "inst✝³ : HasWeakSheafify J A", "inst✝² : HasWeakSheafify K A", "inst✝¹ : G.IsCocontinuous J K", "inst✝ : G.IsContinuous J K", "F : Dᵒᵖ ⥤ A", "adj₁ : (whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op ⊣ G.op.ran := G.op.ranAdjunction A", "adj₂ : presheafToSheaf J A ⊣ sheafToPresheaf J A := sheafificationAdjunction J A", "adj₃ : presheafToSheaf K A ⊣ sheafToPresheaf K A := sheafificationAdjunction K A", "adj₄ : G.sheafPushforwardContinuous A J K ⊣ G.sheafPushforwardCocontinuous A J K := G.sheafAdjunctionCocontinuous A J K", "eq : adj₁.unit.app F ≫ G.op.ran.map (adj₂.unit.app (((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj F)) ≫ G.op.ran.map ((sheafToPresheaf J A).map (((adj₁.comp adj₂).leftAdjointUniq (adj₃.comp adj₄)).hom.app F)) = adj₃.unit.app F ≫ (sheafToPresheaf K A).map (adj₄.unit.app ((presheafToSheaf K A).obj F))"], "goal": "(adj₁.homEquiv F ((sheafToPresheaf J A).obj ((presheafToSheaf K A ⋙ G.sheafPushforwardContinuous A J K).obj F))) (adj₂.unit.app (((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).obj F) ≫ (sheafToPresheaf J A).map (((adj₁.comp adj₂).leftAdjointUniq (adj₃.comp adj₄)).hom.app F)) = (adj₁.homEquiv F ((sheafToPresheaf J A).obj ((presheafToSheaf K A ⋙ G.sheafPushforwardContinuous A J K).obj F))) (((whiskeringLeft Cᵒᵖ Dᵒᵖ A).obj G.op).map (adj₃.unit.app F))"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁹ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : CompleteSpace E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : CompleteSpace F", "G : Type u_4", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace G", "φ : ImplicitFunctionData 𝕜 E F G", "g'inv : G →L[𝕜] E", "hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G", "hg'invf : φ.leftDeriv.comp g'inv = 0", "this : HasStrictFDerivAt (HasStrictFDerivAt.localInverse φ.prodFun (φ.leftDeriv.equivProdOfSurjectiveOfIsCompl φ.rightDeriv ⋯ ⋯ ⋯) φ.pt ⋯) ↑(φ.leftDeriv.equivProdOfSurjectiveOfIsCompl φ.rightDeriv ⋯ ⋯ ⋯).symm (φ.leftFun φ.pt, φ.rightFun φ.pt)"], "goal": "HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "s : Set R", "x : R"], "goal": "x ∈ Subsemiring.closure s ↔ ∃ L, (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (List.map List.prod L).sum = x"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "A : Type u_4", "B : Type u_5", "C : Type u_6", "inst✝⁴ : AddCommMonoid A", "inst✝³ : AddCommMonoid B", "inst✝² : AddCommMonoid C", "t : ι → A → C", "h0 : ∀ (i : ι), t i 0 = 0", "h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y", "s✝ : Finset α", "f✝ : α → ι →₀ A", "i : ι", "g : ι →₀ A", "k : ι → A → γ → B", "x : γ", "β : Type u_7", "M : Type u_8", "M' : Type u_9", "N : Type u_10", "P : Type u_11", "G : Type u_12", "H : Type u_13", "R : Type u_14", "S : Type u_15", "inst✝¹ : AddCommMonoid M", "inst✝ : AddCommMonoid N", "f : Multiset (α →₀ M)", "h : α → M → N", "h₀ : ∀ (a : α), h a 0 = 0", "h₁ : ∀ (a : α) (b₁ b₂ : M), h a (b₁ + b₂) = h a b₁ + h a b₂", "a : α →₀ M", "s : Multiset (α →₀ M)", "ih : s.sum.sum h = (Multiset.map (fun g => g.sum h) s).sum"], "goal": "(a ::ₘ s).sum.sum h = (Multiset.map (fun g => g.sum h) (a ::ₘ s)).sum"}}
+{"state": {"context": ["ι : Type uι", "E : Type uE", "NormedAddCommGroup E", "NormedSpace ℝ E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace ℝ F", "H : Type uH", "TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "TopologicalSpace M", "ChartedSpace H M", "s : Set M", "f : SmoothPartitionOfUnity ι I M s", "g : ι → M → F", "t : Set F", "x : M", "hx : x ∈ s", "hg : ∀ (i : ι), (f i) x ≠ 0 → g i x ∈ t", "ht : Convex ℝ t"], "goal": "∑ᶠ (i : ι), (f i) x • g i x ∈ t"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "v : VitaliFamily μ", "ι : Sort u_2", "p : ι → Prop", "s : ι → Set α", "x : α", "h : (𝓝 x).HasBasis p s"], "goal": "(v.filterAt x).HasBasis p fun i => {t | t ∈ v.setsAt x ∧ t ⊆ s i}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "U : Submodule 𝕜 E"], "goal": "U ⟂ ⊤ ↔ U = ⊥"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "h : H.index = 2", "a b : G", "ha : a ∉ H", "hb : b ∉ H"], "goal": "a * b ∈ H"}}
+{"state": {"context": ["G : Type u_1", "Monoid G", "H : Submonoid G", "x : ↥H"], "goal": "IsOfFinOrder ↑x ↔ IsOfFinOrder x"}}
+{"state": {"context": ["x : ℂ"], "goal": "DifferentiableAt ℂ Complex.sin x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "Module 𝕜 E", "x z : E", "s : Set E", "hs : StarConvex 𝕜 (x + z) s"], "goal": "StarConvex 𝕜 x ((fun x => x + z) ⁻¹' s)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "M : Type u_4", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : AddCommMonoid M", "inst✝⁹ : Module R M", "inst✝⁸ : Semiring A", "inst✝⁷ : Semiring B", "inst✝⁶ : Module A M", "inst✝⁵ : Module B M", "inst✝⁴ : Algebra R A", "inst✝³ : Algebra R B", "inst✝² : IsScalarTower R A M", "inst✝¹ : IsScalarTower R B M", "inst✝ : SMulCommClass A B M", "p : Submodule (A ⊗[R] B) M", "b : B", "m : M", "hm : m ∈ p"], "goal": "b • m = 1 ⊗ₜ[R] b • m"}}
+{"state": {"context": ["ι : Type v", "β : ι → Type w", "inst✝³ : (i : ι) → AddCommMonoid (β i)", "M : Type u_1", "S : Type u_2", "inst✝² : AddCommMonoid M", "inst✝¹ : SetLike S M", "inst✝ : AddSubmonoidClass S M", "A : ι → S", "x : ⨁ (i : ι), ↥(A i)"], "goal": "(support fun i => ↑(x i)).Finite"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.sin x = 0 ↔ ∃ n, ↑n * Real.pi = x"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "X Y : Center C", "U : C"], "goal": "(X.snd.β Y.fst).hom ▷ U ≫ (α_ Y.fst X.fst U).hom ≫ Y.fst ◁ (X.snd.β U).hom ≫ (α_ Y.fst U X.fst).inv ≫ (Y.snd.β U).hom ▷ X.fst ≫ (α_ U Y.fst X.fst).hom = ((α_ X.fst Y.fst U).hom ≫ X.fst ◁ (Y.snd.β U).hom ≫ (α_ X.fst U Y.fst).inv ≫ (X.snd.β U).hom ▷ Y.fst ≫ (α_ U X.fst Y.fst).hom) ≫ U ◁ (X.snd.β Y.fst).hom"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : E → F", "g : F → G", "hg : IsBoundedLinearMap ���� g", "hf : IsBoundedLinearMap 𝕜 f"], "goal": "IsBoundedLinearMap 𝕜 (g ∘ f)"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "R : Type u_1", "inst✝ : NonAssocSemiring R", "f : (k : ℕ) → R →+* ZMod (p ^ k)", "f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1", "ε : ℚ", "hε : 0 < ε", "j : ℕ", "hj : j ≥ 1"], "goal": "padicNorm p (↑(nthHomSeq f_compat 1 - 1) j) < ε"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q✝ : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : α → F", "p q : ℝ", "hq_pos : 0 < q"], "goal": "(∫⁻ (a : α), ↑‖‖f a‖ ^ q‖₊ ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), ↑‖f a‖₊ ^ (p * q) ∂μ) ^ (1 / p * (1 / q) * q)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι : Type u_12", "inst✝¹⁸ : SeminormedRing 𝕜", "inst✝¹⁷ : SeminormedRing 𝕜₂", "inst✝¹⁶ : SeminormedRing 𝕜₃", "σ₁₂ : 𝕜 →+* 𝕜₂", "inst✝¹⁵ : RingHomIsometric σ₁₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "inst✝¹⁴ : RingHomIsometric σ₂₃", "σ₁₃ : 𝕜 →+* 𝕜₃", "inst✝¹³ : RingHomIsometric σ₁₃", "inst✝¹² : AddCommGroup E", "inst✝¹¹ : AddCommGroup E₂", "inst✝¹⁰ : AddCommGroup E₃", "inst✝⁹ : AddCommGroup F", "inst✝⁸ : AddCommGroup G", "inst✝⁷ : Module 𝕜 E", "inst✝⁶ : Module 𝕜₂ E₂", "inst✝⁵ : Module 𝕜₃ E₃", "inst✝⁴ : Module 𝕜 F", "inst✝³ : Module 𝕜 G", "inst✝² : SMul R ℝ", "inst✝¹ : SMul R ℝ≥0", "inst✝ : IsScalarTower R ℝ≥0 ℝ", "p : ι → Seminorm 𝕜 E", "s : Finset ι"], "goal": "s.sup p ≤ ∑ i ∈ s, p i"}}
+{"state": {"context": ["α : Type u_2", "Zero α", "s : Set α", "h : s.Nonempty"], "goal": "s ⊆ 0 ↔ s = 0"}}
+{"state": {"context": ["n k : ℕ"], "goal": "n.descFactorial k = 0 ↔ n < k"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Nat.card α = 0 ↔ IsEmpty α ∨ Infinite α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "f : α → β"], "goal": "MeasureTheory.Memℒp f 1 μ ↔ MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["a b : ℝ", "n✝ m n : ℕ", "hc : Continuous fun u => u ^ n * (1 - u ^ 2) ^ m"], "goal": "-∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n = ∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n"}}
+{"state": {"context": ["R : Type u_1", "M M₁ M₂ M₃ : Type u", "M' : Type v", "inst✝¹¹ : Ring R", "inst✝¹⁰ : AddCommGroup M", "inst✝⁹ : AddCommGroup M₁", "inst✝⁸ : AddCommGroup M₂", "inst✝⁷ : AddCommGroup M₃", "inst✝⁶ : AddCommGroup M'", "inst✝⁵ : Module R M", "inst✝⁴ : Module R M₁", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M'", "inst✝ : HasRankNullity.{u, u_1} R", "s t : Submodule R M"], "goal": "Module.rank R ↥(s ⊔ t) ≤ Module.rank R ↥s + Module.rank R ↥t"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ", "i j : ℤ", "hij : i + 1 = j"], "goal": "(CochainComplex.mappingCone φ).d i j ≫ (CochainComplex.mappingCone.snd φ).v j j ⋯ =\n  (↑(CochainComplex.mappingCone.fst φ)).v i j hij ≫ φ.f j + (CochainComplex.mappingCone.snd φ).v i i ⋯ ≫ G.d i j"}}
+{"state": {"context": ["X : Type u", "x : X", "s : Set X", "TopologicalSpace X", "hs : IsOpen s", "hx : x ∈ s"], "goal": "s ∈ 𝓝 x"}}
+{"state": {"context": ["α : Type u_1", "k n m : ℕ", "H : Raised n m"], "goal": "Raised (n + k) (m + k)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s t : Set E", "f : E → F", "x : E", "y : E", "h : s =ᶠ[𝓝[{y}ᶜ] x] t", "n : ℕ"], "goal": "iteratedFDerivWithin 𝕜 n f s =ᶠ[𝓝 x] iteratedFDerivWithin 𝕜 n f t"}}
+{"state": {"context": ["n : ℕ", "h : 0 < ↑n", "x : ↑I", "h' : ↑n ≠ 0"], "goal": "(fun x => x * ↑n) (∑ k : Fin (n + 1), (↑x - ↑(k/ₙ)) ^ 2 * (bernstein n ↑k) x) = (fun x => x * ↑n) (↑x * (1 - ↑x) / ↑n)"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "k : Type u_2", "Field k", "IsAlgClosed k", "CharP k p", "V : Type u_3", "AddCommGroup V", "WittVector.Isocrystal p k V", "h_dim : FiniteDimensional.finrank (FractionRing (WittVector p k)) V = 1"], "goal": "∃ m, Nonempty (WittVector.IsocrystalEquiv p k (WittVector.StandardOneDimIsocrystal p k m) V)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "exps : HeightOneSpectrum R → ℤ"], "goal": "∀ (x y : FractionalIdeal R⁰ K), count K v x = 0 → count K v y = 0 → count K v (x * y) = 0"}}
+{"state": {"context": ["X₁ X₂ : CategoryTheory.Limits.WidePullbackShape CategoryTheory.Limits.WalkingPair"], "goal": "∀ (a : X₁ ⟶ X₂),\n  CategoryTheory.Limits.walkingSpanOpEquiv.inverse.map a =\n    CategoryTheory.Limits.widePullbackShapeOpMap CategoryTheory.Limits.WalkingPair X₁ X₂ a"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "(𝓟 s).NeBot ↔ s.Nonempty"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ l : List α", "a : α", "h : a ∈ l"], "goal": "⟨a, h⟩ ∈ l.attach"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "κ : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝² : LinearOrderedField 𝕜", "r : α → β → Prop", "inst✝¹ : (a : α) → DecidablePred (r a)", "s✝ s₁ s₂ : Finset α", "t t₁ t₂ : Finset β", "a : α", "b : β", "δ : 𝕜", "inst✝ : DecidableEq β", "s : Finset α", "P : Finpartition t"], "goal": "(interedges r s t).card = ∑ b ∈ P.parts, (interedges r s b).card"}}
+{"state": {"context": ["D : Type u_1", "C : Type u_2", "inst✝³ : Groupoid D", "inst✝² : Category.{?u.4103, u_2} C", "inst✝¹ : MonoidalCategory C", "inst✝ : MonoidalClosed C", "F G : D ⥤ C", "X Y : D", "f : X ⟶ Y"], "goal": "(ihom.coev (F.obj Y)).app (G.obj X) ≫ (ihom (F.obj Y)).map (F.obj Y ◁ G.map f) = (ihom.coev (F.obj Y)).app (G.obj X) ≫ (ihom (F.obj Y)).map (CategoryTheory.inv (F.map f) ▷ G.obj X ≫ F.map f ▷ G.obj X ≫ F.obj Y ◁ G.map f)"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "inst✝⁴ : DivisionRing K", "inst✝³ : AddCommGroup V", "inst✝² : AddCommGroup V'", "inst✝¹ : Module K V", "inst✝ : Module K V'", "v✝ : ι → V", "s t : Set V", "x y z : V", "n : ℕ", "v : Fin n → V"], "goal": "LinearIndependent K (Fin.snoc v x) ↔ LinearIndependent K v ∧ x ∉ span K (range v)"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "CommGroup α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b : α"], "goal": "(a ⊓ b) ^ 2 = a * b / mabs (b / a)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G", "Finite G"], "goal": "Nat.card ↥H ≤ Nat.card G"}}
+{"state": {"context": ["k : Type u_1", "inst✝⁸ : Field k", "K : Type u_2", "inst✝⁷ : Field K", "F : Type u_3", "inst✝⁶ : Field F", "inst✝⁵ : NumberField K", "inst✝⁴ : Algebra k K", "inst✝³ : Algebra k F", "inst✝² : Algebra K F", "inst✝¹ : IsScalarTower k K F", "σ : K ≃ₐ[k] K", "w : InfinitePlace K", "inst✝ : IsGalois k K", "φ : K →+* ℂ", "H : ∀ (σ : K ≃ₐ[k] K), ComplexEmbedding.IsConj φ σ → σ = 1"], "goal": "IsUnramified k (mk φ)"}}
+{"state": {"context": ["R : Type u", "inst✝ : LinearOrder R", "a b : Tropical (WithTop R)"], "goal": "a = 0 ∧ a ≤ b ∨ b = 0 ∧ b ≤ a ↔ a = 0 ∧ b = 0"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "a x y : α"], "goal": "SemiconjBy (MulOpposite.op a) (MulOpposite.op y) (MulOpposite.op x) ↔ SemiconjBy a x y"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "W.HasLeftCalculusOfFractions", "X : C"], "goal": "(CategoryTheory.MorphismProperty.LeftFraction.Localization.Q W).obj X = X"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "inst✝⁵ : AddCommMonoid ι", "inst✝⁴ : DecidableEq ι", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ι → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "x : Submonoid A", "y : NumDenSameDeg 𝒜 x"], "goal": "(-Quotient.mk'' y).val = -val (Quotient.mk'' y)"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a"], "goal": "a ≤ a * a"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "CharZero R"], "goal": "Function.Injective fun n => Polynomial.cyclotomic n R"}}
+{"state": {"context": ["G : Type u_1", "Group G", "IsSimpleGroup G", "H : Subgroup G", "Hn : H.Normal"], "goal": "H = ⊥ ∨ H = ⊤"}}
+{"state": {"context": ["m : Type u", "n : Type v", "α : Type w", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "inst✝ : CommRing α", "A : Matrix n n α", "b : n → α", "this : b = ∑ i : n, b i • Pi.single i 1", "k : n"], "goal": "A.cramer (∑ i : n, b i • Pi.single i 1) k = ((of fun i => Aᵀᵀ.cramer (Pi.single i 1))ᵀ *ᵥ b) k"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : CategoryTheory.Center C", "self : X.Hom Y", "U : C"], "goal": "self.f ▷ U ≫ (Y.snd.β U).hom = (X.snd.β U).hom ≫ U ◁ self.f"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : LinearOrder R", "inst✝ : OrderTop R", "s : Finset S", "f : S → Tropical R"], "goal": "s.inf (untrop ∘ f) = (Multiset.map untrop (Multiset.map f s.val)).inf"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "Neg α", "f : m → n → α"], "goal": "-Matrix.of f = Matrix.of (-f)"}}
+{"state": {"context": ["α : Type u", "f : Filter α"], "goal": "∅ ∈ f ↔ f = ⊥"}}
+{"state": {"context": ["m : ℕ", "n : ℕ"], "goal": "StrictMono (Fin.natAdd n)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "S : Set G"], "goal": "AddSubgroup.IsComplement S Set.univ ↔ ∃ g, S = {g}"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Bool", "l : List α"], "goal": "countP p l = l.length ↔ ∀ (a : α), a ∈ l → p a = true"}}
+{"state": {"context": ["V₁ : Type u_3", "V₂ : Type u_4", "V₃ : Type u_5", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂", "SeminormedAddCommGroup V₃", "W₁ : Type u_6", "W₂ : Type u_7", "W₃ : Type u_8", "SeminormedAddCommGroup W₁", "SeminormedAddCommGroup W₂", "SeminormedAddCommGroup W₃", "f₁ : NormedAddGroupHom V₁ W₁", "f₂ : NormedAddGroupHom V₂ W₂", "f₃ : NormedAddGroupHom V₃ W₃", "φ : NormedAddGroupHom V₁ V₂", "ψ : NormedAddGroupHom W₁ W₂", "φ' : NormedAddGroupHom V₂ V₃", "ψ' : NormedAddGroupHom W₂ W₃", "hf : ψ.comp f₁ = f₂.comp φ", "hf' : ψ'.comp f₂ = f₃.comp φ'"], "goal": "(ψ'.comp ψ).comp f₁ = f₃.comp (φ'.comp φ)"}}
+{"state": {"context": ["K : Type u", "Field K", "G : Type u_1", "Group G", "MulSemiringAction G K", "g : G", "S T : ValuationSubring K"], "goal": "S ≤ g • T ↔ g⁻¹ • S ≤ T"}}
+{"state": {"context": ["ι : Type u_2", "β : ι → Type u_4", "Fintype ι", "(i : ι) → SeminormedAddCommGroup (β i)", "f : (i : ι) → β i"], "goal": "‖(WithLp.equiv ⊤ ((i : ι) → β i)).symm f‖ = ‖f‖"}}
+{"state": {"context": ["P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme", "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop", "AlgebraicGeometry.HasRingHomProperty P Q", "X Y : AlgebraicGeometry.Scheme", "f : X ⟶ Y", "AlgebraicGeometry.IsAffine Y", "𝒰 : X.OpenCover", "∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)", "H : ∀ (i : 𝒰.J), Q (AlgebraicGeometry.Scheme.Hom.app (𝒰.map i ≫ f) ⊤)"], "goal": "P f"}}
+{"state": {"context": ["α : Type u_1", "Zero α", "TopologicalSpace α", "PartialOrder α", "DecidableRel fun x x_1 => x < x_1", "OrderTopology α", "a : α", "h : 0 < a"], "goal": "ContinuousAt (⇑SignType.sign) a"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "x : G₀", "h : x ≠ 0"], "goal": "Function.Injective fun y => y * x"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "s₁ s₂ : Set (Sym2 V)", "h : s₁ ⊆ s₂"], "goal": "G.deleteEdges s₂ ≤ G.deleteEdges s₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Fintype α", "inst✝² : Fintype β", "inst✝¹ : DecidableEq α", "inst✝ : DecidableEq β", "G : SimpleGraph α", "a b : α", "p : G.Walk a b", "hp : p.IsHamiltonian"], "goal": "p.length + 1 = Fintype.card α"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "CompleteSpace E", "p q : Subspace 𝕜 E", "h : IsCompl p q", "hp : IsClosed ↑p", "hq : IsClosed ↑q"], "goal": "⇑(Submodule.linearProjOfClosedCompl p q h hp hq) = ⇑(Submodule.linearProjOfIsCompl p q h)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_1", "inst✝² : Preorder α", "s : Set α", "inst✝¹ : TopologicalSpace β", "f : β → α", "inst✝ : IsLower α"], "goal": "Continuous f ↔ ∀ (a : α), IsClosed (f ⁻¹' Ici a)"}}
+{"state": {"context": ["R : Type uR", "A : Type uA", "CommRing R", "Ring A", "Algebra R A", "s : Set A", "x : A"], "goal": "x ∈ Algebra.adjoin R s ↔ x ∈ Subring.closure (Set.range ⇑(algebraMap R A) ∪ s)"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n m : ℕ", "ω : Ω", "ℱ : Filtration ℕ m0", "inst✝ : IsFiniteMeasure μ", "hf : Submartingale f ℱ μ", "hfN : ∀ (ω : Ω), a ≤ f N ω", "hfzero : 0 ≤ f 0", "hab : a < b"], "goal": "∫ (a_1 : Ω), (b - a) * ↑(upcrossingsBefore a b f N a_1) ∂μ ≤ ∫ (x : Ω), (∑ k ∈ Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : CompleteSpace F", "f : E → F", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r : ℝ≥0∞", "hf : HasFPowerSeriesOnBall f p x r", "h : y ∈ EMetric.ball x r", "this✝ : ↑‖y - x‖₊ < r", "this : HasFPowerSeriesOnBall f (p.changeOrigin (y - x)) (x + (y - x)) (r - ↑‖y - x‖₊)"], "goal": "AnalyticAt 𝕜 f y"}}
+{"state": {"context": ["x : ℝ", "p : ℝ", "hp : 1 < p"], "goal": "HasDerivAt (fun x => ‖x‖ ^ p) (p * ‖x‖ ^ (p - 2) * x) x"}}
+{"state": {"context": ["R : Type u", "Ring R", "M N : ModuleCat R"], "goal": "(M.biprodIsoProd N).inv ≫ CategoryTheory.Limits.biprod.fst = LinearMap.fst R ↑M ↑N"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α✝", "inst✝¹ : TopologicalSpace β", "f : α✝ → β", "α : Type u_3", "inst✝ : TopologicalSpace α"], "goal": "IsQuasiSeparated Set.univ ↔ QuasiSeparatedSpace α"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Field R", "p q : R[X]", "hp : p ≠ 0", "hq : 0 < q.degree", "hq0 : q ≠ 0"], "goal": "(p / q).degree < p.degree"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : ProbabilityTheory.Kernel α γ", "a : α"], "goal": "MeasurableSet (κ.mutuallySingularSetSlice η a)"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "inst✝⁶ : Field K", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝³ : H.IsCartanSubalgebra", "inst✝² : IsTriangularizable K (↥H) L", "inst✝¹ : IsKilling K L", "inst✝ : PerfectField K", "x : L", "hx : x ∈ H", "N S : Module.End K L", "hS₀ : S ∈ Algebra.adjoin K {(ad K L) x}", "hN : _root_.IsNilpotent N", "hS : S.IsSemisimple", "hSN : (ad K L) x = N + S"], "goal": "((ad K L) x).IsSemisimple"}}
+{"state": {"context": ["x : ℝ", "hx : 0 ≤ x"], "goal": "1 ≤ Real.exp x"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a₁ a₂ b₁ b₂ : α", "h₁ : a₁ ≤ b₁"], "goal": "Finset.Icc a₁ b₁ ⊆ Finset.Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁷ : TopologicalSpace H", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "inst✝⁴ : TopologicalSpace H'", "inst✝³ : TopologicalSpace M'", "inst✝² : ChartedSpace H' M'", "inst✝¹ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e✝ e'✝ : PartialHomeomorph M H", "f✝ f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s✝ t : Set M", "x✝ : M", "Q : (H → H) → Set H → H → Prop", "inst✝ : ClosedUnderRestriction G", "s : Set H", "x : H", "f : H → H", "e' : PartialHomeomorph H H", "he'G : e' ∈ G", "a✝ : s ⊆ f ⁻¹' e'.source", "hfx : f x ∈ e'.source", "h : G.IsLocalStructomorphWithinAt f s x", "hx : x ∈ s", "e : PartialHomeomorph H H", "heG : e ∈ G", "hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)", "hex : x ∈ e.source"], "goal": "∃ e ∈ G, EqOn (↑e' ∘ f) (↑e.toPartialEquiv) (s ∩ e.source) ∧ x ∈ e.source"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "AddGroup α", "AddAction α β", "A B : Set β", "a : α"], "goal": "a +ᵥ A ⊆ a +ᵥ B ↔ A ⊆ B"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "e : M₁ ≃SL[σ₁₂] M₂", "b : M₁"], "goal": "↑e b = e b"}}
+{"state": {"context": ["p : ℚ", "q : ℚ≥0", "hp : 0 ≤ p"], "goal": "q ≤ p.toNNRat ↔ ↑q ≤ p"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "S : Submonoid R", "OreLocalization.OreSet S"], "goal": "Subsingleton (OreLocalization S R) ↔ 0 ∈ S"}}
+{"state": {"context": ["R : Type u_3", "CommSemiring R", "S : Submonoid R", "A : Type u_4", "CommSemiring A", "Algebra R A", "IsLocalization S A", "M : Type u_5", "N : Type u_6", "AddCommMonoid M", "Module R M", "Module A M", "IsScalarTower R A M", "AddCommMonoid N", "Module R N", "Module A N", "IsScalarTower R A N", "f : M →ₗ[A] N"], "goal": "LinearMap.extendScalarsOfIsLocalization S A (↑R f) = f"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "M₂ : Type w₁", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "N N' : LieSubmodule R L M", "I J : LieIdeal R L", "N₂ : LieSubmodule R L M₂", "h✝ : ⁅I, N⁆ ⊔ ⁅I, N'⁆ ≤ ⁅I, N ⊔ N'⁆", "m : M", "x : ↥I", "n : M", "hn : n ∈ N ⊔ N'", "h : ⁅↑x, ↑⟨n, hn⟩⁆ = m"], "goal": "∃ y ∈ ⁅I, N⁆, ∃ z ∈ ⁅I, N'⁆, y + z = m"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "OrderedAddCommMonoid α", "OrderedAddCommMonoid β", "SMulZeroClass α β", "CeilDiv α β", "a : α", "b c : β", "ha : 0 < a"], "goal": "b ⌈/⌉ a ≤ c ↔ b ≤ a • c"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝² : CommSemiring R", "inst✝¹ : CommSemiring A", "inst✝ : Algebra R A", "s t : Set A", "z : A"], "goal": "z ∈ ↑(Submonoid.closure (s ∪ t)) ↔ z ∈ ↑(Submonoid.closure s) * ↑(Submonoid.closure t)"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁴ : (i : α) → NormedAddCommGroup (E i)", "I : Type u_3", "𝕜 : Type u_4", "B : I → Type u_5", "inst✝³ : NormedField 𝕜", "inst✝² : (i : I) → NormedRing (B i)", "inst✝¹ : (i : I) → NormedAlgebra 𝕜 (B i)", "inst✝ : ∀ (i : I), NormOneClass (B i)", "k : 𝕜"], "goal": "Memℓp ((algebraMap 𝕜 ((i : I) → B i)) k) ⊤"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : ℕ → α → ℝ≥0∞", "g : α → ℝ≥0∞", "hf : ∀ (i : ℕ), Measurable (f i)", "lim : Filter.Tendsto f Filter.atTop (𝓝 g)"], "goal": "Measurable g"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)"], "goal": "legendreSym p 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : (a : α) → a ∈ s → β", "hf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t", "hinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂", "hst : t.ncard ≤ s.ncard", "ht : t.Finite := by toFinite_tac", "b : β"], "goal": "b ∈ t → ∃ a, ∃ (ha : a ∈ s), b = f a ha"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x✝ : M", "p✝ p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "ι : Sort u_9", "p : ι → Submodule R M", "C : (x : M) → x ∈ ⨆ i, p i → Prop", "mem : ∀ (i : ι) (x : M) (hx : x ∈ p i), C x ⋯", "zero : C 0 ⋯", "add : ∀ (x y : M) (hx : x ∈ ⨆ i, p i) (hy : y ∈ ⨆ i, p i), C x hx → C y hy → C (x + y) ⋯", "x : M", "hx : x ∈ ⨆ i, p i"], "goal": "(fun x => ∃ (hx : x ∈ ⨆ i, p i), C x hx) 0"}}
+{"state": {"context": ["α β : Type u", "n : ℕ"], "goal": "succ ↑n = ↑n + 1"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs : ∀ (n : ℕ), s ≠ -↑n", "hs' : a ≠ 0 ∨ s ≠ 1", "this : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹"], "goal": "completedCosZeta a s * (s.Gammaℂ * Complex.cos (↑π * s / 2) * s.Gammaℝ⁻¹) = s.Gammaℂ * Complex.cos (↑π * s / 2) * (completedCosZeta a s / s.Gammaℝ)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid N", "inst✝¹ : Module R M", "inst✝ : Module R N", "M₁ M₂ : Submodule R M", "N₁ N₂ : Submodule R N", "x : M ⊗[R] N"], "goal": "∃ S, x = S.sum fun n m => m ⊗ₜ[R] n"}}
+{"state": {"context": ["G : Type u_1", "inst✝⁷ : Group G", "inst✝⁶ : TopologicalSpace G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "μ : Measure G", "inst✝² : μ.IsHaarMeasure", "inst✝¹ : LocallyCompactSpace G", "inst✝ : μ.InnerRegular", "E : Set G", "hE : MeasurableSet E", "hEpos : 0 < μ E", "K : Set G", "hKE : K ⊆ E", "hK : IsCompact K", "K_closed : IsClosed K", "hKpos : 0 < μ K", "g : G", "hg : μ (g • K \\ K) < μ K"], "goal": "g ∈ E / E"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "M : Type u_1", "CategoryTheory.Category.{u_2, u_1} M", "CategoryTheory.MonoidalCategory M", "F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)", "m n : M", "h₁ : CategoryTheory.MonoidalCategory.tensorObj m n ≅ 𝟙_ M", "h₂ : CategoryTheory.MonoidalCategory.tensorObj n m ≅ 𝟙_ M", "H :\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerRight h₁.hom m)\n      (CategoryTheory.MonoidalCategory.leftUnitor m).hom =\n    CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator m n m).hom\n      (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft m h₂.hom)\n        (CategoryTheory.MonoidalCategory.rightUnitor m).hom)"], "goal": "(CategoryTheory.equivOfTensorIsoUnit F m n h₁ h₂ H).inverse = F.obj n"}}
+{"state": {"context": ["ι : Type u_6", "α : Type u_8", "DecidableEq ι", "CommSemiring α", "κ : ι → Type u_9", "Fintype ι", "(i : ι) → Fintype (κ i)", "f : (i : ι) → κ i → α"], "goal": "∏ i : ι, ∑ j : κ i, f i j = ∑ x : (i : ι) → κ i, ∏ i : ι, f i (x i)"}}
+{"state": {"context": ["E : Type u_2", "AddCommGroup E", "Module ℝ E", "s : Set E", "x : E"], "goal": "gauge s x = sInf {r | r ∈ Set.Ioi 0 ∧ r⁻¹ • x ∈ s}"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l : List α", "a : α"], "goal": "filter (fun x => a == x) l = replicate (count a l) a"}}
+{"state": {"context": ["o : Ordinal.{u_1}", "ho : o.card.ord = o", "ho' : ω ≤ o", "a : Ordinal.{u_1}", "ha : (aleph' a).ord = o"], "goal": "∃ a, (aleph a).ord = o"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedBooleanAlgebra α", "a b c : α"], "goal": "a ∆ b = c ∆ b ↔ a = c"}}
+{"state": {"context": ["α : Type u_1", "DivisionMonoid α", "a : α", "n : ℕ"], "goal": "a⁻¹ ^ n = (a ^ n)⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "R : Type u_6", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "inst✝ : MeasurableSpace δ", "f g : α → β", "μ ν : Measure α", "h : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y", "hμ : μ ≠ 0"], "goal": "AEMeasurable f μ"}}
+{"state": {"context": ["a b c : EReal", "h₁ : c ≠ ⊥", "h₂ : c ≠ ⊤", "h₃ : c ≠ 0"], "goal": "a * c * (b * c)⁻¹ = a * b⁻¹"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "s : Set α", "a : α", "has : a ∉ s"], "goal": "Set.InjOn f (insert a s) ↔ Set.InjOn f s ∧ f a ∉ f '' s"}}
+{"state": {"context": ["α : Type u_1", "Inhabited α", "l : List α", "n : Nat"], "goal": "l.get! n = l.getD n default"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "f : α → β", "hf : Function.Surjective f"], "goal": "Function.Injective fun g => g ∘ f"}}
+{"state": {"context": ["z τ : ℂ"], "goal": "(starRingEnd ℂ) (jacobiTheta₂ z τ) = jacobiTheta₂ ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s✝ s : Seq α"], "goal": "match (corec (fun x => match x.1.destruct with | none => match x.2.destruct with | none => none | some (a, b) => some (a, nil, b) | some (a, s₁') => some (a, s₁', x.2)) (nil, s)).destruct, s.destruct with | none, none => True | some (a, s), some (a', s') => a = a' ∧ ∃ s_1, s = corec (fun x => match x.1.destruct with | none => match x.2.destruct with | none => none | some (a, b) => some (a, nil, b) | some (a, s₁') => some (a, s₁', x.2)) (nil, s_1) ∧ s' = s_1 | x, x_1 => False"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SMul α β", "s : Set α", "t : Set β"], "goal": "s.Finite → t.Finite → (s • t).Finite"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ", "η : ProbabilityTheory.Kernel (α × β) γ", "ProbabilityTheory.IsSFiniteKernel η", "a : α", "s : Set (β × γ)", "hs : MeasurableSet s", "h2s : ((κ ⊗ₖ η) a) s ≠ ⊤"], "goal": "MeasureTheory.Integrable (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : CommMonoid β", "S : Multiset α", "g1 g2 : α → β", "h : ∀ a ∈ S, g1 a ∣ g2 a"], "goal": "(map g1 S).prod ∣ (map g2 S).prod"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "loc : IsLocalization S P", "I : FractionalIdeal S P"], "goal": "↑I.num ≤ I"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq β", "s : Finset α", "f : α → β", "a : α", "h : a ∈ s"], "goal": "f a ∈ Finset.image f s"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝¹² : NontriviallyNormedField 𝕜", "E : Type v", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace 𝕜 E", "F : Type w", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedSpace 𝕜 F", "F' : Type x", "inst✝⁷ : AddCommGroup F'", "inst✝⁶ : Module 𝕜 F'", "inst✝⁵ : TopologicalSpace F'", "inst✝⁴ : TopologicalAddGroup F'", "inst✝³ : ContinuousSMul 𝕜 F'", "inst✝² : CompleteSpace 𝕜", "inst✝¹ : Nontrivial E", "inst✝ : LocallyCompactSpace E", "v : E", "hv : v ≠ 0", "L : 𝕜 → E := fun t => t • v"], "goal": "LocallyCompactSpace 𝕜"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x✝ y z : X", "ι : Type u_3", "γ✝ : Path x✝ y", "a b : X", "γ : Path a b", "x : ↑I", "this : ↑x ∈ Icc 0 1"], "goal": "(γ.truncate 0 1) x = γ x"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "α : Type u'", "L'' : FirstOrder.Language", "φ : L' →ᴸ L''", "ψ : L →ᴸ L'"], "goal": "(φ.comp ψ).onTerm = φ.onTerm ∘ ψ.onTerm"}}
+{"state": {"context": ["X : Type u_1", "inst✝² : TopologicalSpace X", "𝕜 : Type u_2", "inst✝¹ : RCLike 𝕜", "s : Set 𝕜", "h0 : 0 ∈ s", "inst✝ : CompactSpace ↑s"], "goal": "closure ↑(polynomialFunctions s).starClosure = Set.univ"}}
+{"state": {"context": ["x a : ℝ"], "goal": "0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1"}}
+{"state": {"context": ["m : Type u", "n : Type v", "α : Type w", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "inst✝ : CommRing α", "A : Matrix n n α", "h_card : Fintype.card n - Nat.succ 0 + Nat.succ 0 = Fintype.card n", "A' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ"], "goal": "A'.adjugate.det = A'.det ^ (Fintype.card n - 1)"}}
+{"state": {"context": ["Ω : Type u_1", "MeasurableSpace Ω", "TopologicalSpace Ω", "OpensMeasurableSpace Ω", "μ : MeasureTheory.ProbabilityMeasure Ω", "μs : ℕ → MeasureTheory.ProbabilityMeasure Ω", "h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ Filter.liminf (fun i => (μs i) G) Filter.atTop"], "goal": "Filter.Tendsto (fun i => μs i) Filter.atTop (𝓝 μ)"}}
+{"state": {"context": ["b : ℂ", "hb : 0 < b.re", "c : ℝ", "T : ℝ", "hT : 0 ≤ T"], "goal": "‖GaussianFourier.verticalIntegral b c T‖ ≤ 2 * |c| * rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α"], "goal": "s.encard ≠ 0 ↔ s.Nonempty"}}
+{"state": {"context": ["A B : Grp", "f : A ⟶ B", "b✝ : ↑B", "hb : b✝ ∈ MonoidHom.range f", "b : ↑B := b✝"], "goal": "b • ↑(MonoidHom.range f) = ↑(MonoidHom.range f)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "x y : R", "inst✝ : MonoidWithZero R", "k : ℕ", "s : Set ℕ := {k | x ^ k = 0}", "this : ∀ (k₁ k₂ : ℕ), k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s"], "goal": "nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0"}}
+{"state": {"context": ["a b : ℚ"], "goal": "a ≠ b ↔ ↑a ≠ ↑b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : Ring S", "f : R[X]", "inst✝ : Algebra R S", "h : IsAdjoinRootMonic S f", "i j : Fin f.natDegree"], "goal": "(X ^ ↑i).toFinsupp ↑j = (Finsupp.single i 1) j"}}
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+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m' mΩ : MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "hm' : m' ≤ mΩ", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "β : Type u_5", "m : MeasurableSpace β", "Add β", "MeasurableAdd₂ β", "f : ι → Ω → β", "hf_indep : ProbabilityTheory.iCondIndepFun m' hm' (fun x => m) f μ", "hf_meas : ∀ (i : ι), Measurable (f i)", "i j k : ι", "hik : i ≠ k", "hjk : j ≠ k"], "goal": "ProbabilityTheory.CondIndepFun m' hm' (f i + f j) (f k) μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : LinearOrderedField α", "β : Type u_2", "inst✝² : Field β", "abv : β → α", "inst✝¹ : IsAbsoluteValue abv", "inst✝ : IsComplete β abv", "f✝ : CauSeq β abv", "hf✝ : ¬f✝.LimZero", "hl : f✝.lim ≠ 0", "g f : CauSeq β abv", "hf : ¬f.LimZero"], "goal": "(g - f * f.inv hf * g).LimZero"}}
+{"state": {"context": ["K : Type u_1", "σ : Type u_2", "ι : Type u_3", "inst✝⁵ : Fintype K", "inst✝⁴ : Field K", "inst✝³ : Fintype σ", "inst✝² : DecidableEq σ", "inst✝¹ : DecidableEq K", "p : ℕ", "inst✝ : CharP K p", "s : Finset ι", "f : ι → MvPolynomial σ K", "h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ", "hq : 0 < q - 1", "S : Finset (σ → K) := {x | x ∈ univ ∧ ∀ i ∈ s, (eval x) (f i) = 0}.toFinset", "hS : ∀ (x : σ → K), x ∈ S ↔ ∀ i ∈ s, (eval x) (f i) = 0", "F : MvPolynomial σ K := ∏ i ∈ s, (1 - f i ^ (q - 1))", "hF : ∀ (x : σ → K), (eval x) F = if x ∈ S then 1 else 0", "key : ∑ x : σ → K, (eval x) F = ↑(Fintype.card { x // ∀ i ∈ s, (eval x) (f i) = 0 })"], "goal": "∑ x : σ → K, (eval x) F = 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : InnerProductSpace ℝ V", "inst✝³ : MetricSpace P", "inst✝² : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "ps : Set P", "h : ps ⊆ ↑s", "inst✝¹ : Nonempty ↥s", "n : ℕ", "inst✝ : FiniteDimensional ℝ ↥s.direction", "hd : finrank ℝ ↥s.direction = n", "c : P", "hc✝ : c ∈ s", "r : ℝ", "hcr : ∀ p ∈ ps, dist p c = r", "sx : Simplex ℝ P n", "hsxps : Set.range sx.points ⊆ ps", "hsx : affineSpan ℝ (Set.range sx.points) = s", "hc : c ∈ affineSpan ℝ (Set.range sx.points)"], "goal": "sx.circumcenter = c"}}
+{"state": {"context": [], "goal": "⊤ = PUnit.unit"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "a b c : α"], "goal": "(fun x => a + x) ⁻¹' [[b, c]] = [[b - a, c - a]]"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "f : α → γ", "g : β → γ", "NormedAddCommGroup γ", "BorelSpace γ", "h : ProbabilityTheory.IdentDistrib f g μ ν"], "goal": "ProbabilityTheory.IdentDistrib (fun x => ‖f x‖) (fun x => ‖g x‖) μ ν"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "inst✝⁴ : Field K", "inst✝³ : AddCommGroup V", "inst✝² : Module K V", "inst✝¹ : FiniteDimensional K V", "f : End K V", "inst✝ : PerfectField K"], "goal": "∃ n ∈ adjoin K {f}, ∃ s ∈ adjoin K {f}, IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "R : ℝ≥0", "f : α → ℝ", "hbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R", "hnm : m ≤ m0", "hfint : Integrable f μ", "h : 0 < μ {x | ↑R < |(μ[f|m]) x|}"], "goal": "∫ (x : α) in {x | ↑R < |(μ[f|m]) x|}, |f x| ∂μ ≤ (μ {x | ↑R < |(μ[f|m]) x|}).toReal * ↑R"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : Rel α β", "s : Rel β γ", "t : Set α", "z : γ"], "goal": "(∃ x ∈ t, (r • s) x z) ↔ ∃ x, (∃ x_1 ∈ t, r x_1 x) ∧ s x z"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "a b : α", "c : α", "h : 0 < c"], "goal": "(fun x => x * c) ⁻¹' Set.Icc a b = Set.Icc (a / c) (b / c)"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs' : s ≠ 1 ∨ a ≠ 0"], "goal": "DifferentiableAt ℂ (HurwitzZeta.cosZeta a) s"}}
+{"state": {"context": ["n m k : Nat"], "goal": "n - m - k = n - (m + k)"}}
+{"state": {"context": ["p : ℝ≥0∞", "α : Type u_2", "β : Type u_3", "UniformSpace α", "UniformSpace β"], "goal": "UniformContinuous ⇑(WithLp.equiv p (α × β)).symm"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : OrderedCancelAddCommMonoid α", "inst✝¹ : ExistsAddOfLE α", "inst✝ : LocallyFiniteOrder α", "a b c : α"], "goal": "map (fun x => c + x) (Ico a b) = Ico (c + a) (c + b)"}}
+{"state": {"context": ["a b : ℤ", "ha : 0 < a", "hab : a ∣ b"], "goal": "a ≤ b ↔ 0 < b"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "p : ℝ", "f g : α → ENNReal", "hf : AEMeasurable f μ", "hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ⊤", "hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ⊤", "hp1 : 1 ≤ p"], "goal": "∫⁻ (a : α), (f + g) a ^ p ∂μ < ⊤"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.HasShift C ℤ", "X : Cᵒᵖ"], "goal": "(CategoryTheory.shiftFunctorZero Cᵒᵖ ℤ).inv.app X =\n  CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorZero C ℤ).hom.app (Opposite.unop X)).op\n    ((CategoryTheory.Pretriangulated.shiftFunctorOpIso C 0 0 ⋯).inv.app X)"}}
+{"state": {"context": ["x y : SetTheory.PGame", "hy : y.Numeric", "ihyx : Surreal.Multiplication.IH1 y x", "i : x.LeftMoves", "k : y.LeftMoves", "l : (-y).LeftMoves"], "goal": "Surreal.Multiplication.P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l)"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedAddCommGroup X", "inst✝¹ : NormedSpace ℝ X", "inst✝ : HasContDiffBump E", "c : E", "f : ContDiffBump c", "x✝ : E", "n : ℕ∞", "x : E"], "goal": "↑f (c - x) = ↑f (c + x)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹⁷ : CommRing R", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : CommRing T", "inst✝¹⁴ : Algebra R S", "inst✝¹³ : Algebra R T", "K : Type u_4", "L : Type u_5", "inst✝¹² : Field K", "inst✝¹¹ : Field L", "inst✝¹⁰ : Algebra K L", "ι κ : Type w", "inst✝⁹ : Fintype ι", "F : Type u_6", "inst✝⁸ : Field F", "inst✝⁷ : Algebra R L", "inst✝⁶ : Algebra L F", "inst✝⁵ : Algebra R F", "inst✝⁴ : IsScalarTower R L F", "inst✝³ : Algebra K F", "inst✝² : IsScalarTower K L F", "E : Type u_7", "inst✝¹ : Field E", "inst✝ : Algebra K E", "pb : PowerBasis K L", "hE : Splits (algebraMap K E) (minpoly K pb.gen)", "hfx : IsSeparable K pb.gen", "this✝ : DecidableEq E := Classical.decEq E", "this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb"], "goal": "∀ (x : { x // x ∈ (minpoly K pb.gen).aroots E }), ?f x = _root_.id ↑x"}}
+{"state": {"context": ["n m : ℕ", "j : Fin n", "i : Fin (n + 1)", "hi : i ≠ 0"], "goal": "j < i.pred hi ↔ j.succ < i"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "s : Set G", "t : Set P", "a✝ : P", "inst✝³ : PartialOrder G", "inst✝² : PartialOrder P", "inst✝¹ : SMul G P", "inst✝ : IsOrderedCancelSMul G P", "x y : ↑(s.smulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : P", "h : (s.smulAntidiagonal t a).Infinite", "h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)", "isrfl : IsRefl (G × P) (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "istrns : IsTrans (G × P) (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "g : ℕ ↪o ℕ", "hg : ∀ (m n : ℕ), m ≤ n → (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.smulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.smulAntidiagonal t a) h) (g n))", "m n : ℕ", "mn : m < n", "h2' : (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.smulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.smulAntidiagonal t a) h) (g n))"], "goal": "False"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "G : SimpleGraph V", "a b : V", "f : ι → G.Subgraph"], "goal": "(⨅ i, f i).Adj a b ↔ (∀ (i : ι), (f i).Adj a b) ∧ G.Adj a b"}}
+{"state": {"context": ["a b : ℝ≥0∞", "h : a = ⊤ → b = ⊤", "h_nnreal : a ≠ ⊤ → b ≠ ⊤ → a.toNNReal ≤ b.toNNReal"], "goal": "a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "UniformSpace α"], "goal": "Continuous ↑α"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "s : Set α", "h : ∀ (a : α), a ∈ s"], "goal": "s ∈ f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "S : Set (Set α)", "hS : ∀ (s : ↑S), IsOpen ↑s", "hS' : ∀ (s : ↑S), QuasiSober ↑↑s", "hS'' : ⋃₀ S = ⊤", "t : Set α", "h : IsIrreducible t", "h' : IsClosed t", "x : α", "hx : x ∈ t", "U : Set α", "hU : U ∈ S", "hU' : x ∈ U", "this : QuasiSober ↑U", "H : IsIrreducible (Subtype.val ⁻¹' t)"], "goal": "IsGenericPoint (↑H.genericPoint) t"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Monoid R", "MulAction R M", "p : SubMulAction R M", "G : Type u_1", "Group G", "SMul G R", "MulAction G M", "IsScalarTower G R M", "g : G", "x : M"], "goal": "g • x ∈ p ↔ x ∈ p"}}
+{"state": {"context": ["α : Sort u₁", "β : Sort u₂", "φ : Sort u₃", "g : β → φ", "f : α → β", "hg : Function.Injective g", "hf : Function.Injective f"], "goal": "Function.Injective (g ∘ f)"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "F : CategoryTheory.Pseudofunctor B C"], "goal": "F.toOplax.toPrelaxFunctor = F.toPrelaxFunctor"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "I : Ideal R", "x : R", "this : TopologicalSpace R := I.adicTopology"], "goal": "(𝓝 x).HasBasis (fun _n => True) fun n => (fun y => x + y) '' ↑(I ^ n)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "DecidableEq β", "SMul α β", "s : Finset α", "t : Finset β"], "goal": "(s • t).card ≤ s.card • t.card"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : RCLike 𝕜", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace 𝕜 E", "μ : ℝ", "T : E →ₗ[𝕜] E", "hμ : HasEigenvalue T ↑μ", "hnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫_𝕜"], "goal": "0 < μ"}}
+{"state": {"context": ["C : Type u_1", "D₁ : Type u_5", "D₂ : Type u_6", "CategoryTheory.Category.{u_8, u_1} C", "CategoryTheory.Category.{u_10, u_5} D₁", "CategoryTheory.Category.{u_9, u_6} D₂", "W : CategoryTheory.MorphismProperty C", "L₁ : CategoryTheory.Functor C D₁", "L₁.IsLocalization W", "L₂ : CategoryTheory.Functor C D₂", "L₂.IsLocalization W", "X Y : C", "f : X ⟶ Y"], "goal": "(CategoryTheory.Localization.homEquiv W L₁ L₂) (L₁.map f) = L₂.map f"}}
+{"state": {"context": ["α : Type u", "s : Computation α", "T : s.Terminates", "a : α", "h : a ∈ s"], "goal": "s.Results a s.length"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommSemiring R", "f g : R⟦X⟧", "n : ℕ", "h₁ : n < n + 1", "h₂ : n < n + 1 + 1"], "goal": "(coeff R n) (f * g).derivativeFun = (coeff R n) (f • g.derivativeFun + g • f.derivativeFun)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : CompleteSpace 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : CompleteSpace E", "F : Type u_3", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : FiniteDimensional 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "a : E", "hf : HasStrictFDerivAt f f' a", "hf' : range f' = ⊤", "α : Type u_4", "l : Filter α", "g₁ : α → F", "g₂ : �� → ↥(LinearMap.ker f')", "h₁ : Tendsto g₁ l (𝓝 (f a))", "h₂ : Tendsto g₂ l (𝓝 0)"], "goal": "Tendsto (fun t => implicitFunction f f' hf hf' (g₁ t) (g₂ t)) l (𝓝 a)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι✝ : Type u_12", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : SeminormedRing 𝕝", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module 𝕜 E", "inst✝³ : Module 𝕝 E", "ι : Sort u_13", "inst✝² : UniformSpace E", "inst✝¹ : UniformAddGroup E", "inst✝ : ContinuousConstSMul 𝕜 E", "p' : ι → Prop", "s : ι → Set E", "p : Seminorm 𝕜 E", "hb : (𝓝 0).HasBasis p' s", "h₁ : ∃ r, p.closedBall 0 r ∈ 𝓝 0", "h₂ : ∀ (i : ι), p' i → ∃ r > 0, p.ball 0 r ⊆ s i"], "goal": "𝓝 0 ≤ 𝓝 0"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "f : ι → α → β", "μ : Measure α", "p : α → (ι → β) → Prop", "inst✝ : Countable ι", "hf : ∀ (i : ι), AEMeasurable (f i) μ", "hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x"], "goal": "μ (aeSeqSet hf p)ᶜ = 0"}}
+{"state": {"context": ["r s : ℝ", "hrs : r.IsConjExponent s"], "goal": "{x | ∃ k, beattySeq r k = x}ᶜ = {x | ∃ k, beattySeq' s k = x}"}}
+{"state": {"context": ["a n : ℕ"], "goal": "n.floorRoot a = 0 ↔ n = 0 ∨ a = 0"}}
+{"state": {"context": ["x y : Int", "h : y ≥ x"], "goal": "¬y < x"}}
+{"state": {"context": ["α : Type u_1", "m n : Nat", "x : α", "xs : List α", "hl : (x :: xs).Nodup", "h : m ≤ n"], "goal": "(take m (x :: xs)).Disjoint (drop n (x :: xs))"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α", "b : ↑(Set.Iio a)"], "goal": "Subtype.val '' Set.Iio b = Set.Iio ↑b"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : SMulCommClass ℝ 𝕜 F", "f : 𝓢(E, F)", "k : ℕ", "x₀ : E", "this : ‖x₀‖ ^ k * ‖iteratedFDeriv ℝ 0 (⇑f) x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f"], "goal": "‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f"}}
+{"state": {"context": ["G : Type u_2", "MeasurableSpace G", "Group G", "MeasurableMul G", "μ : MeasureTheory.Measure G", "μ.IsMulRightInvariant", "x : G"], "goal": "Filter.map (fun h => h * x) (MeasureTheory.ae μ) = MeasureTheory.ae μ"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E", "a : E"], "goal": "‖a‖₊ ≠ 0 → a ≠ 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "h : 2 ≤ n"], "goal": "decodeSum n = none"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.SFinite μ"], "goal": "μ.toFiniteAux Set.univ ≤ 1"}}
+{"state": {"context": ["J : Type w", "C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroMorphisms C", "D : Type uD", "inst✝² : Category.{uD', uD} D", "inst✝¹ : HasZeroMorphisms D", "P Q X Y : C", "inst✝ : HasBinaryBiproduct X Y", "b : BinaryBicone X Y", "hb : b.IsBilimit"], "goal": "inl ≫ (BinaryBiproduct.bicone X Y).toCone.π.app { as := WalkingPair.left } = inl ≫ desc b.inl b.inr ≫ b.toCone.π.app { as := WalkingPair.left }"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.Category.{w, v} J", "F : J ⥤ Type u", "CategoryTheory.Limits.HasColimit F", "j j' : J", "x : F.obj j", "f : j ⟶ j'"], "goal": "CategoryTheory.Limits.colimit.ι F j' (F.map f x) = CategoryTheory.Limits.colimit.ι F j x"}}
+{"state": {"context": ["x : EReal"], "goal": "-x = ⊤ ↔ x = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommMonoid α", "TopologicalSpace α", "f : β → α", "h : ¬Multipliable f"], "goal": "∏' (b : β), f b = 1"}}
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+{"state": {"context": ["α : Type u_1", "l₁ l₂ l₃ : List α", "h₁ : l₁ <+ l₂", "h₂ : l₂ <+ l₃"], "goal": "l₁ <+ l₃"}}
+{"state": {"context": ["𝕜 : Type u_1", "F : Type u_5", "NormedAddCommGroup F", "NormedSpace ℝ F", "NormedField 𝕜", "NormedSpace 𝕜 F", "SMulCommClass ℝ 𝕜 F", "k n : ℕ", "f : SchwartzMap ℝ F", "M : ℝ", "hMp : 0 ≤ M", "hM : ∀ (x : ℝ), |x| ^ k * ‖iteratedDeriv n (⇑f) x‖ ≤ M"], "goal": "(SchwartzMap.seminorm 𝕜 k n) f ≤ M"}}
+{"state": {"context": ["ι : Sort u_1", "f g : ι → ℝ≥0∞", "a b c d : ℝ≥0∞", "r p q : ℝ≥0", "s : Set ℝ≥0∞", "hf : ∀ r ∈ s, r ≠ ⊤"], "goal": "(sInf s).toReal = sInf (ENNReal.toReal '' s)"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "o.oangle x (x + y) = ↑(Real.arcsin (‖y‖ / ‖x + y‖))"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "ι : Type u_3", "inst✝³ : CommRing R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "e : ι → M", "ε : ι → Dual R M", "inst✝ : DecidableEq ι", "h : DualBases e ε", "i j : ι"], "goal": "(if j = i then 1 else 0) = if i = j then 1 else 0"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "ι : Type u_3", "γ : Type u_4", "LinearOrderedAddCommMonoid γ", "TopologicalSpace γ", "OrderTopology γ", "ContinuousAdd γ", "f : ι → α → γ", "a : Finset ι", "ha : ∀ i ∈ a, LowerSemicontinuous (f i)"], "goal": "LowerSemicontinuous fun z => ∑ i ∈ a, f i z"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p q : α → Prop", "inst✝² : DecidablePred p", "inst✝¹ : DecidablePred q", "s t : Finset α", "inst✝ : DecidableEq α", "s₁ s₂ : Finset α", "x✝ : α"], "goal": "x✝ ∈ filter p (s₁ ∪ s₂) ↔ x✝ ∈ filter p s₁ ∪ filter p s₂"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NonUnitalNormedRing A", "inst✝⁴ : NormedSpace 𝕜 A", "inst✝³ : SMulCommClass 𝕜 A A", "inst✝² : IsScalarTower 𝕜 A A", "inst✝¹ : StarRing A", "inst✝ : CstarRing A", "a : 𝓜(𝕜, A)", "h0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊) → ‖f‖₊ ≤ C"], "goal": "‖a.toProd.1‖ = ‖a.toProd.2‖"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝ q✝ r : R[X]", "n : ℕ", "p q : R[ℕ]"], "goal": "(p * q) n = ∑ x ∈ antidiagonal n, p x.1 * q x.2"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedLinearOrderedField 𝕜", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "h : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y"], "goal": "StrictConvexSpace ℝ E"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : PseudoEMetricSpace α", "f : α → ℝ≥0∞", "C : ℝ≥0∞", "hC : C ≠ ⊤", "h : ∀ (x y : α), f x ≤ f y + C * edist x y", "x : α", "ε : ℝ≥0∞", "ε0 : ε > 0", "δ : ℝ≥0", "δ0 : δ > 0", "hδ : C * ↑δ < ε"], "goal": "∀ᶠ (x_1 : α) in 𝓝 x, f x_1 ∈ Icc (f x - ε) (f x + ε)"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E", "a b : E"], "goal": "edist a b = ↑‖a / b‖₊"}}
+{"state": {"context": ["α : Type u", "a : α", "s : Stream'.WSeq α"], "goal": "(Stream'.WSeq.cons a s).tail = s"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "Semiring R", "DecidableEq ι", "i i' : ι"], "goal": "(LinearMap.stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "ι : Type u_2", "Fintype ι", "DecidableEq ι", "ne : Nonempty ι", "e : OrthonormalBasis ι ℝ E", "x : Orientation ℝ E ι"], "goal": "Orthonormal ℝ ⇑(e.toBasis.adjustToOrientation x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z", "hxy' : 0 < y - x", "hyz' : 0 < z - y", "hxz' : 0 < z - x"], "goal": "f y ≤ ((z - y) * f x + (y - x) * f z) / (z - x)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x✝ : X", "s s₁ s₂ t : Set X", "p p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X", "x : X", "F : Filter X"], "goal": "AccPt x F ↔ ClusterPt x (𝓟 {x}ᶜ ⊓ F)"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "s : Finset α", "n : ℤ", "f : α → ℤ"], "goal": "(Multiset.map ((fun x => x % n) ∘ fun i => f i) s.val).prod % n = (Multiset.map (fun i => f i % n) s.val).prod % n"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l₁ l₂ : List α"], "goal": "l₁ ~ l₂ ↔ ∀ (a : α), count a l₁ = count a l₂"}}
+{"state": {"context": ["a b c : Cardinal.{u_1}"], "goal": "a ≠ 0 → b ≤ c → a ^ b ≤ a ^ c"}}
+{"state": {"context": ["L : Language", "T : L.Theory", "α : Type w", "n : ℕ", "φ : L.Sentence"], "goal": "T ⊨ᵇ φ ↔ ¬(T ∪ {Formula.not φ}).IsSatisfiable"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "CommRing A", "Field K", "CommRing B", "Field L", "Algebra A K", "Algebra B L", "Algebra A B", "Algebra K L", "Algebra A L", "IsScalarTower A K L", "IsScalarTower A B L", "IsDomain A", "IsFractionRing A K", "IsIntegralClosure B A L", "IsFractionRing B L", "FiniteDimensional K L", "Algebra.IsSeparable K L", "IsIntegrallyClosed A", "IsDedekindDomain B", "I : FractionalIdeal B⁰ L", "hI : I ≠ 0", "NoZeroSMulDivisors A B"], "goal": "↑(differentIdeal A B) ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (Submodule.restrictScalars A ↑I⁻¹) ≤ 1"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop"], "goal": "1 + m ≤ n ↔ m < n"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "K : Type u_4", "A : Type u_5", "A' : Type u_6", "A'' : Type u_7", "V : Type u", "V' : Type u_8", "x : ι → A", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing A'", "inst✝³ : CommRing A''", "inst✝² : Algebra R A", "inst✝¹ : Algebra R A'", "inst✝ : Algebra R A''", "a b : R", "s : Set ι", "this✝ : aeval (x ∘ Subtype.val) = (aeval x).comp (rename Subtype.val)", "this : ∀ (p : MvPolynomial (↑s) R), (rename Subtype.val) p = 0 ↔ p = 0"], "goal": "AlgebraicIndependent R (x ∘ Subtype.val) ↔ ∀ p ∈ supported R s, (aeval x) p = 0 → p = 0"}}
+{"state": {"context": ["Γ✝ : Type u_1", "R : Type u_2", "inst✝² : PartialOrder Γ✝", "inst✝¹ : AddGroup R", "Γ : Type u_3", "inst✝ : LinearOrder Γ", "x y : HahnSeries Γ R", "hxy : x.orderTop < y.orderTop"], "goal": "(x + -y).leadingCoeff = x.leadingCoeff"}}
+{"state": {"context": ["y : ℝ", "hp : 0 < y", "A : Tendsto (fun p => rexp (log p.1 * p.2)) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0)", "B : Tendsto (fun p => p.1 ^ p.2) (𝓝[≠] 0 ×ˢ 𝓝 y) (𝓝 0)"], "goal": "ContinuousAt (fun p => p.1 ^ p.2) (0, y)"}}
+{"state": {"context": ["X Y : LocallyRingedSpace", "f g : X ⟶ Y", "U : Opens ↑↑(coequalizer f.val g.val).toPresheafedSpace", "s : ↑((coequalizer f.val g.val).presheaf.obj (op U))"], "goal": "⇑f.val.base ⁻¹' (imageBasicOpen f g U s).carrier = ⇑g.val.base ⁻¹' (imageBasicOpen f g U s).carrier"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a b : R", "x y : M", "hv : LinearIndependent R v", "H : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0", "l : ι →₀ R", "hl : (Finsupp.total ι M R v) l = 0", "i : ι"], "goal": "l i = 0 i"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Topology.WithLowerSet.toLowerSet.symm = Topology.WithLowerSet.ofLowerSet"}}
+{"state": {"context": ["α : Type u", "β : Type v", "p : List α → Prop", "base : p []", "ind : ∀ (l : List α) (e : α), p l → p (l ++ [e])", "q : List α → Prop := fun l => p l.reverse", "pq : ∀ (l : List α), p l.reverse → q l"], "goal": "∀ (l : List α), p l"}}
+{"state": {"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "TotalComplexShape c₁ c₂ c₁₂", "TotalComplexShape c₂ c₁ c₁₂", "self : TotalComplexShapeSymmetry c₁ c₂ c₁₂", "i₁ : I₁", "i₂ i₂' : I₂", "h₂ : c₂.Rel i₂ i₂'"], "goal": "TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁ i₂ * c₁.ε₂ c₂ c₁₂ (i₁, i₂) =\n  c₂.ε₁ c₁ c₁₂ (i₂, i₁) * TotalComplexShapeSymmetry.σ c₁ c₂ c₁₂ i₁ i₂'"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l₁ l₂ : List α", "r : α → α → Prop", "a b : α", "L : List (List α)"], "goal": "L.join.Nodup ↔ (∀ l ∈ L, l.Nodup) ∧ Pairwise Disjoint L"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "ν : MeasureTheory.Measure α", "MeasureTheory.SigmaFinite ν", "f : α → ℝ≥0∞", "hf : Measurable f"], "goal": "(ν.withDensity f).rnDeriv ν =ᵐ[ν] f"}}
+{"state": {"context": ["𝕜 : Type u_1", "M : Type u_2", "α : Type u_3", "G : Type u_4", "E : Type u_5", "F : Type u_6", "inst✝⁷ : MeasurableSpace G", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "inst✝³ : NormedAddCommGroup F", "μ : Measure G", "f : G → E", "g : G", "inst✝² : Group G", "inst✝¹ : MeasurableMul G", "inst✝ : μ.IsMulLeftInvariant", "hf' : ∀ (x : G), f (g * x) = -f x"], "goal": "∫ (x : G), f x ∂μ = 0"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_4", "f : ι → α", "i : ι"], "goal": "Ordinal.bfamilyOfFamily f (Ordinal.typein WellOrderingRel i) ⋯ = f i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : Equiv.Perm α", "f : Equiv.Perm β"], "goal": "(e.sumCongr f)⁻¹ = e⁻¹.sumCongr f⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝² : Monoid M", "inst✝¹ : Monoid N", "inst✝ : Monoid P", "l l₁ l₂ : List M", "a : M", "x✝¹ : ℕ", "x✝ : M"], "goal": "([].set x✝¹ x✝).prod = ((take x✝¹ []).prod * if x✝¹ < [].length then x✝ else 1) * (drop (x✝¹ + 1) []).prod"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "𝕜 : Type u_3", "inst✝⁹ : _root_.RCLike 𝕜", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : InnerProductSpace 𝕜 E", "E' : Type u_5", "inst✝⁶ : NormedAddCommGroup E'", "inst✝⁵ : InnerProductSpace 𝕜 E'", "F : Type u_6", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace ℝ F", "F' : Type u_7", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : Fintype ι", "b : OrthonormalBasis ι 𝕜 E", "x : E"], "goal": "∑ x_1 : ι, (b.toBasis.repr x) x_1 • b.toBasis x_1 = x"}}
+{"state": {"context": ["R : Type u", "Semiring R", "S T : Ideal R", "x : R"], "goal": "x ∈ S → x ∈ S ⊔ T"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X", "x : X", "h : x ∈ s"], "goal": "𝓝 x ≤ 𝓝ˢ s"}}
+{"state": {"context": [], "goal": "PrimrecRel fun x x_1 => x ≤ x_1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s t u : Set α", "a : ℝ", "this : IsGLB (range Rat.cast ∩ Ioi a) a"], "goal": "MeasurableSet (Ioi a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : ℕ", "n : Fin m → ℕ", "inst✝ : ∀ (i : Fin m), NeZero (n i)", "i : Fin m", "j : Fin (n i)", "x : Fin m", "hx : x ≠ i"], "goal": "↑(Pi.single i j x) * ∏ j : Fin ↑x, n (Fin.castLE ⋯ j) = 0"}}
+{"state": {"context": ["R : Type u_2", "S : Type u_3", "NonUnitalNonAssocSemiring R", "NonUnitalNonAssocSemiring S", "f : R →ₙ+* S", "hf : ∀ (x y : R), Commute (f x) (f y)"], "goal": "⇑(f.toOpposite hf) = MulOpposite.op ∘ ⇑f"}}
+{"state": {"context": ["R : Type u", "Semiring R", "P : R[X]", "hP : P.Monic", "hdeg : 0 < P.degree", "n : ℕ", "hn : n ≠ 0"], "goal": "(∑ i ∈ Finset.range n, P ^ i).Monic"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "S : Set α", "hS : BooleanGenerators S", "T : Set α", "hT : T ⊆ S", "ha : IsAtom (sSup T)", "haS : sSup T ≤ sSup S", "a : α", "haT : a ∈ T", "this : a ≤ sSup T"], "goal": "a ≠ ⊥"}}
+{"state": {"context": ["motive : Nat → Nat → Sort u_1", "zero_zero : motive 0 0", "zero_succ : (n : Nat) → motive 0 (n + 1)", "succ_zero : (m : Nat) → motive (m + 1) 0", "succ_succ : (m n : Nat) → motive (m + 1) (n + 1)", "n : Nat"], "goal": "Nat.casesDiagOn 0 (n + 1) zero_zero zero_succ succ_zero succ_succ = zero_succ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "MetricSpace β", "n : ℕ", "j : ι", "f : ι → α → β", "g : α → β", "Preorder ι", "Countable ι", "hf : ∀ (n : ι), MeasureTheory.StronglyMeasurable (f n)", "hg : MeasureTheory.StronglyMeasurable g"], "goal": "MeasurableSet (MeasureTheory.Egorov.notConvergentSeq f g n j)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "E✝ B I X R✝ J E R : Set α"], "goal": "∀ I ⊆ (loopyOn E ↾ R).E, (loopyOn E ↾ R).Indep I ↔ (loopyOn R).Indep I"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : CosimplicialObject C", "n : ℕ", "i : Fin (n + 1)"], "goal": "X.δ i.castSucc ≫ X.σ i = 𝟙 (X.obj [n])"}}
+{"state": {"context": ["K : Type u", "V : Type v", "Ring K", "StrongRankCondition K", "AddCommGroup V", "Module K V", "Module.Free K V"], "goal": "Module.rank K V ≤ 1 ↔ ⊤.IsPrincipal"}}
+{"state": {"context": ["X : TopCat", "U : TopologicalSpace.Opens ↑X"], "goal": "⋯.functor.obj ⊤ = U"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace Ω", "inst✝³ : StandardBorelSpace Ω", "inst✝² : Nonempty Ω", "inst✝¹ : CountableOrCountablyGenerated α β", "κ : Kernel α (β × Ω)", "inst✝ : IsFiniteKernel κ", "f : β × Ω → ℝ≥0∞", "hf : Measurable f", "a : α", "t : Set Ω", "ht : MeasurableSet t"], "goal": "∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (b : β) in Set.univ, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "Ring k", "AddCommGroup V", "Module k V", "p : Submodule k V", "x : V"], "goal": "x ∈ p.toAffineSubspace ↔ x ∈ p"}}
+{"state": {"context": ["α : Type u_1", "Inv α", "a : α"], "goal": "AddOpposite.op a⁻¹ = (AddOpposite.op a)⁻¹"}}
+{"state": {"context": ["α : Type v", "LinearOrder α", "s : Set α", "a : α", "a_glb : IsGLB s a", "a_mem : a ∈ s"], "goal": "⋃ x ∈ s, Set.Ici x = Set.Ici a"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Preorder α", "Preorder β", "e : α ≃ β", "h₁ : Monotone ⇑e", "h₂ : Monotone ⇑e.symm"], "goal": "⇑(e.toOrderIso h₁ h₂) = ⇑e"}}
+{"state": {"context": ["x : ENNReal", "y : ℝ"], "goal": "(x ^ y).log = ↑y * x.log"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y : C", "a b : X ⟶ Y"], "goal": "a + -b = a - b"}}
+{"state": {"context": ["α : Type u_1", "β : α → Type u_4", "i : α"], "goal": "Function.Injective (Sigma.mk i)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "p : R[X]"], "goal": "(∀ (r : R), r ∣ p.content → IsUnit r) ↔ IsUnit p.content"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "s : Finset ι", "S₁ S₂ : Set ((i : { x // x ∈ s }) → α ↑i)"], "goal": "MeasureTheory.cylinder s S₁ ∩ MeasureTheory.cylinder s S₂ = MeasureTheory.cylinder s (S₁ ∩ S₂)"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β✝", "inst✝¹ : MeasurableSpace γ", "μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "f✝ : α → β✝", "β : Type u_8", "inst✝ : MeasurableSpace α", "mβ : MeasurableSpace β", "f : α → β", "hfi : Injective f", "hf : ∀ (s : Set α), MeasurableSet s → MeasurableSet (f '' s)", "μ : Measure β", "hs : MeasurableSet s"], "goal": "((comapₗ f) μ) s = μ (f '' s)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : AffineSubspace k P", "p : P", "hp : p ∈ s", "p2 : P"], "goal": "p2 -ᵥ p ∈ s.direction ↔ p2 ∈ s"}}
+{"state": {"context": ["α : Type u_1", "I✝ B X E : Set α", "Indep : Set α → Prop", "indep_empty : Indep ∅", "indep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I", "indep_aug : ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)", "indep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Indep J) → Indep I", "subset_ground : ∀ (I : Set α), Indep I → I ⊆ E", "I : Set α", "e : α", "hIe : ¬Indep (insert e I)", "h : ∀ I₀ ⊆ I, I₀.Finite → Indep (insert e I₀)", "J : Set α", "hJss : J ⊆ insert e I", "hJfin : J.Finite"], "goal": "Indep J"}}
+{"state": {"context": ["n a b : ℤ", "m : ℤ"], "goal": "a ≡ b [ZMOD n * m] → a ≡ b [ZMOD n]"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop", "ha : 2 ≤ a"], "goal": "Injective fun x => a ^ x"}}
+{"state": {"context": ["ι : Type u_1", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "c : ComplexShape ι", "K L : HomologicalComplex C c", "f : K ⟶ L", "i : ι", "K.HasHomology i", "L.HasHomology i"], "goal": "QuasiIsoAt f i ↔ CategoryTheory.IsIso (HomologicalComplex.homologyMap f i)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "G : CategoryTheory.Functor D C", "adj : F ⊣ G", "J : Type u", "CategoryTheory.Category.{v, u} J", "K : CategoryTheory.Functor J C", "c : CategoryTheory.Limits.Cocone K"], "goal": "((adj.functorialityUnit K).app c).hom = adj.unit.app c.pt"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝⁴ : MeasurableSpace Ω", "R : Type u_2", "inst✝³ : SMul R ℝ≥0", "inst✝² : SMul R ℝ≥0∞", "inst✝¹ : IsScalarTower R ℝ≥0 ℝ≥0∞", "inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞", "μ : FiniteMeasure Ω", "A : Set Ω"], "goal": "μ.restrict A = 0 ↔ μ A = 0"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "N : Type w₁", "CommRing R", "LieRing L", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "LieRingModule L M", "LieRingModule L N", "e : M ≃ₗ⁅R,L⁆ N"], "goal": "Function.Surjective ⇑e"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : PartialEquiv α β"], "goal": "Set.LeftInvOn (↑e.symm) (↑e) e.source"}}
+{"state": {"context": ["c c' : Cardinal.{u_1}", "hc : c ≤ Cardinal.aleph0", "hc' : c' ≤ Cardinal.aleph0"], "goal": "Cardinal.toPartENat c = Cardinal.toPartENat c' ↔ c = c'"}}
+{"state": {"context": ["x : ℝ", "m : ℤ"], "goal": "Irrational (x ^ m) → Irrational x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝ : TopologicalSpace α", "s t u v : Set α", "x : α"], "goal": "(connectedComponent x).Subsingleton ↔ connectedComponent x = {x}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Primcodable α", "Primcodable β", "r s : α → β → Prop", "(a : α) → (b : β) → Decidable (r a b)", "(a : α) → (b : β) → Decidable (s a b)", "hr : PrimrecRel r", "H : ∀ (a : α) (b : β), r a b ↔ s a b"], "goal": "PrimrecRel s"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Monoid M", "Monoid N", "f : M → N", "hf : IsMonoidHom f", "x : Mˣ"], "goal": "↑((Units.map' hf) x) = f ↑x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : _root_.RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "inst✝ : HasOrthogonalProjection K", "u : ↥K", "v : E"], "goal": "⟪↑((orthogonalProjection K) v), ↑u⟫_𝕜 = ⟪↑((orthogonalProjection K) v), ↑u⟫_𝕜 + ⟪v - ↑((orthogonalProjection K) v), ↑u⟫_𝕜"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : OrderedCommGroup α", "a b : α", "m n : ℤ", "hmn : m ≠ n", "x : α", "hx : x ∈ Ico (a * b ^ m) (a * b ^ (m + 1)) ∩ Ico (a * b ^ n) (a * b ^ (n + 1))"], "goal": "m = n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : Stream' (Option α)", "al : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a"], "goal": "↑(map id ⟨f, al⟩) = ↑⟨f, al⟩"}}
+{"state": {"context": ["R : Type u_1", "m : ℕ", "this : ∑ i ∈ range (m + 1), (2 * m + 1).choose (2 * m + 1 - i) = ∑ i ∈ range (m + 1), (2 * m + 1).choose i"], "goal": "m + 1 ≤ 2 * m + 1 + 1"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "i : σ", "j : σ →₀ ℕ", "r : R"], "goal": "(r = 0 ∨ Finsupp.single i 1 ≤ j) ∧ 1 ∣ r ↔ r = 0 ∨ j i ≠ 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "n : ℕ", "s : Affine.Simplex ℝ P n"], "goal": "(s.circumsphere.center ∈ affineSpan ℝ (Set.range s.points) ∧\n    Set.range s.points ⊆ Metric.sphere s.circumsphere.center s.circumsphere.radius) ∧\n  ∀ (cs : EuclideanGeometry.Sphere P),\n    cs.center ∈ affineSpan ℝ (Set.range s.points) ∧ Set.range s.points ⊆ Metric.sphere cs.center cs.radius →\n      cs = s.circumsphere"}}
+{"state": {"context": ["α : Type u_1", "SeminormedRing α", "a : α", "n : ℕ", "h : 0 < n"], "goal": "‖a ^ n‖ ≤ ‖a‖ ^ n"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M", "s : Set M"], "goal": "Submonoid.closure (Subtype.val ⁻¹' s) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝² : TopologicalSpace α", "β : Type u_3", "inst✝¹ : TopologicalSpace β", "inst✝ : PolishSpace β", "s : Set β", "hs : IsOpen s", "f : β → α", "f_cont : Continuous f"], "goal": "AnalyticSet (f '' s)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "s t : Set α", "a : α", "hs : s.Finite", "ht : t.Finite", "inst✝¹ : Fintype α", "p : α → Prop", "inst✝ : DecidablePred p", "h : {x | p x}.Finite", "a✝ : α"], "goal": "a✝ ∈ h.toFinset ↔ a✝ ∈ Finset.filter p Finset.univ"}}
+{"state": {"context": ["X : Type u_5", "TopologicalSpace X", "s t : Set X"], "goal": "DiscreteTopology ↑s → t ⊆ s → DiscreteTopology ↑t"}}
+{"state": {"context": ["α : Type u", "s t : Set α", "h : ∀ (u : Set α), s ⊆ u ↔ t ⊆ u"], "goal": "s = t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_5", "One β", "s : ι → Set α", "f : α → β", "a : α"], "goal": "Filter.Tendsto (fun n => (⋃ i ∈ n, s i).mulIndicator f a) Filter.atTop (pure ((Set.iUnion s).mulIndicator f a))"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹⁶ : CommSemiring R", "R' : Type u_2", "inst✝¹⁵ : Monoid R'", "R'' : Type u_3", "inst✝¹⁴ : Semiring R''", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "S : Type u_8", "T : Type u_9", "inst✝¹³ : AddCommMonoid M", "inst✝¹² : AddCommMonoid N", "inst✝¹¹ : AddCommMonoid P", "inst✝¹⁰ : AddCommMonoid Q", "inst✝⁹ : AddCommMonoid S", "inst✝⁸ : AddCommMonoid T", "inst✝⁷ : Module R M", "inst✝⁶ : Module R N", "inst✝⁵ : Module R P", "inst✝⁴ : Module R Q", "inst✝³ : Module R S", "inst✝² : Module R T", "inst✝¹ : DistribMulAction R' M", "inst✝ : Module R'' M", "g : P ≃ₗ[R] Q", "f : N ≃ₗ[R] P", "m : M", "n : N", "p : P", "x : M ⊗[R] N", "y : N ⊗[R] M"], "goal": "(rTensor M (refl R N)) y = y"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝³ : LinearOrder ι", "f : Filtration ι m", "τ : Ω → ι", "inst✝² : TopologicalSpace ι", "inst✝¹ : OrderTopology ι", "inst✝ : FirstCountableTopology ι", "hτ : IsStoppingTime f τ", "i i' : ι", "hi'_lub : IsLUB (Set.Iio i) i'", "h_Iio_eq_Iic : Set.Iio i = Set.Iic i'"], "goal": "MeasurableSet {ω | τ ω < i}"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Finset α", "h : t ⊆ s"], "goal": "s \\ (s \\ t) = t"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "CategoryTheory.Limits.HasEqualizers C", "f : X ⟶ Y", "CategoryTheory.Limits.HasImage f", "Y' : C", "h : Y ⟶ Y'", "CategoryTheory.IsIso h"], "goal": "(CategoryTheory.Limits.imageSubobjectCompIso f h).hom ≫ (CategoryTheory.Limits.imageSubobject f).arrow =\n  (CategoryTheory.Limits.imageSubobject (f ≫ h)).arrow ≫ CategoryTheory.inv h"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_13", "SeminormedRing R", "u : Rˣ", "f : α → R", "l : Filter α"], "goal": "Asymptotics.IsBigOWith ‖↑u⁻¹‖ l f fun x => ↑u * f x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "x : X", "hx : x ∈ e.source"], "goal": "∀ᶠ (y : Y) in 𝓝 (↑e x), ↑e (↑e.symm y) = y"}}
+{"state": {"context": ["α : Type u", "l : List α", "a : α", "n : ℕ", "hl : n < (a :: l).length"], "goal": "(a :: l).nthLe n hl = if hn : n = 0 then a else l.nthLe (n - 1) ⋯"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "OrderedSemiring 𝕜", "AddCommMonoid E", "AddCommMonoid F", "AddCommMonoid G", "Module 𝕜 E", "Module 𝕜 F", "Module 𝕜 G", "g : F →ₗ[𝕜] G", "f : E →ₗ[𝕜] F", "S : ConvexCone 𝕜 G"], "goal": "ConvexCone.comap f (ConvexCone.comap g S) = ConvexCone.comap (g ∘ₗ f) S"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x y : α", "s t : Set α", "Φ : α → β", "ε : ℝ≥0", "εpos : 0 < ε", "h : infEdist x (closure s) < ⊤", "ε0 : 0 < ↑ε / 2"], "goal": "infEdist x s ≤ infEdist x (closure s) + ↑ε"}}
+{"state": {"context": ["R✝ : Type u", "inst✝⁵ : CommRing R✝", "inst✝⁴ : IsDomain R✝", "inst✝³ : DiscreteValuationRing R✝", "R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : LocalRing R", "inst✝ : IsDomain R", "ϖ : R", "hϖ : ϖ ≠ 0", "h : maximalIdeal R = span {ϖ}", "h2 : ¬IsUnit ϖ", "a b : R", "hab : ϖ = ϖ * (ϖ * (a * b))"], "goal": "ϖ * (a * b) ≠ 1"}}
+{"state": {"context": ["w x✝ y z : ℝ", "x : ℝ≥0"], "goal": "√↑x = ↑x ^ (1 / 2)"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{?u.414921, u_1} C", "n m : ℕ", "F G : ComposableArrows C n", "f g : ComposableArrows C 5", "app₀ : f.obj' 0 ⋯ ⟶ g.obj' 0 ⋯", "app₁ : f.obj' 1 ⋯ ⟶ g.obj' 1 ⋯", "app₂ : f.obj' 2 ⋯ ⟶ g.obj' 2 ⋯", "app₃ : f.obj' 3 ⋯ ⟶ g.obj' 3 ⋯", "app₄ : f.obj' 4 ⋯ ⟶ g.obj' 4 ⋯", "app₅ : f.obj' 5 ⋯ ⟶ g.obj' 5 ⋯", "w₀ : f.map' 0 1 ⋯ ⋯ ≫ app₁ = app₀ ≫ g.map' 0 1 ⋯ ⋯", "w₁ : f.map' 1 2 ⋯ ⋯ ≫ app₂ = app₁ ≫ g.map' 1 2 ⋯ ⋯", "w₂ : f.map' 2 3 ⋯ ⋯ ≫ app₃ = app₂ ≫ g.map' 2 3 ⋯ ⋯", "w₃ : f.map' 3 4 ⋯ ⋯ ≫ app₄ = app₃ ≫ g.map' 3 4 ⋯ ⋯", "w₄ : f.map' 4 5 ⋯ ⋯ ≫ app₅ = app₄ ≫ g.map' 4 5 ⋯ ⋯"], "goal": "2 < 5 + 1"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "K L L' M : Set V", "Gc : G.Preconnected", "hK : K.Nonempty", "v : V", "vnK : v ∉ K", "C : G.ComponentCompl K := G.componentComplMk vnK", "dis : K ∩ ↑C ⊆ ∅ := Set.disjoint_iff.mp (ComponentCompl.disjoint_right C)", "h : ∀ (ck : V × V), ck.1 ∈ G.componentComplMk vnK → ck.2 ∈ K → ¬G.Adj ck.1 ck.2"], "goal": "∀ (x : V), x ∈ ↑C"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasureSpace Ω", "inst✝ : IsProbabilityMeasure ℙ", "X : ℕ → Ω → ℝ", "hint : Integrable (X 0) ℙ", "hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ", "hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ", "hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω", "c : ℝ", "c_one : 1 < c", "ω : Ω", "hω : (fun n => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop] fun n => ↑⌊c ^ n⌋₊", "A : Tendsto (fun n => ⌊c ^ n⌋₊) atTop atTop"], "goal": "(fun n => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ↑⌊c ^ n⌋₊ * ∫ (a : Ω), X 0 a) = fun x => (∑ i ∈ range ⌊c ^ x⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ x⌋₊, truncation (X i) ↑i) a) + ((fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) ∘ fun n => ⌊c ^ n⌋₊) x"}}
+{"state": {"context": ["R✝ : Type u", "S✝ : Type v", "F : Type u_1", "inst✝¹⁰ : Semiring R✝", "inst✝⁹ : Semiring S✝", "inst✝⁸ : FunLike F R✝ S✝", "rc : RingHomClass F R✝ S✝", "f : F", "I✝ J : Ideal R✝", "K L : Ideal S✝", "G : Type u_2", "inst✝⁷ : FunLike G S✝ R✝", "rcg : RingHomClass G S✝ R✝", "ι : Sort u_3", "R : Type u_4", "S : Type u_5", "M : Type u_6", "inst✝⁶ : CommSemiring R", "inst✝⁵ : CommSemiring S", "inst✝⁴ : Algebra R S", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : Module S M", "inst✝ : IsScalarTower R S M", "I : Ideal R", "N : Submodule S M"], "goal": "Submodule.restrictScalars R (⨆ b ∈ ↑I, (algebraMap R S) b • N) = ⨆ a ∈ ↑I, Submodule.restrictScalars R ((algebraMap R S) a • N)"}}
+{"state": {"context": ["T : ℝ", "g : ℝ → ℝ", "hg : Function.Periodic g T", "h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂", "hT : 0 < T", "t : ℝ"], "goal": "∫ (x : ℝ) in 0 ..t, g x ≤ sSup ((fun t => ∫ (x : ℝ) in 0 ..t, g x) '' Set.Icc 0 T) + ⌊t / T⌋ • ∫ (x : ℝ) in 0 ..T, g x"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Zero α", "f : α → β"], "goal": "Filter.map f 0 = pure (f 0)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "X Y : Cᵒᵖ", "f : X ⟶ Y"], "goal": "(-f).unop = -f.unop"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.LocallyRingedSpace", "self : X.Hom Y", "x : ↑↑X.toPresheafedSpace"], "goal": "IsLocalRingHom (AlgebraicGeometry.PresheafedSpace.Hom.stalkMap self.val x)"}}
+{"state": {"context": ["M : Type u_3", "N : Type u_4"], "goal": "∀ {x : Mul M} {x_1 : CommSemigroup N} (f g : M →ₙ* N), ⇑f * ⇑g = fun x_2 => f x_2 * g x_2"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "P Q : C", "f : P ⟶ Q"], "goal": "(∀ (a : CategoryTheory.Abelian.Pseudoelement P), CategoryTheory.Abelian.Pseudoelement.pseudoApply f a = 0) → 0 = f"}}
+{"state": {"context": ["R A B : Type u", "CommRing R", "CommRing A", "CommRing B", "Algebra R A", "Algebra R B"], "goal": "(CommRingCat.pushoutCocone R A B).pt = CommRingCat.of (A ⊗[R] B)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasBinaryBiproducts C", "X₁ X₂ Y₁ Y₂ : C", "f₁₁ : X₁ ⟶ Y₁", "f₁₂ : X₁ ⟶ Y₂", "f₂₁ : X₂ ⟶ Y₁", "f₂₂ : X₂ ⟶ Y₂"], "goal": "CategoryTheory.Limits.biprod.inl ≫ CategoryTheory.Biprod.ofComponents f₁₁ f₁₂ f₂₁ f₂₂ =\n  f₁₁ ≫ CategoryTheory.Limits.biprod.inl + f₁₂ ≫ CategoryTheory.Limits.biprod.inr"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "MulOneClass M", "MulOneClass N", "f : M →* N"], "goal": "f.comp (MonoidHom.id M) = f"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "δ : Type u_4", "β : Type u_5", "inst✝ : CommMonoid β", "u a b : β", "hu : IsUnit u"], "goal": "a * u ~ᵤ b ↔ a ~ᵤ b"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_5", "OrderedSemiring 𝕜", "AddCommMonoid E", "OrderedAddCommMonoid β", "Module 𝕜 E", "Module 𝕜 β", "s : Set E", "f : E → β"], "goal": "ConvexOn 𝕜 s f ↔\n  Convex 𝕜 s ∧\n    ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (a • x + b • y) ≤ a • f x + b • f y"}}
+{"state": {"context": [], "goal": "Embedding Real.toEReal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : MeasurableSpace α", "s✝ : SignedMeasure α", "μ ν : Measure α", "inst✝¹ : IsFiniteMeasure μ", "inst✝ : IsFiniteMeasure ν", "s : SignedMeasure α"], "goal": "∃ i, ∃ (hi₁ : MeasurableSet i) (hi₂ : VectorMeasure.restrict 0 i ≤ VectorMeasure.restrict s i) (hi₃ : VectorMeasure.restrict s iᶜ ≤ VectorMeasure.restrict 0 iᶜ), s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂ ∧ s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "LinearOrder α", "f : Fin n → α", "P : (Fin n → α) → Prop", "hf : P f", "h :\n  ∀ (σ : Equiv.Perm (Fin n)) (i j : Fin n),\n    i < j → (f ∘ ⇑σ) j < (f ∘ ⇑σ) i → P (f ∘ ⇑σ) → P (f ∘ ⇑σ ∘ ⇑(Equiv.swap i j))"], "goal": "P (f ∘ ⇑(Tuple.sort f))"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "inst✝⁵ : IsDomain R", "inst✝⁴ : NormalizedGCDMonoid R", "S : Type u_2", "inst✝³ : Ring S", "inst✝² : IsDomain S", "inst✝¹ : Algebra R S", "inst✝ : NoZeroSMulDivisors R S", "p : R[X]", "s : S", "hpzero : p ≠ 0", "hp : (aeval s) p = 0"], "goal": "(aeval s) p.primPart = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t : Finset α", "P : Finpartition s", "hP : P.IsEquipartition", "ht : t ∈ P.parts"], "goal": "s.card / P.parts.card ≤ t.card"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "x : MvPolynomial σ R", "i : σ"], "goal": "(x * MvPolynomial.X i).modMonomial (Finsupp.single i 1) = 0"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B B' I J D X : Set α", "e f : α", "x✝ : M.Indep B ∧ ∀ (I : Set α), M.Indep I → B ⊆ I → B = I", "h : M.Indep B", "h' : ∀ (I : Set α), M.Indep I → B ⊆ I → B = I"], "goal": "M.Base B"}}
+{"state": {"context": ["w i : Nat", "x : BitVec w"], "goal": "BitVec.truncate (i + 1) x = BitVec.cons (x.getLsb i) (BitVec.truncate i x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_5", "ι : Type u_8", "ι' : Type u_9", "Nonempty ι", "Nonempty ι'", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : SeminormFamily 𝕜 E ι", "q : SeminormFamily 𝕜 E ι'", "t : TopologicalSpace E", "hp : WithSeminorms p", "hpq : Seminorm.IsBounded p q LinearMap.id", "hqp : Seminorm.IsBounded q p LinearMap.id"], "goal": "WithSeminorms q"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Type u_4", "s s' : Set α", "x : α", "p✝ : Filter ι✝", "g : ι✝ → α", "inst✝ : UniformSpace β", "𝔖 : Set (Set α)", "ι : Sort u_5", "p : ι → Prop", "V : ι → Set (β × β)", "h : (𝓤 β).HasBasis p V"], "goal": "𝓤 (α →ᵤ[𝔖] β) = ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 (UniformOnFun.gen 𝔖 s (V i))"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "AddGroup α", "ContinuousConstVAdd αᵃᵒᵖ α", "s : Set α", "a : α", "b : α", "h : s ∈ 𝓝 b"], "goal": "s + {a} ∈ 𝓝 (b + a)"}}
+{"state": {"context": ["X : Type u_1", "R : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : Semiring R", "inst✝¹ : TopologicalSpace R", "inst✝ : TopologicalSemiring R", "s : Set X", "f : C(X, R)"], "goal": "(∃ x ∈ sᶜ, f x ≠ 0) ↔ ∃ x ∈ sᶜ, f x ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "σ : Type u_2", "inst✝ : CommRing R", "r : R", "I : Ideal R", "f : MvPolynomial σ R"], "goal": "∀ (p q : MvPolynomial σ R), eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) ((Ideal.Quotient.mk (Ideal.map C I)) p)) = (Ideal.Quotient.mk (Ideal.map C I)) p → eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) ((Ideal.Quotient.mk (Ideal.map C I)) q)) = (Ideal.Quotient.mk (Ideal.map C I)) q → eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map C I)).comp C) ⋯) (fun i => (Ideal.Quotient.mk (Ideal.map C I)) (X i)) ((Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯) ((Ideal.Quotient.mk (Ideal.map C I)) (p + q))) = (Ideal.Quotient.mk (Ideal.map C I)) (p + q)"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeSup α", "a b c : α", "hac : a ⩿ c", "hbc : b ⩿ c", "hab : a ≠ b"], "goal": "a ⊔ b = c"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a", "i n : ℕ", "npos : 0 < n", "lem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))", "ij : i < n + 1", "j2n : n + 1 ≤ 2 * n", "jnn : n + 1 ≠ n", "ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ n + 1 = 2)", "o : n = n ∨ n < n", "lin : i < n"], "goal": "xn a1 (n - 1) + xn a1 i % xn a1 n < xn a1 n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Sort u_4", "ι'✝ : Sort u_5", "l✝ l' : Filter α", "p✝ : ι✝ → Prop", "s✝ : ι✝ → Set α", "t✝ : Set α", "i : ι✝", "p' : ι'✝ → Prop", "s' : ι'✝ → Set α", "i' : ι'✝", "ι : Type u_6", "ι' : Type u_7", "inst✝ : Nonempty ι", "l : ι → Filter α", "s : ι → ι' → Set α", "p : ι → ι' → Prop", "hl : ∀ (i : ι), (l i).HasBasis (p i) (s i)", "h : Directed (fun x x_1 => x ≥ x_1) l", "t : Set α"], "goal": "t ∈ ⨅ i, l i ↔ ∃ i, p i.1 i.2 ∧ s i.1 i.2 ⊆ t"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁸ : NormedRing A", "inst✝⁷ : StarRing A", "inst✝⁶ : CstarRing A", "inst✝⁵ : CompleteSpace A", "inst✝⁴ : NormedAlgebra ℂ A", "inst✝³ : StarModule ℂ A", "inst✝² : PartialOrder A", "inst✝¹ : StarOrderedRing A", "inst✝ : Nontrivial A", "a : A", "ha : autoParam (IsSelfAdjoint a) _auto✝"], "goal": "‖a‖ ∈ spectrum ℝ a ∨ -‖a‖ ∈ spectrum ℝ a"}}
+{"state": {"context": ["m n : ℕ"], "goal": "Nat.fib (m + n + 1) = Nat.fib m * Nat.fib n + Nat.fib (m + 1) * Nat.fib (n + 1)"}}
+{"state": {"context": ["d₁ d₂ d₃ : ManyOneDegree"], "goal": "d₁ ≤ d₂ → d₂ ≤ d₃ → d₁ ≤ d₃"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "LinearOrderedSemifield α", "l : Filter β", "f : β → α", "r : α", "hr : 0 < r", "hf : Filter.Tendsto f l Filter.atTop"], "goal": "Filter.Tendsto (fun x => f x * r) l Filter.atTop"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : MetricSpace E", "f : ℕ → α → E", "g : α → E", "hfg : TendstoInMeasure μ f atTop g", "h_lt_ε_real : ∀ (ε : ℝ), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε"], "goal": "∃ ns, StrictMono ns ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i => f (ns i) x) atTop (𝓝 (g x))"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_4", "E : B → Type u_6", "NontriviallyNormedField 𝕜", "EB : Type u_7", "NormedAddCommGroup EB", "NormedSpace 𝕜 EB", "HB : Type u_8", "TopologicalSpace HB", "IB : ModelWithCorners 𝕜 EB HB", "TopologicalSpace B", "ChartedSpace HB B", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module 𝕜 (E x)", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "TopologicalSpace (Bundle.TotalSpace F E)", "(x : B) → TopologicalSpace (E x)", "FiberBundle F E", "VectorBundle 𝕜 F E", "SmoothVectorBundle F E IB", "e e' : Trivialization F Bundle.TotalSpace.proj", "MemTrivializationAtlas e", "MemTrivializationAtlas e'"], "goal": "SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun b => ↑(Trivialization.coordChangeL 𝕜 e e' b)) (e.baseSet ∩ e'.baseSet)"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → Preorder (α i)", "x✝ y✝ : (i : ι) → α i", "inst✝ : DecidableEq ι", "x y : (i : ι) → α i", "i₀ : ι", "m : α i₀", "z : (i : ι) → α i", "h₁ : z ∈ univ.pi fun i => Ioc (x i) (update y i₀ m i)", "h₂ : z ∈ univ.pi fun i => Ioc (update x i₀ m i) (y i)"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "α' : Type u_1", "β✝ : Type v", "β' : Type u_2", "γ✝ : Type u_3", "φ : Type u_4", "inst✝⁵ : OmegaCompletePartialOrder α", "inst✝⁴ : OmegaCompletePartialOrder β✝", "inst✝³ : OmegaCompletePartialOrder γ✝", "inst✝² : OmegaCompletePartialOrder φ", "inst✝¹ : OmegaCompletePartialOrder α'", "inst✝ : OmegaCompletePartialOrder β'", "β γ : Type v", "f : β → γ", "g : α → Part β", "hg : Continuous' g"], "goal": "Continuous' fun x => f <$> g x"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "j : MeasureTheory.JordanDecomposition α"], "goal": "∃ S,\n  MeasurableSet S ∧\n    MeasureTheory.VectorMeasure.restrict j.toSignedMeasure S ≤ MeasureTheory.VectorMeasure.restrict 0 S ∧\n      MeasureTheory.VectorMeasure.restrict 0 Sᶜ ≤ MeasureTheory.VectorMeasure.restrict j.toSignedMeasure Sᶜ ∧\n        j.posPart S = 0 ∧ j.negPart Sᶜ = 0"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝ : LinearOrder α", "s✝ : Finset α", "H : s✝.Nonempty", "x : α", "s : Finset αᵒᵈ"], "goal": "WithTop.map (⇑ofDual) s.min = s.sup (WithBot.some ∘ ⇑ofDual)"}}
+{"state": {"context": ["V₁ : Type u_2", "V₂ : Type u_3", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂"], "goal": "‖0‖ = 0"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝³ : LinearOrder ι", "f : Filtration ι m", "τ : Ω → ι", "inst✝² : TopologicalSpace ι", "inst✝¹ : OrderTopology ι", "inst✝ : FirstCountableTopology ι", "hτ : IsStoppingTime f τ", "i : ι", "this : {ω | i ≤ τ ω} = {ω | τ ω < i}ᶜ"], "goal": "MeasurableSet {ω | i ≤ τ ω}"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : NonAssocRing R", "inst✝¹ : Pow R ℕ", "inst✝ : BinomialRing R", "n k : ℕ"], "goal": "(fun x => (n + k + 1).factorial • x) (multichoose (-↑n) (n + k + 1)) = (fun x => (n + k + 1).factorial • x) 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : AddMonoidWithOne α", "n : ℕ"], "goal": "↑↑n = ↑n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace α", "a : α", "f : α → ℝ≥0∞", "s : Set α", "inst✝¹ : MeasurableSingletonClass α", "inst✝ : Decidable (a ∈ s)"], "goal": "∫⁻ (x : α) in s, f x ∂dirac a = if a ∈ s then f a else 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝ : CommRing R", "p : R[X]", "h : p.leadingCoeff ∈ R⁰", "x : R[X]", "hx : x * p = 0"], "goal": "x.leadingCoeff = 0"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u", "x : α (Fin.last n)", "p : (i : Fin n) → α i.castSucc"], "goal": "Fin.snoc p x (Fin.last n) = x"}}
+{"state": {"context": ["z : ℍ"], "goal": "‖cexp (2 * ↑π * Complex.I * ↑z)‖ < 1"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝³ : PseudoEMetricSpace Ω", "inst✝² : MeasurableSpace Ω", "inst✝¹ : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝ : SFinite μ", "s : Set Ω", "a b : ℝ", "hab : a < b", "mbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))"], "goal": "∃ r ∈ Ioo a b, μ (frontier (Metric.thickening r s)) = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : ClosedIciTopology α", "inst✝ : TopologicalSpace β", "a b c : α", "f : α → β", "h : a < b"], "goal": "ContinuousWithinAt f (Icc a b) a ↔ ContinuousWithinAt f (Ici a) a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "LocallyFiniteOrder α", "a b : α"], "goal": "b ∈ Multiset.Ioc a b ↔ a < b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "I✝ J : Ideal R", "I : Ideal S", "f : R →+* S", "p : Ideal R", "H : p ∈ (comap f I).minimalPrimes", "this : p.IsPrime", "f' : R →+* S ⧸ I := (Quotient.mk I).comp f", "e : RingHom.ker f' = comap f I"], "goal": "∃ p', p'.IsPrime ∧ I ≤ p' ∧ comap f p' = p"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "X : C"], "goal": "CategoryTheory.Sieve.functorPushforward F ⊥ = ⊥"}}
+{"state": {"context": ["R : Type u_1", "Rₛ : Type u_2", "CommSemiring R", "S : Submonoid R", "CommSemiring Rₛ", "Algebra R Rₛ", "hT : IsLocalization S Rₛ", "M : Type u_3", "AddCommMonoid M", "Module R M", "Module Rₛ M", "IsScalarTower R Rₛ M", "ι : Type u_5", "b : ι → M", "hli : LinearIndependent R b"], "goal": "LinearIndependent Rₛ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "f g : ι → α → β", "p : ℝ≥0∞", "hf : UnifTight f p μ", "ε : ℝ≥0∞", "hε : ε ≠ 0", "hε_top : ¬ε = ⊤", "s : Set α", "hμs : μ s ≠ ⊤", "hfs : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal"], "goal": "∀ᶠ (s : Set α) in μ.cofinite.smallSets, ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ ε"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "CommMonoid α", "CommMonoid β", "CommMonoid γ", "A : Set α", "B : Set β", "C : Set γ", "f : α → β", "g : β → γ", "n : ℕ", "hg : IsMulFreimanHom n B C g", "hf : IsMulFreimanHom n A B f"], "goal": "IsMulFreimanHom n A C (g ∘ f)"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "p : Type u_5", "q : Type u_6", "m' : o → Type u_7", "n' : o → Type u_8", "p' : o → Type u_9", "R : Type u_10", "S : Type u_11", "α : Type u_12", "β : Type u_13", "inst✝³ : DecidableEq o", "inst✝² : NonUnitalNonAssocSemiring α", "inst✝¹ : (i : o) → Fintype (n' i)", "inst✝ : Fintype o", "M : (i : o) → Matrix (m' i) (n' i) α", "N : (i : o) → Matrix (n' i) (p' i) α", "k : o", "i : m' k", "k' : o", "j : p' k'"], "goal": "∀ (x : o), x ≠ k → (∑ x_1 : n' x, (if h : k = x then M k i (cast ⋯ x_1) else 0) * if h : x = k' then N x x_1 (cast ⋯ j) else 0) = 0"}}
+{"state": {"context": ["a b c d m n✝ : ℤ", "n : ℕ", "hn : n ≠ 0"], "goal": "0 < ↑n"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V", "h : ⟪x, y⟫_ℝ = 0", "h0 : x = 0 ∨ y ≠ 0"], "goal": "‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "inst✝⁷ : NontriviallyNormedField 𝕜₁", "inst✝⁶ : SeminormedRing 𝕜₂", "σ₁₂ : 𝕜₁ →+* 𝕜₂", "M₁ : Type u_3", "M₂ : Type u_4", "M₃ : Type u_5", "inst✝⁵ : SeminormedAddCommGroup M₁", "inst✝⁴ : TopologicalSpace M₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : NormedSpace 𝕜₁ M₁", "inst✝¹ : Module 𝕜₂ M₂", "inst✝ : ContinuousConstSMul 𝕜₂ M₂", "f : M₁ →ₛₗ[σ₁₂] M₂", "hf : IsCompactOperator ⇑f", "S : Set M₁", "hS : Bornology.IsBounded S"], "goal": "IsVonNBounded 𝕜₁ S"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "b b' c : R", "a a' : ℕ"], "goal": "a = a' → b = b' → a' • b' = c → a • b = c"}}
+{"state": {"context": ["f : ℕ → ENNReal", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m"], "goal": "∑' (k : ℕ), 2 ^ k * f (2 ^ k) ≤ f 1 + 2 * ∑' (k : ℕ), f k"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β", "c : α", "AddCommMonoid α", "DivisionSemiring γ", "Module γ α", "h : Function.Periodic f c", "a : γ"], "goal": "Function.Periodic (fun x => f (a⁻¹ • x)) (a • c)"}}
+{"state": {"context": ["n m : ℕ", "hm : n ≠ m"], "goal": "(1 &&& 2 ^ n / 2 ^ m != 0) = false"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedAddCommGroup G", "g : E → F", "c : ℝ≥0", "𝕜 : Type u_5", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace 𝕜 F", "K : Type u_6", "inst✝ : RCLike K", "f : α → K"], "goal": "Memℒp (fun x => RCLike.re (f x)) p μ ∧ Memℒp (fun x => RCLike.im (f x)) p μ ↔ Memℒp f p μ"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "inst✝ : SFinite ν", "s : Set (α × β)", "h : (μ.prod ν) s = 0", "t : Set (α × β)", "hst : s ⊆ t", "mt : MeasurableSet t", "ht : (fun x => ν (Prod.mk x ⁻¹' t)) =ᶠ[ae μ] 0"], "goal": "(fun x => ν (Prod.mk x ⁻¹' s)) ≤ᶠ[ae μ] 0 ∧ 0 ≤ᶠ[ae μ] fun x => ν (Prod.mk x ⁻¹' s)"}}
+{"state": {"context": ["n : ZNum"], "goal": "n + n + 1 = n.bit1"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedSemiring α", "a : α", "Invertible a"], "goal": "0 ≤ ⅟a ↔ 0 ≤ a"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : Dist E", "l : Filter ι", "f f' : ι → α → E", "g g' : α → E", "h_left : ∀ᶠ (i : ι) in l, f i =ᶠ[ae μ] f' i", "h_right : g =ᶠ[ae μ] g'", "h_tendsto : TendstoInMeasure μ f l g", "ε : ℝ", "hε : 0 < ε", "i : ι", "h_ae_eq : f i =ᶠ[ae μ] f' i", "x : α", "hxf : f i x = f' i x", "hxg : g x = g' x"], "goal": "{x | ε ≤ dist (f' i x) (g' x)} x ↔ {x | ε ≤ dist (f i x) (g x)} x"}}
+{"state": {"context": ["R : Type u", "inst✝⁷ : Ring R", "Q : Type v", "inst✝⁶ : AddCommGroup Q", "inst✝⁵ : Module R Q", "M : Type u_1", "N : Type u_2", "inst✝⁴ : AddCommGroup M", "inst✝³ : AddCommGroup N", "inst✝² : Module R M", "inst✝¹ : Module R N", "i : M →ₗ[R] N", "f : M →ₗ[R] Q", "inst✝ : Fact (Function.Injective ⇑i)", "h : Baer R Q", "y : N"], "goal": "∃ y_1 ∈ (extensionOfMax i f).domain, ∃ z ∈ Submodule.span R {?y}, y_1 + z = y"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝³ : DecidableEq ι", "inst✝² : (i : ι) → AddZeroClass (β i)", "inst✝¹ : (i : ι) → AddZeroClass (β₁ i)", "inst✝ : (i : ι) → AddZeroClass (β₂ i)", "f : (i : ι) → β₁ i →+ β₂ i", "f₂ : (i : ι) → β i →+ β₁ i"], "goal": "∀ (i : ι), (fun i x => (f i) x) i 0 = 0"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "s : Submodule ℝ E", "hs : s ≠ ⊤", "x : E", "hx : x ∉ s", "c : ℝ", "cpos : 0 < c", "cone : c < 1"], "goal": "μ ↑s = 0"}}
+{"state": {"context": ["F : Type u", "inst✝³ : Field F", "K : Type v", "inst✝² : Field K", "S : Type u_1", "inst✝¹ : CommRing S", "inst✝ : Nontrivial S", "f : F →+* S", "p : F[X]", "H : (map f p).Separable", "m : Ideal S", "hm : m.IsMaximal"], "goal": "p.Separable"}}
+{"state": {"context": ["G : Type u_1", "inst✝³ : MeasurableSpace G", "μ : Measure G", "g✝ : G", "inst✝² : Group G", "inst✝¹ : MeasurableMul G", "inst✝ : μ.IsMulLeftInvariant", "f : G → ℝ≥0∞", "g : G"], "goal": "∫⁻ (x : G), f (g * x) ∂μ = ∫⁻ (x : G), f x ∂μ"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝ : CommRing R", "H : ∀ (P : Ideal R), P.IsPrime → IsPrincipal P", "J : Ideal R", "hJ : J ∈ nonPrincipals R", "I : Ideal R", "Ibad : I ∈ nonPrincipals R", "Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I", "Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J", "hI1 : ¬I = ⊤", "x y : R", "hxy : x * y ∈ I", "hy : y ∉ I", "a : R", "ha : I ⊔ Ideal.span {y} = Submodule.span R {a}"], "goal": "x ∈ I"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "a : α", "n : ℕ", "h : ⌊a⌋₊ < n"], "goal": "a < ↑n"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "e : PartialHomeomorph M H"], "goal": "e.symm ≫ₕ e ∈ analyticGroupoid I"}}
+{"state": {"context": ["a : EReal"], "goal": "1 ≤ a.exp ↔ 0 ≤ a"}}
+{"state": {"context": ["f : CircleDeg1Lift", "τ' : ℝ", "h : Filter.Tendsto f.transnumAuxSeq Filter.atTop (𝓝 τ')"], "goal": "f.translationNumber = τ'"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "x y : α", "p : α → Prop", "hx : p x", "hy : p y"], "goal": "p (min x y)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹² : CommRing R", "M : Type u_2", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R M", "N : Type u_3", "P : Type u_4", "inst✝⁹ : AddCommGroup N", "inst✝⁸ : Module R N", "inst✝⁷ : AddCommGroup P", "inst✝⁶ : Module R P", "ι : Type u_5", "inst✝⁵ : Module.Free R M", "inst✝⁴ : Module.Finite R M", "inst✝³ : Module.Free R N", "inst✝² : Module.Finite R N", "inst✝¹ : Module.Free R P", "inst✝ : Module.Finite R P", "f : M →ₗ[R] M", "g : N →ₗ[R] N", "h : (trace R (M ⊗[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g"], "goal": "(trace R (M ⊗[R] N)) (TensorProduct.map f g) = (trace R M) f * (trace R N) g"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "inst✝⁴ : Fintype ι", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : CompleteSpace E", "f : (ι → ℝ) → E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "I : Box ι", "hc : ContinuousOn f (Box.Icc I)", "l : IntegrationParams", "y : E", "hy : HasIntegral I l f μ.toBoxAdditive.toSMul y"], "goal": "∫ (x : ι → ℝ) in ↑I, f x ∂μ = y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "s t : Set α", "hs : MeasurableSet s", "hsμ : μ s ≠ ⊤", "ht : MeasurableSet t", "htμ : μ t ≠ ⊤", "c : E", "hc : c ≠ 0"], "goal": "MeasureTheory.indicatorConstLp p hs hsμ c = MeasureTheory.indicatorConstLp p ht htμ c ↔ s =ᵐ[μ] t"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α"], "goal": "Metric.hausdorffDist ∅ s = 0"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "S : Subsemiring R"], "goal": "algebraMap (↥S) R = S.subtype"}}
+{"state": {"context": ["L : List ℕ"], "goal": "L.headI + L.tail.sum = L.sum"}}
+{"state": {"context": ["I : Type u", "inst✝³ : LinearOrder I", "inst✝² : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "l : Products I", "J K : I → Prop", "h : ∀ a ∈ ↑l, J a", "inst✝¹ : (j : I) → Decidable (J j)", "inst✝ : (j : I) → Decidable (K j)", "hJK : ∀ (i : I), J i → K i"], "goal": "⇑(eval (π C J) l) ∘ ProjRestricts C hJK = ⇑(eval (π C K) l)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "f✝ f₁ f₂ : Filter α", "g✝ g₁ g₂ : Set α → Filter β", "f : Filter α", "g h : Set α → Filter β"], "goal": "(f.lift fun x => g x ⊓ h x) = f.lift g ⊓ f.lift h"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "inst✝⁹ : NormedSpace 𝕜 E", "inst✝⁸ : SMulCommClass ℝ 𝕜 E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ F", "inst✝⁵ : CompleteSpace F", "G : Type u_5", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedSpace ℝ G", "f✝ g✝ : α → E", "m : MeasurableSpace α", "μ : Measure α", "X : Type u_6", "inst✝² : TopologicalSpace X", "inst✝¹ : FirstCountableTopology X", "ν : Measure α", "β : Type u_7", "inst✝ : MeasurableSpace β", "φ : α → β", "hφ : AEMeasurable φ μ", "f : β → G", "hfm : AEStronglyMeasurable f (Measure.map φ μ)", "g : β → G := AEStronglyMeasurable.mk f hfm"], "goal": "Measure.map φ μ = Measure.map (AEMeasurable.mk φ hφ) μ"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ : C"], "goal": "(α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom ≫ X₁ ◁ ((β_ (𝟙_ C) X₂).hom ≫ (ρ_ X₂).hom) = ((α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv ≫ X₁ ◁ (β_ (𝟙_ C) X₂).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).hom ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).inv) ≫ (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom ≫ (ρ_ (X₁ ⊗ X₂)).hom"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "DecidableEq α", "A B : Finset α", "hAB : ∀ A' ⊆ A, (A + B).card * A'.card ≤ (A' + B).card * A.card", "n : ℕ"], "goal": "↑(A + n • B).card ≤ (↑(A + B).card / ↑A.card) ^ n * ↑A.card"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "a b c : α"], "goal": "max (a - c) (b - c) = max a b - c"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "k n : ℕ", "f : 𝓢(E, F)", "x : E"], "goal": "‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖"}}
+{"state": {"context": ["R : Type u_3", "Γ₀ : Type u_4", "LinearOrderedAddCommGroupWithTop Γ₀", "Ring R", "v : AddValuation R Γ₀", "x y : R", "hx : v y < v x"], "goal": "v (x - y) = v y"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "AddCommMonoid α", "f : β → α", "g : γ → α", "h_eq : ∀ (u : Finset γ), ∃ v, ∀ (v' : Finset β), v ⊆ v' → ∃ u', u ⊆ u' ∧ ∑ x ∈ u', g x = ∑ b ∈ v', f b"], "goal": "Filter.map (fun s => ∑ b ∈ s, f b) Filter.atTop ≤ Filter.map (fun s => ∑ x ∈ s, g x) Filter.atTop"}}
+{"state": {"context": ["r r₁ r₂ : ℝ≥0", "x y : ℝ", "n : ℕ"], "goal": "↑(↑n).toNNReal = ↑↑n"}}
+{"state": {"context": ["α : Type u", "s t u : Set α", "hs : t ∩ u ⊆ s", "ht : s ∩ u ⊆ t"], "goal": "s ∩ u = t ∩ u"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrderedField α", "f g : CauSeq α abs", "h✝ : (g - f).LimZero ∨ (-(g - f)).Pos", "h : (-(g - f)).Pos"], "goal": "g < f"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M"], "goal": "IsCompl (CliffordAlgebra.evenOdd Q 0) (CliffordAlgebra.evenOdd Q 1)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "p : PrimeSpectrum R"], "goal": "p ∈ support R M ↔ ∃ m, (Submodule.span R {m}).annihilator ≤ p.asIdeal"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α", "n : ℤ"], "goal": "toIcoMod hp a b = b ↔ b ∈ Set.Ico a (a + p)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : Part α", "g : α → Part β", "b : β"], "goal": "b ∈ f.bind g ↔ ∃ a ∈ f, b ∈ g a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α", "h : a ⋖ b"], "goal": "b ≠ a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u", "CategoryTheory.Category.{v, u} D", "E : Type u", "CategoryTheory.Category.{v, u} E", "X Y : C", "f : Sum.inl X ⟶ Sum.inl Y"], "goal": "(CategoryTheory.sum.inverseAssociator C D E).map f = f"}}
+{"state": {"context": ["R : Type u", "inst✝³ : CommRing R", "R' : Type v", "inst✝² : CommMonoid R'", "C : Type v", "inst✝¹ : CommMonoid C", "n : ℕ", "inst✝ : NeZero n", "ψ : AddChar (ZMod n) C"], "goal": "(∃ x, ψ x ≠ 1) → ψ 1 ≠ 1"}}
+{"state": {"context": ["α : Type u_4", "CommMonoid α", "TopologicalSpace α", "β : Type u_5", "f : β → α"], "goal": "tprod f = if h : Multipliable f then if (Function.mulSupport f).Finite then finprod f else Exists.choose h else 1"}}
+{"state": {"context": ["α : Type u_1", "cut : α → Ordering", "cmp : α → α → Ordering", "f : α → α", "t : Batteries.RBNode α", "H : Batteries.RBNode.OnRoot (fun x => Batteries.RBNode.cmpEq cmp (f x) x) (Batteries.RBNode.zoom cut t).fst", "h : Batteries.RBNode.Ordered cmp t"], "goal": "Batteries.RBNode.Ordered cmp (Batteries.RBNode.modify cut f t)"}}
+{"state": {"context": ["K : Type u_2", "Field K", "s : ℕ"], "goal": "Valued.v ((HahnSeries.ofPowerSeries ℤ K) PowerSeries.X ^ s) = ↑(Multiplicative.ofAdd (-↑s))"}}
+{"state": {"context": ["x y : ℝ"], "goal": "↑(tan x) = ↑(sin x / cos x)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "Fintype G", "IsAddCyclic G", "d : ℕ", "a b : ZMod d", "hGcard : Fintype.card G = d", "h : ∀ (t : G), a.val • t = b.val • t"], "goal": "a = b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : CommSemiring S", "p : R[X]", "f : R →+* S", "z : S", "hz : eval₂ f z p = 0", "inj : ∀ (x : R), f x = 0 → x = 0"], "goal": "f p.leadingCoeff ^ (p.natDegree - 1) * eval₂ f z p = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G"], "goal": "CochainComplex.HomComplex.Cochain.ofHom (-φ) = -CochainComplex.HomComplex.Cochain.ofHom φ"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_4", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "Submodule.span R Set.univ = ⊤"}}
+{"state": {"context": ["K : Type v", "Field K", "n : ℕ", "f : K[X]"], "goal": "Polynomial.SplittingFieldAux (n + 1) f = Polynomial.SplittingFieldAux n f.removeFactor"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "Fact (FiniteDimensional.finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "a x y : E"], "goal": "⟪a, x⟫_ℝ * ⟪a, y⟫_ℝ + (o.areaForm a) x * (o.areaForm a) y = ‖a‖ ^ 2 * ⟪x, y⟫_ℝ"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semigroup R", "a b : R", "ab : IsLeftRegular (a * b)"], "goal": "Function.Injective ((fun x => a * x) ∘ fun x => b * x)"}}
+{"state": {"context": ["E : Type u_2", "NNNorm E", "x : E"], "goal": "‖Additive.ofMul x‖₊ = ‖x‖₊"}}
+{"state": {"context": ["n : ℕ", "i : Fin (n + 1)", "m : Fin n", "h : m.castSucc < i"], "goal": "(finSuccEquiv' i) m.castSucc = some m"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁶ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "E : Type w", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L : Filter 𝕜", "𝕜' : Type u_1", "inst✝¹ : NontriviallyNormedField 𝕜'", "inst✝ : NormedAlgebra 𝕜 𝕜'", "c d : 𝕜 → 𝕜'", "c' d' : 𝕜'", "hc : HasStrictDerivAt c c' x", "hd : HasStrictDerivAt d d' x", "hx : d x ≠ 0"], "goal": "HasStrictDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x"}}
+{"state": {"context": ["p : ℂ × ℂ", "hp : p.1 ∈ slitPlane"], "goal": "HasStrictFDerivAt (fun x => x.1 ^ x.2) ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℂ ℂ ℂ + (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "f : α → Option β", "g : α → β", "a : α", "l : List α", "ih : filterMap f l = map g l → ∀ (x : α), x ∈ l → f x = some (g x)", "b : β", "ha : f a = some b"], "goal": "b = g a → filterMap f l = map g l → b = g a ∧ ∀ (a : α), a ∈ l → f a = some (g a)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝³ : CommRing R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "x : M", "f : Dual R M", "y : M", "g : Dual R M", "inst✝ : NoZeroSMulDivisors ℤ M", "Φ : Set M", "hΦ : _root_.Finite ↑Φ", "hfx : f x = 2", "hgy : g y = 2", "hgx : g x = 2", "hfy : f y = 2", "hxfΦ : MapsTo (⇑(preReflection x f)) Φ Φ", "hygΦ : MapsTo (⇑(preReflection y g)) Φ Φ", "hyΦ : y ∈ Φ"], "goal": "x = y"}}
+{"state": {"context": ["A₁ : Type u_8", "B₁ : Type u_9", "A₂ : Type u_10", "B₂ : Type u_11", "CommRing A₁", "Ring B₁", "CommRing A₂", "Ring B₂", "Algebra A₁ B₁", "Algebra A₂ B₂", "e₁ : A₁ ≃+* A₂", "e₂ : B₁ ≃+* B₂", "he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)", "x : B₁"], "goal": "(Algebra.norm A₁) x = e₁.symm ((Algebra.norm A₂) (e₂ x))"}}
+{"state": {"context": ["M₀ : Type u_1", "M₀' : Type u_2", "inst✝² : MulZeroOneClass M₀", "inst✝¹ : Nontrivial M₀", "G₀ : Type u_3", "inst✝ : GroupWithZero G₀", "a : G₀", "h : a ≠ 0", "a_eq_0 : a⁻¹ = 0", "this : a * a⁻¹ = 1"], "goal": "False"}}
+{"state": {"context": ["x y : EReal", "h : x < y", "z : ℝ"], "goal": "x + ↑z < y + ↑z"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "p : G.Walk u v"], "goal": "List.map (fun x => x.toProd.1) p.darts = p.support.dropLast"}}
+{"state": {"context": ["α : Type u_1"], "goal": "↑0 = 0"}}
+{"state": {"context": ["n : ℕ"], "goal": "Cardinal.toENat ↑n = ↑n"}}
+{"state": {"context": ["m n : Nat"], "goal": "m < n + 1 ↔ m < n ∨ m = n"}}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re", "n : ℕ", "hn : n ≠ 0", "this : ∀ (x : ℝ), x = x / ↑n * ↑n"], "goal": "↑n ^ s * s.betaIntegral (↑n + 1) = ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1)"}}
+{"state": {"context": ["f✝ f g : ℝ → ℝ", "hf✝¹ : GrowsPolynomially f", "hg✝¹ : GrowsPolynomially g", "b : ℝ", "hb : b ∈ Set.Ioo 0 1", "c₁ : ℝ", "hc₁_mem : c₁ > 0", "c₂ : ℝ", "hc₂_mem : c₂ > 0", "hf✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, (fun x => |f x|) u ∈ Set.Icc (c₁ * (fun x => |f x|) x) (c₂ * (fun x => |f x|) x)", "c₃ : ℝ", "hc₃_mem : c₃ > 0", "c₄ : ℝ", "hc₄_mem : c₄ > 0", "hg✝ : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, (fun x => |g x|) u ∈ Set.Icc (c₃ * (fun x => |g x|) x) (c₄ * (fun x => |g x|) x)", "x : ℝ", "hf : ∀ u ∈ Set.Icc (b * x) x, |f u| ∈ Set.Icc (c₁ * |f x|) (c₂ * |f x|)", "hg : ∀ u ∈ Set.Icc (b * x) x, |g u| ∈ Set.Icc (c₃ * |g x|) (c₄ * |g x|)"], "goal": "∀ u ∈ Set.Icc (b * x) x, |f u| * |g u| ∈ Set.Icc (c₁ * c₃ * (|f x| * |g x|)) (c₂ * c₄ * (|f x| * |g x|))"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "a₂ c₂ : R", "ea₁ b c₁ : ℕ", "xa₁ : R"], "goal": "ea₁ * b = c₁ → a₂ ^ b = c₂ → (xa₁ ^ ea₁ * a₂) ^ b = xa₁ ^ c₁ * c₂"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁶ : MetricSpace α", "β : Type u", "inst✝⁵ : SecondCountableTopology α", "inst✝⁴ : MeasurableSpace α", "inst✝³ : OpensMeasurableSpace α", "inst✝² : HasBesicovitchCovering α", "μ : Measure α", "inst✝¹ : SFinite μ", "inst✝ : μ.OuterRegular", "ε : ℝ≥0∞", "hε : ε ≠ 0", "f : α → Set ℝ", "s : Set α", "hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty", "u : Set α", "su : u ⊇ s", "u_open : IsOpen u", "μu : μ u ≤ μ s + ε / 2", "R : α → ℝ", "hR : ∀ x ∈ s, R x > 0 ∧ ball x (R x) ⊆ u", "t0 : Set α", "r0 : α → ℝ", "t0_count : t0.Countable", "t0s : t0 ⊆ s", "hr0 : ∀ x ∈ t0, r0 x ∈ f x ∩ Ioo 0 (R x)", "μt0 : μ (s \\ ⋃ x ∈ t0, closedBall x (r0 x)) = 0", "t0_disj : t0.PairwiseDisjoint fun x => closedBall x (r0 x)", "s' : Set α := s \\ ⋃ x ∈ t0, closedBall x (r0 x)", "s's : s' ⊆ s", "N : ℕ", "τ : ℝ", "hτ : 1 < τ", "H : IsEmpty (SatelliteConfig α N τ)", "v : Set α", "s'v : v ⊇ s'", "v_open : IsOpen v", "μv : μ v ≤ μ s' + ε / 2 / ↑N", "r1 : α → ℝ", "hr1 : ∀ x ∈ s', r1 x ∈ f x ∩ Ioo 0 1 ∧ closedBall x (r1 x) ⊆ v", "q : BallPackage (↑s') α := { c := fun x => ↑x, r := fun x => r1 ↑x, rpos := ⋯, r_bound := 1, r_le := ⋯ }", "S : Fin N → Set ↑s'", "S_disj : ∀ (i : Fin N), (S i).PairwiseDisjoint fun j => closedBall (q.c j) (q.r j)", "hS : range q.c ⊆ ⋃ i, ⋃ j ∈ S i, ball (q.c j) (q.r j)", "S_count : ∀ (i : Fin N), (S i).Countable", "r : α → ℝ := fun x => if x ∈ s' then r1 x else r0 x", "r_t0 : ∀ x ∈ t0, r x = r0 x"], "goal": "∃ t r, t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x) ∧ s ⊆ ⋃ x ∈ t, closedBall x (r x) ∧ ∑' (x : ↑t), μ (closedBall (↑x) (r ↑x)) ≤ μ s + ε"}}
+{"state": {"context": ["M : Type u_1", "AddCommMonoid M", "S : AddSubmonoid M", "N : Type u_2", "AddCommMonoid N", "P : Type u_3", "AddCommMonoid P", "f : S.LocalizationMap N", "g : M →+ P", "Q : Type u_4", "AddCommMonoid Q", "hg : Function.Surjective ⇑g", "k : (AddSubmonoid.map g S).LocalizationMap Q"], "goal": "Function.Surjective ⇑(f.map ⋯ k)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "δ : Type u_7", "MeasurableSpace δ", "μa : MeasureTheory.Measure α", "μb : MeasureTheory.Measure β", "μc : MeasureTheory.Measure γ", "μd : MeasureTheory.Measure δ", "MeasureTheory.SFinite μa", "MeasureTheory.SFinite μc", "f : α → β", "hf : MeasureTheory.MeasurePreserving f μa μb", "g : α → γ → δ", "hgm : Measurable (Function.uncurry g)", "hg : ∀ᵐ (x : α) ∂μa, MeasureTheory.Measure.map (g x) μc = μd"], "goal": "MeasureTheory.MeasurePreserving (fun p => (f p.1, g p.1 p.2)) (μa.prod μc) (μb.prod μd)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I X J : Set α", "e : α", "hI : M.Basis I X", "hJ : M.Basis J X", "hIJ : I \\ J = {e}"], "goal": "∃ f ∈ J \\ I, J = insert f I \\ {e}"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "k : 𝓞 K", "hk : S.a + ↑η ^ 2 * S.b = λ ^ 2 * k", "k' : 𝓞 K", "hk' : S.a + S.b = λ ^ 2 * k'"], "goal": "S.b = λ * ((k - k') * -↑η)"}}
+{"state": {"context": ["A : Type u_1", "AddCommGroup A", "Module ℂ A", "StarAddMonoid A", "StarModule ℂ A", "a : A"], "goal": "imaginaryPart (Complex.I • a) = realPart a"}}
+{"state": {"context": ["p q : ℕ", "hp : Fact (Nat.Prime p)", "hq0 : ↑q ≠ 0"], "goal": "∑ a ∈ Ico 1 (p / 2).succ, a * q / p + ∑ a ∈ Ico 1 (q / 2).succ, a * p / q = p / 2 * (q / 2)"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "Field A", "Ring B", "Algebra A B", "Nontrivial B", "a : A"], "goal": "minpoly A ((algebraMap A B) a) = Polynomial.X - Polynomial.C a"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "m : Type u_3", "n : Type u_4", "Fintype m", "M : Matrix m n R"], "goal": "LinearMap.range M.vecMulLinear = Submodule.span R (Set.range M)"}}
+{"state": {"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "f g : PreTilt K v O hv p", "m n k : ℕ", "hm : (coeff (ModP K v O hv p) p (max (max m n) k)) f ≠ 0", "hn : (coeff (ModP K v O hv p) p (max (max m n) k)) g ≠ 0", "hk : (coeff (ModP K v O hv p) p (max (max m n) k)) (f + g) ≠ 0"], "goal": "ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (max (max m n) k)) f + (coeff (ModP K v O hv p) p (max (max m n) k)) g) ^ p ^ max (max m n) k ≤ max (ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (max (max m n) k)) f) ^ p ^ max (max m n) k) (ModP.preVal K v O hv p ((coeff (ModP K v O hv p) p (max (max m n) k)) g) ^ p ^ max (max m n) k)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "f : R[X]"], "goal": "f.eraseLead.degree ≤ f.degree"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom", "F : CategoryTheory.Arrow C", "X : CategoryTheory.CosimplicialObject.Augmented C", "G : F ⟶ CategoryTheory.CosimplicialObject.Augmented.toArrow.obj X", "x : SimplexCategory"], "goal": "(CategoryTheory.CosimplicialObject.equivalenceRightToLeft F X G).right.app x =\n  CategoryTheory.Limits.WidePushout.desc (G.left ≫ X.hom.app x)\n    (fun i => G.right ≫ X.right.map ((SimplexCategory.mk 0).const x i)) ⋯"}}
+{"state": {"context": ["a y k : ℕ", "hy0 : y ≠ 0", "hk0 : k ≠ 0", "hyk : y ^ k < a", "hya : y < a"], "goal": "↑a ≤ ↑a ^ 2 - (↑a - 1) ^ 2 - 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_3", "TopologicalSpace B", "NontriviallyNormedField 𝕜", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "EB : Type u_4", "NormedAddCommGroup EB", "NormedSpace 𝕜 EB", "HB : Type u_5", "TopologicalSpace HB", "ChartedSpace HB B", "IB : ModelWithCorners 𝕜 EB HB", "e : PartialHomeomorph (B × F) (B × F)", "U : Set B", "hU : e.source = U ×ˢ Set.univ", "h :\n  ∀ x ∈ U,\n    ∃ φ u,\n      ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ :\n        SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) u),\n        (e.restr (u ×ˢ Set.univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)"], "goal": "∃ Φ U,\n  ∃ (hU₀ : IsOpen U) (hΦ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(Φ x)) U) (h2Φ :\n    SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(Φ x).symm) U), e.EqOnSource (FiberwiseLinear.partialHomeomorph Φ hU₀ ⋯ ⋯)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x : X"], "goal": "IsPreirreducible (irreducibleComponent x) ∧\n  {x} ⊆ irreducibleComponent x ∧\n    ∀ (u : Set X), IsPreirreducible u → irreducibleComponent x ⊆ u → u = irreducibleComponent x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t : Set α", "f : α → β", "hf : AEMeasurable f μ", "s : Set β", "hs : MeasurableSet s"], "goal": "s ∈ ae (Measure.map f μ) ↔ f ⁻¹' s ∈ ae μ"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "f g : PadicSeq p", "hfg : f ≈ g", "hf : ¬f ≈ 0"], "goal": "f.norm = g.norm"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "F : Type u_3", "AddCommGroup F", "Module R F", "f : E →ₗ.[R] F", "x : ↥f.domain"], "goal": "f.toFun x = ↑f x"}}
+{"state": {"context": ["a x z : ℂ", "r : ℝ", "hr : 0 < r", "θ : ℝ"], "goal": "(↑r * (cos ↑θ + sin ↑θ * I)).arg = toIocMod Real.two_pi_pos (-π) θ"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : Group G", "inst✝ : Finite G", "x y : G", "h : ¬IsOfFinOrder x", "n m : ℤ", "hnm : (fun n => x ^ n) n = (fun n => x ^ n) m"], "goal": "n = m"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "I : CategoryTheory.Limits.MulticospanIndex C", "K : CategoryTheory.Limits.Multifork I", "lift : (E : CategoryTheory.Limits.Multifork I) → E.pt ⟶ K.pt", "fac : ∀ (E : CategoryTheory.Limits.Multifork I) (i : I.L), CategoryTheory.CategoryStruct.comp (lift E) (K.ι i) = E.ι i", "uniq :\n  ∀ (E : CategoryTheory.Limits.Multifork I) (m : E.pt ⟶ K.pt),\n    (∀ (i : I.L), CategoryTheory.CategoryStruct.comp m (K.ι i) = E.ι i) → m = lift E", "E : CategoryTheory.Limits.Multifork I"], "goal": "(CategoryTheory.Limits.Multifork.IsLimit.mk K lift fac uniq).lift E = lift E"}}
+{"state": {"context": ["α : Type u", "f₁ f₂ : Filter α"], "goal": "f₁ ≤ f₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ f₁ → ↑g ≤ f₂"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α"], "goal": "∀ (a : (TopologicalSpace.Opens α)ᵒᵈ),\n  (RelIso.symm (TopologicalSpace.Closeds.complOrderIso α)) a = (TopologicalSpace.Opens.compl ∘ ⇑OrderDual.ofDual) a"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E"], "goal": "rexp ∘ Neg.neg '' univ = Ioi 0"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W✝ : Affine R", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "hx : x₁ = x₂", "hy : y₁ ≠ W.negY x₂ y₂"], "goal": "y₁ - W.negY x₁ y₁ = evalEval x₁ y₁ W.polynomialY"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "i : ℕ", "x : ExteriorAlgebra R M", "hx : x ∈ ⋀[R]^i M"], "goal": "(liftι R M) ↑⟨x, hx⟩ = (DirectSum.of (fun i => ↥(⋀[R]^i M)) i) ⟨x, hx⟩"}}
+{"state": {"context": ["x y : ℂ", "h : I * sin (x * I) = -sinh x"], "goal": "sin (x * I) = sinh x * I"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "CommSemiring k", "Monoid G", "Monoid H", "A : Type u₃", "Semiring A", "Algebra k A", "e : G ≃* H"], "goal": "(MonoidAlgebra.domCongr k A e).symm = MonoidAlgebra.domCongr k A e.symm"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "x : R"], "goal": "(↑x).im = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LT α", "inst✝ : LT β", "s t : Set α", "l : List α", "a : α"], "goal": "s.chainHeight = 0 ↔ s = ∅"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Finite ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x y : ι → ℝ", "δ : ℝ", "hs : IsClosed s", "hs' : BddAbove s", "val✝ : Fintype ι"], "goal": "IsClosed ↑(lowerClosure s)"}}
+{"state": {"context": ["l m r : List Char", "s : Substring"], "goal": "Substring.ValidFor l m r s → s.toString = { data := m }"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.EnoughInjectives Cᵒᵖ"], "goal": "CategoryTheory.EnoughProjectives C"}}
+{"state": {"context": ["α : Type u_1", "a b b' : α", "LE α", "h : a ≤ b", "eq : b = b'"], "goal": "a ≤ b'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace Ω", "inst✝³ : StandardBorelSpace Ω", "inst✝² : Nonempty Ω", "inst✝¹ : CountableOrCountablyGenerated α β", "ρ : Measure (β × Ω)", "inst✝ : IsFiniteMeasure ρ", "f : β × Ω → ℝ≥0∞", "hf : Measurable f", "s : Set β", "hs : MeasurableSet s"], "goal": "∫⁻ (b : β) in s, ∫⁻ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in Set.univ, f (b, ω) ∂ρ.condKernel b ∂ρ.fst"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "P Q : Cᵒᵖ ⥤ D", "η : P ⟶ Q", "hQ : Presheaf.IsSheaf J Q", "γ : J.sheafify P ⟶ Q", "h : J.toSheafify P ≫ γ = η"], "goal": "γ = J.sheafifyLift η hQ"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : RCLike 𝕜", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace 𝕜 E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "v : E", "hv : v ∈ ⨅ i, (eigenspace T i)ᗮ"], "goal": "T v ∈ ⨅ i, (eigenspace T i)ᗮ"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : Fintype α", "s t : Finset α", "a b : α", "inst✝ : Nonempty α"], "goal": "univ.dens = 1"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "p✝ : ℝ≥0∞", "q : ℝ", "μ : Measure α", "f g : α → E", "p δ : ℝ≥0∞", "hδ : δ ≠ 0", "this : Tendsto (fun η => LpAddConst p * (η + η)) (𝓝[>] 0) (𝓝 (LpAddConst p * (0 + 0)))"], "goal": "∃ η, 0 < η ∧ ∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → eLpNorm f p μ ≤ η → eLpNorm g p μ ≤ η → eLpNorm (f + g) p μ < δ"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "G : D ⥤ C", "adj : CoreUnitCounit F G", "X : C", "Y : D", "f : F.obj X ⟶ Y", "e_4✝ : (𝟭 D).obj (F.obj X) = F.obj X"], "goal": "F.map (adj.unit.app X) ≫ adj.counit.app (F.obj X) = 𝟙 (F.obj X)"}}
+{"state": {"context": [], "goal": "∅.toSet = ∅"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β i)", "inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)", "s : Set ι", "f : Π₀ (i : ι), β i"], "goal": "(∀ (x : ι), ¬f x = 0 → x ∈ s) ↔ ∀ i ∉ s, f i = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIcoMod hp a b ≤ toIocMod hp a b"}}
+{"state": {"context": ["α β : Type u", "a : Cardinal.{u}", "h✝ : lift.{v, u} (succ a) > succ (lift.{v, u} a)", "b : Cardinal.{u}", "e : lift.{v, u} b = succ (lift.{v, u} a)", "h : b < succ a"], "goal": "False"}}
+{"state": {"context": ["x : ℂ", "hx : Complex.abs x ≤ 1"], "goal": "Complex.abs (Complex.exp x - 1 - x) ≤ Complex.abs x ^ 2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s t✝ : Set α", "a b : α", "f : α ↪ β", "t : Finset β", "ht : t.card ≤ 1"], "goal": "(SimpleGraph.map f G).IsClique ↑t ↔ t.card ≤ 1 ∨ ∃ s, G.IsClique ↑s ∧ map f s = t"}}
+{"state": {"context": ["R : Type u", "S✝ : Type u_1", "σ : Type v", "M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S✝", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "S : Type u_2", "inst✝¹ : CommRing S", "inst✝ : Finite σ", "ϕ : MvPolynomial σ R →+* S", "p : MvPolynomial σ R"], "goal": "ϕ p = eval₂ (ϕ.comp C) (fun s => ϕ (X s)) p"}}
+{"state": {"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "dst : 2 * ε ≤ ↑(G.edgeDensity s t)", "dsu : 2 * ε ≤ ↑(G.edgeDensity s u)", "dtu : 2 * ε ≤ ↑(G.edgeDensity t u)", "utu : G.IsUniform ε t u", "x : α", "hxY : ↑t.card * (↑(G.edgeDensity s t) - ε) ≤ ↑(filter (G.Adj x) t).card", "hsu : ↑u.card * (↑(G.edgeDensity s u) - ε) ≤ ↑(filter (G.Adj x) u).card", "hY : ↑t.card * ε ≤ ↑(filter (G.Adj x) t).card", "hZ : ↑u.card * ε ≤ ↑(filter (G.Adj x) u).card"], "goal": "ε ^ 3 * ↑t.card * ↑u.card ≤ ↑(filter (fun x => match x with | (y, z) => G.Adj y z) (filter (G.Adj x) t ×ˢ filter (G.Adj x) u)).card"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "inst✝ : DecidableRel r", "a b : α", "l l₁ l₂ : List α", "h : (a :: b :: l).split = (l₁, l₂)", "l₁' l₂' : List α", "e : l.split = (l₁', l₂')", "fst_eq✝ : a :: l₁' = l₁", "snd_eq✝ : b :: l₂' = l₂"], "goal": "l₁.length < (a :: b :: l).length ∧ l₂.length < (a :: b :: l).length"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_2", "Group M", "MulAction M α", "s : Set α", "a : α", "a_in_s : a ∈ s"], "goal": "MulAction.orbit (↥(fixingSubgroup M sᶜ)) a ⊆ s"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "m m0 : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "μ : Measure α", "f : α → E", "s : Set α", "hs : MeasurableSet s", "hf : f =ᶠ[ae (μ.restrict sᶜ)] 0", "hm : m ≤ m0", "hsf_zero : ∀ (g : α → E), g =ᶠ[ae (μ.restrict sᶜ)] 0 → s.indicator g =ᶠ[ae μ] g"], "goal": "μ[s.indicator f|m] =ᶠ[ae μ] s.indicator (μ[f|m])"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : Semiring α", "I : Ideal α", "a b x y : α", "hy : IsUnit y"], "goal": "y * x ∈ I ↔ x ∈ I"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E", "ι : Type u_6", "Fintype ι", "F' : ι → Type u_7", "(i : ι) → NormedAddCommGroup (F' i)", "(i : ι) → NormedSpace 𝕜 (F' i)", "φ : (i : ι) → E → F' i", "h : ∀ (i : ι), DifferentiableAt 𝕜 (φ i) x"], "goal": "fderiv 𝕜 (fun x i => φ i x) x = ContinuousLinearMap.pi fun i => fderiv 𝕜 (φ i) x"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕝 : Type u_3", "𝕝₂ : Type u_4", "E : Type u_5", "F : Type u_6", "G : Type u_7", "ι : Type u_8", "ι' : Type u_9", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : Module 𝕜 E", "inst✝¹ : Nonempty ι", "p : SeminormFamily 𝕜 E ι", "inst✝ : TopologicalSpace E", "s : Set E", "hp : WithSeminorms p"], "goal": "(∀ i ∈ p.basisSets, Absorbs 𝕜 (id i) s) ↔ ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r"}}
+{"state": {"context": ["K : Type u_1", "g : GenContFract K", "n : ℕ", "DivisionRing K", "terminatedAt_n : g.TerminatedAt n"], "goal": "g.contsAux (n + 2) = g.contsAux (n + 1)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NormedCommRing R", "inst✝ : CompleteSpace R", "f : ℕ → R", "hf₁ : f 1 = 1", "hmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n", "hsum : Summable fun x => ‖f x‖", "hf₀ : f 0 = 0", "this : Tendsto (fun n => ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p => ∑' (e : ℕ), f (p ^ e)) i) atTop (𝓝 (∑' (n : ℕ), f n))"], "goal": "Tendsto (fun n => ∏ p ∈ n.primesBelow, ∑' (e : ℕ), f (p ^ e)) atTop (𝓝 (∑' (n : ℕ), f n))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ"], "goal": "κ.withDensity 0 = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_3", "x y : E", "b : Basis ι 𝕜 E", "h : ∀ (i : ι), ⟪x, b i⟫_𝕜 = ⟪y, b i⟫_𝕜"], "goal": "x = y"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "Zero X", "A : Type u_2", "NonUnitalRing A", "StarRing A", "Module ℝ A", "φ : ContinuousMapZero X NNReal →⋆ₙₐ[NNReal] A", "f : ContinuousMapZero X NNReal"], "goal": "φ.realContinuousMapZeroOfNNReal\n    ({ toFun := NNReal.toReal, continuous_toFun := NNReal.continuous_coe, map_zero' := ⋯ }.comp f) =\n  φ f"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedSemifield α", "a b c d e : α", "m n : ℤ", "hab : a ≤ b", "hc : 0 ≤ c"], "goal": "a / c ≤ b / c"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "M : Type u_4", "AddMonoid M", "CategoryTheory.HasShift C M", "X Y : C", "Z : C", "CategoryTheory.Preadditive C", "∀ (a : M), (CategoryTheory.shiftFunctor C a).PreservesZeroMorphisms", "a : M", "β : CategoryTheory.ShiftedHom X Y a", "b c : M", "h : b + a = c"], "goal": "β.comp 0 h = 0"}}
+{"state": {"context": ["α : Type u", "G : Type u_1", "Group G", "f : G →* Function.End α"], "goal": "∀ (a : G), ⇑(f.toHomPerm a) = f a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : CompleteSpace E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "μ ν : Measure α", "s✝ t✝ : Set α", "f✝ g f : α → E", "s t : Set α", "hd : AEDisjoint μ s t", "ht : NullMeasurableSet t μ", "hsμ : μ s ≠ ⊤", "htμ : μ t ≠ ⊤", "hfs : IntegrableOn f s μ", "hft : IntegrableOn f t μ", "this✝ : Fact (μ s < ⊤)", "this : Fact (μ t < ⊤)"], "goal": "⨍ (x : α) in s ∪ t, f x ∂μ = ((μ s).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ (x : α) in s, f x ∂μ + ((μ t).toReal / ((μ s).toReal + (μ t).toReal)) • ⨍ (x : α) in t, f x ∂μ"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ✝ : MeasurableSpace Ω", "μ✝ : Measure Ω", "g : Ω → ℝ≥0∞", "X Y : Ω → ℝ", "Mf mΩ : MeasurableSpace Ω", "μ : Measure Ω", "hMf : Mf ≤ mΩ", "c : ℝ≥0∞", "T : Set Ω", "h_meas_T : MeasurableSet T", "h_ind : IndepSets {s | MeasurableSet s} {T} μ", "h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a => g a * T.indicator (fun x => c) a"], "goal": "∀ ⦃f g : Ω → ℝ≥0∞⦄, Disjoint (Function.support f) (Function.support g) → Measurable f → Measurable g → ∫⁻ (ω : Ω), f ω * T.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x => c) ω ∂μ → ∫⁻ (ω : Ω), g ω * T.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), g ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x => c) ω ∂μ → ∫⁻ (ω : Ω), (f + g) ω * T.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), (f + g) ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x => c) ω ∂μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : SMulCommClass ℝ 𝕜 F", "m : ℕ × ℕ", "k n : ℕ", "hk : k ≤ m.1", "hn : n ≤ m.2", "f : 𝓢(E, F)", "x : E"], "goal": "∑ i ∈ Finset.range (k + 1), ‖x‖ ^ i * ↑(k.choose i) * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ 2 ^ m.1 * ((Finset.Iic m).sup fun m => SchwartzMap.seminorm 𝕜 m.1 m.2) f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w"], "goal": "DiscreteTopology ↥(AddSubgroup.zmultiples 1)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "P : Type u_3", "CommSemiring P", "IsLocalization M S", "g : R →+* P", "T : Submonoid P", "Q : Type u_4", "CommSemiring Q", "hy : M ≤ Submonoid.comap g T", "Algebra P Q", "IsLocalization T Q", "j : S →+* Q", "hj : ∀ (x : R), j ((algebraMap R S) x) = (algebraMap P Q) (g x)"], "goal": "IsLocalization.map Q g hy = j"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "c g' : Perm α", "hd : c.Disjoint g'", "hc : c.IsCycle", "hn2 : Fintype.card α < c.support.card + 2", "hng : c.support.card ∈ (c * g').cycleType"], "goal": "(c * g').cycleType = {c.support.card}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "pf : FormalMultilinearSeries 𝕜 E F", "x : E", "r : ℝ≥0∞", "hf : HasFPowerSeriesOnBall f pf x r"], "goal": "HasFPowerSeriesOnBall (-f) (-pf) x r"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "(i : ι) → MulZeroClass (α i)", "DecidableEq ι", "i : ι", "f : (i : ι) → α i", "a : α i"], "goal": "Pi.single i (f i * a) = f * Pi.single i a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "K : Type u_3", "inst✝ : DivisionSemiring α", "a b c d : α", "h : b ≠ 0"], "goal": "(a + b) / b = a / b + 1"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f : α → β → β", "op : α → α → α", "inst✝² : Monoid α", "inst✝¹ : Monoid β", "inst✝ : FunLike F α β", "s : Multiset α", "comm : {x | x ∈ s}.Pairwise Commute", "y : α", "h : ∀ x ∈ s, Commute y x"], "goal": "Commute y (s.noncommProd comm)"}}
+{"state": {"context": ["a r : ℝ"], "goal": "MeasureTheory.volume (Metric.closedBall a r) = ENNReal.ofReal (2 * r)"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "f : α → β → α → γ", "la : List α", "lb : List β"], "goal": "List.zipWith3 f la lb la = List.zipWith (fun a b => f a b a) la lb"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a b c : α", "hab : Relation.ReflTransGen r a b", "hbc : Relation.TransGen r b c"], "goal": "Relation.TransGen r a c"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v w : V", "n : ℕ", "p : G.Walk v w", "h : G.Adj u v"], "goal": "(SimpleGraph.Walk.cons h p).getVert (n + 1) = p.getVert n"}}
+{"state": {"context": ["a b c : Int", "H1 : b ≠ 0", "H2 : a = b * c"], "goal": "a.div b = c"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "hdomain : IsDomain R", "r s : R", "p q : R[X]", "f : RatFunc R"], "goal": "(laurent 0) f = f"}}
+{"state": {"context": ["n d : PosNum"], "goal": "↑(n.pred' / Num.pos d).succ' = -↑n / -↑d"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "l : Filter α", "s : Set β"], "goal": "Filter.Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "inst✝ : AddGroup β", "f g : α → β", "l : Filter α", "h : f =ᶠ[l] g"], "goal": "f - g =ᶠ[l] 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "s : Set E", "hs : MeasureTheory.NullMeasurableSet s μ", "r : ℝ"], "goal": "MeasureTheory.NullMeasurableSet (r • s) μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : DecidableEq α", "inst✝¹ : Fintype α", "γ : α → Type u_2", "δ : α → Type u_3", "s : (a : α) → Finset (γ a)", "β : Type u_4", "inst✝ : DecidableEq β", "a : α"], "goal": "image (fun f => f a) (piFinset fun _i => ∅) = ∅"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "a : α", "n : ℕ", "n.AtLeastTwo"], "goal": "Int.fract (a + OfNat.ofNat n) = Int.fract a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "TopologicalSpace H", "TopologicalSpace M", "f : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "s : Set M", "y : M", "hy : y ∈ f.source", "hs : s ⊆ f.source"], "goal": "Filter.map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I) '' s] ↑(f.extend I) y"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "DecidableEq α", "A : Finset α", "hA : A.Nonempty", "B : Finset α", "n : ℕ"], "goal": "↑(n • B).card ≤ (↑(A - B).card / ↑A.card) ^ n * ↑A.card"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "Quiver.IsThin C", "X Y : C", "f : X ⟶ Y", "g : Y ⟶ X"], "goal": "X ≈ Y"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s✝ s₁ t✝ u✝ : Set E", "f✝ f₁ : E → F", "g✝ : F → G", "x✝ x₀ : E", "c : F", "b : E × F → G", "m✝ n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "s : Set E", "t : Set F", "g : F → G", "f : E → F", "x : E", "hg : ContDiffWithinAt 𝕜 n g t (f x)", "hf : ContDiffWithinAt 𝕜 n f s x", "st : s ⊆ f ⁻¹' t", "m : ℕ", "hm : ↑m ≤ n", "u : Set F", "u_nhd : u ∈ 𝓝[insert (f x) t] f x", "left✝ : u ⊆ insert (f x) t", "hu : ContDiffOn 𝕜 (↑m) g u", "v : Set E", "v_nhd : v ∈ 𝓝[insert x s] x", "vs : v ⊆ insert x s", "hv : ContDiffOn 𝕜 (↑m) f v"], "goal": "∃ u ∈ 𝓝[insert x s] x, ∃ p, HasFTaylorSeriesUpToOn (↑m) (g ∘ f) p u"}}
+{"state": {"context": ["R : Type u", "CommRing R", "V W : HopfAlgebraCat R"], "goal": "Function.Injective HopfAlgebraCat.Hom.toBialgHom"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{?u.123089, u_2} D", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "inst✝ : HasHomotopyCofiber φ", "K : CochainComplex C ℤ", "n m : ℤ", "α : Cochain F K m", "β : Cochain G K n", "h : m + 1 = n", "p₁ p₂ : ℤ", "h₁₂ : p₁ + n = p₂"], "goal": "(inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h₁₂"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_4", "Semiring R", "AddCommMonoid M", "Module R M", "Module.Finite R M"], "goal": "∃ n f, Function.Surjective ⇑f"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "D : Type u'", "inst✝¹ : Category.{v', u'} D", "inst✝ : HasZeroMorphisms C", "X : C", "h : 𝟙 X = 0", "Y : C", "f : Y ⟶ X"], "goal": "0 = default"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X : Ω → ℝ", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "hX : Memℒp X 2 μ"], "goal": "variance X μ = ∫ (x : Ω), ((X - fun x => ∫ (x : Ω), X x ∂μ) ^ 2) x ∂μ"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "NonUnitalNonAssocSemiring α", "NonUnitalNonAssocSemiring β", "f : α →ₙ+* β"], "goal": "f.comp (NonUnitalRingHom.id α) = f"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R"], "goal": "Monotone NonUnitalSubring.toSubsemigroup"}}
+{"state": {"context": ["R : Type u_1", "m : Type u_2", "n₁ : Type u_6", "n₂ : Type u_7", "A : Matrix m (n₁ ⊕ n₂) R", "i : m", "j : n₂"], "goal": "A.toColumns₂ i j = A i (Sum.inr j)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq β", "inst✝² : SemilatticeSup α", "inst✝¹ : OrderBot α", "inst✝ : DecidableRel Disjoint", "s s₁ s₂ t t₁ t₂ u : Finset α", "a b c : α", "p : α → Prop"], "goal": "(∀ (c : α), (∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = c) → p c) ↔ ∀ a ∈ s, ∀ b ∈ t, Disjoint a b → p (a ⊔ b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s t u : Set α", "m' : MeasurableSpace β", "inst✝ : BooleanAlgebra β", "h : Measurable compl", "f : α → β"], "goal": "MeasurableSpace.comap (fun a => (f a)ᶜ) inferInstance = MeasurableSpace.comap f inferInstance"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : NormalSpace Y", "inst✝ : Nonempty X", "f : X →ᵇ ℝ", "e : X → Y", "he : ClosedEmbedding e", "inhabited_h : Inhabited X", "a : ℝ", "ha : IsGLB (range ⇑f) a", "b : ℝ", "hb : IsLUB (range ⇑f) b", "hmem : ∀ (x : X), f x ∈ Icc a b", "hle : a ≤ b", "hlt : a < b", "c : ℝ := (a + b) / 2"], "goal": "∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑f"}}
+{"state": {"context": ["R : Type u₁", "S : Type u₂", "CommRing R", "CommRing S", "f : R →+* S", "M₁ M₂ M₃ : ModuleCat R", "l₁₂ : M₁ ⟶ M₂", "l₂₃ : M₂ ⟶ M₃"], "goal": "ModuleCat.ExtendScalars.map' f (l₁₂ ≫ l₂₃) = ModuleCat.ExtendScalars.map' f l₁₂ ≫ ModuleCat.ExtendScalars.map' f l₂₃"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R", "s : Set R", "sm : Subsemigroup R", "hm : ↑sm = s", "sa : AddSubgroup R", "ha : ↑sa = s"], "goal": "(NonUnitalSubring.mk' s sm sa hm ha).toSubsemigroup = sm"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "f✝ g : Filter α", "s✝ t : Set α", "f : ι → Filter α", "h : Directed (fun x x_1 => x ≥ x_1) f", "inst✝ : Nonempty ι", "s : Set α"], "goal": "s ∈ iInf f ↔ ∃ i, s ∈ f i"}}
+{"state": {"context": ["α : Type u_1", "PseudoMetricSpace α", "s : Set α", "hs : s.Nontrivial"], "goal": "s.einfsep ≠ ⊤"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "m : ℕ", "f : α → ℕ", "h₁ : ∀ x ∈ s, f x = m"], "goal": "∑ x ∈ s, f x = ∑ _x ∈ s, m"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "G : K ⥤ D", "h : CategoryTheory.Limits.Cocone G ≌ CategoryTheory.Limits.Cocone F", "c : CategoryTheory.Limits.Cocone G", "P : CategoryTheory.Limits.IsColimit (h.functor.obj c)", "s : CategoryTheory.Limits.Cocone G"], "goal": "((CategoryTheory.Limits.IsColimit.ofCoconeEquiv h) P).desc s =\n  (h.unit.app c).hom ≫ (h.inverse.map (P.descCoconeMorphism (h.functor.obj s))).hom ≫ (h.unitInv.app s).hom"}}
+{"state": {"context": ["ι : Type uι", "E : Type uE", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : FiniteDimensional ℝ E", "F : Type uF", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ F", "H : Type uH", "inst✝⁵ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝⁴ : TopologicalSpace M", "inst✝³ : ChartedSpace H M", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "t : M → Set F", "n : ℕ∞", "s : Set M", "hs : IsOpen s", "f : SmoothPartitionOfUnity M I M", "hf : f.IsSubordinate fun x => (chartAt H x).source", "g : M → H → ℝ", "g_supp : ∀ (c : M), support (g c) = (chartAt H c).target ∩ ↑(chartAt H c).symm ⁻¹' s", "g_diff : ∀ (c : M), Smooth I 𝓘(ℝ, ℝ) (g c)", "hg : ∀ (c : M), range (g c) ⊆ Icc 0 1"], "goal": "∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ ∀ (x : M), 0 ≤ f x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "n : ℕ∞", "Eu : Type u", "NormedAddCommGroup Eu", "NormedSpace 𝕜 Eu", "Fu : Type u", "NormedAddCommGroup Fu", "NormedSpace 𝕜 Fu", "Gu : Type u", "NormedAddCommGroup Gu", "NormedSpace 𝕜 Gu", "s : Set Eu", "t : Set Fu", "g : Fu → Gu", "f : Eu → Fu", "hg : ContDiffOn 𝕜 n g t", "hf : ContDiffOn 𝕜 n f s", "st : s ⊆ f ⁻¹' t"], "goal": "ContDiffOn 𝕜 n (g ∘ f) s"}}
+{"state": {"context": ["n k : ℕ", "hk : k ≠ 0"], "goal": "(n ^ k).minFac = n.minFac"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroObject C", "A : C", "n : ℕ"], "goal": "HomologicalComplex.ExactAt ((CochainComplex.single₀ C).obj A) (n + 1)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ✝ μ : Measure α", "p q : ℝ", "hpq : p.IsConjExponent q", "f g : α → ℝ≥0∞", "hf : AEMeasurable f μ", "hg : AEMeasurable g μ", "hf_zero : ¬∫⁻ (a : α), f a ^ p ∂μ = 0", "hg_zero : ¬∫⁻ (a : α), g a ^ q ∂μ = 0", "hf_top : ¬∫⁻ (a : α), f a ^ p ∂μ = ⊤", "hg_top : ¬∫⁻ (a : α), g a ^ q ∂μ = ⊤"], "goal": "∫⁻ (a : α), (f * g) a ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ q ∂μ) ^ (1 / q)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "UniformSpace α", "UniformSpace β", "UniformSpace γ", "g : β → γ", "hg : UniformEmbedding g", "f : α → β", "hf : UniformEmbedding f"], "goal": "UniformEmbedding (g ∘ f)"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "s : ι → MeasurableSpace Ω", "m m0 : MeasurableSpace Ω", "κ : Kernel α Ω", "μα : Measure α", "μ : Measure Ω", "inst✝⁴ : IsMarkovKernel κ", "inst✝³ : IsProbabilityMeasure μ", "inst✝² : SemilatticeInf ι", "inst✝¹ : NoMinOrder ι", "inst✝ : Nonempty ι", "h_le : ∀ (n : ι), s n ≤ m0", "h_indep : iIndep s κ μα", "ns : ι → Set ι := Set.Ici", "hnsp : ∀ (i : ι), BddBelow (ns i)"], "goal": "∀ (n : ι), ∃ a, n ∈ ns a"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "LinearOrder α", "OrderTopology α", "DenselyOrdered α", "a b : α", "h : a < b"], "goal": "Filter.map Subtype.val Filter.atBot = 𝓝[>] a"}}
+{"state": {"context": ["n : ℕ", "i j : Fin n", "h : i ≠ j"], "goal": "↑i ≠ ↑j"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type u_1", "c : Chain (Part α)", "a b : α", "i : ℕ", "ha : c i = some a", "j : ℕ", "hb : some b = c j"], "goal": "a = b"}}
+{"state": {"context": ["m n k : ℕ"], "goal": "(n * k + m).Coprime n ↔ m.Coprime n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "r p q : α → α → Prop", "f g : ι → α", "s t u : Set α", "a b : α", "hr : Symmetric r", "ha : a ∉ s"], "goal": "(insert a s).Pairwise r ↔ s.Pairwise r ∧ ∀ b ∈ s, r a b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "a : α", "l : List α"], "goal": "l.sublists' ++ map (cons a) l.sublists' ~ l.sublists ++ map (cons a) l.sublists"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁ ⊆ l₂"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "DecidableEq ι", "(i : ι) → DecidableEq (α i)", "(i : ι) → PartialOrder (α i)", "(i : ι) → Zero (α i)", "(i : ι) → LocallyFiniteOrder (α i)", "f g : Π₀ (i : ι), α i"], "goal": "(Finset.Icc f g).card = ∏ i ∈ f.support ∪ g.support, (Finset.Icc (f i) (g i)).card"}}
+{"state": {"context": ["T : Type uT", "N : Type uN", "r : ContextFreeRule T N", "v w p x y : List (Symbol T N)", "hxy : v = x ++ [Symbol.nonterminal r.input] ++ y ∧ w = x ++ r.output ++ y"], "goal": "∃ p_1 q, p ++ v = p_1 ++ [Symbol.nonterminal r.input] ++ q ∧ p ++ w = p_1 ++ r.output ++ q"}}
+{"state": {"context": ["α : Type u_1", "EuclideanDomain α", "DecidableEq α", "x y : α"], "goal": "Ideal.span {EuclideanDomain.gcd x y} = Ideal.span {x, y}"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "f : CategoryTheory.WithTerminal.star ⟶ CategoryTheory.WithTerminal.of X"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "SMul α ℂ", "IsScalarTower α ℂ ℂ", "k : ℤ", "A : ↥GL(2, ℝ)⁺", "f : ℍ → ℂ", "c : α"], "goal": "ModularForm.slash k A (c • f) = c • ModularForm.slash k A f"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "LinearOrder α", "OrderTopology α", "a l' : α", "s : Set α", "hl' : l' < a"], "goal": "s ∈ 𝓝[≤] a ↔ ∃ l ∈ Set.Ico l' a, Set.Ioc l a ⊆ s"}}
+{"state": {"context": ["x : ℝ≥0"], "goal": "Real.nnabs ↑x = x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f✝ f₁ f₂ : Filter α", "g g₁ g₂ : Filter β", "m : α → β", "m' : β → γ", "s : Set α", "t : Set β", "f : α → β", "F : Filter α"], "goal": "(map f F).NeBot ↔ F.NeBot"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "s : Set X"], "goal": "e.restr (e.source ∩ s) = e.restr s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R✝ : Type u_3", "R' : Type u_4", "S✝ : Type u_5", "S' : Type u_6", "T : Type u_7", "T' : Type u_8", "R : Type u_9", "S : Type u_10", "inst✝⁴ : Ring R", "inst✝³ : Ring S", "inst✝² : IsDomain (R × S)", "inst✝¹ : Nontrivial R", "inst✝ : Nontrivial S", "this : 0 = 0 ∧ 1 = 0 ∨ 1 = 0 ∧ 0 = 0"], "goal": "False"}}
+{"state": {"context": ["s : Finset ℕ", "m n p✝ p k : ℕ", "hp : Prime p", "hk : 0 < k", "h : Squarefree (p ^ k)"], "goal": "Prime (p ^ k)"}}
+{"state": {"context": ["u✝ : Lean.Level", "arg : Q(Type u✝)", "sα✝ : Q(CommSemiring «$arg»)", "u : Lean.Level", "α : Q(Type u)", "sα : Q(CommSemiring «$α»)", "R✝ : Type u_1", "inst✝¹ : CommSemiring R✝", "a a' a₁✝ a₂✝ a₃ b b' b₁ b₂ b₃ c c₁ c₂ : R✝", "R : Type u_2", "inst✝ : Ring R", "a₁ a₂ : R"], "goal": "-(a₁ + a₂) = -a₁ + -a₂"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[ℕ]"], "goal": "{ toFinsupp := p }.support = p.support"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : OrderTopology α", "inst✝ : DenselyOrdered α", "a b : α", "s : Set α", "l : Filter β", "f : α → β", "h : a < b"], "goal": "Tendsto (fun x => f ↑x) atBot l ↔ Tendsto (f ∘ Subtype.val) atBot l"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.cos x ^ 2 = 1 - Real.sin x ^ 2"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r R : ℝ≥0", "k l : ℕ", "s : Finset (Fin (k + l))", "hs : s.card = l"], "goal": "(fun s => ⟨s.fst - s.snd.card, ⟨s.snd.card, ⟨Finset.map (finCongr ⋯).toEmbedding s.snd, ⋯⟩⟩⟩) ((fun s => ⟨s.fst + s.snd.fst, ↑s.snd.snd⟩) ⟨k, ⟨l, ⟨s, hs⟩⟩⟩) = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : AddCommSemigroup α", "inst✝⁴ : PartialOrder α", "inst✝³ : ExistsAddOfLE α", "inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a b c d : α", "hc : AddLECancellable c", "h : c ≤ b", "h' : a < b - c"], "goal": "a + c ≠ b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι✝ : Type u_1", "π : ι✝ → Type u_2", "inst✝ : TopologicalSpace α", "s✝ t✝ u v : Set α", "ι : Type u_3", "t : Set ι", "s : ι → Set α", "H : ∀ i ∈ t, IsPreconnected (s i)", "K : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j", "R : ι → ι → Prop := fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t", "P : ∀ i ∈ t, ∀ j ∈ t, ReflTransGen R i j → ∃ p ⊆ t, i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j)"], "goal": "IsPreconnected (⋃ n ∈ t, s n)"}}
+{"state": {"context": ["z : ℂ", "this : HasSum (fun x => ((z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)! + (-z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)!) / 2) ((NormedSpace.exp ℂ (z * I) + NormedSpace.exp ℂ (-z * I)) / 2)", "k : ℕ"], "goal": "(z * I) ^ (2 * k) / ↑(2 * k)! = ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 + ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "m : M"], "goal": "LieModuleEquiv.refl m = m"}}
+{"state": {"context": ["α : Type u_1", "Nonempty α", "DecidableEq α", "s : Finset α", "f : α → ℤ", "n : ℕ", "h : ∑ i ∈ s, (f i).natAbs ≤ n"], "goal": "∃ β x sgn g,\n  (∀ (b : β), g b ∉ s → sgn b = 0) ∧ Fintype.card β = n ∧ ∀ a ∈ s, (∑ i : β, if g i = a then ↑(sgn i) else 0) = f a"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : SemilatticeInf α", "inst✝ : OrderTop α", "s : Multiset α", "a : α"], "goal": "∀ (a_1 : α) (s : Multiset α), (a ≤ s.inf ↔ ∀ b ∈ s, a ≤ b) → (a ≤ (a_1 ::ₘ s).inf ↔ ∀ b ∈ a_1 ::ₘ s, a ≤ b)"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v} → Ordinal.{max u v}", "H : ∀ (i : ι), Ordinal.IsNormal (f i)"], "goal": "Set.Unbounded (fun x x_1 => x < x_1) (⋂ i, Function.fixedPoints (f i))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "inst✝⁵ : MeasurableSpace Ω", "inst✝⁴ : StandardBorelSpace Ω", "inst✝³ : Nonempty Ω", "ρ : Measure (β × Ω)", "inst✝² : IsFiniteMeasure ρ", "E : Type u_4", "f : β × Ω → E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hf : Integrable f ρ"], "goal": "∫ (b : β), ∫ (ω : Ω), f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω), f x ∂ρ.fst ⊗ₘ ρ.condKernel"}}
+{"state": {"context": ["α : Type u", "inst✝¹ : LinearOrderedCommGroup α", "a b c : α", "inst✝ : Nontrivial α", "y : α", "hy : y ≠ 1", "h : 1 < y"], "goal": "∃ a, 1 < a"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_2", "DecidableEq ι", "Zero M", "l : List (ι →₀ M)", "x : ι"], "goal": "x ∈ List.foldr (fun x x_1 => x.support ⊔ x_1) ∅ l ↔ ∃ f ∈ l, x ∈ f.support"}}
+{"state": {"context": [], "goal": "ℵ₀ < aleph 1"}}
+{"state": {"context": ["a b : ℕ", "hb : 0 < b"], "goal": "a < a / b * b + b"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X✝ : Ω → ℝ", "μ : Measure Ω", "inst✝¹ : MeasureSpace Ω", "inst✝ : IsProbabilityMeasure ℙ", "X : Ω → ℝ", "hm : AEStronglyMeasurable X ℙ", "hX : ¬Memℒp X 2 ℙ"], "goal": "0 ≤ᶠ[ae ℙ] fun ω => (X ω - ∫ (x : Ω), X x) ^ 2"}}
+{"state": {"context": ["n : Type u_2", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "S : Type w", "CommRing S", "f : R ≃+* S", "M : Matrix n n R"], "goal": "f M.det = (f.mapMatrix M).det"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝¹ : LE α", "inst✝ : LE β", "s t : Set α", "a : α", "h : IsLowerSet sᶜ"], "goal": "s = sᶜᶜ"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "AddCommMonoid ι", "DecidableEq ι", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ι → Submodule R A", "GradedAlgebra 𝒜", "x : Submonoid A", "c1 c2 : HomogeneousLocalization.NumDenSameDeg 𝒜 x"], "goal": "↑(c1 + c2).den = ↑c1.den * ↑c2.den"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "Ring k", "AddCommGroup V", "Module k V", "ι : Type u_6", "s : Finset ι", "w : ι → k", "p : ι → V", "hw : s.sum w = 0"], "goal": "(s.weightedVSub p) w = ∑ i ∈ s, w i • p i"}}
+{"state": {"context": ["R : Type u", "inst✝²⁵ : CommRing R", "S : Type v", "inst✝²⁴ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝²³ : Algebra R S", "K : Type u_1", "inst✝²² : Field K", "inst✝²¹ : Algebra R K", "hRK : IsFractionRing R K", "L : Type u_2", "inst✝²⁰ : Field L", "inst✝¹⁹ : Algebra S L", "inst✝¹⁸ : IsFractionRing S L", "V : Type u_3", "V' : Type u_4", "V'' : Type u_5", "inst✝¹⁷ : AddCommGroup V", "inst✝¹⁶ : Module R V", "inst✝¹⁵ : Module K V", "inst✝¹⁴ : IsScalarTower R K V", "inst✝¹³ : AddCommGroup V'", "inst✝¹² : Module R V'", "inst✝¹¹ : Module S V'", "inst✝¹⁰ : IsScalarTower R S V'", "inst✝⁹ : AddCommGroup V''", "inst✝⁸ : Module R V''", "inst✝⁷ : IsDomain S", "inst✝⁶ : IsDedekindDomain R", "inst✝⁵ : Algebra K L", "inst✝⁴ : Algebra R L", "inst✝³ : IsScalarTower R K L", "inst✝² : IsScalarTower R S L", "inst✝¹ : IsIntegralClosure S R L", "hp : p.IsMaximal", "inst✝ : IsNoetherian R S", "this : Field (R ⧸ p) := Quotient.field p", "ι : Type v := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)", "b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Module.Free.chooseBasis (R ⧸ p) (S ⧸ map (algebraMap R S) p)", "b' : ι → S := fun i => ⋯.choose", "b_eq_b' : ⇑b = ⇑(↑R (Submodule.mkQ (map (algebraMap R S) p))) ∘ b'"], "goal": "finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "n : ℕ", "ifp_n : IntFractPair K", "seq_nth_eq : IntFractPair.stream v n = some ifp_n", "left✝ : ifp_n.fr ≠ 0", "stream_succ_nth_eq : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr⁻¹)", "succ_nth_fr_eq_zero : (IntFractPair.of ifp_n.fr⁻¹).fr = 0"], "goal": "ifp_n.fr⁻¹ = ↑⌊ifp_n.fr⁻¹⌋"}}
+{"state": {"context": ["α : Type u_1", "Rₐ : α → α → Prop", "l : List α"], "goal": "List.Forall₂ Rₐ l l ↔ ∀ (x : α), x ∈ l → Rₐ x x"}}
+{"state": {"context": ["p : ℝ≥0∞ → Prop"], "goal": "(∀ (a : ℝ≥0∞), a ≠ ⊤ → p a) ↔ ∀ (r : ℝ≥0), p ↑r"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "DecidableEq α", "s : Sym α n", "a : α", "h : a ∈ a ::ₛ s := ⋯"], "goal": "(a ::ₛ s).erase a h = s"}}
+{"state": {"context": ["R : Type u", "M : Type v", "CommRing R", "AddCommGroup M", "Module R M", "Module.Free R M", "Module.Finite R M", "f : M →ₗ[R] M"], "goal": "IsIntegral R f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : PartialOrder α", "inst✝² : PartialOrder β", "inst✝¹ : BoundedOrder α", "inst✝ : BoundedOrder β", "x y : α × β"], "goal": "IsCompl x y ↔ IsCompl x.1 y.1 ∧ IsCompl x.2 y.2"}}
+{"state": {"context": ["x y : SetTheory.PGame", "r : x.Relabelling y"], "goal": "y ≤ x"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "inst✝³ : Group G", "inst✝² : MulAction G α", "M : Type u_3", "inst✝¹ : Monoid M", "inst✝ : MulAction M α", "s : Set α", "g : G", "s_ss_fixedBy : s ⊆ fixedBy α g"], "goal": "s ∈ fixedBy (Set α) g"}}
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+{"state": {"context": ["x : ℝ", "hx : |x| ≤ Real.pi"], "goal": "Real.cos x ≤ 1 - 2 / Real.pi ^ 2 * x ^ 2"}}
+{"state": {"context": ["L : Type u_2", "Field L", "F E : Subfield L", "h : F ≤ E", "x : L"], "goal": "x ∈ Subfield.extendScalars h ↔ x ∈ E"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "x : E", "r : ℝ", "hr : r ≠ 0", "y : E", "hy : y ∈ closedBall x r", "this : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1"], "goal": "MapsTo (fun c => c • (y - x) + x) (Ico 0 1) (ball x r)"}}
+{"state": {"context": ["R : Type u", "a✝ b : R", "m n : ℕ", "inst✝ : DivisionRing R", "a : ℚ", "f : R[X]"], "goal": "a • f = C ↑a * f"}}
+{"state": {"context": ["Γ : Type u_1", "Inhabited Γ", "L R : Turing.ListBlank Γ"], "goal": "Turing.Tape.move Turing.Dir.left (Turing.Tape.mk' L R) = Turing.Tape.mk' L.tail (Turing.ListBlank.cons L.head R)"}}
+{"state": {"context": ["G : Type w", "inst✝ : TopologicalSpace G", "μ : Content G", "H : μ.ContentRegular", "K : Compacts G", "ε : ℝ≥0", "hε : ε ≠ 0", "hc : ∀ (x : Compacts G), K.carrier ⊆ interior x.carrier → ↑(μ.toFun K) + ↑ε < ↑(μ.toFun x)"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "LinearOrder α", "a b c : α"], "goal": "min a (min b c) = min b (min a c)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s✝ t K : Set X", "l : Filter Y", "s : Set (X × Y)", "hK : IsCompact K", "hs : ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l", "x : X", "hx : x ∈ K"], "goal": "∃ t ∈ 𝓝[K] x, s ∈ 𝓝ˢ t ×ˢ l"}}
+{"state": {"context": [], "goal": "(treesOfNumNodesEq 0).card = catalan 0"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : HasZeroMorphisms C", "S S₁ S₂ S₃ : ShortComplex C", "h : S.RightHomologyData", "inst✝ : S.HasRightHomology"], "goal": "h.rightHomologyIso.hom ≫ h.ι = S.rightHomologyι ≫ h.opcyclesIso.hom"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "S : Type u_2", "hp : Fact (Nat.Prime p)", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "n j : ℕ", "hj : j < p ^ n"], "goal": "p ^ (n - v p ⟨j + 1, ⋯⟩) ∣ (p ^ n).choose (j + 1)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "MonoidWithZero R", "Zero M", "self : MulActionWithZero R M", "m : M"], "goal": "0 • m = 0"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "f : R", "U : Opens ↑(PrimeSpectrum.Top R)", "hu : ∀ x ∈ U, f ∈ x.asIdeal.primeCompl", "x : ↥(unop (op U))"], "goal": "(algebraMap R (Localizations R ↑x)) 0 = ↑0 x * (algebraMap R (Localizations R ↑x)) ↑⟨f, ⋯⟩"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁸ : OrderedRing R", "inst✝⁷ : AddCommGroup V", "inst✝⁶ : Module R V", "inst✝⁵ : AddTorsor V P", "inst✝⁴ : AddCommGroup V'", "inst✝³ : Module R V'", "inst✝² : AddTorsor V' P'", "inst✝¹ : NoZeroDivisors R", "inst✝ : NoZeroSMulDivisors R V", "ι : Type u_6", "p : ι → P", "ha : AffineIndependent R p", "w w₁ w₂ : ι → R", "s : Finset ι", "hw : ∑ i ∈ s, w i = 1", "hw₁ : ∑ i ∈ s, w₁ i = 1", "hw₂ : ∑ i ∈ s, w₂ i = 1", "i : ι", "his : i ∈ s", "r : R", "hs : w₁ i ≠ w₂ i ∧ r ∈ Set.Ioo 0 1", "hr : ∀ i ∈ s, w i = (r • (w₂ - w₁) + w₁) i", "hz : (s.weightedVSub p) (w₁ - w₂) = 0", "hw₁w₂ : ∑ i ∈ s, (w₁ - w₂) i = 0", "ha' : (w₁ - w₂) i = 0"], "goal": "w₁ i = w₂ i"}}
+{"state": {"context": ["n : ℕ", "a : Fin (n + 1)"], "goal": "a.succ ≠ Fin.last (n + 1) ↔ a ≠ Fin.last n"}}
+{"state": {"context": ["p : ℝ × ℝ"], "goal": "Complex.equivRealProdCLM.symm p = ↑p.1 + ↑p.2 * Complex.I"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{u_3, u_1} C", "ι : Type u_2", "c : ComplexShape ι", "inst✝² : Preadditive C", "inst✝¹ : CategoryWithHomology C", "inst✝ : (HomologicalComplex.quasiIso C c).HasLocalization", "K L : HomologicalComplex C c", "f : { as := K } ⟶ { as := L }"], "goal": "quasiIso C c f ↔ (HomologicalComplex.quasiIso C c).map (quotient C c) f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Ring α", "n e1 e2 t1 t2 : α", "h1 : n * e1 = t1", "h2 : n * e2 = t2"], "goal": "n * (e1 - e2) = t1 - t2"}}
+{"state": {"context": ["i : Nat", "xs : Lean.Omega.Coeffs", "h : xs.length ≤ i"], "goal": "xs.get i = 0"}}
+{"state": {"context": ["F : Type v", "Field F", "W : WeierstrassCurve.Jacobian F", "P Q : Fin 3 → F", "hP : W.Equation P", "hQ : W.Equation Q", "hPz : P z ≠ 0", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy : P y * Q z ^ 3 ≠ Q y * P z ^ 3"], "goal": "IsUnit (W.dblZ P)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "m0 : MeasurableSpace α", "μ : Measure α", "s✝ t s : Set α", "ht : NullMeasurableSet t μ", "s' : Set α", "hsub : s ⊆ s'", "hs'm : MeasurableSet s'", "hs' : μ s' = μ s"], "goal": "μ (s ∩ t) + μ (s \\ t) ≤ μ s"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : ι → Type u_3", "f : ι' ≃ ι", "s : Set ι'", "t : (i : ι) → Set (α i)"], "goal": "∀ (x : (a : ι') → α (f a)), x ∈ ⇑(piCongrLeft α f) ⁻¹' (⇑f '' s).pi t ↔ x ∈ s.pi fun i => t (f i)"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "S : StarSubalgebra R A"], "goal": "S.val = S.subtype.toAlgHom"}}
+{"state": {"context": ["α : Type u", "AddMonoidWithOne α", "n : ℕ"], "goal": "↑n ≠ ⊤"}}
+{"state": {"context": ["ι : Type u", "β : ι → Type v", "DecidableEq ι", "(i : ι) → Zero (β i)", "p : ι → Prop", "DecidablePred p", "i : ι", "x : β i", "h : p i"], "goal": "DFinsupp.filter p (DFinsupp.single i x) = DFinsupp.single i x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "F : Type u_4", "AddCommGroup E", "UniformSpace E", "UniformAddGroup E", "AddCommGroup F", "UniformSpace F", "UniformAddGroup F", "NontriviallyNormedField 𝕜", "Module 𝕜 E", "Module 𝕜 F", "ContinuousSMul 𝕜 E", "f : E →ₗ[𝕜] F", "h : ∃ V ∈ 𝓝 0, Bornology.IsVonNBounded 𝕜 (⇑f '' V)", "x : E"], "goal": "(f.clmOfExistsBoundedImage h) x = f x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : CommRing S", "inst✝⁷ : CommRing T", "inst✝⁶ : Algebra R S", "inst✝⁵ : Algebra R T", "ι κ : Type w", "inst✝⁴ : Fintype ι", "b : Basis ι R S", "inst✝³ : Module.Free R S", "inst✝² : Module.Free R T", "inst✝¹ : Module.Finite R S", "inst✝ : Module.Finite R T", "x : S × T", "a✝ : Nontrivial R", "f : S × T →ₗ[R] Module.End R S × Module.End R T := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap", "this : (lmul R (S × T)).toLinearMap = prodMapLinear R S T S T R ∘ₗ f"], "goal": "(LinearMap.trace R (S × T) ∘ₗ prodMapLinear R S T S T R ∘ₗ f) x = (LinearMap.trace R S ∘ₗ (lmul R S).toLinearMap) x.1 + (LinearMap.trace R T ∘ₗ (lmul R T).toLinearMap) x.2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type x", "UniformSpace β", "F : ι → α → β", "f : α → β", "p : Filter ι", "TopologicalSpace α"], "goal": "TendstoLocallyUniformly F f p ↔ ∀ (x : α), Filter.Tendsto (fun y => (f y.2, F y.1 y.2)) (p ×ˢ 𝓝 x) (𝓤 β)"}}
+{"state": {"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "gp_n : Pair K", "nth_s_eq : (of v).s.get? n = some gp_n", "nth_partDen_eq : (of v).partDens.get? n = some gp_n.b", "ifp_n : IntFractPair K", "succ_nth_stream_eq : IntFractPair.stream v (n + 1) = some ifp_n", "ifp_n_b_eq_gp_n_b : ↑ifp_n.b = gp_n.b"], "goal": "1 ≤ gp_n.b"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "T : Monad C", "J✝ : Type u", "inst✝ : Category.{v, u} J✝", "D : J✝ ⥤ T.Algebra", "c : Cone (D ⋙ T.forget)", "t : IsLimit c", "s : Cone D", "m : s.pt ⟶ (liftedCone D c t).pt", "J : ∀ (j : J✝), m ≫ (liftedCone D c t).π.app j = s.π.app j", "j : J✝"], "goal": "m.f ≫ c.π.app j = ((fun s => { f := t.lift (T.forget.mapCone s), h := ⋯ }) s).f ≫ c.π.app j"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "Algebra R A", "StarRing A", "StarModule R A", "x : A"], "goal": "x ∈ ⊥ ↔ x ∈ Set.range ⇑(algebraMap R A)"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "p q : Set α → Prop", "H : μ.InnerRegularWRT p q", "c : ℝ≥0∞"], "goal": "(c • μ).InnerRegularWRT p q"}}
+{"state": {"context": ["x y : SetTheory.PGame"], "goal": "x.Numeric → y.Numeric → (x + y).Numeric"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "h : s.Nonempty → ∃ x, s = {x}", "a b : α", "has : a ∈ s", "hbs : b ∈ s"], "goal": "a = b"}}
+{"state": {"context": ["k : Type u", "CommRing k", "G : Type u", "Group G", "V : Type v", "AddCommGroup V", "Module k V", "Module (MonoidAlgebra k G) V", "IsScalarTower k (MonoidAlgebra k G) V", "W : Type w", "AddCommGroup W", "Module k W", "Module (MonoidAlgebra k G) W", "IsScalarTower k (MonoidAlgebra k G) W", "π : W →ₗ[k] V", "Fintype G", "v : W"], "goal": "(LinearMap.sumOfConjugatesEquivariant G π) v = ∑ g : G, (π.conjugate g) v"}}
+{"state": {"context": ["X : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : RegularSpace X", "inst✝ : SecondCountableTopology X", "B : Set (Set X)", "hBc : B.Countable", "hB : IsTopologicalBasis B", "s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.1 ⊆ UV.2}", "this✝¹ : Encodable ↑s", "this✝ : TopologicalSpace ↑s := ⊥", "this : DiscreteTopology ↑s", "hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).1) (↑UV).2ᶜ", "ε : ↑s → ℝ", "ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1", "hε : Tendsto ε cofinite (𝓝 0)", "f : ↑s → C(X, ℝ)", "hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1", "hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x => ε UV) (↑UV).2ᶜ", "hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)"], "goal": "∃ f, Inducing f"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Algebra R S", "K : Type u_4", "L : Type u_5", "F : Type u_6", "inst✝⁵ : Field K", "inst✝⁴ : Field L", "inst✝³ : Field F", "inst✝² : Algebra K L", "inst✝¹ : Algebra K F", "ι : Type w", "inst✝ : Fintype ι", "b : Basis ι R S", "x : R", "this : DecidableEq ι"], "goal": "((toMatrix b b) ((lsmul R R S) x)).det = x ^ Fintype.card ι"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "Fact (1 ≤ p)", "f : ↥(MeasureTheory.Lp.simpleFunc E p μ)"], "goal": "‖f‖ = (MeasureTheory.eLpNorm (↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f)) p μ).toReal"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "σ τ : Perm α", "hσ : σ.IsCycle", "hτ : τ.IsCycle", "h : σ.support.card = τ.support.card", "x : α", "hx : x ∈ ↑σ.support"], "goal": "↑(hτ.zpowersEquivSupport ((zpowersEquivZPowers ⋯) (hσ.zpowersEquivSupport.symm ⟨σ x, ⋯⟩))) = τ ↑(hτ.zpowersEquivSupport ((zpowersEquivZPowers ⋯) (hσ.zpowersEquivSupport.symm ⟨x, hx⟩)))"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Field R", "p q : R[X]", "hq0 : q ≠ 0", "h : p / q = 0", "this : q * (p / q) + p % q = p"], "goal": "p.degree < q.degree"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "σ α : Type u_1", "Monad m", "f : σ → α × σ", "s : σ"], "goal": "(modifyGet f).run s = pure ((f s).fst, (f s).snd)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq β", "inst✝ : Monoid α", "s t : Finset α", "a : α", "m n✝ : ℕ", "hst : s ⊆ t", "n : ℕ"], "goal": "s ^ (n + 1) ⊆ t ^ (n + 1)"}}
+{"state": {"context": ["α : Type u_1", "SeminormedAddCommGroup α", "n : ℤ", "a : α"], "goal": "‖n • a‖₊ ≤ ‖n‖₊ * ‖a‖₊"}}
+{"state": {"context": ["x : SetTheory.PGame", "hx : x.Numeric"], "goal": "0 ≤ Surreal.mk x hx ↔ 0 ≤ x"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α", "P : Finpartition s", "a b : α", "ha : a ∈ s", "hb : b ∈ s"], "goal": "a ∈ P.part b ↔ P.part a = P.part b"}}
+{"state": {"context": ["ι : Sort u_1", "𝕜 : Type u_2", "E : Type u_3", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "s t u : Set E", "x y : E", "hs : s.Nonempty", "ht : t.Nonempty"], "goal": "(convexHull 𝕜) (s ∪ t) = convexJoin 𝕜 ((convexHull 𝕜) s) ((convexHull 𝕜) t)"}}
+{"state": {"context": ["G : Type u_3", "Group G", "a : G", "n : ���"], "goal": "a * a ^ n = a ^ (n + 1)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : IsLocallyFiniteMeasure μ", "inst✝ : IsUnifLocDoublingMeasure μ", "p : ℕ → Prop", "s : ℕ → Set α", "M : ℝ", "hM : 0 < M", "r : ℕ → ℝ", "hr : Tendsto r atTop (𝓝 0)", "hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i"], "goal": "blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "L : Filter ι", "inst✝³ : MeasurableSpace Ω", "inst✝² : TopologicalSpace Ω", "inst✝¹ : HasOuterApproxClosed Ω", "inst✝ : OpensMeasurableSpace Ω", "μ : FiniteMeasure Ω", "μs : ι → FiniteMeasure Ω", "μs_lim : Tendsto μs L (𝓝 μ)", "F : Set Ω", "F_closed : IsClosed F", "hne : L.NeBot"], "goal": "∀ (ε : ℝ≥0), 0 < ε → ↑μ F < ⊤ → limsup (fun i => ↑(μs i) F) L ≤ ↑μ F + ↑ε"}}
+{"state": {"context": ["𝕜 : Type u_1", "F : Type u_2", "inst✝³ : RCLike 𝕜", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace 𝕜 F", "inst✝ : CompleteSpace F", "f : F → 𝕜", "f' x : F", "s : Set F", "L : Filter F", "frechet : F →L[𝕜] 𝕜"], "goal": "HasFDerivAt f frechet x ↔ HasGradientAt f ((toDual 𝕜 F).symm frechet) x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "b : β", "p : β → Prop", "s : (x : β) → x = b ∨ p x → Set α"], "goal": "⋂ x, ⋂ (h : x = b ∨ p x), s x h = s b ⋯ ∩ ⋂ x, ⋂ (h : p x), s x ⋯"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s✝ t s : Set X", "H : s ∈ irreducibleComponents X"], "goal": "IsClosed s"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "ι₁ : Type u_3", "s₁ : Finset ι₁", "w₁ : ι₁ → ℝ", "p₁ : ι₁ → P", "h₁ : ∑ i ∈ s₁, w₁ i = 0", "ι₂ : Type u_4", "s₂ : Finset ι₂", "w₂ : ι₂ → ℝ", "p₂ : ι₂ → P", "h₂ : ∑ i ∈ s₂, w₂ i = 0"], "goal": "⟪(s₁.weightedVSub p₁) w₁, (s₂.weightedVSub p₂) w₂⟫_ℝ = (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (dist (p₁ i₁) (p₂ i₂) * dist (p₁ i₁) (p₂ i₂))) / 2"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "n : ℕ"], "goal": "(PowerSeries.coeff R n) PowerSeries.X = if n = 1 then 1 else 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "OmegaCompletePartialOrder α", "CompleteLinearOrder β", "f g : α →o β", "hf : OmegaCompletePartialOrder.Continuous f", "hg : OmegaCompletePartialOrder.Continuous g"], "goal": "OmegaCompletePartialOrder.Continuous (f ⊓ g)"}}
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+{"state": {"context": ["n : Type u_1", "p : Type u_2", "q : Type u_3", "l : Type u_4", "R : Type u₂", "inst✝⁶ : DecidableEq n", "inst✝⁵ : DecidableEq p", "inst✝⁴ : DecidableEq q", "inst✝³ : DecidableEq l", "inst✝² : CommRing R", "inst✝¹ : Fintype n", "inst✝ : Nontrivial R", "h : 1 < Fintype.card n", "j i : n", "hij : j ≠ i", "A : ↥(sl n R) := Eb R i j hij", "B : ↥(sl n R) := Eb R j i ⋯"], "goal": "¬IsLieAbelian ↥(sl n R)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s₁ s₂ : Set P", "h : affineSpan k s₁ ∥ affineSpan k s₂"], "goal": "vectorSpan k s₁ = vectorSpan k s₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e e' : PartialEquiv α β", "h : e ≈ e'", "s : e.source = Set.univ", "t : e.target = Set.univ"], "goal": "e = e'"}}
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+{"state": {"context": ["α : Type u_1", "Preorder α", "p : α → Prop"], "goal": "IsUpperSet {a | p a} ↔ Monotone p"}}
+{"state": {"context": ["G : Type u_1", "LeftCancelMonoid G", "x : G", "n m : ℕ"], "goal": "x ^ n = x ^ m ↔ n % orderOf x = m % orderOf x"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α"], "goal": "∀ (x : α), Levenshtein.defaultCost.insert x = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PseudoMetricSpace α", "PseudoMetricSpace β", "x : α", "y : β", "r : ℝ"], "goal": "Metric.ball x r ×ˢ Metric.ball y r = Metric.ball (x, y) r"}}
+{"state": {"context": ["q r : ℚ", "s : ℚ := (q.num * ↑r.den + r.num * ↑q.den) /. (↑q.den * ↑r.den)"], "goal": "(q + r).num * ↑q.den * ↑r.den = (q.num * ↑r.den + r.num * ↑q.den) * ↑(q + r).den"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "γ' : Type u_6", "δ : Type u_7", "δ' : Type u_8", "ε : Type u_9", "ε' : Type u_10", "m✝ : α → β → γ", "f f₁ f₂ : Filter α", "g g₁ g₂ : Filter β", "h h₁ h₂ : Filter γ", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "u : Set γ", "v : Set δ", "a : α", "b : β", "c : γ", "m : α → γ → δ", "n : β → γ"], "goal": "map₂ m f (map n g) = map₂ (fun a b => m a (n b)) f g"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "Add α"], "goal": "(Matrix.transposeAddEquiv m n α).symm = Matrix.transposeAddEquiv n m α"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "Nontrivial R", "g : Matrix.SpecialLinearGroup n R", "i : n"], "goal": "↑g i ≠ 0"}}
+{"state": {"context": ["G₀ : Type u_1", "α : Type u_2", "GroupWithZero G₀", "Bornology G₀", "MulAction G₀ α", "s t u : Set α", "hs : Absorbs G₀ s u", "ht : Absorbs G₀ t u"], "goal": "Absorbs G₀ (s ∩ t) u"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "x : α", "m : ℕ", "hm : Function.IsPeriodicPt f m x", "n : ℕ"], "goal": "Function.IsPeriodicPt f (n * m) x"}}
+{"state": {"context": ["a : ℝ"], "goal": "UniqueDiffWithinAt ℝ (Set.Iio a) a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "g : E → F", "c : ℝ≥0", "hg : LipschitzWith c g", "g0 : g 0 = 0", "f f' : ↥(Lp E p μ)"], "goal": "∀ᵐ (x : α) ∂μ, ‖↑↑(hg.compLp g0 f - hg.compLp g0 f') x‖ ≤ ↑c * ‖↑↑(f - f') x‖"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝⁷ : Field K", "inst✝⁶ : CharZero K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : IsKilling K L", "inst✝² : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝¹ : H.IsCartanSubalgebra", "inst✝ : IsTriangularizable K (↥H) L", "α✝ β✝ : Weight K (↥H) L", "hα : α✝.IsNonZero", "α β : { α // α.IsNonZero }", "e : ∃ x, x • Weight.toLinear K (↥H) L ↑α = Weight.toLinear K (↥H) L ↑β"], "goal": "(rootSystem H).root α = (rootSystem H).root β ∨ (rootSystem H).root α = -(rootSystem H).root β"}}
+{"state": {"context": ["α : Type u", "EMetricSpace α", "s : Set α", "hs : IsClosed s"], "goal": "IsClosed {t | ↑t ⊆ s}"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α"], "goal": "s.diag.card = s.card"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "S : Set (Set α)", "hS : ∀ (s : ↑S), IsOpen ↑s", "hS' : ∀ (s : ↑S), QuasiSober ↑↑s", "hS'' : ⋃₀ S = ⊤", "t : Set α", "h : IsIrreducible t", "h' : IsClosed t", "x : α", "hx : x ∈ t", "U : Set α", "hU : U ∈ S", "hU' : x ∈ U", "this✝ : QuasiSober ↑U", "H : IsIrreducible (Subtype.val ⁻¹' t)", "this : H.genericPoint ∈ Subtype.val ⁻¹' t"], "goal": "t ∩ ↑⟨U, hU⟩ ⊆ Subtype.val '' closure (Subtype.val ⁻¹' t)"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "Monoid G", "M : ModuleCat (MonoidAlgebra k G)"], "goal": "CoeSort.coe (Rep.ofModuleMonoidAlgebra.obj M) = RestrictScalars k (MonoidAlgebra k G) ↑M"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "SuccOrder α", "a b : α", "hb : ¬IsMax b"], "goal": "Set.Ioo a (Order.succ b) = Set.Ioc a b"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "inst✝⁸ : Fintype n", "inst✝⁷ : Fintype o", "inst✝⁶ : CommRing R", "inst✝⁵ : Finite m", "inst✝⁴ : DecidableEq n", "M₁ : Type u_6", "M₂ : Type u_7", "inst✝³ : AddCommGroup M₁", "inst✝² : AddCommGroup M₂", "inst✝¹ : Module R M₁", "inst✝ : Module R M₂", "A : Matrix m n R", "v₁ : Basis m R M₁", "v₂ : Basis n R M₂", "val✝ : Fintype m", "e₁ : (m → R) ≃ₗ[R] M₁ := (Pi.basisFun R m).equiv v₁ (Equiv.refl m)", "e₂ : (n → R) ≃ₗ[R] M₂ := (Pi.basisFun R n).equiv v₂ (Equiv.refl n)", "range_e₂ : LinearMap.range e₂ = ⊤"], "goal": "LinearMap.range (↑e₁ ∘ₗ A.mulVecLin) = LinearMap.range ((toLin v₂ v₁) A ∘ₗ ↑e₂)"}}
+{"state": {"context": ["K : Type u", "inst✝¹ : Field K", "A✝ A : ValuationSubring K", "P : Ideal ↥A", "inst✝ : P.IsPrime", "x : ↥A"], "goal": "optParam (LocalRing ↥(A.ofPrime P)) ⋯"}}
+{"state": {"context": ["α : Type u_1", "F' : Type u_4", "NormedAddCommGroup F'", "NormedSpace ℝ F'", "CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "hm : m ≤ m0", "MeasureTheory.SigmaFinite (μ.trim hm)", "f : ↥(MeasureTheory.lpMeas F' ℝ m 1 μ)"], "goal": "(MeasureTheory.condexpL1CLM F' hm μ) ↑f = ↑f"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "CategoryTheory.EnoughProjectives C", "X Y : C", "f : X ⟶ Y"], "goal": "(CategoryTheory.ShortComplex.mk (CategoryTheory.Projective.d f) f ⋯).Exact"}}
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+{"state": {"context": ["G : Type u_1", "inst✝³ : Group G", "X : Type u_2", "inst✝² : MulAction G X", "H : Type u_3", "Y : Type u_4", "inst✝¹ : Group H", "inst✝ : MulAction H Y", "φ : G →* H", "j : X →ₑ[⇑φ] Y", "hφ : Function.Surjective ⇑φ", "hj : Function.Injective ⇑j", "B : Set X", "hB : IsBlock G B", "g g' : G", "h : g • B = g' • B"], "goal": "⇑j '' (g • B) = ⇑j '' (g' • B) ∨ Disjoint (⇑j '' (g • B)) (⇑j '' (g' • B))"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂"], "goal": "fst R M M₂ ⊓ snd R M M₂ = ⊥"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f : CategoryTheory.Arrow C", "CategoryTheory.Limits.HasImage f.hom", "CategoryTheory.Limits.HasImageMap (𝟙 f)"], "goal": "CategoryTheory.Limits.image.map (𝟙 f) = 𝟙 (CategoryTheory.Limits.image f.hom)"}}
+{"state": {"context": ["X : Type u", "α : Type v", "TopologicalSpace X", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "s : Set X", "hs : IsPreconnected s", "l₁ l₂ : Filter X", "l₁.NeBot", "l₂.NeBot", "hl₁ : l₁ ≤ 𝓟 s", "hl₂ : l₂ ≤ 𝓟 s", "f g : X → α", "hf : ContinuousOn f s", "hg : ContinuousOn g s", "he₁ : f ≤ᶠ[l₁] g", "he₂ : g ≤ᶠ[l₂] f"], "goal": "∃ x ∈ s, f x = g x"}}
+{"state": {"context": ["R : Type u_1", "NormedRing R", "CompleteSpace R", "f g : ℕ → R", "hf : Summable fun x => ‖f x‖", "hg : Summable fun x => ‖g x‖"], "goal": "(∑' (n : ℕ), f n) * ∑' (n : ℕ), g n = ∑' (n : ℕ), ∑ kl ∈ Finset.antidiagonal n, f kl.1 * g kl.2"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "R X Y : C", "f : X ⟶ Y", "a : R ⟶ X", "CategoryTheory.IsIso a", "CategoryTheory.Mono f"], "goal": "CategoryTheory.IsKernelPair f a a"}}
+{"state": {"context": ["m n : ℕ", "α : Fin (n + 1) → Type u", "x✝ : α 0", "q : (i : Fin (n + 1)) → α i", "p : (i : Fin n) → α i.succ", "i : Fin n", "y : α i.succ", "z : α 0", "P : ((i : Fin n.succ) → α i) → Sort v", "h : (x₀ : α 0) → (x : (i : Fin n) → α i.succ) → P (cons x₀ x)", "x₀ : α 0", "x : (i : Fin n) → α i.succ"], "goal": "h (cons x₀ x 0) (tail (cons x₀ x)) = h x₀ x"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "OrderBot α", "a b : α", "hab : Disjoint a b"], "goal": "b ≤ a → b = ⊥"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "𝓕 : Type u_10", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "FunLike 𝓕 E E₂", "SemilinearIsometryClass 𝓕 σ₁₂ E E₂", "f : 𝓕"], "goal": "EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDomain R", "self : DiscreteValuationRing R"], "goal": "LocalRing.maximalIdeal R ≠ ⊥"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "inst✝⁵ : AddCommMonoid ι", "inst✝⁴ : DecidableEq ι", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ι → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "x : Submonoid A", "y : NumDenSameDeg 𝒜 x", "n : ℕ"], "goal": "(Quotient.mk'' y ^ n).val = val (Quotient.mk'' y) ^ n"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "R S T U : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "L : Bimod T U"], "goal": "(α_ R.X (P.X ⊗ Q.X) L.X).inv ≫ (α_ R.X P.X Q.X).inv ▷ L.X ≫ (((α_ (R.X ⊗ P.X) Q.X L.X).hom ≫ P.actLeft ▷ (Q.X ⊗ L.X)) ≫ P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫ P.X ◁ TensorBimod.actLeft Q L) = R.X ◁ (α_ P.X Q.X L.X).hom ≫ (((α_ R.X P.X (Q.X ⊗ L.X)).inv ≫ (R.X ⊗ P.X) ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫ P.actLeft ▷ (Q.tensorBimod L).X) ≫ coequalizer.π (P.actRight ▷ (Q.tensorBimod L).X) ((α_ P.X S.X (Q.tensorBimod L).X).hom ≫ P.X ◁ (Q.tensorBimod L).actLeft)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "t : ℕ → Set α", "ht : ∀ (n : ℕ), MeasurableSet (t n)", "n : ℕ", "s : Set α", "hs : s ∈ memPartition t n"], "goal": "MeasurableSet s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : PseudoEMetricSpace α", "x y z : α", "s t : Set α", "hxy : x ≠ y"], "goal": "{x, y}.einfsep = edist x y"}}
+{"state": {"context": ["α : Type u_1", "n : Nat", "a b : Array α", "hsz₁ : a.size = n", "hsz₂ : b.size = n", "h : ∀ (i : Nat) (hi : i < n), a.getLit i hsz₁ hi = b.getLit i hsz₂ hi"], "goal": "a = b"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s : Set E", "i j : ι", "c : R", "t : Finset ι", "w : ι → R", "z : ι → E"], "goal": "(∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s) → Convex R s"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u_2", "inst✝¹ : Category.{?u.44478, u_2} V", "c : ComplexShape ι", "inst✝ : HasZeroMorphisms V", "X : (HomologicalComplex V c)ᵒᵖ", "i✝ : ι"], "goal": "((opFunctor V c).map ((opUnitIso V c).hom.app X) ≫ (opCounitIso V c).hom.app ((opFunctor V c).obj X)).f i✝ = (𝟙 ((opFunctor V c).obj X)).f i✝"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s : Finset ι", "v : ι → F", "i j : ι", "hij : i ≠ j"], "goal": "Lagrange.basis {i, j} v i = basisDivisor (v i) (v j)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "UniformSpace α", "l : Filter β", "l.NeBot", "f g : β → α", "a b : α", "ha : Filter.Tendsto f l (𝓝 a)", "hb : Filter.Tendsto g l (𝓝 b)"], "goal": "Inseparable a b ↔ Filter.Tendsto (fun x => (f x, g x)) l (𝓤 α)"}}
+{"state": {"context": ["E : Type u_1", "ι : Type u_2", "K : Type u_3", "inst✝⁴ : NormedLinearOrderedField K", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace K E", "b : Basis ι K E", "inst✝¹ : FloorRing K", "inst✝ : Fintype ι", "m : E", "h : m ∈ span ℤ (Set.range ⇑b)", "i : ι"], "goal": "↑⌈(b.repr m) i⌉ = (b.repr m) i"}}
+{"state": {"context": ["K : Type u", "inst✝ : Field K"], "goal": "intDegree 1 = 0"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Sort u_3", "S : ι → Set α", "f : (i : ι) → ↑(S i) → β", "hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩", "T : Set α", "hT' : T = Set.iUnion S", "dir : Directed (fun x x_1 => x ≤ x_1) S", "op : ↑T → ↑T → ↑T", "opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)", "hopi : ∀ (i : ι) (x y : ↑(S i)), Set.inclusion ⋯ (opi i x y) = op (Set.inclusion ⋯ x) (Set.inclusion ⋯ y)", "opβ : β → β → β", "h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)", "x y : ↑T"], "goal": "Set.iUnionLift S f hf T ⋯ (op x y) = opβ (Set.iUnionLift S f hf T ⋯ x) (Set.iUnionLift S f hf T ⋯ y)"}}
+{"state": {"context": ["z : ℂ", "A : Tendsto (fun n => ((↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) * ∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (2 * n)) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ (2 * n))) atTop (𝓝 (Complex.sin (↑π * z)))", "this : 𝓝 (Complex.sin (↑π * z)) = 𝓝 (Complex.sin (↑π * z) * 1)"], "goal": "Tendsto (fun n => ↑π * z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)) atTop (𝓝 (Complex.sin (↑π * z)))"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hC : IsClosed C", "hsC : contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1"], "goal": "Set.range (sum_to C ho) = GoodProducts (π C fun x => ord I x < o) ∪ MaxProducts C ho"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "L₂ : Type w", "LieRing L₂", "LieAlgebra R L₂", "f : L →ₗ⁅R⁆ L₂"], "goal": "Function.Surjective ⇑f.rangeRestrict"}}
+{"state": {"context": ["�� : Type u_1", "inst✝ : Preorder α", "t : Ordnode α", "h : t.Valid"], "goal": "t.eraseMax.Valid"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁵ : Field F", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Algebra F K", "inst✝¹ : Algebra F L", "inst✝ : Algebra.IsAlgebraic F K", "splits : ∀ (x : K), Splits (algebraMap F L) (minpoly F x)", "x : ↥(⨆ f, f.fieldRange)", "x✝ : x ∈ ⊤", "f : K →ₐ[F] L"], "goal": "f.fieldRange ≤ ⨆ i, IntermediateField.map (⨆ f, f.fieldRange).val i.fieldRange"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.NonPreadditiveAbelian C", "X Y : C", "f : X ⟶ Y"], "goal": "CategoryTheory.Limits.prod.lift (𝟙 X) 0 ≫ CategoryTheory.Limits.prod.map f f =\n  f ≫ CategoryTheory.Limits.prod.lift (𝟙 Y) 0"}}
+{"state": {"context": ["R : Type u_1", "n✝ x y p : ℕ", "hp : Fact (Nat.Prime p)", "hp1 : Odd p", "hyx : y < x", "hxy : p ∣ x - y", "hx : ¬p ∣ x", "n : ℕ", "hn : n ≠ 0"], "goal": "padicValNat p (x ^ n - y ^ n) = padicValNat p (x - y) + padicValNat p n"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝⁹ : CommSemiring ι", "inst✝⁸ : Module ι (Additive ℤˣ)", "inst✝⁷ : DecidableEq ι", "inst✝⁶ : CommRing R", "inst✝⁵ : Ring A", "inst✝⁴ : Ring B", "inst✝³ : Algebra R A", "inst✝² : Algebra R B", "𝒜 : ι → Submodule R A", "ℬ : ι → Submodule R B", "inst✝¹ : GradedAlgebra 𝒜", "inst✝ : GradedAlgebra ℬ", "i₂ : ι", "a₁ : A", "a₂ : ↥(𝒜 i₂)", "b₂ : B"], "goal": "b₂ = ↑GradedMonoid.GOne.one * b₂"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝¹ : Semiring R", "inst✝ : Nontrivial R"], "goal": "¬Module.Finite R R[X]"}}
+{"state": {"context": ["a b c : ℤ", "ha : a ≠ 0", "Hgcd : {a, b, c}.gcd id = 1", "h3a : 3 ∣ a", "HF : a ^ 3 + b ^ 3 + c ^ 3 = 0", "H : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3"], "goal": "3 ∣ b"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "inst✝¹ : Group G", "H K : Subgroup G", "hH : IsPGroup p ↥H", "hK : IsPGroup p ↥K", "inst✝ : K.Normal"], "goal": "IsPGroup p ↥(QuotientGroup.mk' K).ker"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "N : Type u_4", "inst✝⁷ : CommRing R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup N", "inst✝³ : Module R N", "inst✝² : _root_.Finite ι", "P : RootPairing ι R M N", "i j : ι", "inst✝¹ : CharZero R", "inst✝ : NoZeroSMulDivisors R M", "P₁ P₂ : RootPairing ι R M N", "he : P₁.toLin = P₂.toLin", "hr : P₁.root = P₂.root", "hc : range ⇑P₁.coroot = range ⇑P₂.coroot", "hp : P₁.coroot = P₂.coroot → P₁.reflection_perm = P₂.reflection_perm"], "goal": "P₁.coroot = P₂.coroot"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_4, u_1} C", "inst✝³ : Category.{u_3, u_2} D", "G : C ⥤ D", "A : Type w", "inst✝² : Category.{w', w} A", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "inst✝¹ : G.IsCocontinuous J K", "inst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F", "F : Cᵒᵖ ⥤ A", "hF : Presheaf.IsSheaf J F", "R : Dᵒᵖ ⥤ A", "α : G.op ⋙ R ⟶ F", "hR : (Functor.RightExtension.mk R α).IsPointwiseRightKanExtension", "X : D", "S : K.Cover X", "s : Multifork (S.index R)", "i : S.Arrow", "j : StructuredArrow (op i.Y) G.op", "eq : lift hF hR s ≫ R.map (i.f.op ≫ j.hom) ≫ α.app j.right = liftAux hF α s (j.hom.unop ≫ i.f)"], "goal": "(lift hF hR s ≫ R.map i.f.op) ≫ R.map j.hom ≫ α.app j.right = s.ι i ≫ R.map j.hom ≫ α.app j.right"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝⁴ : Ring R", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "ι✝ : Type w", "ι' : Type w'", "inst✝¹ : StrongRankCondition R", "ι : Type w", "b : Basis ι R M", "inst✝ : Infinite ι", "κ : Type w", "v : κ → M", "i : LinearIndependent R v", "m : i.Maximal"], "goal": "#ι ≤ #κ"}}
+{"state": {"context": ["M : Type u", "X : Type w", "PseudoMetricSpace X", "VAdd M X", "IsometricVAdd M X", "c : M", "x y : X"], "goal": "nndist (c +ᵥ x) (c +ᵥ y) = nndist x y"}}
+{"state": {"context": ["w x✝ y✝ z✝ : ℝ", "x y : ℝ≥0", "z : ℝ", "hz : 0 < z"], "goal": "x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "N : ι → Submodule R M", "inst✝² : DecidableEq ι", "h✝ : IsInternal N", "κ : ι → Type u_4", "inst✝¹ : (i : ι) → Fintype (κ i)", "inst✝ : (i : ι) → DecidableEq (κ i)", "s : Finset ι", "h : IsInternal fun i => N ↑i", "b : (i : { x // x ∈ s }) → Basis (κ ↑i) R ↥(N ↑i)", "σ : ι → ι", "hσ : ∀ (i : ι), σ i ≠ i", "f : Module.End R M", "hf : ∀ (i : ι), MapsTo ⇑f ↑(N i) ↑(N (σ i))", "hN : ∀ i ∉ s, N i = ⊥", "i : { x // x ∈ s }", "k : κ ↑i"], "goal": "((h.collectedBasis b).repr (f ↑((b i) k))) ⟨i, k⟩ = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : UniformSpace X", "s : Set X", "inst✝ : (𝓤 X).IsCountablyGenerated", "hs : IsSeqCompact s", "l : Filter X", "hl : Cauchy l", "hls : l ≤ 𝓟 s", "this : l.NeBot", "V : ℕ → Set (X × X)", "hV : (𝓤 X).HasAntitoneBasis V", "W : ℕ → Set (X × X)", "hW : ∀ (n : ℕ), W n ∈ 𝓤 X", "hWV : ∀ (n : ℕ), W n ○ W n ⊆ V n", "hWV' : ∀ (n : ℕ), W n ⊆ V n", "t : ℕ → Set X", "ht_anti : Antitone t", "htl : ∀ (n : ℕ), t n ∈ l", "htW : ∀ (n : ℕ), t n ×ˢ t n ⊆ W n", "hts : ∀ (n : ℕ), t n ⊆ s", "u : ℕ → X", "hu : ∀ (n : ℕ), u n ∈ t n"], "goal": "∃ x ∈ s, l ≤ 𝓝 x"}}
+{"state": {"context": ["n : ℕ", "i : Fin (n + 1)", "h : i < Fin.last n", "h' : ¬i = Fin.last n := ⋯"], "goal": "i.castLT h = i.castPred h'"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α", "p : α → Prop", "DecidablePred p"], "goal": "ComputablePred p ↔ RePred p ∧ RePred fun a => ¬p a"}}
+{"state": {"context": ["S : Type u", "Semiring S", "n : ℕ"], "goal": "ascPochhammer S (n + 1) = Polynomial.X * (ascPochhammer S n).comp (Polynomial.X + 1)"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "κ : ι → Type u_5"], "goal": "∀ {x : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {m : (i : ι) → MeasurableSpace (κ i)} {f : (x : ι) → Ω → κ x},\n  ProbabilityTheory.iIndepFun m f μ → ProbabilityTheory.iIndep (fun x => MeasurableSpace.comap (f x) (m x)) μ"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "X' : Type u_4", "Y : Type u_5", "Z : Type u_6", "α : Type u_7", "α' : Type u_8", "β : Type u_9", "β' : Type u_10", "γ : Type u_11", "𝓕 : Type u_12", "tX : TopologicalSpace X", "tY : TopologicalSpace Y", "tZ : TopologicalSpace Z", "uα : UniformSpace α", "uβ : UniformSpace β", "uγ : UniformSpace γ", "F : ι → X → α", "x₀ : X", "u : α → β", "hu : UniformInducing u", "this : Inducing (⇑UniformFun.ofFun ∘ (fun x => u ∘ x) ∘ ⇑toFun)"], "goal": "EquicontinuousAt F x₀ ↔ EquicontinuousAt ((fun x => u ∘ x) ∘ F) x₀"}}
+{"state": {"context": ["α : Type u_1", "OrderedAddCommMonoid α", "s : Set α", "x : α", "hs : IsUpperSet s", "hx : 0 ≤ x"], "goal": "x +ᵥ s ⊆ s"}}
+{"state": {"context": ["C : Type u'", "CategoryTheory.Category.{v', u'} C", "J : CategoryTheory.GrothendieckTopology C", "R : CategoryTheory.Sheaf J RingCat", "CategoryTheory.HasWeakSheafify J AddCommGrp", "J.WEqualsLocallyBijective AddCommGrp", "J.HasSheafCompose (CategoryTheory.forget₂ RingCat AddCommGrp)", "M N P : SheafOfModules R", "σ : M.GeneratingSections", "p : M ⟶ N", "q : N ⟶ P", "CategoryTheory.Epi p", "CategoryTheory.Epi q"], "goal": "σ.ofEpi (p ≫ q) = (σ.ofEpi p).ofEpi q"}}
+{"state": {"context": ["R : Type u", "Ring R"], "goal": "Function.Injective fun s => s.toSubmonoid"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I X : Set α", "N : Matroid α", "hI : N.Basis I X", "hNM : N.Restriction M"], "goal": "M.Basis I X"}}
+{"state": {"context": ["ι : Type u_3", "π : ι → Type u_4", "(i : ι) → Bornology (π i)"], "goal": "Bornology.cobounded ((i : ι) → π i) = Filter.coprodᵢ fun i => Bornology.cobounded (π i)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "X Y : C"], "goal": "∀ (a : X ⟶ Y), CategoryTheory.Groupoid.invEquiv a = CategoryTheory.Groupoid.inv a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ l₃ : List α", "a✝ b : α", "m n : ℕ", "s✝¹ : List α", "a : α", "t : List α", "o : s✝¹ = [] ∨ ∃ l, l ∈ t.inits ∧ a :: l = s✝¹", "s✝ r : List α", "hr : r ∈ t.inits", "hs : a :: r = s✝", "s : List α", "ht : r ++ s = t"], "goal": "s✝ <+: a :: t"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "hdomain : IsDomain R", "r s : R", "p q : R[X]", "f : RatFunc R"], "goal": "(laurent r) ((laurent s) f) = (laurent (r + s)) f"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "Preorder G", "CovariantClass G G (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "CovariantClass G G (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "s : Set G", "h : BddBelow s"], "goal": "BddAbove (-s)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X", "hs : IsOpen s", "s' : Set X", "hs' : IsOpen s'"], "goal": "(PartialHomeomorph.ofSet s hs).trans (PartialHomeomorph.ofSet s' hs') = PartialHomeomorph.ofSet (s ∩ s') ⋯"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "a b : α", "h1 : -b ≤ a", "h2 : a ≤ b"], "goal": "a ^ 2 ≤ b ^ 2"}}
+{"state": {"context": ["R✝ : Type uR", "inst✝⁴ : Semiring R✝", "S : Type uS", "inst✝³ : CommSemiring S", "T : Type uT", "A : Type uA", "inst✝² : Semiring A", "inst✝¹ : Algebra S A", "R : Type uR", "inst✝ : Ring R", "r : R → R → Prop", "a b : R", "h : Rel r a b"], "goal": "Rel r (-a) (-b)"}}
+{"state": {"context": ["ι : Type u_1", "inst✝⁴ : Fintype ι", "M : Type u_2", "inst✝³ : AddCommGroup M", "R : Type u_3", "inst✝² : CommRing R", "inst✝¹ : Module R M", "I : Ideal R", "b : ι → M", "hb : Submodule.span R (Set.range b) = ⊤", "inst✝ : DecidableEq ι", "A : Matrix ι ι R", "j : ι"], "goal": "((fromMatrix R b) A) (Pi.single j 1) = ∑ i : ι, A i j • b i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p : Set γ", "f f₁ f₂ f₃ : α → β", "g✝ g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a : α", "b : β", "hf : SurjOn f s t", "g : β → γ"], "goal": "g '' t ⊆ g '' (f '' s)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_15", "𝕜' : Type u_16", "NormedDivisionRing 𝕜", "NormedDivisionRing 𝕜'", "c : ℝ", "l : Filter α", "f : α → 𝕜", "g : α → 𝕜'", "h : Asymptotics.IsBigOWith c l f g", "h₀ : ∀ᶠ (x : α) in l, f x = 0 → g x = 0"], "goal": "Asymptotics.IsBigOWith c l (fun x => (g x)⁻¹) fun x => (f x)⁻¹"}}
+{"state": {"context": ["o e : ONote", "n : ℕ+", "a : ONote", "m : ℕ", "inst✝ : o.NF", "h : o.split = (e.oadd n a, m)", "h₁ : (e.oadd n a).NF", "h₂ : o.repr = (e.oadd n a).repr + ↑m", "e0 : e.repr ≠ 0", "d : ω ∣ a.repr"], "goal": "ω ≤ ω ^ e.repr"}}
+{"state": {"context": ["a b c d m n✝ k : ℕ", "p q : ℕ → Prop", "h : m = n✝", "n : ℕ"], "goal": "m ∣ n ↔ n✝ ∣ n"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "Ring R", "Invertible 2", "AddCommGroup V", "Module R V", "AddTorsor V P", "x y : P"], "goal": "(Equiv.pointReflection (midpoint R x y)) y = x"}}
+{"state": {"context": ["Ω✝ : Type u_1", "inst✝⁷ : PseudoEMetricSpace Ω✝", "inst✝⁶ : MeasurableSpace Ω✝", "inst✝⁵ : OpensMeasurableSpace Ω✝", "Ω : Type u_2", "ι : Type u_3", "L : Filter ι", "inst✝⁴ : L.NeBot", "inst✝³ : MeasurableSpace Ω", "inst✝² : PseudoEMetricSpace Ω", "inst✝¹ : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "μs : ι → Measure Ω", "h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i => (μs i) E) L (𝓝 (μ E))", "F : Set Ω", "F_closed : IsClosed F", "ex : ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) F)) = 0", "rs : ℕ → ℝ := Classical.choose ex", "rs_lim : Tendsto rs atTop (𝓝 0)", "rs_pos : ∀ (n : ℕ), 0 < rs n", "rs_null : ∀ (n : ℕ), μ (frontier (Metric.thickening (rs n) F)) = 0"], "goal": "limsup (fun i => (μs i) F) L ≤ μ F"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "CommRing R", "AddCommGroup M", "Module R M"], "goal": "RingTheory.Sequence.IsWeaklyRegular M []"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹¹ : TopologicalSpace M", "inst✝¹⁰ : ChartedSpace H M", "E' : Type u_5", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁶ : TopologicalSpace M'", "inst✝⁵ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝¹ : TopologicalSpace M''", "inst✝ : ChartedSpace H'' M''", "f f₀ f₁ : M → M'", "x : M", "s t : Set M", "g : M' → M''", "u : Set M'", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "I''s : SmoothManifoldWithCorners I'' M''", "f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)", "g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))", "n : ℕ∞", "h : HasMFDerivWithinAt I I' f s x f'", "h₁ : f₁ =ᶠ[𝓝[s] x] f", "hx : f₁ x = f x"], "goal": "writtenInExtChartAt I I' x f₁ =ᶠ[𝓝[↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I] ↑(extChartAt I x) x] writtenInExtChartAt I I' x f"}}
+{"state": {"context": ["R : Type u", "CommRing R", "E : EllipticCurve R", "C : WeierstrassCurve.VariableChange R"], "goal": "(E.variableChange C).a₃ = ↑C.u⁻¹ ^ 3 * (E.a₃ + C.r * E.a₁ + 2 * C.t)"}}
+{"state": {"context": ["V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P", "𝕜 : Type u_6", "NormedField 𝕜", "NormedSpace 𝕜 V", "Invertible 2", "p₁ p₂ : P"], "goal": "dist (midpoint 𝕜 p₁ p₂) p₂ = ‖2‖⁻¹ * dist p₁ p₂"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "c : ℝ≥0∞"], "goal": "(μ.withDensity fun x => c) = c • μ"}}
+{"state": {"context": ["n p : ℕ"], "goal": "eval (↑n) (bernoulli p.succ) - _root_.bernoulli p.succ = (↑p + 1) * ∑ k ∈ range n, ↑k ^ p"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module ℝ E", "inst✝³ : TopologicalSpace E", "inst✝² : T1Space E", "inst✝¹ : TopologicalAddGroup E", "inst✝ : ContinuousSMul ℝ E", "s✝ t✝ s t : Set E", "hsc : Convex ℝ s", "hs₀ : s ∈ 𝓝 0", "hsb : IsVonNBounded ℝ s", "htc : Convex ℝ t", "ht₀ : t ∈ 𝓝 0", "htb : IsVonNBounded ℝ t"], "goal": "⇑(gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb) '' closure s = closure t"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁶ : NormedAddCommGroup E", "inst✝¹⁵ : NormedAddCommGroup E'", "inst✝¹⁴ : NormedAddCommGroup E''", "inst✝¹³ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x x' : G", "y y' : E", "inst✝¹² : RCLike 𝕜", "inst✝¹¹ : NormedSpace 𝕜 E", "inst✝¹⁰ : NormedSpace 𝕜 E'", "inst✝⁹ : NormedSpace 𝕜 E''", "inst✝⁸ : NormedSpace ℝ F", "inst✝⁷ : NormedSpace 𝕜 F", "inst✝⁶ : CompleteSpace F", "inst✝⁵ : MeasurableSpace G", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : BorelSpace G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup P", "inst✝ : NormedSpace 𝕜 P", "μ : Measure G", "L : E →L[𝕜] E' →L[𝕜] F", "n : ℕ∞", "hcg : HasCompactSupport g", "hf : LocallyIntegrable f μ", "hg : ContDiff 𝕜 n g", "k : Set G", "hk : IsCompact k", "h'k : ∀ x ∉ k, g x = 0"], "goal": "ContDiff 𝕜 n (f ⋆[L, μ] g)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "H : Type u_2", "TopologicalSpace H", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_4", "Mul G", "TopologicalSpace G", "ChartedSpace H G", "SmoothMul I G", "g h : G"], "goal": "(smoothRightMul I g) h = h * g"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝ : Ring R", "a✝ : Nontrivial R", "hR : IsField R[X]", "p : R[X]", "hp : X * p = 1", "hp0 : p ≠ 0", "this : p = 0"], "goal": "False"}}
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+{"state": {"context": ["M : Type u_5", "N : Type u_6", "P : Type u_7", "Mul M", "Mul N", "Mul P", "f : M →ₙ* N", "g : M →ₙ* P"], "goal": "(MulHom.fst N P).comp (f.prod g) = f"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "EuclideanDomain R", "CommRing S", "IsDomain S", "Algebra R S", "abv : AbsoluteValue R ℤ", "ι : Type u_5", "DecidableEq ι", "Fintype ι", "bS : Basis ι R S", "a : S", "y : ℤ", "hy : ∀ (k : ι), abv ((bS.repr a) k) ≤ y"], "goal": "abv ((Algebra.norm R) a) ≤ ClassGroup.normBound abv bS * y ^ Fintype.card ι"}}
+{"state": {"context": ["a b c : ℤ", "h : Fermat42 a b c", "a0 b0 c0 : ℤ", "hf : Minimal a0 b0 c0"], "goal": "∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1"}}
+{"state": {"context": ["x : ℝ���0∞", "hx : x ≠ ⊤"], "goal": "ContinuousAt ENNReal.toReal x"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "c : CategoryTheory.Limits.Cone F", "X : C", "f : X ⟶ c.pt"], "goal": "(c.extend f).pt = X"}}
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+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeInf α", "OrderTop α", "s : Finset β", "h : s.Nonempty", "c : α"], "goal": "(s.inf fun x => c) = c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "E I✝ s : Set α", "M : Matroid α", "N : Matroid β", "I X : Set α", "h : N.Basis (f '' I) (f '' X) ∧ InjOn f I ∧ I ⊆ X", "e : α", "x✝ : e ∈ X \\ I", "heX : e ∈ X", "heI : e ∉ I", "hIE : insert e I ⊆ (N.comap f).E"], "goal": "N.Indep (insert (f e) (f '' I)) → ∃ x ∈ I, f x = f e"}}
+{"state": {"context": ["r : ℝ", "hp : 0 < r"], "goal": "r.sign = -1 ∨ r.sign = 0 ∨ r.sign = 1"}}
+{"state": {"context": ["K : Type u_2", "Field K", "φ : K →+* ℂ"], "goal": "NumberField.InfinitePlace.mk (NumberField.ComplexEmbedding.conjugate φ) = NumberField.InfinitePlace.mk φ"}}
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+{"state": {"context": ["x : Cardinal.{v}", "m : ℕ∞"], "goal": "↑m < lift.{u, v} x ↔ ↑m < x"}}
+{"state": {"context": ["S : Type u_1", "Semiring S", "a b : ℕ"], "goal": "↑(a.descFactorial b) = Polynomial.eval (↑(a - (b - 1))) (ascPochhammer S b)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "F : CategoryTheory.Functor C D", "G : CategoryTheory.Functor D E", "G.Final", "G.Full", "G.Faithful"], "goal": "F.Final ↔ (F.comp G).Final"}}
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+{"state": {"context": ["α : Type u_1", "σ : Type u_5", "Primcodable α", "Primcodable σ", "f : α → σ", "hf : Primrec f"], "goal": "Primrec (Option.map f)"}}
+{"state": {"context": ["r : ℝ", "hr : r ≤ 0"], "goal": "‖r‖ = -r"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : HeytingAlgebra α", "a b c : α"], "goal": "⊥ᶜ = ⊤"}}
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+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "p : Equiv.Perm α", "x : α", "n : ℕ", "hn : n < (p.toList x).length"], "goal": "(p.toList x).nthLe n hn = (p ^ n) x"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "TopologicalSpace α", "TopologicalSpace β", "f : SpectralMap α β"], "goal": "f.comp (SpectralMap.id α) = f"}}
+{"state": {"context": ["L : Language", "α : Type w", "M : Type w'", "n : ℕ", "inst✝³ : L.IsOrdered", "inst✝² : L.Structure M", "inst✝¹ : Preorder M", "inst✝ : L.OrderedStructure M", "t₁ t₂ : L.Term (α ⊕ Fin n)", "v : α → M", "xs : Fin n → M"], "goal": "(t₁.lt t₂).Realize v xs ↔ realize (Sum.elim v xs) t₁ < realize (Sum.elim v xs) t₂"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E", "hf : AnalyticAt 𝕜 f x"], "goal": "AnalyticAt 𝕜 (-f) x"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α"], "goal": "∅.max = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)", "s t u : Finset α", "a b : α", "r : ℕ"], "goal": "{ ofColex := s } < { ofColex := t } ↔ ∃ w ∈ t \\ s, filter (fun x => w < x) s = filter (fun x => w < x) t"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f✝ f₀ f₁ : E → F", "f'✝ : F", "s t : Set E", "x v : E", "L : E →L[𝕜] F", "f : E → F", "f' : F", "x₀ : E", "hf : HasLineDerivAt 𝕜 f f' x₀ v", "C : ℝ", "hC₀ : 0 ≤ C", "hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"], "goal": "‖f'‖ ≤ C * ‖v‖"}}
+{"state": {"context": ["a t : ℝ", "ht : 0 < t", "this : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑(n + 1)) + cexp (-(2 * ↑π * I * ↑a * ↑(n + 1)))) * ↑(rexp (-π * ↑(n + 1) ^ 2 * t))) (↑(cosKernel (↑a) t) - 1)", "n : ℕ"], "goal": "2 * Complex.cos (2 * ↑π * ↑a * (↑n + 1)) * cexp (-↑π * (↑n + 1) ^ 2 * ↑t) = (cexp (2 * ↑π * I * ↑a * (↑n + 1)) + cexp (-(2 * ↑π * I * ↑a * (↑n + 1)))) * cexp (-↑π * (↑n + 1) ^ 2 * ↑t)"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "ι : Type u_3", "α : Type u_4", "inst✝ : SeminormedAddCommGroup α", "f : ℕ → α", "r : ℝ", "hr : 1 < r", "hf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0", "N₀ : ℕ", "hN₀ : ∀ b ≥ N₀, r * ‖f b‖ ≤ ‖f (b + 1)‖"], "goal": "¬Summable f"}}
+{"state": {"context": [], "goal": "Finset.univ = ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_4", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_6", "TopologicalSpace M", "ChartedSpace H M", "F : Type u_11", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "n : ℕ∞", "V : M → Type u_12", "TopologicalSpace (Bundle.TotalSpace F V)", "(x : M) → AddCommGroup (V x)", "(x : M) → Module 𝕜 (V x)", "(x : M) → TopologicalSpace (V x)", "FiberBundle F V", "VectorBundle 𝕜 F V", "s : Cₛ^n⟮I; F, V⟯", "z : ℤ"], "goal": "⇑(z • s) = z • ⇑s"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace ℂ E", "V : Type u_2", "inst✝⁹ : NormedAddCommGroup V", "inst✝⁸ : InnerProductSpace ℝ V", "inst✝⁷ : MeasurableSpace V", "inst✝⁶ : BorelSpace V", "inst✝⁵ : FiniteDimensional ℝ V", "W : Type u_3", "inst✝⁴ : NormedAddCommGroup W", "inst✝³ : InnerProductSpace ℝ W", "inst✝² : MeasurableSpace W", "inst✝¹ : BorelSpace W", "inst✝ : FiniteDimensional ℝ W", "f : V → E", "w : V"], "goal": "𝓕⁻ f w = ∫ (v : V), Complex.exp (↑(2 * π * ⟪v, w⟫_ℝ) * Complex.I) • f v"}}
+{"state": {"context": ["α : Type u", "e : Equiv.Perm α"], "goal": "e * Equiv.refl α = e"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "k : Type u_1", "Field k", "IsAlgClosed k", "a₁ a₂ : WittVector p k"], "goal": "∃ x, x ^ (p - 1) = a₂.coeff 0 / a₁.coeff 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "X : C"], "goal": "(CategoryTheory.Limits.zeroProdIso X).hom = CategoryTheory.Limits.prod.snd"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "inst✝ : FiniteDimensional ℝ ↥s.direction", "hd : finrank ℝ ↥s.direction = 2", "c₁ c₂ p₁ p₂ : P", "hc₁s : c₁ ∈ s", "hc₂s : c₂ ∈ s", "hp₁s : p₁ ∈ s", "hp₂s : p₂ ∈ s", "r₁ r₂ : ℝ", "hc : c₁ ≠ c₂", "hp : p₁ ≠ p₂", "hp₁c₁ : dist p₁ c₁ = r₁", "hp₂c₁ : dist p₂ c₁ = r₁", "hp₁c₂ : dist p₁ c₂ = r₂", "hp₂c₂ : dist p₂ c₂ = r₂", "ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫_ℝ = 0", "b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]", "hb : LinearIndependent ℝ b", "hbs : Submodule.span ℝ (Set.range b) = s.direction", "hv : ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)", "t₁ t₂ : ℝ", "hop : t₁ = 0", "hps : t₂ • (p₂ -ᵥ p₁) +ᵥ p₁ ∈ s", "hpc₁ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁", "hpc₂ : dist (t₂ • (p₂ -ᵥ p₁) +ᵥ p₁) c₂ = r₂", "hp' : p₂ -ᵥ p₁ ≠ 0", "hp₂ : dist (1 • (p₂ -ᵥ p₁) +ᵥ p₁) c₁ = r₁"], "goal": "t₂ • (p₂ -ᵥ p₁) +ᵥ p₁ = p₁ ∨ t₂ • (p₂ -ᵥ p₁) +ᵥ p₁ = p₂"}}
+{"state": {"context": ["M' : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "φ : M → N", "ψ : N → P", "χ : M → P", "X : Type u_5", "inst✝⁵ : SMul M X", "inst✝⁴ : SMul M' X", "Y : Type u_6", "inst✝³ : SMul N Y", "inst✝² : SMul M' Y", "Z : Type u_7", "inst✝¹ : SMul P Z", "φ' : N → M", "Y₁ : Type u_8", "inst✝ : SMul M Y₁", "f : X →ₑ[φ] Y", "g : Y → X", "k₂ : Function.RightInverse φ' φ", "h₁ : Function.LeftInverse g ⇑f", "h₂ : Function.RightInverse g ⇑f", "x : Y"], "goal": "f ((f.inverse' g k₂ h₁ h₂) x) = x"}}
+{"state": {"context": ["R✝ : Type u₁", "S✝ : Type u₂", "inst✝³ : CommRing R✝", "inst✝² : CommRing S✝", "f✝ : R✝ →+* S✝", "R : Type u₁", "S : Type u₂", "inst✝¹ : Ring R", "inst✝ : Ring S", "f : R →+* S", "X : ModuleCat R", "Y : ModuleCat S", "g : (restrictScalars f).obj Y ⟶ X", "y : ↑Y", "r : R", "s : S"], "goal": "g ((r • s) • y) = g (r • s • y)"}}
+{"state": {"context": ["a b c d : Ordinal.{u}"], "goal": "d < a ⨳ b ⨳ c ↔\n  ∃ a' < a,\n    ∃ b' < b,\n      ∃ c' < c, d ♯ a' ⨳ b' ⨳ c ♯ a' ⨳ b ⨳ c' ♯ a ⨳ b' ⨳ c' ≤ a' ⨳ b ⨳ c ♯ a ⨳ b' ⨳ c ♯ a ⨳ b ⨳ c' ♯ a' ⨳ b' ⨳ c'"}}
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+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "F.Final", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "G : CategoryTheory.Functor D E", "t : CategoryTheory.Limits.ColimitCocone (F.comp G)"], "goal": "(CategoryTheory.Functor.Final.colimitCoconeOfComp F t).cocone = CategoryTheory.Functor.Final.extendCocone.obj t.cocone"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ : Cocycle K L n", "a n' : ℤ", "hn' : n + a = n'"], "goal": "δ n' (n' + 1) ((↑γ).leftShift a n' hn') = 0"}}
+{"state": {"context": ["J : Type v", "inst✝¹ : SmallCategory J", "inst✝ : IsCofiltered J", "F : J ⥤ Profinite", "C : Cone F", "hC : IsLimit C", "f : LocallyConstant (↑C.pt.toTop) (Fin 2)", "U : Set ↑C.pt.toTop := ⇑f ⁻¹' {0}", "hU : IsClopen U", "j : J", "V : Set ↑(F.obj j).toTop", "hV : IsClopen V", "h : U = ⇑(C.π.app j) ⁻¹' V"], "goal": "⇑f ⁻¹' {0} = ⇑(C.π.app j) ⁻¹' V"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v', u'} D"], "goal": "∀ {X Y : C ⥤ D} (η : X ⟶ Y) (A : CategoryTheory.SimplicialObject.Augmented C),\n  (((CategoryTheory.SimplicialObject.Augmented.whiskering C D).map η).app A).right =\n    η.app (CategoryTheory.SimplicialObject.Augmented.point.obj A)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G"], "goal": "H.Characteristic ↔ ∀ (ϕ : G ≃+ G), AddSubgroup.map ϕ.toAddMonoidHom H ≤ H"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s t : WSeq α", "h : s ~ʷ t"], "goal": "Computation.LiftRel Equiv (s.destruct.bind (Computation.pure ∘ fun o => Option.rec nil Prod.snd o)) (t.destruct.bind (Computation.pure ∘ fun o => Option.rec nil Prod.snd o))"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Inf α", "Top α", "Inf β", "Top β", "f g : InfTopHom α β", "h : ∀ (a : α), f a = g a"], "goal": "f = g"}}
+{"state": {"context": ["M : Type u_1", "inst✝³ : CommMonoid M", "inst✝² : TopologicalSpace M", "m m' : M", "G : Type u_2", "inst✝¹ : CommGroup G", "g g' : G", "inst✝ : ContinuousMul M", "f : ℕ → M", "k : ℕ", "h : HasProd (fun n => f (n + k)) m"], "goal": "HasProd f ((∏ i ∈ range k, f i) * m)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "CommRing R", "Ring A", "Ring B", "Algebra R A", "Algebra R B", "c₁ c₂ : R", "q : QuaternionAlgebra.Basis A c₁ c₂", "F : A →ₐ[R] B"], "goal": "(q.compHom F).i = F q.i"}}
+{"state": {"context": ["ι₁ : Type u_6", "ι₂ : Type u_7", "Y₁ : ι₁ → Type u_8", "Y₂ : ι₂ → Type u_9", "(i₁ : ι₁) → TopologicalSpace (Y₁ i₁)", "(i₂ : ι₂) → TopologicalSpace (Y₂ i₂)", "e : ι₁ ≃ ι₂", "F : (i₁ : ι₁) → Y₁ i₁ ≃ₜ Y₂ (e i₁)"], "goal": "(Homeomorph.piCongr e F).toEquiv = (Equiv.piCongrRight fun i => (F i).toEquiv).trans (Equiv.piCongrLeft Y₂ e)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤"], "goal": "DenseEmbedding Subtype.val"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "CommMonoid α", "m : Multiset α"], "goal": "m.prod = ∏ x : m.ToType, x.fst"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "n i i' : ℤ", "hi : n + i = i'", "X : CochainComplex C ℤ"], "goal": "(CochainComplex.shiftEval C n i i' hi).inv.app X = (HomologicalComplex.XIsoOfEq X ⋯).inv"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r : ℚ", "x : ℤ_[p]", "m n : ℕ", "hm : x - ↑m ∈ Ideal.span {↑p}", "hn : x - ↑n ∈ Ideal.span {↑p}", "this : (ZMod.castHom ⋯ (ZMod p)) ↑↑m = (ZMod.castHom ⋯ (ZMod p)) ↑↑n"], "goal": "↑m = ↑n"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "x : α", "ι : Sort u_3", "δ : Type u_4", "CompleteLinearOrder δ", "p : ι → Prop", "f : (i : ι) → p i → α → δ", "h : ∀ (i : ι) (hi : p i), UpperSemicontinuousAt (f i hi) x"], "goal": "UpperSemicontinuousAt (fun x' => ⨅ i, ⨅ (hi : p i), f i hi x') x"}}
+{"state": {"context": ["K : Type u_1", "NormedLinearOrderedField K", "HasSolidNorm K", "FloorRing K", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace K E", "FiniteDimensional K E", "ProperSpace E", "L : AddSubgroup E", "DiscreteTopology ↥L", "ι : Type u_3", "hs : IsZlattice K L", "b : Basis ι ℤ ↥L"], "goal": "(Submodule.span ℤ (Set.range ⇑(Basis.ofZlatticeBasis K L b))).toAddSubgroup = L"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing S", "W : WeierstrassCurve R", "inst✝⁸ : Algebra R S", "A : Type u", "inst✝⁷ : CommRing A", "inst✝⁶ : Algebra R A", "inst✝⁵ : Algebra S A", "inst✝⁴ : IsScalarTower R S A", "B : Type v", "inst✝³ : CommRing B", "inst✝² : Algebra R B", "inst✝¹ : Algebra S B", "inst✝ : IsScalarTower R S B", "f : A →ₐ[S] B", "n : ℤ"], "goal": "(W.baseChange B).Ψ n = Polynomial.map (mapRingHom ↑f) ((W.baseChange A).Ψ n)"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "A : Type u_2", "AddCommMonoid A", "u : σ →₀ ℕ", "r : R", "b : (σ →₀ ℕ) → R → A", "w : b u 0 = 0"], "goal": "Finsupp.sum ((MvPolynomial.monomial u) r) b = b u r"}}
+{"state": {"context": ["I : Type u", "AddMonoid I", "C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.MonoidalCategory C", "X₁ X₂ : CategoryTheory.GradedObject I C", "X₁.HasTensor X₂", "A : C", "k : I", "f : (i₁ i₂ : I) → i₁ + i₂ = k → (CategoryTheory.MonoidalCategory.tensorObj (X₁ i₁) (X₂ i₂) ⟶ A)", "i₁ i₂ : I", "hi : i₁ + i₂ = k"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.Monoidal.ιTensorObj X₁ X₂ i₁ i₂ k hi)\n    (CategoryTheory.GradedObject.Monoidal.tensorObjDesc f) =\n  f i₁ i₂ hi"}}
+{"state": {"context": ["α : Type v", "Small.{u, v} α", "f : Ordinal.{u} → α"], "goal": "¬Function.Injective f"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D"], "goal": "e.inverse.IsRightAdjoint"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs : ∀ (n : ℕ), s ≠ 1 - ↑n", "this : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹"], "goal": "completedHurwitzZetaEven a s * (s.Gammaℂ * Complex.cos (↑π * s / 2) * s.Gammaℝ⁻¹) = s.Gammaℂ * Complex.cos (↑π * s / 2) * (completedHurwitzZetaEven a s / s.Gammaℝ)"}}
+{"state": {"context": ["α : Type u", "inst✝¹ : DecidableEq α", "β : Type v", "inst✝ : Fintype α", "s : Perm α →* ℤˣ", "hs : Surjective ⇑s", "f : Perm α", "x✝ : f.IsSwap", "x y : α", "hxy : x ≠ y", "hxy' : f = swap x y", "h : ¬s (swap x y) = -1"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Function.Bijective Language.reverse"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.arccos x = π / 2 ↔ x = 0"}}
+{"state": {"context": ["α : Type u", "CompleteLattice α", "f : α →o α", "x : α"], "goal": "Antitone (OrdinalApprox.gfpApprox f x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "f : β → γ", "g✝ : γ → α", "η : Kernel β γ", "κ : Kernel α β", "a : α", "g : γ → ℝ≥0∞", "hg : Measurable g"], "goal": "∫⁻ (c : γ), g c ∂(η ∘ₖ κ) a = ∫⁻ (b : β), ∫⁻ (c : γ), g c ∂η b ∂κ a"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "t :\n  CategoryTheory.Bicategory.LeftExtension f (CategoryTheory.CategoryStruct.comp g (CategoryTheory.CategoryStruct.id c))"], "goal": "t.ofCompId.right = t.extension"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "s : Set α", "t : Set β"], "goal": "t ⊆ f '' s ↔ ∃ u ⊆ s, f '' u = t"}}
+{"state": {"context": ["R✝ : Type u_1", "A✝ : Type u_2", "inst✝¹¹ : CommSemiring R✝", "inst✝¹⁰ : NonUnitalRing A✝", "inst✝⁹ : Module R✝ A✝", "inst✝⁸ : IsScalarTower R✝ A✝ A✝", "inst✝⁷ : SMulCommClass R✝ A✝ A✝", "R : Type u_3", "S : Type u_4", "A : Type u_5", "inst✝⁶ : Semifield R", "inst✝⁵ : Semifield S", "inst✝⁴ : Ring A", "inst✝³ : Algebra R S", "inst✝² : Algebra R A", "inst✝¹ : Algebra S A", "a✝ : A", "f✝ : S → R", "inst✝ : IsScalarTower R S A", "h✝ : SpectrumRestricts a✝ f✝", "a b : A", "f : S → R", "ha : SpectrumRestricts a f", "h : spectrum S a = spectrum S b"], "goal": "Set.RightInvOn f (⇑(algebraMap R S)) (quasispectrum S a)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "PartialOrder α", "PartialOrder β", "OrderBot α", "OrderBot β", "l : α → β", "u : β → α", "gi : GaloisCoinsertion l u", "a : α", "hb : IsAtom (l a)"], "goal": "IsAtom a"}}
+{"state": {"context": ["R : Type u", "Semiring R", "r : R"], "goal": "Polynomial.X * Polynomial.C r = Polynomial.C r * Polynomial.X"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹² : CommRing R", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R M", "I : Ideal R", "S : Type u_3", "inst✝⁹ : CommSemiring S", "inst✝⁸ : Algebra S R", "inst✝⁷ : Module S M", "inst✝⁶ : IsScalarTower S R M", "R' : Type u_4", "M' : Type u_5", "inst✝⁵ : CommRing R'", "inst✝⁴ : AddCommGroup M'", "inst✝³ : Module R' M'", "inst✝² : Algebra R R'", "inst✝¹ : Module R M'", "inst✝ : IsScalarTower R R' M'", "f : M →ₗ[R] M'", "p : R[X]", "q : PolynomialModule R M"], "goal": "(map R' f) (p • q) = Polynomial.map (algebraMap R R') p • (map R' f) q"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NormedDivisionRing 𝕜", "SeminormedAddCommGroup E", "Module 𝕜 E", "BoundedSMul 𝕜 E", "c : 𝕜", "hc : c ≠ 0", "s : Set E", "x : E"], "goal": "Metric.infDist (c • x) (c • s) = ‖c‖ * Metric.infDist x s"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{u₂, u₁} C", "inst✝¹ : GaloisCategory C", "F : C ⥤ FintypeCat", "inst✝ : FiberFunctor F", "f g : End F", "A : PointedGaloisObject F"], "goal": "f.app A.obj (g.app A.obj A.pt) = (f.app A.obj ≫ F.map ((AutGalois.π F A) ((endEquivAutGalois F) g)).hom) A.pt"}}
+{"state": {"context": ["M : Type u_1", "M₂ : Type u_2", "AddCommGroup M", "Module ℚ M", "AddCommGroup M₂", "Module ℚ M₂"], "goal": "Function.Injective AddMonoidHom.toRatLinearMap"}}
+{"state": {"context": ["n : ℕ", "n_pos : 0 < n"], "goal": "⊤ ^ n = ⊤"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedAddCommGroup E'", "inst✝¹² : NormedAddCommGroup E''", "inst✝¹¹ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x x' : G", "y y' : E", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "inst✝⁹ : NormedSpace 𝕜 E", "inst✝⁸ : NormedSpace 𝕜 E'", "inst✝⁷ : NormedSpace 𝕜 E''", "inst✝⁶ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝⁵ : MeasurableSpace G", "μ ν : Measure G", "inst✝⁴ : AddCommGroup G", "inst✝³ : MeasurableNeg G", "inst✝² : μ.IsAddLeftInvariant", "inst✝¹ : MeasurableAdd₂ G", "inst✝ : SFinite μ", "x₀ : G", "s : Set G", "hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => x₀ - t) ⁻¹' s))", "hs : MeasurableSet s", "h2s : (support fun t => (L (f t)) (g (x₀ - t))) ⊆ s", "hf : IntegrableOn f s μ", "hmg : AEStronglyMeasurable g μ"], "goal": "BddAbove ((fun i => ‖g i‖) '' ((fun t => -t + x₀) ⁻¹' s))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "P Q : C", "c : CategoryTheory.Limits.BinaryBicone P Q"], "goal": "c.toCocone.ι.app { as := CategoryTheory.Limits.WalkingPair.left } = c.inl"}}
+{"state": {"context": ["x : ℝ*"], "goal": "x⁻¹⁻¹.Infinitesimal ∧ x⁻¹⁻¹ < 0 ↔ x.Infinitesimal ∧ x < 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "s : Set α", "h : s.Countable", "default : α", "n : ℕ"], "goal": "enumerateCountable h default n ∈ insert default s"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹¹ : AddCommGroup E", "inst✝¹⁰ : UniformSpace E", "inst✝⁹ : UniformAddGroup E", "inst✝⁸ : AddCommGroup F", "inst✝⁷ : UniformSpace F", "inst✝⁶ : FirstCountableTopology E", "inst✝⁵ : RCLike 𝕜", "inst✝⁴ : Module 𝕜 E", "inst✝³ : ContinuousSMul 𝕜 E", "inst✝² : RCLike 𝕜'", "inst✝¹ : Module 𝕜' F", "inst✝ : ContinuousSMul 𝕜' F", "σ : 𝕜 →+* 𝕜'", "f : E →ₛₗ[σ] F", "hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)", "b : ℕ → Set E", "bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)", "bE : (𝓝 0).HasAntitoneBasis fun i => b i", "bE' : (𝓝 0).HasBasis (fun x => x ≠ 0) fun n => (↑n)⁻¹ • b n", "V : Set F", "hV : V ∈ 𝓝 0", "u : ℕ → E", "hu : ∀ (ia : ℕ), ia ≠ 0 → u ia ∈ (↑ia)⁻¹ • b ia", "hu' : ∀ (ia : ℕ), ia ≠ 0 → f (u ia) ∉ V", "h_tendsto : Tendsto (fun n => ↑n • u n) atTop (𝓝 0)", "h_bounded : IsVonNBounded 𝕜 (Set.range fun n => ↑n • u n)", "r : ℝ", "hr : r > 0", "h' : ∀ (c : 𝕜'), r ≤ ‖c‖ → (⇑f '' Set.range fun n => ↑n • u n) ⊆ c • V", "n : ℕ", "hn : r < ↑n", "h1 : r ≤ ‖↑n‖", "hn' : ¬n = 0"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "r : ↥R⁰", "m n : R_hat R K"], "goal": "∃ a, a • ↑↑r = m • ↑↑r * n • ↑↑r"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝ : Semiring R", "p q r : R[X]", "hp : p.Monic", "h : p.natDegree = 0", "this : p = C (p.coeff 0)"], "goal": "C (p.coeff 0) = 1"}}
+{"state": {"context": ["α : Sort u_1", "a' : α", "motive : (a : α) → a' = a → Sort u_2", "p : motive a' ⋯", "a : α", "t : a' = a"], "goal": "HEq (t ▸ p) p"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "p✝ : ℝ", "hp✝ : 1 ≤ p✝", "p : ℝ", "hp : 1 ≤ p", "h₁ : 0 < p", "h₂ : ∀ (x : ι → ℂ), 0 ≤ ∑ i : ι, ‖x i‖ ^ p", "eq_norm : ∀ (x : ι → ℂ), ‖x‖ = (∑ x_1 : ι, ‖x x_1‖ ^ p) ^ (1 / p)", "this : Fact (1 ≤ ENNReal.ofReal p)", "nm_zero : ‖0‖ = 0"], "goal": "volume {x | ∑ i : ι, ‖x i‖ ^ p < 1} = ENNReal.ofReal ((π * Real.Gamma (2 / p + 1)) ^ card ι / Real.Gamma (2 * ↑(card ι) / p + 1))"}}
+{"state": {"context": ["X✝ : Scheme", "X : Scheme", "hX : IsReduced X", "U : X.Opens", "s : ↑Γ(X, U)", "hs : X.basicOpen s = ⊥"], "goal": "∀ (x : ↥U), (X.sheaf.presheaf.germ x) s = (X.sheaf.presheaf.germ x) 0"}}
+{"state": {"context": ["n : ℕ", "G : Type u_3", "Group G", "f : Fin (n + 1) → G"], "goal": "(f 0 • Fin.partialProd fun i => (f ↑↑i)⁻¹ * f i.succ) = f"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "M A : Comon_ C"], "goal": "A.comul.op ▷ op A.X ≫ A.comul.op = (α_ (op A.X) (op A.X) (op A.X)).hom ≫ op A.X ◁ A.comul.op ≫ A.comul.op"}}
+{"state": {"context": ["α : Type u", "s₁ s₂ t₁ t₂ : Set α", "h₁ : s₁ ⊆ s₂", "h₂ : t₁ ⊆ t₂"], "goal": "s₁ ∪ t₁ ⊆ s₂ ∪ t₂"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Balanced C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "h₂ : S₂.Exact", "CategoryTheory.Epi S₁.g", "CategoryTheory.Epi S₂.g", "CategoryTheory.Epi φ.τ₁", "CategoryTheory.Epi φ.τ₃"], "goal": "CategoryTheory.Epi φ.τ₂"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H₁ H₂ : AddSubgroup G", "h : Disjoint H₁ H₂"], "goal": "Function.Injective fun g => ↑g.1 + ↑g.2"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : MeasurableSpace α'", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace β'", "inst✝³ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : SFinite ν", "f : α → β → E", "hf : StronglyMeasurable (uncurry f)", "hE : CompleteSpace E", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "this : SeparableSpace ↑(range (uncurry f) ∪ {0})"], "goal": "StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedRing A", "inst✝¹ : NormedAlgebra 𝕜 A", "inst✝ : CompleteSpace A", "a : A", "r : ℝ≥0", "hr : ↑r < (spectralRadius 𝕜 a)⁻¹", "z : 𝕜", "z_mem : z ∈ Metric.closedBall 0 ↑r"], "goal": "DifferentiableAt 𝕜 (fun z => Ring.inverse (1 - z • a)) z"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "β : ι → Type u_3", "Fintype ι", "(i : ι) → DecidableEq (β i)", "(i : ι) → SMul α (β i)", "k : α", "x y : (i : ι) → β i", "hk : ∀ (i : ι), IsSMulRegular (β i) k"], "goal": "hammingDist (k • x) (k • y) = hammingDist x y"}}
+{"state": {"context": ["α β : Type u", "t : Type u → Type u", "Traversable t", "LawfulTraversable t", "f : α → β", "xs : t α"], "goal": "Traversable.toList (f <$> xs) = f <$> Traversable.toList xs"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "inst✝⁹ : AddCommMonoid ι", "inst✝⁸ : DecidableEq ι", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing A", "inst✝⁵ : Algebra R A", "𝒜 : ι → Submodule R A", "inst✝⁴ : GradedAlgebra 𝒜", "x : Submonoid A", "α : Type u_4", "inst✝³ : SMul α R", "inst✝² : SMul α A", "inst✝¹ : IsScalarTower α R A", "inst✝ : IsScalarTower α A A", "n : ℕ", "y : HomogeneousLocalization 𝒜 x"], "goal": "(n • y).val = n • y.val"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "∀ (a : α →₀ M) (a_1 : α), Finsupp.lcoeFun a a_1 = a a_1"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.Scheme", "f : X.Hom Y", "H : AlgebraicGeometry.IsOpenImmersion f"], "goal": "f.opensFunctor.obj ⊤ = f.opensRange"}}
+{"state": {"context": ["α : Type u_1", "DistribLattice α", "OrderBot α", "Fintype α", "DecidablePred SupIrred", "a : α"], "goal": "OrderEmbedding.birkhoffSet a = ↑(OrderIso.lowerSetSupIrred a)"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "C₃ : Type u_3", "C₄ : Type u_4", "C₅ : Type u_5", "C₆ : Type u_6", "inst✝⁵ : Category.{u_8, u_1} C₁", "inst✝⁴ : Category.{u_9, u_2} C₂", "inst✝³ : Category.{u_10, u_3} C₃", "inst✝² : Category.{u_7, u_4} C₄", "inst✝¹ : Category.{?u.29733, u_5} C₅", "inst✝ : Category.{?u.29737, u_6} C₆", "T✝ : C₁ ⥤ C₂", "L✝ : C₁ ⥤ C₃", "R✝ : C₂ ⥤ C₄", "B✝ : C₃ ⥤ C₄", "T : C₁ ⥤ C₂", "L : C₁ ≌ C₃", "R : C₂ ≌ C₄", "B : C₃ ⥤ C₄", "h : CatCommSq T L.functor R.functor B", "X : C₁"], "goal": "iso'.hom.app X = iso'.hom.app X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_9", "Preorder α", "CompleteLattice β", "f : ι → α → β", "hf : ∀ (i : ι), Monotone (f i)"], "goal": "Monotone (⨆ i, f i)"}}
+{"state": {"context": ["ι : Type u_1", "α✝ : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "inst✝³ : Monoid M", "inst✝² : Monoid N", "inst✝¹ : Monoid P", "l l₁ l₂ : List M", "a : M", "α : Type u_8", "inst✝ : CommMonoid α", "m : α", "L : List α", "ih : IsUnit L.prod → ∀ m ∈ L, IsUnit m", "h : IsUnit m ∧ IsUnit L.prod"], "goal": "∀ m_1 ∈ m :: L, IsUnit m_1"}}
+{"state": {"context": ["G : Type u_2", "Group G", "n : ℕ", "v : ↑(Equiv.Perm.vectorsProdEqOne G n)"], "goal": "Equiv.Perm.VectorsProdEqOne.rotate v n = v"}}
+{"state": {"context": ["α : Type u_1", "σ : Type u_4", "Primcodable α", "Primcodable σ", "s : Part σ", "Decidable s.Dom"], "goal": "Partrec fun x => s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "γ₂ : Type u_4", "δ : Type u_5", "ι : Sort y", "s t u : Set α", "inst✝⁹ : TopologicalSpace α", "inst✝⁸ : MeasurableSpace α", "inst✝⁷ : BorelSpace α", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : BorelSpace β", "inst✝³ : TopologicalSpace γ", "inst✝² : MeasurableSpace γ", "inst✝¹ : BorelSpace γ", "inst✝ : MeasurableSpace δ"], "goal": "MeasurableSpace.comap Prod.snd (borel β) ≤ borel (α × β)"}}
+{"state": {"context": ["a b : Ordinal.{u}", "ha : a ≠ 0", "hb : Ordinal.Principal (fun x x_1 => x * x_1) b", "hb₂ : 2 < b"], "goal": "a * b = b ^ Order.succ (Ordinal.log b a)"}}
+{"state": {"context": ["S : Type u_1", "Ring S", "a : ℕ"], "goal": "↑(a.descFactorial 2) = ↑a * (↑a - 1)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "E : Set α", "hE : E.Finite", "f : Matroid α → Set α × Set (Set α) := fun M => (M.E, {B | M.Base B})", "hf : Function.Injective f"], "goal": "(f '' {M | M.E ⊆ E}).Finite"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F✝ : Type u_3", "𝕜 : Type u_4", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : SMulCommClass ℝ 𝕜 E", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace ℝ F✝", "inst✝⁴ : CompleteSpace F✝", "G : Type u_5", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace ℝ G", "f✝ g : α → E", "m : MeasurableSpace α", "μ✝ : Measure α", "X : Type u_6", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X", "μ : Measure α", "f : ℕ → α → ℝ", "F : α → ℝ", "hf : ∀ (n : ℕ), Integrable (f n) μ", "hF : Integrable F μ", "h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x", "h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))", "f' : ℕ → α → ℝ := fun n x => f n x - f 0 x", "hf'_nonneg : ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, 0 ≤ f' i a", "hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ", "hF_ge : 0 ≤ᶠ[ae μ] fun x => (F - f 0) x", "h_cont : ContinuousAt ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ)"], "goal": "∀ (n : ℕ), AEMeasurable (fun a => ENNReal.ofReal (f' n a)) μ"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y", "s : Set Y"], "goal": "⇑h ⁻¹' closure s = closure (⇑h ⁻¹' s)"}}
+{"state": {"context": ["R : Type u", "EuclideanDomain R", "DecidableEq R", "x y : R"], "goal": "EuclideanDomain.xgcd x y = (EuclideanDomain.gcdA x y, EuclideanDomain.gcdB x y)"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "a b : ℝ", "f✝ : ℝ → E", "μ : Measure ℝ", "inst✝ : Countable ι", "f : ι → C(ℝ, E)", "hf_sum : Summable fun i => ‖ContinuousMap.restrict (↑{ carrier := uIcc a b, isCompact' := ⋯ }) (f i)‖", "hE : CompleteSpace E"], "goal": "HasSum (fun i => ∫ (x : ℝ) in a..b, (f i) x) (∫ (x : ℝ) in a..b, ∑' (i : ι), (f i) x)"}}
+{"state": {"context": ["a b : ℝ"], "goal": "frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "S : Type u_2", "hp : Fact (Nat.Prime p)", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "n j : ℕ", "hj : j < p ^ n"], "goal": "↑(n - v p ⟨j + 1, ⋯⟩) ≤ ↑(n - (multiplicity p j.succ).get ⋯)"}}
+{"state": {"context": ["Ω : Type u_2", "ι : Type u_3", "s : ι → MeasurableSpace Ω", "m m0 : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "SemilatticeInf ι", "NoMinOrder ι", "Nonempty ι", "StandardBorelSpace Ω", "Nonempty Ω", "hm : m ≤ m0", "MeasureTheory.IsFiniteMeasure μ", "h_le : ∀ (n : ι), s n ≤ m0", "h_indep : ProbabilityTheory.iCondIndep m hm s μ", "t : Set Ω", "ht_tail : MeasurableSet t"], "goal": "∀ᵐ (ω : Ω) ∂μ, (μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1"}}
+{"state": {"context": ["R₁ : Type u_3", "CommRing R₁", "K : Type u_4", "Field K", "Algebra R₁ K", "IsFractionRing R₁ K", "IsDomain R₁", "J : FractionalIdeal R₁⁰ K", "d : K"], "goal": "J / FractionalIdeal.spanSingleton R₁⁰ d = FractionalIdeal.spanSingleton R₁⁰ d⁻¹ * J"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type u_1", "δ : Type u_2", "inst✝² : LinearOrder α", "inst✝¹ : PartialOrder β", "inst✝ : OrderBot β", "f : α → β", "H : StrictMono f", "a : α", "h_bot : f a = ⊥", "x : α", "p : β"], "goal": "⊥ ≤ p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Nonempty α", "f : α → β", "hf : Function.Injective f", "h : Nat.card α = 0"], "goal": "Nat.card β = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "P : C → Prop", "CategoryTheory.MonoidalCategory.MonoidalPredicate P", "X : CategoryTheory.FullSubcategory P"], "goal": "(CategoryTheory.MonoidalCategory.leftUnitor X).hom = (CategoryTheory.MonoidalCategory.leftUnitor X.obj).hom"}}
+{"state": {"context": ["β : Type u_1", "ι : Type u_2", "t : ι → TopologicalSpace β", "T : ι → Set (Set β)", "h_basis : ∀ (i : ι), TopologicalSpace.IsTopologicalBasis (T i)"], "goal": "TopologicalSpace.IsTopologicalBasis {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}"}}
+{"state": {"context": ["I : Type u", "f : I → Type v₁", "g : I → Type v₂", "F : (i : I) → f i → g i", "hF : ∀ (i : I), Function.Bijective (F i)"], "goal": "Function.Bijective fun x i => F i (x i)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "T2Space M", "Zero M'", "n : ℕ∞", "U : TopologicalSpace.Opens M", "f : ↥U → M'", "supp : HasCompactSupport f", "diff : ContMDiff I I' n f"], "goal": "ContMDiff I I' n (Function.extend Subtype.val f 0)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Ring R", "x : R", "hx : IsUnit (1 + x)", "this : (1 + x) * ↑hx.unit⁻¹ - 1 = 1 - 1"], "goal": "↑hx.unit⁻¹ - 1 + x + x * (↑hx.unit⁻¹ - 1) = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "G : Type u_6", "k : Type u_7", "R : Type u_8", "inst✝¹ : CommMonoid M", "inst✝ : OrderedCommMonoid N", "f g : ι → N", "s t : Finset ι", "i j : ι", "hf : ∀ i ∈ s, 1 ≤ f i", "hi : i ∈ s", "hj : j ∈ s", "hne : i ≠ j"], "goal": "cons i {j} ⋯ ⊆ s"}}
+{"state": {"context": ["n : ℕ", "n.AtLeastTwo"], "goal": "(𝓝[>] OfNat.ofNat n).NeBot"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedAddCommGroup α", "LinearOrderedAddCommGroup β", "f : ι → α", "g₁ g₂ : ι → β", "h₁ : Monovary f g₁", "h₂ : Monovary f g₂"], "goal": "Monovary f (g₁ + g₂)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "a : α", "s : Finmap β"], "goal": "a ∉ erase a s"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "AddGroup G", "AddAction G α", "Nonempty α"], "goal": "AddAction.IsPretransitive G α ↔ Nonempty (Unique (AddAction.orbitRel.Quotient G α))"}}
+{"state": {"context": ["α : Type u", "p : α → Prop", "x : α"], "goal": "{x | p x} x ↔ p x"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "f g : CauSeq α abs"], "goal": "f.Pos → g.Pos → (f * g).Pos"}}
+{"state": {"context": ["α : Type u", "Group α", "LT α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a : α"], "goal": "a⁻¹ < 1 ↔ 1 < a"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝¹ : Semiring R", "inst✝ : CommSemiring K", "i : R →+* K", "a✝ b : R", "bi : K", "N : ℕ", "a : R", "bu : bi * i b = 1", "n : ℕ", "r : R", "nN : n ≤ N"], "goal": "i (r * a ^ n * b ^ (N - n)) = i b ^ N * (i r * eval (i a * bi) X ^ n)"}}
+{"state": {"context": ["a : ℤ", "n : ℕ", "h : jacobiSym a n = -1"], "goal": "∃ p, Nat.Prime p ∧ p ∣ n ∧ jacobiSym a p = -1"}}
+{"state": {"context": ["x✝ y x : ℂ", "hx : cos x ≠ 0"], "goal": "(1 + sin x ^ 2 / cos x ^ 2)⁻¹ = cos x ^ 2"}}
+{"state": {"context": ["x✝ y x : ℝ", "h : 0 ≤ x"], "goal": "arcsin x = arccos √(1 - x ^ 2)"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "a✝ b✝ c✝ d✝ x y z : P", "r R : ℝ", "a b c d : P", "hb : b ≠ a", "hc : c ≠ a", "hd : d ≠ a", "H : 1 / (dist b a * dist d a) * dist b d ≤ 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d"], "goal": "dist a c * dist b d ≤ dist a b * dist c d + dist b c * dist a d"}}
+{"state": {"context": ["V : SemiNormedGrp"], "goal": "⇑(𝟙 V) = id"}}
+{"state": {"context": ["x : ℝ≥0∞", "z : ℝ"], "goal": "x.toReal ^ z = (x ^ z).toReal"}}
+{"state": {"context": ["X : TopCat", "ι : Type w", "U : ι → TopologicalSpace.Opens ↑X", "Y : TopologicalSpace.Opens ↑X", "hY : Y = iSup U"], "goal": "∀ {x x_1 : TopCat.Presheaf.SheafCondition.OpensLeCover U} (g : x ⟶ x_1),\n  (TopCat.Presheaf.generateEquivalenceOpensLe_inverse' U hY).map g = CategoryTheory.Over.homMk g ⋯"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "NormalizedGCDMonoid α", "a : α"], "goal": "{a}.lcm = normalize a"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "self : Bialgebra R A"], "goal": "Coalgebra.counit 1 = 1"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X : Ω → ℝ", "μ : MeasureTheory.Measure Ω"], "goal": "0 ≤ ProbabilityTheory.variance X μ"}}
+{"state": {"context": ["R : Type uR", "M : Type uM", "CommRing R", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "A : Type uA", "Ring A", "Algebra R A", "f : CliffordAlgebra.EvenHom Q A", "m₁ m₂ : M"], "goal": "(CliffordAlgebra.even.lift.aux f) (((CliffordAlgebra.even.ι Q).bilin m₁) m₂) = (f.bilin m₁) m₂"}}
+{"state": {"context": ["α : Type u", "a : α"], "goal": "List.nil.getLastD a = a"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "C D : ℝ≥0", "f g✝ : E → ℝ", "s✝ : Set E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "ι : Type u_3", "s : Finset ι", "a : ι → ℝ", "v : ι → E", "g : E → ℝ", "g_smooth : ContDiff ℝ ⊤ g", "g_comp : HasCompactSupport g"], "goal": "∫ (x : E), g x • lineDeriv ℝ f x (∑ i ∈ s, a i • v i) ∂μ = ∫ (x : E), ∑ x_1 ∈ s, g x • (a x_1 • fun x => lineDeriv ℝ f x (v x_1)) x ∂μ"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "Fintype ι", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : MeasureTheory.SimpleFunc (ι → ℝ) E", "μ : MeasureTheory.Measure (ι → ℝ)", "MeasureTheory.IsLocallyFiniteMeasure μ", "I : BoxIntegral.Box ι", "l : BoxIntegral.IntegrationParams", "hl : l.bRiemann = false"], "goal": "BoxIntegral.HasIntegral I l (↑f) μ.toBoxAdditive.toSMul (MeasureTheory.SimpleFunc.integral (μ.restrict ↑I) f)"}}
+{"state": {"context": ["x : ZFSet"], "goal": "∅ ⊆ x"}}
+{"state": {"context": ["ι : Type u_1", "G : Type u_2", "Group G", "ConditionallyCompleteLattice G", "CovariantClass G G (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "Nonempty ι", "f : ι → G", "hf : BddAbove (Set.range f)", "a : G"], "goal": "(⨆ i, f i) / a = ⨆ i, f i / a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "hC : CategoryTheory.Pretriangulated C", "T : CategoryTheory.Pretriangulated.Triangle C", "hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles", "h : T.mor₂ = 0"], "goal": "CategoryTheory.Mono T.mor₃"}}
+{"state": {"context": ["α : Type u", "Semiring α", "LinearOrder α", "a b : α", "PosMulStrictMono α", "ha : 0 < a"], "goal": "0 ≤ a * b ↔ 0 ≤ b"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "E : Type u_3", "F : Type u_4", "mα : MeasurableSpace α", "inst✝⁶ : MeasurableSpace Ω", "inst✝⁵ : StandardBorelSpace Ω", "inst✝⁴ : Nonempty Ω", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "ρ : Measure (α × Ω)", "inst✝ : IsFiniteMeasure ρ", "f : α × Ω → F", "hf_int : Integrable f ρ"], "goal": "Integrable (fun x => ∫ (y : Ω), ‖f (x, y)‖ ∂ρ.condKernel x) ρ.fst"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁴ : CommSemiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid N", "inst✝¹ : Module R M", "inst✝ : Module R N", "Q : QuadraticMap R M N"], "goal": "¬Q.Anisotropic ↔ ∃ x, x ≠ 0 ∧ Q x = 0"}}
+{"state": {"context": ["ι : Type u_1", "π : ι → Type u_2", "x : (i : ι) → π i", "inst✝ : DecidableEq ι", "s t : Finset ι", "hst : Disjoint s t", "y : (i : { x // x ∈ s }) → π ↑i", "z : (i : { x // x ∈ t }) → π ↑i", "e : { x // x ∈ s } ⊕ { x // x ∈ t } ≃ { x // x ∈ s ∪ t } := Finset.union s t hst", "i : ι", "his : i ∉ s", "hit : i ∈ t"], "goal": "z ⟨i, ⋯⟩ = (piFinsetUnion π hst) (y, z) ⟨i, ⋯⟩"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E", "s : Set E", "hxs : UniqueDiffWithinAt 𝕜 s x", "c : F"], "goal": "fderivWithin 𝕜 (fun y => c - f y) s x = -fderivWithin 𝕜 f s x"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝⁴ : Monoid K", "inst✝³ : (i : ι) → Group (G i)", "inst✝² : Group H", "φ : (i : ι) → H →* G i", "d : Transversal φ", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → DecidableEq (G i)", "h : H", "w : NormalWord d"], "goal": "MulAction.toEndHom h w = { toWord := w.toWord, head := h * w.head, normalized := ⋯ }"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "CommMonoid M", "f : α → M", "s : Set α", "hs : s.Finite"], "goal": "∏ᶠ (i : α) (_ : i ∈ s), f i = ∏ i ∈ hs.toFinset, f i"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "n : ℕ"], "goal": "((IntFractPair.stream v n).bind fun ap_n => if ap_n.fr = 0 then none else some (IntFractPair.of ap_n.fr⁻¹)) = none ↔ IntFractPair.stream v n = none ∨ ∃ ifp, IntFractPair.stream v n = some ifp ∧ ifp.fr = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "f : α → ℝ", "g : ℝ → ℝ", "s : Set α", "μ : Measure α", "f_nn : 0 ≤ f", "f_mble : Measurable f", "g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t", "g_mble : Measurable g", "g_nn : ∀ t > 0, 0 ≤ g t", "f_nonneg : ∀ (ω : α), 0 ≤ f ω", "H1 : ¬g =ᶠ[ae (volume.restrict (Ioi 0))] 0", "H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0 ..s, g t → μ {a | s < f a} ≠ ⊤", "M_bdd : BddAbove {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}", "M : ℝ := sSup {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}", "zero_mem : 0 ∈ {s | g =ᶠ[ae (volume.restrict (Ioc 0 s))] 0}", "M_nonneg : 0 ≤ M", "hgM : g =ᶠ[ae (volume.restrict (Ioc 0 M))] 0", "ν : Measure α := μ.restrict {a | M < f a}", "this : SigmaFinite ν"], "goal": "∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝⁵ : Category.{max v u, w} D", "inst✝⁴ : ConcreteCategory D", "inst✝³ : PreservesLimits (forget D)", "inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)", "P : Cᵒᵖ ⥤ D", "hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y", "X : C", "S : J.Cover X", "s : Meq (J.plusObj P) S", "inj : ∀ (X : C), Function.Injective ⇑((J.toPlus P).app (op X))", "T : (I : S.Arrow) → J.Cover I.Y", "t : (I : S.Arrow) → Meq P (T I)", "ht : ∀ (I : S.Arrow), ↑s I = mk (t I)", "B : J.Cover X := S.bind T", "Z : B.Arrow → C", "e1 : (I : B.Arrow) → I.Y ⟶ Z I", "e2 : (I : B.Arrow) → Z I ⟶ X", "he2 : ∀ (I : B.Arrow), S.sieve.arrows (e2 I)", "h✝ : ∀ (I : B.Arrow), (fun Y f h => ((fun Y f hf => (T { Y := Y, f := f, hf := hf }).sieve) Y f h).arrows) (Z I) (e2 I) ⋯ (e1 I)", "a✝ : ∀ (I : B.Arrow), e1 I ≫ e2 I = I.f", "w : Meq P B := meqOfSep P hsep X S s T t ht", "I : S.Arrow", "II : (T I).Arrow"], "goal": "∃ W h1 h2, ((w.pullback I.f).pullback II.f).refine h1 = ((t I).pullback II.f).refine h2"}}
+{"state": {"context": ["x✝ x : ℝ", "h1 : 0 < x", "h2 : x < π / 2", "U : Set ℝ := Ico 0 (π / 2)", "intU : interior U = Ioo 0 (π / 2)", "half_pi_pos : 0 < π / 2", "cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y", "sin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y", "tan_cts_U : ContinuousOn tan U", "tan_minus_id_cts : ContinuousOn (fun y => tan y - y) U", "deriv_pos : ∀ y ∈ interior U, 0 < deriv (fun y' => tan y' - y') y"], "goal": "x < tan x"}}
+{"state": {"context": ["a b : ℕ", "hab : a.Coprime b"], "goal": "(a * b).primeFactorsList.Perm (a.primeFactorsList ++ b.primeFactorsList)"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "κ✝ η✝ : Kernel α γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : Kernel α γ", "inst✝¹ : IsFiniteKernel κ", "inst✝ : IsFiniteKernel η", "a : α", "h_eq_add : ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a)ᶜ = (κ a) (κ.mutuallySingularSetSlice η a)ᶜ"], "goal": "(η.withDensity (κ.rnDeriv η)) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a)ᶜ = 0"}}
+{"state": {"context": ["n : ℕ", "hn : n ≠ 0"], "goal": "∑ i ∈ n.divisors, i = ∑ i ∈ n.properDivisors, i + n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : LinearOrder α", "inst✝⁴ : LinearOrder β", "inst✝³ : DenselyOrdered β", "inst✝² : NoMinOrder β", "inst✝¹ : NoMaxOrder β", "inst✝ : Nonempty β", "f : PartialIso α β", "a : α", "h : ¬∃ b, (a, b) ∈ ↑f", "this : ∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f), ∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), x < y", "b : β", "hb : (∀ x ∈ Finset.image Prod.snd (Finset.filter (fun p => p.1 < a) ↑f), x < b) ∧ ∀ y ∈ Finset.image Prod.snd (Finset.filter (fun p => a < p.1) ↑f), b < y"], "goal": "∀ p ∈ ↑f, cmp p.1 a = cmp p.2 b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrderedField α", "f g h : CauSeq α abs", "fg : f ≈ g", "gh : g < h"], "goal": "f < h"}}
+{"state": {"context": ["β : Type u_2", "AddCommMonoid β", "f : Fin 7 → β"], "goal": "∑ i : Fin 7, f i = f 0 + f 1 + f 2 + f 3 + f 4 + f 5 + f 6"}}
+{"state": {"context": ["n m k : ℕ", "h : m ∈ n.smoothNumbers", "h' : k ∣ m"], "goal": "k ∈ n.smoothNumbers"}}
+{"state": {"context": ["a b c : Cardinal.{u_1}", "h : a + b = c + b", "hb : b < ℵ₀"], "goal": "a = c"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ι → Submodule R A", "x : Submonoid A", "i : HomogeneousLocalization.NumDenSameDeg 𝒜 x"], "goal": "HomogeneousLocalization.mk (-i) = -HomogeneousLocalization.mk i"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "inst✝¹⁹ : Field R", "inst✝¹⁸ : AddCommGroup V", "inst✝¹⁷ : TopologicalSpace R", "inst✝¹⁶ : TopologicalSpace V", "inst✝¹⁵ : TopologicalRing R", "inst✝¹⁴ : TopologicalAddGroup V", "inst✝¹³ : Module R V", "inst✝¹² : SeparatingDual R V", "𝕜 : Type u_3", "E : Type u_4", "F : Type u_5", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace 𝕜 F", "inst✝⁶ : SeparatingDual 𝕜 E", "inst✝⁵ : Nontrivial E", "ι : Type u_6", "inst✝⁴ : Finite ι", "M : ι → Type u_7", "inst✝³ : (i : ι) → NormedAddCommGroup (M i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (M i)", "inst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i)", "inst✝ : CompleteSpace (ContinuousMultilinearMap 𝕜 M F)", "m : (i : ι) → M i", "hm : ∀ (i : ι), m i ≠ 0", "f : ℕ → F", "hf : CauchySeq f", "this : ∀ (i : ι), ∃ φ, φ (m i) = 1"], "goal": "∃ a, Tendsto f atTop (𝓝 a)"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m✝ : MeasurableSpace Ω", "inst✝¹ : ConditionallyCompleteLinearOrder ι", "u : ι → Ω → β", "s : Set β", "n i : ι", "ω : Ω", "inst✝ : IsWellOrder ι fun x x_1 => x < x_1", "m : ι", "h_exists : ∃ j ∈ Set.Icc n m, u j ω ∈ s", "h_nonempty : (Set.Icc n m ∩ {i | u i ω ∈ s}).Nonempty"], "goal": "u (sInf (Set.Icc n m ∩ {i | u i ω ∈ s})) ω ∈ s"}}
+{"state": {"context": ["α : Type u", "CompleteLattice α", "f : α →o α", "a : α", "ha : OrderHom.lfp f ≤ a"], "goal": "OrderHom.lfp f ≤ f a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "R : Type x", "inst✝ : Ring α", "a✝ b✝ a b : α", "u : αˣ"], "goal": "a /ₚ u - b /ₚ u = (a - b) /ₚ u"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "P : WeakFEPair E"], "goal": "Set.EqOn (fun x => P.f_modif x - P.f x + P.f₀)\n  (((Set.Ioo 0 1).indicator fun x => P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀) + {1}.indicator fun x => P.f₀ - P.f 1)\n  (Set.Ioi 0)"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n : ℕ", "M : Fin n.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝¹¹ : Semiring R", "inst✝¹⁰ : (i : Fin n.succ) → AddCommMonoid (M i)", "inst✝⁹ : (i : ι) → AddCommMonoid (M₁ i)", "inst✝⁸ : AddCommMonoid M₂", "inst✝⁷ : AddCommMonoid M₃", "inst✝⁶ : AddCommMonoid M'", "inst✝⁵ : (i : Fin n.succ) → Module R (M i)", "inst✝⁴ : (i : ι) → Module R (M₁ i)", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M'", "f f' : MultilinearMap R M₁ M₂", "inst✝ : DecidableEq ι", "m : (i : ι) → M₁ i", "t : Finset ι"], "goal": "∀ ⦃a : ι⦄ {s : Finset ι}, a ∉ s → (∀ (m' : (i : ι) → M₁ i), f (s.piecewise (m + m') m') = ∑ s ∈ s.powerset, f (s.piecewise m m')) → ∀ (m' : (i : ι) → M₁ i), f ((insert a s).piecewise (m + m') m') = ∑ s ∈ (insert a s).powerset, f (s.piecewise m m')"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : ConcreteCategory C", "X Y : C", "f : X ⟶ Y", "inst✝ : IsIso f"], "goal": "Function.Bijective ((forget C).map f)"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁸ : DivisionRing K", "inst✝⁷ : AddCommGroup V", "inst✝⁶ : Module K V", "W : Type u_1", "A : Type u_2", "inst✝⁵ : Semiring A", "inst✝⁴ : Module A V", "inst✝³ : AddCommGroup W", "inst✝² : Module K W", "inst✝¹ : Module A W", "inst✝ : LinearMap.CompatibleSMul V W K A", "h : finrank K W = 1", "f : V →ₗ[A] W", "w : f ≠ 0", "v : V", "n : f v ≠ 0 v", "z : W"], "goal": "∃ a, (↑K f) a = z"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝ : NonUnitalSeminormedRing α", "x y : α"], "goal": "‖(AddMonoidHom.mulRight x) y‖ ≤ ‖x‖ * ‖y‖"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x✝ y z✝ : E", "δ ε : ℝ", "x z : E", "a b : ℝ", "ha : 0 ≤ a", "hb : 0 ≤ b", "hab : a + b = 1"], "goal": "∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z"}}
+{"state": {"context": ["α : Type u", "DecidableEq α", "β : Type v", "Fintype α", "DecidableEq β", "Fintype β", "e : α ≃ β", "p : Equiv.Perm α"], "goal": "Equiv.Perm.sign (e.permCongr p) = Equiv.Perm.sign p"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "F : Type u_3", "AddCommGroup F", "Module R F", "g : Submodule R (E × F)", "hg : ∀ x ∈ g, x.1 = 0 → x.2 = 0"], "goal": "LinearMap.range g.toLinearPMap.toFun = Submodule.map (LinearMap.snd R E F) g"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "α : Type u_9", "TopologicalSpace α", "e : M₁ ≃SL[σ₁₂] M₂", "f : α → M₁"], "goal": "Continuous (⇑e ∘ f) ↔ Continuous f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_4", "F : Type u_5", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F", "RCLike 𝕜", "NormedSpace 𝕜 F", "SMulCommClass ℝ 𝕜 F", "n : ℕ", "m : Fin (n + 1) → E", "f : SchwartzMap E F"], "goal": "(SchwartzMap.iteratedPDeriv 𝕜 m) f =\n  (SchwartzMap.iteratedPDeriv 𝕜 (Fin.init m)) ((SchwartzMap.pderivCLM 𝕜 (m (Fin.last n))) f)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.HasShift C ℤ", "X : Cᵒᵖ", "n : ℤ"], "goal": "(CategoryTheory.shiftFunctor C n).map\n    ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence C n).counitIso.inv.app X).unop =\n  ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence C n).unitIso.hom.app\n      (Opposite.op ((CategoryTheory.shiftFunctor C n).obj (Opposite.unop X)))).unop"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type u_1", "P : Type u_2", "L.Structure M", "L.Structure N", "L.Structure P", "MN : L.ElementarilyEquivalent M N", "NP : L.ElementarilyEquivalent N P"], "goal": "L.ElementarilyEquivalent M P"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "a b : α", "s : Set α", "ho : Set.Ioo a b ⊆ s", "hc : s ⊆ Set.Icc a b"], "goal": "s ∈ {Set.Icc a b, Set.Ico a b, Set.Ioc a b, Set.Ioo a b}"}}
+{"state": {"context": ["α : Type u_1", "E' : Type u_2", "F : Type u_3", "F' : Type u_4", "𝕜 : Type u_5", "p : ℝ≥0∞", "inst✝¹⁰ : RCLike 𝕜", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : InnerProductSpace 𝕜 E'", "inst✝⁷ : CompleteSpace E'", "inst✝⁶ : NormedSpace ℝ E'", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "ι : Type u_6", "m m0 : MeasurableSpace α", "μ : Measure α", "hm : m ≤ m0", "f : ↥(lpMeasSubgroup F m p μ)"], "goal": "↑↑(lpMeasSubgroupToLpTrim F p μ hm (-f)) =ᶠ[ae μ] -↑↑(lpMeasSubgroupToLpTrim F p μ hm f)"}}
+{"state": {"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "z w : K"], "goal": "z ≤ 0 ↔ re z ≤ 0 ∧ im z = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "f : α → β", "a : α", "s : Set α", "l : Filter β", "h : Filter.Tendsto f (𝓝 a) l"], "goal": "Filter.Tendsto f (𝓝[s] a) l"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "R : Type u_3", "inst✝⁶ : Zero X", "inst✝⁵ : Zero Y", "inst✝⁴ : Zero R", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace R", "inst✝ : T1Space R"], "goal": "IsClosed {f | f 0 = 0}"}}
+{"state": {"context": [], "goal": "LipschitzWith 1 fun p => min p.1 p.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : TopologicalSpace α", "inst✝¹ : LinearOrderedAddCommGroup α", "inst✝ : OrderTopology α", "l : Filter β", "f g : β → α", "a : α"], "goal": "𝓝 a = ⨅ r, ⨅ (_ : r > 0), 𝓟 {b | |a - b| < r}"}}
+{"state": {"context": ["S : Type u", "inst✝ : Semiring S", "n : ℕ"], "goal": "ascPochhammer S (n + 1) = ascPochhammer S n * (X + ↑n)"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A"], "goal": "GaloisConnection (Algebra.adjoin R) SetLike.coe"}}
+{"state": {"context": ["G✝ : Type u", "H : Type v", "inst✝³ : Mul G✝", "inst✝² : Mul H", "G : Type u_1", "inst✝¹ : Group G", "inst✝ : UniqueProds G", "A B : Finset G", "hc : A.Nonempty ∧ B.Nonempty ∧ (1 < A.card ∨ 1 < B.card)", "a : G", "ha : a ∈ A", "b : G", "hb : b ∈ B", "hu : UniqueMul A B a b"], "goal": "∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s✝ : Set E", "i j : ι", "c : R", "t : Finset ι", "w✝ : ι → R", "z : ι → E", "s : Finset E", "x : E", "w : E → R", "hw : ∑ y ∈ s, w y = 1"], "goal": "s.centerMass w id = x ↔ ∑ y ∈ s, w y • y = x"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "d : G.Dart"], "goal": "d.edge ∈ G.edgeSet"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "S : Type u_1", "SetLike S K", "h : SubfieldClass S K", "s : S", "q : ℚ≥0", "x : ↥s"], "goal": "↑(q • x) = q • ↑x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝³ : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "p : ℝ≥0∞", "μ : Measure α", "G : Type u_7", "inst✝ : NormedLatticeAddCommGroup G", "hp : Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤", "g : { g // 0 ≤ g }", "this✝¹ : MeasurableSpace G := borel G", "this✝ : BorelSpace G"], "goal": "∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)"}}
+{"state": {"context": ["x : ℝ", "hx : x ∈ Ioo (-(π / 2)) (π / 2)", "y : ℝ", "hy : y ∈ Ioo (-(π / 2)) (π / 2)", "hlt : x < y"], "goal": "tan x < tan y"}}
+{"state": {"context": ["f : ℕ → ℂ", "h : ∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C"], "goal": "∃ C, ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ 0"}}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re"], "goal": "Filter.Tendsto (fun X => s.partialGamma X) Filter.atTop (𝓝 s.GammaIntegral)"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "J : GrothendieckTopology C", "F F' F'' : Cᵒᵖ ⥤ Type w", "G G' : Subpresheaf F", "h : G ≤ G'", "hF : Presieve.IsSheaf J F", "hG' : Presieve.IsSheaf J G'.toPresheaf", "U : Cᵒᵖ", "x : F.obj U", "hx : x ∈ (sheafify J G).obj U", "V : C", "i : V ⟶ unop U", "hi : (G.sieveOfSection x).arrows i", "this : (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG') = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe h)"], "goal": "↑((G.sheafifyLift (homOfLe h) hG').app (op V) ((sheafify J G).toPresheaf.map i.op ⟨x, hx⟩)) = (fun f => ↑(f.app (op V) ⟨F.map i.op x, hi⟩)) (homOfLe ⋯ ≫ G.sheafifyLift (homOfLe h) hG')"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁷ : Ring R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : AddCommGroup M''", "inst✝³ : Module R M", "inst✝² : Module R M'", "inst✝¹ : Module R M''", "a b : R", "x y : M", "inst✝ : Nontrivial R", "hv : LinearIndependent R v", "f : ι' ↪ ι", "hss : range v ⊆ ↑(span R (range (v ∘ ⇑f)))", "i : ι", "repr : ↥(span R (range (v ∘ ⇑f))) → ι' →₀ R := ⇑⋯.repr", "l : ι →₀ R := Finsupp.mapDomain (⇑f) (repr ⟨v i, ⋯⟩)", "h_total_l : (Finsupp.total ι M R v) l = v i", "h_total_eq : (Finsupp.total ι M R v) l = (Finsupp.total ι M R v) (Finsupp.single i 1)"], "goal": "∃ a, f a = i"}}
+{"state": {"context": ["G : Type u_1", "Group G", "M : Type u_7", "MulOneClass M", "f : G →* M"], "goal": "f.ker = ⊥ ↔ Function.Injective ⇑f"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f g : ℝ × ℝ → E", "f' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E", "a b : ℝ × ℝ", "hle : a ≤ b", "s : Set (ℝ × ℝ)", "hs : s.Countable", "Hcf : ContinuousOn f (Set.Icc a b)", "Hcg : ContinuousOn g (Set.Icc a b)", "Hdf : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \\ s, HasFDerivAt f (f' x) x", "Hdg : ∀ x ∈ Set.Ioo a.1 b.1 ×ˢ Set.Ioo a.2 b.2 \\ s, HasFDerivAt g (g' x) x", "Hi : IntegrableOn (fun x => (f' x) (1, 0) + (g' x) (0, 1)) (Set.Icc a b) volume", "e : (ℝ × ℝ) ≃L[ℝ] Fin 2 → ℝ := (ContinuousLinearEquiv.finTwoArrow ℝ ℝ).symm"], "goal": "MeasurePreserving (⇑e) volume volume"}}
+{"state": {"context": ["x y : ℝ", "a b : Cauchy abs"], "goal": "({ cauchy := a } - { cauchy := b }).cauchy = { cauchy := a }.cauchy - { cauchy := b }.cauchy"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A : CategoryTheory.Functor Cᵒᵖ (Type v)", "h : CategoryTheory.Limits.IsIndObject A"], "goal": "CategoryTheory.IsFiltered (CategoryTheory.CostructuredArrow CategoryTheory.yoneda A)"}}
+{"state": {"context": ["E : Type u_6", "NormedGroup E", "a : E"], "goal": "‖a‖ ≤ 0 ↔ a = 1"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : CosimplicialObject C", "X✝ Y✝ : Augmented C", "η : X✝ ⟶ Y✝"], "goal": "η.left ≫ Y✝.hom.app [0] = X✝.hom.app [0] ≫ η.right.app [0]"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p : ℕ", "ExpChar R p", "n : ℕ"], "goal": "(frobenius R p) ↑n = ↑n"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "sq sq₁ sq₂ : Square C", "h : sq₁.IsPushout", "e : sq₁ ≅ sq₂"], "goal": "sq₁.f₃₄ ≫ (evaluation₄.mapIso e).hom = (evaluation₃.mapIso e).hom ≫ sq₂.f₃₄"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Monoid α", "a b : α", "u : αˣ", "h : a ∣ b"], "goal": "a * ↑u * ↑u⁻¹ = a"}}
+{"state": {"context": ["c : Cardinal.{u_1}", "hc : ℵ₀ ≤ c"], "goal": "Ordinal.Principal (fun x x_1 => x + x_1) c.ord"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "P : Prop", "inst✝ : IsDomain R", "h_pos : ∀ (p : ℕ), Nat.Prime p → CharP R p → P", "h_equal : Algebra ℚ R → P", "h_mixed : ∀ (p : ℕ), Nat.Prime p → MixedCharZero R p → P"], "goal": "∀ (p : ℕ), p ≠ 0 → CharP R p → P"}}
+{"state": {"context": ["G : Type u_2", "AddGroup G", "ι : Sort u_5", "S : ι → AddSubgroup G", "C : G → Prop", "x : G", "hx : x ∈ ⨆ i, S i", "mem : ∀ (i : ι), ∀ x ∈ S i, C x", "one : C 0", "mul : ∀ (x y : G), C x → C y → C (x + y)"], "goal": "C x"}}
+{"state": {"context": ["M : Type u_3", "AddCommMonoid M", "f : ℕ → ℕ → M", "n : ℕ"], "goal": "∑ ij ∈ Finset.antidiagonal n, f ij.1 ij.2 = ∑ k ∈ Finset.range n.succ, f k (n - k)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H' H : AddSubgroup G", "h : H' ≤ H"], "goal": "∃ S, ↑H' + S = ↑H ∧ Nat.card ↥H' * Nat.card ↑S = Nat.card ↥H"}}
+{"state": {"context": ["n : ℕ", "p : Fin (n + 1) → Type u", "τ : Type u", "t : τ", "x : p 0"], "goal": "Function.FromTypes.const p t x = Function.FromTypes.const (Matrix.vecTail p) t"}}
+{"state": {"context": ["α : Type u", "J : Type w", "inst✝⁴ : SmallCategory J", "inst✝³ : FinCategory J", "inst✝² : SemilatticeInf α", "inst✝¹ : OrderTop α", "ι : Type u", "inst✝ : Fintype ι", "f : ι → α"], "goal": "Finset.univ.inf (f ∘ ⇑discreteEquiv.toEmbedding) = Fintype.elems.inf f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "NormedAddCommGroup γ", "f : α → β", "g : α → γ", "hf : MeasureTheory.Integrable f μ", "hg : MeasureTheory.Integrable g μ"], "goal": "MeasureTheory.Integrable (fun x => (f x, g x)) μ"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "P : Type u → Prop", "of_equiv : ∀ {α β : Type u}, α ≃ β → P α → P β", "h_empty : P PEmpty.{u + 1}", "h_option : ∀ {α : Type u} [inst : Fintype α], P α → P (Option α)", "α : Type u", "inst✝ : Finite α", "val✝ : Fintype α"], "goal": "P α"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "f✝¹ : ℝ → E", "g' g φ : ℝ → ℝ", "a b : ℝ", "inst✝³ : CompleteSpace E", "f✝ f'✝ : ℝ → E", "inst✝² : RCLike 𝕜", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : IsScalarTower ℝ 𝕜 E", "f f' : 𝕜 → E", "z₀ z₁ : 𝕜", "hcont : ContinuousOn (fun t => f' (z₀ + t • z₁)) (Icc 0 1)", "hderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f' (z₀ + t • z₁)) (z₀ + t • z₁)", "γ : ℝ → 𝕜 := fun t => z₀ + t • z₁", "hint : IntervalIntegrable (z₁ • f' ∘ γ) volume 0 1", "hderiv' : ∀ t ∈ [[0, 1]], HasDerivAt (f ∘ γ) (z₁ • (f' ∘ γ) t) t"], "goal": "z₁ • ∫ (t : ℝ) in 0 ..1, f' (z₀ + t • z₁) = ∫ (y : ℝ) in 0 ..1, z₁ • (f' ∘ γ) y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : α ≃ β", "f' : PartialEquiv β γ"], "goal": "(e.transPartialEquiv f').target = f'.target"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "G' : Type u_5", "NormedAddCommGroup G'", "NormedSpace 𝕜 G'", "B : E →L[𝕜] F →L[𝕜] G", "f : G' → E", "g : G' → F", "f' : G' →L[𝕜] E", "g' : G' →L[𝕜] F", "x : G'", "hf : HasStrictFDerivAt f f' x", "hg : HasStrictFDerivAt g g' x"], "goal": "HasStrictFDerivAt (fun y => (B (f y)) (g y))\n  (((ContinuousLinearMap.precompR G' B) (f x)) g' + ((ContinuousLinearMap.precompL G' B) f') (g x)) x"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop"], "goal": "AntisymmRel (Function.swap r) = AntisymmRel r"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "TopologicalSpace α", "μ : MeasureTheory.Measure α", "μ.OuterRegular", "A : Set α", "hA : MeasurableSet A", "hA' : μ A ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0"], "goal": "∃ U ⊇ A, IsOpen U ∧ μ U < ⊤ ∧ μ (U \\ A) < ε"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "LinearOrderedSemifield α", "LinearOrderedSemifield β", "f : ι → α", "g : ι → β", "hf : StrongLT 0 f"], "goal": "Antivary f⁻¹ g ↔ Monovary f g"}}
+{"state": {"context": ["p : ℕ → Prop", "inst✝ : DecidablePred p", "n : ℕ", "hp : ∀ n' < n, ¬p n'"], "goal": "∀ x ∈ range n, ¬p x"}}
+{"state": {"context": ["Ω : Type u_1", "Ω' : Type u_2", "MeasurableSpace Ω", "MeasurableSpace Ω'", "ν : MeasureTheory.FiniteMeasure Ω", "f : Ω → Ω'"], "goal": "↑(ν.map f) = MeasureTheory.Measure.map f ↑ν"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "ι' : Type v'", "E : ι → Type wE", "E₁ : ι → Type wE₁", "E' : ι' → Type wE'", "G : Type wG", "G' : Type wG'", "inst✝¹² : Fintype ι", "inst✝¹¹ : Fintype ι'", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)", "inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)", "inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)", "inst✝⁵ : (i : ι') → SeminormedAddCommGroup (E' i)", "inst✝⁴ : (i : ι') → NormedSpace 𝕜 (E' i)", "inst✝³ : SeminormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : SeminormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f : MultilinearMap 𝕜 E G", "hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0", "ε : ι → ℝ", "C : ℝ", "hε : ��� (i : ι), 0 < ε i", "c : ι → 𝕜", "hc : ∀ (i : ι), 1 < ‖c i‖", "hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖", "m : (i : ι) → E i", "hm : ∀ (i : ι), ‖m i‖ ≠ 0", "δ : ι → 𝕜", "hδ0 : ∀ (i : ι), δ i ≠ 0", "hδm_lt : ∀ (i : ι), ‖δ i • m i‖ < ε i", "hle_δm : ∀ (i : ι), ε i / ‖c i‖ ≤ ‖δ i • m i‖", "a✝ : ∀ (i : ι), ‖δ i‖⁻¹ ≤ (ε i)⁻¹ * ‖c i‖ * ‖m i‖"], "goal": "‖f m‖ ≤ C * ∏ i : ι, ‖m i‖"}}
+{"state": {"context": ["n : Type u_2", "𝕜 : Type u_4", "Fintype n", "RCLike 𝕜", "DecidableEq n", "A : Matrix n n 𝕜", "hA : A.PosSemidef", "i : n"], "goal": "0 ≤ ⋯.eigenvalues i"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "F : J ⥤ C", "G : C ⥤ D"], "goal": "∀ {X Y : CategoryTheory.Limits.Cone F} (f : X ⟶ Y),\n  ((CategoryTheory.Limits.Cones.functoriality F G).map f).hom = G.map f.hom"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I J : Box ι", "i : ι", "x : ℝ", "inst✝ : Finite ι", "s : Finset (Box ι)"], "goal": "∀ᶠ (t : Finset (ι × ℝ)) in atTop, ∀ (I J : Box ι), J ∈ s → ∀ J' ∈ splitMany I t, ¬Disjoint ↑J ↑J' → J' ≤ J"}}
+{"state": {"context": ["α : Type u_2", "InvolutiveInv α", "a : α"], "goal": "{a}⁻¹ = {a⁻¹}"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommSemiring R", "inst✝⁵ : TopologicalSpace R", "inst✝⁴ : TopologicalSemiring R", "A : Type u_2", "inst✝³ : Semiring A", "inst✝² : TopologicalSpace A", "inst✝¹ : TopologicalSemiring A", "inst✝ : Algebra R A", "f : A →A[R] A"], "goal": "↑f = AlgHom.id R A ↔ f = ContinuousAlgHom.id R A"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "(j : α) → Decidable (j ∈ s)", "f f' g : α → β", "t t' : Set α", "h : Set.EqOn f g (t ∩ s)", "h' : Set.EqOn f' g (t' ∩ sᶜ)"], "goal": "Set.EqOn (s.piecewise f f') g (s.ite t t')"}}
+{"state": {"context": ["J : Type v'", "CategoryTheory.Category.{u', v'} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "F G : CategoryTheory.Functor J C", "α : F ⟶ G", "CategoryTheory.IsIso α", "c : CategoryTheory.Limits.Cocone G", "hc : CategoryTheory.IsUniversalColimit c"], "goal": "CategoryTheory.IsUniversalColimit ((CategoryTheory.Limits.Cocones.precompose α).obj c)"}}
+{"state": {"context": ["A B : TopCat"], "goal": "∀ (X Y : ↑(π.obj (TopCat.of (↑A × ↑B)))) (f : X ⟶ Y), HEq (((projLeft A B).prod' (projRight A B) ⋙ prodToProdTop A B).map f) ((𝟭 ↑(π.obj (TopCat.of (↑A × ↑B)))).map f)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid N", "inst✝¹ : Module R M", "inst✝ : Module R N", "M₁ M₂ : Submodule R M", "N₁ N₂ : Submodule R N", "x : M ⊗[R] N", "S : M →₀ N", "h : x = S.sum fun m n => m ⊗ₜ[R] n"], "goal": "∑ a ∈ S.support, a ⊗ₜ[R] S a = ∑ i ∈ S.graph, i.1 ⊗ₜ[R] i.2"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "⊥ = ⟨a, ⋯⟩"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕝 : Type u_3", "𝕝₂ : Type u_4", "E : Type u_5", "F : Type u_6", "G : Type u_7", "ι : Type u_8", "ι' : Type u_9", "inst✝¹⁰ : NormedField 𝕜", "inst✝⁹ : AddCommGroup E", "inst✝⁸ : Module 𝕜 E", "inst✝⁷ : NormedField 𝕜₂", "inst✝⁶ : AddCommGroup F", "inst✝⁵ : Module 𝕜₂ F", "σ₁₂ : 𝕜 →+* 𝕜₂", "inst✝⁴ : RingHomIsometric σ₁₂", "inst✝³ : TopologicalSpace F", "κ : ι → Type u_10", "inst✝² : Nonempty ((i : ι) × κ i)", "inst✝¹ : ∀ (i : ι), Nonempty (κ i)", "p : (i : ι) → SeminormFamily 𝕜 E (κ i)", "t : ι → TopologicalSpace E", "inst✝ : ∀ (i : ι), TopologicalAddGroup E", "hp : ∀ (i : ι), WithSeminorms (p i)"], "goal": "WithSeminorms (SeminormFamily.sigma p)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : LinearOrderedAddCommGroup α", "a b c d : α", "h : a ≤ b", "δ : α", "inst✝ : Archimedean α", "hδ : 0 < δ"], "goal": "∃ m, ∀ n ≥ m, ↑(addNSMul h δ n) = b"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x : E"], "goal": "∀ (v : E), ⟪(-o).rightAngleRotation x, v⟫_ℝ = ⟪-o.rightAngleRotation x, v⟫_ℝ"}}
+{"state": {"context": ["R : Type u_1", "m : Type u_2", "m₁ : Type u_3", "m₂ : Type u_4", "n : Type u_5", "n₁ : Type u_6", "n₂ : Type u_7", "inst✝¹¹ : Fintype m", "inst✝¹⁰ : Fintype m₁", "inst✝⁹ : Fintype m₂", "inst✝⁸ : Fintype n", "inst✝⁷ : Fintype n₁", "inst✝⁶ : Fintype n₂", "inst✝⁵ : DecidableEq m", "inst✝⁴ : DecidableEq m₁", "inst✝³ : DecidableEq m₂", "inst✝² : DecidableEq n", "inst✝¹ : DecidableEq n₁", "inst✝ : DecidableEq n₂", "x1 x2 : Matrix m₁ n R", "y1 y2 : Matrix m₂ n R"], "goal": "x1.fromRows y1 = x2.fromRows y2 → x1 = x2 ∧ y1 = y2"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "p k : ℕ", "hn : Fact (Nat.Prime p)"], "goal": "eval 1 (cyclotomic (p ^ (k + 1)) R) = ↑p"}}
+{"state": {"context": ["ι : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "Z : Type u_5", "inst✝² : TopologicalSpace B", "inst✝¹ : TopologicalSpace F", "proj : Z → B", "e✝ : Pretrivialization F proj", "x : Z", "e' : Pretrivialization F TotalSpace.proj", "x' : TotalSpace F E", "b✝ : B", "y✝ : E b✝", "inst✝ : (x : B) → Zero (E x)", "e : Pretrivialization F TotalSpace.proj", "b : B", "hb : b ∈ e.baseSet", "y : F"], "goal": "{ proj := b, snd := e.symm b y } = ↑e.symm (b, y)"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : ProbabilityTheory.Kernel α γ", "ProbabilityTheory.IsFiniteKernel κ", "ProbabilityTheory.IsFiniteKernel η", "a : α"], "goal": "(η.withDensity (κ.rnDeriv η)) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a)ᶜ = 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "a : R"], "goal": "(if a = 0 then ⊥ else sum 0 fun x e => e) = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "CategoryTheory.MonoidalCategory E", "F : CategoryTheory.LaxMonoidalFunctor C D", "G : CategoryTheory.LaxMonoidalFunctor C E"], "goal": "(F.prod' G).ε = (F.ε, G.ε)"}}
+{"state": {"context": ["α : Type u_1", "f g h✝ : Perm α", "h : f.Disjoint g", "x : α"], "goal": "f x = x ∨ g x = x"}}
+{"state": {"context": [], "goal": "@zipWithLeft = @zipWithLeftTR"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "R : CommRingCatᵒᵖ"], "goal": "AlgebraicGeometry.ΓSpec.adjunction.homEquiv X R =\n  AlgebraicGeometry.ΓSpec.locallyRingedSpaceAdjunction.homEquiv X.toLocallyRingedSpace R"}}
+{"state": {"context": ["G : Type u_1", "Group G", "a : G", "m n : ℤ"], "goal": "Commute (a ^ m) (a ^ n)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "e : E →L[𝕜] F"], "goal": "Differentiable 𝕜 ⇑e"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁴ : Category.{u_2, u_1} C", "inst✝³ : HasZeroMorphisms C", "inst✝² : HasFiniteCoproducts C", "inst✝¹ : HasCokernels C", "inst✝ : NormalEpiCategory C", "X Y : C", "f : X ⟶ Y", "Z : C", "l : IsLimit (KernelFork.ofι 0 ⋯)", "Z✝ : C", "u v : Z✝ ⟶ X", "huv : u ≫ f = v ≫ f", "W : C", "w : W ⟶ X", "hw : w ≫ coequalizer.π u v = 0", "hl : IsColimit (CokernelCofork.ofπ (coequalizer.π u v) hw)", "m : coequalizer u v ⟶ Y", "hm : coequalizer.π u v ≫ m = f", "reassoced : ∀ {W_1 : C} (h : coequalizer u v ⟶ W_1), w ≫ coequalizer.π u v ≫ h = 0 ≫ h", "hwf : w ≫ f = 0", "n : W ⟶ (KernelFork.ofι 0 ⋯).pt", "hn : 0 = w", "this : IsIso (coequalizer.π u v)"], "goal": "u ≫ coequalizer.π u v = v ≫ coequalizer.π u v"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "x : E", "p : ℝ", "hf : HasStrictFDerivAt f f' x", "h : f x ≠ 0 ∨ 1 ≤ p"], "goal": "HasStrictFDerivAt (fun x => f x ^ p) ((p * f x ^ (p - 1)) • f') x"}}
+{"state": {"context": ["n : ℕ", "NeZero n", "a : ZMod n"], "goal": "↑a.valMinAbs.natAbs = if a.val ≤ n / 2 then a else -a"}}
+{"state": {"context": ["α : Type u", "xs ys : List (List α)", "a : α", "hxs : xs ≠ []", "hys : ys ≠ []"], "goal": "(List.zipWith (fun x x_1 => x ++ x_1) (List.modifyHead (List.cons a) xs) ys).join =\n  a :: (List.zipWith (fun x x_1 => x ++ x_1) xs ys).join"}}
+{"state": {"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "x : ModP K v O hv p"], "goal": "v ↑p < preVal K v O hv p x ↔ x ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "ConditionallyCompleteLinearOrder α", "f : ι → α"], "goal": "sInf (⋃ i, Set.Ici (f i)) = ⨅ i, f i"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝³ : SeminormedGroup E", "inst✝² : SeminormedGroup F", "inst✝¹ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "inst✝ : FunLike 𝓕 E F", "x : E"], "goal": "x ∈ closure {1} ↔ ‖x‖ = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A B : C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.Exponentiable A"], "goal": "CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.Limits.prod.map (CategoryTheory.CategoryStruct.id A) ((CategoryTheory.exp.coev A).app B))\n    ((CategoryTheory.exp.ev A).app (A ⨯ B)) =\n  CategoryTheory.CategoryStruct.id (A ⨯ B)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type w", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "𝕜' : Type u_1", "NontriviallyNormedField 𝕜'", "NormedAlgebra 𝕜 𝕜'", "h₂ : 𝕜' → 𝕜'", "h₂' y : 𝕜'", "f : E → 𝕜'", "f' : E →L[𝕜] 𝕜'", "s : Set E", "t : Set 𝕜'", "x : E", "hh : HasDerivWithinAt h₂ h₂' t y", "hf : HasFDerivWithinAt f f' s x", "hst : Set.MapsTo f s t", "hy : y = f x"], "goal": "HasFDerivWithinAt (h₂ ∘ f) (h₂' • f') s x"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y : C", "f : X ⟶ Y", "g : (CokernelCofork.ofπ σ ⋯).pt ⟶ Y", "hg : Cofork.π (CokernelCofork.ofπ σ ⋯) ≫ g = prod.map f f ≫ σ"], "goal": "f = g"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁹ : LinearOrder ι", "inst✝⁸ : MeasurableSpace ι", "inst✝⁷ : TopologicalSpace ι", "inst✝⁶ : OrderTopology ι", "inst✝⁵ : SecondCountableTopology ι", "inst✝⁴ : BorelSpace ι", "inst✝³ : TopologicalSpace β", "u : ι → Ω → β", "τ : Ω → ι", "f : Filtration ι m", "inst✝² : MetrizableSpace β", "inst✝¹ : MeasurableSpace β", "inst✝ : BorelSpace β", "hf_prog : ProgMeasurable f u", "hτ : IsStoppingTime f τ", "h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i)"], "goal": "Measurable (stoppedValue u τ)"}}
+{"state": {"context": ["x nx z : ℕ"], "goal": "Mathlib.Meta.NormNum.IsNat x nx → nx.sqrt = z → Mathlib.Meta.NormNum.IsNat x.sqrt z"}}
+{"state": {"context": ["z : ℂ", "n : ℕ", "hn : 2 ≤ n", "hz : z ≠ 0"], "goal": "∫ (x : ℝ) in 0 ..π / 2, Complex.cos (2 * z * ↑x) * ↑(Real.cos x) ^ n =\n  ↑n / (2 * z) * ∫ (x : ℝ) in 0 ..π / 2, Complex.sin (2 * z * ↑x) * ↑(Real.sin x) * ↑(Real.cos x) ^ (n - 1)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "W : CategoryTheory.MorphismProperty C", "L₁ L₂ : CategoryTheory.Functor C D", "e : L₁ ≅ L₂", "L₁.IsLocalization W"], "goal": "L₂.IsLocalization W"}}
+{"state": {"context": ["G : Type u_1", "Group G", "x : G", "n : ℤ"], "goal": "x ^ n = 1 ↔ n ≡ 0 [ZMOD ↑(orderOf x)]"}}
+{"state": {"context": ["α β : Type u_1", "s : Set α", "f : α → Set β", "h : s.Finite", "hf : ∀ a ∈ s, (f a).Finite"], "goal": "(s >>= f).Finite"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : AddCommGroup G", "inst✝ : Fintype G", "A B : Finset (G × G)", "a b c d x y : G", "n : ℕ", "ε : ℝ"], "goal": "(∃ a_1 b_1, (a_1, b_1) ∈ A ∧ a = a_1 ∧ b = b_1 ∧ c = a_1 + b_1) ↔ (a, b) ∈ A ∧ c = a + b"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a✝ a₁ a₂ b✝ b₁ b₂ : E", "r r₁ r₂ : ℝ", "a b c : E"], "goal": "‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖"}}
+{"state": {"context": ["R : Type u_1", "AddMonoidWithOne R", "p : ℕ", "CharP R p", "a b : ℕ", "h : a ≡ b [MOD p]"], "goal": "↑a = ↑b"}}
+{"state": {"context": ["E : Type u_2", "AddCommGroup E", "F : Type u_3", "AddCommGroup F", "K : Type u_5", "DivisionRing K", "Module K E", "Module K F", "f : E →ₗ.[K] F", "x : E", "y : F", "hx : x ∉ f.domain"], "goal": "(f.supSpanSingleton x y hx).domain = f.domain ⊔ Submodule.span K {x}"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsArtinianRing R", "Jac : Ideal R := ⊥.jacobson", "f : ℕ →o (Ideal R)ᵒᵈ := { toFun := fun n => Jac ^ n, monotone' := ⋯ }", "n : ℕ", "hn : ∀ (m : ℕ), n ≤ m → Jac ^ n = Jac ^ m", "J : Ideal R := annihilator (Jac ^ n)", "hJ : J ≠ ⊤", "J' : Submodule R R", "hJJ' : J < J'", "hJ' : ∀ (I : Ideal R), J < I → ¬I < J'"], "goal": "False"}}
+{"state": {"context": ["ι : Type u", "X : Type v", "inst✝⁶ : TopologicalSpace X", "E : Type u_1", "inst✝⁵ : AddCommMonoid E", "inst✝⁴ : SMulWithZero ℝ E", "inst✝³ : TopologicalSpace E", "inst✝² : ContinuousSMul ℝ E", "s✝ : Set X", "f : PartitionOfUnity ι X s✝", "s : Set X", "ρ : PartitionOfUnity ι X s", "x₀✝ x₀ : X", "M : Type u_2", "inst✝¹ : AddCommGroup M", "inst✝ : Module ℝ M", "φ : ι → X → M"], "goal": "∑ i �� ρ.finsupport x₀, (ρ i) x₀ • φ i x₀ = ∑ᶠ (i : ι), (ρ i) x₀ • φ i x₀"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_2", "Group M", "MulAction M α", "ι : Sort u_3", "s : ι → Set α"], "goal": "fixingSubgroup M (⋃ i, s i) = ⨅ i, fixingSubgroup M (s i)"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "s : Set G", "t : Set P", "a : P", "SMul G P", "IsCancelSMul G P", "x y : ↑(s.smulAntidiagonal t a)", "h : (↑x).1 = (↑y).1"], "goal": "x = y"}}
+{"state": {"context": ["x y : SimplexCategory", "f : x ⟶ y", "hyp_f_mono : Mono f"], "goal": "x.len ≤ y.len"}}
+{"state": {"context": ["p n k : ℕ", "hp : Nat.Prime p", "hkn : k ≤ p ^ n", "hk0 : k ≠ 0"], "goal": "multiplicity p ((p ^ n).choose k) = ↑(n - (multiplicity p k).get ⋯)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t✝ : Set X", "l : Filter X", "inst✝ : CountableInterFilter l", "hs : IsLindelof s", "H : ∀ x ∈ s, Disjoint (𝓝 x) l", "U : X → Set X", "hUl : ∀ x ∈ s, (U x)ᶜ ∈ l", "hxU : ∀ x ∈ s, x ∈ U x", "hUo : ∀ x ∈ s, IsOpen (U x)", "t : Set X", "htc : t.Countable", "hts : ∀ x ∈ t, x ∈ s", "hst : s ⊆ ⋃ x ∈ t, U x"], "goal": "⋂ i ∈ t, (U i)ᶜ ∈ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι✝ : Type u_5", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "ι : Type u_8", "inst✝⁴ : LinearOrder ι", "inst✝³ : TopologicalSpace ι", "inst✝² : OrderTopology ι", "inst✝¹ : DenselyOrdered ι", "inst✝ : FirstCountableTopology ι", "s : ι → Set α", "a : ι", "hs : ∀ r > a, MeasurableSet (s r)", "hm : ∀ (i j : ι), a < i → i ≤ j → s i ⊆ s j", "hf : ∃ r > a, μ (s r) ≠ ⊤", "L : ℝ≥0∞", "hL : L > μ (⋂ r, ⋂ (_ : r > a), s r)", "u : ℕ → ι", "u_anti : StrictAnti u", "u_pos : ∀ (n : ℕ), a < u n", "u_lim : Tendsto u atTop (𝓝 a)", "A : Tendsto (⇑μ ∘ s ∘ u) atTop (𝓝 (μ (⋂ r, ⋂ (_ : r > a), s r)))", "B : ⋂ n, s (u n) = ⋂ r, ⋂ (_ : r > a), s r", "n : ℕ", "hn : μ (s (u n)) < L"], "goal": "∀ᶠ (b : ι) in 𝓝[>] a, (⇑μ ∘ s) b < L"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "c : Cardinal.{u}"], "goal": "lift.{max (u + 1) v, u} c < univ.{u, v}"}}
+{"state": {"context": ["V : Type u_1", "α : Type u_2", "A : Matrix V V α", "MulZeroOneClass α", "Nontrivial α", "h : A.IsAdjMatrix", "i j : V"], "goal": "¬A i j = 1 ↔ A i j = 0"}}
+{"state": {"context": ["r : ℝ", "hr : 0 < r"], "goal": "Filter.Tendsto (fun x => Real.log x * x ^ r) (𝓝[>] 0) (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : α → β", "hf : Memℒp f 1 μ", "hmeas✝ : StronglyMeasurable f", "ε : ℝ", "hε : 0 < ε", "htendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M => {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) atTop (𝓝 0)", "hmeas : ∀ (M : ℕ), AEStronglyMeasurable ({x | ↑M ≤ ↑‖f x‖₊}.indicator f) μ", "hbound : HasFiniteIntegral (fun x => ‖f x‖) μ", "this : ∀ ε > 0, ∃ N, ∀ n ≥ N, ∫⁻ (a : α), ENNReal.ofReal ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a - 0‖ ∂μ ≤ ε", "M : ℕ", "hM : ∀ n ≥ M, ∫⁻ (a : α), ENNReal.ofReal ‖{x | ↑n ≤ ‖f x‖₊}.indicator f a‖ ∂μ ≤ ENNReal.ofReal ε"], "goal": "∫⁻ (x : α), ↑‖{x | ↑M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε"}}
+{"state": {"context": ["S : Type u_2", "R : Type u", "M : Type v", "Monoid S", "AddMonoid M", "SMul S R", "DistribMulAction S M", "s : S", "r : R"], "goal": "TrivSqZeroExt.inl (s •> r) = s •> TrivSqZeroExt.inl r"}}
+{"state": {"context": ["α : Type u_1", "dec : DecidableEq α", "a : α"], "goal": "RegularExpression.deriv 0 a = 0"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "ι₁ : Type u_4", "ι₂ : Type u_5", "CommRing R", "AddCommGroup M₁", "AddCommGroup M₂", "Module R M₁", "Module R M₂", "Fintype ι₁", "Finite ι₂", "DecidableEq ι₁", "b₁ : Basis ι₁ R M₁", "b₂ : Basis ι₂ R M₂", "f : M₁ →ₗ[R] M₂", "i : ι₂"], "goal": "(LinearMap.toMvPolynomial b₁ b₂ f i).IsHomogeneous 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s : α → Set β", "t : α → Set γ", "inst✝ : Preorder α", "x : α"], "goal": "⋃ y, ⋃ (_ : y ≤ x), s y ⊆ ⋃ y, ⋃ (_ : y ≤ x), Accumulate s y"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "ι : Type u_4", "V₂ : Type u_5", "P₂ : Type u_6", "AddCommGroup V₂", "Module k V₂", "AffineSpace V₂ P₂", "p : ι → P", "f : P →ᵃ[k] P₂", "hf : Function.Injective ⇑f"], "goal": "AffineIndependent k (⇑f ∘ p) ↔ AffineIndependent k p"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : CommMonoid α", "inst✝ : CommMonoid β", "s✝ t : Multiset α", "a : α", "m : Multiset ι", "f g : ι → α", "s : Multiset α"], "goal": "foldl (fun x y => y * x) ⋯ 1 s = foldl (fun x x_1 => x * x_1) ⋯ 1 s"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "β : Type u_2", "γ : Type u_3", "a : α", "v : Vector α (n + 1)", "i : ℕ", "hi : i < n + 1", "j : ℕ", "hj : j < n + 2"], "goal": "(List.insertNth j a ↑v).eraseIdx ↑(if i < j then ⟨i, ⋯⟩ else ⟨i + 1, ⋯⟩) = List.insertNth (↑(if h : i < j then ⟨j, hj⟩.pred ⋯ else ⟨j, hj⟩.castPred ⋯)) a ↑(match v with | ⟨l, p⟩ => ⟨l.eraseIdx i, ⋯⟩)"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalSemiring A", "StarRing A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A"], "goal": "(NonUnitalStarSubalgebra.center R A).toNonUnitalSubalgebra = NonUnitalSubalgebra.center R A"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "inst✝ : SigmaFinite μ", "f : α → ℝ≥0∞", "hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤", "this : f =ᶠ[ae μ] fun x => ↑(f x).toNNReal"], "goal": "SigmaFinite (μ.withDensity fun x => ↑(f x).toNNReal)"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "p : α → Prop", "DecidablePred p", "x₀ : { a // p a } → β", "x : { a // ¬p a } → β", "a : α", "h : p a"], "goal": "↑((Equiv.subtypePreimage p x₀).symm x) a = x₀ ⟨a, h⟩"}}
+{"state": {"context": ["k : Type u_1", "E : Type u_2", "Field k", "AddCommGroup E", "Module k E", "f : k → E", "a b : k", "h : a ≠ b"], "goal": "slope (fun x => (x - a) • f x) a b = f b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "DecidableEq γ", "f : α → β → γ", "s : Finset α", "t : Finset β"], "goal": "Finset.image₂ f s t = ∅ ↔ s = ∅ ∨ t = ∅"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝¹ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "inst✝ : IsFiniteKernel ν", "a : α", "s : Set β", "hs : MeasurableSet s", "A : Set γ", "hA : MeasurableSet A", "this : IsFiniteKernel κ"], "goal": "0 ≤ᶠ[ae ((ν a).restrict A)] fun x => κ.density ν a x s"}}
+{"state": {"context": ["R₁ : Type u_3", "CommRing R₁", "K : Type u_4", "Field K", "Algebra R₁ K", "IsFractionRing R₁ K", "ι : Type u_5", "s : Finset ι", "f : ι → K"], "goal": "↑(FractionalIdeal.spanFinset R₁ s f) = Submodule.span R₁ (f '' ↑s)"}}
+{"state": {"context": ["m k n : Nat", "h₁ : m + k ≤ n", "h₂ : 0 < k"], "goal": "n - (m + k) < n - m"}}
+{"state": {"context": ["M : Type u_1", "MeasurableSpace M", "Sup M", "self : MeasurableSup M", "c : M"], "goal": "Measurable fun x => c ⊔ x"}}
+{"state": {"context": ["α : Type u", "l : List α", "r : α → α → Prop", "hl : l.Nodup", "h : ∀ a ∈ l, ∀ b ∈ l, a ≠ b → r a b"], "goal": "List.Pairwise r l"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ"], "goal": "HurwitzZeta.sinZeta a s = (HurwitzZeta.expZeta a s - HurwitzZeta.expZeta (-a) s) / (2 * Complex.I)"}}
+{"state": {"context": ["f : CircleDeg1Lift", "n : ℕ"], "goal": "Function.Commute ⇑f fun x => x + ↑n"}}
+{"state": {"context": ["x : ℝ*", "hi : x.Infinitesimal", "r : ℝ", "hr : r < 0"], "goal": "↑r < x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁶ : MeasurableSpace δ", "inst✝⁵ : NormedAddCommGroup β", "inst✝⁴ : NormedAddCommGroup γ", "𝕜 : Type u_5", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : CompleteSpace 𝕜", "E : Type u_6", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f : α → 𝕜", "c : E", "hc : c ≠ 0", "a✝ : AEStronglyMeasurable f μ"], "goal": "∫⁻ (a : α), ↑‖f a‖₊ * ↑‖c‖₊ ∂μ < ⊤ ↔ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y Z : CategoryTheory.Center C", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "(f ≫ g).f = f.f ≫ g.f"}}
+{"state": {"context": ["α : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s t : Set α", "h : μ t = 0"], "goal": "MeasureTheory.AEDisjoint μ s t"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "m0 : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "ℱ : MeasureTheory.Filtration ℕ m0", "MeasureTheory.SigmaFiniteFiltration μ ℱ", "f g : ℕ → Ω → E", "hf : MeasureTheory.Martingale f ℱ μ", "hg : MeasureTheory.Adapted ℱ fun n => g (n + 1)", "hg0 : g 0 = 0", "hgint : ∀ (n : ℕ), MeasureTheory.Integrable (g n) μ", "n : ℕ"], "goal": "MeasureTheory.martingalePart (f + g) ℱ μ n =ᵐ[μ] f n"}}
+{"state": {"context": ["R : Type u", "Ring R", "r : ↥⊤"], "goal": "Subring.topEquiv r = ↑r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_6", "ConditionallyCompleteLattice β", "ConditionallyCompleteLattice γ", "f : Filter α", "u : α → β", "g : β ≃o γ", "hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault", "hu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f u := by isBoundedDefault", "hgu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f fun x => g (u x) := by isBoundedDefault", "hgu_co : Filter.IsCoboundedUnder (fun x x_1 => x ≥ x_1) f fun x => g (u x) := by isBoundedDefault"], "goal": "g (Filter.liminf u f) = Filter.liminf (fun x => g (u x)) f"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁸ : MeasurableSpace X", "inst✝⁷ : TopologicalSpace X", "inst✝⁶ : OpensMeasurableSpace X", "μ : Measure X", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : SecondCountableTopology E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : IsProbabilityMeasure μ", "f : X →ᵇ E"], "goal": "‖f‖ = (μ Set.univ).toReal * ‖f‖"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "p : ℝ≥0∞", "q : ℝ", "μ : Measure α", "f g : α → E"], "goal": "0 ∉ Set.Ioo 0 1"}}
+{"state": {"context": ["R : Type u", "Semiring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "AddCommMonoid M", "Module R M", "A : ι → Submodule R M", "h : DirectSum.IsInternal A", "i j : ι", "hij : i ≠ j", "x : M", "hx : x ∈ A i"], "goal": "((LinearEquiv.ofBijective (DirectSum.coeLinearMap A) h).symm x) j = 0"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "r : β → β → Prop", "f : α → β", "h : Transitive r"], "goal": "Transitive (InvImage r f)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "CompleteSpace E", "U : Submodule 𝕜 E", "CompleteSpace ↥U"], "goal": "IsSelfAdjoint (U.subtypeL.comp (orthogonalProjection U))"}}
+{"state": {"context": ["R : Type u", "M : Type v", "N : Type w", "inst✝⁴ : Ring R", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : M →ₗ[R] N", "hf : Injective ⇑f"], "goal": "finrank R ↥(range f) = finrank R M"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X Y : C", "f : X ⟶ Y"], "goal": "CategoryTheory.CategoryStruct.comp f (CategoryTheory.MonoidalCategory.leftUnitor Y).inv =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor X).inv\n    (CategoryTheory.MonoidalCategory.whiskerLeft (𝟙_ C) f)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "CompleteSpace E", "f : E → F", "s : Set E", "c : ℝ≥0", "f' : E →L[𝕜] F", "hf : ApproximatesLinearOn f f' s c", "f'symm : f'.NonlinearRightInverse", "hs : IsOpen s", "hc : Subsingleton F ∨ c < f'symm.nnnorm⁻¹"], "goal": "IsOpen (f '' s)"}}
+{"state": {"context": ["x : ℂ", "y : ℝ"], "goal": "Complex.abs (x ^ ↑y) = Complex.abs x ^ y"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "t : TopologicalSpace α", "B : Set (Set α)", "s : Set α", "inst✝¹ : SeparableSpace α", "inst✝ : Nonempty α"], "goal": "∃ u, DenseRange u"}}
+{"state": {"context": ["A : Type u_2", "K : Type u_3", "CommRing A", "Field K", "IsDomain A", "Algebra A K", "IsFractionRing A K", "h : IsDedekindDomainInv A", "I : FractionalIdeal A⁰ K", "hI : I ≠ 0"], "goal": "I⁻¹ * I = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LE α", "S : Set (UpperSet α)", "s t : UpperSet α", "a : α"], "goal": "Codisjoint ↑s ↑t ↔ Disjoint s t"}}
+{"state": {"context": ["ι : Type u_1", "a s : ι → ℕ", "l l' : List ι", "hl : l.Perm l'", "hs : ∀ i ∈ l, s i ≠ 0", "co : List.Pairwise (Coprime on s) l", "z : { k // ∀ i ∈ l', k ≡ a i [MOD s i] } := chineseRemainderOfList a s l' ⋯"], "goal": "↑(chineseRemainderOfList a s l co) = ↑(chineseRemainderOfList a s l' ⋯)"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "x : Fix F α"], "goal": "(mk ∘ dest) x = x"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Ring R", "p : ℕ", "inst✝¹ : Fact (Nat.Prime p)", "inst✝ : CharP R p", "x y : R", "h : Commute x y"], "goal": "(x - y + y) ^ p = x ^ p"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y : C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "b : CategoryTheory.Limits.BinaryBicone X Y", "h : b.IsBilimit"], "goal": "CategoryTheory.IsPullback b.inl 0 b.snd 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α", "M : Type u_3", "inst✝² : AddCommMonoid M", "inst✝¹ : TopologicalSpace M", "inst✝ : OrderedAddCommMonoid M", "s : SignedMeasure α", "i j : Set α", "hi : ↑s i < 0", "hi₁ : MeasurableSet i", "h : ¬restrict s i ≤ restrict 0 i", "hn : ∀ (n : ℕ), ¬restrict s (i \\ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \\ ⋃ l, ⋃ (_ : l < n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "A : Set α := i \\ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "hA : A = i \\ ⋃ l, MeasureTheory.SignedMeasure.restrictNonposSeq s i l", "bdd : ℕ → ℕ := fun n => MeasureTheory.SignedMeasure.findExistsOneDivLT s (i \\ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)", "hn' : ∀ (n : ℕ), ¬restrict s (i \\ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l) ≤ restrict 0 (i \\ ⋃ l, ⋃ (_ : l ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "h₁ : ↑s i = ↑s A + ∑' (l : ℕ), ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i l)", "h₂ : ↑s A ≤ ↑s i", "h₃' : Summable fun n => 1 / (↑(bdd n) + 1)", "h₃ : Tendsto (fun n => ↑(bdd n) + 1) atTop atTop", "h₄ : Tendsto (fun n => ↑(bdd n)) atTop atTop", "A_meas : MeasurableSet A"], "goal": "∃ j, MeasurableSet j ∧ j ⊆ i ∧ restrict s j ≤ restrict 0 j ∧ ↑s j < 0"}}
+{"state": {"context": ["R S : Type u_1", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "f : R →+* S", "s : Set R", "hs : Ideal.span s = ⊤", "H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)", "this : Algebra R S := f.toAlgebra", "x : S", "l : ↑s →₀ R", "hl : (Finsupp.total (↑s) R R Subtype.val) l = 1"], "goal": "∀ (i : ↑s), ∃ n, f ↑i ^ n • x ∈ (Algebra.ofId R S).range"}}
+{"state": {"context": ["X Y : TopCat", "f : ↑X ≃ₜ ↑Y", "x✝ : (forget TopCat).obj X"], "goal": "(f.toContinuousMap ≫ f.symm.toContinuousMap) x✝ = (𝟙 X) x✝"}}
+{"state": {"context": ["α : Type u", "Preorder α", "NoMinOrder α"], "goal": "lowerBounds Set.univ = ∅"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "a b : A", "n : ℕ"], "goal": "∑ x ∈ Finset.range n.succ, a ^ x * (algebraMap ℚ A) (1 / ↑x !) * (b ^ (n - x) * (algebraMap ℚ A) (1 / ↑(n - x)!)) = ∑ x ∈ Finset.range (n + 1), a ^ x * b ^ (n - x) * ↑(n.choose x) * (algebraMap ℚ A) (1 / ↑n !)"}}
+{"state": {"context": ["m : Type u_1", "DecidableEq m", "Fintype m", "R : Type v", "CommRing R", "StarRing R", "M : Matrix m m R"], "goal": "M.conjTranspose.det = star M.det"}}
+{"state": {"context": ["R : Type u", "I : Type v", "CommSemiring R", "x : R", "s : I → R", "t : Finset I"], "goal": "IsCoprime x (∏ i ∈ t, s i) ↔ ∀ i ∈ t, IsCoprime x (s i)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R r : α → α → Prop", "l l₁✝ l₂✝ : List α", "a b : α", "l₁ l₂ : List α", "L : List (List α)", "hL : ¬[] ∈ l₁ :: l₂ :: L"], "goal": "Chain' R (l₁ :: l₂ :: L).join ↔ (∀ (l : List α), l ∈ l₁ :: l₂ :: L → Chain' R l) ∧ Chain' (fun l₁ l₂ => ∀ (x : α), x ∈ l₁.getLast? → ∀ (y : α), y ∈ l₂.head? → R x y) (l₁ :: l₂ :: L)"}}
+{"state": {"context": ["α : Type u_1", "CompleteSemilatticeSup α", "s t : Set α", "h : ∀ x ∈ s, ∃ y ∈ t, x ≤ y"], "goal": "sSup s ≤ sSup t"}}
+{"state": {"context": ["R : Type u", "S : Type v", "K : Type w", "inst✝² : CommRing R", "f : R[X]", "inst✝¹ : CommRing S", "i : R →+* S", "a : S", "h : eval₂ i a f = 0", "inst✝ : Algebra R S", "ϕ : AdjoinRoot f →ₐ[R] S"], "goal": "(aeval (ϕ (root f))) f = 0"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "MulOneClass M", "MulOneClass N", "Monoid P", "f : M × N →* P"], "goal": "(f.comp (MonoidHom.inl M N)).noncommCoprod (f.comp (MonoidHom.inr M N)) ⋯ = f"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹² : TopologicalSpace α", "R : Type u_2", "inst✝¹¹ : CommSemiring R", "A : Type u_3", "inst✝¹⁰ : TopologicalSpace A", "inst✝⁹ : Semiring A", "inst✝⁸ : Algebra R A", "inst✝⁷ : TopologicalSemiring A", "A₂ : Type u_4", "inst✝⁶ : TopologicalSpace A₂", "inst✝⁵ : Semiring A₂", "inst✝⁴ : Algebra R A₂", "inst✝³ : TopologicalSemiring A₂", "𝕜 : Type u_5", "inst✝² : TopologicalSpace 𝕜", "s✝ : Set C(α, 𝕜)", "f✝ : ↑s✝", "x✝ : α", "inst✝¹ : Field 𝕜", "inst✝ : TopologicalRing 𝕜", "s : Subalgebra 𝕜 C(α, 𝕜)", "h : s.SeparatesPoints", "v : α → 𝕜", "x y : α", "n : ¬x = y", "f : C(α, 𝕜)", "hf : f ∈ ↑s", "hxy : f x - f y ≠ 0"], "goal": "∃ f ∈ ↑s, f x = v x ∧ f y = v y"}}
+{"state": {"context": ["M : Type u_1", "inst✝¹ : OrderedCancelAddCommMonoid M", "inst✝ : ExistsAddOfLE M", "a b c d : M"], "goal": "(fun x => a + x) '' Ico b c = Ico (a + b) (a + c)"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "InnerProductSpaceable E", "y z : E"], "goal": "‖RCLike.I • y - 2 • z‖ * ‖RCLike.I • y - 2 • z‖ =\n  2 * (‖RCLike.I • y - z‖ * ‖RCLike.I • y - z‖ + ‖z‖ * ‖z‖) - ‖RCLike.I • y‖ * ‖RCLike.I • y‖"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "Field K", "CharZero K", "hp : Fact (Nat.Prime ↑p)", "IsCyclotomicExtension {p ^ (k + 1)} ℚ K"], "goal": "NumberField.discr K = (-1) ^ (↑p ^ k * (↑p - 1) / 2) * ↑↑p ^ (↑p ^ k * ((↑p - 1) * (k + 1) - 1))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : Module 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "f : QuadraticMap 𝕜 E F", "a b : E"], "goal": "HasLineDerivAt 𝕜 (⇑f) (polar (⇑f) a b) a b"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x y : α", "s t u : Set α", "Φ : α → β"], "goal": "hausdorffEdist s u ≤ hausdorffEdist s t + hausdorffEdist t u"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "n : ℕ", "f : Fin n → α", "g : α → β"], "goal": "List.map g (List.ofFn f) = List.ofFn (g ∘ f)"}}
+{"state": {"context": ["n : Type u_2", "R : Type u_4", "Semiring R", "Fintype n", "v : n → R"], "goal": "(∀ (w : n → R), Matrix.dotProduct v w = 0) ↔ v = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹² : RCLike 𝕜", "inst✝¹¹ : NormedAddCommGroup F", "inst✝¹⁰ : NormedSpace 𝕜 F", "inst✝⁹ : NormedAddCommGroup F'", "inst✝⁸ : NormedSpace 𝕜 F'", "inst✝⁷ : NormedSpace ℝ F'", "inst✝⁶ : CompleteSpace F'", "inst✝⁵ : NormedAddCommGroup G", "inst✝⁴ : NormedAddCommGroup G'", "inst✝³ : NormedSpace ℝ G'", "inst✝² : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "inst✝¹ : NormedSpace ℝ G", "hm : m ≤ m0", "inst✝ : SigmaFinite (μ.trim hm)", "hs : MeasurableSet s", "ht : MeasurableSet t", "hμs : μ s ≠ ⊤", "hμt : μ t ≠ ⊤", "hst : s ∩ t = ∅", "x : G"], "goal": "↑↑(condexpIndL1Fin hm ⋯ ⋯ x) =ᶠ[ae μ] ↑↑(condexpIndL1Fin hm hs hμs x + condexpIndL1Fin hm ht hμt x)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "m✝ m0✝ : MeasurableSpace α", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : CompleteSpace E", "μ✝ : Measure α", "f : α → E", "s : Set α", "m m₂ m0 : MeasurableSpace α", "μ : Measure α", "hm : m ≤ m0", "hm₂ : m₂ ≤ m0", "inst✝¹ : SigmaFinite (μ.trim hm)", "inst✝ : SigmaFinite (μ.trim hm₂)", "hs_m : MeasurableSet s", "hs : ∀ (t : Set α), MeasurableSet (s ∩ t) ↔ MeasurableSet (s ∩ t)", "hs_m₂ : MeasurableSet s", "hf_int : Integrable f μ"], "goal": "AEStronglyMeasurable' m₂ (μ[s.indicator f|m]) μ"}}
+{"state": {"context": ["m n✝ n : ℕ", "u : (ZMod (n + 1))ˣ"], "goal": "↑((↑u).val * (↑u⁻¹).val) = ↑(↑(u * u⁻¹)).val"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "p : ι → Prop", "inst✝¹ : DecidablePred p", "inst✝ : AddCommMonoid γ", "S : ι → AddSubmonoid γ"], "goal": "⨆ i, ⨆ (_ : p i), S i ≤ AddMonoidHom.mrange ((sumAddHom fun i => (S i).subtype).comp (filterAddMonoidHom (fun i => ↥(S i)) p))"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Monoid M", "Monoid N", "x : N"], "goal": "Monoid.Coprod.snd (Monoid.Coprod.inr x) = x"}}
+{"state": {"context": ["F : Type u_1", "K✝ : Type u_2", "inst✝¹³ : Field F", "inst✝¹² : Field K✝", "inst✝¹¹ : Algebra F K✝", "K₁ : Type u_3", "K₂ : Type u_4", "K₃ : Type u_5", "inst✝¹⁰ : Field F", "inst✝⁹ : Field K₁", "inst✝⁸ : Field K₂", "inst✝⁷ : Field K₃", "inst✝⁶ : Algebra F K₁", "inst✝⁵ : Algebra F K₂", "inst✝⁴ : Algebra F K₃", "ϕ : K₁ →ₐ[F] K₂", "χ : K₁ ≃ₐ[F] K₂", "ψ : K₂ →ₐ[F] K₃", "ω : K₂ ≃ₐ[F] K₃", "K : Type u_6", "L : Type u_7", "inst✝³ : Field K", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : Normal K L", "x y : L", "hy : IsAlgebraic K y", "h_ev : (Polynomial.aeval x) (minpoly K y) = 0", "hx : IsAlgebraic K x"], "goal": "∃ σ, σ x = y"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasStrictTerminalObjects C", "I : C", "hI : CategoryTheory.Limits.IsTerminal I", "A : C", "f : I ⟶ A"], "goal": "CategoryTheory.IsIso f"}}
+{"state": {"context": ["x y : SetTheory.PGame"], "goal": "x ≤ y ↔ (∀ (i : x.LeftMoves), (x.moveLeft i).LF y) ∧ ∀ (j : y.RightMoves), x.LF (y.moveRight j)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VSub α β", "DecidableEq α", "s t : Finset β", "b c : β"], "goal": "b ∈ s → c ∈ t → b -ᵥ c ∈ s -ᵥ t"}}
+{"state": {"context": ["y z : ℝ", "hyz : y < z", "x : ℝ≥0", "hx : 1 < ↑x"], "goal": "↑x ^ y < ↑x ^ z"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "F : Type u_5", "NormedDivisionRing F", "f g : α → F", "hint : MeasureTheory.Integrable g μ", "hm : MeasureTheory.AEStronglyMeasurable f μ", "hfbdd : ∃ C, ∀ (x : α), ‖f x‖ ≤ C"], "goal": "MeasureTheory.Integrable (fun x => f x * g x) μ"}}
+{"state": {"context": ["num den : Int"], "goal": "-num /. -den = num /. den"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_4", "Semiring R", "AddCommMonoid M", "Module R M", "n : ℕ", "H : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → s.card ≤ n", "s : Set M"], "goal": "LinearIndependent (ι := { x // x ∈ s }) R Subtype.val → #↑s ≤ ↑n"}}
+{"state": {"context": ["E : Type u_1", "V : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℂ E", "f : V → E", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : MeasurableSpace V", "inst✝³ : BorelSpace V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : FiniteDimensional ℝ V", "inst✝ : CompleteSpace E", "w : V", "hw : w ≠ 0", "hiw : ⟪i w, w⟫_ℝ = 1 / 2", "this✝ : (fun v => 𝐞 (-⟪v, w⟫_ℝ) • f (v + i w)) = fun v => (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w)", "this : ∫ (x : V), -(𝐞 (-⟪x + i w, w⟫_ℝ) • f (x + i w)) = ∫ (x : V), -(𝐞 (-⟪x, w⟫_ℝ) • f x)"], "goal": "∫ (v : V), (fun x => -(𝐞 (-⟪x, w⟫_ℝ) • f x)) (v + i w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o✝ : Ordinal.{u}", "ho✝ : o✝.IsLimit", "hsC✝ : contained C o✝", "o : Ordinal.{u}", "ho : o.IsLimit", "hsC : contained C o"], "goal": "LinearIndependent ℤ (eval C) ↔ LinearIndependent ℤ fun (p : ↑(⋃ e, smaller C ↑e)) => ↑p"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "Z : Type u_5", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "e : PartialHomeomorph X Y", "e' : PartialHomeomorph Y Z", "h : e.target = e'.source"], "goal": "∀ (a : Z), ↑(e.trans' e' h).symm a = ↑e.symm (↑e'.symm a)"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n m : ℕ", "ω : Ω", "ℱ : Filtration ℕ m0", "this : upcrossings a b f ω < ⊤ ↔ ∃ k, upcrossings a b f ω ≤ ↑k"], "goal": "(∃ k, ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k) ↔ ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω ≤ k"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : StrictConvexSpace ℝ E", "x : E"], "goal": "ThreeAPFree (sphere x 0)"}}
+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "inst✝ : Group G", "A : Rep k G", "f : ↥(oneCocycles A)"], "goal": "↑f 1 = 0"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α", "hs : Dense s", "x : α", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ y ∈ s, dist x y < ε"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α"], "goal": "(SimpleGraph.hasse α).Adj a b ↔ a ⋖ b ∨ b ⋖ a"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "S : Type u_2", "T : Type u_3", "hp : Fact (Nat.Prime p)", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "inst✝ : CommRing T", "α : Type u_4", "β : Type u_5", "f : α → β", "hf : Surjective f", "x : 𝕎 β", "n : ℕ"], "goal": "(mapFun f (mk p fun n => Classical.choose ⋯)).coeff n = x.coeff n"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "ζ : R", "n : ℕ", "h : IsPrimitiveRoot ζ n"], "goal": "(Polynomial.cyclotomic' n R).degree = ↑n.totient"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ inst✝² : Group α", "inst✝¹ : MulAction α β", "x : β", "G : Type u_1", "inst✝ : Group G", "H : Subgroup G", "x✝ : G"], "goal": "x✝ ∈ stabilizer G ↑1 ↔ x✝ ∈ H"}}
+{"state": {"context": ["R : Type u", "inst✝⁷ : Ring R", "Q : Type v", "inst✝⁶ : AddCommGroup Q", "inst✝⁵ : Module R Q", "M : Type u_1", "N : Type u_2", "inst✝⁴ : AddCommGroup M", "inst✝³ : AddCommGroup N", "inst✝² : Module R M", "inst✝¹ : Module R N", "i : M →ₗ[R] N", "f : M →ₗ[R] Q", "inst✝ : Fact (Function.Injective ⇑i)", "h : Baer R Q", "y : N"], "goal": "y ∈ (extensionOfMax i f).domain"}}
+{"state": {"context": ["m : ℕ+", "v : PrimeMultiset", "h : m.factorMultiset ≤ v.prod.factorMultiset ↔ m ∣ v.prod := factorMultiset_le_iff"], "goal": "m.factorMultiset �� v ↔ m ∣ v.prod"}}
+{"state": {"context": ["U : Type u_1", "Quiver U", "V : Type u_2", "Quiver V", "φ : U ⥤q V", "hφ : ∀ (u : U), Function.Bijective (φ.star u)", "u : U"], "goal": "Function.Bijective (φ.pathStar u)"}}
+{"state": {"context": ["p : ℕ → Prop", "DecidablePred p", "h : ∃ n, p n", "n : ℕ"], "goal": "n < Nat.find h ↔ ∀ (m : ℕ), m ≤ n → ¬p m"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : HasFiniteCoproducts C", "X : SimplicialObject C"], "goal": "Γ₂N₁.natTrans.app X = Γ₂N₂ToKaroubiIso.inv.app X ≫ 𝟙 (Γ₂.obj (N₂.obj { X := X, p := 𝟙 X, idem := ⋯ })) ≫ (Γ₂N₂ToKaroubiIso.hom.app X ≫ Γ₂N₁.natTrans.app X) ≫ 𝟙 { X := X, p := 𝟙 X, idem := ⋯ }"}}
+{"state": {"context": ["V : Type", "G : SimpleGraph V", "e : ↑G.end", "K : (Finset V)ᵒᵖ", "L : Finset V", "h : Opposite.unop K ⊆ Opposite.unop (Opposite.op L)"], "goal": "∃ D, ComponentCompl.hom h D = ↑e K"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "f : X → Y", "U₁ Z₁ : Set X", "hU₁ : IsOpen U₁", "hZ₁ : IsClosed Z₁", "U₂ Z₂ : Set X", "hU₂ : IsOpen U₂", "hZ₂ : IsClosed Z₂"], "goal": "IsLocallyClosed (U₁ ∩ Z₁ ∩ (U₂ ∩ Z₂))"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "r : HomRel C", "D : Type u_3", "CategoryTheory.Category.{u_4, u_3} D", "F G : CategoryTheory.Functor (CategoryTheory.Quotient r) D", "τ : (CategoryTheory.Quotient.functor r).comp F ≅ (CategoryTheory.Quotient.functor r).comp G"], "goal": "(CategoryTheory.Quotient.natIsoLift r τ).inv = CategoryTheory.Quotient.natTransLift r τ.inv"}}
+{"state": {"context": ["R : Type u_1", "Monoid R", "StarMul R", "A₀ A₁ B₀ B₁ : R", "self : IsCHSHTuple A₀ A₁ B₀ B₁"], "goal": "A₀ * B₁ = B₁ * A₀"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "hf : Differentiable ℝ f", "hx : ∀ (x : E), f x ≠ 0"], "goal": "Differentiable ℝ fun x => Real.log (f x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "n : ℕ", "h : 1 < n"], "goal": "(fun x => x ^ n) =o[𝓝 0] fun x => x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "TopologicalSpace α", "Preorder β", "Preorder γ", "f : α → β", "a : α", "hf : IsLocalMax f a", "g : β → γ", "hg : Monotone g"], "goal": "IsLocalMax (g ∘ f) a"}}
+{"state": {"context": ["α✝ : Type u_1", "α : Type u_2", "inst✝ : DecidableEq α", "l : List α", "l' : List (Multiset α)", "H : ∀ ⦃x : Multiset α × Multiset α⦄, x ∈ l'.revzip → x.1 + x.2 = ↑l"], "goal": "l'.revzip = List.map (fun x => (x, ↑l - x)) l'"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "Countable β", "f : α ↪ β"], "goal": "Countable α"}}
+{"state": {"context": ["α : Type u_1", "CompleteLattice α", "self : CompleteSublattice α", "s : Set α"], "goal": "s ⊆ self.carrier → sInf s ∈ self.carrier"}}
+{"state": {"context": ["ι✝ : Type ?u.54871", "R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁶ : LinearOrder ι✝", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : AddCommGroup N", "inst✝² : Module R M", "inst✝¹ : Module R N", "inst✝ : Module.Free R M", "Q : QuadraticMap R M N", "ι : Type u_2", "b : Basis ι R M", "this : LinearOrder ι := IsWellOrder.linearOrder WellOrderingRel"], "goal": "∃ a, a.toQuadraticMap = Q"}}
+{"state": {"context": ["α : Type u_2", "r : α → α → Prop"], "goal": "∅.WellFoundedOn r"}}
+{"state": {"context": ["n : Type u'", "α : Type v", "Fintype n", "DecidableEq n", "CommRing α", "A : (Matrix n n α)ˣ"], "goal": "↑A⁻¹ = (↑A)⁻¹"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X Y : C"], "goal": "(CategoryTheory.MonoidalCategory.rightUnitor (CategoryTheory.MonoidalCategory.tensorObj X Y)).hom =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Y (𝟙_ C)).hom\n    (CategoryTheory.MonoidalCategory.whiskerLeft X (CategoryTheory.MonoidalCategory.rightUnitor Y).hom)"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "CommRing R", "LieRing L", "LieAlgebra R L", "x : L", "D₁ D₂ : LieDerivation R L L"], "goal": "⁅x, ⁅D₁, D₂⁆⁆ = ⁅⁅x, D₁⁆, D₂⁆ + ⁅D₁, ⁅x, D₂⁆⁆"}}
+{"state": {"context": ["R : Type u", "M : Type v", "CommRing R", "AddCommGroup M", "Module R M", "Small.{v, u} R"], "goal": "Module.Flat R M ↔\n  ∀ ⦃N N' N'' : Type v⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N'']\n    [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄ ⦃g : N' →ₗ[R] N''⦄,\n    Function.Exact ⇑f ⇑g → Function.Exact ⇑(LinearMap.lTensor M f) ⇑(LinearMap.lTensor M g)"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "CategoryTheory.Category.{u_5, u_1} C₁", "CategoryTheory.Category.{u_6, u_2} C₂", "W₁ : CategoryTheory.MorphismProperty C₁", "W₂ : CategoryTheory.MorphismProperty C₂", "Φ : CategoryTheory.LocalizerMorphism W₁ W₂", "X₂ : C₂ᵒᵖ", "L : Φ.op.LeftResolution X₂"], "goal": "L.unop.w = L.w.unop"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddCommGroup E", "a b : E", "r : ℝ", "c : E"], "goal": "a + c ∈ Metric.ball (b + c) r ↔ a ∈ Metric.ball b r"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeSup α", "OrderTop α", "a b : α"], "goal": "Codisjoint a b → ⊤ ≤ a ⊔ b"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "c : F"], "goal": "(Differentiable 𝕜 fun y => c + f y) ↔ Differentiable 𝕜 f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "M : Type u_5", "M' : Type u_6", "N : Type u_7", "P : Type u_8", "G : Type u_9", "H : Type u_10", "R : Type u_11", "S : Type u_12", "inst✝² : Zero M", "inst✝¹ : Zero N", "inst✝ : Zero P", "f : M → N → P", "hf : f 0 0 = 0", "a : α", "m : M", "n : N", "a' : α"], "goal": "f ((single a m) a') ((single a n) a') = (single a (f m n)) a'"}}
+{"state": {"context": ["E : Type u_2", "Field E", "S : Set E", "F : Subfield E", "T : Set E", "HT : T ⊆ ↑F"], "goal": "T ⊆ ↑(IntermediateField.adjoin (↥F) S)"}}
+{"state": {"context": ["xl xr : Type u"], "goal": "0 ⧏ star"}}
+{"state": {"context": ["α : Type u_5", "self : ConditionallyCompleteLattice α", "s : Set α", "a : α"], "goal": "BddBelow s → a ∈ s → sInf s ≤ a"}}
+{"state": {"context": ["δ : Type u_4", "π : δ → Type u_6", "(a : δ) → MeasurableSpace (π a)", "p : δ → Prop", "DecidablePred p"], "goal": "Measurable ⇑(Equiv.piEquivPiSubtypeProd p π).symm"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝¹⁰ : Category.{?u.21810, u_1} C", "inst✝⁹ : Category.{?u.21814, u_2} D", "inst✝⁸ : Category.{?u.21818, u_3} E", "inst✝⁷ : HasShift C ℤ", "inst✝⁶ : HasShift D ℤ", "inst✝⁵ : HasShift E ℤ", "F : C ⥤ D", "inst✝⁴ : F.CommShift ℤ", "G : D ⥤ E", "inst✝³ : G.CommShift ℤ", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "inst✝ : F.Additive", "n : ℤ", "T : Triangle C"], "goal": "(F.map (n.negOnePow • (shiftFunctor C n).map T.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app T.obj₁) ≫ (F.commShiftIso 1).hom.app ((shiftFunctor C n).obj T.obj₁)) ≫ (shiftFunctor D 1).map ((F.commShiftIso n).hom.app T.obj₁) = (F.commShiftIso n).hom.app T.obj₃ ≫ (n.negOnePow • (shiftFunctor D n).map (F.map T.mor₃ ≫ (F.commShiftIso 1).hom.app T.obj₁) ≫ (shiftFunctorComm D 1 n).hom.app (F.obj T.obj₁))"}}
+{"state": {"context": ["n : ℕ"], "goal": "(n ^ 2).sqrt = n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "sq : CategoryTheory.Square C"], "goal": "(CategoryTheory.Square.toArrowArrowFunctor'.obj sq).right.hom = sq.f₃₄"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VAdd α β", "s₁ s₂ : Set α", "t₁ t₂ : Set β"], "goal": "s₁ ∩ s₂ +ᵥ (t₁ ∪ t₂) ⊆ s₁ +ᵥ t₁ ∪ (s₂ +ᵥ t₂)"}}
+{"state": {"context": ["a : ℤ"], "goal": "(-a.succ).succ = -a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y : F"], "goal": "⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = -1 ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommSemiring R", "CommSemiring S", "Algebra R S", "A B : Subalgebra R S"], "goal": "B.mulMap A = (A.mulMap B).comp ↑(Algebra.TensorProduct.comm R ↥B ↥A)"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "T1Space X", "T35Space X"], "goal": "Set.SeparatesPoints Continuous"}}
+{"state": {"context": ["L : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'", "t : L.Term α", "h : listDecode ([t].bind listEncode) = [t]"], "goal": "(fun l => (do let a ← (listDecode l).head? pure (some a)).join) t.listEncode = some t"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_4", "R : Type u_13", "Norm F", "SeminormedRing R", "g : α → F", "l : Filter α", "f : α → R", "h : f =O[l] g", "c' : R"], "goal": "(fun x => c' * f x) =O[l] g"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "Sub α", "x : α", "v : Fin n → α", "w : Fin n.succ → α"], "goal": "Matrix.vecCons x v - w = Matrix.vecCons (x - Matrix.vecHead w) (v - Matrix.vecTail w)"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Fintype ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x y✝ : ι → ℝ", "δ : ℝ", "hs : IsUpperSet s", "hx : x ∈ closure s", "hδ : 0 < δ", "y : ι → ℝ", "hy : y ∈ s", "z : ι → ℝ", "hz : ‖x + const ι (3 / 4 * δ) - z‖ ≤ δ / 4", "i : ι", "hxy : ‖(x - y) i‖ ≤ δ / 4"], "goal": "y i < z i"}}
+{"state": {"context": ["u✝ : Lean.Level", "arg : Q(Type u✝)", "sα✝ : Q(CommSemiring «$arg»)", "u : Lean.Level", "α : Q(Type u)", "sα : Q(CommSemiring «$α»)", "R : Type u_1", "inst✝ : CommSemiring R", "a a' a₁✝ a₂✝ a₃ b b' b₁ b₂ b₃ c₁ c₂ a₁ : R", "a₂ : ℕ"], "goal": "a₁ ^ a₂ * a₃ * b = a₁ ^ a₂ * (a₃ * b)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "M : Mon_ (Comon_ C)"], "goal": "(Bimon_.ofMon_Comon_obj C M).X = (Comon_.forgetMonoidal C).mapMon.obj M"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M"], "goal": "(AddSubmonoid.center M).toAddSubsemigroup = AddSubsemigroup.center M"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "s : Set E", "A : Type u_3", "NormedRing A", "NormedAlgebra 𝕜 A", "f g : E → A", "N : ℕ∞", "hf : ContDiffOn 𝕜 N f s", "hg : ContDiffOn 𝕜 N g s", "hs : UniqueDiffOn 𝕜 s", "x : E", "hx : x ∈ s", "n : ℕ", "hn : ↑n ≤ N"], "goal": "‖iteratedFDerivWithin 𝕜 n (fun y => f y * g y) s x‖ ≤\n  ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖"}}
+{"state": {"context": ["ιA : Type u_1", "ιB : Type u_2", "A : ιA → Type u_3", "M : ιB → Type u_4", "AddMonoid ιA", "VAdd ιA ιB", "(i : ιA) → AddCommMonoid (A i)", "(i : ιB) → AddCommMonoid (M i)", "DecidableEq ιA", "DecidableEq ιB", "GradedMonoid.GMonoid A", "DirectSum.Gmodule A M", "x : ⨁ (i : ιB), M i"], "goal": "1 • x = x"}}
+{"state": {"context": ["α : Type u_1", "s : ℕ", "l : Ordnode α", "x : α", "r : Ordnode α", "H : (Ordnode.node s l x r).Sized"], "goal": "(Ordnode.node s l x r).size = l.size + r.size + 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "MeasureTheory.SFinite ν", "f g : α × β → γ", "h : f =ᵐ[μ.prod ν] g"], "goal": "∀ᵐ (x : α) ∂μ, Function.curry f x =ᵐ[ν] Function.curry g x"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝ : ConditionallyCompleteLinearOrderBot ι", "a b : ℝ", "f : ι → Ω → ℝ", "N : ι", "n m : ℕ", "ω : Ω"], "goal": "lowerCrossingTime a b f N n ω ≤ N"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Preadditive C", "G : C"], "goal": "(preadditiveCoyoneda.obj (op G) ⋙ forget AddCommGrp).Faithful ↔ (preadditiveCoyoneda.obj (op G)).Faithful"}}
+{"state": {"context": ["P : Type u_1", "inst✝¹ : LE P", "I✝ J✝ s t : Ideal P", "x y : P", "inst✝ : IsDirected P fun x x_1 => x ≥ x_1", "I J : Ideal P", "a : P", "ha : a ∈ ↑I", "b : P", "hb : b ∈ ↑J"], "goal": "(↑I ∩ ↑J).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "MeasurableSpace G", "MeasurableSpace α", "CommGroup G", "MeasurableMul₂ G", "MeasurableInv G", "μ : MeasureTheory.Measure α", "f g : α → G", "hf : AEMeasurable f μ"], "goal": "AEMeasurable (f * g) μ ↔ AEMeasurable g μ"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "x y z : 𝒰.J"], "goal": "𝒰.gluedCoverT' x y z ≫\n    𝒰.gluedCoverT' y z x ≫\n      𝒰.gluedCoverT' z x y ≫\n        CategoryTheory.Limits.pullback.snd (CategoryTheory.Limits.pullback.fst (𝒰.map x) (𝒰.map y))\n          (CategoryTheory.Limits.pullback.fst (𝒰.map x) (𝒰.map z)) =\n  CategoryTheory.Limits.pullback.snd (CategoryTheory.Limits.pullback.fst (𝒰.map x) (𝒰.map y))\n    (CategoryTheory.Limits.pullback.fst (𝒰.map x) (𝒰.map z))"}}
+{"state": {"context": ["I : Type u", "LinearOrder I", "IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hsC : Profinite.NobelingProof.contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1"], "goal": "(⇑(Profinite.NobelingProof.Linear_CC' C hsC ho) ∘ fun l => Profinite.NobelingProof.Products.eval C ↑l) = fun l =>\n  Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.C' C ho) (↑l).Tail"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "x : CliffordAlgebra Q", "n : ZMod 2"], "goal": "CliffordAlgebra.reverse x ∈ CliffordAlgebra.evenOdd Q n ↔ x ∈ CliffordAlgebra.evenOdd Q n"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "IsEmpty ↑s ↔ s = ∅"}}
+{"state": {"context": ["a : ENNReal"], "goal": "a⁻¹ ≤ 1 ↔ 1 ≤ a"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_3", "Fintype ι", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "SecondCountableTopology E", "b : Basis ι ℝ E", "μ : MeasureTheory.Measure E", "MeasureTheory.SigmaFinite μ", "μ.IsAddLeftInvariant"], "goal": "b.addHaar = μ ↔ μ ↑b.parallelepiped = 1"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "p : R⟦X⟧", "n d : ℕ"], "goal": "∀ b ∈ antidiagonal (d + n), b ≠ (d, n) → (coeff R b.1) p * (coeff R b.2) (X ^ n) = 0"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace ℝ E", "hE : CompleteSpace E", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "c : E"], "goal": "∫ (x : α), c ∂μ = (μ Set.univ).toReal • c"}}
+{"state": {"context": ["n : ℕ", "n_large : 512 ≤ n"], "goal": "∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n"}}
+{"state": {"context": ["R S T : Type u", "CommRing R", "CommRing S", "CommRing T", "Algebra R S", "Algebra R T"], "goal": "(AlgebraicGeometry.pullbackSpecIso R S T).inv ≫\n    CategoryTheory.Limits.pullback.fst (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R S)))\n      (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R T))) =\n  AlgebraicGeometry.Spec.map (CommRingCat.ofHom Algebra.TensorProduct.includeLeftRingHom)"}}
+{"state": {"context": ["ι : Type u_1", "B : Type u_2", "F : Type u_3", "TopologicalSpace B", "TopologicalSpace F", "Z : FiberBundleCore ι B F", "i : ι"], "goal": "Z.baseSet i = (Z.localTriv i).baseSet"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁹ : LinearOrderedField R", "inst✝⁸ : LinearOrderedField R'", "inst✝⁷ : AddCommGroup E", "inst✝⁶ : AddCommGroup F", "inst✝⁵ : LinearOrderedAddCommGroup α", "inst✝⁴ : Module R E", "inst✝³ : Module R F", "inst✝² : Module R α", "inst✝¹ : OrderedSMul R α", "s✝ : Set E", "i j : ι", "c : R", "t : Finset ι", "w✝ : ι → R", "z✝ : ι → E", "s : Set E", "x : E", "inst✝ : Fintype ι", "w : ι → R", "z : ι → E", "hw₀ : ∀ (i : ι), 0 ≤ w i", "hw₁ : ∑ i : ι, w i = 1", "hz : ∀ (i : ι), z i ∈ s", "hx : ∑ i : ι, w i • z i = x"], "goal": "x ∈ (convexHull R) s"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "CommRing R", "S : Type u_2", "CommRing S", "f : R →+* S", "n : ℕ"], "goal": "(MvPolynomial.map f) (W_ R n) = W_ S n"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a b : α", "h : a > b"], "goal": "¬a ≤ b"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C"], "goal": "∀ {X Y Z : CategoryTheory.Limits.Cocone F} (f : X ⟶ Y) (g : Y ⟶ Z), (f ≫ g).hom = f.hom ≫ g.hom"}}
+{"state": {"context": ["F✝ : Type u_1", "X : Type u", "Y : Type v", "Z : Type w", "Z' : Type x", "ι : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "f₀ f₁ : C(X, Y)", "F : f₀.Homotopy f₁", "t : ℝ", "ht : 1 ≤ t", "x : X"], "goal": "(F.extend t) x = F (1, x)"}}
+{"state": {"context": ["q : ℚ", "n₁ n₂ d₂ : ℤ"], "goal": "n₁ /. 0 * (n₂ /. d₂) = n₁ * n₂ /. (0 * d₂)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : HeytingAlgebra α", "a✝ a b : α", "h : IsCompl a b"], "goal": "a ⇔ b = ⊥"}}
+{"state": {"context": ["α : Type u", "a b c : Set α"], "goal": "a ∩ b ⊆ c ↔ a ⊆ bᶜ ∪ c"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : TopologicalSpace E", "inst✝³ : AddCommGroup E", "inst✝² : TopologicalAddGroup E", "inst✝¹ : Module ℝ E", "inst✝ : ContinuousSMul ℝ E", "s : Set E", "hs₀ : 0 ∈ s", "hs₁ : Convex ℝ s", "hs₂ : IsOpen s", "x₀ : E", "hx₀ : x₀ ∉ s", "f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯", "x : E", "hx : x ∈ f.domain"], "goal": "↑f ⟨x, hx⟩ ≤ gauge s ↑⟨x, hx⟩"}}
+{"state": {"context": ["α : Type u_1", "MulOneClass α", "Preorder α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c : α", "ha : 1 < a", "hbc : b < c"], "goal": "b < a * c"}}
+{"state": {"context": ["δ : Type u_1", "δ' : Type u_2", "π : δ → Type u_3", "inst✝³ : (x : δ) → MeasurableSpace (π x)", "μ : (i : δ) → Measure (π i)", "inst✝² : ∀ (i : δ), SigmaFinite (μ i)", "inst✝¹ : DecidableEq δ", "s✝ t : Finset δ", "f✝ g✝ : ((i : δ) → π i) → ℝ≥0∞", "x y : (i : δ) → π i", "i : δ", "inst✝ : Fintype δ", "s : Finset δ", "f g : ((i : δ) → π i) → ℝ≥0∞", "hf : Measurable f", "hg : Measurable g", "hfg : ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s, g ∂μ", "h : IsEmpty ((i : δ) → π i)"], "goal": "∫⁻ (x : (i : δ) → π i), f x ∂Measure.pi μ = ∫⁻ (x : (i : δ) → π i), g x ∂Measure.pi μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r"], "goal": "(fun z => -deriv ε z) =o[atTop] fun x => x⁻¹"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : GCDMonoid α", "a b c : α", "hab : IsUnit (gcd b a)", "k : ℕ", "h✝ : Nontrivial α", "ha : ¬a = 0", "hb : ¬b = 0", "hk : k > 0", "hc✝ : c ∣ a * b", "d₁ d₂ : α", "hd₁ : d₁ ∣ a", "hd₂ : d₂ ∣ b", "hc : c = d₂ * d₁", "h0₁ : d₁ ^ k ≠ 0", "a' : α", "ha' : a = d₁ ^ k * a'", "h0₂ : d₂ ^ k ≠ 0", "b' : α", "h : d₂ ^ k * b' * (d₁ ^ k * a') = d₂ ^ k * d₁ ^ k", "hb' : b = d₂ ^ k * b'"], "goal": "Associated (d₁ ^ k) a"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hC : IsClosed C", "hsC : contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1", "l : ↑(MaxProducts C ho)", "h₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x => ord I x < o)))", "this : Inhabited I", "m : Products I →₀ ℤ", "hmmem : m ∈ Finsupp.supported ℤ ℤ {m | m < (↑l).Tail}", "hmsum : (m.sum fun i a => a • Products.eval (C' C ho) i) = (Linear_CC' C hsC ho) (Products.eval C ↑l)"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁✝ l₂✝ l₁ l₂ : List α", "h' : ∀ (n : ℕ), n < max l₁.length l₂.length → l₁.get? n = l₂.get? n", "n : ℕ", "hn : n ≥ max l₁.length l₂.length"], "goal": "l₁.get? n = l₂.get? n"}}
+{"state": {"context": ["K : Type u_1", "Field K", "NumberField K", "IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : FermatLastTheoremForThreeGen.Solution hζ"], "goal": "IsCoprime (FermatLastTheoremForThreeGen.Solution.y S) (FermatLastTheoremForThreeGen.Solution.z S)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Finset α"], "goal": "Finset.Icc s t = Finset.filter (fun x => s ⊆ x) t.powerset"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "N : ι → Submodule R M", "inst✝⁶ : DecidableEq ι", "h : IsInternal N", "inst✝⁵ : ∀ (i : ι), Module.Finite R ↥(N i)", "inst✝⁴ : ∀ (i : ι), Module.Free R ↥(N i)", "inst✝³ : IsDomain R", "inst✝² : IsPrincipalIdealRing R", "inst✝¹ : Module.Free R M", "inst✝ : Module.Finite R M", "f g : Module.End R M", "h_comm : Commute f g", "hf : ⨆ μ, ⨆ k, (f.genEigenspace μ) k = ⊤", "hg : ∀ (μ : R), (trace R ↥(⨆ k, (f.genEigenspace μ) k)) (restrict g ⋯) = 0"], "goal": "(trace R M) (g ∘ₗ f) = 0"}}
+{"state": {"context": ["α : Type u", "l : List α", "n : ℕ"], "goal": "l.rotate n = [] ↔ l = []"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Mul α", "inst✝² : LinearOrder α", "a b c d : α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "h : a * b < c * d"], "goal": "min a b < max c d"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H K : AddSubgroup G"], "goal": "↑(H.addSubgroupOf K) = ⇑K.subtype ⁻¹' ↑H"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "f : α → β", "WellFoundedLT β", "hf : StrictMono f"], "goal": "WellFoundedLT α"}}
+{"state": {"context": ["S : Type uᵒᵖ ⥤ Type u", "hs : IsSheaf typesGrothendieckTopology S", "α : Type u", "f : α → S.obj (op PUnit.{u + 1})"], "goal": "eval S α (typesGlue S hs α f) = f"}}
+{"state": {"context": ["E : ℕ → Type u_1", "inst✝¹ : (n : ℕ) → TopologicalSpace (E n)", "inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)", "s : Set ((n : ℕ) → E n)", "hs : IsClosed s", "hne : s.Nonempty", "x : (n : ℕ) → E n", "hx : x ∉ s", "A : ∃ n, Disjoint s (cylinder x n)", "B : Nat.find A - 1 < Nat.find A", "y : (n : ℕ) → E n", "ys : y ∈ s", "hy : x ∈ cylinder y (Nat.find A - 1)"], "goal": "(s ∩ cylinder x (Nat.find A - 1)).Nonempty"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "ConditionallyCompleteLinearOrder ι", "u : ι → Ω → β", "s : Set β", "n m : ι", "ω : Ω"], "goal": "MeasureTheory.hitting u s n m ω ≤ m"}}
+{"state": {"context": ["x y z : ℝ"], "goal": "0 ≤ sinh x ↔ 0 ≤ x"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "α : Type u_3", "β : Type u_1", "Monad m", "l : List α", "f : α → β → m β", "b : β"], "goal": "List.foldrM f b l.reverse = List.foldlM (fun x y => f y x) b l"}}
+{"state": {"context": ["a b r : ℝ≥0", "hr : r ≠ 0"], "goal": "r * a ≤ b ↔ a ≤ r⁻¹ * b"}}
+{"state": {"context": ["α✝ β : Type u", "c : Cardinal.{?u.236298}", "α : Type u_1", "l : List α", "h : ∀ (z : α), z ∈ l"], "goal": "#α ≤ ↑l.length"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : InnerProductSpace 𝕜 E", "inst✝⁴ : InnerProductSpace 𝕜 F", "inst✝³ : InnerProductSpace 𝕜 G", "inst✝² : CompleteSpace E", "inst✝¹ : CompleteSpace G", "inst✝ : CompleteSpace F", "A : E →L[𝕜] F", "x : E"], "goal": "‖A x‖ ^ 2 = re ⟪((adjoint A).comp A) x, x⟫_𝕜"}}
+{"state": {"context": ["α : Type u_3", "E : Type u_6", "SeminormedAddGroup E", "l : Filter α", "f : α → E", "h : Filter.Tendsto (fun y => ‖f y‖) l Filter.atTop", "x : E"], "goal": "∀ᶠ (y : α) in l, f y ≠ x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "κ : Type u_7", "inst✝ : TopologicalSpace X", "T : (i : ι) → TopologicalSpace (π i)", "f : X → (i : ι) → π i", "i : Set ι", "s : (a : ι) → Set (π a)", "hi : i.Finite", "hs : ∀ a ∈ i, IsOpen (s a)"], "goal": "IsOpen (⋂ a ∈ i, eval a ⁻¹' s a)"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "Preorder M", "CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1"], "goal": "↑(Submonoid.oneLE M) = Set.Ici 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : MeasurableSpace α"], "goal": "toJordanDecomposition 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "self : NormedLatticeAddCommGroup α", "a b : α"], "goal": "a ≤ b → ∀ (c : α), c + a ≤ c + b"}}
+{"state": {"context": ["G : Type u_1", "inst✝⁸ : TopologicalSpace G", "inst✝⁷ : LocallyCompactSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "μ : Measure G", "inst✝² : IsFiniteMeasureOnCompacts μ", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "g : G → E", "hg : Continuous g", "h'g : HasCompactSupport g", "k : Set G := tsupport g", "k_comp : IsCompact k", "x₀ : G", "t : Set G", "t_comp : IsCompact t", "ht : t ∈ 𝓝 x₀", "k' : Set G := t • k⁻¹", "k'_comp : IsCompact k'", "A : ContinuousOn (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) t"], "goal": "ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j k : ι", "hi : c.prev j = i", "hk : c.next j = k", "K.HasHomology j", "(K.sc' i j k).HasHomology"], "goal": "K.toCycles i j ≫ (K.cyclesIsoSc' i j k hi hk).hom = (K.sc' i j k).toCycles"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "f : α ≃o β", "a : α"], "goal": "(LowerSet.map f) (LowerSet.Iic a) = LowerSet.Iic (f a)"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "Fintype V", "DecidableRel G.Adj", "v : V"], "goal": "G.degree v < Fintype.card V"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w₁ w₂ : W"], "goal": "cs.length (w₁ * w₂) % 2 = (cs.length w₁ + cs.length w₂) % 2"}}
+{"state": {"context": ["x y z : ZFSet"], "goal": "x ∈ {y, z} ↔ x = y ∨ x = z"}}
+{"state": {"context": ["a✝ b✝ c✝ d✝ : Ordinal.{u}", "a b c d : Ordinal.{u_1}", "hd : d < a ⨳ b"], "goal": "d ♯ a ⨳ c < a ⨳ (b ♯ c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "op : α → α → α", "hc : Std.Commutative op", "ha : Std.Associative op", "b : α", "l : List α"], "goal": "List.foldl (fun x y => op y x) b l = List.foldl op b l"}}
+{"state": {"context": ["m n : ℕ", "hn : 1 < n", "h : n ! = m !", "hnm : n < m"], "goal": "n = m"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s t : Set α", "x : α", "h : s ⊆ t", "hs : s.Nonempty"], "goal": "Metric.infDist x t ≤ Metric.infDist x s"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "I✝ J I : Box ι", "r : (ι → ℝ) → ↑(Set.Ioi 0)", "π : Prepartition I"], "goal": "∃ π', π'.toPrepartition ≤ π ∧ π'.IsHenstock ∧ π'.IsSubordinate r ∧ π'.distortion = π.distortion ∧ π'.iUnion = π.iUnion"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ", "h : f =ᶠ[Filter.atTop] g"], "goal": "LSeriesSummable f s ↔ LSeriesSummable g s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ✝ : Type x", "γ : Type u_1", "P : γ → Set γ → Prop", "this : Nonempty γ", "c : Set γ → γ", "hc : ∀ (t : Set γ), t.Finite → P (c t) t", "f : (n : ℕ) → ((m : ℕ) → m < n → γ) → γ := fun n g => c (range fun k => g ↑k ⋯)", "u : ℕ → γ := fun n => n.strongRecOn' f", "n : ℕ"], "goal": "u n = c (range fun x => u ↑x)"}}
+{"state": {"context": ["G₀ : Type u_1", "α : Type u_2", "inst✝⁵ : GroupWithZero G₀", "inst✝⁴ : Bornology G₀", "inst✝³ : MulAction G₀ α", "s✝ t u : Set α", "S : Set (Set α)", "inst✝² : (cobounded G₀).NeBot", "E : Type u_3", "inst✝¹ : AddMonoid E", "inst✝ : DistribMulAction G₀ E", "s : Set E"], "goal": "Absorbs G₀ s 0 ↔ 0 ∈ s"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a b c : α"], "goal": "a > b → b > c → a > c"}}
+{"state": {"context": ["α : Type u_1", "m : Multiset α", "z : Sym2 α"], "goal": "z ∈ m.sym2 ↔ ∀ y ∈ z, y ∈ m"}}
+{"state": {"context": ["R : Type u", "Ring R", "p q : R[X]", "S : Type u_1", "Ring S", "f : R →+* S"], "goal": "Polynomial.map f (p - q) = Polynomial.map f p - Polynomial.map f q"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L"], "goal": "lieIdealSubalgebra R L ⊤ = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "μ✝ : Measure α", "κ η : Kernel α β", "μ : Measure β", "inst✝ : IsFiniteMeasure μ"], "goal": "dirac () ⊗ₘ Kernel.const Unit μ = map (Prod.mk ()) μ"}}
+{"state": {"context": ["x y : ℂ"], "goal": "sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : ↥(MeasureTheory.Lp E p μ)"], "goal": "‖f‖₊ = (MeasureTheory.eLpNorm (↑↑f) p μ).toNNReal"}}
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+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "(o.oangle (x - y) x).sin = ‖y‖ / ‖x - y‖"}}
+{"state": {"context": ["R : Type u_1", "N : Type u_2", "M : Type u_3", "Semiring R", "AddCommMonoid N", "Module R N", "AddCommMonoid M", "Module R M", "f i : N →ₗ[R] M", "K : Submodule R N", "m : ℕ", "heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K", "hf : Function.Surjective ⇑f", "hi : Function.Injective ⇑i", "n : ℕ"], "goal": "f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : C ⥤ D", "G : D ⥤ C", "adj : F ⊣ G"], "goal": "F.PreservesEpimorphisms"}}
+{"state": {"context": ["K : Type u_1", "Field K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3"], "goal": "(hζ.toInteger - 1) ^ 2 = -3 * ↑⋯.unit"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω✝ : Type u_3", "F✝ : Type u_4", "inst✝⁵ : MeasurableSpace Ω✝", "inst✝⁴ : StandardBorelSpace Ω✝", "inst✝³ : Nonempty Ω✝", "inst✝² : NormedAddCommGroup F✝", "mα : MeasurableSpace α", "μ✝ : Measure α", "inst✝¹ : IsFiniteMeasure μ✝", "X✝ : α → β", "Y : α → Ω✝", "mβ : MeasurableSpace β", "s : Set Ω✝", "t : Set β", "f✝ : β × Ω✝ → F✝", "Ω : Type u_5", "F : Type u_6", "mΩ : MeasurableSpace Ω", "X : Ω → β", "μ : Measure Ω", "inst✝ : TopologicalSpace F", "f : Ω → F", "hf : AEStronglyMeasurable f μ"], "goal": "Measure.QuasiMeasurePreserving Prod.snd (Measure.map (fun ω => (X ω, ω)) μ) μ"}}
+{"state": {"context": ["α : Sort u_2", "β : Sort u_1", "a b : α", "C : α → Sort u_1", "x : β", "y : C a", "e : a = b"], "goal": "HEq x (e ▸ y) ↔ HEq x y"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "ι : Type u_2", "s : Finset ι", "v : ι → R", "inst✝ : Nontrivial R"], "goal": "(nodal s v).natDegree = s.card"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I J X : Set α", "hI : M.Basis' I X", "hIJ : I ⊆ J", "hJ : M.Indep J"], "goal": "J ∩ X = I"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "X Y : C", "inst���¹ : Simple Y", "f : X ⟶ Y", "inst✝ : Mono f", "w : IsIso f → False"], "goal": "f = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "s : Set R", "a✝ a b : R", "dvd : a ∣ b"], "goal": "{b} ⊆ ↑(span R {a})"}}
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+{"state": {"context": ["n : ℕ", "a b : Fin (n + 1)", "ha : a.castSucc ≠ 0"], "goal": "a.castSucc.pred ha < b ↔ a ≤ b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝² : CommRing R", "inst✝¹ : Ring S", "f : R →+* S", "I : Type u_3", "inst✝ : Fintype I", "e : I → R", "he : CompleteOrthogonalIdempotents e"], "goal": "∏ i : I, (1 - e i) = 0"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "s : Set E", "hs : NullMeasurableSet s μ", "r : ℝ", "hs' : s.Nonempty", "hr : r ≠ 0"], "goal": "NullMeasurableSet (r • s) μ"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x✝ y✝ x y : V", "hx : x ≠ 0", "hy : y ≠ 0", "h : ‖x + y‖ = ‖x‖ + ‖y‖"], "goal": "⟪x, y⟫_ℝ = ‖x‖ * ‖y‖"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁴ : Group G", "inst✝¹³ : MeasurableSpace G", "inst✝¹² : TopologicalSpace G", "inst✝¹¹ : TopologicalGroup G", "inst✝¹⁰ : BorelSpace G", "inst✝⁹ : PolishSpace G", "Γ : Subgroup G", "inst✝⁸ : Countable ↥Γ", "inst✝⁷ : Γ.Normal", "inst✝⁶ : T2Space (G ⧸ Γ)", "inst✝⁵ : SecondCountableTopology (G ⧸ Γ)", "μ : Measure (G ⧸ Γ)", "ν : Measure G", "inst✝⁴ : SigmaFinite ν", "inst✝³ : ν.IsHaarMeasure", "inst✝² : ν.IsMulRightInvariant", "𝓕 : Set G", "h𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 ν", "inst✝¹ : μ.IsMulLeftInvariant", "inst✝ : SigmaFinite μ", "V : Set (G ⧸ Γ)", "hV : (interior V).Nonempty", "meas_V : MeasurableSet V", "hμK : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)", "neTopV : μ V ≠ ⊤"], "goal": "μ V = ν (QuotientGroup.mk ⁻¹' V ∩ 𝓕)"}}
+{"state": {"context": ["N : ℕ", "χ : DirichletCharacter ℂ N", "s : ℂ", "hs : 1 < s.re"], "goal": "LSeriesSummable (fun n => χ ↑n) s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "inst✝⁸ : Semifield R", "inst✝⁷ : Semifield S", "inst✝⁶ : Ring A", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra R A", "inst✝³ : Algebra S A", "inst✝² : IsScalarTower R S A", "inst✝¹ : TopologicalSpace R", "inst✝ : TopologicalSpace S", "a : A", "f : C(S, R)", "h : SpectrumRestricts a ⇑f", "h_cpct : CompactSpace ↑(spectrum S a)"], "goal": "CompactSpace ↑(spectrum R a)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasProducts C", "X : TopCat", "F : Presheaf C X", "ι : Type v'", "U : ι → Opens ↑X", "x✝ : Discrete (ι × ι)"], "goal": "F.map (leSupr U x✝.as.1).op ≫ F.map ((U x✝.as.1).infLELeft (U x✝.as.2)).op = F.map (leSupr U x✝.as.2).op ≫ F.map ((U x✝.as.1).infLERight (U x✝.as.2)).op"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "r : α → β → Prop", "self : Concept α β r"], "goal": "extentClosure r self.toProd.2 = self.toProd.1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_3", "FunLike F (Set α) ℝ≥0∞", "MeasureTheory.OuterMeasureClass F α", "μ : F", "f : α → β"], "goal": "f =ᵐ[μ] f"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g✝ : α → β", "inst✝¹ : CommMonoid β", "s : Finset α", "p : α → Prop", "hp : DecidablePred p", "inst✝ : DecidablePred fun x => ¬p x", "f : (x : α) → p x → γ", "g : (x : α) → ¬p x → γ", "h : γ → β", "x : { x // x ∈ filter p s }"], "goal": "p ↑x"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommMonoidWithZero R", "n : ℕ", "χ : DirichletCharacter R n", "m : ℕ", "inst✝ : NeZero m", "hm : n ∣ m", "a₁✝ a₂✝ : DirichletCharacter R n", "h : (changeLevel hm) a₁✝ = (changeLevel hm) a₂✝", "z : (ZMod m)ˣ"], "goal": "a₁✝ ↑((ZMod.unitsMap hm) z) = a₂✝ ↑((ZMod.unitsMap hm) z)"}}
+{"state": {"context": ["R : Type u_2", "LinearOrderedAddCommGroup R", "x y z : R"], "goal": "|x - y| ≤ z ↔ y ∈ Set.Icc (x - z) (x + z)"}}
+{"state": {"context": ["R : Type u", "CommRing R"], "goal": "(AlgebraicGeometry.StructureSheaf.globalSectionsIso R).inv =\n  CategoryTheory.inv (AlgebraicGeometry.StructureSheaf.toOpen R ⊤)"}}
+{"state": {"context": ["α : Type u_1", "LT α"], "goal": "WellFoundedLT αᵒᵈ ↔ WellFoundedGT α"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "p : ℝ≥0∞", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup G", "f : ↥(MeasureTheory.Lp G p μ)", "hp_ne_zero : p ≠ 0", "hp_ne_top : p ≠ ⊤"], "goal": "MeasureTheory.FinStronglyMeasurable (↑↑f) μ"}}
+{"state": {"context": ["ι : Type uι", "inst✝⁵ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝¹ : SeminormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "f : ContinuousMultilinearMap 𝕜 E F", "x : ⨂[𝕜] (i : ι), E i", "G : Type (max (max (max uE u𝕜) uι) (max uE uι) u𝕜) := (⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)", "G' : Submodule 𝕜 F := LinearMap.range (lift f.toMultilinearMap)", "e : ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) ≃ₗ[𝕜] ↥(LinearMap.range (lift f.toMultilinearMap)) := (lift f.toMultilinearMap).quotKerEquivRange", "this✝ : SeminormedAddCommGroup G := SeminormedAddCommGroup.induced G (↥G') e", "this : NormedSpace 𝕜 G := NormedSpace.induced 𝕜 G (↥G') e", "f'₀ : MultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := lift.symm (↑e.symm ∘ₗ (lift f.toMultilinearMap).rangeRestrict)", "hf'₀ : ∀ (x : (i : ι) → E i), ‖f'₀ x‖ ≤ ‖f‖ * ∏ i : ι, ‖x i‖", "f' : ContinuousMultilinearMap 𝕜 E ((⨂[𝕜] (i : ι), E i) ⧸ LinearMap.ker (lift f.toMultilinearMap)) := f'₀.mkContinuous ‖f‖ hf'₀", "hnorm : ‖f'‖ ≤ ‖f‖", "heq : ↑(e ((lift f'.toMultilinearMap) x)) = (lift f.toMultilinearMap) x", "hle : (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G) ≤ injectiveSeminorm"], "goal": "‖(lift f'.toMultilinearMap) x‖ ≤ ‖f'‖ * (fun f => ⇑f) ((normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)) x"}}
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+{"state": {"context": ["α : Type u_1", "C : Set (Set α)", "self : MeasureTheory.IsSetSemiring C"], "goal": "∅ ∈ C"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "x y : α", "ε : ℝ≥0∞"], "goal": "y ∈ EMetric.closedBall x ε ↔ edist x y ≤ ε"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w : ι → k", "p₂ : ι → P", "p₁ : P", "h : ∑ i ∈ s, w i = 0"], "goal": "∑ i ∈ s, w i • (p₁ -ᵥ p₂ i) = -(s.weightedVSub p₂) w"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕝 : Type u_2", "E : Type u_3", "NormedField 𝕜", "NormedRing 𝕝", "NormedSpace 𝕜 𝕝", "AddCommGroup E", "Module 𝕜 E", "SMulWithZero 𝕝 E", "IsScalarTower 𝕜 𝕝 E", "s : Set E", "hs : Balanced 𝕝 s", "a : 𝕝", "b : 𝕜", "h : ‖a‖ ≤ ‖b‖"], "goal": "a • s ⊆ b • s"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "DecidableEq ι", "(i : ι) → Zero (β i)", "(i : ι) → (x : β i) → Decidable (x ≠ 0)", "AddCommMonoid γ", "f : Π₀ (i : ι), β i", "h : (i : ι) → β i → γ", "hyp : ∀ (i : ι), h i (f i) = 0"], "goal": "f.sum h = 0"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "Polynomial.natDegree 0 = 0"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "X : Type u_3", "Y : Type u_4", "Z : Type u_5", "inst✝ : TopologicalSpace X", "f g : X → Y", "A : Set X", "x : X", "P : (x : X) → (𝓝 x).Germ Y → Prop", "hf : RestrictGermPredicate P A x ↑f", "h : ∀ᶠ (z : X) in 𝓝ˢ A, g z = f z", "hx : x ∈ A", "y : X", "hy : P y ↑f", "h'y : ∀ᶠ (x : X) in 𝓝 y, g x = f x"], "goal": "P y ↑g"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n : ℕ", "inst✝ : Ring R", "p q : R[X]"], "goal": "(p - q).natDegree = (q - p).natDegree"}}
+{"state": {"context": ["β : Type v", "inst✝ : NormedField β", "f : ℕ → β", "hf : CauchySeq f"], "goal": "IsCauSeq norm f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s✝ t : Set X", "f : X → Y", "hf : IsOpenMap f", "s : Set Y"], "goal": "f ⁻¹' closure s ⊆ closure (f ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "f : α → E", "l : Filter α", "g₁ g₂ : α → F", "h : f =O[l] g₁", "hg : ∀ (x : α), g₁ x = g₂ x"], "goal": "f =O[l] g₂"}}
+{"state": {"context": ["R : Type u_6", "S : Type u_7", "CommRing R", "CommRing S", "f : R →+* S", "hf : f.IsIntegral", "M : Submonoid R"], "goal": "(IsLocalization.map (Localization (Submonoid.map (↑f) M)) f ⋯).IsIntegral"}}
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+{"state": {"context": ["x y : ℝ", "hxy : x < y"], "goal": "Set.Icc x y ∈ (IsUnifLocDoublingMeasure.vitaliFamily MeasureTheory.volume 1).setsAt x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "F : CategoryTheory.LaxMonoidalFunctor C D"], "goal": "∀ {X Y : Mon_ C} (f : X ⟶ Y), (F.mapMon.map f).hom = F.map f.hom"}}
+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "t✝ t₁ t₂ : TopologicalSpace α✝", "t' : TopologicalSpace β✝", "f : α✝ → β✝", "g : β✝ → α✝", "α : Type u_4", "β : Type u_5", "e : α ≃ β", "t : TopologicalSpace α", "U : Set β"], "goal": "(∃ t_1, IsOpen t_1 ∧ ⇑e.symm ⁻¹' t_1 = U) ↔ IsOpen (⇑e ⁻¹' U)"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "s : Set α", "h : IsClosed s"], "goal": "↑{ carrier := s, closed' := h } = s"}}
+{"state": {"context": ["α : Type u", "s : Set α", "Finite ↑s"], "goal": "s.Finite"}}
+{"state": {"context": ["α : Type u_2", "_m0 : MeasurableSpace α", "s : Set α", "μ : MeasureTheory.Measure α", "hs : ∀ᵐ (x : α) ∂μ, x ∈ s"], "goal": "μ.restrict s = μ"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "R X Y Z : C", "f : X ⟶ Y", "a b : R ⟶ X", "f₁ : X ⟶ Y", "f₂ : Y ⟶ Z", "inst✝ : Mono f₂", "small_k : IsKernelPair f₁ a b"], "goal": "∀ (s : PullbackCone (f₁ ≫ f₂) (f₁ ≫ f₂)) (m : s.pt ⟶ (PullbackCone.mk a b ⋯).pt), (∀ (j : WalkingCospan), m ≫ (PullbackCone.mk a b ⋯).π.app j = s.π.app j) → m = (fun s => small_k.lift s.fst s.snd ⋯) s"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a n' : ℤ", "hn' : n' + a = n", "x : R", "p q : ℤ", "hpq : p + n' = q"], "goal": "((x • γ).rightShift a n' hn').v p q hpq = x • (γ.rightShift a n' hn').v p q hpq"}}
+{"state": {"context": ["q : ℕ+", "n : ℕ"], "goal": "(↑n).1 = ↑n"}}
+{"state": {"context": ["X : Type u", "inst✝⁵ : MetricSpace X", "inst✝⁴ : CompactSpace X", "inst✝³ : Nonempty X", "Y : Type v", "inst✝² : MetricSpace Y", "inst✝¹ : CompactSpace Y", "inst✝ : Nonempty Y"], "goal": "NonemptyCompacts.kuratowskiEmbedding X ≈ NonemptyCompacts.kuratowskiEmbedding Y → Nonempty (X ≃ᵢ Y)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "a b : α"], "goal": "|a| ≤ |b| ↔ a * a ≤ b * b"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "f' : F", "x : 𝕜", "s : Set 𝕜", "h : HasDerivWithinAt f f' s x"], "goal": "DifferentiableWithinAt 𝕜 f s x"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "x y : MvPolynomial R ℚ"], "goal": "WittVector.teichmullerFun p (x * y) = WittVector.teichmullerFun p x * WittVector.teichmullerFun p y"}}
+{"state": {"context": ["R : Type u_1", "C : Type u_2", "Semiring R", "CategoryTheory.Category.{u_3, u_2} C", "CategoryTheory.Preadditive C", "CategoryTheory.Linear R C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "γ : CategoryTheory.ShortComplex.HomologyMapData φ h₁ h₂", "a : R"], "goal": "(γ.smul a).right = γ.right.smul a"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a a₁ a₂ : α", "h₁ : r a₁ a", "h₂ : r a₂ a"], "goal": "Relation.CutExpand r {a₁, a₂} {a}"}}
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+{"state": {"context": ["m : Type u_1", "n : Type u_2", "p : Type u_3", "R : Type u_4", "inst✝³ : Fintype m", "inst✝² : Fintype n", "inst✝¹ : Fintype p", "inst✝ : LinearOrderedRing R", "v : n → R"], "goal": "(∀ i ∈ Finset.univ, v i * v i = 0) ↔ v = 0"}}
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+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "hS : S.Exact", "hS' : S.ShortExact", "v : ι → ↑S.X₁", "w : ι' → ↑S.X₃", "hv : ⊤ ≤ span R (range v)", "hw : ⊤ ≤ span R (range w)", "hE : Epi S.g"], "goal": "Sum.elim (⇑S.f ∘ v) (Function.invFun S.g.toFun ∘ w) ∘ Sum.inl = ⇑S.f ∘ v"}}
+{"state": {"context": ["p : ℝ≥0∞", "𝕜 : Type u_1", "ι : Type u_2", "α : ι → Type u_3", "β✝ : ι → Type u_4", "inst✝² : Fact (1 ≤ p)", "inst✝¹ : Fintype ι", "β : ι → Type u_5", "inst✝ : (i : ι) → SeminormedAddCommGroup (β i)", "x : PiLp 2 β"], "goal": "‖x‖ = √(∑ x_1 : ι, ‖x x_1‖ ^ 2)"}}
+{"state": {"context": ["P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme", "Q : AlgebraicGeometry.AffineTargetMorphismProperty", "AlgebraicGeometry.HasAffineProperty P Q", "hP' : Q.StableUnderBaseChange", "X Y S : AlgebraicGeometry.Scheme", "f : X ⟶ S", "g : Y ⟶ S", "AlgebraicGeometry.IsAffine S", "H : Q g"], "goal": "P (CategoryTheory.Limits.pullback.fst f g)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "MeasurableSpace.CountablyGenerated α"], "goal": "(MeasurableSpace.countableGeneratingSet α).Nonempty"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "t : Finset (α × β × γ)", "SimpleGraph.TripartiteFromTriangles.ExplicitDisjoint t", "SimpleGraph.TripartiteFromTriangles.NoAccidental t"], "goal": "(SimpleGraph.TripartiteFromTriangles.graph t).LocallyLinear"}}
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+{"state": {"context": ["a : UInt16"], "goal": "a ≤ a"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "b : Basis ι R M", "i : ι"], "goal": "b.sumCoords (b i) = 1"}}
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+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "f : C(X, Y)", "hf : QuotientMap ⇑f", "g : C(X, Z)", "h : Function.FactorsThrough ⇑g ⇑f"], "goal": "∀ (a : Y), (hf.lift g h) a = ((fun i => i.liftOn' ⇑g ⋯) ∘ ⇑hf.homeomorph.symm) a"}}
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+{"state": {"context": ["l : Type u_1", "m : Type u_2", "α : Type u_12", "DecidableEq l", "DecidableEq m", "Zero α", "v : l ⊕ m → α"], "goal": "(Matrix.diagonal v).toBlocks₂₂ = Matrix.diagonal fun i => v (Sum.inr i)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "G : Type u_2", "inst✝ : Group G", "n : ℕ", "v : ↑(vectorsProdEqOne G (n + 1))"], "goal": "(↑v).head * (↑v).tail.toList.prod = 1"}}
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+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "rι : ι → ι → Prop", "rπ : (i : ι) → π i → π i → Prop", "f : γ → ι", "g : (i : ι) → γ → π i", "s : Set γ", "hι : WellFounded (rι on f)", "hπ : ∀ (i : ι), (f ⁻¹' {i}).WellFoundedOn (rπ i on g i)", "c c' : γ", "h : (Sigma.Lex rι rπ on fun c => ⟨f c, g (f c) c⟩) c c'"], "goal": "((PSigma.Lex (fun a b => rι ↑a ↑b) fun a a_1 b => (rπ ↑a on g ↑a) ↑a_1 ↑b) on fun c => ⟨⟨f c, ⋯⟩, ⟨c, ⋯⟩⟩) c c'"}}
+{"state": {"context": ["K : Type u", "inst✝² : CommRing K", "p : ℕ", "inst✝¹ : Fact (Nat.Prime p)", "inst✝ : CharP K p", "x1 x2 y : ℕ × K", "H : R K p x1 x2", "n : ℕ", "x : K"], "goal": "R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2) ((n + y.1).succ, (frobenius K p) ((⇑(frobenius K p))^[y.1] x + (⇑(frobenius K p))^[n] y.2))"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "Subsingleton (Subgroup G) ↔ Subsingleton G"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "s t : Set α", "hs : IsSublattice s", "ht : IsSublattice t"], "goal": "IsSublattice (s ∩ t)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "G : CategoryTheory.Functor D C", "adj : F ⊣ G", "∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)", "∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)", "X : C"], "goal": "adj.toEquivalence.unitIso.hom.app X = adj.unit.app X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "m n✝ : ℕ", "s : Finset α", "n : ℕ", "f : ⊤ ↪g G", "v' : Fin n", "hv : f.toEmbedding v' ∈ ↑(map f.toEmbedding univ)", "w' : Fin n", "hw : f.toEmbedding w' ∈ ↑(map f.toEmbedding univ)"], "goal": "(induce (↑(map f.toEmbedding univ)) G).Adj ⟨f.toEmbedding v', hv⟩ ⟨f.toEmbedding w', hw⟩ ↔ ⊤.Adj ⟨f.toEmbedding v', hv⟩ ⟨f.toEmbedding w', hw⟩"}}
+{"state": {"context": [], "goal": "ConvexOn ℝ (Ioi 0) (log ∘ doublingGamma)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝³ : LinearOrderedField α", "inst✝² : ConditionallyCompleteLinearOrderedField β", "inst✝¹ : ConditionallyCompleteLinearOrderedField γ", "inst✝ : Archimedean α", "a : α", "b : β", "q : ℚ", "x y : α"], "goal": "inducedMap α β (x + y) = inducedMap α β x + inducedMap α β y"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "n : ℤ", "z₁ z₂ : CochainComplex.HomComplex.Cochain F G n", "h : z₁ = z₂", "p q : ℤ", "hpq : p + n = q"], "goal": "z₁.v p q hpq = z₂.v p q hpq"}}
+{"state": {"context": ["p q y : ℝ", "r : ℚ", "m✝ : ℤ", "n✝ : ℕ", "hp : 1 < p", "M : ℤ", "h : LiouvilleWith p ↑M", "n : ℕ", "hn : 0 < n", "m : ℤ", "hne : ¬↑M * ↑n = ↑m", "hlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)", "hn' : 0 < ↑n"], "goal": "M * ↑n ≠ m"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝⁴ : CommMonoid M", "inst✝³ : CommMonoid N", "inst✝² : DivisionCommMonoid G", "k l : ℕ", "inst✝¹ : CommRing R", "ζ✝ : Rˣ", "h✝ : IsPrimitiveRoot ζ✝ k", "inst✝ : IsDomain R", "ζ : R", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n"], "goal": "Fintype.card ↥(rootsOfUnity n R) = ↑n"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "κ : ι → Sort u_7", "s : (i : ι) → κ i → Set α", "t : Set α"], "goal": "(⋂ i, ⋂ j, s i j) ∪ t = ⋂ i, ⋂ j, s i j ∪ t"}}
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+{"state": {"context": ["x✝ y x : ℝ", "hx₁ : -1 ≤ x", "hx₂ : x ≤ 1"], "goal": "cos (arcsin x) = √(1 - x ^ 2)"}}
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+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "E : Type u_4", "F : Type u_6", "SeminormedAddCommGroup E", "SeminormedAddCommGroup F", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜₂", "NormedSpace 𝕜 E", "NormedSpace 𝕜₂ F", "σ₁₂ : 𝕜 →+* 𝕜₂", "RingHomIsometric σ₁₂"], "goal": "𝓤 (E →SL[σ₁₂] F) = ⨅ r, ⨅ (_ : r > 0), 𝓟 {f | ‖f.1 - f.2‖ < r}"}}
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+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "l : Filter ι", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "OpensMeasurableSpace α", "a b : ι → α", "A B : α", "MeasureTheory.NoAtoms μ", "ha : Filter.Tendsto a l (𝓝 A)", "hb : Filter.Tendsto b l (𝓝 B)"], "goal": "MeasureTheory.AECover (μ.restrict (Set.Icc A B)) l fun i => Set.Icc (a i) (b i)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Preorder α", "Preorder β", "l : α → β", "u : β → α", "self : GaloisInsertion l u", "x : β"], "goal": "x ≤ l (u x)"}}
+{"state": {"context": ["R : Type u", "ι : Type w", "s : Finset ι", "inst✝ : CommSemiring R", "f : ι → R[X]", "t : Multiset R[X]", "h : ∀ f ∈ t, f.Monic", "a✝ : Nontrivial R"], "goal": "t.prod.natDegree = (Multiset.map natDegree t).sum"}}
+{"state": {"context": ["K : Type u", "Field K", "p : K[X]", "q : K[X]", "hq : q ≠ 0"], "goal": "((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q).denom =\n  Polynomial.C (q / gcd p q).leadingCoeff⁻¹ * (q / gcd p q)"}}
+{"state": {"context": ["α : Type u_1", "a : Array α", "i j : Fin a.size"], "goal": "(a.swap i j).size = a.size"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "H : Type u_5", "inst✝¹⁵ : CommRing 𝕜", "inst✝¹⁴ : AddCommGroup E", "inst✝¹³ : AddCommGroup F", "inst✝¹² : AddCommGroup G", "inst✝¹¹ : Module 𝕜 E", "inst✝¹⁰ : Module 𝕜 F", "inst✝⁹ : Module 𝕜 G", "inst✝⁸ : TopologicalSpace E", "inst✝⁷ : TopologicalSpace F", "inst✝⁶ : TopologicalSpace G", "inst✝⁵ : TopologicalAddGroup E", "inst✝⁴ : ContinuousConstSMul 𝕜 E", "inst✝³ : TopologicalAddGroup F", "inst✝² : ContinuousConstSMul 𝕜 F", "inst✝¹ : TopologicalAddGroup G", "inst✝ : ContinuousConstSMul 𝕜 G", "p : FormalMultilinearSeries 𝕜 E F", "n : ℕ", "hn : 0 < n", "v : Fin n → E", "i : ℕ", "hi2 : i < n", "j_val : ℕ", "j_property : j_val < (Composition.single n hn).length", "hi1 : i < (Composition.single n hn).blocksFun ⟨j_val, j_property⟩", "this : j_val = 0"], "goal": "((Composition.single n hn).embedding ⟨j_val, j_property⟩) ⟨i, hi1⟩ = ((Composition.single n hn).embedding 0) ⟨i, hi2⟩"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)", "r : ℝ≥0", "hr : ↑r < p.radius"], "goal": "↑r ≤ liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "F.Final", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "G : CategoryTheory.Functor D E"], "goal": "(CategoryTheory.Functor.Final.coconesEquiv F G).counitIso =\n  CategoryTheory.NatIso.ofComponents\n    (fun c =>\n      CategoryTheory.Limits.Cocones.ext\n        (CategoryTheory.Iso.refl\n          (((CategoryTheory.Limits.Cocones.whiskering F).comp CategoryTheory.Functor.Final.extendCocone).obj c).pt)\n        ⋯)\n    ⋯"}}
+{"state": {"context": ["X Y : Type u_1", "Add X", "Add Y", "e : X ≃+ Y"], "goal": "⇑↑e = ⇑e"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "s : Stream' (Option α)", "al : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, s n = some a → s (n + 1) = some a"], "goal": "↑(map f (think ⟨s, al⟩)) = ↑(map f ⟨s, al⟩).think"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "∀ (i : 𝒰.J), CategoryTheory.Limits.HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "AlgebraicGeometry.Scheme.Pullback.t' 𝒰 f g i j k ≫\n    AlgebraicGeometry.Scheme.Pullback.t' 𝒰 f g j k i ≫\n      AlgebraicGeometry.Scheme.Pullback.t' 𝒰 f g k i j ≫\n        CategoryTheory.Limits.pullback.fst (AlgebraicGeometry.Scheme.Pullback.fV 𝒰 f g i j)\n            (AlgebraicGeometry.Scheme.Pullback.fV 𝒰 f g i k) ≫\n          CategoryTheory.Limits.pullback.fst (CategoryTheory.Limits.pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j) ≫\n            CategoryTheory.Limits.pullback.snd (𝒰.map i ≫ f) g =\n  CategoryTheory.Limits.pullback.fst (AlgebraicGeometry.Scheme.Pullback.fV 𝒰 f g i j)\n      (AlgebraicGeometry.Scheme.Pullback.fV 𝒰 f g i k) ≫\n    CategoryTheory.Limits.pullback.fst (CategoryTheory.Limits.pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j) ≫\n      CategoryTheory.Limits.pullback.snd (𝒰.map i ≫ f) g"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "CompactSpace X", "A : Subalgebra ℝ C(X, ℝ)", "f g : ↥A"], "goal": "↑f ⊓ ↑g ∈ A.topologicalClosure"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : CoheytingAlgebra α", "a✝ b✝ c a b : α"], "goal": "Codisjoint (a ⊔ b) (￢(a ⊔ b))"}}
+{"state": {"context": ["σ : Type u_1"], "goal": "Finsupp.degree = ⇑(Finsupp.weight 1)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "I : Ideal R", "M : Type u_2", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "m : M", "i : ℕ", "a : R", "a_in : a ∈ I ^ i"], "goal": "a ∈ (fun x => x • m) ⁻¹' ↑(I ^ i • ⊤)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : X ≃ₜ Y", "s : Set X", "hs : IsOpen s", "t : Set Y", "h : ⇑e '' s = t"], "goal": "↑(e.toPartialHomeomorphOfImageEq s hs t h).symm = ⇑e.symm"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "A : Type u_1", "B : Type u_2", "CategoryTheory.Category.{u_4, u_1} A", "CategoryTheory.Category.{u_3, u_2} B", "F : CategoryTheory.Functor A B", "CategoryTheory.HasWeakSheafify J A", "CategoryTheory.HasWeakSheafify J B", "J.HasSheafCompose F", "J.PreservesSheafification F", "P : CategoryTheory.Functor Cᵒᵖ A"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.whiskerRight (CategoryTheory.toSheafify J P) F)\n    (CategoryTheory.sheafifyComposeIso J F P).inv =\n  CategoryTheory.toSheafify J (P.comp F)"}}
+{"state": {"context": ["R : Type uR", "CommRing R", "A : Type uA", "AddCommGroup A", "B : Type uB", "AddCommGroup B", "Module R A", "Module R B", "c : CharacterModule (TensorProduct R A B)"], "goal": "∀ (x : A), (CharacterModule.curry c) x = AddMonoidHom.comp c ↑((TensorProduct.mk R A B) x)"}}
+{"state": {"context": ["x y : Bool"], "goal": "(x || y) = (y || x)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace ℝ F", "inst✝⁶ : NormedAddCommGroup F'", "inst✝⁵ : NormedSpace ℝ F'", "inst✝⁴ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "T T' T'' : Set α → E →L[ℝ] F", "C C' C'' : ℝ", "hT : DominatedFinMeasAdditive μ T C", "hT' : DominatedFinMeasAdditive μ T' C'", "hT'' : DominatedFinMeasAdditive μ T'' C''", "h_add : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → T'' s = T s + T' s", "f✝ : ↥(Lp E 1 μ)", "f : ↥(simpleFunc E 1 μ)"], "goal": "(setToL1 hT) ↑f + (setToL1 hT') ↑f = (setToL1SCLM α E μ hT'') f"}}
+{"state": {"context": ["α : Type u_1", "NormedLatticeAddCommGroup α", "a b c d : α"], "goal": "‖a ⊓ b - c ⊓ d‖ ≤ ‖a - c‖ + ‖b - d‖"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "h : S₁.RightHomologyData", "CategoryTheory.Epi φ.τ₁", "CategoryTheory.IsIso φ.τ₂", "CategoryTheory.Mono φ.τ₃"], "goal": "(CategoryTheory.ShortComplex.RightHomologyData.ofEpiOfIsIsoOfMono φ h).H = h.H"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p q : ℝ≥0"], "goal": "ℝ≥0∞ ≃o ↑(Iic 1)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "x : X", "p : Y → Prop", "s : Set X", "hx : x ∈ e.source"], "goal": "(∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)"}}
+{"state": {"context": ["n m : Nat", "h₁ : n ≤ m", "h₂ : m < n + 1"], "goal": "m = n"}}
+{"state": {"context": ["α : Type u", "Small.{v, u} α"], "goal": "Cardinal.lift.{max u w, v} (Cardinal.mk (Shrink.{v, u} α)) = Cardinal.lift.{max v w, u} (Cardinal.mk α)"}}
+{"state": {"context": ["a b r q : Int", "h : 0 < b"], "goal": "a / b = q ∧ a % b = r ↔ r + b * q = a ∧ 0 ≤ r ∧ r < b"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "β : Type u_2", "MeasurableSpace β", "μ : MeasureTheory.FiniteMeasure α", "ν : MeasureTheory.FiniteMeasure β"], "goal": "(μ.prod ν).map Prod.fst = ν Set.univ • μ"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "s : Set G", "t : Set P", "a✝ : P", "inst✝³ : PartialOrder G", "inst✝² : PartialOrder P", "inst✝¹ : SMul G P", "inst✝ : IsOrderedCancelSMul G P", "x y : ↑(s.smulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : P", "h : (s.smulAntidiagonal t a).Infinite", "h1 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.smulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "SuccOrder α", "PredOrder α", "a : α"], "goal": "Order.pred (Order.succ a) ≤ a"}}
+{"state": {"context": ["V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P", "v v' : V", "p p' : P"], "goal": "edist (v +ᵥ p) (v' +ᵥ p') ≤ edist v v' + edist p p'"}}
+{"state": {"context": ["R : Type u", "M₁ : Type v", "M₂ : Type w", "CommRing R", "AddCommGroup M₁", "Module R M₁", "AddCommGroup M₂", "Module R M₂", "e : M₁ ≃ₗ[R] M₂", "f : Module.End R M₁"], "goal": "e.lieConj f = e.conj f"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A"], "goal": "NonUnitalAlgebra.adjoin R ∅ = ⊥"}}
+{"state": {"context": ["a b m n✝ p : ℕ", "β : Type u_1", "inst✝ : CommMonoid β", "n : ℕ", "f : ℕ → β"], "goal": "(gcd 0 n).primeFactors.prod f * (0 * n).primeFactors.prod f = (primeFactors 0).prod f * n.primeFactors.prod f"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u_3", "B : Type u_4", "inst✝³¹ : CommRing A", "inst✝³⁰ : CommRing B", "inst✝²⁹ : Algebra A B", "inst✝²⁸ : Field K", "inst✝²⁷ : Field L", "inst✝²⁶ : Algebra A K", "inst✝²⁵ : IsFractionRing A K", "inst✝²⁴ : Algebra B L", "inst✝²³ : Algebra K L", "inst✝²² : Algebra A L", "inst✝²¹ : IsScalarTower A B L", "inst✝²⁰ : IsScalarTower A K L", "inst✝¹⁹ : IsIntegralClosure B A L", "inst✝¹⁸ : FiniteDimensional K L", "Aₘ : Type ?u.592608", "Bₘ : Type ?u.592611", "inst✝¹⁷ : CommRing Aₘ", "inst✝¹⁶ : CommRing Bₘ", "inst✝¹⁵ : Algebra Aₘ Bₘ", "inst✝¹⁴ : Algebra A Aₘ", "inst✝¹³ : Algebra B Bₘ", "inst✝¹² : Algebra A Bₘ", "inst✝¹¹ : IsScalarTower A Aₘ Bₘ", "inst✝¹⁰ : IsScalarTower A B Bₘ", "M : Submonoid A", "inst✝⁹ : IsLocalization M Aₘ", "inst✝⁸ : IsLocalization (algebraMapSubmonoid B M) Bₘ", "inst✝⁷ : IsIntegrallyClosed A", "inst✝⁶ : IsDomain A", "inst✝⁵ : IsIntegrallyClosed A", "inst✝⁴ : IsDomain B", "inst✝³ : IsIntegrallyClosed B", "inst✝² : Module.Finite A B", "inst✝¹ : NoZeroSMulDivisors A B", "inst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)", "this✝ : IsIntegralClosure B A (FractionRing B)", "this : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)"], "goal": "(intNorm A B) 0 = 0"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "c : CategoryTheory.Limits.Cone F"], "goal": "(CategoryTheory.Limits.Cones.eta c).inv.hom = 𝟙 c.pt"}}
+{"state": {"context": ["a x z✝ z : ℂ"], "goal": "z.arg = 0 ↔ 0 ≤ z"}}
+{"state": {"context": ["M : Type u_1", "inst✝³ : Monoid M", "ι : Type u_2", "inst✝² : DecidableEq ι", "inst✝¹ : Fintype ι", "N : ι → Type u_3", "inst✝ : (i : ι) → Monoid (N i)", "ϕ : (i : ι) → N i →* M", "hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)", "f g : (i : ι) → N i", "i : ι", "y : N i"], "goal": "(ϕ i) y * ?m ^ (Finset.univ.erase i).card = (ϕ i) y"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l₁ l₂✝ : List α", "r : α → α → Prop", "a b : α", "l₂ : List β", "d₁ : l₁.Nodup", "d₂ : l₂.Nodup", "a₁ : α", "b₁ : β", "left✝ : b₁ ∈ l₂", "h₁ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂", "n : a₁ ≠ a₁", "h₂ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂", "mb₂ : b₁ ∈ l₂"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "p : Associates α", "s : Associates.FactorSet α", "h : p ∈ s"], "goal": "Irreducible p"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : BooleanAlgebra α", "a b c d : α"], "goal": "bᶜ \\ aᶜ ⊔ aᶜ \\ bᶜ = a ∆ b"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "s : Set E", "hs : Convex ℝ s", "hspan : (interior s).Nonempty"], "goal": "μ (frontier s) = 0"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_3", "m : MeasurableSpace Ω", "LinearOrder ι", "f : MeasureTheory.Filtration ι m", "τ : Ω → ι", "hτ : MeasureTheory.IsStoppingTime f τ", "s : Set Ω", "i : ι"], "goal": "MeasurableSet (s ∩ {ω | τ ω ≤ i}) ↔ MeasurableSet (s ∩ {ω | τ ω ≤ i})"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : DecidableEq R", "x✝² : (p : R[X]) ×' p ≠ 0", "a✝¹ : ∀ (y : (p : R[X]) ×' p ≠ 0), (invImage (fun x => PSigma.casesOn x fun p x => p.natDegree) instWellFoundedRelationOfSizeOf).1 y x✝² → ∃ s, ↑(Multiset.card s) ≤ y.1.degree ∧ ∀ (a : R), count a s = rootMultiplicity a y.1", "p✝ : R[X]", "x✝¹ : p✝ ≠ 0", "a✝ : ∀ (y : (p : R[X]) ×' p ≠ 0), (invImage (fun x => PSigma.casesOn x fun p x => p.natDegree) instWellFoundedRelationOfSizeOf).1 y ⟨p✝, x✝¹⟩ → ∃ s, ↑(Multiset.card s) ≤ y.1.degree ∧ ∀ (a : R), count a s = rootMultiplicity a y.1", "p : R[X]", "hp : p ≠ 0", "x✝ : ∀ (y : (p : R[X]) ×' p ≠ 0), (invImage (fun x => PSigma.casesOn x fun p x => p.natDegree) instWellFoundedRelationOfSizeOf).1 y ⟨p, hp⟩ → ∃ s, ↑(Multiset.card s) ≤ y.1.degree ∧ ∀ (a : R), count a s = rootMultiplicity a y.1", "h : ∃ x, p.IsRoot x", "x : R", "hx : p.IsRoot x", "hpd : 0 < p.degree", "hd0 : p /ₘ (X - C x) ≠ 0", "wf : (p /ₘ (X - C x)).degree < p.degree"], "goal": "↑(p /ₘ (X - C x)).natDegree < ↑p.natDegree"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "f₀ f₁ : C(X, Y)", "P : C(X, Y) → Prop", "F : f₀.HomotopyWith f₁ P", "x : X"], "goal": "F (1, x) = f₁ x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "𝕜' : Type u_3", "NonUnitalNormedRing 𝕜'", "NormedSpace 𝕜 𝕜'", "IsScalarTower 𝕜 𝕜' 𝕜'", "SMulCommClass 𝕜 𝕜' 𝕜'", "RegularNormedAlgebra 𝕜 𝕜'", "Nontrivial 𝕜'"], "goal": "‖ContinuousLinearMap.mul 𝕜 𝕜'‖ = 1"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "p : ℕ → Prop", "n : ℕ", "hn : n ≠ 0", "N : ℕ → ℕ", "hN : ∀ k < n, ∀ b ≥ N k, p (n * b + k)", "b : ℕ", "hb : b ≥ (Finset.range n).sup fun x => n * N x", "hlt : b % n < n"], "goal": "b / n ≥ N (b % n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t : Set α", "a : α", "s : Set α"], "goal": "(insert a s).ncard ≤ s.ncard + 1"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : α → β → β", "op : α → α → α", "inst✝³ : Monoid α", "inst✝² : Monoid β", "inst✝¹ : FunLike F α β", "inst✝ : MonoidHomClass F α β", "s : Multiset α", "comm : {x | x ∈ s}.Pairwise Commute", "f : F"], "goal": "{x | ∃ a ∈ s, f a = x}.Pairwise Commute"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "G : SimpleGraph α", "H : SimpleGraph β", "a : α", "b₁ b₂ : β"], "goal": "(G □ H).Adj (a, b₁) (a, b₂) ↔ H.Adj b₁ b₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "m : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "f : α → β → γ", "hf : Measurable (Function.uncurry f)", "y : β"], "goal": "Measurable fun x => f x y"}}
+{"state": {"context": ["R : Type u₁", "S : Type u₂", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "f : R →+* S", "M✝ M : ModuleCat R", "x : ↑(obj' f M)"], "goal": "(LinearMap.baseChange S (𝟙 M)) x = x"}}
+{"state": {"context": ["α : Type u", "σ : Type v", "M : NFA α σ", "S : Set σ", "a : α"], "goal": "M.evalFrom S [a] = M.stepSet S a"}}
+{"state": {"context": ["R : Type u", "L : Type v", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "M : Type u_1", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : LieRingModule L M", "inst✝³ : LieModule R L M", "M' : Type u_2", "inst✝² : AddCommGroup M'", "inst✝¹ : Module R M'", "inst✝ : LieRingModule L M'", "e : M ≃ₗ⁅R,L⁆ M'"], "goal": "Function.Surjective ⇑e.toLieModuleHom"}}
+{"state": {"context": ["B : Type u₁", "inst✝⁴ : Category.{v₁, u₁} B", "C : Type u₂", "inst✝³ : Category.{v₂, u₂} C", "D : Type u₃", "inst✝² : Category.{v₃, u₃} D", "E : Type u₄", "inst✝¹ : Category.{v₄, u₄} E", "H : Type u₅", "inst✝ : Category.{v₅, u₅} H", "F : C ⥤ D ⥤ E", "X✝ Y✝ Z✝ : C × D", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝"], "goal": "(F.map f.1).app X✝.2 ≫ ((F.obj Y✝.1).map f.2 ≫ (F.map g.1).app Y✝.2) ≫ (F.obj Z✝.1).map g.2 = (F.map f.1).app X✝.2 ≫ (F.obj Y✝.1).map f.2 ≫ (F.map g.1).app Y✝.2 ≫ (F.obj Z✝.1).map g.2"}}
+{"state": {"context": ["G : Type u", "α : Type w", "m : MeasurableSpace α", "Group G", "MulAction G α", "MeasurableSpace G", "MeasurableSMul G α", "μ : MeasureTheory.Measure α"], "goal": "[MeasureTheory.SMulInvariantMeasure G α μ, ∀ (c : G) (s : Set α), MeasurableSet s → μ ((fun x => c • x) ⁻¹' s) = μ s,\n    ∀ (c : G) (s : Set α), MeasurableSet s → μ (c • s) = μ s, ∀ (c : G) (s : Set α), μ ((fun x => c • x) ⁻¹' s) = μ s,\n    ∀ (c : G) (s : Set α), μ (c • s) = μ s, ∀ (c : G), MeasureTheory.Measure.map (fun x => c • x) μ = μ,\n    ∀ (c : G), MeasureTheory.MeasurePreserving (fun x => c • x) μ μ].TFAE"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "N : Type u_6", "AddGroup N", "H : AddSubgroup G", "f : N →+ G", "hf : Function.Injective ⇑f", "h : H.normalizer ≤ f.range"], "goal": "AddSubgroup.comap f H.normalizer = (AddSubgroup.comap f H).normalizer"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "ps : Set P", "n : ℕ", "FiniteDimensional ℝ V", "hd : FiniteDimensional.finrank ℝ V = n", "hc : EuclideanGeometry.Cospherical ps"], "goal": "∃ c, ∀ (sx : Affine.Simplex ℝ P n), Set.range sx.points ⊆ ps → sx.circumcenter = c"}}
+{"state": {"context": ["F : Type u", "Field F", "f g : F[X]", "h1 : f ∣ g", "h2 : g ≠ 0"], "goal": "f.natSepDegree ≤ g.natSepDegree"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V"], "goal": "Function.Surjective SimpleGraph.Walk.reverse"}}
+{"state": {"context": ["p : ℕ", "hp : Nat.Prime p"], "goal": "p.primeFactors = {p}"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α"], "goal": "Nat.card { x // x ∈ s } = s.card"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : MeasurableSpace α'", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace β'", "inst✝³ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝² : NormedAddCommGroup E", "inst✝¹ : SFinite ν", "inst✝ : SFinite μ", "s : Set (α × β)", "hs : MeasurableSet s"], "goal": "(μ.prod ν) s = ∫⁻ (y : β), μ ((fun x => (x, y)) ⁻¹' s) ∂ν"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : Abelian C", "S : ShortComplex C"], "goal": "Epi (imageToKernel' S.f S.g ⋯) ↔ Epi (imageToKernel S.f S.g ⋯)"}}
+{"state": {"context": ["p : ℕ+", "K : Type u", "Field K", "CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p} ℚ K", "hζ : IsPrimitiveRoot ζ ↑p"], "goal": "hζ.toInteger - 1 ∣ ↑↑p"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : C ⥤ D", "F.ReflectsMonomorphisms", "X Y : C", "f : X ⟶ Y", "h : CategoryTheory.Mono (F.map f)"], "goal": "CategoryTheory.Mono f"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "P : (H → H') → Set H → H → Prop", "s✝ t u : Set H", "x✝ : H", "hG : G.LocalInvariantProp G' P", "s : Set H", "x : H", "f : H → H'", "e : PartialHomeomorph H H", "he : e ∈ G", "hxe : x ∈ e.source", "h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)", "this : P ((f ∘ ↑e.symm) ∘ ↑e) (↑e ⁻¹' (↑e.symm ⁻¹' s)) x"], "goal": "↑e ⁻¹' (↑e.symm ⁻¹' s) =ᶠ[𝓝 x] s"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroMorphisms D", "F : C ⥤ D", "F.PreservesZeroMorphisms", "J : Type w₁", "f : J → C", "CategoryTheory.Limits.HasBiproduct f", "CategoryTheory.Limits.PreservesBiproduct f F", "W : C", "g : (j : J) → f j ⟶ W"], "goal": "(F.mapBiproduct f).inv ≫ F.map (CategoryTheory.Limits.biproduct.desc g) =\n  CategoryTheory.Limits.biproduct.desc fun j => F.map (g j)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : α ≃ β", "s : Set α", "t : Set β", "h : ⇑e '' s = t"], "goal": "(e.toPartialEquivOfImageEq s t h).source = s"}}
+{"state": {"context": ["x : ZFSet", "h : x.IsTransitive"], "goal": "x.powerset.IsTransitive"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Finite ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x✝ y : ι → ℝ", "δ : ℝ", "hs : IsClosed s", "hs' : BddBelow s", "val✝ : Fintype ι", "f : ℕ → ι → ℝ", "x : ι → ℝ", "hx : Filter.Tendsto f Filter.atTop (nhds x)", "g : ℕ → ι → ℝ", "hg : ∀ (n : ℕ), g n ∈ s", "hgf : ∀ (n : ℕ), g n ≤ f n", "a : ι → ℝ", "ha : a ∈ upperBounds (range f)"], "goal": "x ∈ ↑(upperClosure s)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "xs : LazyList α", "ys : Thunk (LazyList α)", "f : α → LazyList β"], "goal": "(xs.append ys).bind f = (xs.bind f).append { fn := fun x => ys.get.bind f }"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t✝ : Set α", "f : α → β", "s t : Set α", "h : (s ∪ t).Finite"], "goal": "(s ∪ t).ncard ≤ s.ncard + t.ncard"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹⁷ : CommRing R", "M : Submonoid R", "S : Type u_2", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : Algebra R S", "P : Type u_3", "inst✝¹⁴ : CommRing P", "A : Type u_4", "K : Type u_5", "inst✝¹³ : CommRing A", "inst✝¹² : IsDomain A", "L : Type u_6", "inst✝¹¹ : Field K", "inst✝¹⁰ : Field L", "inst✝⁹ : Algebra A K", "inst✝⁸ : Algebra A L", "inst✝⁷ : IsFractionRing A K", "C : Type u_7", "inst✝⁶ : CommRing C", "inst✝⁵ : IsDomain C", "inst✝⁴ : Algebra C L", "inst✝³ : IsIntegralClosure C A L", "inst✝² : Algebra A C", "inst✝¹ : IsScalarTower A C L", "inst✝ : Algebra.IsAlgebraic A L", "inj : ∀ (x : A), (algebraMap A L) x = 0 → x = 0", "z : L", "x : ↥(integralClosure A L)", "y : A", "hy : y ≠ 0", "hxy : z * (algebraMap A L) y = ↑x", "h : (algebraMap A C) y = 0"], "goal": "(algebraMap A L) y = 0"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "AddMonoidWithOne α", "Invertible 2", "a : α", "Invertible a"], "goal": "⅟(SymAlg.sym a) = SymAlg.sym ⅟a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : PseudoMetricSpace α", "x y z : α", "s✝ t : Set α", "inst✝ : DecidableEq α", "s : Finset α"], "goal": "(↑s).infsep = if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "l l' : Filter α", "p : ι → Prop", "s✝ : ι → Set α", "t✝ : Set α", "i : ι", "p' : ι' → Prop", "s' : ι' → Set α", "i' : ι'", "s : β → Set α", "S : Set β", "h : DirectedOn (s ⁻¹'o fun x x_1 => x ≥ x_1) S", "ne : S.Nonempty", "t : Set α"], "goal": "t ∈ ⨅ i ∈ S, 𝓟 (s i) ↔ ∃ i ∈ S, s i ⊆ t"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α"], "goal": "EMetric.diam s = ⊤ ↔ ¬Bornology.IsBounded s"}}
+{"state": {"context": ["α : Type u", "s : Set α", "inst✝ : DivisionRing α", "a : ↥(closure s)", "x✝ : a ∈ ⊤"], "goal": "↑a ∈ Subtype.val '' ↑(Subfield.rangeOfWType s)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : CommSemiring α", "s : Multiset ι", "f g : ι → α"], "goal": "(map (fun i => f i + g i) 0).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) (antidiagonal 0)).sum"}}
+{"state": {"context": ["X : TopCat", "T : Type v", "U V : (TopologicalSpace.Opens ↑X)ᵒᵖ", "i : U ⟶ V", "f : (X.presheafToType T).obj U"], "goal": "(X.presheafToType T).map i f = f ∘ fun x => ⟨↑x, ⋯⟩"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "ϖ : R", "hϖ : Irreducible ϖ", "hR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x", "p : R", "hp : Irreducible p"], "goal": "Associated p ϖ"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "p : α → Prop", "x y : α", "hx : y ≤ x → p x", "hy : x ≤ y → p y"], "goal": "p (max x y)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "la : Filter α", "lb : Filter β", "p : α → β → Prop", "h : ∃ᶠ (x : α) in la, ∃ᶠ (y : β) in lb, p x y"], "goal": "∃ᶠ (xy : α × β) in la ×ˢ lb, p xy.1 xy.2"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹² : MeasurableSpace X", "G : Type u_5", "𝕜 : Type u_6", "inst✝¹¹ : TopologicalSpace X", "inst✝¹⁰ : TopologicalSpace Y", "inst✝⁹ : MeasurableSpace Y", "inst✝⁸ : OpensMeasurableSpace Y", "μ : Measure Y", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : NormedSpace 𝕜 E", "L : F →L[𝕜] G →L[𝕜] E", "f : X → Y → G", "s : Set X", "k : Set Y", "g : Y → F", "hk : IsCompact k", "hf : ContinuousOn (uncurry f) (s ×ˢ univ)", "hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0", "hg : IntegrableOn g k μ"], "goal": "ContinuousOn (fun x => ∫ (y : Y), (L (g y)) (f x y) ∂μ) s"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "κ : Type u_3", "κ' : Type u_4", "R : Type u_5", "M : Type u_6", "inst✝⁶ : CommSemiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "R₂ : Type u_7", "M₂ : Type u_8", "inst✝³ : CommRing R₂", "inst✝² : AddCommGroup M₂", "inst✝¹ : Module R₂ M₂", "e : Basis ι R M", "v : ι' → M", "i : ι", "j : ι'", "inst✝ : DecidableEq ι'", "x : M", "i' : ι", "k : ι'"], "goal": "e.toMatrix (update v j x) i' k = (e.toMatrix v).updateColumn j (⇑(e.repr x)) i' k"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "E' : Type u_3", "F : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹⁸ : RCLike 𝕜", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : InnerProductSpace 𝕜 E", "inst✝¹⁵ : CompleteSpace E", "inst✝¹⁴ : NormedAddCommGroup E'", "inst✝¹³ : InnerProductSpace 𝕜 E'", "inst✝¹² : CompleteSpace E'", "inst✝¹¹ : NormedSpace ℝ E'", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace 𝕜 F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup G'", "inst✝⁶ : NormedSpace ℝ G'", "inst✝⁵ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t✝ : Set α", "E'' : Type u_8", "𝕜' : Type u_9", "inst✝⁴ : RCLike 𝕜'", "inst✝³ : NormedAddCommGroup E''", "inst✝² : InnerProductSpace 𝕜' E''", "inst✝¹ : CompleteSpace E''", "inst✝ : NormedSpace ℝ E''", "hm : m ≤ m0", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "x : E'", "t : Set α", "ht : MeasurableSet t", "hμt : μ t ≠ ⊤"], "goal": "∫⁻ (a : α) in t, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ * ↑‖x‖₊ ∂μ = (∫⁻ (a : α) in t, ↑‖↑↑↑((condexpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1)) a‖₊ ∂μ) * ↑‖x‖₊"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "r : ℝ", "hr : r ≠ 0", "x : ↑(Metric.sphere 0 r)"], "goal": "↑x ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "β : Type u_2", "DivisionRing β", "abv : β → α", "IsAbsoluteValue abv"], "goal": "0 ≠ 1"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹¹ : CommRing R", "inst✝¹⁰ : AddCommGroup E", "inst✝⁹ : AddCommGroup F", "inst✝⁸ : Module R E", "inst✝⁷ : Module R F", "inst✝⁶ : TopologicalSpace E", "inst✝⁵ : TopologicalSpace F", "inst✝⁴ : ContinuousAdd E", "inst✝³ : ContinuousAdd F", "inst✝² : TopologicalSpace R", "inst✝¹ : ContinuousSMul R E", "inst✝ : ContinuousSMul R F", "f : E →ₗ.[R] F", "hf : f.IsClosable", "x✝¹ x✝ : E →ₗ.[R] F", "hy₁ : f.graph.topologicalClosure = x✝¹.graph", "hy₂ : f.graph.topologicalClosure = x✝.graph"], "goal": "x✝¹.graph = x✝.graph"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : Finite β"], "goal": "closure {σ | σ.IsCycle} = ⊤"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝³ : Field K", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : Algebra.IsIntegral K L"], "goal": "IsTotallyDisconnected Set.univ"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝²³ : NormedAddCommGroup E", "inst✝²² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝²¹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝²⁰ : TopologicalSpace M", "inst✝¹⁹ : ChartedSpace H M", "Is : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹⁸ : NormedAddCommGroup E'", "inst✝¹⁷ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁶ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹⁵ : TopologicalSpace M'", "inst✝¹⁴ : ChartedSpace H' M'", "I's : SmoothManifoldWithCorners I' M'", "F : Type u_8", "inst✝¹³ : NormedAddCommGroup F", "inst✝¹² : NormedSpace 𝕜 F", "G : Type u_9", "inst✝¹¹ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N✝ : Type u_10", "inst✝¹⁰ : TopologicalSpace N✝", "inst✝⁹ : ChartedSpace G N✝", "Js : SmoothManifoldWithCorners J N✝", "F' : Type u_11", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace 𝕜 F'", "G' : Type u_12", "inst✝⁶ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_13", "inst✝⁵ : TopologicalSpace N'", "inst✝⁴ : ChartedSpace G' N'", "J's : SmoothManifoldWithCorners J' N'", "F₁ : Type u_14", "inst✝³ : NormedAddCommGroup F₁", "inst✝² : NormedSpace 𝕜 F₁", "F₂ : Type u_15", "inst✝¹ : NormedAddCommGroup F₂", "inst✝ : NormedSpace 𝕜 F₂", "f f₁ : M → M'", "s s₁ t : Set M", "x✝ : M", "m n : ℕ∞", "x : M", "v : TangentSpace I x", "N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))", "A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x", "B : (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))) v = (v, 0)"], "goal": "(fderivWithin 𝕜 (fun x_1 => (↑I (↑(ChartedSpace.chartAt x) (↑(ChartedSpace.chartAt x).symm (↑I.symm x_1))), 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))) v = (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))) v"}}
+{"state": {"context": [], "goal": "doublingGamma 1 = 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hε₁ : ε ≤ 1", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "hUVne : U ≠ V", "hUV : ¬G.IsUniform ε U V"], "goal": "3 / 4 * ε ≤ |(∑ ab ∈ (star hP G ε hU V).product (star hP G ε hV U), ↑(G.edgeDensity ab.1 ab.2)) / (↑(star hP G ε hU V).card * ↑(star hP G ε hV U).card) - (∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / 16 ^ P.parts.card|"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b"], "goal": "FermatLastTheoremForThreeGen.Solution.x S * FermatLastTheoremForThreeGen.Solution.y S * FermatLastTheoremForThreeGen.Solution.z S = ↑S.u * FermatLastTheoremForThreeGen.Solution.w S ^ 3"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l₁ l₂ : List α", "r : α → α → Prop", "a b : α", "inst✝ : DecidableEq α", "f : α → β", "x : α", "y : β", "hd : α", "tl : List α", "ihl : tl.Nodup → map (update f x y) tl = if x ∈ tl then (map f tl).set (indexOf x tl) y else map f tl", "hl : hd ∉ tl ∧ tl.Nodup"], "goal": "(update f x y hd :: if x ∈ tl then (map f tl).set (indexOf x tl) y else map f tl) = if x = hd ∨ x ∈ tl then (f hd :: map f tl).set (indexOf x (hd :: tl)) y else f hd :: map f tl"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝¹ : Lattice α", "inst✝ : OrderBot α", "s t : Finset ι", "f : ι → α", "i : ι", "hs : s.SupIndep f", "hts : t ⊆ s", "hi : i ∈ s", "hf : ∀ (i : ι), f i ≠ ⊥", "h : f i ≤ t.sup f"], "goal": "i ∈ t"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "M : (i : ι) → Matroid (α i)"], "goal": "(Matroid.sigma M).E = Set.univ.sigma fun i => (M i).E"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "Preorder α", "h : Set.Unbounded (fun x x_1 => x ≥ x_1) s"], "goal": "Set.Unbounded (fun x x_1 => x > x_1) s"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : Set α"], "goal": "s ⊆ ↑(upperClosure s)"}}
+{"state": {"context": ["α : Type u_7", "β : Type u_8", "Lattice α", "Lattice β", "self : LatticeHom α β", "a b : α"], "goal": "self.toFun (a ⊓ b) = self.toFun a ⊓ self.toFun b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s t : Finset α", "h : s ⊆ t", "u : Finset α"], "goal": "s ≤ u ∧ u ≤ t ↔ ∃ a ⊆ t \\ s, s ∪ a = u"}}
+{"state": {"context": ["α : Type u_1", "CompleteLattice α", "One α"], "goal": "sSup 1 = 1"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "s : ι → Set α", "hs : IndexedPartition s", "β : Type u_3", "f : ι → α → β", "h_injOn : ∀ (i : ι), InjOn (f i) (s i)", "h_disjoint : univ.PairwiseDisjoint fun i => f i '' s i", "x y : α", "h : Disjoint (f (hs.index x) '' s (hs.index x)) (f (hs.index y) '' s (hs.index y))"], "goal": "hs.piecewise f x ≠ hs.piecewise f y"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "V₂ : Type u_5", "P : Type u_10", "P₂ : Type u_11", "NormedField 𝕜", "SeminormedAddCommGroup V", "NormedSpace 𝕜 V", "PseudoMetricSpace P", "NormedAddTorsor V P", "SeminormedAddCommGroup V₂", "NormedSpace 𝕜 V₂", "PseudoMetricSpace P₂", "NormedAddTorsor V₂ P₂", "f : P →ᵃⁱ[𝕜] P₂"], "goal": "LipschitzWith 1 ⇑f"}}
+{"state": {"context": ["Γ : Subgroup SL(2, ℤ)", "z : ℤ"], "goal": "⇑↑z = ↑z"}}
+{"state": {"context": ["d : ℕ", "a : ℤ√↑d", "n : ℕ", "ha : a.Nonneg"], "goal": "(↑↑n * a).Nonneg"}}
+{"state": {"context": ["x y : ℤ"], "goal": "toComplex { re := x, im := y } = ↑x + ↑y * I"}}
+{"state": {"context": ["F : Type u", "E : Type v", "Field F", "Field E", "Algebra F E", "Algebra.IsAlgebraic F E", "hdeg : Field.finSepDegree F E = 1"], "goal": "IsPurelyInseparable F E"}}
+{"state": {"context": ["m n a b c d : ℕ", "h : a ≡ b [MOD n.gcd m]", "hn : n ≠ 0", "hm : m ≠ 0"], "goal": "↑(if hn : n = 0 then ⟨a, ⋯⟩ else if hm : m = 0 then ⟨b, ⋯⟩ else ⟨((↑n * (n.xgcd m).1 * ↑b + ↑m * (n.xgcd m).2 * ↑a) / ↑(n.gcd m) % ↑(n.lcm m)).toNat, ⋯⟩) < n.lcm m"}}
+{"state": {"context": ["X W : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : W ⟶ X", "x : ↑↑W.toPresheafedSpace"], "goal": "(𝒰.pullbackCover' f).f x = 𝒰.f (f.val.base x)"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_2", "F : Type u_3", "CommRing R", "AddCommGroup E", "AddCommGroup F", "Module R E", "Module R F", "TopologicalSpace E", "TopologicalSpace F", "ContinuousAdd E", "ContinuousAdd F", "TopologicalSpace R", "ContinuousSMul R E", "ContinuousSMul R F", "f : E →ₗ.[R] F", "hf : f.IsClosable"], "goal": "f.graph.topologicalClosure = f.closure.graph"}}
+{"state": {"context": ["n : ℕ", "p i : Fin n", "h : p < i", "hi : i.castSucc ≠ 0 := ⋯"], "goal": "p.predAbove i.castSucc = i.castSucc.pred hi"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_4", "Zero M", "t : Set α", "s : Set M"], "goal": "t.indicator 0 ⁻¹' s ∈ {Set.univ, ∅}"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : ℝ → F", "K : Set F", "L : F", "r ε δ : ℝ", "h : ε ≤ δ", "x r' : ℝ", "r'r : r' ∈ Ioc (r / 2) r", "hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r"], "goal": "x ∈ A f L r δ"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "DecidableEq α", "s : Finset ι", "f : ι → Finset α", "hs : (↑s).PairwiseDisjoint f"], "goal": "s.card ≤ (s.biUnion f).card + (Finset.filter (fun i => f i = ∅) s).card"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝⁴ : Ring R", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : IsArtinian R M", "inst✝ : IsNoetherian R M", "f : M →ₗ[R] M", "k : ℕ", "hk : ∀ b ≥ k, IsCompl (ker (f ^ b)) (range (f ^ b)) ∧ ⨅ m, range (f ^ m) = range (f ^ b) ∧ ⨆ m, ker (f ^ m) = ker (f ^ b)", "h₁ : IsCompl (ker (f ^ k)) (range (f ^ k))", "h₂ : ⨅ m, range (f ^ m) = range (f ^ k)", "h₃ : ⨆ m, ker (f ^ m) = ker (f ^ k)"], "goal": "IsCompl (⨆ n, ker (f ^ n)) (⨅ n, range (f ^ n))"}}
+{"state": {"context": ["G : Type u_1", "Add G", "IsRightCancelAdd G", "g h : G"], "goal": "(addRightEmbedding g) h = h + g"}}
+{"state": {"context": ["α : Type u_1", "One α", "a : α"], "goal": "SymAlg.sym a ≠ 1 ↔ a ≠ 1"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "A : Type z", "a b : R", "n✝ : ℕ", "inst✝ : Semiring R", "p : R[X]", "n : ℕ"], "goal": "(p.coeff (n + 1) * ↑(n + 1) * if n = n + 1 - 1 then 1 else 0) = p.coeff (n + 1) * (↑n + 1)"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedHeytingAlgebra α", "a b : α"], "goal": "a ⊔ b ⇨ a = b ⇨ a"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "f g : E →ₛₗᵢ[σ₁₂] E₂"], "goal": "f.toContinuousLinearMap = g.toContinuousLinearMap ↔ f = g"}}
+{"state": {"context": [], "goal": "StrictMonoOn Real.arcsin (Set.Icc (-1) 1)"}}
+{"state": {"context": ["ι : Sort u_1", "𝕜 : Type u_2", "E : Type u_3", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "s t u : Set E", "x y : E", "hs : Convex 𝕜 s", "ht : Convex 𝕜 t", "x₁ : E", "hx₁ : x₁ ∈ s", "y₁ : E", "hy₁ : y₁ ∈ t", "a₁ b₁ : 𝕜", "ha₁ : 0 ≤ a₁", "hb₁ : 0 ≤ b₁", "hab₁ : a₁ + b₁ = 1", "x₂ : E", "hx₂ : x₂ ∈ s", "y₂ : E", "hy₂ : y₂ ∈ t", "a₂ b₂ : 𝕜", "ha₂ : 0 ≤ a₂", "hb₂ : 0 ≤ b₂", "hab₂ : a₂ + b₂ = 1", "p q : 𝕜", "hp : 0 ≤ p", "hq : 0 ≤ q", "hpq : p + q = 1"], "goal": "∃ i, ∃ (_ : i ∈ s), ∃ i_1, ∃ (_ : i_1 ∈ t), p • (a₁ • x₁ + b₁ • y₁) + q • (a₂ • x₂ + b₂ • y₂) ∈ [i-[𝕜]i_1]"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁹ : CommRing R", "inst✝⁸ : IsDomain R", "inst✝⁷ : CommRing S", "L : Type u_3", "inst✝⁶ : Field L", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra S L", "inst✝³ : Algebra R L", "inst✝² : IsScalarTower R S L", "inst✝¹ : IsIntegralClosure S R L", "inst✝ : Algebra.IsAlgebraic R L", "inj : Function.Injective ⇑(algebraMap R L)", "a b : S", "hb : b ≠ 0", "c : ↥(integralClosure R L)", "d : R", "d_ne : d ≠ 0", "hx : (algebraMap S L) a / (algebraMap S L) b * (algebraMap R L) d = ↑c"], "goal": "∃ c d, d ≠ 0 ∧ d • a = b * c"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "Group G", "MulAction G α", "a b : α"], "goal": "Setoid.r a b ↔ a ∈ MulAction.orbit G b"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedBooleanAlgebra α", "a b c : α"], "goal": "a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "f : M → M'", "x : M"], "goal": "MDifferentiableAt I I' f x ↔\n  ContinuousAt f x ∧ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (Set.range ↑I) (↑(extChartAt I x) x)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "b x : α"], "goal": "x ∈ Set.Iic b ↔ x ≤ b"}}
+{"state": {"context": ["x y : PGame", "i : x.LeftMoves", "j : y.LeftMoves"], "goal": "⟦x.moveLeft i * y.moveLeft j⟧ = ⟦y.moveLeft j * x.moveLeft i⟧"}}
+{"state": {"context": ["R₁ : Type u_3", "M₁ : Type u_4", "CommRing R₁", "AddCommGroup M₁", "Module R₁ M₁", "B : LinearMap.BilinForm R₁ M₁"], "goal": "(-B).IsSymm ↔ B.IsSymm"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "AddCommMonoid α", "AddCommMonoid β", "A : Set α", "B : Set β", "n : ℕ", "EquivLike F α β", "AddEquivClass F α β", "f : F", "hfAB : Set.BijOn (⇑f) A B"], "goal": "IsAddFreimanIso n A B ⇑f"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_6", "SMul R ENNReal", "IsScalarTower R ENNReal ENNReal", "_m : MeasurableSpace α", "c : R", "μ : MeasureTheory.Measure α", "s : Set α"], "goal": "(c • μ) s = c • μ s"}}
+{"state": {"context": ["G : Type u_1", "A : Type u_2", "H : Type u_3", "Group G", "MulAction G A", "Monoid H", "x : G"], "goal": "∀ (a : A → H) (a_1 : A), (MulEquiv.symm (mulAutArrow x)) a a_1 = (x⁻¹ • a) a_1"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "P Q R : C", "f : P ⟶ Q", "g : Q ⟶ R", "inst✝¹ : StrongMono (f ≫ g)", "X Y : C", "z : X ⟶ Y", "inst✝ : Epi z", "u : X ⟶ P", "v : Y ⟶ Q", "sq : CommSq u z f v"], "goal": "sq.HasLift"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b₁ b₂ : α", "h : b₁ ≤ b₂"], "goal": "Set.Ioc a b₁ ⊆ Set.Ioc a b₂"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "ncs : CauSeq ℤ_[p] norm", "F : Polynomial ℤ_[p]", "hnorm : Filter.Tendsto (fun i => ‖Polynomial.eval (↑ncs i) F‖) Filter.atTop (𝓝 0)"], "goal": "Polynomial.eval ncs.lim F = 0"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Fintype ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x y : ι → ℝ", "δ : ℝ", "hs : IsLowerSet s", "hx : x ∈ closure s", "hδ : 0 < δ"], "goal": "δ / 4 + dist (x - const ι (3 / 4 * δ)) x ≤ δ"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ"], "goal": "term (f - g) s = term f s - term g s"}}
+{"state": {"context": [], "goal": "decide False = false"}}
+{"state": {"context": ["N : Type u_2", "Monoid N", "p : N", "n : ℕ", "h : IsIdempotentElem p"], "goal": "IsIdempotentElem (p ^ n)"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "x : WittVector p (ZMod p)"], "goal": "WittVector.frobenius x = x"}}
+{"state": {"context": ["α : Type u", "σ : Type v", "M : DFA α σ", "s : σ", "x : List α", "a : α"], "goal": "M.evalFrom s (x ++ [a]) = M.step (M.evalFrom s x) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : Filter α", "g : Set α → Set β", "h : Set β → Filter γ", "hg : Monotone g", "hh : Monotone h"], "goal": "(f.lift' g).lift h = f.lift fun s => h (g s)"}}
+{"state": {"context": ["C : Type u₁"], "goal": "Function.Injective CategoryTheory.MonoidalOpposite.unmop"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddZeroClass M", "AddZeroClass N", "F : Type u_4", "FunLike F M N", "mc : AddMonoidHomClass F M N", "f : F", "S : AddSubmonoid M", "x : M", "hx : x ∈ S"], "goal": "f x ∈ AddSubmonoid.map f S"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f : α →ₘ[μ] β"], "goal": "MeasureTheory.StronglyMeasurable ↑f"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "P : Type u_3", "N : Type u_4", "Ring R", "AddCommGroup M", "AddCommGroup P", "AddCommGroup N", "Module R M", "Module R P", "Module R N", "IsArtinian R M", "IsArtinian R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "hf : Function.Injective ⇑f", "hg : Function.Surjective ⇑g", "h : LinearMap.range f = LinearMap.ker g"], "goal": "IsArtinian R N"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasZeroMorphisms C", "S S₁ S₂ S₃ S₄ : ShortComplex C", "φ : S₁ ⟶ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "A : C", "inst✝ : S.HasHomology", "k : A ⟶ S.X₂", "x : A ⟶ S.X₁", "hx : k = x ≫ S.f"], "goal": "S.liftCycles k ⋯ ≫ S.leftHomologyπ ≫ S.leftHomologyIso.hom = 0"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "inst✝³ : Fintype ι", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "l : IntegrationParams", "hl : l.bRiemann = false", "s : Set (ι → ℝ)", "hs : MeasurableSet s", "I : Box ι", "y : E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "ε : ℝ≥0", "ε0 : 0 < ε", "A : μ (s ∩ Box.Icc I) ≠ ⊤", "B : μ (s ∩ ↑I) ≠ ⊤", "F : Set (ι → ℝ)", "hFs : F ⊆ s ∩ Box.Icc I", "hFc : IsClosed F", "hμF : μ ((s ∩ Box.Icc I) \\ F) < ↑ε", "U : Set (ι → ℝ)", "hsU : s ∩ Box.Icc I ⊆ U", "hUo : IsOpen U", "hUt : μ U < ⊤", "hμU : μ (U \\ (s ∩ Box.Icc I)) < ↑ε", "rs : (ι → ℝ) → ↑(Set.Ioi 0)", "hrsU : ∀ x ∈ s ∩ Box.Icc I, closedBall x ↑(rs x) ⊆ U", "rs' : (ι → ℝ) → ↑(Set.Ioi 0)", "hrs'F : ∀ x ∈ Box.Icc I \\ s, closedBall x ↑(rs' x) ⊆ Fᶜ", "r : (ι → ℝ) → ↑(Set.Ioi 0) := s.piecewise rs rs'", "c : ℝ≥0", "π : TaggedPrepartition I", "hπ : l.MemBaseSet I c ((fun x => r) c) π", "hπp : π.IsPartition"], "goal": "|(μ (π.filter fun i => π.tag i ∈ s).iUnion).toReal - (μ (s ∩ ↑I)).toReal| * ‖y‖ ≤ ↑ε * ‖y‖"}}
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+{"state": {"context": ["α : Type u_1", "Mul α", "LinearOrder α", "a b c d : α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "h : a * b < c * d"], "goal": "min a b < max c d"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_1", "Preorder α", "TopologicalSpace α", "Topology.IsLower α", "TopologicalSpace β", "f : β → α"], "goal": "Continuous f ↔ ∀ (a : α), IsClosed (f ⁻¹' Set.Ici a)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Category.{?u.83140, u_2} D", "inst✝¹ : Preadditive C", "inst✝ : Preadditive D", "S : ShortComplex C", "s s' : S.Splitting", "h : s.s = s'.s", "this : Mono S.f", "eq : s.r ≫ S.f + S.g ≫ s.s = 𝟙 S.X₂"], "goal": "s = s'"}}
+{"state": {"context": ["J : Type u₁", "inst✝¹ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝ : Category.{v₂, u₂} K", "h : ∀ (j₁ j₂ : J), ∃ l, List.Chain Zag j₁ l ∧ (j₁ :: l).getLast ⋯ = j₂"], "goal": "∀ (j₁ j₂ : J), Zigzag j₁ j₂"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "ι : Type u_2", "H : ι → AddSubgroup G", "g : ι → G", "s : Finset ι", "hcovers : ⋃ i ∈ s, g i +ᵥ ↑(H i) = Set.univ", "DecidablePred AddSubgroup.FiniteIndex"], "goal": "⋃ k ∈ Finset.filter (fun i => (H i).FiniteIndex) s, g k +ᵥ ↑(H k) = Set.univ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Fintype β", "r : α → β → Prop", "(a : α) → DecidablePred (r a)"], "goal": "(∀ (A : Finset α), A.card ≤ (Finset.filter (fun b => ∃ a ∈ A, r a b) Finset.univ).card) ↔\n  ∃ f, Function.Injective f ∧ ∀ (x : α), r x (f x)"}}
+{"state": {"context": ["R✝ : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R✝", "m n✝ : ℕ", "inst✝² : CommSemiring R✝", "p q : R✝[X]", "x✝ : R✝", "inst✝¹ : CommSemiring S", "f : R✝ →+* S", "R : Type u_1", "inst✝ : CommSemiring R", "n : ℕ", "x : R"], "goal": "eval x (∑ i ∈ range n, X ^ i) = ∑ i ∈ range n, x ^ i"}}
+{"state": {"context": ["α : Type", "Finite α", "Z : α → Stonean", "a : α"], "goal": "OpenEmbedding ⇑(CategoryTheory.Limits.Sigma.ι Z a)"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero R", "inst✝³ : UniqueFactorizationMonoid R", "inst✝² : CancelCommMonoidWithZero α", "inst✝¹ : UniqueFactorizationMonoid α", "β : Type u_3", "inst✝ : CancelCommMonoidWithZero β", "P : α → Prop", "a : α", "h0 : P 0", "h1 : ∀ {x : α}, IsUnit x → P x", "hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)", "hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)", "this✝¹ : DecidableEq α := Classical.decEq α", "P_of_associated : ∀ {x y : α}, x ~ᵤ y → P x → P y", "ha0 : ¬a = 0", "this✝ : Nontrivial α", "this : NormalizationMonoid α := UniqueFactorizationMonoid.normalizationMonoid"], "goal": "P (normalizedFactors a).prod"}}
+{"state": {"context": ["𝕝 : Type u_6", "E : Type u_7", "SeminormedRing 𝕝", "AddCommGroup E", "Module 𝕝 E", "TopologicalSpace E", "p : Seminorm 𝕝 E", "hp : Continuous ⇑p", "r : ℝ", "hr : 0 < r"], "goal": "p.ball 0 r ∈ 𝓝 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "R : Type u_2", "V : Type u_3", "W : Type u_4", "W₂ : Type u_5", "P : Type u_6", "Q : Type u_7", "Q₂ : Type u_8", "inst✝¹⁶ : NormedAddCommGroup V", "inst✝¹⁵ : MetricSpace P", "inst✝¹⁴ : NormedAddTorsor V P", "inst✝¹³ : NormedAddCommGroup W", "inst✝¹² : MetricSpace Q", "inst✝¹¹ : NormedAddTorsor W Q", "inst✝¹⁰ : NormedAddCommGroup W₂", "inst✝⁹ : MetricSpace Q₂", "inst✝⁸ : NormedAddTorsor W₂ Q₂", "inst✝⁷ : NormedField R", "inst✝⁶ : NormedSpace R V", "inst✝⁵ : NormedSpace R W", "inst✝⁴ : NormedSpace R W₂", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedSpace 𝕜 V", "inst✝¹ : NormedSpace 𝕜 W", "inst✝ : NormedSpace 𝕜 W₂", "f : V →ᵃ[R] W", "h : Continuous f.toFun"], "goal": "⇑{ toAffineMap := f, cont := h } = ⇑{ toAffineMap := f, cont := h }.contLinear + Function.const V ({ toAffineMap := f, cont := h } 0)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "f : Polynomial R", "n : ℕ"], "goal": "f.reverse.coeff n = f.coeff ((Polynomial.revAt f.natDegree) n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝ : Countable β", "f : β → α → ℝ≥0∞", "hf : ∀ (b : β), Measurable (f b)", "h_directed : Directed (fun x x_1 => x ≤ x_1) f", "val✝ : Encodable β", "h✝ : Nonempty β", "inhabited_h : Inhabited β", "this : ∀ (a : α), ⨆ b, f b a = ⨆ n, f (Directed.sequence f h_directed n) a"], "goal": "∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "CommRing S", "f : R →+* S", "n : ℤ"], "goal": "Polynomial.map f (Polynomial.Chebyshev.T R n) = Polynomial.Chebyshev.T S n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p✝ q r : R[X]", "inst✝ : Semiring S", "f : R →+* S", "p : R[X]", "h : p.natDegree = 0"], "goal": "(map f p).natDegree = p.natDegree ↔ f p.leadingCoeff ≠ 0 ∨ p.natDegree = 0"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W✝ : Affine R", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "h₁ : y₁ ^ 2 + W.a₁ * x₁ * y₁ + W.a₃ * y₁ = x₁ ^ 3 + W.a₂ * x₁ ^ 2 + W.a₄ * x₁ + W.a₆", "h₂ : y₂ ^ 2 + W.a₁ * x₂ * y₂ + W.a₃ * y₂ = x₂ ^ 3 + W.a₂ * x₂ ^ 2 + W.a₄ * x₂ + W.a₆", "hx : x₁ = x₂"], "goal": "y₁ = y₂ ∨ y₁ = W.negY x₂ y₂"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "I : MultispanIndex C", "inst✝¹ : HasCoproduct I.left", "inst✝ : HasCoproduct I.right", "K₁ K₂ : Multicofork I", "f : K₁ ⟶ K₂"], "goal": "K₁.toSigmaCofork.ι.app WalkingParallelPair.one ≫ f.hom = K₂.toSigmaCofork.ι.app WalkingParallelPair.one"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "Add G", "Add H", "a : G", "b : H", "f : AddConstMap G H a b", "a' : G", "b' : H", "ha : a = a'", "hb : b = b'"], "goal": "⇑(f.replaceConsts a' b' ha hb) = ⇑f"}}
+{"state": {"context": ["c : ℝ", "f g : ℕ → Bool", "n : ℕ", "h : f n = g n"], "goal": "Cardinal.cantorFunctionAux c f n = Cardinal.cantorFunctionAux c g n"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Submonoid R", "S : Type u_2", "CommRing S", "Algebra R S", "IsLocalization M S"], "goal": "Submodule.span R ↑(IsLocalization.invSubmonoid M S) = ⊤"}}
+{"state": {"context": ["V : Type u_1", "inst✝ : DecidableEq V", "G : SimpleGraph V", "s t : V", "ha : G.Adj s t", "e : Sym2 V"], "goal": "∀ (x y : V), s(x, y) ∈ (G.replaceVertex s t).edgeSet ↔ s(x, y) ∈ (G.edgeSet \\ G.incidenceSet t ∪ (fun x => s(x, t)) '' G.neighborSet s) \\ {s(t, t)}"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "F : Ctop.Realizer α", "s : Set α", "a : α"], "goal": "a ∈ interior s ↔ ∃ b, a ∈ F.F.f b ∧ F.F.f b ⊆ s"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "P : Type u_3", "CommSemiring P", "IsLocalization M S", "T : Submonoid P", "Q : Type u_4", "CommSemiring Q", "Algebra P Q", "IsLocalization T Q", "j : R ≃+* P", "H : Submonoid.map j.toMonoidHom M = T", "x : R", "y : ↥M"], "goal": "(IsLocalization.ringEquivOfRingEquiv S Q j H) (IsLocalization.mk' S x y) = IsLocalization.mk' Q (j x) ⟨j ↑y, ⋯⟩"}}
+{"state": {"context": ["α : Type", "f g : (α → ℕ) → ℕ", "df : Dioph.DiophFn f", "dg : Dioph.DiophFn g"], "goal": "Dioph fun v => f v ∣ g v"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "inst✝² : NonUnitalNonAssocSemiring R", "M : Subsemigroup R", "inst✝¹ : NonUnitalNonAssocSemiring S", "inst✝ : NonUnitalNonAssocSemiring T", "s : Set R", "x : R", "hx : x ∈ closure ↑(AddSubmonoid.closure s)"], "goal": "x ∈ closure s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "f : α → β", "s : Set α", "H : IsQuasiSeparated s", "h : Embedding f", "U V : Set β", "hU : U ⊆ f '' s", "hU' : IsOpen U", "hU'' : IsCompact U", "hV : V ⊆ f '' s", "hV' : IsOpen V", "hV'' : IsCompact V"], "goal": "f ⁻¹' V ⊆ s"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "I : BoxIntegral.Box ι"], "goal": "(MeasureTheory.volume ↑I).toReal = ∏ i : ι, (I.upper i - I.lower i)"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "t t' : Set (PrimeSpectrum R)"], "goal": "PrimeSpectrum.vanishingIdeal t ⊔ PrimeSpectrum.vanishingIdeal t' ≤ PrimeSpectrum.vanishingIdeal (t ∩ t')"}}
+{"state": {"context": ["s : ℂ", "hs : 1 < s.re"], "goal": "Filter.Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) Filter.atTop (𝓝 (riemannZeta s))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "f : α → β", "hf : Antitone f"], "goal": "Antitone fun x => Set.Iio (f x)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "E : EllipticCurve R", "C : WeierstrassCurve.VariableChange R"], "goal": "(E.variableChange C).j = E.j"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "n : ℕ"], "goal": "b ∈ Multiset.replicate n a ↔ n ≠ 0 ∧ b = a"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : Preadditive C", "F : C ⥤ C", "inst✝ : F.Additive", "A₁ A₂ : Algebra F", "a✝ b✝ : A₁ ⟶ A₂"], "goal": "(a✝ - b✝).f = (a✝ + -b✝).f"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "inst✝¹ : Mul G", "inst✝ : Mul H", "A B : Finset G", "a0 b0 : G", "x✝ : ∃ g, ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = g", "a b : G", "hab : (a, b) ∈ A ×ˢ B", "h : ∃! ab, ab ∈ A ×ˢ B ∧ ab.1 * ab.2 = (a, b).1 * (a, b).2", "ha : (a, b).1 ∈ A", "hb : (a, b).2 ∈ B"], "goal": "∃ a0 b0, a0 ∈ A ∧ b0 ∈ B ∧ UniqueMul A B a0 b0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "I₁ : Type u_2", "I₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "X : HomologicalComplex₂ C c₂ c₁", "i : I₂"], "goal": "∀ (i_1 : I₁), (((HomologicalComplex₂.flipEquivalenceCounitIso C c₁ c₂).inv.app X).f i).f i_1 = 𝟙 ((X.X i).X i_1)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "β : Type u_2", "Ring β", "abv : β → α", "IsAbsoluteValue abv", "γ : Type u_3", "SMul γ β", "IsScalarTower γ β β", "c : γ", "f : CauSeq β abv"], "goal": "c • CauSeq.Completion.mk f = CauSeq.Completion.mk (c • f)"}}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "inst✝³ : CommSemiring R", "inst✝² : LocalRing R", "S : Type v", "inst✝¹ : CommSemiring S", "inst✝ : LocalRing S", "f : R →+* S", "this : IsLocalRingHom f ↔ Ideal.comap f (maximalIdeal S) = maximalIdeal R"], "goal": "Ideal.comap f (maximalIdeal S) = maximalIdeal R ↔ ((PrimeSpectrum.comap f) (closedPoint S)).asIdeal = (closedPoint R).asIdeal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : BooleanRing α", "inst✝¹ : BooleanRing β", "inst✝ : BooleanRing γ", "f : α →+* β", "a b : AsBoolAlg α"], "goal": "toBoolAlg (f (ofBoolAlg a + ofBoolAlg b + ofBoolAlg a * ofBoolAlg b)) = toBoolAlg (f (ofBoolAlg a)) ⊔ toBoolAlg (f (ofBoolAlg b))"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "inst✝⁵ : IsDedekindDomain R", "inst✝⁴ : Module.Free ℤ R", "inst✝³ : Module.Finite ℤ R", "K : Type u_2", "inst✝² : CommRing K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "I : FractionalIdeal R⁰ K", "a : ↥R⁰", "I₀ : Ideal R", "h : a • ↑I = Submodule.map (Algebra.linearMap R K) I₀"], "goal": "↑(Ideal.absNorm I.num) * ↑|(Algebra.norm ℤ) ↑a| = ↑(Ideal.absNorm I₀) * ↑|(Algebra.norm ℤ) ↑I.den|"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "r : ℝ", "hr : 0 < r"], "goal": "o.oangle (r • x) y = o.oangle x y"}}
+{"state": {"context": ["F : Type u_1", "K✝ : Type u_2", "L : Type u_3", "inst✝⁵ : Field F", "inst✝⁴ : Field K✝", "inst✝³ : Field L", "inst✝² : Algebra F K✝", "inst✝¹ : Algebra F L", "K K' : IntermediateField F L", "inst✝ : Normal F L", "h : K ≤ K'"], "goal": "⨆ f, map f K ≤ ⨆ f, map f K'"}}
+{"state": {"context": ["a b : ℕ+"], "goal": "Fintype.card ↑(Set.Icc a b) = ↑b + 1 - ↑a"}}
+{"state": {"context": ["R : Type u_1", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "E : Type u_7", "E₂ : Type u_8", "SeminormedRing 𝕜", "SeminormedRing 𝕜₂", "σ₁₂ : 𝕜 →+* 𝕜₂", "RingHomIsometric σ₁₂", "AddCommGroup E", "AddCommGroup E₂", "Module 𝕜 E", "Module 𝕜₂ E₂", "SMul R ℝ", "SMul R ℝ≥0", "IsScalarTower R ℝ≥0 ℝ", "p : Seminorm 𝕜₂ E₂", "f : E →ₛₗ[σ₁₂] E₂", "c : R"], "goal": "(c • p).comp f = c • p.comp f"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "s✝ t : Set α", "a b x y : α", "s : Finset α", "hf : f.IsCycleOn ↑s", "ha : a ∈ s", "n : ℕ", "hs : s.Nontrivial"], "goal": "(f ^ n) a = a ↔ s.card ∣ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ✝ : Kernel α β", "f✝ : α → β → ℝ≥0∞", "inst✝¹ : Countable ι", "κ : Kernel α β", "inst✝ : IsSFiniteKernel κ", "f : ι → α → β → ℝ≥0∞", "hf : ∀ (i : ι), Measurable (Function.uncurry (f i))"], "goal": "κ.withDensity (∑' (n : ι), f n) = Kernel.sum fun n => κ.withDensity (f n)"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "κ✝ η✝ : Kernel α γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : Kernel α γ", "inst✝¹ : IsFiniteKernel κ", "inst✝ : IsFiniteKernel η", "a : α", "s : Set γ", "hs : MeasurableSet s", "hm : MeasurableSet (κ.mutuallySingularSetSlice η a)"], "goal": "((η.withDensity (κ.rnDeriv η)) a) (s ∩ κ.mutuallySingularSetSlice η a) + ((η.withDensity (κ.rnDeriv η)) a) (s \\ κ.mutuallySingularSetSlice η a) + (((κ.singularPart η) a) (s ∩ κ.mutuallySingularSetSlice η a) + ((κ.singularPart η) a) (s \\ κ.mutuallySingularSetSlice η a)) = (κ a) (s ∩ κ.mutuallySingularSetSlice η a) + (κ a) (s \\ κ.mutuallySingularSetSlice η a)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝³ : Ring R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "inst✝ : IsArtinian R M", "f : ℕ →o (Submodule R M)ᵒᵈ"], "goal": "EventuallyConst (⇑f) atTop"}}
+{"state": {"context": ["R : Type u_1", "CommMonoid R", "R' : Type u_2", "CommMonoidWithZero R'", "χ : MulChar R R'"], "goal": "χ.IsNontrivial ↔ χ ≠ 1"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "l : List (Sigma β)"], "goal": "List.dlookup a l = none ↔ a ∉ l.keys"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_3, u_2} D", "CategoryTheory.Preadditive D", "F G : CategoryTheory.Functor C D", "X : C", "α : F ⟶ G", "n : ℤ"], "goal": "(n • α).app X = n • α.app X"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "s : Set X", "hs : s.Subsingleton"], "goal": "IsCompact s"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "z✝ : ℂ", "f✝ : ℂ → E", "f : ℂ → ℂ", "z : ℂ", "h : fderiv ℝ f z ≠ 0", "h_diff : DifferentiableAt ℝ f ((starRingEnd ℂ) ((starRingEnd ℂ) z))", "g : ℂ →L[ℂ] ℂ", "this : fderiv ℝ (⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z) = ↑conjCLE"], "goal": "(fderiv ℝ f z).comp ↑conjCLE = fderiv ℝ (f ∘ ⇑(starRingEnd ℂ)) ((starRingEnd ℂ) z)"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p✝ q✝ : ℝ≥0", "p q : ℝ≥0∞", "hpq : p ≤ q", "hp : 0 < p", "hq : q < ⊤"], "goal": "0 < p.toReal ∧ 0 < q.toReal ∧ p.toReal ≤ q.toReal"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "f : α → E", "g : α → F"], "goal": "f =O[⊤] g ↔ ∃ C, ∀ (x : α), ‖f x‖ ≤ C * ‖g x‖"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b c : α", "hab : a ⋖ b", "hbc : AntisymmRel (fun x x_1 => x ≤ x_1) b c"], "goal": "a ⋖ c"}}
+{"state": {"context": ["a b : Int"], "goal": "a < 0 → 0 < b → a.fdiv b < 0"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_5", "NormedAddCommGroup G", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "NormedSpace ℝ G", "hm : m ≤ m0", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "x : G"], "goal": "MeasureTheory.AEStronglyMeasurable' m (↑↑(MeasureTheory.condexpIndSMul hm hs hμs x)) μ"}}
+{"state": {"context": ["F : Type u_1", "R : outParam (Type u_2)", "S : outParam (Type u_3)", "Semiring R", "Semiring S", "σ : outParam (R →+* S)", "σ' : outParam (S →+* R)", "RingHomInvPair σ σ'", "RingHomInvPair σ' σ", "M : outParam (Type u_4)", "TopologicalSpace M", "AddCommMonoid M", "M₂ : outParam (Type u_5)", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R M", "Module S M₂", "EquivLike F M M₂", "self : ContinuousSemilinearEquivClass F σ M M₂", "f : F"], "goal": "Continuous (EquivLike.inv f)"}}
+{"state": {"context": ["f : ℝ → ℝ", "hf' : Differentiable ℝ f", "hf'' : Differentiable ℝ (deriv f)", "hf''_nonpos : ∀ (x : ℝ), deriv^[2] f x ≤ 0"], "goal": "ConcaveOn ℝ Set.univ f"}}
+{"state": {"context": ["α : Type u_5", "E : Type u_6", "R : Type u_7", "AddCommMonoid E", "DivisionSemiring R", "Monoid α", "Module R E", "DistribMulAction α E", "n : ℕ", "s : α", "x : E"], "goal": "(↑n)⁻¹ • s • x = s • (↑n)⁻¹ • x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝¹ : Semiring R", "inst✝ : Nontrivial R", "p q : R[X]", "n : ℕ", "hp : p ≠ 0"], "goal": "(p * X ^ n).natDegree = p.natDegree + n"}}
+{"state": {"context": ["I : Type u", "C : Set (I → Bool)", "J : I → Prop", "(i : I) → Decidable (J i)"], "goal": "Function.Bijective ⇑(Profinite.NobelingProof.iso_map C J)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s₁ s₂ : Set α", "t₁ t₂ : Set β", "f : α → β", "h₁ : Set.SurjOn f s₁ t₁", "h₂ : Set.SurjOn f s₂ t₂"], "goal": "Set.SurjOn f (s₁ ∪ s₂) (t₁ ∪ t₂)"}}
+{"state": {"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ≤ 0"], "goal": "cosKernel a x = 0"}}
+{"state": {"context": ["G : Type u_3", "DivInvMonoid G", "g : ConjAct G", "h : G"], "goal": "g • h = ConjAct.ofConjAct g * h * (ConjAct.ofConjAct g)⁻¹"}}
+{"state": {"context": ["z : ℂ"], "goal": "(-z).im = -z.im"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "E : Type u₃", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : Category.{v₂, u₂} D", "inst✝¹ : Category.{v₃, u₃} E", "i : D ⥤ C", "inst✝ : Reflective i", "X Y : i.EssImageSubcategory", "f : X ⟶ Y", "h : (reflectorAdjunction i).unit.app X.obj ≫ (reflector i ⋙ i).map f = (𝟭 C).map f ≫ (reflectorAdjunction i).unit.app Y.obj"], "goal": "i.essImageInclusion.map (((i.essImageInclusion ⋙ reflector i) ⋙ i.toEssImage).map f ≫ (equivEssImageOfReflective_counitIso_app Y).hom) = i.essImageInclusion.map ((equivEssImageOfReflective_counitIso_app X).hom ≫ (𝟭 i.EssImageSubcategory).map f)"}}
+{"state": {"context": ["n : ℕ", "P : MvPFunctor.{u} (n + 1)", "α β : TypeVec.{u} n", "g : α ⟹ β", "a : P.mp.A", "f : P.mp.B a ⟹ α"], "goal": "dest P (g <$$> ⟨a, f⟩) = (g ::: fun x => g <$$> x) <$$> dest P ⟨a, f⟩"}}
+{"state": {"context": ["α : Type u_3", "Monoid α", "s : List (Additive α)"], "goal": "Additive.toMul s.sum = (List.map (⇑Additive.toMul) s).prod"}}
+{"state": {"context": ["n : Type v", "α : Type w", "DecidableEq n", "Fintype n", "CommRing α", "A : Matrix n n α"], "goal": "A.adjugate.det = A.det ^ (Fintype.card n - 1)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MetricSpace α", "β : Type u", "inst✝ : Nonempty β", "p : TauPackage β α", "N : ℕ", "hN : IsEmpty (SatelliteConfig α N p.τ)", "i : Ordinal.{u}", "IH : ∀ k < i, k < p.lastStep → p.color k < N", "hi : i < p.lastStep", "A : Set ℕ := ⋃ j, ⋃ (_ : (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {p.color ↑j}", "color_i : p.color i = sInf (univ \\ A)", "N_mem : N ∈ univ \\ A", "Inf_eq_N : sInf (univ \\ A) = N", "g : ℕ → Ordinal.{u}", "hg : ∀ k < N, g k < i ∧ (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color (g k)", "G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n", "color_G : ∀ n ≤ N, p.color (G n) = n", "G_lt_last : ∀ n ≤ N, G n < p.lastStep", "fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))", "Gab : ∀ (a b : Fin N.succ), G ↑a < G ↑b → p.r (p.index (G ↑a)) ≤ dist (p.c (p.index (G ↑a))) (p.c (p.index (G ↑b))) ∧ p.r (p.index (G ↑b)) ≤ p.τ * p.r (p.index (G ↑a))", "sc : SatelliteConfig α N p.τ := { c := fun k => p.c (p.index (G ↑k)), r := fun k => p.r (p.index (G ↑k)), rpos := ⋯, h := ⋯, hlast := ⋯, inter := ⋯ }"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "p : ℝ≥0∞", "NormedAddCommGroup F", "m m0 : MeasurableSpace α", "one_le_p : Fact (1 ≤ p)", "NormedSpace ℝ F", "hm : m ≤ m0", "s : Set α", "μ : MeasureTheory.Measure α", "hs : MeasurableSet s", "hμs : (μ.trim hm) s ≠ ⊤", "c : F"], "goal": "↑((MeasureTheory.lpMeasToLpTrimLie F ℝ p μ hm).symm (MeasureTheory.indicatorConstLp p hs hμs c)) =\n  MeasureTheory.indicatorConstLp p ⋯ ⋯ c"}}
+{"state": {"context": ["x : Int", "m : Nat", "p : x % ↑m < (↑m + 1) / 2"], "goal": "x.bmod m = x % ↑m"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α", "hs : IsCompact s"], "goal": "Disjoint (𝓝ˢ s) (Bornology.cobounded α)"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "V₂ : Type u_5", "V₃ : Type u_6", "P : Type u_10", "P₂ : Type u_11", "P₃ : Type u_12", "NormedField 𝕜", "SeminormedAddCommGroup V", "NormedSpace 𝕜 V", "PseudoMetricSpace P", "NormedAddTorsor V P", "SeminormedAddCommGroup V₂", "NormedSpace 𝕜 V₂", "PseudoMetricSpace P₂", "NormedAddTorsor V₂ P₂", "SeminormedAddCommGroup V₃", "NormedSpace 𝕜 V₃", "PseudoMetricSpace P₃", "NormedAddTorsor V₃ P₃", "e₁ : P ≃ᵃⁱ[𝕜] P₂", "e₂ : P₂ ≃ᵃⁱ[𝕜] P₃"], "goal": "⇑(e₁.trans e₂) = ⇑e₂ ∘ ⇑e₁"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : PseudoMetricSpace α", "x y z : Completion α"], "goal": "dist x z ≤ dist x y + dist y z"}}
+{"state": {"context": ["K : Type u_1", "Field K", "NumberField K", "IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : FermatLastTheoremForThreeGen.Solution hζ", "DecidableRel fun a b => a ∣ b"], "goal": "IsCoprime (FermatLastTheoremForThreeGen.Solution.x S) (FermatLastTheoremForThreeGen.Solution.y S)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "l✝ : List α", "x a b : α", "l : List α"], "goal": "(a :: b :: l).reverse.formPerm = (a :: b :: l).formPerm⁻¹"}}
+{"state": {"context": ["R : Type u_3", "LinearOrderedSemiring R", "a b : R", "ExistsAddOfLE R", "h : a ^ 2 ≤ b"], "goal": "a ^ 4 ≤ b ^ 2"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "inst✝¹⁹ : Field R", "inst✝¹⁸ : AddCommGroup V", "inst✝¹⁷ : TopologicalSpace R", "inst✝¹⁶ : TopologicalSpace V", "inst✝¹⁵ : TopologicalRing R", "inst✝¹⁴ : TopologicalAddGroup V", "inst✝¹³ : Module R V", "inst✝¹² : SeparatingDual R V", "𝕜 : Type u_3", "E : Type u_4", "F : Type u_5", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace 𝕜 F", "inst✝⁶ : SeparatingDual 𝕜 E", "inst✝⁵ : Nontrivial E", "ι : Type u_6", "inst✝⁴ : Finite ι", "M : ι → Type u_7", "inst✝³ : (i : ι) → NormedAddCommGroup (M i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (M i)", "inst✝¹ : ∀ (i : ι), SeparatingDual 𝕜 (M i)", "inst✝ : CompleteSpace (ContinuousMultilinearMap 𝕜 M F)", "m : (i : ι) → M i", "hm : ∀ (i : ι), m i ≠ 0", "f : ℕ → F", "hf : CauchySeq f", "φ : (i : ι) → M i →L[𝕜] 𝕜", "hφ : ∀ (i : ι), (φ i) (m i) = 1", "val✝ : Fintype ι", "g : ℕ → ContinuousMultilinearMap 𝕜 M F := fun n => (compContinuousLinearMapL φ) (((smulRightL 𝕜 (fun x => 𝕜) F) (ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι 𝕜)) (f n))", "this : CauchySeq g", "a : ContinuousMultilinearMap 𝕜 M F", "ha : Tendsto g atTop (𝓝 a)"], "goal": "Tendsto f atTop (𝓝 (a m))"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : SMulCommClass ℝ 𝕜 E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : CompleteSpace F", "G : Type u_5", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace ℝ G", "f✝ g : α → E", "m : MeasurableSpace α", "μ : Measure α", "X : Type u_6", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X", "ν : Measure α", "f : α → G", "c : ℝ≥0∞", "hG : CompleteSpace G", "hc : c ≠ ⊤"], "goal": "setToFun (c • μ) (weightedSMul (c • μ)) ⋯ f = setToFun μ (fun s => c.toReal • weightedSMul μ s) ⋯ f"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "C D : ℝ≥0", "f g : E → ℝ", "s : Set E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "hg : LipschitzWith D g", "h'g : HasCompactSupport g", "v : E", "A : Tendsto (fun t => ∫ (x : E), t⁻¹ • (f (x + t • v) - f x) * g x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ f x v * g x ∂μ))", "B : Tendsto (fun t => ∫ (x : E), t⁻¹ • (g (x + t • -v) - g x) * f x ∂μ) (𝓝[>] 0) (𝓝 (∫ (x : E), lineDeriv ℝ g x (-v) * f x ∂μ))", "t : ℝ", "S3 : ∫ (x : E), f (x + t • v) * g x ∂μ = ∫ (x : E), f x * g (x + t • -v) ∂μ"], "goal": "Integrable (fun x => f (x + t • v) * g x) μ"}}
+{"state": {"context": ["F : Type u", "E✝ : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E✝", "inst✝⁵ : Algebra F E✝", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "E : Type v", "inst✝² : Field E", "inst✝¹ : Algebra F E", "inst✝ : Algebra.IsSeparable F E", "hfd : FiniteDimensional F E"], "goal": "finSepDegree F E = finrank F E"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Group α", "AddMonoid β", "DistribMulAction α β", "a : α", "x : β"], "goal": "a • x = 0 ↔ x = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : LinearOrder α", "inst✝² : TopologicalSpace α", "inst✝¹ : OrderTopology α", "a : α", "inst✝ : (𝓝[>] a).NeBot", "this : Nonempty ↑(Ioi a)", "c : α", "hc : a < c"], "goal": "∃ i', a < i' ∧ Iic i' ⊆ Iio c"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s✝ s₁ s��� t : Set X", "p p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X", "s : Set (Set X)", "hs : s.Finite", "x✝ : ∀ t ∈ ∅, IsOpen t"], "goal": "IsOpen (⋂₀ ∅)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "W.HasLeftCalculusOfFractions", "X Y Z : C", "z₁ : W.LeftFraction X Y", "z₂ : W.LeftFraction Y Z", "z₃ : W.LeftFraction z₁.Y' z₂.Y'", "h₃ : CategoryTheory.CategoryStruct.comp z₂.f z₃.s = CategoryTheory.CategoryStruct.comp z₁.s z₃.f"], "goal": "z₁.comp z₂ = CategoryTheory.MorphismProperty.LeftFraction.Localization.Hom.mk (z₁.comp₀ z₂ z₃)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "x : E", "s : Set E", "x✝ : ℝ"], "goal": "logDeriv cos x✝ = (-tan) x✝"}}
+{"state": {"context": ["x : ℝ", "hc : ContinuousAt tan x", "h₀ : cos x = 0"], "goal": "False"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Abelian C", "S : ShortComplex C", "hS : S.Exact", "b' : Pseudoelement S.X₂", "b : Over S.X₂", "hb : pseudoApply S.g ⟦b⟧ = 0", "hb' : b.hom ≫ S.g = 0", "c : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι S.f) ⋯).pt", "hc : c ≫ Fork.ι (KernelFork.ofι (image.ι S.f) ⋯) = b.hom"], "goal": "pseudoApply S.f (Quot.mk (PseudoEqual S.X₁) (Over.mk (pullback.fst (Abelian.factorThruImage S.f) c))) = ⟦b⟧"}}
+{"state": {"context": ["p : ℕ → Prop", "hf : (setOf p).Finite", "n : ℕ", "hn : n < hf.toFinset.card"], "goal": "nth p n = (hf.toFinset.orderEmbOfFin ⋯) ⟨n, hn⟩"}}
+{"state": {"context": ["a b : Bool"], "goal": "((a || b) = true) = (a = true ∨ b = true)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Field R", "p q : R[X]", "h : (C 0).degree = 0", "this : p.degree ≤ 0", "hc : p.coeff 0 = 0"], "goal": "False"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "a : α"], "goal": "round (a - 1) = round a - 1"}}
+{"state": {"context": ["R : Type u_2", "Ring R"], "goal": "-0 = 0"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "J : GrothendieckTopology C", "F F' F'' : Cᵒᵖ ⥤ Type w", "G G' : Subpresheaf F", "h : Presieve.IsSheaf J F"], "goal": "G = sheafify J G ↔ sheafify J G ≤ G"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "PseudoEMetricSpace X", "PseudoEMetricSpace Y", "e : X ≃ᵈ Y"], "goal": "(DilationEquiv.refl X).trans e = e"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p1 p2 p3 : P", "h : EuclideanGeometry.angle p1 p2 p3 = 0"], "goal": "dist p1 p3 = |dist p1 p2 - dist p3 p2|"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : PartialOrder α", "inst✝ : BoundedOrder α", "n : ℕ", "f : Fin (n + 1) → α", "h0 : f 0 = ⊥", "hlast : f (Fin.last n) = ⊤", "hcovBy : ∀ (k : Fin n), f k.castSucc ⩿ f k.succ", "hmono : Monotone f", "t : Set α", "htc : IsChain (fun x x_1 => x ≤ x_1) t", "hbt : range f ⊆ t", "x : α", "hx : x ∈ t", "h : ∀ (y : Fin (n + 1)), f y ≠ x", "k : Fin (n + 1)"], "goal": "f k < x"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "Archimedean α"], "goal": "⋃ n, Set.Ico (↑n) (↑n + 1) = Set.univ"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "A : Type u_3", "T : Type u_4", "B : Type u_5", "ι : Type u_6", "inst✝⁵ : Semiring R", "inst✝⁴ : Ring R'", "inst✝³ : AddZeroClass A", "inst✝² : SemilatticeSup B", "inst✝¹ : Add B", "inst✝ : OrderBot B", "D : A → B", "f g : R'[A]"], "goal": "supDegree D (f - g) ≤ supDegree D f ⊔ supDegree D g"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "x : E", "r : ℝ", "hr✝ : r ≠ 0", "hr : 0 < r"], "goal": "interior (closedBall x r) = ball x r"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "F.Faithful", "X Y : C", "f g : X ⟶ Y"], "goal": "F.map f = F.map g ↔ f = g"}}
+{"state": {"context": ["C : Type u_1", "H : Type u_3", "D : Type u_5", "CategoryTheory.Category.{u_7, u_1} C", "CategoryTheory.Category.{u_8, u_3} H", "CategoryTheory.Category.{u_9, u_5} D", "L : CategoryTheory.Functor C D", "F F' : CategoryTheory.Functor C H", "iso₂ : F ≅ F'"], "goal": "L.HasLeftKanExtension F ↔ L.HasLeftKanExtension F'"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x✝ : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "x : M"], "goal": "span R {x} = ⊤ ↔ ∀ (v : M), ∃ r, r • x = v"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "x y : R", "h : IsRelPrime x y", "z : R"], "goal": "IsRelPrime x (x * z + y)"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "PartialOrder Γ", "Zero R", "Zero Γ", "x : HahnSeries Γ R"], "goal": "0 ≤ x.orderTop ↔ 0 ≤ x.order"}}
+{"state": {"context": ["α : Type u_1", "V : Type u_2", "P : Type u_3", "W : Type u_4", "Q : Type u_5", "inst✝⁸ : SeminormedAddCommGroup V", "inst✝⁷ : PseudoMetricSpace P", "inst✝⁶ : NormedAddTorsor V P", "inst✝⁵ : NormedAddCommGroup W", "inst✝⁴ : MetricSpace Q", "inst✝³ : NormedAddTorsor W Q", "𝕜 : Type u_6", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedSpace 𝕜 V", "inst✝ : NormedSpace 𝕜 W", "p₁ p₂ : P", "c : 𝕜"], "goal": "dist p₂ ((homothety p₁ c) p₂) = ‖1 - c‖ * dist p₁ p₂"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β₁ : Type u_3", "β₂ : Type u_4", "γ₁ : Type u_5", "γ₂ : Type u_6", "f : α₁ → β₁ → Finset γ₁", "g : α₂ → β₂ → Finset γ₂", "a : α₁ ⊕ α₂", "b : β₁ ⊕ β₂"], "goal": "Finset.sumLift₂ f g a b = ∅ ↔\n  (∀ (a₁ : α₁) (b₁ : β₁), a = Sum.inl a₁ → b = Sum.inl b₁ → f a₁ b₁ = ∅) ∧\n    ∀ (a₂ : α₂) (b₂ : β₂), a = Sum.inr a₂ → b = Sum.inr b₂ → g a₂ b₂ = ∅"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "f : (i : α) → E i", "hf : Summable fun i => ‖f i‖ ^ ENNReal.toReal 0"], "goal": "Memℓp f 0"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Ring R", "DecidableEq σ", "n : σ →₀ ℕ", "a : R", "φ : MvPowerSeries σ R"], "goal": "(MvPowerSeries.coeff R n) (MvPowerSeries.inv.aux a φ) =\n  if n = 0 then a\n  else\n    -a *\n      ∑ x ∈ Finset.antidiagonal n,\n        if x.2 < n then (MvPowerSeries.coeff R x.1) φ * (MvPowerSeries.coeff R x.2) (MvPowerSeries.inv.aux a φ) else 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : ℕ → ℝ≥0", "r : ℝ≥0"], "goal": "HasSum f r ↔ Tendsto (fun n => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r)"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "r : ι → ι → Prop", "s : (i : ι) → α i → α i → Prop", "inst✝¹ : IsStrictTotalOrder ι r", "inst✝ : Finite ι", "hs : ∀ (i : ι), WellFounded (s i)"], "goal": "WellFounded (Pi.Lex r fun {i} => s i)"}}
+{"state": {"context": ["α✝ : Sort u", "β : Sort v", "γ : Sort w", "α : Type u_1", "inst✝ : DecidableEq α", "a b : α", "s : Set α", "ha : a ∈ s", "hb : b ∉ s", "x : α", "hx : x ∈ insert b (s \\ {a})"], "goal": "x ∈ ⇑(swap a b) '' s"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X", "h : IsPreirreducible s", "u : Set X", "hu : IsOpen u", "v : Set X", "hv : IsOpen v", "x✝ : X", "hxs✝ : x✝ ∈ s", "hxu✝ : ⟨x✝, hxs✝⟩ ∈ val ⁻¹' u", "y : X", "hys : y ∈ s", "hyv : ⟨y, hys⟩ ∈ val ⁻¹' v", "x : X", "hxs : x ∈ s", "hxu : x ∈ u", "hxv : x ∈ v"], "goal": "(univ ∩ (val ⁻¹' u ∩ val ⁻¹' v)).Nonempty"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "M : Type u_1", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : Module S M", "inst✝ : IsScalarTower R S M", "x : S ⊗[R] S", "hx : x ∈ ideal R S", "this : x ∈ Submodule.span S (Set.range fun s => 1 ⊗ₜ[R] s - s ⊗ₜ[R] 1)"], "goal": "∀ (x y : S ⊗[R] S), (∃ (hx : x ∈ ideal R S), (ideal R S).toCotangent ⟨x, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))) → (∃ (hx : y ∈ ideal R S), (ideal R S).toCotangent ⟨y, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))) → ∃ (hx : x + y ∈ ideal R S), (ideal R S).toCotangent ⟨x + y, hx⟩ ∈ Submodule.span S (Set.range ⇑(D R S))"}}
+{"state": {"context": ["𝕜 : Type u_1", "LinearOrderedField 𝕜", "x y z : 𝕜", "h : x < y"], "goal": "z ∈ Set.Ioc x y ↔ ∃ a b, 0 ≤ a ∧ 0 < b ∧ a + b = 1 ∧ a * x + b * y = z"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "Semiring k", "AddZeroClass G", "f : AddMonoidAlgebra k G", "r : k", "x : G"], "goal": "(AddMonoidAlgebra.single 0 r * f) x = r * f x"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDomain R", "p : Polynomial R"], "goal": "(Multiset.map (fun a => Polynomial.X - Polynomial.C a) p.roots).prod.Monic"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "Preorder β", "u : β → α", "h : CauchySeq u"], "goal": "Filter.Tendsto (Prod.map u u) Filter.atTop (𝓤 α)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s✝ t✝ u v : Finset α", "a✝ b : α", "s t : Finset α", "a : α"], "goal": "s.erase a \\ t = (s \\ t).erase a"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "s t : Finset α", "h : s ⊆ t"], "goal": "Finset.powersetCard n s ⊆ Finset.powersetCard n t"}}
+{"state": {"context": ["f : StieltjesFunction", "a : ℝ", "u : ℕ → ℝ", "u_mono : StrictMono u", "u_lt_a : ∀ (n : ℕ), u n < a", "u_lim : Tendsto u atTop (𝓝 a)", "A : {a} = ⋂ n, Ioc (u n) a", "L1 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a}))", "L2 : Tendsto (fun n => f.measure (Ioc (u n) a)) atTop (𝓝 (ofReal (↑f a - leftLim (↑f) a)))"], "goal": "f.measure {a} = ofReal (↑f a - leftLim (↑f) a)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s✝ t : Set X", "s : Set ι", "f : ι → Set X", "hs : s.Countable", "hf : ∀ i ∈ s, IsLindelof (f i)", "i : Type u", "U : i → Set X", "hU : ∀ (i : i), IsOpen (U i)", "hUcover : ⋃ i ∈ s, f i ⊆ ⋃ i, U i"], "goal": "∃ t, t.Countable ∧ ⋃ i ∈ s, f i ⊆ ⋃ i_1 ∈ t, U i_1"}}
+{"state": {"context": ["α : Type u_1", "C : Set (Set α)", "s t : Set α", "hC : IsSetSemiring C", "hs : s ∈ C", "ht : t ∈ C", "h : (↑(hC.diffFinset hs ht)).PairwiseDisjoint id", "u : Set α", "hu : u ∈ ↑(hC.diffFinset hs ht)"], "goal": "Disjoint t u"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "DecidableEq ι", "(i : ι) → PartialOrder (α i)", "(i : ι) → Zero (α i)", "i : ι", "b : α i"], "goal": "Pi.single i '' Set.Ico 0 b = Set.Ico 0 (Pi.single i b)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : Semiring α", "inst✝⁴ : NoZeroDivisors α", "m n✝ n : ℕ", "R : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : IsDomain R", "inst✝¹ : DecidableEq R", "inst✝ : NormalizedGCDMonoid R", "hn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n", "a b c : R", "ha : a ≠ 0", "hb : b ≠ 0", "hc : c ≠ 0", "habc : a ^ n + b ^ n = c ^ n", "s : Finset R := {a, b, c}", "d : R := s.gcd id", "A : R", "hA : a = d * A", "B : R", "hB : b = d * B", "C : R", "hC : c = d * C"], "goal": "False"}}
+{"state": {"context": ["U : TopCat", "X : AlgebraicGeometry.Scheme", "f : U ⟶ TopCat.of ↑↑X.toPresheafedSpace", "h : OpenEmbedding ⇑f"], "goal": "↑(X.restrict h).toPresheafedSpace = U"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝ : CommRing R", "p q : R[X]", "x : R", "hpq : (p.comp (X + C x)).trailingCoeff * (q.comp (X + C x)).trailingCoeff ≠ 0"], "goal": "rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u₂", "CategoryTheory.Category.{w, u₂} D", "F : C ⥤ D", "A A' B B' : C", "CategoryTheory.Limits.HasBinaryCoproduct A B", "CategoryTheory.Limits.HasBinaryCoproduct A' B'", "CategoryTheory.Limits.HasBinaryCoproduct (F.obj A) (F.obj B)", "CategoryTheory.Limits.HasBinaryCoproduct (F.obj A') (F.obj B')", "f : A ⟶ A'", "g : B ⟶ B'", "CategoryTheory.IsIso (CategoryTheory.Limits.coprodComparison F A B)", "CategoryTheory.IsIso (CategoryTheory.Limits.coprodComparison F A' B')"], "goal": "CategoryTheory.inv (CategoryTheory.Limits.coprodComparison F A B) ≫\n    CategoryTheory.Limits.coprod.map (F.map f) (F.map g) =\n  F.map (CategoryTheory.Limits.coprod.map f g) ≫ CategoryTheory.inv (CategoryTheory.Limits.coprodComparison F A' B')"}}
+{"state": {"context": ["f : ℝ → ℝ", "x f' : ℝ", "s : Set ℝ", "hf : HasDerivWithinAt f f' Set.univ x", "hx : f x ≠ 0"], "goal": "HasDerivWithinAt (fun y => Real.log (f y)) (f' / f x) Set.univ x"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "x : α", "S Z : Set α", "h : IsGenericPoint x S", "hZ : IsClosed Z"], "goal": "x ∈ Z ↔ S ⊆ Z"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "J : GrothendieckTopology C", "A : Type u'", "inst✝ : Category.{v', u'} A", "F G : Cᵒᵖ ⥤ A", "X : C", "S : Sieve X", "hG : ⦃Y : C⦄ → (f : Y ⟶ X) → IsLimit (G.mapCone (Sieve.pullback f S).arrows.cocone.op)", "x : Presieve.FamilyOfElements (presheafHom F G) S.arrows", "hx : x.Compatible"], "goal": "∃ x_1, x.IsAmalgamation x_1"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "DecidableEq β", "SMul α β", "a : α", "s : Set β", "Fintype ↑s", "Fintype ↑(a • s)"], "goal": "(a • s).toFinset = a • s.toFinset"}}
+{"state": {"context": ["α : Type u", "β : Type v", "s : Set α", "Fintype ↑s", "f : α → β", "Fintype ↑(f '' s)", "H : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y"], "goal": "Fintype.card ↑(f '' s) = Fintype.card ↑s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c✝ c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "c : E''", "hc : c ≠ 0"], "goal": "(fun _x => c) =o[l] g ↔ (fun _x => 1) =o[l] g"}}
+{"state": {"context": ["F : Type u_1", "inst✝³ : Field F", "p q : F[X]", "E : Type u_2", "inst✝² : Field E", "inst✝¹ : Algebra F E", "inst✝ : CharZero F", "p_irr : Irreducible p", "p_deg : Nat.Prime p.natDegree"], "goal": "p.natDegree ∣ finrank F p.SplittingField"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "𝒜 : Finset (Finset α)", "s : Finset α", "a : α"], "goal": "s ∈ Down.compression a 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁴ : TopologicalSpace α", "inst✝³ : LinearOrder α", "inst✝² : OrderTopology α", "inst✝¹ : NoMinOrder α", "inst✝ : DenselyOrdered α", "a : α", "s : Set α"], "goal": "s ∈ 𝓝[<] a ↔ ∃ l ∈ Iio a, Ico l a ⊆ s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝³ : Mul β", "inst✝² : Preorder β", "inst✝¹ : CovariantClass β β (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "inst✝ : CovariantClass β β (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "l : Filter α", "f₁ f₂ g₁ g₂ : α → β", "hf : f₁ ≤ᶠ[l] f₂", "hg : g₁ ≤ᶠ[l] g₂"], "goal": "f₁ * g₁ ≤ᶠ[l] f₂ * g₂"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n✝ : ℕ", "inst✝¹ : Semiring R", "p q r✝ : R[X]", "inst✝ : Semiring S", "f : R →+* S", "x : S", "n : ℕ", "r : R"], "goal": "eval₂ f x ((monomial n) r) = f r * x ^ n"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "m n : ℕ∞", "f₂ : 𝕜 → F", "s₂ : Set 𝕜", "hf : ContDiffOn 𝕜 n f₂ s₂", "hs : IsOpen s₂", "hmn : m + 1 ≤ n"], "goal": "ContDiffOn 𝕜 m (deriv f₂) s₂"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝¹⁵ : CommRing R", "inst✝¹⁴ : CommRing A", "inst✝¹³ : Field K", "inst✝¹² : IsDomain A", "inst✝¹¹ : Algebra A K", "inst✝¹⁰ : IsFractionRing A K", "L : Type u_4", "inst✝⁹ : Field L", "C : Type u_5", "inst✝⁸ : CommRing C", "inst✝⁷ : Algebra K L", "inst✝⁶ : Algebra A L", "inst✝⁵ : IsScalarTower A K L", "inst✝⁴ : Algebra C L", "inst✝³ : IsIntegralClosure C A L", "inst✝² : Algebra A C", "inst✝¹ : IsScalarTower A C L", "inst✝ : FiniteDimensional K L", "this✝ : DecidableEq L := Classical.decEq L", "this : IsNoetherian K L := IsNoetherian.iff_fg.mpr inferInstance", "s' : Finset L := IsNoetherian.finsetBasisIndex K L", "bs' : Basis { x // x ∈ IsNoetherian.finsetBasisIndex K L } K L := IsNoetherian.finsetBasis K L", "y : A", "hy : y ≠ 0", "his' : ∀ x ∈ Finset.image (⇑bs') Finset.univ, IsIntegral A (y • x)", "hy' : (algebraMap A L) y ≠ 0"], "goal": "∃ s b, ∀ (x : { x // x ∈ s }), IsIntegral A (b x)"}}
+{"state": {"context": ["α : Type u_2", "LT α", "h : WellFounded fun x x_1 => x < x_1", "s : Set α"], "goal": "s.IsWF"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "CategoryTheory.Limits.HasBinaryProducts C"], "goal": "(CategoryTheory.coalgebraEquivOver X).counitIso =\n  CategoryTheory.NatIso.ofComponents\n    (fun f =>\n      CategoryTheory.Over.isoMk\n        (CategoryTheory.Iso.refl\n          (((CategoryTheory.overToCoalgebra X).comp (CategoryTheory.coalgebraToOver X)).obj f).left)\n        ⋯)\n    ⋯"}}
+{"state": {"context": ["p : Prop", "a b c : ℕ", "self : Mathlib.Meta.NormNum.IsNatPowT p a b c"], "goal": "p → a.pow b = c"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "ℱ : Filtration ℕ m0", "a b : ℝ", "f : ℕ → Ω → ℝ", "ω : Ω", "R : ℝ≥0", "hab : a < b", "hω : ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k"], "goal": "¬((∃ᶠ (n : ℕ) in atTop, f n ω < a) ∧ ∃ᶠ (n : ℕ) in atTop, b < f n ω)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : ℤ → R", "f : R →+* S", "b c d : R", "n : ℤ"], "goal": "preNormEDS b c d (-n) = -preNormEDS b c d n"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a : α"], "goal": "¬a > a"}}
+{"state": {"context": ["R : Type u_2", "NonAssocSemiring R", "S : Set (Subsemiring Rᵐ��ᵖ)"], "goal": "(sInf S).unop = sInf (Subsemiring.op ⁻¹' S)"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "inst✝¹ : Category.{v, u} V", "inst✝ : Preadditive V", "c : ComplexShape ι", "C D E : HomologicalComplex V c", "f g : C ⟶ D", "h k : D ⟶ E", "i✝ : ι", "f₁ g₁ f₂ g₂ : C ⟶ D", "h₁ : Homotopy f₁ g₁", "h₂ : Homotopy f₂ g₂", "i : ι"], "goal": "(dNext i) h₁.hom + (prevD i) h₁.hom + g₁.f i + ((dNext i) h₂.hom + (prevD i) h₂.hom + g₂.f i) = (dNext i) h₁.hom + (dNext i) h₂.hom + ((prevD i) h₁.hom + (prevD i) h₂.hom) + (g₁.f i + g₂.f i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : EDist α", "x y : α", "s t : Set α", "x✝ : ℝ≥0∞"], "goal": "x✝ ≤ s.einfsep ↔ x✝ ≤ ⨅ d, uncurry edist ↑d"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "a : (i : ℕ) → M [⋀^Fin i]→ₗ[R] N"], "goal": "ExteriorAlgebra.liftAlternatingEquiv a = ExteriorAlgebra.liftAlternating a"}}
+{"state": {"context": ["x : ℝ", "hx : 0 ≤ x", "y : ℝ"], "goal": "x⁻¹ ^ y = (x ^ y)⁻¹"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "𝒜 ℬ : Finset (Finset α)", "s t : Finset α"], "goal": "s ∈ 𝒜 → t ∈ ℬ → s ∩ t ∈ 𝒜 ⊼ ℬ"}}
+{"state": {"context": ["a b : UInt8"], "goal": "a + b = { val := a.val + b.val }"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "M' : Type u_3", "N : Type u_4", "N' : Type u_5", "P : Type u_6", "P' : Type u_7", "inst✝⁸ : AddGroup M", "inst✝⁷ : AddGroup N", "inst✝⁶ : AddGroup P", "f : M →+ N", "g : N →+ P", "X₁ : Type u_8", "X₂ : Type u_9", "X₃ : Type u_10", "Y₁ : Type u_11", "Y₂ : Type u_12", "Y₃ : Type u_13", "inst✝⁵ : AddCommMonoid X₁", "inst✝⁴ : AddCommMonoid X₂", "inst✝³ : AddCommMonoid X₃", "inst✝² : AddCommMonoid Y₁", "inst✝¹ : AddCommMonoid Y₂", "inst✝ : AddCommMonoid Y₃", "e₁ : X₁ ≃+ Y₁", "e₂ : X₂ ≃+ Y₂", "e₃ : X₃ ≃+ Y₃", "f₁₂ : X₁ →+ X₂", "f₂₃ : X₂ →+ X₃", "g₁₂ : Y₁ →+ Y₂", "g₂₃ : Y₂ →+ Y₃", "comm₁₂ : g₁₂.comp ↑e₁ = (↑e₂).comp f₁₂", "comm₂₃ : g₂₃.comp ↑e₂ = (↑e₃).comp f₂₃", "H : Exact ⇑f₁₂ ⇑f₂₃", "h₁₂ : ∀ (x : X₁), g₁₂ (e₁ x) = e₂ (f₁₂ x)", "h₂₃ : ∀ (x : X₂), g₂₃ (e₂ x) = e₃ (f₂₃ x)"], "goal": "⇑g₂₃ ∘ ⇑g₁₂ = 0"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "𝕜 : Type u_4", "RCLike 𝕜", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "f : ↥(MeasureTheory.Lp E 2 μ)"], "goal": "∫ (a : α), ⟪↑↑f a, ↑↑f a⟫_𝕜 ∂μ = ↑(∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ).toReal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "f g : ι → α → β", "p : ℝ≥0∞", "hf : UnifTight f p μ", "hfg : ∀ (n : ι), f n =ᶠ[ae μ] g n", "ε : ℝ≥0", "hε : 0 < ε", "s : Set α", "hμs : μ s ≠ ⊤", "hfε : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε", "n : ι"], "goal": "sᶜ.indicator (g n) =ᶠ[ae μ] sᶜ.indicator (f n)"}}
+{"state": {"context": ["n : ℕ", "p : Fin n → Prop", "DecidablePred p", "i : Fin n"], "goal": "i ∈ Fin.find p → p i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddCommGroup α", "UniformSpace α", "UniformAddGroup α", "f : β → α", "CompleteSpace α"], "goal": "Summable f ↔ ∀ e ∈ 𝓝 0, ∃ s, ∀ (t : Finset β), Disjoint t s → ∑ b ∈ t, f b ∈ e"}}
+{"state": {"context": ["R : Type u_1", "Semiring R"], "goal": "LaurentSeries.powerSeriesPart 0 = 0"}}
+{"state": {"context": ["p : ℝ≥0"], "goal": "⊤ / ↑p = ⊤"}}
+{"state": {"context": ["X : Type u_1", "MeasurableSpace X", "μ : MeasureTheory.Measure X", "f : X → ℝ", "s : Set X", "hs : ∀ x ∈ s, f x ≤ 1", "h's : ∀ x ∈ sᶜ, f x ≤ 0"], "goal": "ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "M N P : Mon_ C"], "goal": "((λ_ (𝟙_ C)).inv ≫ (M.one ⊗ N.one) ⊗ P.one) ≫ (α_ M.X N.X P.X).hom = M.one ⊗ (λ_ (𝟙_ C)).inv ≫ (N.one ⊗ P.one)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : DecidableEq α", "A : Finset α", "P : Finpartition A", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε δ : 𝕜", "u v s t : Finset α", "hA : A.Nonempty", "hε : 0 < ε", "hP : P.IsEquipartition", "hG : P.IsUniform G ε"], "goal": "↑((P.nonUniforms G ε).biUnion fun x => match x with | (U, V) => U ×ˢ V).card < 4 * ε * ↑A.card ^ 2"}}
+{"state": {"context": [], "goal": "⊥.cells = ∅"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "h₁ : min a b < max c d", "h₂ : min c d < max a b"], "goal": "Icc a b ∪ Icc c d = Icc (min a c) (max b d)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Ring R", "p q : R[X]", "h : q.degree < p.degree"], "goal": "(p - q).degree = p.degree"}}
+{"state": {"context": ["α : Type u_1", "s : Multiset α", "a : α"], "goal": "a ∈ s → ∃ t, s = a ::ₘ t"}}
+{"state": {"context": ["n k : ℕ"], "goal": "n.ascFactorial k.succ = (n + k) * n.ascFactorial k"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "inst✝ : FiniteDimensional ℝ V", "s : Set V", "hs : Convex ℝ s"], "goal": "(interior s).Nonempty ↔ affineSpan ℝ s = ⊤"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_4", "AddCommMonoid M", "TopologicalSpace M", "n : MeasurableSpace α", "v : MeasureTheory.VectorMeasure α M"], "goal": "v.trim ⋯ = v"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Ω : Type u_4", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝⁴ : CountablyGenerated γ", "inst✝³ : MeasurableSpace Ω", "inst✝² : StandardBorelSpace Ω", "inst✝¹ : Nonempty Ω", "κ : Kernel α (γ × ℝ)", "inst✝ : IsFiniteKernel κ", "a : α", "s : ℕ → ℚ", "hs_anti : Antitone s", "hs_tendsto : Tendsto s atTop atBot", "s' : ℕ → Set ℝ := fun n => Iic ↑(s n)"], "goal": "⋂ i, s' i = ∅"}}
+{"state": {"context": ["P : Type u_1", "inst✝¹ : SemilatticeSup P", "inst✝ : IsDirected P fun x x_1 => x ≥ x_1", "x : P", "I J K s t : Ideal P", "hx : x ∉ I", "h : I = I ⊔ principal x"], "goal": "x ∈ I"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ", "h : 0 ∉ [[a, b]]"], "goal": "∫ (x : ℝ) in a..b, 1 / x = Real.log (b / a)"}}
+{"state": {"context": ["Γ : Type u_1", "Λ : Type u_2", "σ : Type u_3", "S : Finset Λ", "q₁ q₂ : Turing.TM1.Stmt Γ Λ σ", "h : q₁ ∈ Turing.TM1.stmts₁ q₂", "hs : Turing.TM1.SupportsStmt S q₂"], "goal": "Turing.TM1.SupportsStmt S q₁"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.HasShift C ℤ", "X : CategoryTheory.Pretriangulated.Triangle C"], "goal": "((CategoryTheory.Functor.mapTriangleIdIso C).hom.app X).hom₁ = CategoryTheory.CategoryStruct.id X.obj₁"}}
+{"state": {"context": ["α : Type u_1", "OrderedCommGroup α", "a c d : α"], "goal": "a⁻¹ ∈ Set.Ioc c d ↔ a ∈ Set.Ico d⁻¹ c⁻¹"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : SeminormedAddCommGroup G", "H : Type u_2", "inst✝¹ : SeminormedAddCommGroup H", "K : Type u_3", "inst✝ : SeminormedAddCommGroup K", "f : NormedAddGroupHom G H", "g : G", "h₀ : f g = 0"], "goal": "(toCompl.comp (incl f.ker)).toAddMonoidHom ⟨g, h₀⟩ ∈ f.completion.ker"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "K : Type u_4", "inst✝¹ : Semiring R", "inst✝ : Semiring S", "p : R[X]", "s : R", "this : (a : Prop) → Decidable a", "hp : ¬p = 0"], "goal": "(p.scaleRoots s).coeff p.natDegree ≠ 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "OrderedCommMonoid α", "TopologicalSpace α", "OrderClosedTopology α", "f : ι → α", "a : α", "s : Finset ι", "hs : ∀ i ∉ s, 1 ≤ f i", "hf : HasProd f a"], "goal": "∏ i ∈ s, f i ≤ a"}}
+{"state": {"context": ["X : Type u_2", "EMetricSpace X", "s : Set X", "h : s.Subsingleton"], "goal": "dimH s = 0"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s s' : Finset α", "P : Prop", "Decidable P"], "goal": "s ∩ s' ⊆ if P then s else s'"}}
+{"state": {"context": ["δ : Type u_3", "AddCommMonoid δ", "f : ℕ → δ", "k : ℕ", "m n : ℕ", "h : m ≤ n + 1"], "goal": "∑ j ∈ Finset.Ico k m, f (n - j) = ∑ j ∈ Finset.Ico (n + 1 - m) (n + 1 - k), f j"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : ℝ → E", "s : Set ℝ", "l : ℝ≥0", "h : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))", "x : ℝ", "xs : x ∈ s", "y : ℝ", "ys : y ∈ s", "xy : x ≤ y"], "goal": "eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "s✝ s₁ s₂ : Set α", "t t₁ t₂ : Set β", "a : α", "b : β", "s : Set α"], "goal": "s ×ˢ univ = Prod.fst ⁻¹' s"}}
+{"state": {"context": ["α : Type u_1", "𝒜 : Finset (Finset α)", "r : ℕ"], "goal": "𝒜 # r ⊆ 𝒜"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "m0 : MeasurableSpace α", "μ : Measure α", "v : VitaliFamily μ", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : SecondCountableTopology α", "inst✝² : BorelSpace α", "inst✝¹ : IsLocallyFiniteMeasure μ", "ρ : Measure α", "inst✝ : IsLocallyFiniteMeasure ρ", "hρ : ρ ≪ μ", "p : ℝ≥0", "s : Set α", "h : s ⊆ {x | v.limRatioMeas hρ x < ↑p}", "t : Set α := {x | Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatioMeas hρ x))}", "A : μ tᶜ = 0", "x : α", "hx : x ∈ s ∩ t", "I : ∀ᶠ (b : Set α) in v.filterAt x, ρ b / μ b < ↑p", "a : Set α", "ha : ρ a / μ a < ↑p"], "goal": "ρ a ≤ ↑p * μ a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "TopologicalSpace α", "LinearOrder α", "OrderClosedTopology α", "f g : β → α", "TopologicalSpace β", "TopologicalSpace γ", "(x : β) → Decidable (f x ≤ g x)", "f' g' : β → γ", "hf' : Continuous f'", "hg' : Continuous g'", "hf : Continuous f", "hg : Continuous g", "hfg : ∀ (x : β), f x = g x → f' x = g' x"], "goal": "Continuous fun x => if f x ≤ g x then f' x else g' x"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "inst✝⁵ : CommRing F", "inst✝⁴ : Ring K", "inst✝³ : Algebra F K", "E : Type u_3", "inst✝² : Ring E", "inst✝¹ : Algebra F E", "e : K ≃ₐ[F] E", "inst✝ : Algebra.IsSeparable F K", "x✝ : E"], "goal": "IsSeparable F x✝"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.ConcreteCategory D", "CategoryTheory.Preregular C", "CategoryTheory.FinitaryExtensive C", "F G : CategoryTheory.Sheaf (CategoryTheory.coherentTopology C) D", "f : F ⟶ G", "CategoryTheory.Limits.PreservesFiniteProducts (CategoryTheory.forget D)"], "goal": "CategoryTheory.Sheaf.IsLocallySurjective f ↔\n  CategoryTheory.Presheaf.IsLocallySurjective (CategoryTheory.regularTopology C) f.val"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "T2Space X", "S : Set X", "h : IsPreirreducible S"], "goal": "S.Subsingleton"}}
+{"state": {"context": ["α : Type u", "L L₁ L₂ L₃ L₄ L1 L2 : List (α × Bool)", "x : α", "b : Bool"], "goal": "L1.length + L2.length + 2 = L1.length + ((x, b) :: (x, !b) :: L2).length"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s s' : Finset α", "t t' : Finset β", "a : α", "b : β", "inst✝ : DecidableEq β", "i : β"], "goal": "i ∈ image Prod.snd (s ×ˢ t) → i ∈ t"}}
+{"state": {"context": ["ι : Type u_1", "I J J₁ J₂ : Box ι", "π π₁ π₂ : Prepartition I", "x : ι → ℝ"], "goal": "π₁.iUnion = ∅ ↔ π₁ = ⊥"}}
+{"state": {"context": ["n✝ m : ℕ", "p : Fin (n✝ + 1)", "i✝ j✝ : Fin n✝", "n : ℕ", "i : Fin (n + 1)", "j : Fin n", "h : i < j.succ"], "goal": "i.succ.succAbove j.succ = (i.succAbove j).succ"}}
+{"state": {"context": ["C : Type u", "inst✝¹⁰ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w₁", "inst✝⁹ : Category.{max v u, w₁} D", "E : Type w₂", "inst✝⁸ : Category.{max v u, w₂} E", "F✝ : D ⥤ E", "inst✝⁷ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D", "inst✝⁶ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E", "inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝⁴ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E", "inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝", "inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝", "P✝ P : Cᵒᵖ ⥤ D", "F : D ⥤ E", "inst✝¹ : (F : D ⥤ E) → (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F", "inst✝ : (F : D ⥤ E) → (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F"], "goal": "(𝟙 (J.plusObj (J.plusObj (P ⋙ F))) ≫ J.plusMap (J.plusCompIso F P).inv) ≫ (J.plusCompIso F (J.plusObj P)).inv = J.plusMap (J.plusCompIso F P).inv ≫ (J.plusCompIso F (J.plusObj P)).inv"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝ : MonoidWithZero M₀", "x y z : M₀", "h : IsUnit z", "h1 : x * z = y"], "goal": "x = y * inverse z"}}
+{"state": {"context": ["z : ℂ"], "goal": "HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((NormedSpace.exp ℂ (z * I) + NormedSpace.exp ℂ (-z * I)) / 2)"}}
+{"state": {"context": ["a b c d m n✝ : ℤ", "n : ℕ"], "goal": "0 + ↑n = ofNat n"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "PartialOrder α", "ContravariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c : α", "h : a * b = c * b"], "goal": "a = c"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝⁷ : Semiring R", "inst✝⁶ : StarMul R", "inst✝⁵ : TrivialStar R", "inst✝⁴ : AddCommGroup A", "inst✝³ : Module R A", "inst✝² : StarAddMonoid A", "inst✝¹ : StarModule R A", "inst✝ : Invertible 2"], "goal": "A ≃ₗ[R] ↥(selfAdjoint A) × ↥(skewAdjoint A)"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "𝔹 : Type u_3", "inst✝⁴ : NontriviallyNormedField 𝕂", "inst✝³ : NormedRing 𝔸", "inst✝² : NormedRing 𝔹", "inst✝¹ : NormedAlgebra 𝕂 𝔸", "inst✝ : NormedAlgebra 𝕂 𝔹", "x : 𝔸", "hx : x ∈ EMetric.ball 0 (expSeries 𝕂 𝔸).radius"], "goal": "Summable (norm ∘ fun n => (↑n !)⁻¹ • x ^ n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "m n : ℕ", "h : m < n"], "goal": "(fun x => x ^ n) =o[𝓝 0] fun x => x ^ m"}}
+{"state": {"context": ["E : Type u_3", "NormedAddCommGroup E", "NormedSpace ℝ E", "a b : ℝ", "f g : ℝ → E", "μ : MeasureTheory.Measure ℝ", "hf : IntervalIntegrable f μ a b", "hg : IntervalIntegrable g μ a b"], "goal": "∫ (x : ℝ) in a..b, f x - g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in a..b, g x ∂μ"}}
+{"state": {"context": ["α : Type u", "Group α", "LT α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α"], "goal": "a⁻¹ < b / c ↔ c < a * b"}}
+{"state": {"context": ["α : Type u_2", "SemilatticeSup α", "s t : Set α"], "goal": "Set.image2 (fun x x_1 => x ⊔ x_1) s t = s ⊻ t"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "Group G", "H : Type u_2", "Group H", "P : Sylow p H", "f : H →* G", "hf : IsPGroup p ↥f.ker"], "goal": "∃ Q, Subgroup.comap f ↑Q = ↑P"}}
+{"state": {"context": ["m : ℕ", "hx : ↑m ≠ ⊤", "n : ℕ", "h : ↑m + 1 ≤ ↑n"], "goal": "m < n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ✝ : Type u_3", "δ : Type u_4", "inst✝⁸ : Monoid γ✝", "inst✝⁷ : TopologicalSpace α", "inst✝⁶ : AddCommMonoid α", "inst✝⁵ : DistribMulAction γ✝ α", "inst✝⁴ : ContinuousConstSMul γ✝ α", "f : β → α", "γ : Type u_5", "inst✝³ : Group γ", "inst✝² : DistribMulAction γ α", "inst✝¹ : ContinuousConstSMul γ α", "inst✝ : T2Space α", "g : γ", "hf : ¬Summable f", "mul_g : α ≃+ α := DistribMulAction.toAddEquiv α g"], "goal": "¬Summable (⇑mul_g ∘ f)"}}
+{"state": {"context": ["α : Type u", "inst✝³ : Group α", "inst✝² : LT α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α"], "goal": "a⁻¹ < b / c ↔ c < a * b"}}
+{"state": {"context": ["Ω : Type u_1", "f : ℕ → Ω → ℝ", "i : ℕ", "r : ℝ", "ω : Ω"], "goal": "MeasureTheory.leastGE f r i ω ≤ i"}}
+{"state": {"context": ["R : Type u_1", "P : Cubic R", "Semiring R"], "goal": "P.toPoly.coeff 1 = P.c"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_3", "inst✝ : OrderedCommGroup α", "m : ℤ", "a b : α", "ha : 1 < a", "n : ℕ", "hn : 0 < ↑n"], "goal": "1 < a ^ ↑n"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "e : E ≃ₛₗᵢ[σ₁₂] E₂"], "goal": "Function.Injective ⇑e"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L : Filter 𝕜", "hxs : UniqueDiffWithinAt 𝕜 s x", "c : F"], "goal": "derivWithin (fun y => f y - c) s x = derivWithin f s x"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P✝ : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "P Q : Cᵒᵖ ⥤ D", "η : P ⟶ Q", "x✝ : Cᵒᵖ"], "goal": "η.app x✝ ≫ Multiequalizer.lift (⊤.index Q) (Q.obj (op (unop x✝))) (fun I => Q.map I.f.op) ⋯ ≫ colimit.ι (J.diagram Q (unop x✝)) (op ⊤) = Multiequalizer.lift (⊤.index P) (P.obj (op (unop x✝))) (fun I => P.map I.f.op) ⋯ ≫ (J.diagramNatTrans η (unop x✝)).app (op ⊤) ≫ colimit.ι (J.diagram Q (unop x✝)) (op ⊤)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "n : ℕ", "κ : ProbabilityTheory.Kernel α β", "a : α"], "goal": "(n • κ) a = n • κ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrderedField α", "Ring β", "abv : β → α", "IsAbsoluteValue abv", "f : CauSeq β abv", "x : α"], "goal": "∃ r > x, ∀ (i : ℕ), abv (↑f i) < r"}}
+{"state": {"context": ["x : ℝ"], "goal": "Irrational x⁻¹ ↔ Irrational x"}}
+{"state": {"context": ["α✝ β : Type u", "c : Cardinal.{?u.214930}", "α : Type u", "s : Set α", "a : α", "h : a ∉ s"], "goal": "#↑(insert a s) = #↑s + 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "f : α → ℝ", "g : ℝ → ℝ", "s : Set α", "μ : Measure α", "f_nn : 0 ≤ᶠ[ae μ] f", "f_mble : AEMeasurable f μ", "cst : ℝ → ℝ := fun x => 1", "cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t", "key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, cst t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (cst t)"], "goal": "∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "AddGroup G", "AddAction G α", "a b : α"], "goal": "AddAction.orbit G a = AddAction.orbit G b ↔ a ∈ AddAction.orbit G b"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K : Set G", "hK : IsCompact K", "V : Set G", "hV : (interior V).Nonempty", "g : G", "s : Finset G", "h1s : K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index K V"], "goal": "(fun h => g * h) '' ⋃ g ∈ s, (fun h => g * h) ⁻¹' V ⊆ ⋃ g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s, (fun h => g_1 * h) ⁻¹' V"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹¹ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝⁸ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝⁷ : TopologicalSpace M", "inst✝⁶ : ChartedSpace H M", "E' : Type u_5", "inst✝⁵ : NormedAddCommGroup E'", "inst✝⁴ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝³ : TopologicalSpace H'", "N : Type u_7", "inst✝² : TopologicalSpace N", "inst✝¹ : ChartedSpace H' N", "J : ModelWithCorners 𝕜 E' H'", "inst✝ : SmoothManifoldWithCorners J N", "x : M", "y : N"], "goal": "(I.prod J).boundary (M × N) = univ.prod (J.boundary N) ∪ (I.boundary M).prod univ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.IsFiltered Cᵒᵖ"], "goal": "CategoryTheory.IsCofiltered C"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Type u_2", "CommRing S", "Algebra R S", "K : Type u_5", "L : Type u_6", "IsDomain R", "IsDomain S", "Field K", "Field L", "Algebra R K", "Algebra R L", "Algebra S L", "IsIntegralClosure S R L", "IsFractionRing S L", "Algebra K L", "IsScalarTower R S L", "IsScalarTower R K L", "a : S", "b : Set S", "Algebra.IsAlgebraic R L", "inj : Function.Injective ⇑(algebraMap R L)", "h : ↑(Ideal.span {a}) ⊆ ↑(Submodule.span R b)"], "goal": "↑(Ideal.span {(algebraMap S L) a}) ⊆ ↑(Submodule.span K (⇑(algebraMap S L) '' b))"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Ring R", "Γ₀ : Type u_2", "inst✝¹ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "inst✝ : v.RankOne", "x : Γ₀", "hx : (hom v) x = 0", "h_lt : 0 < x"], "goal": "False"}}
+{"state": {"context": ["m n✝ : ℕ", "α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : AddCommMonoid α", "n : ℕ", "a : Fin (n + 2) → α"], "goal": "sum a = ∑ i : Fin (n + 2), a i"}}
+{"state": {"context": ["m n : Nat", "k : Nat", "hmn : m.Coprime n"], "goal": "m.Coprime (k.gcd n)"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "R : ℝ", "c w : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "hw : w ∈ ball c R", "hc : ContinuousOn f (closedBall c R)", "hd : ∀ x ∈ ball c R \\ s, DifferentiableAt ℂ f x", "hR : 0 < R", "t : Set ℂ", "ht : t ∈ 𝓝 w", "g : ℝ → ℂ := fun x => w + ofReal x", "this : Tendsto g (𝓝 0) (𝓝 w)", "l u : ℝ", "hlu₀ : 0 ∈ Ioo l u", "hlu_sub : Ioo l u ⊆ g ⁻¹' (t ∩ ball c R)"], "goal": "(t ∩ (ball c R \\ s)).Nonempty"}}
+{"state": {"context": ["a : ℝ"], "goal": "interior {z | z.im ≤ a} = {z | z.im < a}"}}
+{"state": {"context": ["G : Type u_1", "Group G", "s t : Set G", "ht : IsNormalSubgroup t"], "goal": "s ⊆ t ↔ Group.normalClosure s ⊆ t"}}
+{"state": {"context": ["α : Type u_1", "m : Multiset α", "r : α → α → Prop"], "goal": "(∀ x ∈ m, r x x) → Multiset.Rel r m m"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ : Measure α", "p q : ℝ", "hpq : p.IsConjExponent q", "f g : α → ℝ≥0∞", "hf : AEMeasurable f μ", "hg : AEMeasurable g μ", "hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤", "hf_zero_rpow : ¬(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) = 0", "hf_top_rpow : (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) ≠ ⊤"], "goal": "(∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "c c₁ c₂ c₃ s t : Set α", "a b x y : α", "hc₁ : ChainClosure r c₁", "hc₂ : ChainClosure r c₂", "hc : SuccChain r c₂ = c₂"], "goal": "c₁ ⊆ c₂"}}
+{"state": {"context": ["L : Language", "T : L.Theory", "α : Type w", "p : T.CompleteType α", "φ : L[[α]].Sentence", "hf : Formula.not φ ∈ p", "ht : φ ∈ p", "h : ¬{φ, Formula.not φ}.IsSatisfiable"], "goal": "{φ, Formula.not φ} ⊆ ↑p"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C", "f₁ : X₁ ⟶ Y₁", "f₂ : X₂ ⟶ Y₂", "g₁ : U₁ ⟶ V₁", "g₂ : U₂ ⟶ V₂"], "goal": "(α_ X₁ X₂ (U₁ ⊗ U₂)).hom ≫ ((((𝟙 X₁ ⊗ (α_ X₂ U₁ U₂).inv) ≫ (f₁ ⊗ (f₂ ⊗ g₁) ⊗ g₂)) ≫ (𝟙 Y₁ ⊗ (β_ Y₂ V₁).hom ⊗ 𝟙 V₂)) ≫ (𝟙 Y₁ ⊗ (α_ V₁ Y₂ V₂).hom)) ≫ (α_ Y₁ V₁ (Y₂ ⊗ V₂)).inv = ((α_ X₁ X₂ (U₁ ⊗ U₂)).hom ≫ (𝟙 X₁ ⊗ (α_ X₂ U₁ U₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ U₁).hom ⊗ 𝟙 U₂) ≫ (𝟙 X₁ ⊗ (α_ U₁ X₂ U₂).hom) ≫ (α_ X₁ U₁ (X₂ ⊗ U₂)).inv) ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂)"}}
+{"state": {"context": ["m n k : ℕ+"], "goal": "k.Coprime n → (k * m).gcd n = m.gcd n"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : SemilatticeInf α", "inst✝ : OrderBot α", "s t : Set α", "a✝ b✝ c : α", "h : (s.Pairwise fun a b => ¬Disjoint a b) ∧ s ≠ {⊥}", "a : α", "ha : a ∈ s", "b : α", "hb : ⊥ ∈ s", "hab : b = ⊥", "this : a = b"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "l : Filter α", "f : α → R", "r : R", "inst✝¹ : LinearOrderedSemiring R", "inst✝ : Archimedean R", "hr : 0 < r", "hf : Tendsto f l atTop", "b : R"], "goal": "∀ᶠ (a : α) in l, b ≤ f a * r"}}
+{"state": {"context": ["K : Type uK", "V : Type uV", "CommRing K", "AddCommGroup V", "Module K V", "Module.Free K V", "v : V"], "goal": "(∀ (φ : Module.Dual K V), φ v = 0) ↔ v = 0"}}
+{"state": {"context": ["p q : ℝ", "h : p.IsConjExponent q"], "goal": "0 < p - 1"}}
+{"state": {"context": ["z : ℂ"], "goal": "z ∈ Complex.slitPlane ↔ 0 < z.re ∨ z.im ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝ : LinearOrderedRing R", "a b : R", "n : ℕ", "hb : b ≠ 0", "he : Even n"], "goal": "-b ^ n < 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder β", "l : Filter α", "a : α", "b : β"], "goal": "IsExtrFilter (fun x => b) l a"}}
+{"state": {"context": ["ι✝ : Type u_1", "α : Type u_2", "M : Matroid α", "F X Y : Set α", "e : α", "ι : Sort u_3", "I J B : Set α", "f x y : α", "hI : M.Indep I", "heI : e ∈ I"], "goal": "¬M.Dep (insert e I)"}}
+{"state": {"context": ["R : Type u_4", "S : Type u_5", "Mul R", "Mul S", "Add R", "Add S"], "goal": "Function.Bijective RingEquiv.symm"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X : Ω → ℝ", "μ : MeasureTheory.Measure Ω", "t : ℝ", "MeasureTheory.IsFiniteMeasure μ", "ε : ℝ", "ht : 0 ≤ t", "h_int : MeasureTheory.Integrable (fun ω => rexp (t * X ω)) μ"], "goal": "(μ {ω | ε ≤ X ω}).toReal ≤ rexp (-t * ε) * ProbabilityTheory.mgf X μ t"}}
+{"state": {"context": ["n : ℕ"], "goal": "(↑n.succ).pred = ↑n"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : CompleteSpace E", "T : E →L[𝕜] E", "hT : IsSelfAdjoint T", "x₀ : E", "hx₀ : x₀ ≠ 0", "hextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀"], "goal": "HasEigenvector (↑T) (↑(⨅ x, T.rayleighQuotient ↑x)) x₀"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommSemiring R", "CommSemiring S", "Algebra R S", "M N : Submodule R S", "H : M.LinearDisjoint N"], "goal": "N.LinearDisjoint M"}}
+{"state": {"context": ["𝕜 : Type u_1", "β : Type u_4", "OrderedSemiring 𝕜", "LinearOrderedAddCommMonoid β", "Module 𝕜 β", "OrderedSMul 𝕜 β", "r s : β"], "goal": "Convex 𝕜 (Set.uIcc r s)"}}
+{"state": {"context": ["f : ℕ → ℂ", "s : ℂ"], "goal": "LSeriesSummable (-f) s ↔ LSeriesSummable f s"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : Type u_1", "inst✝ : Category.{?u.23232, u_1} D", "P : MorphismProperty C"], "goal": "P.map (𝟭 C) ≤ P.isoClosure"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "One M", "f : α → M", "s : Set α"], "goal": "Function.mulSupport f = s ↔ (∀ x ∈ s, f x ≠ 1) ∧ ∀ x ∉ s, f x = 1"}}
+{"state": {"context": ["X : Type u_1", "α : Type u_2", "TopologicalSpace X", "AddMonoid α", "f g : X → α"], "goal": "(tsupport fun x => f x + g x) ⊆ tsupport f ∪ tsupport g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PseudoEMetricSpace α", "PseudoEMetricSpace β", "K : ℝ≥0", "f : α → β", "s : Set α", "hf : AntilipschitzWith K (s.restrict f)", "g : β → α", "t : Set β", "g_maps : Set.MapsTo g t s", "g_inv : Set.RightInvOn g f t"], "goal": "LipschitzWith K (t.restrict g)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : Type u_1", "inst✝¹ : CommSemiring R", "inst✝ : Nontrivial R", "hf : ¬IsField R"], "goal": "∃ x, ∃ (_ : x ≠ 0), ¬IsUnit x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "AddCommMonoid M", "DecidableEq β", "f : α → β", "hf : Function.Injective f", "s : α →₀ M"], "goal": "(Finsupp.mapDomain f s).support = Finset.image f s.support"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X"], "goal": "T2Space X ↔ IsClosed (diagonal X)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I J : Ideal R", "h : I ≤ J"], "goal": "Ideal.map (Ideal.Quotient.mk J) I = ⊥"}}
+{"state": {"context": ["α : Type u", "UniformSpace α"], "goal": "DenseEmbedding CauchyFilter.pureCauchy"}}
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+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α", "a : α", "h : a ∉ s"], "goal": "s ⊂ insert a s"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "TopologicalSpace M", "ContinuousMul M", "f : ℤ → M", "hf₁ : Multipliable fun n => f ↑n", "hf₂ : Multipliable fun n => f (-(↑n + 1))"], "goal": "Multipliable f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p✝ : Set γ", "f f₁ f₂ f₃ : α → β", "g g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a✝ : α", "b : β", "p : β → Prop", "hf : BijOn f s t", "a : α", "ha : a ∈ s", "h : p (f a)"], "goal": "∃ b ∈ t, p b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : AddCommMonoid M", "inst✝⁹ : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝⁸ : CommRing R₁", "inst✝⁷ : AddCommGroup M₁", "inst✝⁶ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝⁵ : Field K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M₂' : Type u_7", "inst✝² : AddCommMonoid M₂'", "inst✝¹ : Module R M₂'", "n : Type w", "inst✝ : Nontrivial R", "B : BilinForm R M", "v : Basis n R M", "h : B.iIsOrtho ⇑v", "hB : B.Nondegenerate", "i : n", "ho : B.IsOrtho (v i) (v i)"], "goal": "False"}}
+{"state": {"context": ["F : Type u_1", "Field F", "Fintype F", "DecidableEq F"], "goal": "quadraticCharFun F 1 = 1"}}
+{"state": {"context": ["M : Type u_6", "AddMonoid M", "a : AddUnits M"], "goal": "Function.Bijective fun x => x + ↑a"}}
+{"state": {"context": ["R : Type uR", "A₁ : Type uA₁", "A₂ : Type uA₂", "CommSemiring R", "Semiring A₁", "Semiring A₂", "Algebra R A₁", "Algebra R A₂", "e : A₁ ≃ₐ[R] A₂", "x : A₁", "n : ℕ"], "goal": "e (x ^ n) = e x ^ n"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "J : GrothendieckTopology C", "F F' F'' : Cᵒᵖ ⥤ Type w", "G G' : Subpresheaf F", "h : Presieve.IsSheaf J F", "hG : Presieve.IsSheaf J G.toPresheaf", "U : Cᵒᵖ", "s : F.obj U", "hs : s ∈ (sheafify J G).obj U"], "goal": "∀ ⦃Y : C⦄ ⦃f : Y ⟶ unop U⦄, (G.sieveOfSection s).arrows f → F.map f.op ↑(⋯.amalgamate (G.familyOfElementsOfSection s) ⋯) = F.map f.op s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "CompactSpace α", "MetricSpace β", "f g : C(α, β)", "C : ℝ", "Nonempty α"], "goal": "dist f g < C ↔ ∀ (x : α), dist (f x) (g x) < C"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝¹ : DecidableEq α", "inst✝ : CommSemiring R", "f✝ : α → R", "s✝ s : Finset α", "f : α → R", "n : ℕ"], "goal": "(∑ i ∈ s, f i) ^ n = ∑ x ∈ s.piAntidiag n, ↑(multinomial s x) * s.noncommProd (fun i => f i ^ x i) ⋯"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "CommRing R", "LieRing L", "LieAlgebra R L", "LieRing L'", "LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L"], "goal": "LieIdeal.map f (I ⊔ f.ker) = LieIdeal.map f I"}}
+{"state": {"context": ["n : ℕ", "a b : Fin n"], "goal": "Fintype.card ↑(Set.Icc a b) = ↑b + 1 - ↑a"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y f' : ℝ", "hfc : StrictConvexOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hf' : HasDerivAt f f' x"], "goal": "f' < slope f x y"}}
+{"state": {"context": ["E : Type u_1", "V : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℂ E", "f : V → E", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : MeasurableSpace V", "inst✝³ : BorelSpace V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : FiniteDimensional ℝ V", "inst✝ : CompleteSpace E", "hfi : Integrable f volume", "ε : ℝ", "hε : ε > 0", "g : V → E", "hg_supp : HasCompactSupport g", "hfg : ∫ (x : V), ‖f x - g x‖ ≤ ε / 2", "hg_cont : Continuous g", "w : V", "hI : ‖(∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • g v) - 0‖ < ε / 2"], "goal": "‖(∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v) - 0‖ < ε"}}
+{"state": {"context": ["R : Type uR", "A : Type uA", "M : Type uM", "ι : Type uι", "CommSemiring R", "Semiring A", "Algebra R A", "AddCommMonoid M", "Module R M", "b : Basis ι R M", "a : A", "m : M"], "goal": "(Algebra.TensorProduct.basisAux A b) (a ⊗ₜ[R] m) = a • Finsupp.mapRange ⇑(algebraMap R A) ⋯ (b.repr m)"}}
+{"state": {"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝⁴ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝³ : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝² : Ring R", "inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀", "inst✝ : LinearOrderedCommMonoidWithZero Γ'₀", "v : Valuation R Γ₀", "x y z : R", "ι : Type u_7", "s✝ : Finset ι", "f : ι → R", "g : Γ₀", "hf✝ : ∀ i ∈ s✝, v (f i) ≤ g", "a : ι", "s : Finset ι", "has : a ∉ s", "ih : (∀ i ∈ s, v (f i) ≤ g) → v (∑ i ∈ s, f i) ≤ g", "hf : ∀ i ∈ insert a s, v (f i) ≤ g"], "goal": "v (∑ i ∈ insert a s, f i) ≤ g"}}
+{"state": {"context": ["R : Type u", "Rack R", "a b : Rack.PreEnvelGroup R"], "goal": "Rack.PreEnvelGroupRel R a b → Rack.PreEnvelGroupRel R b a"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type u_1", "L.Structure M", "L.Structure N", "f : L.Hom M N"], "goal": "Monotone (FirstOrder.Language.Substructure.map f)"}}
+{"state": {"context": ["α : Type u", "inst✝ : CanonicallyLinearOrderedCommMonoid α", "a b c : α", "hb : b ≤ a"], "goal": "min a (b * c) = min a (min a b * min a c)"}}
+{"state": {"context": ["α : Type u_3", "LinearOrderedSemiring α", "a b : ℕ"], "goal": "↑(max a b) = max ↑a ↑b"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "M : Type u_1", "CategoryTheory.Category.{u_2, u_1} M", "CategoryTheory.MonoidalCategory M", "F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)", "m n : M", "h : CategoryTheory.MonoidalCategory.tensorObj m n ≅ 𝟙_ M", "X : C"], "goal": "(CategoryTheory.unitOfTensorIsoUnit F m n h).inv.app X =\n  CategoryTheory.CategoryStruct.comp (F.ε.app X)\n    (CategoryTheory.CategoryStruct.comp ((F.map h.inv).app X) ((F.μIso m n).inv.app X))"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Limits.MonoCoprod C", "I : Type u_2", "J : Type u_3", "X : I → C", "ι : J → I", "hι : Function.Injective ι", "c : CategoryTheory.Limits.Cofan X", "c₁ : CategoryTheory.Limits.Cofan (X ∘ ι)", "hc : CategoryTheory.Limits.IsColimit c", "hc₁ : CategoryTheory.Limits.IsColimit c₁", "CategoryTheory.Limits.HasCoproduct fun k => X ↑k"], "goal": "CategoryTheory.Mono (CategoryTheory.Limits.Cofan.IsColimit.desc hc₁ fun i => c.inj (ι i))"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddZeroClass M", "AddZeroClass N", "f : M →+ N", "g : N → M", "h : Function.LeftInverse g ⇑f"], "goal": "∀ (a : ↥(AddMonoidHom.mrange f)), (AddEquiv.ofLeftInverse' f h).symm a = g ↑a"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedAddCommGroup F", "inst✝² : CompleteSpace E", "f : ℝ → E", "g : ℝ → F", "k : Set ℝ", "l : Filter ℝ", "inst✝¹ : l.NeBot", "inst✝ : TendstoIxxClass Icc l l", "hl : k ∈ l", "hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x", "hf : Tendsto (fun x => ‖f x‖) l atTop", "hfg : deriv f =O[l] g", "C : ℝ", "hC₀ : 0 ≤ C", "s : Set ℝ", "hsl : s ∈ l", "hgi : IntegrableOn (fun x => C * ‖g x‖) k volume", "c : ℝ", "hc : c ∈ s", "d : ℝ", "hd : d ∈ s", "hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖", "hsub : [[c, d]] ⊆ k", "hfd : ∀ z ∈ [[c, d]], DifferentiableAt ℝ f z", "hg : ∀ x ∈ Ι c d, ‖deriv f x‖ ≤ C * ‖g x‖", "hg_ae : ∀ᵐ (x : ℝ) ∂volume.restrict (Ι c d), ‖deriv f x‖ ≤ C * ‖g x‖", "hsub' : Ι c d ⊆ k", "hfi : IntervalIntegrable (deriv f) volume c d"], "goal": "‖f d‖ - ‖f c‖ ≤ ∫ (y : ℝ) in k, C * ‖g y‖"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "S : Type u_2", "Ring S", "f : R ≃+* S", "IsNoetherianRing R"], "goal": "IsNoetherianRing S"}}
+{"state": {"context": ["A : Type u_2", "CommRing A", "IsDedekindDomain A", "x : A", "n : ℕ", "I : Ideal A"], "goal": "Associates.mk I ^ n ≤ Associates.mk (Ideal.span {x}) ↔ x ∈ I ^ n"}}
+{"state": {"context": ["α : Type u_2", "PartialOrder α", "LocallyFiniteOrderTop α", "DecidableEq α", "a : α"], "goal": "(Finset.Ici a).erase a = Finset.Ioi a"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "e : M₁ ≃SL[σ₁₂] M₂", "s : Set M₂"], "goal": "⇑e.symm '' s = ⇑e ⁻¹' s"}}
+{"state": {"context": ["α : Type u_1", "s : Multiset α", "a : α", "m : a ∉ s", "n : s.Nodup"], "goal": "(a ::ₘ s).Nodup"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : CommRing R", "M : Submonoid R", "S : Type u_2", "inst✝⁷ : CommRing S", "inst✝⁶ : Algebra R S", "T : Type u_3", "inst✝⁵ : CommRing T", "inst✝⁴ : Algebra R T", "inst✝³ : Algebra S T", "inst✝² : IsScalarTower R S T", "g : S →+* T", "f : R →+* S", "inst✝¹ : LocalRing S", "inst✝ : IsLocalRingHom f", "H : f.SurjectiveOnStalks", "x : S"], "goal": "∃ a, f a = x"}}
+{"state": {"context": ["α : Type u_1", "m : Set (MeasureTheory.OuterMeasure α)", "t : Set α"], "goal": "MeasureTheory.OuterMeasure.sInfGen m t = ⨅ μ ∈ m, μ t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : MeasureTheory.Measure α", "NormedAddCommGroup β", "f : α → β", "h : MeasureTheory.HasFiniteIntegral f (μ + ν)"], "goal": "MeasureTheory.HasFiniteIntegral f ν"}}
+{"state": {"context": ["α : Type u_1", "x : α", "z : Sym2 α"], "goal": "Sym2.Mem x z ↔ x ∈ z"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.Scheme", "AlgebraicGeometry.IsAffine X", "AlgebraicGeometry.IsAffine Y", "f : X ⟶ Y", "h : Function.Surjective ⇑(AlgebraicGeometry.Scheme.Γ.map f.op)"], "goal": "AlgebraicGeometry.IsClosedImmersion f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝¹ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "inst✝ : IsFiniteKernel ν", "a : α", "s : Set β", "hs : MeasurableSet s"], "goal": "Submartingale (fun n x => κ.densityProcess ν n a x s) (countableFiltration γ) (ν a)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "f : R", "s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))", "ι : Type u := ↥(PrimeSpectrum.basicOpen f)", "t : Finset ι", "a h : ι → R", "iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f", "ah_ha : ∀ i ∈ t, ∀ j ∈ t, a i * h j = h i * a j", "s_eq : ∀ i ∈ t, ((structureSheaf R).val.map (iDh i).op) s = const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯", "ht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ i ∈ t, ↑(PrimeSpectrum.basicOpen (h i))", "n : ℕ", "b : ι →₀ R", "b_supp : b ∈ Finsupp.supported R R ↑t", "hb : ∑ i ∈ t, b i • h i = f ^ (n + 1)"], "goal": "∃ a, (toBasicOpen R f) a = s"}}
+{"state": {"context": ["m n : Nat"], "goal": "Int.negOfNat m + Int.negOfNat n = Int.negOfNat (m + n)"}}
+{"state": {"context": ["b x✝ y✝ : ℝ", "b_pos : 0 < b", "b_lt_one : b < 1", "x : ℝ", "hx : x < 0", "y : ℝ", "hy : y < 0", "hxy : x < y"], "goal": "logb b x < logb b y"}}
+{"state": {"context": ["n✝ m n : ℕ", "H : ∀ (k : ℕ), Prime k → k ∣ m → ¬k ∣ n", "g2 : ¬m.gcd n = 1"], "goal": "False"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "inst✝⁴ : Fintype ι", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : CompleteSpace E", "μ : Measure (ι → ℝ)", "inst✝ : IsLocallyFiniteMeasure μ", "I : Box ι", "l : IntegrationParams", "hl : l.bRiemann = false", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "g : (ι → ℝ) → E", "hg : StronglyMeasurable g", "this : SeparableSpace ↑(Set.range g ∪ {0})", "hgi : IntegrableOn g (↑I) μ", "f : ℕ → SimpleFunc (ι → ℝ) E := SimpleFunc.approxOn g ⋯ (Set.range g ∪ {0}) 0 ⋯", "hfi : ∀ (n : ℕ), IntegrableOn (↑(f n)) (↑I) μ", "hfi' : ∀ (n : ℕ), BoxIntegral.Integrable I l (↑(f n)) μ.toBoxAdditive.toSMul", "hfg_mono : ∀ (x : ι → ℝ) {m n : ℕ}, m ≤ n → ‖↑(f n) x - g x‖ ≤ ‖↑(f m) x - g x‖", "ε : ℝ≥0", "ε0 : 0 < ε", "ε0' : 0 < ↑ε", "N₀ : ℕ", "hN₀ : ∫ (x : ι → ℝ) in ↑I, ‖↑(f N₀) x - g x‖ ∂μ ≤ ↑ε", "Nx : (ι → ℝ) → ℕ", "hNx : ∀ (x : ι → ℝ), N₀ ≤ Nx x", "hNxε : ∀ (x : ι → ℝ), dist (↑(f (Nx x)) x) (g x) ≤ ↑ε", "δ : ℕ → ℝ≥0", "δ0 : ∀ (i : ℕ), 0 < δ i", "c✝ : ℝ≥0", "hδc : HasSum δ c✝", "hcε : c✝ < ε", "r : ℝ≥0 → (ι → ℝ) → ↑(Set.Ioi 0) := fun c x => ⋯.convergenceR (↑(δ (Nx x))) c x", "c : ℝ≥0", "π : TaggedPrepartition I", "hπ : l.MemBaseSet I c (r c) π", "hπp : π.IsPartition"], "goal": "dist (integralSum g μ.toBoxAdditive.toSMul π) (∑ J ∈ π.boxes, (μ ↑J).toReal • ↑(f (Nx (π.tag J))) (π.tag J)) + dist (∑ J ∈ π.boxes, (μ ↑J).toReal • ↑(f (Nx (π.tag J))) (π.tag J)) (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, ↑(f (Nx (π.tag J))) x ∂μ) + dist (∑ J ∈ π.boxes, ∫ (x : ι → ℝ) in ↑J, ↑(f (Nx (π.tag J))) x ∂μ) (∫ (a : ι → ℝ) in ↑I, g a ∂μ) ≤ ((μ ↑I).toReal + 1 + 1) * ↑ε"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝ : Semiring R", "n : ℕ", "f : R[X]"], "goal": "⊥ < ↑n"}}
+{"state": {"context": ["α : Type u_1", "OrderedCommGroup α", "s t : UpperSet α"], "goal": "↑(s * t) = ↑s * ↑t"}}
+{"state": {"context": ["R : Type u", "S : Type u'", "M : Type v", "N✝ : Type v'", "inst✝¹⁵ : CommRing R", "inst✝¹⁴ : CommRing S", "inst✝¹³ : AddCommGroup M", "inst✝¹² : AddCommGroup N✝", "inst✝¹¹ : Module R M", "inst✝¹⁰ : Module R N✝", "inst✝⁹ : Algebra R S", "inst✝⁸ : Module S N✝", "inst✝⁷ : IsScalarTower R S N✝", "p : Submonoid R", "inst✝⁶ : IsLocalization p S", "f✝ : M →ₗ[R] N✝", "inst✝⁵ : IsLocalizedModule p f✝", "hp : p ≤ R⁰", "N : Type v", "inst✝⁴ : AddCommGroup N", "inst✝³ : Module R N", "inst✝² : Module S N", "inst✝¹ : IsScalarTower R S N", "f : M →ₗ[R] N", "inst✝ : IsLocalizedModule p f"], "goal": "Module.rank S N = Module.rank R M"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.Limits.HasCoequalizers C", "(X : C) →\n    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)", "(X : C) →\n    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}\n      (CategoryTheory.MonoidalCategory.tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N : Bimod Y Z"], "goal": "(M.associatorBimod (Bimod.regular Y) N).hom ≫ M.whiskerLeft N.leftUnitorBimod.hom =\n  Bimod.whiskerRight M.rightUnitorBimod.hom N"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "e : α ≃o β", "s : Set β"], "goal": "BddBelow (⇑e ⁻¹' s) ↔ BddBelow s"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : InnerProductSpace ℝ V", "inst✝³ : MetricSpace P", "inst✝² : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "inst✝¹ : Nonempty ↥s", "inst✝ : HasOrthogonalProjection s.direction", "p : P", "this : ∀ (a : ↥s) (b : V), (_root_.orthogonalProjection s.direction) b = 0 → (reflection s) ((reflection s) (b +ᵥ ↑a)) = b +ᵥ ↑a"], "goal": "(reflection s) ((reflection s) (p -ᵥ ↑((orthogonalProjection s) p) +ᵥ ↑((orthogonalProjection s) p))) = p -ᵥ ↑((orthogonalProjection s) p) +ᵥ ↑((orthogonalProjection s) p)"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M : Type u_5", "M₁ : Type u_6", "M₂ : Type u_7", "M₃ : Type u_8", "Mₗ₁ : Type u_9", "Mₗ₁' : Type u_10", "Mₗ₂ : Type u_11", "Mₗ₂' : Type u_12", "K : Type u_13", "K₁ : Type u_14", "K₂ : Type u_15", "V : Type u_16", "V₁ : Type u_17", "V₂ : Type u_18", "n : Type u_19", "inst✝⁶ : CommRing R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : AddCommGroup M₁", "inst✝² : Module R M₁", "inst✝¹ : AddCommGroup M₂", "inst✝ : Module R M₂", "B F : M →ₗ[R] M →ₗ[R] M₂", "e : M₁ ≃ₗ[R] M", "f : Module.End R M", "hₗ : F.compl₁₂ ↑e ↑e ∘ₗ e.symm.conj f = (F ∘ₗ f).compl₁₂ ↑e ↑e", "hᵣ : (B.compl₁₂ ↑e ↑e).compl₂ (e.symm.conj f) = (B.compl₂ f).compl₁₂ ↑e ↑e"], "goal": "B.IsPairSelfAdjoint F f ↔ (B.compl₁₂ ↑e ↑e).IsPairSelfAdjoint (F.compl₁₂ ↑e ↑e) (e.symm.conj f)"}}
+{"state": {"context": ["n : ℕ"], "goal": "0 ≤ ↑n"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "Field K", "Field L", "Field L'", "Algebra K L", "Algebra K L'", "f : L →ₐ[K] L'"], "goal": "f.fieldRange.toSubfield = (↑f).fieldRange"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "NormedAddCommGroup E", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedSpace ℝ E", "f : ↥(MeasureTheory.Lp.simpleFunc E 1 μ)"], "goal": "‖MeasureTheory.L1.SimpleFunc.integral f‖ ≤ ‖f‖"}}
+{"state": {"context": ["G : Type uG", "E' : Type uE'", "inst✝¹¹ : NormedAddCommGroup E'", "g : G → E'", "inst✝¹⁰ : MeasurableSpace G", "μ : Measure G", "inst✝⁹ : NormedSpace ℝ E'", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedSpace ℝ G", "inst✝⁶ : HasContDiffBump G", "inst✝⁵ : CompleteSpace E'", "φ✝ : ContDiffBump 0", "x₀✝ : G", "inst✝⁴ : BorelSpace G", "inst✝³ : IsLocallyFiniteMeasure μ", "inst✝² : μ.IsOpenPosMeasure", "inst✝¹ : FiniteDimensional ℝ G", "inst✝ : μ.IsAddLeftInvariant", "ι : Type u_1", "φ : ι → ContDiffBump 0", "l : Filter ι", "K : ℝ", "hφ : Tendsto (fun i => (φ i).rOut) l (𝓝 0)", "h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn", "hg : LocallyIntegrable g μ", "this✝ : μ.IsAddHaarMeasure", "x₀ : G", "h₀ : Tendsto (fun a => ⨍ (y : G) in a, ‖g y - g x₀‖ ∂μ) ((Besicovitch.vitaliFamily μ).filterAt x₀) (𝓝 0)", "hφ' : Tendsto (fun i => (φ i).rOut) l (𝓝[>] 0)", "this : Tendsto (fun x => ⨍ (y : G) in closedBall x₀ (φ x).rOut, ‖g y - g x₀‖ ∂μ) l (𝓝 0)"], "goal": "∀ᶠ (i : ι) in l, (support fun y => (φ i).normed μ (x₀ - y)) ⊆ closedBall x₀ (φ i).rOut"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "𝕜 : Type u_4", "E : Type u_5", "inst✝¹ : DecidableEq α", "inst✝ : CancelCommMonoid α", "s : Finset α", "a : α"], "goal": "mulRothNumber s ≤ mulRothNumber (map (mulLeftEmbedding a) s)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "S : Type v", "CommRing S", "J : Ideal R", "I : Ideal S", "f : R →+* S", "H : J ≤ Ideal.comap f I", "h : Ideal.comap f I ≤ J"], "goal": "Function.Injective ⇑(Ideal.quotientMap I f H)"}}
+{"state": {"context": ["α : Type u_1", "Semiring α", "Invertible 2", "a : α"], "goal": "SymAlg.sym (a * a) = SymAlg.sym a * SymAlg.sym a"}}
+{"state": {"context": ["G : Type u_1", "M : Type u_2", "N : Type u_3", "α : Sort u_4", "β : Sort u_5", "ι : Sort u_6", "inst✝¹ : CommMonoid M", "inst✝ : CommMonoid N", "f : α → M", "s : Finset (PLift α)", "hs : mulSupport (f ∘ PLift.down) ⊆ ↑s", "x : PLift α", "hx : x ∈ mulSupport (f ∘ PLift.down)"], "goal": "x ∈ s"}}
+{"state": {"context": ["n : ℕ"], "goal": "(∑ i ∈ Finset.range n, i) * 2 = n * (n - 1)"}}
+{"state": {"context": ["a✝ b n✝ n a : ℕ"], "goal": "(if n = 0 ∨ a = 0 then 0 else (a.factorization ⌈/⌉ n).prod fun x x_1 => x ^ x_1).factorization = a.factorization ⌈/⌉ n"}}
+{"state": {"context": ["α : Type u_1", "N : α → Type u_2", "inst✝² : DecidableEq α", "inst✝¹ : (a : α) → DecidableEq (N a)", "inst✝ : (a : α) → Zero (N a)", "f g : Π₀ (a : α), N a"], "goal": "f.neLocus g = g.neLocus f"}}
+{"state": {"context": ["α : Type u_1", "as : Array α", "start stop : Nat", "h : stop ≤ start"], "goal": "as.extract start stop = #[]"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁴ : Group G", "inst✝¹³ : MeasurableSpace G", "inst✝¹² : TopologicalSpace G", "inst✝¹¹ : TopologicalGroup G", "inst✝¹⁰ : BorelSpace G", "inst✝⁹ : PolishSpace G", "Γ : Subgroup G", "inst✝⁸ : Countable ↥Γ", "inst✝⁷ : Γ.Normal", "inst✝⁶ : T2Space (G ⧸ Γ)", "inst✝⁵ : SecondCountableTopology (G ⧸ Γ)", "μ : Measure (G ⧸ Γ)", "ν : Measure G", "inst✝⁴ : ν.IsMulLeftInvariant", "inst✝³ : ν.IsMulRightInvariant", "inst✝² : SigmaFinite ν", "inst✝¹ : μ.IsMulLeftInvariant", "inst✝ : SigmaFinite μ", "s : Set G", "fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν", "V : Set (G ⧸ Γ)", "meas_V : MeasurableSet V", "neZeroV : μ V ≠ 0", "hV : μ V = ν (QuotientGroup.mk ⁻¹' V ∩ s)", "neTopV : μ V ≠ ⊤", "U : Set (G ⧸ Γ)", "a✝ : MeasurableSet U", "meas_π : Measurable QuotientGroup.mk", "μ' : Measure (G ⧸ Γ) := map QuotientGroup.mk (ν.restrict s)", "has_fund : HasFundamentalDomain (↥Γ.op) G ν", "i : QuotientMeasureEqMeasurePreimage ν μ'", "this✝ : μ'.IsMulLeftInvariant", "this : SigmaFinite μ'"], "goal": "(μ' V / ν (QuotientGroup.mk ⁻¹' V ∩ s)) • μ = μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Semiring α", "inst✝ : NoZeroDivisors α", "m n✝ n : ℕ", "p : ℤ", "hp : Prime p", "a b c : ℤ", "hpa : p ∣ a", "hpb : p ∣ b", "HF : a ^ n + b ^ n + c ^ n = 0"], "goal": "p ∣ c"}}
+{"state": {"context": ["a b p q : ℝ≥0", "h : p.IsConjExponent q"], "goal": "p / q = p - 1"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Preorder α", "inst✝¹ : LocallyFiniteOrder α", "inst✝ : Preorder β", "f : α → β", "h : ∀ (a b : α), a ⋖ b → f a < f b", "a b : α", "hab : a < b", "this : a < b → f a < f b"], "goal": "f a < f b"}}
+{"state": {"context": ["B : Type u", "inst✝ : Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "s t : LeftExtension f g", "i : s ⟶ t", "x : B", "h : c ⟶ x"], "goal": "(s.whisker h).hom ≫ (precomp x f).map (i.right ▷ h) = (t.whisker h).hom"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NonUnitalNormedRing E", "inst✝⁵ : StarRing E", "inst✝⁴ : NormedStarGroup E", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : IsScalarTower 𝕜 E E", "inst✝¹ : SMulCommClass 𝕜 E E", "inst✝ : RegularNormedAlgebra 𝕜 E", "a : E"], "goal": "‖(mul 𝕜 E) a⋆‖ ≤ ‖(mul 𝕜 E).flip a‖"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "σ : Type v", "M : Type w", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I : Ideal (MvPolynomial σ R)", "p : MvPolynomial σ R", "hcoe : ∀ (m : σ →₀ ℕ), coeff m p ∈ Ideal.comap C I"], "goal": "∀ m ∈ p.support, (monomial m) (coeff m p) ∈ I"}}
+{"state": {"context": ["α : Type u", "n : ℕ"], "goal": "Function.Injective List.ofFn"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "β : Type u_7", "inst✝ : SeminormedAddCommGroup β", "T T' : Set α → β", "C C' : ℝ", "c : ℝ≥0∞", "hc_ne_top : c ≠ ⊤", "hT : DominatedFinMeasAdditive (c • μ) T C"], "goal": "DominatedFinMeasAdditive μ T (c.toReal * C)"}}
+{"state": {"context": ["b : ℝ"], "goal": "∫ (x : ℝ) in Set.Ioi 0, rexp (-b * x ^ 2) = √(π / b) / 2"}}
+{"state": {"context": ["α : Type u_4", "LinearOrderedSemiring α", "FloorSemiring α"], "goal": "Filter.Tendsto (fun x => ⌈x⌉₊) Filter.atTop Filter.atTop"}}
+{"state": {"context": ["R : Type u", "inst✝¹⁰ : CommSemiring R", "A : Type u", "inst✝⁹ : CommSemiring A", "inst✝⁸ : Algebra R A", "B : Type u", "inst✝⁷ : Semiring B", "inst✝⁶ : Algebra R B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsScalarTower R A B", "inst✝³ : FormallyUnramified R A", "inst✝² : FormallyUnramified A B", "C : Type u", "inst✝¹ : CommRing C", "inst✝ : Algebra R C", "I : Ideal C", "hI : I ^ 2 = ⊥", "f₁ f₂ : B →ₐ[R] C", "e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂"], "goal": "f₁ = f₂"}}
+{"state": {"context": ["L : Language", "M : Type u_1", "inst✝¹ : L.Structure M", "N : Type u_2", "inst✝ : L.Structure N", "f : M ↪[L] N", "s : L.Substructure M", "t : Set N", "h1 : t.Countable", "h2 : (closure L).toFun t = Substructure.map f.toHom s", "hf : Function.Injective ⇑f.toHom", "x : N", "hx : x ∈ t", "h' : x ∈ ↑((closure L).toFun t)"], "goal": "x ∈ range ⇑f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "u : ι → UniformSpace γ"], "goal": "UniformFun.filter α γ (𝓤 γ) = 𝓤 (α →ᵤ γ)"}}
+{"state": {"context": ["ι : Sort u_1", "f : ι → ℕ"], "goal": "⨆ i, ↑(f i) ≠ ⊤ ↔ BddAbove (Set.range f)"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "inst✝⁴ : Semiring R", "φ : ι → Type u_3", "inst✝³ : (i : ι) → AddCommMonoid (φ i)", "inst✝² : (i : ι) → Module R (φ i)", "inst✝¹ : DecidableEq ι", "inst✝ : Finite ι", "val✝ : Fintype ι"], "goal": "univ ⊆ ↑?intro.convert_1 ∪ ∅"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "β : Type u_7", "inst✝ : AddCommMonoid β", "T T' : Set α → β", "c : ℝ≥0∞", "hc_ne_zero : c ≠ 0", "hT : FinMeasAdditive μ T", "s : Set α", "x✝ : MeasurableSet s", "hμs : μ s = ⊤"], "goal": "μ s = ⊤ ∨ c = ⊤ ∧ ¬μ s = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "r : α → α → Prop", "r' : β → β → Prop", "f : α → β", "s t : Set α", "a : α", "inst✝¹ : IsRefl α r", "inst✝ : IsSymm α r"], "goal": "s.PartiallyWellOrderedOn r ↔ ∀ t ⊆ s, IsAntichain r t → t.Finite"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n : ℕ"], "goal": "((↑k + 1) ^ 2)⁻¹ + (↑k + 1)⁻¹ ≤ 1 / (↑k + 1) + 1 / (↑k + 1)"}}
+{"state": {"context": ["G : Type u_1", "inst✝ : Group G", "H K : Subgroup G", "S T : Set G", "h✝ : H.IsComplement' K", "h : ↑↑H", "k : ↑↑K", "x✝ : (fun x => ↑x.1 * ↑x.2) (h, k) ∈ ⊤"], "goal": "(fun x => ↑x.1 * ↑x.2) (h, k) ∈ H ⊔ K"}}
+{"state": {"context": ["ι : Type u_2", "f : ι → Cardinal.{u_2}", "c : Cardinal.{u_2}", "hc : c.IsRegular", "hι : #ι < c"], "goal": "(∀ (i : ι), f i < c) → iSup f < c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LT α", "LT β", "a b : α"], "goal": "toLex (Sum.inl a) < toLex (Sum.inl b) ↔ a < b"}}
+{"state": {"context": ["α : Type u_2", "AddSemigroup α", "DecidableEq α", "a : α", "s t : Finset α"], "goal": "AddOpposite.op a +ᵥ s + t = s + (a +ᵥ t)"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "n : ℕ"], "goal": "∑ i ∈ antidiagonal n, bernoulli' i.1 / ↑i.1! * ((↑i.2 + 1) * ↑i.2!)⁻¹ * ↑n ! = ∑ k ∈ antidiagonal n, ↑((k.1 + k.2).choose k.2) / (↑k.2 + 1) * bernoulli' k.1"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V"], "goal": "↑⊥ = ⊥"}}
+{"state": {"context": ["m : ℤ", "n : ℕ", "hn : Even n", "hm : m ∉ Ico 0 ↑n"], "goal": "0 < ∏ k ∈ Finset.range n, (m - ↑k)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "Preorder α", "Preorder β", "Preorder γ", "f : α → β → γ", "s : Set α", "t : Set β", "h₀ : ∀ (b : β), Monotone (Function.swap f b)", "h₁ : ∀ (a : α), Monotone (f a)"], "goal": "BddAbove s → BddAbove t → BddAbove (Set.image2 f s t)"}}
+{"state": {"context": ["α✝ : Type u_1", "r : α✝ → α✝ → Prop", "α : Type u_2", "β : Type u_3", "inst✝ : Infinite β", "f : α → Finset β", "w : ⋃ a, ↑(f a) = ⊤", "k : Infinite ↑(range f)"], "goal": "#β ≤ #↑(range f)"}}
+{"state": {"context": ["Ω✝ : Type u_1", "inst✝⁷ : PseudoEMetricSpace Ω✝", "inst✝⁶ : MeasurableSpace Ω✝", "inst✝⁵ : OpensMeasurableSpace Ω✝", "Ω : Type u_2", "ι : Type u_3", "L : Filter ι", "inst✝⁴ : L.NeBot", "inst✝³ : MeasurableSpace Ω", "inst✝² : PseudoEMetricSpace Ω", "inst✝¹ : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "μs : ι → Measure Ω", "h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i => (μs i) E) L (𝓝 (μ E))", "F : Set Ω", "F_closed : IsClosed F"], "goal": "limsup (fun i => (μs i) F) L ≤ μ F"}}
+{"state": {"context": ["R : Type u_1", "Rₛ : Type u_2", "inst✝¹¹ : CommSemiring R", "S : Submonoid R", "inst✝¹⁰ : CommSemiring Rₛ", "inst✝⁹ : Algebra R Rₛ", "hT : IsLocalization S Rₛ", "M : Type u_3", "M' : Type u_4", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : Module R M", "inst✝⁶ : Module Rₛ M", "inst✝⁵ : IsScalarTower R Rₛ M", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : Module R M'", "inst✝² : Module Rₛ M'", "inst✝¹ : IsScalarTower R Rₛ M'", "f : M →ₗ[R] M'", "inst✝ : IsLocalizedModule S f", "v : Set M", "hv : span R v = ⊤", "x : M'", "x✝ : x ∈ ⊤"], "goal": "x ∈ span Rₛ (⇑f '' v)"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M", "s : Set M", "S : Submonoid M"], "goal": "Submonoid.closure s ≤ S ↔ s ⊆ ↑S"}}
+{"state": {"context": ["α : Type u_2", "Monoid α", "a : α", "n : ℕ"], "goal": "IsSquare a → IsSquare (a ^ n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "F✝ : ℕ → α → β", "f✝ : α → β", "bound✝ : α → ℝ", "F : ℕ → α → β", "f : α → β", "bound : α → ℝ", "F_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ", "bound_hasFiniteIntegral : HasFiniteIntegral bound μ", "h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a", "h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))", "f_measurable : AEStronglyMeasurable f μ"], "goal": "Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) atTop (𝓝 0)"}}
+{"state": {"context": ["K : Type u", "V : Type v", "DivisionRing K", "AddCommGroup V", "Module K V", "n : ℕ", "hn : Fact (FiniteDimensional.finrank K V = n + 1)"], "goal": "FiniteDimensional K V"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "S : Set α", "hS : BooleanGenerators S", "h : sSup S = ⊤", "_i : DistribLattice α := hS.distribLattice_of_sSup_eq_top h", "_i₁ : IsAtomistic α"], "goal": "ComplementedLattice α"}}
+{"state": {"context": ["ι : Type u_4", "R : Type u_1", "M : Type u_2", "N : Type u_3", "LinearOrder ι", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "Q : QuadraticMap R M N", "bm : Basis ι R M", "i j : ι"], "goal": "((Q.toBilin bm) (bm i)) (bm j) = if i = j then Q (bm i) else if i < j then QuadraticMap.polar (⇑Q) (bm i) (bm j) else 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : IsLocallyFiniteMeasure μ", "inst✝ : IsUnifLocDoublingMeasure μ", "p : ℕ → Prop", "s : ℕ → Set α", "hs : ∀ (i : ℕ), IsClosed (s i)", "r₁ r₂ : ℕ → ℝ", "hrp : 0 ≤ r₁", "M : ℝ", "hM : 0 < M", "hM' : M < 1", "Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)", "Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)", "Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j", "i : ℕ", "W : Set α := blimsup Y₁ atTop p \\ Z i", "contra : ¬μ W = 0", "d : α", "hd' : ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ), Tendsto δ l (𝓝[>] 0) → (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) → Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)", "hd : d ∈ blimsup Y₁ atTop p", "f : ℕ → ℕ", "hf₁ : ∀ (x : ℕ), p (f x)", "hf₂ : ∀ (j : ℕ), j ≤ f j", "hf₃ : Tendsto f atTop atTop", "hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)", "hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)", "w : ℕ → α", "hw : ∀ (j : ℕ), w j ∈ s (f j)", "hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))", "C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹", "hC : C ≠ 0", "b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))", "B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))", "h₁ : ∀ (j : ℕ), b j ⊆ B j", "h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j", "h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)", "h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)", "j : ℕ", "hj₀ : Disjoint (b j) (W ∩ B j) ∧ μ (B j) ≤ ↑C * μ (b j)", "hB : μ (B j) ≠ ⊤"], "goal": "μ (W ∩ B j) ≤ μ (B j) - ↑C⁻¹ * μ (B j)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "p : ℝ≥0∞", "f : MeasureTheory.SimpleFunc α E", "hp_pos : p ≠ 0", "hp_ne_top : p ≠ ⊤"], "goal": "MeasureTheory.Memℒp (↑f) p μ ↔ f.FinMeasSupp μ"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "α₁ : Type u_6", "α₂ : Type u_7", "β₁ : Type u_8", "β₂ : Type u_9", "inst✝¹ : Preorder β₁", "inst✝ : Preorder β₂", "u₁ : β₁ → α₁", "u₂ : β₂ → α₂"], "goal": "map u₁ atTop ×ˢ map u₂ atTop = map (Prod.map u₁ u₂) atTop"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s : Set α", "a b : α", "hs : s.Pairwise r", "ha : a ∈ s", "hb : b ∈ s", "h : ¬r a b"], "goal": "a = b"}}
+{"state": {"context": ["a b : Nat"], "goal": "compare a b = Ordering.gt ↔ b < a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "Module 𝕜 E", "s t : Set E", "ht : Convex 𝕜 t"], "goal": "(convexHull 𝕜) s ⊆ t ↔ s ⊆ t"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "δ : Type u_2", "π : ι → Type u_3", "r : α → α → Prop", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "s : Set α", "x y : α"], "goal": "(∃ a b c, a ≤ b ∧ b ≤ c ∧ (f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c)) → (∃ x x_1 h, f x_1 < f x) ∧ ∃ x x_1 h, f x < f x_1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c'✝ c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "hfg : IsBigOWith c l f g", "hc : 0 < c", "hgk : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l g k", "c' : ℝ", "c'pos : 0 < c'"], "goal": "IsBigOWith c' l f k"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.Limits.HasPullbacks C", "X : CategoryTheory.Dial C"], "goal": "(CategoryTheory.MonoidalCategory.rightUnitor X).inv.F =\n  CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : DiscreteValuationRing R", "a b : R", "hp : Prime (Classical.choose ⋯)"], "goal": "(addVal R) a ≤ (addVal R) b ↔ a ∣ b"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "n : ℕ", "hn : ¬n = 0", "hn' : 0 < n", "hn'' : X ^ n - 1 ≠ 0", "σ τ : (X ^ n - 1).Gal", "a : (X ^ n - 1).SplittingField", "ha : a ^ n = 1"], "goal": "(σ * τ) a = (τ * σ) a"}}
+{"state": {"context": ["G : Type u_2", "AddGroup G", "H : AddSubgroup G", "x g : G", "h : ↥H.op"], "goal": "h +ᵥ (g + x) = g + (h +ᵥ x)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "self : CategoryTheory.Subgroupoid C", "c d e : C", "p : c ⟶ d"], "goal": "p ∈ self.arrows c d → ∀ {q : d ⟶ e}, q ∈ self.arrows d e → CategoryTheory.CategoryStruct.comp p q ∈ self.arrows c e"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A F : CategoryTheory.Functor Cᵒᵖ (Type v)", "η : F ⟶ A", "X : C", "s : CategoryTheory.yoneda.obj X ⟶ A", "u : F.obj (Opposite.op X)", "h : CategoryTheory.OverPresheafAux.MakesOverArrow η s u"], "goal": "CategoryTheory.OverPresheafAux.OverArrows.val ⟨u, h⟩ = u"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_6", "N : Type u_7", "AddZeroClass M", "AddZeroClass N", "f : M →+ N", "s : Set α", "g : α → M", "x : α"], "goal": "f (s.indicator g x) = s.indicator (⇑f ∘ g) x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "A : Comon_ (C ⥤ D)", "X : C"], "goal": "((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.functorObj A).obj X).X = A.X.obj X"}}
+{"state": {"context": ["α : Type u_2"], "goal": "Function.Injective PiNat.res"}}
+{"state": {"context": ["R : Type u", "inst₁ inst₂ : NonUnitalRing R", "h_add : HAdd.hAdd = HAdd.hAdd", "h_mul : HMul.hMul = HMul.hMul"], "goal": "inst₁ = inst₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "𝔖 : Set (Set α)", "Zero β"], "goal": "(UniformOnFun.toFun 𝔖) 0 = 0"}}
+{"state": {"context": [], "goal": "Differentiable ℂ fun s => (Complex.Gamma s)⁻¹"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "D1 D2 : LieDerivation R L M", "s : Set L", "hs : LieSubalgebra.lieSpan R L s = ⊤", "h : Set.EqOn (⇑D1) (⇑D2) s"], "goal": "D1 = D2"}}
+{"state": {"context": ["R : Type u_1", "F : Type u", "K : Type v", "L : Type w", "inst✝² : CommRing K", "inst✝¹ : Field L", "inst✝ : Field F", "i✝ : K →+* L", "f : K[X]", "hs : Splits i✝ f", "hf0 : (map i✝ f).degree ≠ 0", "this : DecidableEq L := Classical.decEq L", "hf0' : ¬map i✝ f = 0", "g : L[X]", "hg : Irreducible g ∧ g ∣ map i✝ f", "x : L", "hx : g.IsRoot x", "i : L[X]", "hi : map i✝ f = g * i"], "goal": "eval₂ i✝ x f = 0"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : Semigroup M", "inst✝¹ : TopologicalSpace M", "inst✝ : T2Space M", "continuous_mul_left : ∀ (r : M), Continuous fun x => x * r", "s : Set M", "snemp : s.Nonempty", "s_compact : IsCompact s", "s_add : ∀ x ∈ s, ∀ y ∈ s, x * y ∈ s", "M' : Type u_1 := { m // m ∈ s }", "this✝ : Semigroup M' := Semigroup.mk ⋯", "this : CompactSpace M'"], "goal": "∃ m ∈ s, m * m = m"}}
+{"state": {"context": ["α : Type u_2", "M : Type u_4", "LE M", "One M", "s : Set α", "f : α → M", "a : α", "y : M", "hfg : a ∈ s → f a ≤ y", "hg : a ∉ s → 1 ≤ y"], "goal": "s.mulIndicator f a ≤ y"}}
+{"state": {"context": ["Γ : Type u_1", "Inhabited Γ", "l : Turing.ListBlank Γ"], "goal": "Turing.ListBlank.cons l.head l.tail = l"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "R : Type u_5", "inst✝⁸ : MeasurableSpace X", "inst✝⁷ : TopologicalSpace X", "inst✝⁶ : MeasurableSpace Y", "inst✝⁵ : TopologicalSpace Y", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "f✝ g✝ : X → E", "μ : Measure X", "s : Set X", "inst✝² : NormedSpace ℝ E", "inst✝¹ : OpensMeasurableSpace X", "inst✝ : T2Space X", "f : X → ℝ", "hf : LocallyIntegrable f μ", "g : X → E", "hg : Continuous g", "h'g : HasCompactSupport g", "K : Set X := tsupport g", "hK : IsCompact K"], "goal": "Integrable (fun x => f x • g x) μ"}}
+{"state": {"context": ["G : Type u_2", "Group G", "H : Subgroup G"], "goal": "↑H.op = MulOpposite.unop ⁻¹' ↑H"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "hs : MeasurableSet s"], "goal": "μ.restrict s + μ.restrict sᶜ = μ"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "K : Type u_2", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "r : R", "s : ↥(nonZeroDivisors R)"], "goal": "v.valuation (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation ↑s"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "SMul 𝕜 E", "p : Seminorm 𝕜 E", "x y : E", "r : ℝ"], "goal": "y ∈ p.closedBall x r ↔ p (y - x) ≤ r"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π : BoxIntegral.Prepartition I", "πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J"], "goal": "(π.biUnion πi).iUnion = ⋃ J ∈ π, (πi J).iUnion"}}
+{"state": {"context": ["cmp✝ : ?m.4485 → ?m.4485 → Ordering", "inst✝¹ : TransCmp cmp✝", "x✝ : Sort ?u.4483", "cmp : x✝ → x✝ → Ordering", "inst✝ : TransCmp cmp", "x y z : x✝", "h₁ : cmp y x = Ordering.lt", "h₂ : cmp z y = Ordering.lt"], "goal": "cmp z x = Ordering.lt"}}
+{"state": {"context": ["J : Type u_1", "CategoryTheory.Category.{u_2, u_1} J", "K : CategoryTheory.Functor J (Type u)", "c : CategoryTheory.Limits.Cocone K", "hc : CategoryTheory.Limits.IsColimit c", "lc : CategoryTheory.Limits.Cocone (K.comp CategoryTheory.uliftFunctor.{v, u})", "x : lc.pt", "j : J", "y : K.obj j"], "goal": "c.ι.app j y ∈ CategoryTheory.Limits.Types.descSet hc {x} ↔ lc.ι.app j { down := y } = x"}}
+{"state": {"context": ["Γ : Type u_1", "inst✝² : Inhabited Γ", "Λ : Type u_2", "inst✝¹ : Inhabited Λ", "σ : Type u_3", "inst✝ : Inhabited σ", "n : ℕ", "enc : Γ → Vector Bool n", "dec : Vector Bool n → Γ", "enc0 : enc default = Vector.replicate n false", "M : Λ → Stmt₁", "encdec : ∀ (a : Γ), dec (enc a) = a", "enc₀ : enc default = Vector.replicate n false", "x✝ : Cfg₁", "v : σ", "L : ListBlank Γ", "q : Stmt₁", "R : ListBlank Γ"], "goal": "Reaches (step (tr enc dec M)) (stepAux (trNormal dec q) v (trTape' enc0 L R)) (trCfg enc enc0 (stepAux q v (Tape.mk' L R)))"}}
+{"state": {"context": ["α : Type v", "M : Type u", "Monoid M", "MulAction M α", "exp_pos : 0 < Monoid.exponent M", "m : M", "a : α"], "goal": "0 < MulAction.period m a"}}
+{"state": {"context": ["J : Type u₁", "inst✝³ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝² : Category.{v₂, u₂} K", "C : Type u", "inst✝¹ : Category.{v, u} C", "F : J ⥤ C", "inst✝ : HasLimitsOfShape J C", "X✝ Y✝ Z✝ : J ⥤ C", "α : X✝ ⟶ Y✝", "β : Y✝ ⟶ Z✝", "j : J"], "goal": "{ obj := fun F => limit F, map := fun {X Y} α => limMap α }.map (α ≫ β) ≫ limit.π Z✝ j = ({ obj := fun F => limit F, map := fun {X Y} α => limMap α }.map α ≫ { obj := fun F => limit F, map := fun {X Y} α => limMap α }.map β) ≫ limit.π Z✝ j"}}
+{"state": {"context": ["G : Type w", "inst✝² : TopologicalSpace G", "μ : Content G", "inst✝¹ : R1Space G", "S : MeasurableSpace G", "inst✝ : BorelSpace G", "U : Set G", "hU : U ∈ {s | IsOpen s}", "U' : Opens G", "this✝ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }", "L : Compacts G", "L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }", "hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U", "hL'U : ↑L' ⊆ U", "hL'U' : ↑L' ⊆ ↑U'", "this : ↑U' \\ U ⊆ ↑U' \\ ↑L'"], "goal": "↑(μ.toFun L') + ⨆ x, ↑(μ.toFun ↑x) ≤ μ.outerMeasure ↑U'"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "s : Set E", "f : E → 𝕜", "hf : DiffContOnCl 𝕜 f s", "h₀ : ∀ x ∈ closure s, f x ≠ 0"], "goal": "DiffContOnCl 𝕜 f⁻¹ s"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroupWithOne α", "m n : ℤ"], "goal": "↑(m * n) = m • ↑n"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "p : Type u_5", "q : Type u_6", "m' : o → Type u_7", "n' : o → Type u_8", "p' : o → Type u_9", "R : Type u_10", "S : Type u_11", "α : Type u_12", "β : Type u_13", "inst✝ : Star α", "A : Matrix n l α", "B : Matrix n m α", "C : Matrix o l α", "D : Matrix o m α"], "goal": "(fromBlocks A B C D)ᴴ = fromBlocks Aᴴ Cᴴ Bᴴ Dᴴ"}}
+{"state": {"context": ["x : ℂ", "n : ℕ", "n.AtLeastTwo"], "goal": "x ^ OfNat.ofNat n = x ^ OfNat.ofNat n"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "CommRing 𝕜", "AddCommGroup E", "AddCommGroup F", "Module 𝕜 E", "Module 𝕜 F", "TopologicalSpace E", "TopologicalSpace F", "TopologicalAddGroup E", "ContinuousConstSMul 𝕜 E", "TopologicalAddGroup F", "ContinuousConstSMul 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "n : ℕ", "hn : 0 < n", "v : Fin n → E"], "goal": "p.applyComposition (Composition.single n hn) v = fun _j => (p n) v"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "c : ClosureOperator α", "x y : α", "hy : c.IsClosed y"], "goal": "c x ≤ y ↔ x ≤ y"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "ι : Type u_1", "f : ι → R[X]", "m : Multiset ι", "hm : m.Nodup"], "goal": "{ val := m, nodup := hm }.prod f ≠ 0 → ({ val := m, nodup := hm }.prod f).roots = { val := m, nodup := hm }.val.bind fun i => (f i).roots"}}
+{"state": {"context": ["R : Type u", "CommRing R", "n : Type v", "DecidableEq n", "Fintype n", "M : Matrix n n R"], "goal": "(M.charpoly - ∏ i : n, (Polynomial.X - Polynomial.C (M i i))).degree < ↑(Fintype.card n - 1)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "F : CategoryTheory.Functor C (CategoryTheory.Idempotents.Karoubi D)"], "goal": "∀ {X Y : CategoryTheory.Idempotents.Karoubi C} (f : X ⟶ Y),\n  ((CategoryTheory.Idempotents.FunctorExtension₁.obj F).map f).f = (F.map f.f).f"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "DiscreteValuationRing R", "a : R"], "goal": "(DiscreteValuationRing.addVal R) a = ⊤ ↔ a = 0"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Monoid M", "Monoid N"], "goal": "(MonoidHom.fst M N).comp Monoid.Coprod.toProd = Monoid.Coprod.fst"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "x : α", "γ : Type u_4", "LinearOrderedAddCommMonoid γ", "TopologicalSpace γ", "OrderTopology γ", "ContinuousAdd γ", "f g : α → γ", "hf : UpperSemicontinuousAt f x", "hg : UpperSemicontinuousAt g x"], "goal": "UpperSemicontinuousAt (fun z => f z + g z) x"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_3", "CommSemiring R", "m : M", "a : AddMonoidAlgebra R M"], "goal": "a ∈ AddMonoidAlgebra.grade R m ↔ a.support ⊆ {m}"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Computation α", "H : ¬s.Terminates", "n : ℕ", "val✝ : α", "h : ↑s n = some val✝"], "goal": "s.Terminates"}}
+{"state": {"context": ["a b c : Prop"], "goal": "((a ↔ b) ↔ c) ↔ ((a ↔ c) ↔ b)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "s : Setoid α"], "goal": "Measurable Quotient.mk''"}}
+{"state": {"context": ["α : Type u_1", "X : Type u_2", "Y : Type u_3", "Z : Type u_4", "T : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace T", "K✝ : Set X", "U✝ : Set Y", "this : ∀ {a : Y} {K : Set X} {U : Set (Y × X)}, MapsTo (⇑(coev X Y a)) K U ↔ {a} ×ˢ K ⊆ U", "x : Y", "K : Set X", "hK : IsCompact K", "U : Set (Y × X)", "hU : IsOpen U", "hKU : {x} ×ˢ K ⊆ U", "V : Set Y", "W : Set X", "hV : IsOpen V", "hxV : {x} ⊆ V", "hKW : K ⊆ W", "hVWU : V ×ˢ W ⊆ U"], "goal": "∀ᶠ (a : Y) in 𝓝 x, {a} ×ˢ K ⊆ U"}}
+{"state": {"context": ["R : Type u", "a : R", "Semiring R", "p : R[X]"], "goal": "p.IsRoot a ↔ Polynomial.eval a p = 0"}}
+{"state": {"context": ["α✝ : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α✝", "s : ι → MeasurableSpace Ω", "m m0 : MeasurableSpace Ω", "κ : Kernel α✝ Ω", "μα : Measure α✝", "μ : Measure Ω", "inst✝⁴ : IsMarkovKernel κ", "inst✝³ : IsProbabilityMeasure μ", "α : Type u_4", "p : Set ι → Prop", "f : Filter ι", "ns : α → Set ι", "inst✝² : StandardBorelSpace Ω", "inst✝¹ : Nonempty Ω", "hm : m ≤ m0", "inst✝ : IsFiniteMeasure μ", "h_le : ∀ (n : ι), s n ≤ m0", "h_indep : iCondIndep m hm s μ", "hf : ∀ (t : Set ι), p t → tᶜ ∈ f", "hns : Directed (fun x x_1 => x ≤ x_1) ns", "hnsp : ∀ (a : α), p (ns a)", "hns_univ : ∀ (n : ι), ∃ a, n ∈ ns a", "t : Set Ω", "ht_tail : MeasurableSet t", "h : ∀ᵐ (x : Ω) ∂μ, ((condexpKernel μ m) x) t = 0 ∨ ((condexpKernel μ m) x) t = 1", "ht : MeasurableSet t", "ω : Ω", "hω_eq : (((condexpKernel μ m) ω) t).toReal = (μ[t.indicator fun ω => 1|m]) ω", "hω : ((condexpKernel μ m) ω) t = 0 ∨ ((condexpKernel μ m) ω) t = 1"], "goal": "(μ[t.indicator fun ω => 1|m]) ω = 0 ∨ (μ[t.indicator fun ω => 1|m]) ω = 1"}}
+{"state": {"context": ["α : Type u_1", "β : α → Type u_2", "γ : Type u_3", "fγ : (a : α) × (β a → γ) → γ", "fγ_injective : Function.Injective fγ", "a₁ : α", "f₁ f₂ : β a₁ → WType β", "h✝ : elim γ fγ (mk a₁ f₁) = elim γ fγ (mk a₁ f₂)", "h : HEq (fun b => elim γ fγ (f₁ b)) fun b => elim γ fγ (f₂ b)", "x : β a₁"], "goal": "f₁ x = f₂ x"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "p : α → Prop", "a : { x // p x }"], "goal": "Subtype.val ⁻¹' Set.Ici ↑a = Set.Ici a"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "_mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "s₁ s₂ : Set (Set Ω)", "h : IndepSets s₁ s₂ κ μ", "t1 t2 : Set Ω", "ht1 : t1 ∈ s₂", "ht2 : t2 ∈ s₁"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝³ : DivisionRing K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "s t : Submodule K V", "hdim : finrank K ↥s + finrank K ↥t = finrank K V", "hdisjoint : Disjoint s t", "h_finrank_inf : finrank K ↥(s ⊓ t) = 0"], "goal": "finrank K ↥(s ⊔ t) = finrank K ↥s + finrank K ↥t"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B : Set α", "hB : M.Indep B", "hBmax : ∀ e ∈ M.E \\ B, ¬M.Indep (insert e B)"], "goal": "M.Base B"}}
+{"state": {"context": ["M : Type w", "A : Set M", "L : FirstOrder.Language", "L.Structure M", "α : Type u₁", "f g : Set (α → M)", "hf : A.Definable L f", "hg : A.Definable L g"], "goal": "A.Definable L (f ∩ g)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "l : Filter β", "f : β → α"], "goal": "Cauchy (Filter.map f l) ↔ l.NeBot ∧ Filter.Tendsto (fun p => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "n : ℕ", "p : ℕ → ℝ", "hp : ∀ (k : ℕ), 0 ≤ p k", "r a : ℝ", "hr : 0 ≤ r", "ha : 0 ≤ a"], "goal": "∑ x ∈ Ico 2 (n + 1), ∑ i ∈ {c | 1 < c.length}.toFinset, a ^ x * (r ^ i.length * ∏ j : Fin i.length, p (i.blocksFun j)) = ∑ k ∈ Ico 2 (n + 1), ∑ c ∈ {c | 1 < c.length}.toFinset, ∏ j : Fin c.length, r * (a ^ c.blocksFun j * p (c.blocksFun j))"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w : W", "hw : w ≠ 1"], "goal": "∃ i, cs.IsLeftDescent w i"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "x y z : α", "ε ε₁ ε₂ : ℝ≥0∞", "s t : Set α"], "goal": "y ∈ ball x ε ↔ edist x y < ε"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s t : Finset α"], "goal": "(Iio s).card = 2 ^ s.card - 1"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "δ : ℝ", "E : Set α"], "goal": "Metric.cthickening (max 0 δ) E = Metric.cthickening δ E"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Preorder α", "Preorder β", "MulZeroOneClass α", "MulZeroOneClass β", "f : α →*₀o β", "f' : α → β", "h : f' = ⇑f"], "goal": "f.copy f' h = ↑f"}}
+{"state": {"context": ["α✝ : Type u", "β✝ : Type u_1", "γ : Type u_2", "r✝ : α✝ → α✝ → Prop", "s✝ : β✝ → β✝ → Prop", "t : γ → γ → Prop", "a b : Ordinal.{v}", "α : Type v", "r : α → α → Prop", "x✝¹ : IsWellOrder α r", "β : Type v", "s : β → β → Prop", "x✝ : IsWellOrder β s"], "goal": "lift.{u, v} (type r) ≤ lift.{u, v} (type s) ↔ type r ≤ type s"}}
+{"state": {"context": ["𝕜 : Type u_2", "E : Type u_3", "LinearOrderedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "s t u v : Set E"], "goal": "convexJoin 𝕜 (convexJoin 𝕜 s t) (convexJoin 𝕜 u v) = convexJoin 𝕜 (convexJoin 𝕜 s u) (convexJoin 𝕜 t v)"}}
+{"state": {"context": ["α : Type u", "l : List α", "a : α", "ha : a ∈ l.getLast?"], "goal": "a ∈ l"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝¹² : RCLike 𝕜", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedAddCommGroup G", "inst✝⁸ : InnerProductSpace 𝕜 E", "inst✝⁷ : InnerProductSpace 𝕜 F", "inst✝⁶ : InnerProductSpace 𝕜 G", "H : Type u_5", "inst✝⁵ : NormedAddCommGroup H", "inst✝⁴ : InnerProductSpace 𝕜 H", "inst✝³ : CompleteSpace H", "K : Type u_6", "inst✝² : NormedAddCommGroup K", "inst✝¹ : InnerProductSpace 𝕜 K", "inst✝ : CompleteSpace K", "e : H ≃ₗᵢ[𝕜] K", "e' : H →L[𝕜] K := ↑{ toLinearEquiv := e.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }"], "goal": "adjoint e' = (adjoint e').comp (e'.comp ↑{ toLinearEquiv := e.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ })"}}
+{"state": {"context": ["α : Type u_1", "a b c d : ℝ≥0∞", "r p q : ℝ≥0", "inst✝ : CompleteLattice α", "f : ℝ≥0∞ → α"], "goal": "⨅ x, ⨅ (_ : x ≠ ⊤), f x = ⨅ x, f ↑x"}}
+{"state": {"context": ["p : ℕ+", "hp : Fact (Nat.Prime ↑p)"], "goal": "IsIntegralClosure (CyclotomicRing p ℤ ℚ) ℤ (CyclotomicField p ℚ)"}}
+{"state": {"context": ["n✝¹ : ℕ+", "S T : Set ℕ+", "A : Type u", "B : Type v", "K : Type w", "L : Type z", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : Algebra A B", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Algebra K L", "ζ : B", "n : ℕ+", "h : IsPrimitiveRoot ζ ↑n", "n✝ : ℕ+", "hi : n✝ = n"], "goal": "∃ r, IsPrimitiveRoot r ↑n✝"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β i)", "inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)", "s : Finset ι", "f : (i : { x // x ∈ s }) → { x // x ≠ 0 }", "x✝ : { x // x ∈ s }"], "goal": "f ⟨↑x✝, ⋯⟩ = f x✝"}}
+{"state": {"context": ["β : Type u_1", "AddCommGroup β", "b : β", "V : Type u_2", "CategoryTheory.Category.{u_3, u_2} V", "CategoryTheory.Limits.HasZeroMorphisms V"], "goal": "(HomologicalComplex.dgoEquivHomologicalComplex b V).functor = HomologicalComplex.dgoToHomologicalComplex b V"}}
+{"state": {"context": ["R : Type u", "S : Type v", "A : Type w", "B : Type u₁", "M : Type v₁", "inst✝⁶ : Semiring R", "inst✝⁵ : Semiring S", "inst✝⁴ : AddCommMonoid A", "inst✝³ : Module R S", "inst✝² : Module S A", "inst✝¹ : Module R A", "inst✝ : IsScalarTower R S A", "s : Set S", "hs : span R s = ⊤", "t : Set A", "k : S", "x : A", "hx : x ∈ span R t"], "goal": "k • x ∈ restrictScalars R (span S t)"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "AddGroup G", "AddAction G α"], "goal": "∀ {x : MeasurableSpace α} (μ : MeasureTheory.Measure α) [inst : MeasureTheory.VAddInvariantMeasure G α μ] (s : Set α),\n  ↑(AddAction.aestabilizer G μ s) = {g | g +ᵥ s =ᵐ[μ] s}"}}
+{"state": {"context": ["α : Type u_2", "Zero α"], "goal": "↑0 = 0"}}
+{"state": {"context": ["α : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure μ", "r : ℝ≥0∞", "s : ℕ → Set α", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "hr₀ : r ≠ 0", "hr : ∀ (n : ℕ), r ≤ μ (s n)"], "goal": "∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "s : Set α", "t : Set β", "p : α × β"], "goal": "p ∈ s ×ˢ t ↔ p.fst ∈ s ∧ p.snd ∈ t"}}
+{"state": {"context": ["E : Type u_2", "AddCommGroup E", "Module ℝ E", "s : Set E", "x : E", "TopologicalSpace E", "TopologicalAddGroup E", "ContinuousSMul ℝ E", "hc : Convex ℝ s", "hs₀ : s ∈ 𝓝 0"], "goal": "ContinuousAt (gauge s) x"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : LinearOrder α", "E : Type u_2", "inst✝ : PseudoEMetricSpace E", "f : α → E", "s t : Set α", "hst : t ⊆ s", "n : ℕ", "u : ℕ → α", "hu : Monotone u", "ut : ∀ (i : ℕ), u i ∈ t"], "goal": "∑ i ∈ Finset.range (n, ⟨u, ⋯⟩).1, edist (f (↑(n, ⟨u, ⋯⟩).2 (i + 1))) (f (↑(n, ⟨u, ⋯⟩).2 i)) ≤ eVariationOn f s"}}
+{"state": {"context": ["α : Type u", "l p : Nat", "a : Array α", "h : a.size ≤ l"], "goal": "a.extract p l = a.extract p a.size"}}
+{"state": {"context": ["ι : Type u₁", "k : Type u₂", "V : Type u₃", "P : Type u₄", "inst✝⁶ : AddCommGroup V", "inst✝⁵ : AffineSpace V P", "inst✝⁴ : Ring k", "inst✝³ : Module k V", "b : AffineBasis ι k P", "ι' : Type u_1", "inst✝² : Fintype ι", "inst✝¹ : Finite ι'", "inst✝ : DecidableEq ι'", "p : ι' → P", "A : Matrix ι ι' k", "hA : b.toMatrix p * A = 1"], "goal": "AffineIndependent k p"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_4", "Zero M", "s : Set α", "f : α → M", "DecidablePred fun x => x ∈ s"], "goal": "s.piecewise f 0 = s.indicator f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝⁶ : ConditionallyCompleteLinearOrder α", "inst✝⁵ : TopologicalSpace α", "inst✝⁴ : OrderTopology α", "inst✝³ : ConditionallyCompleteLinearOrder β", "inst✝² : TopologicalSpace β", "inst✝¹ : OrderTopology β", "inst✝ : Nonempty γ", "a b : α", "s : Set α", "hs : IsClosed (s ∩ Icc a b)", "hab : a ≤ b", "hgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty", "S : Set α := s ∩ Icc a b", "ha : a ∈ S", "Sbd : BddAbove S", "c : α := sSup (s ∩ Icc a b)", "c_mem : c ∈ S"], "goal": "b ∈ s"}}
+{"state": {"context": ["α : Type u_1", "a : α", "n : ℕ"], "goal": "n • {a} = Multiset.replicate n a"}}
+{"state": {"context": ["a b : ℕ", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "(a.lcm b).factorization = a.factorization ⊔ b.factorization"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : 𝓢(E, F)", "k : ℕ", "d : ℝ", "left✝ : 0 < d", "hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d"], "goal": "⇑f =O[cocompact E] fun x => ‖x‖ ^ (-↑k)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "FunLike F (Set α) ℝ≥0∞", "MeasureTheory.OuterMeasureClass F α", "μ : F", "s : Set α", "hs : s.Countable"], "goal": "μ s = 0 ↔ ∀ x ∈ s, μ {x} = 0"}}
+{"state": {"context": ["ι : Type u", "X : ι → Type", "(i : ι) → TopologicalSpace (X i)", "C : Set ((i : ι) → X i)", "a b : ↑C", "h :\n  ∀ (J : Finset ι),\n    (Profinite.IndexFunctor.π_app C fun x => x ∈ J) a = (Profinite.IndexFunctor.π_app C fun x => x ∈ J) b"], "goal": "a = b"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ₁ φ₂ : S₁ ⟶ S₂", "h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂"], "goal": "h.neg.h₀ = -h.h₀"}}
+{"state": {"context": ["α : Type u_1", "AddCommSemigroup α", "PartialOrder α", "ExistsAddOfLE α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Sub α", "OrderedSub α", "a b c d : α", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "hba : b ≤ a", "hdc : d ≤ c"], "goal": "a - b + (c - d) = a + c - (b + d)"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁸ : NormedAddCommGroup E", "inst✝¹⁷ : NormedAddCommGroup E'", "inst✝¹⁶ : NormedAddCommGroup E''", "inst✝¹⁵ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x x' : G", "y y' : E", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "inst✝¹³ : NormedSpace 𝕜 E", "inst✝¹² : NormedSpace 𝕜 E'", "inst✝¹¹ : NormedSpace 𝕜 E''", "inst✝¹⁰ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝⁹ : MeasurableSpace G", "μ ν : Measure G", "inst✝⁸ : NormedSpace ℝ F", "inst✝⁷ : AddCommGroup G", "inst✝⁶ : μ.IsAddLeftInvariant", "inst✝⁵ : μ.IsNegInvariant", "inst✝⁴ : TopologicalSpace G", "inst✝³ : TopologicalAddGroup G", "inst✝² : BorelSpace G", "inst✝¹ : FirstCountableTopology G", "inst✝ : SecondCountableTopologyEither G E", "hbf : BddAbove (range fun x => ‖f x‖)", "hf : Continuous f", "hg : Integrable g μ"], "goal": "Continuous (g ⋆[L.flip, μ] f)"}}
+{"state": {"context": [], "goal": "ClosedEmbedding Int.cast"}}
+{"state": {"context": ["B : Type u₁", "inst✝¹ : Bicategory B", "C : Type u₂", "inst✝ : Bicategory C", "F G H : OplaxFunctor B C", "η : StrongOplaxNatTrans F G", "θ : StrongOplaxNatTrans G H", "a b c : B", "a' : C", "f : a' ⟶ G.obj a", "g h : a ⟶ b", "β : g ⟶ h"], "goal": "f ◁ G.map₂ β ▷ θ.app b ≫ f ◁ (θ.naturality h).hom = f ◁ (θ.naturality g).hom ≫ f ◁ θ.app a ◁ H.map₂ β"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "X Y : Type u", "x₀ : X", "f : X → Y", "g : Y → F (α ::: X)", "R : Cofix F α → Cofix F α → Prop := fun a b => ∃ x, a = Cofix.corec (g ∘ f) x ∧ b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) ∘ g) (f x)", "a b : Cofix F α", "x : X", "Ha : a = Cofix.corec (g ∘ f) x", "Hb : b = Cofix.corec (MvFunctor.map (TypeVec.id ::: f) ∘ g) (f x)"], "goal": "∀ (x : X), R (Cofix.corec (g ∘ f) x) ((Cofix.corec (MvFunctor.map (TypeVec.id ::: f) ∘ g) ∘ f) x)"}}
+{"state": {"context": ["x : ℝ", "n : ℤ"], "goal": "Real.sin (x - ↑n * (2 * Real.pi)) = Real.sin x"}}
+{"state": {"context": ["α : Type u", "s : Stream'.WSeq α"], "goal": "s.think ~ʷ s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v, u'} D", "CategoryTheory.Limits.HasLimits D", "CategoryTheory.WellPowered D", "𝒢 : Set D", "Small.{v, u'} ↑𝒢", "h𝒢 : CategoryTheory.IsCoseparating 𝒢", "G : CategoryTheory.Functor D C", "CategoryTheory.Limits.PreservesLimits G"], "goal": "G.IsRightAdjoint"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "CommRing A", "Field K", "CommRing B", "Field L", "Algebra A K", "Algebra B L", "Algebra A B", "Algebra K L", "Algebra A L", "IsScalarTower A K L", "IsScalarTower A B L", "IsDomain A", "IsFractionRing A K", "IsIntegralClosure B A L", "IsFractionRing B L", "FiniteDimensional K L", "Algebra.IsSeparable K L", "IsIntegrallyClosed A", "IsDedekindDomain B", "I : Ideal B", "hI : I ≠ ⊥", "NoZeroSMulDivisors A B"], "goal": "differentIdeal A B ≤ I ↔ Submodule.map (↑A (Algebra.trace K L)) (Submodule.restrictScalars A ↑(↑I)⁻¹) ≤ 1"}}
+{"state": {"context": ["o : Ordinal.{u_2}"], "goal": "(Order.succ o).cof = 1"}}
+{"state": {"context": ["α : Type u", "UniformSpace α", "CompleteSpace α", "f g : CauchyFilter α"], "goal": "Inseparable (lim ↑f) (lim ↑g) ↔ Inseparable f g"}}
+{"state": {"context": ["α : Type u_3", "Preorder α", "a : α"], "goal": "{b | a ≤ b} ∈ Filter.atTop"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "G' : G.Subgraph", "s : Set V"], "goal": "G'.deleteVerts (G'.verts ∩ s) = G'.deleteVerts s"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : C", "p : CategoryTheory.ExactPairing X Y"], "goal": "CategoryTheory.rightDualIso p p = CategoryTheory.Iso.refl Y"}}
+{"state": {"context": ["α : Type u", "HasSubset α", "HasSSubset α", "IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1", "a b : α", "h₁ : a ⊆ b", "h₂ : ¬b ⊆ a"], "goal": "a ⊂ b"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "R : Type u_2", "inst✝⁶ : CommSemiring k", "inst✝⁵ : Monoid G", "inst✝⁴ : Monoid H", "A : Type u₃", "inst✝³ : Semiring A", "inst✝² : Algebra k A", "B : Type u_3", "inst✝¹ : Semiring B", "inst✝ : Algebra k B", "F : G →* A", "f : MonoidAlgebra k G"], "goal": "((lift k G A) F) f = sum f fun a b => b • F a"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "f : ℕ → Ω → ℝ", "ℱ : MeasureTheory.Filtration ℕ m0", "MeasureTheory.IsFiniteMeasure μ", "hf : MeasureTheory.Submartingale f ℱ μ", "a b : ℝ", "N : ℕ"], "goal": "MeasureTheory.Submartingale (fun n => ∑ k ∈ Finset.range n, MeasureTheory.upcrossingStrat a b f N k * (f (k + 1) - f k))\n  ℱ μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝ : Countable β", "f : β → α → ℝ≥0∞", "hf : ∀ (b : β), AEMeasurable (f b) μ", "h_directed : Directed (fun x x_1 => x ≤ x_1) f", "p : α → (β → ℝ≥0∞) → Prop := fun x f' => Directed LE.le f'", "hp : ∀ᵐ (x : α) ∂μ, p x fun i => f i x"], "goal": "∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ"}}
+{"state": {"context": ["X Y : Scheme", "U : X.Opens", "hU : IsAffineOpen U", "f : ↑Γ(X, U)", "x : ↥U"], "goal": "(((↑U).isoSpec.hom ≫ Spec.map (X.presheaf.map (eqToHom ⋯).op)) ≫ Spec.map (X.presheaf.map (eqToHom ⋯).op) ≫ (↑U).isoSpec.inv ≫ U.ι).val.base x = ↑x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "β : Type u_4", "inst✝⁴ : OrderedSemiring 𝕜", "inst✝³ : AddCommMonoid E", "inst✝² : AddCommMonoid F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "s : Set E", "x : E", "t : Set E", "hs : Convex 𝕜 s", "ht : Convex 𝕜 t"], "goal": "Convex 𝕜 ((fun x => x.1 + x.2) '' s ×ˢ t)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "R1Space X", "WeaklyLocallyCompactSpace X", "x : X"], "goal": "∃ U, IsOpen U ∧ x ∈ U ∧ IsCompact (closure U)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "h : p.degree = 0"], "goal": "p = Polynomial.C (p.coeff 0)"}}
+{"state": {"context": ["x : Nat", "z : Nat", "H : 0 < z"], "goal": "(x + z) / z = x / z + 1"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "k✝ : ℕ", "N : LieSubmodule R L M", "M₂ : Type w₁", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "x y : L", "m : M", "k : ℕ", "ih : (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k)", "hk : 2 * k.succ = 2 * k + 1 + 1"], "goal": "(⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k + 1] m ∈ lowerCentralSeries R L M (2 * (k + 1))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α ↪ β", "s : Finset α"], "goal": "↑(Finset.map f s) = ⇑f '' ↑s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasProducts C", "X : TopCat", "F : TopCat.Presheaf C X", "ι : Type v'", "U : ι → TopologicalSpace.Opens ↑X"], "goal": "(TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquiv F U).unitIso =\n  TopCat.Presheaf.SheafConditionPairwiseIntersections.coneEquivUnitIso F U"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : CommRing S", "inst✝² : IsDomain S", "Rₘ : Type u_3", "Sₘ : Type u_4", "inst✝¹ : CommRing Rₘ", "inst✝ : CommRing Sₘ", "hR : IsJacobson R", "I : Ideal R[X]", "hI : I.IsPrime", "R' : Subring (R[X] ⧸ I) := ((Quotient.mk I).comp C).range", "i : R →+* ↥R' := ((Quotient.mk I).comp C).rangeRestrict", "hi : Function.Surjective ⇑i", "hi' : RingHom.ker (mapRingHom i) ≤ I", "R'_jacob : IsJacobson ↥R'", "J : Ideal (↥R')[X] := map (mapRingHom i) I"], "goal": "I.jacobson = I"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝³ : OrderedCancelAddCommMonoid α", "inst✝² : ExistsAddOfLE α", "inst✝¹ : LocallyFiniteOrder α", "inst✝ : DecidableEq α", "a b c : α"], "goal": "image (fun x => x + c) (Ioc a b) = Ioc (a + c) (b + c)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "S : CategoryTheory.Subgroupoid C", "self : S.IsNormal", "c d : C", "p : c ⟶ d", "γ : c ⟶ c"], "goal": "γ ∈ S.arrows c c →\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.Groupoid.inv p) (CategoryTheory.CategoryStruct.comp γ p) ∈\n    S.arrows d d"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝¹⁹ : CommRing R", "inst✝¹⁸ : LieRing L", "inst✝¹⁷ : LieAlgebra R L", "inst✝¹⁶ : LieAlgebra.IsNilpotent R L", "inst✝¹⁵ : AddCommGroup M", "inst✝¹⁴ : Module R M", "inst✝¹³ : LieRingModule L M", "inst✝¹² : LieModule R L M", "M₁ : Type u_5", "M₂ : Type u_6", "M₃ : Type u_7", "inst✝¹¹ : AddCommGroup M₁", "inst✝¹⁰ : Module R M₁", "inst✝⁹ : LieRingModule L M₁", "inst✝⁸ : LieModule R L M₁", "inst✝⁷ : AddCommGroup M₂", "inst✝⁶ : Module R M₂", "inst✝⁵ : LieRingModule L M₂", "inst✝⁴ : LieModule R L M₂", "inst✝³ : AddCommGroup M₃", "inst✝² : Module R M₃", "inst✝¹ : LieRingModule L M₃", "inst✝ : LieModule R L M₃", "g : M₁ ⊗[R] M₂ →ₗ⁅R,L⁆ M₃", "χ₁ χ₂ : R", "x : L", "t : ↥𝕎(M₁, χ₁, x) ⊗[R] ↥𝕎(M₂, χ₂, x)", "F : Module.End R M₃ := (toEnd R L M₃) x - (χ₁ + χ₂) • 1", "m₁ : M₁", "hm₁ : m₁ ∈ 𝕎(M₁, χ₁, x)", "m₂ : M₂", "hm₂ : m₂ ∈ 𝕎(M₂, χ₂, x)", "f₁ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.rTensor M₂ ((toEnd R L M₁) x - χ₁ • 1)", "f₂ : Module.End R (M₁ ⊗[R] M₂) := LinearMap.lTensor M₁ ((toEnd R L M₂) x - χ₂ • 1)", "h_comm_square : F ∘ₗ ↑g = ↑g ∘ₗ (f₁ + f₂)"], "goal": "∃ k, ((f₁ + f₂) ^ k) (m₁ ⊗ₜ[R] m₂) = 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "A : Type z", "a b : R", "n✝ : ℕ", "inst✝¹ : Semiring R", "inst✝ : Semiring S", "p : R[X]", "f : R →+* S", "n : ℕ := max p.natDegree (map f p).natDegree"], "goal": "∀ (n : ℕ), C (0 * ↑n) * X ^ (n - 1) = 0"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a b : α"], "goal": "BddBelow (Set.Ioo a b)"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w₁ w₂ : W"], "goal": "cs.length w₁ - cs.length w₂ ≤ cs.length (w₁ * w₂)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder β", "f : α → β", "l : List α", "m : α", "DecidableEq α"], "goal": "m ∈ List.argmin f l ↔\n  m ∈ l ∧ (∀ (a : α), a ∈ l → f m ≤ f a) ∧ ∀ (a : α), a ∈ l → f a ≤ f m → List.indexOf m l ≤ List.indexOf a l"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeSup α", "s : Finset β", "H : s.Nonempty", "a : α"], "goal": "(s.sup' H fun x => a) = a"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π₁ π₂ : BoxIntegral.Prepartition I", "M : Type u_2", "AddCommMonoid M", "h : Disjoint π₁.iUnion π₂.iUnion", "f : BoxIntegral.Box ι → M"], "goal": "∑ J ∈ π₁.boxes ∪ π₂.boxes, f J = ∑ J ∈ π₁.boxes, f J + ∑ J ∈ π₂.boxes, f J"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_2", "PseudoMetricSpace α", "F : ι → β → α", "f : β → α", "p : Filter ι", "s : Set β"], "goal": "TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ (n : ι) in p, ∀ x ∈ s, dist (f x) (F n x) < ε"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n✝ m : σ", "s : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q a : MvPolynomial σ R", "n : ℕ", "x✝¹ x✝ : σ →₀ ℕ"], "goal": "((x✝¹ + x✝).sum fun x e => e) = (x✝¹.sum fun x e => e) + x✝.sum fun x e => e"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "ι : Sort u_1", "hι : Nonempty ι", "S : ι → Subfield K", "hS : Directed (fun x x_1 => x ≤ x_1) S", "x : K"], "goal": "x ∈ ⨆ i, S i ↔ ∃ i, x ∈ S i"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "H : Type u_2", "TopologicalSpace H", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "TopologicalSpace G", "ChartedSpace H G", "AddGroup G", "LieAddGroup I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "n : ℕ∞", "f : M → G", "s : Set M", "hf : ContMDiffOn I' I n f s"], "goal": "ContMDiffOn I' I n (fun x => -f x) s"}}
+{"state": {"context": ["M : Type u_1", "AddCommMonoid M", "S : AddSubmonoid M", "N : Type u_2", "AddCommMonoid N", "P : Type u_3", "AddCommMonoid P", "f : S.LocalizationMap N", "k : N ≃+ P"], "goal": "f.addEquivOfLocalizations (f.ofAddEquivOfLocalizations k) = k"}}
+{"state": {"context": ["α : Type u_1", "s : List α", "x : α", "n : ℕ", "hn : n < (List.permutations'Aux x s).length"], "goal": "(List.permutations'Aux x s)[n] = List.insertNth n x s"}}
+{"state": {"context": ["α : Type u_2"], "goal": "Function.Injective (@Preorder.toLE α)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "𝔖 : Set (Set α)", "S : Set α", "V : Set (β × β)"], "goal": "UniformOnFun.gen 𝔖 S V =\n  Prod.map (S.restrict ∘ ⇑UniformFun.toFun) (S.restrict ∘ ⇑UniformFun.toFun) ⁻¹' UniformFun.gen (↑S) β V"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "s ∪ t = (sᶜ ∩ tᶜ)ᶜ"}}
+{"state": {"context": ["R : Type u", "Ring R", "S : CategoryTheory.ShortComplex (ModuleCat R)"], "goal": "CategoryTheory.CategoryStruct.comp S.moduleCatCyclesIso.hom (LinearMap.ker S.g).subtype = S.iCycles"}}
+{"state": {"context": ["a b : Ordinal.{u_4}"], "goal": "(a + b).IsLimit ↔ b.IsLimit ∨ b = 0 ∧ a.IsLimit"}}
+{"state": {"context": ["𝕜 : Type u_1", "OrderedSemiring 𝕜", "E : Type u_2", "AddCommMonoid E", "TopologicalSpace E", "T1Space E", "Module 𝕜 E"], "goal": "↑↑0 = 0"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "inst✝¹⁸ : CommRing A", "inst✝¹⁷ : Field K", "inst✝¹⁶ : CommRing B", "inst✝¹⁵ : Field L", "inst✝¹⁴ : Algebra A K", "inst✝¹³ : Algebra B L", "inst✝¹² : Algebra A B", "inst✝¹¹ : Algebra K L", "inst✝¹⁰ : Algebra A L", "inst✝⁹ : IsScalarTower A K L", "inst✝⁸ : IsScalarTower A B L", "inst✝⁷ : IsDomain A", "inst✝⁶ : IsFractionRing A K", "inst✝⁵ : IsIntegralClosure B A L", "inst✝⁴ : IsFractionRing B L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra.IsSeparable K L", "inst✝¹ : IsIntegrallyClosed A", "inst✝ : IsDedekindDomain B", "I J : FractionalIdeal B⁰ L", "hI✝ : I ≠ 0", "hJ : J ≠ 0", "hI : ¬I = 0"], "goal": "I = (fun J => J * (dual A K I)⁻¹) ((fun x => I * x) (dual A K I))"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : Preadditive C", "M : Mat_ C", "i : M.ι"], "goal": "𝟙 M i i = 𝟙 (M.X i)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "ι : Type u_3", "I : ι → FractionalIdeal R⁰ K", "hS : ∀ i ∈ ∅, I i ≠ 0"], "goal": "count K v (∏ i ∈ ∅, I i) = ∑ i ∈ ∅, count K v (I i)"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝³ : TopologicalSpace G", "inst✝² : Group G", "inst✝¹ : TopologicalGroup G", "inst✝ : NoncompactSpace G", "K L : Set G", "hK : IsCompact K", "hL : IsCompact L", "A : ¬K * L⁻¹ = univ", "g : G", "hg : g ∉ K * L⁻¹"], "goal": "Disjoint K (g • L)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "f✝ : α → β", "inst✝ : MeasurableSpace α", "f : α → β", "μ : Measure β", "hfi : Injective f", "hf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ", "hs : NullMeasurableSet s (comap f μ)"], "goal": "(comap f μ) s = μ (f '' s)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "𝕜 : Type u_5", "NormedDivisionRing 𝕜", "Module 𝕜 F", "BoundedSMul 𝕜 F", "c : 𝕜", "f : α → F"], "goal": "MeasureTheory.eLpNorm (c • f) p μ = ↑‖c‖₊ * MeasureTheory.eLpNorm f p μ"}}
+{"state": {"context": ["n : ℕ", "a b : Fin n"], "goal": "(uIcc a b).card = (↑↑b - ↑↑a).natAbs + 1"}}
+{"state": {"context": ["n' : Type u_1", "inst✝² : DecidableEq n'", "inst✝¹ : Fintype n'", "R : Type u_2", "inst✝ : CommRing R", "A X Y : M", "hx : IsUnit X.det", "hy : IsUnit Y.det", "h : SemiconjBy A X Y", "n : ℕ", "hx' : IsUnit (X ^ n.succ).det", "hy' : IsUnit (Y ^ n.succ).det"], "goal": "A * ↑hx'.unit⁻¹ • (X ^ n.succ).adjugate = ↑hy'.unit⁻¹ • (Y ^ n.succ).adjugate * A"}}
+{"state": {"context": ["ι✝ : Type u_1", "α : Type u_2", "M : Matroid α", "F X Y : Set α", "e : α", "ι : Sort u_3", "I J B : Set α", "f x y : α", "hI : M.Indep I"], "goal": "x ∈ M.closure I ↔ M.Dep (insert x I) ∨ x ∈ I"}}
+{"state": {"context": ["R : Type u", "NonAssocSemiring R", "S : Subsemiring R", "s : Set R", "hs : s = ↑S"], "goal": "S.copy s hs = S"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "x : ↥(IsUnit.submonoid M)"], "goal": "↑x⁻¹ = ↑(IsUnit.unit ⋯)⁻¹"}}
+{"state": {"context": ["α : Type u", "n : ℕ", "a : α", "h : List.replicate n a ≠ []"], "goal": "(List.replicate n a).head h = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁹ : NontriviallyNormedField 𝕜", "H : Type u_2", "inst✝¹⁸ : TopologicalSpace H", "E : Type u_3", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "F : Type u_4", "inst✝¹⁵ : NormedAddCommGroup F", "inst✝¹⁴ : NormedSpace 𝕜 F", "J : ModelWithCorners 𝕜 F F", "G : Type u_5", "inst✝¹³ : TopologicalSpace G", "inst✝¹² : ChartedSpace H G", "inst✝¹¹ : Group G", "inst✝¹⁰ : LieGroup I G", "E' : Type u_6", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_7", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H' M", "E'' : Type u_9", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_10", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M' : Type u_11", "inst✝¹ : TopologicalSpace M'", "inst✝ : ChartedSpace H'' M'", "n : ℕ∞", "f g : M → G", "x₀ : M", "hf : ContMDiffAt I' I n f x₀", "hg : ContMDiffAt I' I n g x₀"], "goal": "ContMDiffAt I' I n (fun x => f x * (g x)⁻¹) x₀"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p n : ℕ", "ExpChar R p"], "goal": "⇑(iterateFrobenius R p n) = (⇑(frobenius R p))^[n]"}}
+{"state": {"context": ["X : Type u_2", "Y : Type u_3", "EMetricSpace X", "EMetricSpace Y", "K : ℝ≥0", "f : X → Y", "h : LipschitzWith K f"], "goal": "dimH (Set.range f) ≤ dimH Set.univ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "n : ℕ", "x✝ : α × List α", "a : α", "l : List α"], "goal": "Tendsto (fun p => insertNth n p.1 p.2) (𝓝 a ×ˢ 𝓝 l) (𝓝 (insertNth n (a, l).1 (a, l).2))"}}
+{"state": {"context": ["R : Type u", "S : Type v", "K : Type w", "inst✝ : CommRing R", "I : Ideal R", "f p : R[X]"], "goal": "(quotQuotEquivComm I f) ((Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) f})) (Polynomial.map (Ideal.Quotient.mk I) p)) = (Ideal.Quotient.mk (span {(Ideal.Quotient.mk (Ideal.map C I)) f})) ((Ideal.Quotient.mk (Ideal.map C I)) p)"}}
+{"state": {"context": ["X Y : TopCat", "f g : C(↑X, ↑Y)", "H : f.Homotopy g", "x₀ x₁ : ↑X", "p : fromTop x₀ ⟶ fromTop x₁"], "goal": "(π.map f).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧ ⟦H.evalAt x₀⟧ ≫ (π.map g).map p = H.diagonalPath' p"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Sup α", "Sup β", "p q : α × β"], "goal": "p ⊔ q = (p.1 ⊔ q.1, p.2 ⊔ q.2)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "K L L' M : Set V", "C : G.ComponentCompl L"], "goal": "ConnectedComponent.map (induceHom Hom.id ⋯) C = C"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "σ : Perm α"], "goal": "σ.cycleType.lcm = orderOf σ"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "K L : Set V", "v : V", "vnL : v ∉ L", "h : K ⊆ L"], "goal": "SimpleGraph.ComponentCompl.hom h (G.componentComplMk vnL) = G.componentComplMk ⋯"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "Polynomial.X.natTrailingDegree ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_8", "N : Type u_10", "P : Type u_11", "Zero M", "AddCommMonoid N", "AddCommMonoid P", "H : Type u_16", "FunLike H N P", "AddMonoidHomClass H N P", "h : H", "f : α →₀ M", "g : α → M → N"], "goal": "h (f.sum g) = f.sum fun a b => h (g a b)"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "R : Type u_3", "α : Type u_4", "A : Type u_5", "inst✝⁵ : AddCommMonoid A", "inst✝⁴ : HasAntidiagonal A", "inst✝³ : TopologicalSpace α", "inst✝² : NonUnitalNonAssocSemiring α", "f g : A → α", "inst✝¹ : T3Space α", "inst✝ : TopologicalSemiring α", "hf : Summable f", "hg : Summable g", "hfg : Summable fun x => f x.1 * g x.2"], "goal": "(∑' (n : A), f n) * ∑' (n : A), g n = ∑' (x : A) (b : ↑↑(antidiagonal x)), f (↑b).1 * g (↑b).2"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : PartialOrder α", "inst✝ : SuccOrder α", "a b✝ : α", "C : α → Sort u_2", "hs : (a : α) → ¬IsMax a → C (succ a)", "hl : (a : α) → IsSuccLimit a → C a", "b : α", "hb : ¬IsMax b", "hb' : ¬IsSuccLimit (succ b)", "H : ¬IsMax (Classical.choose ⋯) ∧ succ (Classical.choose ⋯) = succ b"], "goal": "HEq (hs (Classical.choose ⋯) ⋯) (hs b hb)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "hf : S.f = 0", "hg : S.g = 0"], "goal": "(CategoryTheory.ShortComplex.LeftHomologyData.ofZeros S hf hg).π = CategoryTheory.CategoryStruct.id S.X₂"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "M : G.Subgraph", "h : M.IsMatching", "c : G.ConnectedComponent"], "goal": "(M.induce (M.verts ∩ c.supp)).IsMatching"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι"], "goal": "addHaarMeasure (piIcc01 ι) = volume"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x✝ y✝ f' : ℝ", "hfc : ConvexOn ℝ S f", "hfd : ∀ x ∈ S, DifferentiableAt ℝ f x", "x : ℝ", "hx : x ∈ S", "y : ℝ", "hy : y ∈ S", "hxy : x ≤ y", "hxy' : x < y"], "goal": "deriv f x ≤ deriv f y"}}
+{"state": {"context": ["c : ℝ", "f✝ g : ℕ → Bool", "n : ℕ", "f : ℕ → Bool", "h1 : 0 ≤ c", "h2 : c < 1"], "goal": "(bif f 0 then 1 else 0) + c * ∑' (x : ℕ), cantorFunctionAux c (fun n => f (n + 1)) x = (bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "SeminormedAddCommGroup E", "NormedSpace 𝕜 E", "s : Set (WeakDual 𝕜 E)", "hb : Bornology.IsBounded (⇑NormedSpace.Dual.toWeakDual ⁻¹' s)", "hc : IsClosed s"], "goal": "IsClosed (DFunLike.coe '' s)"}}
+{"state": {"context": ["C : Type u₁", "inst✝ : Category.{v₁, u₁} C", "G : Comonad C", "A B : G.Coalgebra", "h : A.A ≅ B.A", "w : autoParam (A.a ≫ G.map h.hom = h.hom ≫ B.a) _auto✝"], "goal": "B.a ≫ G.map h.inv = h.inv ≫ A.a"}}
+{"state": {"context": ["α : Type u_1", "BEq α", "LawfulBEq α", "l₁ l₂ : List α"], "goal": "l₁.diff l₂ ⊆ l₁"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_4", "LinearOrderedField 𝕜", "AddCommGroup E", "LinearOrderedAddCommGroup β", "Module 𝕜 E", "Module 𝕜 β", "OrderedSMul 𝕜 β", "s : Set E", "f : E → β", "hf : ConcaveOn 𝕜 ((convexHull 𝕜) s) f", "x : E", "hx : x ∈ (convexHull 𝕜) s"], "goal": "∃ y ∈ s, f y ≤ f x"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Zero M", "r : R"], "goal": "(TrivSqZeroExt.inl r).snd = 0"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{u_4, u_2} D", "G : C ⥤ D", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "A : Type w", "inst✝² : Category.{w', w} A", "inst✝¹ : ∀ (X : Dᵒᵖ), Limits.HasLimitsOfShape (StructuredArrow X G.op) A", "inst✝ : IsDenseSubsite J K G", "Y : Sheaf J A", "U : C", "X : A"], "goal": "IsIso ((yoneda.map ((G.op.ranCounit.app Y.val).app (op U))).app (op X))"}}
+{"state": {"context": ["a b : ℝ", "ha : a ≠ 0", "hc : ↑↑⟨!![a, b; 0, a⁻¹], ⋯⟩ 1 0 = 0", "z : ℂ", "hz : 0 < z.im"], "goal": "↑(⟨!![a, b; 0, a⁻¹], ⋯⟩ • ⟨z, hz⟩) = ↑(((fun x => b * a +ᵥ x) ∘ fun x => ⟨a * a, ⋯⟩ • x) ⟨z, hz⟩)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "ι : Type u_3", "hm : MeasurableSpace α", "μ : Measure α", "inst✝⁴ : TopologicalSpace α", "inst✝³ : BorelSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "g : α → E", "l : Filter ι", "x₀ : α", "s t✝ : Set α", "φ : ι → α → ℝ", "a : E", "inst✝ : CompleteSpace E", "t : Set α", "ht : MeasurableSet t", "h'ts : t ∈ 𝓝 x₀", "h't : μ t ≠ ⊤", "hnφ : ∀ᶠ (i : ι) in l, ∀ (x : α), 0 ≤ φ i x", "hlφ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l uᶜ", "hiφ : Tendsto (fun i => ∫ (x : α) in t, φ i x ∂μ) l (𝓝 1)", "h'iφ : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (φ i) μ", "hmg : Integrable g μ", "hcg : Tendsto g (𝓝 x₀) (𝓝 a)"], "goal": "Tendsto (fun i => ∫ (x : α) in univ, φ i x • g x ∂μ) l (𝓝 a)"}}
+{"state": {"context": ["A : Type u_5", "a b : ℝ", "NormedRing A", "NormedAlgebra ℝ A", "CompleteSpace A", "u v u' v' : ℝ → A", "hu : ∀ x ∈ [[a, b]], HasDerivWithinAt u (u' x) [[a, b]] x", "hv : ∀ x ∈ [[a, b]], HasDerivWithinAt v (v' x) [[a, b]] x", "hu' : IntervalIntegrable u' MeasureTheory.volume a b", "hv' : IntervalIntegrable v' MeasureTheory.volume a b"], "goal": "∫ (x : ℝ) in a..b, u x * v' x = u b * v b - u a * v a - ∫ (x : ℝ) in a..b, u' x * v x"}}
+{"state": {"context": ["a b : ℝ", "f f' g : ℝ → ℝ", "hf : ContinuousOn f [[a, b]]", "hff' : ∀ x ∈ Set.Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Set.Ioi x) x", "hg_cont : ContinuousOn g (f '' Set.Ioo (min a b) (max a b))", "hg1 : MeasureTheory.IntegrableOn g (f '' [[a, b]]) MeasureTheory.volume", "hg2 : MeasureTheory.IntegrableOn (fun x => (g ∘ f) x * f' x) [[a, b]] MeasureTheory.volume"], "goal": "∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u"}}
+{"state": {"context": ["X : Compactum", "A B : Set X.A"], "goal": "Compactum.basic (A ∩ B) = Compactum.basic A ∩ Compactum.basic B"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "inst✝¹¹ : Field F", "inst✝¹⁰ : Field K", "inst✝⁹ : Algebra F K", "E : Type u_3", "inst✝⁸ : Field E", "inst✝⁷ : Algebra F E", "inst✝⁶ : Algebra K E", "inst✝⁵ : IsScalarTower F K E", "E' : Type u_4", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Algebra F E", "inst✝¹ : Field E'", "inst✝ : Algebra F E'", "f : E ≃ₐ[F] E'", "x : E'", "h : IsIntegral F (f.symm x) ∧ Splits (algebraMap F E) (minpoly F (f.symm x))"], "goal": "IsIntegral F (f (f.symm x)) ∧ Splits ((↑↑f).comp (algebraMap F E)) (minpoly F (f.symm x))"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M"], "goal": "Submodule.span R (Set.range fun x => (ιMulti R x.fst) x.snd) = ⊤"}}
+{"state": {"context": ["α : Type u", "inst✝ : Ring α", "x : α", "m n : ℕ", "hmn : m ≤ n"], "goal": "(∑ i ∈ Ico m n, x ^ i) * (1 - x) = x ^ m - x ^ n"}}
+{"state": {"context": ["α : Type u_1", "δ : α → Type u_7", "(i : α) → Preorder (δ i)", "s : Set α", "(j : α) → Decidable (j ∈ s)", "f₁ f₂ g : (i : α) → δ i", "h₁ : ∀ i ∈ s, g i ≤ f₁ i", "h₂ : ∀ i ∉ s, g i ≤ f₂ i"], "goal": "g ≤ s.piecewise f₁ f₂"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "ConditionallyCompleteLinearOrderBot α", "f : �� → α"], "goal": "⨆ x, ↑(f x) = ⊤ ↔ ¬BddAbove (Set.range f)"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a", "n : ℕ", "IH : n < a ^ n", "this : a ^ n + a ^ n ≤ a ^ n * a"], "goal": "n + 1 < a ^ n + a ^ n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → α", "n✝ n : ℕ", "ihn : id^[n] = id"], "goal": "id^[n.succ] = id"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X : QuadraticModuleCat R"], "goal": "(CategoryTheory.Iso.refl X).toIsometryEquiv = QuadraticMap.IsometryEquiv.refl X.form"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "Mₗ : Type u_9", "Nₗ : Type u_10", "Pₗ : Type u_11", "Qₗ : Type u_12", "Qₗ' : Type u_13", "AddCommMonoid Mₗ", "AddCommMonoid Nₗ", "AddCommMonoid Pₗ", "AddCommMonoid Qₗ", "AddCommMonoid Qₗ'", "Module R Mₗ", "Module R Nₗ", "Module R Pₗ", "Module R Qₗ", "Module R Qₗ'", "f₁ f₂ : Mₗ →ₗ[R] Nₗ →ₗ[R] Pₗ", "g : Qₗ →ₗ[R] Mₗ", "g' : Qₗ' →ₗ[R] Nₗ", "hₗ : Function.Surjective ⇑g", "hᵣ : Function.Surjective ⇑g'"], "goal": "f₁.compl₁₂ g g' = f₂.compl₁₂ g g' ↔ f₁ = f₂"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "Field K", "Field L", "Field L'", "Algebra K L", "Algebra K L'", "f : L →ₐ[K] L'", "y : L'"], "goal": "y ∈ f.fieldRange ↔ ∃ x, f x = y"}}
+{"state": {"context": ["l✝ : List ℕ", "a✝ m n a : ℕ", "l : List ℕ", "hl hl' : (a :: l).IsZeckendorfRep"], "goal": "(map fib (a :: l)).sum.zeckendorf = a :: l"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : WithBot α", "b : α"], "goal": "a ≤ ↑(WithBot.unbot' b a)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "DecidableRel fun x x_1 => x ≤ x_1", "x : α", "t : Ordnode α", "a₁ : WithBot α", "a₂ : WithTop α", "h : Ordnode.Valid' a₁ t a₂"], "goal": "Ordnode.Valid' a₁ (Ordnode.erase x t) a₂ ∧ Ordnode.Raised (Ordnode.erase x t).size t.size"}}
+{"state": {"context": ["o : Ordinal.{u_1}", "ho : o.card.ord = o", "ho' : ω ≤ o"], "goal": "∃ a, (aleph a).ord = o"}}
+{"state": {"context": ["s m n step : Nat"], "goal": "List.range' s m step ++ List.range' (s + step * m) n step = List.range' s (n + m) step"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "self : f.NeBot"], "goal": "f ≠ ⊥"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{?u.156646, u_2} D", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "inst✝ : HasHomotopyCofiber φ", "K : CochainComplex C ℤ", "α : Cochain F K (-1)", "β : G ⟶ K", "eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)", "p q : ℤ", "hpq : p + 1 = q"], "goal": "(desc φ α β eq).f p = (↑(fst φ)).v p q hpq ≫ α.v q p ⋯ + (snd φ).v p p ⋯ ≫ β.f p"}}
+{"state": {"context": ["b x y : ℝ", "hb : 1 < b", "hx : 0 < x", "hxy : x < y"], "goal": "log x < log y"}}
+{"state": {"context": [], "goal": "I ^ 4 = 1"}}
+{"state": {"context": ["ι✝ : Type u", "s✝ : Finset ι✝", "ι : Type u_1", "s : Finset ι", "w z : ι → ℝ", "hw : ∀ i ∈ s, 0 ≤ w i", "hw' : 0 < ∑ i ∈ s, w i", "hz : ∀ i ∈ s, 0 ≤ z i"], "goal": "(∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / ∑ i ∈ s, w i"}}
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+{"state": {"context": ["α : Type u_1", "Group α", "s : Subgroup α", "a : α", "ha : a ∈ s"], "goal": "MulAction.orbit (↥s) a = ↑s"}}
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+{"state": {"context": ["n : ℕ"], "goal": "0 < n.size ↔ 0 < n"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "AddCommSemigroup α", "a b : α"], "goal": "max a b + min a b = a + b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "s : Set α", "h_inf : s.Infinite", "h_fin : (f '' s).Finite"], "goal": "¬Set.InjOn f s"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "LE α", "a : α", "f : ι → UpperSet α"], "goal": "a ∈ ⨆ i, f i ↔ ∀ (i : ι), a ∈ f i"}}
+{"state": {"context": ["x y : ℂ", "hx₀ : x ≠ 0", "hy₀ : y ≠ 0"], "goal": "log (x * y) = log x + log y ↔ x.arg + y.arg ∈ Set.Ioc (-π) π"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_7", "ι' : Type u_8", "S : Type u_12", "Semiring S", "Module S N", "SMulCommClass R S N", "σ : ι ≃ ι'"], "goal": "↑(AlternatingMap.domDomCongrₗ S σ) = AlternatingMap.domDomCongrEquiv σ"}}
+{"state": {"context": ["a b m n✝ p✝ d n : ℕ", "hd : d ≠ 0", "hdn : d ≤ n", "hd_lt_n : d < n", "h1 : n / d ≠ 0", "h : (n / d).factorization = n.factorization - d.factorization", "h2 : ¬n ≤ n / d * d", "p : ℕ", "hp : (n / d * d).factorization p < n.factorization p"], "goal": "False"}}
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+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "X : CategoryTheory.Functor C D", "P : CategoryTheory.Idempotents.Karoubi C"], "goal": "(((CategoryTheory.Idempotents.functorExtension₂ C D).obj X).obj P).p = X.map P.p"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I : Ideal R", "M : Type u", "AddCommGroup M", "Module R M", "IsNoetherianRing R", "ι : Type", "Fintype ι", "f : (ι → R) →ₗ[R] M", "hf : Function.Surjective ⇑f"], "goal": "Function.Exact ⇑(AdicCompletion.map I (LinearMap.ker f).subtype) ⇑(AdicCompletion.map I f)"}}
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+{"state": {"context": ["P : Prop", "Q : P → Prop", "R : ¬P → Prop", "Decidable P"], "goal": "dite P Q R ↔ (∃ p, Q p) ∨ ∃ p, R p"}}
+{"state": {"context": ["n k p : ℕ", "Fact (Nat.Prime p)"], "goal": "↑(n.choose k) ≡ ↑((n % p).choose (k % p)) * ↑((n / p).choose (k / p)) [ZMOD ↑p]"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w", "CommRing R", "LieRing L", "LieAlgebra R L", "LieRing L'", "LieAlgebra R L'", "self : L ≃ₗ⁅R⁆ L'"], "goal": "Function.LeftInverse self.invFun (↑self.toLieHom).toFun"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "w z : ι → ℝ", "hw : ∀ i ∈ s, 0 ≤ w i", "hw' : ∑ i ∈ s, w i = 1", "hz : ∀ i ∈ s, 0 ≤ z i", "p : ℝ", "hp : 1 ≤ p", "this : 0 < p"], "goal": "0 ≤ ∑ i ∈ s, w i * z i"}}
+{"state": {"context": ["α : Type u_1", "G H : SimpleGraph α", "s : Set α", "n : ℕ", "hGH : G ≤ H", "hH : H.CliqueFreeOn s n"], "goal": "G.CliqueFreeOn s n"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p q : R[X]"], "goal": "(p * q).natDegree ≤ p.natDegree + q.natDegree"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeSup α", "OrderBot α", "s₁ s₂ : Finset β", "f g : β → α", "hs : s₁ = s₂", "hfg : ∀ a ∈ s₂, f a = g a"], "goal": "s₁.sup f = s₂.sup g"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "self : { x // x ∈ Set.univ }"], "goal": "(AffineSubspace.topEquiv k V P) self = ↑self"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁹ : LinearOrder ι", "inst✝⁸ : MeasurableSpace ι", "inst✝⁷ : TopologicalSpace ι", "inst✝⁶ : OrderTopology ι", "inst✝⁵ : SecondCountableTopology ι", "inst✝⁴ : BorelSpace ι", "inst✝³ : TopologicalSpace β", "u : ι → Ω → β", "τ : Ω → ι", "f : Filtration ι m", "inst✝² : MetrizableSpace β", "inst✝¹ : MeasurableSpace β", "inst✝ : BorelSpace β", "hf_prog : ProgMeasurable f u", "hτ : IsStoppingTime f τ", "h_str_meas : ∀ (i : ι), StronglyMeasurable (stoppedValue u fun ω => min (τ ω) i)", "t : Set β", "ht : MeasurableSet t", "i : ι", "ω : Ω"], "goal": "τ ω ≤ i → (u (τ ω) ω ∈ t ↔ u (min (τ ω) i) ω ∈ t)"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n : ℕ", "inst✝ : Semiring R", "p✝ q : R[X]", "h : 0 = 1", "p : R[X]"], "goal": "p = 0"}}
+{"state": {"context": ["S : Type v", "T : Type w", "CommRing T", "CommRing S", "IsDomain S", "Algebra T S", "p : Polynomial T", "n : ℕ"], "goal": "(p ^ n).aroots S = n • p.aroots S"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "β : Type u_2", "Preorder β", "x : α", "s : Set α", "y : β", "Zero β", "hs : IsOpen s", "hy : 0 ≤ y"], "goal": "LowerSemicontinuousAt (s.indicator fun _x => y) x"}}
+{"state": {"context": ["α✝ : Type u", "β✝ : Type v", "X : Type u_1", "ι : Type u_2", "inst✝² : PseudoMetricSpace α✝", "γ : Type w", "inst✝¹ : MetricSpace γ", "α : Type u", "inst✝ : MetricSpace α", "H : ∀ ε > 0, ∃ β x F, ∀ (x y : α), F x = F y → dist x y ≤ ε", "ε : ℝ", "ε0 : ε > 0", "β : Type u_3", "fβ : Encodable β", "F : α → β", "hF : ∀ (x y : α), F x = F y → dist x y ≤ ε", "Finv : ↑(range F) → α := rangeSplitting F", "x : α"], "goal": "∃ y ∈ range Finv, dist x y ≤ ε"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "inst✝⁴ : Semiring α", "inst✝³ : PartialOrder α", "a b c d : α", "inst✝² : ZeroLEOneClass α", "inst✝¹ : PosMulStrictMono α", "H : 0 < a", "inst✝ : Nontrivial α"], "goal": "0 < 1"}}
+{"state": {"context": ["β : Type u_4", "OrderedAddCommGroup β"], "goal": "Filter.map Neg.neg Filter.atBot = Filter.atTop"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "Semiring R", "TopologicalSpace F", "TopologicalSpace B", "TopologicalSpace (Bundle.TotalSpace F E)", "AddCommMonoid F", "Module R F", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module R (E x)", "e e' : Trivialization F Bundle.TotalSpace.proj", "Trivialization.IsLinear R e", "Trivialization.IsLinear R e'", "b : B", "hb : b ∈ e.baseSet ∩ e'.baseSet", "y : F"], "goal": "(Trivialization.coordChangeL R e e' b) y = (↑e' { proj := b, snd := e.symm b y }).2"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "J : CategoryTheory.GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : CategoryTheory.Sheaf J RingCat", "α : R₀ ⟶ R.val", "CategoryTheory.Presheaf.IsLocallyInjective J α", "CategoryTheory.Presheaf.IsLocallySurjective J α", "M₀ : PresheafOfModules R₀", "A : CategoryTheory.Sheaf J AddCommGrp", "φ : M₀.presheaf ⟶ A.val", "CategoryTheory.Presheaf.IsLocallyInjective J φ", "CategoryTheory.Presheaf.IsLocallySurjective J φ", "X : Cᵒᵖ", "r : ↑(R.val.obj X)"], "goal": "PresheafOfModules.Sheafify.smul α φ r 0 = 0"}}
+{"state": {"context": ["G : Type u_1", "Group G", "s : Set G", "hs : IsNormalSubgroup s", "a b : G", "hab : a * b ∈ s"], "goal": "b * a ∈ s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "A B : Subalgebra R S", "inst✝³ : Module.Free R ↥A", "inst✝² : Module.Free R ↥B", "inst✝¹ : Module.Free ↥A ↥(Algebra.adjoin ↥A ↑B)", "inst✝ : Module.Free ↥B ↥(Algebra.adjoin ↥B ↑A)", "h✝ : Nontrivial R"], "goal": "Module.rank R ↥(A ⊔ B) = Module.rank R ↥A * Module.rank ↥A ↥(Algebra.adjoin ↥A ↑B)"}}
+{"state": {"context": ["α : Type u_2", "s : Set α", "t : Set ↑s"], "goal": "𝓟 t = Filter.comap Subtype.val (𝓟 (Subtype.val '' t))"}}
+{"state": {"context": ["E : ℕ → Type u_1", "x y : (n : ℕ) → E n", "h : x ≠ y"], "goal": "x (firstDiff x y) ≠ y (firstDiff x y)"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G", "a : G"], "goal": "a - a = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n : ℕ", "v : Fin n.succ → R"], "goal": "(Fin.cons (fun j => v 0 ^ ↑j) fun i => Fin.cons 1 fun j => Fin.tail v i * vandermonde (Fin.tail v) i j) = Fin.cons (fun j => v 0 ^ ↑j) fun i => Fin.cons 1 fun j => v i.succ * vandermonde (Fin.tail v) i j"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "Zero M", "a a' : α", "b : M"], "goal": "Finsupp.single a ((Finsupp.single a' b) a) = (Finsupp.single a' (Finsupp.single a' b)) a"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst��³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : 𝓢(E, F)", "k : ℕ", "d : ℝ", "left✝ : 0 < d", "hd' : ∀ (x : E), ‖x‖ ^ k * ‖f x‖ ≤ d", "x : E", "hx : x ≠ 0"], "goal": "0 < ‖x‖ ^ ↑k"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "f : β → α → β", "H : RightCommutative f", "x : β", "q : α → Prop", "p : β → Prop", "s : Multiset α", "hpqf : ∀ (a : α) (b : β), q a → p b → p (f b a)", "px : p x", "q_s : ∀ a ∈ s, q a"], "goal": "p (foldr (fun x y => f y x) ⋯ x s)"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G", "a : G", "m n : ℤ"], "goal": "(m + n) • a = m • a + n • a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t u : Set α", "f : α → β", "p q : Prop"], "goal": "q ∈ {p}ᶜ ↔ q ∈ {¬p}"}}
+{"state": {"context": ["K : Type u_1", "R✝ : Type u_2", "R : Type u_3", "inst✝³ : CommRing R", "inst✝² : IsDomain R", "p : ℕ", "inst✝¹ : NeZero p", "inst✝ : CharP R p", "x : ℤ"], "goal": "∃ a b, ↑a ^ 2 + ↑b ^ 2 = ↑x"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "v : ℝ → E → E", "K : ℝ≥0", "f g : ℝ → E", "a b : ℝ", "hv : ∀ (t : ℝ), LipschitzWith K (v t)", "hf : ContinuousOn f (Set.Icc a b)", "hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (v t (f t)) (Set.Ici t) t", "hg : ContinuousOn g (Set.Icc a b)", "hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (v t (g t)) (Set.Ici t) t", "ha : f a = g a"], "goal": "Set.EqOn f g (Set.Icc a b)"}}
+{"state": {"context": ["K : Type u", "inst✝³ : Field K", "V : Type v", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)", "j : ↑(Basis.ofVectorSpaceIndex K V)"], "goal": "∑ x : ↑(Basis.ofVectorSpaceIndex K V), (Basis.ofVectorSpace K V) x ⊗ₜ[K] ((Basis.ofVectorSpace K V).coord x) ((Basis.ofVectorSpace K V) j) = (Basis.ofVectorSpace K V) j ⊗ₜ[K] 1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.HasShift C ℤ", "J : Type u_1", "T : J → CategoryTheory.Pretriangulated.Triangle C", "CategoryTheory.Limits.HasProduct fun j => (T j).obj₁", "CategoryTheory.Limits.HasProduct fun j => (T j).obj₂", "CategoryTheory.Limits.HasProduct fun j => (T j).obj₃", "CategoryTheory.Limits.HasProduct fun j => (CategoryTheory.shiftFunctor C 1).obj (T j).obj₁", "T' : CategoryTheory.Pretriangulated.Triangle C", "φ : (j : J) → T' ⟶ T j"], "goal": "(CategoryTheory.Pretriangulated.productTriangle.lift T φ).hom₁ = CategoryTheory.Limits.Pi.lift fun j => (φ j).hom₁"}}
+{"state": {"context": ["K : Type u_1", "Field K", "nf : NumberField K"], "goal": "¬IsField (NumberField.RingOfIntegers K)"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : TopCat", "F : Presheaf C X", "ι : Type w", "U : ι → Opens ↑X", "Y : Opens ↑X", "hY : Y = iSup U"], "goal": "({ hom := F.map (eqToHom ⋯), w := ⋯ } ≫ { hom := F.map (eqToHom ⋯), w := ⋯ }).hom = (𝟙 (Functor.mapCone F (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op)).hom"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ", "f : ℝ → ℝ", "μ ν : Measure ℝ", "inst✝ : IsLocallyFiniteMeasure μ", "c d s t : ℝ", "ht : 0 < t", "h : IntegrableOn (fun x => x ^ s) (Set.Ioo 0 t) volume"], "goal": "-1 < s"}}
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+{"state": {"context": ["R : Type u_1", "M : Type u_2", "ι : Type u_3", "inst✝³ : CommRing R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "e : ι → M", "ε : ι → Dual R M", "inst✝ : DecidableEq ι", "h : DualBases e ε", "H : Set ι", "i : ι", "l : ι →₀ R", "supp_l : l ∈ Finsupp.supported R R H", "hmem : (Finsupp.total ι M R e) l ∈ Submodule.span R (e '' H)", "hi : (ε i) ((Finsupp.total ι M R e) l) ≠ 0"], "goal": "i ∈ H"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ S₃ : CategoryTheory.ShortComplex C", "φ₁₂ : S₁ ⟶ S₂", "φ₂₃ : S₂ ⟶ S₃"], "goal": "(CategoryTheory.CategoryStruct.comp φ₁₂ φ₂₃).τ₁ = CategoryTheory.CategoryStruct.comp φ₁₂.τ₁ φ₂₃.τ₁"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "n : ℕ", "a : R", "φ : PowerSeries R"], "goal": "(PowerSeries.coeff R n) (PowerSeries.inv.aux a φ) =\n  if n = 0 then a\n  else\n    -a *\n      ∑ x ∈ Finset.antidiagonal n,\n        if x.2 < n then (PowerSeries.coeff R x.1) φ * (PowerSeries.coeff R x.2) (PowerSeries.inv.aux a φ) else 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_4", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "H : Type u_5", "TopologicalSpace H", "G : Type u_7", "TopologicalSpace G", "I : ModelWithCorners 𝕜 E H", "J : ModelWithCorners 𝕜 F G", "M : Type u_9", "TopologicalSpace M", "ChartedSpace H M", "N : Type u_11", "TopologicalSpace N", "ChartedSpace G N", "n : ℕ∞", "h : M ≃ₘ^n⟮I, J⟯ N", "x : N"], "goal": "h (h.symm x) = x"}}
+{"state": {"context": ["X : Type u_1", "X' : Type u_2", "Y : Type u_3", "Y' : Type u_4", "Z : Type u_5", "Z' : Type u_6", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : TopologicalSpace X'", "inst✝⁴ : TopologicalSpace Y", "inst✝³ : TopologicalSpace Y'", "inst✝² : TopologicalSpace Z", "inst✝¹ : TopologicalSpace Z'", "f : X → Y", "h : OpenEmbedding f", "inst✝ : Nonempty X", "x : Y", "hx : x ∈ range f"], "goal": "f (↑(toPartialHomeomorph f h).symm x) = x"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ"], "goal": "(fun t => inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t => ‖t‖ ^ 2"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : CommSemiring R", "A : Type u", "inst✝⁷ : TopologicalSpace A", "inst✝⁶ : Semiring A", "inst✝⁵ : Algebra R A", "inst✝⁴ : TopologicalSemiring A", "s : Subalgebra R A", "B : Type u_2", "inst✝³ : TopologicalSpace B", "inst✝² : Ring B", "inst✝¹ : TopologicalRing B", "inst✝ : Algebra R B", "f : B →ₐ[R] A", "f' : B ≃ₜ A", "w : ⇑f = ⇑f'"], "goal": "↑(comap f s.topologicalClosure) = ↑(comap f s).topologicalClosure"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝⁸ : CommRing R", "inst✝⁷ : CommRing S", "inst✝⁶ : Algebra R S", "P✝ : Generators R S", "T✝ : Type ?u.93774", "inst✝⁵ : CommRing T✝", "inst✝⁴ : Algebra R T✝", "inst✝³ : Algebra S T✝", "inst✝² : IsScalarTower R S T✝", "T : Type ?u.94985", "inst✝¹ : CommRing T", "inst✝ : Algebra R T", "P : Generators R S", "x : T ⊗[R] S"], "goal": "∃ a, (aeval fun x => 1 ⊗ₜ[R] P.val x) a = x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C", "S : CategoryTheory.ShortComplex.SnakeInput C"], "goal": "CategoryTheory.ShortComplex.SnakeInput.functorL₁'.obj S = S.L₁'"}}
+{"state": {"context": ["z : ℂ", "hz : ‖z‖ < 1"], "goal": "HasSum (fun n => (-1) ^ (n + 1) * z ^ n / ↑n) (Complex.log (1 + z))"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "h : 1 < Module.rank ℝ E", "x : E", "r : ℝ"], "goal": "IsPreconnected (Metric.sphere x r)"}}
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+{"state": {"context": ["ι : Type u_3", "π : ι → Type u_4", "(i : ι) → Bornology (π i)", "s : Set ((i : ι) → π i)", "hs : Bornology.IsBounded s", "i : ι"], "goal": "Bornology.IsBounded (Function.eval i '' s)"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "𝒢 : Filtration ℕ m0", "f : ℕ → Ω → ℝ", "τ π : Ω → ℕ", "inst✝ : IsFiniteMeasure μ", "hsub : Submartingale f 𝒢 μ", "ε : ℝ≥0", "n : ℕ", "hn : Set.Icc 0 n = {k | k ≤ n}", "this : ∀ (ω : Ω), (↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω) → ↑ε ≤ stoppedValue f (hitting f {y | ↑ε ≤ y} 0 n) ω"], "goal": "ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω} ≤ ENNReal.ofReal (∫ (ω : Ω) in {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k => f k ω}, stoppedValue f (hitting f {y | ↑ε ≤ y} 0 n) ω ∂μ)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "AddGroup α", "SubtractionMonoid β", "FunLike F α β", "AddMonoidHomClass F α β", "m : F", "s t : Set α"], "goal": "⇑m '' (s - t) = ⇑m '' s - ⇑m '' t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝ : SeminormedRing α", "l : List α", "hl : l ≠ []"], "goal": "(map norm l).prod = (fun a => ↑a) (map nnnorm l).prod"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "M : Type u_4", "CommRing R", "Field K", "Field L", "CommRing M", "Algebra R K", "Algebra R M", "Algebra K M", "IsScalarTower R K M", "x : M", "int : IsIntegral R x", "f : K →+* L", "h : Polynomial.Splits (f.comp (algebraMap R K)) (minpoly R x)"], "goal": "Polynomial.Splits f (minpoly K x)"}}
+{"state": {"context": ["α : Type u_2", "Ring α", "a b : α", "ha : Odd a", "hb : Odd b"], "goal": "Even (a - b)"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommRing R", "S : Type v", "inst✝¹ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝ : IsDedekindDomain S", "hp0 : map f p ≠ ⊥", "hP : P.IsPrime", "le : map f p ≤ P", "hP0 : P ≠ ⊥", "hPirr : Irreducible P"], "goal": "Multiset.count P (normalizedFactors (map f p)) ≠ 0"}}
+{"state": {"context": ["α : Type u", "l l' : List α", "β : Type u_1", "l₁ : List α", "f : α → β", "n : ℕ"], "goal": "List.map f l₁ ~r List.map f (l₁.rotate n)"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y z : P", "h : Sbtw R x y z", "hx : x ∉ s", "hy : y ∈ s"], "goal": "s.SOppSide x z"}}
+{"state": {"context": ["R : Type u_1", "M M₁ M₂ M₃ : Type u", "M' : Type v", "inst✝¹² : Ring R", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : AddCommGroup M₁", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : AddCommGroup M₃", "inst✝⁷ : AddCommGroup M'", "inst✝⁶ : Module R M", "inst✝⁵ : Module R M₁", "inst✝⁴ : Module R M₂", "inst✝³ : Module R M₃", "inst✝² : Module R M'", "inst✝¹ : HasRankNullity.{u, u_1} R", "inst✝ : StrongRankCondition R", "n : ℕ", "v : Fin n → M", "hv : LinearIndependent R v", "h : ↑n < Module.rank R M", "x : M", "hx : LinearIndependent R (Fin.cons x v)"], "goal": "∃ x, LinearIndependent R fun i => Fin.cons x v ((finRotate (n + 1)) i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq α", "DecidableEq β", "f : α → β", "s : Multiset α"], "goal": "(Multiset.map f s.dedup).dedup = (Multiset.map f s).dedup"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "A : C", "hA : IsSubterminal A", "inst✝ : HasBinaryProduct A A"], "goal": "prod.fst ≫ diag A = 𝟙 (A ⨯ A)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝³ : DivisionRing k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "s : Set P", "p₁ : P", "hp₁ : p₁ ∈ s", "v : V", "h : ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁", "r₂ : k", "hp₂ : r₂ • v +ᵥ p₁ ∈ s", "hp₁p₂ : p₁ ≠ r₂ • v +ᵥ p₁", "r₃ : k", "hp₃ : r₃ • v +ᵥ p₁ ∈ s", "h₂ : r₂ ≠ 0"], "goal": "(r₃ / r₂) • (r₂ • v +ᵥ p₁ -ᵥ p₁) = r₃ • v"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "AddZeroClass α", "s : Finset α"], "goal": "Finset.coeAddMonoidHom s = ↑s"}}
+{"state": {"context": ["α : Type u_1"], "goal": "none.isNone = true"}}
+{"state": {"context": ["G : Type v", "X : Type w", "PseudoMetricSpace X", "Group G", "MulAction G X", "IsometricSMul G X", "c : G", "x : X", "r : ℝ"], "goal": "(fun x => c • x) ⁻¹' Metric.ball x r = Metric.ball (c⁻¹ • x) r"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "p : α → Prop", "inst✝ : DecidablePred p", "s t : Multiset α", "h : s ≤ t"], "goal": "countP p s ≤ countP p t"}}
+{"state": {"context": ["G : Type u_1", "TopologicalSpace G", "Group G", "TopologicalGroup G", "MeasurableSpace G", "BorelSpace G", "LocallyCompactSpace G", "μ' μ : MeasureTheory.Measure G", "μ.IsHaarMeasure", "MeasureTheory.IsFiniteMeasureOnCompacts μ'", "μ'.IsMulLeftInvariant", "μ.InnerRegularCompactLTTop", "s : Set G", "hs : MeasurableSet s", "h's : IsCompact (closure s)"], "goal": "μ'.haarScalarFactor μ • μ s ≤ μ' s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "s : Set α", "SemilatticeSup α", "SemilatticeInf β", "h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊔ y) = f x ⊓ f y"], "goal": "AntitoneOn f s"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝⁵ : Ring R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "ι✝ : Type w", "ι'✝ : Type w'", "inst✝² : StrongRankCondition R", "ι : Type u_1", "inst✝¹ : Fintype ι", "b : Basis ι R M", "ι' : Type u_2", "inst✝ : Fintype ι'", "v : ι' → M", "hv : LinearIndependent R v"], "goal": "Fintype.card ι' ≤ Fintype.card ι"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "v : Fin 2 → R", "hab : IsCoprime (v 0) (v 1)", "A : SL(2, R)"], "goal": "IsCoprime ((↑A *ᵥ v) 0) ((↑A *ᵥ v) 1)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a a' : α", "b' : β a'", "s : AList β"], "goal": "a ∈ AList.insert a' b' s ↔ a = a' ∨ a ∈ s"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_3", "E : Type u_4", "F : Type u_5", "RCLike 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "CompleteSpace F", "u : α → ℝ", "NormedSpace 𝕜 F", "f : α → E → F", "f' : α → E → E →L[𝕜] F", "s : Set E", "x₀ x : E", "hu : Summable u", "hs : IsOpen s", "h's : IsPreconnected s", "hf : ∀ (n : α), ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x", "hf' : ∀ (n : α), ∀ x ∈ s, ‖f' n x‖ ≤ u n", "hx₀ : x₀ ∈ s", "hf0 : Summable fun n => f n x₀", "hx : x ∈ s"], "goal": "HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "R : Type u_1", "NonAssocSemiring R", "f : (k : ℕ) → R →+* ZMod (p ^ k)", "f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1", "r s : R"], "goal": "PadicInt.limNthHom f_compat (r * s) = PadicInt.limNthHom f_compat r * PadicInt.limNthHom f_compat s"}}
+{"state": {"context": ["α : Type u", "p q : α → Prop"], "goal": "{a | p a} ⊆ {a | q a} ↔ ∀ (a : α), p a → q a"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "CommMonoid β", "σ : Equiv.Perm α", "s : Finset α", "f : α → β", "hs : {a | σ a ≠ a} ⊆ ↑s"], "goal": "∏ x ∈ s, f (σ x) = ∏ x ∈ s, f x"}}
+{"state": {"context": ["m n : Nat"], "goal": "Int.negSucc m * Int.negSucc n = Int.ofNat (m.succ * n.succ)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "𝕜 : Type u_3", "inst✝⁹ : _root_.RCLike 𝕜", "E : Type u_4", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : InnerProductSpace 𝕜 E", "E' : Type u_5", "inst✝⁶ : NormedAddCommGroup E'", "inst✝⁵ : InnerProductSpace 𝕜 E'", "F : Type u_6", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace ℝ F", "F' : Type u_7", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : InnerProductSpace ℝ F'", "inst✝ : Fintype ι", "v : OrthonormalBasis (Fin 2) ℝ F", "f : F ≃ₗᵢ[ℝ] F'"], "goal": "orthonormalBasisOneI.repr.trans (v.repr.symm.trans f) = (orthonormalBasisOneI.repr.trans v.repr.symm).trans f"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Function.Injective FreeAddMonoid.of"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "h :\n  ∀ (J : Type w) {𝒥 : CategoryTheory.SmallCategory J},\n    CategoryTheory.FinCategory J → CategoryTheory.Limits.HasLimitsOfShape J C"], "goal": "CategoryTheory.Limits.HasFiniteLimits C"}}
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+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type v", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "LocallyCompactSpace 𝕜", "FiniteDimensional 𝕜 E"], "goal": "ProperSpace E"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : CompleteSpace F", "u : α → ℝ", "f : α → β → F", "hu : Summable u", "s : Set β", "hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n", "ε : ℝ", "εpos : ε > 0", "t : Finset α", "ht : ∑' (a : { x // x ∉ t }), u ↑a < ε", "x : β", "hx : x ∈ s"], "goal": "dist (∑' (n : α), f n x) (∑ n ∈ t, f n x) < ε"}}
+{"state": {"context": ["α : Type u_1", "SubtractionCommMonoid α", "a b c : α"], "goal": "a + (b - c) = b + (a - c)"}}
+{"state": {"context": ["x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame", "ih' : ∀ (a : Args), ArgsRel a (Args.P24 x₁ x₂ y) → P124 a", "z w : PGame", "h : w.IsOption y"], "goal": "z.IsOption x₁ → P2 z x₂ w"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedCommGroupWithZero α", "a b c d : α", "m n : ℕ", "h : a < c * b"], "goal": "a * b⁻¹ < c"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a : α"], "goal": "IsLUB {a} a"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "_mΩ : MeasurableSpace Ω", "κ : ProbabilityTheory.Kernel α Ω", "ProbabilityTheory.IsMarkovKernel κ", "μ : MeasureTheory.Measure α", "f : ι → Set Ω", "hf : ∀ (i : ι), MeasurableSet (f i)"], "goal": "ProbabilityTheory.Kernel.iIndepSet f κ μ ↔ ProbabilityTheory.Kernel.iIndepSets (fun i => {f i}) κ μ"}}
+{"state": {"context": ["o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}", "a b : Ordinal.{max v u}"], "goal": "o.nfpBFamily f a ≤ b ↔ ∀ (l : List (Quotient.out o).α), List.foldr (o.familyOfBFamily f) a l ≤ b"}}
+{"state": {"context": ["α : Type u", "Sup α", "a b : αᵒᵈ"], "goal": "OrderDual.ofDual (a ⊓ b) = OrderDual.ofDual a ⊔ OrderDual.ofDual b"}}
+{"state": {"context": ["q : ℚ"], "goal": "q = 0 ↔ q.num = 0"}}
+{"state": {"context": ["o : Type u_4", "m' : o → Type u_7", "n' : o → Type u_8", "α : Type u_12", "AddZeroClass α", "M N : Matrix ((i : o) × m' i) ((i : o) × n' i) α"], "goal": "(M + N).blockDiag' = M.blockDiag' + N.blockDiag'"}}
+{"state": {"context": ["J : Type v'", "inst✝⁴ : Category.{u', v'} J", "C : Type u", "inst✝³ : Category.{v, u} C", "K : Type u_1", "inst✝² : Category.{?u.288379, u_1} K", "D : Type u_2", "inst✝¹ : Category.{?u.288386, u_2} D", "n : ℕ", "f : Fin (n + 1) → C", "c₁ : Cofan fun i => f i.succ", "c₂ : BinaryCofan (f 0) c₁.pt", "t₁ : IsUniversalColimit c₁", "t₂ : IsUniversalColimit c₂", "inst✝ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i", "F : Discrete (Fin (n + 1)) ⥤ C", "c : Cocone F", "α : F ⟶ Discrete.functor f", "i : c.pt ⟶ (extendCofan c₁ c₂).pt", "e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i", "hα : NatTrans.Equifibered α", "H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)", "F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk", "this : F = Discrete.functor F'", "H₁ : IsColimit (Cofan.mk (pullback c₂.inr i) fun j => pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)", "t₂' : Nonempty (IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (pullback.snd c₂.inr i)))"], "goal": "Nonempty (IsColimit c)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LocallyFiniteOrder α", "a a₁ a₂ b b₁ b₂ c x : α"], "goal": "Set.uIcc a b = Set.Icc a b ∪ Set.Icc b a"}}
+{"state": {"context": ["x y z : ℝ"], "goal": "∀ (x : ℝ), 0 < cosh x"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "l : Filter α", "p : ι → Prop", "s : ι → Set α", "t : Set α", "hl : l.HasBasis p s"], "goal": "t ∈ l ↔ ∃ i, p i ∧ s i ⊆ t"}}
+{"state": {"context": ["n : ℕ", "hn₀ : n ≠ 0", "hn : Even n"], "goal": "AddSubmonoid.closure (Set.range fun x => x ^ n) = AddSubmonoid.nonneg ℚ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝² : _root_.RCLike 𝕜", "inst✝¹ : AddCommGroup F", "inst✝ : Module 𝕜 F", "c : Core 𝕜 F", "x : F"], "goal": "I * ((starRingEnd 𝕜) ⟪x, x⟫_𝕜 - ⟪x, x⟫_𝕜) / 2 = ↑0"}}
+{"state": {"context": ["f : ℕ →. ℕ"], "goal": "Partrec f ↔ Nat.Partrec f"}}
+{"state": {"context": ["p : Bool → Prop", "b : Bool"], "goal": "(∀ (x : Bool), p x) ↔ p b ∧ p !b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a : α", "f : β → γ → ℝ≥0∞", "hf : Measurable (Function.uncurry f)", "F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)", "h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b", "h_mono : Monotone F", "this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b)", "h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂η (a, b)"], "goal": "∀ (n : ℕ), Measurable fun b => ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : Fintype α", "p : α → Prop", "inst✝¹ : DecidablePred p", "inst✝ : CommMonoid β", "f : (a : α) → p a → β", "g : (a : α) → ¬p a → β"], "goal": "(∏ x : { x // x ∈ filter p univ }, f ↑x ⋯) * ∏ x : { x // x ∈ filter (fun x => ¬p x) univ }, g ↑x ⋯ = (∏ a : { a // p a }, f ↑a ⋯) * ∏ a : { a // ¬p a }, g ↑a ⋯"}}
+{"state": {"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "inst✝⁴ : Preadditive C", "σ τ : Type", "inst✝³ : Finite τ", "S : σ → C", "inst✝² : HasBiproduct S", "T : τ → C", "inst✝¹ : HasBiproduct T", "s : σ", "f : ⨁ S ⟶ ⨁ T", "inst✝ : IsIso f", "val✝ : Fintype τ", "z : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0", "reassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h"], "goal": "𝟙 (S s) = 0"}}
+{"state": {"context": ["α : Type u", "Group α", "LE α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α"], "goal": "(OrderIso.mulRight a).symm = OrderIso.mulRight a⁻¹"}}
+{"state": {"context": ["p : Polynomial ℤ", "hp : p.IsUnitTrinomial", "h : ∀ (z : ℂ), ¬((Polynomial.aeval z) p = 0 ∧ (Polynomial.aeval z) p.mirror = 0)"], "goal": "Irreducible p"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hC : IsClosed C", "hsC : contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1"], "goal": "∀ (i : I), Continuous fun a => if ord I i = o then true else a i"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : NormedLatticeAddCommGroup α", "this : {x | 0 ≤ x} = negPart ⁻¹' {0}"], "goal": "IsClosed {x | 0 ≤ x}"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m m₂ m0 : MeasurableSpace α", "μ : Measure α", "inst✝¹ : TopologicalSpace E", "inst✝ : Zero E", "hm : m ≤ m0", "s : Set α", "f : α → E", "hs_m : MeasurableSet s", "hs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)", "hf : AEStronglyMeasurable' m f μ", "hf_zero : f =ᶠ[ae (μ.restrict sᶜ)] 0", "h_ind_eq : s.indicator (mk f hf) =ᶠ[ae μ] f", "hf_ind : StronglyMeasurable (s.indicator (mk f hf))"], "goal": "StronglyMeasurable (s.indicator (mk f hf))"}}
+{"state": {"context": ["α✝ β α : Type u", "h : #α = ℵ₀"], "goal": "Nonempty (Denumerable α)"}}
+{"state": {"context": ["J K : Type v", "inst✝³ : SmallCategory J", "inst✝² : SmallCategory K", "C : Type u", "inst✝¹ : Category.{v, u} C", "F : J ⥤ K ⥤ C", "G : J × K ⥤ C", "inst✝ : HasColimits C", "j : J", "k : K"], "goal": "((colimit.ι ((curry.obj (Prod.swap K J ⋙ G)).obj k) j ≫ colimit.ι (curry.obj (Prod.swap K J ⋙ G) ⋙ colim) k ≫ (colimitIsoColimitCurryCompColim (Prod.swap K J ⋙ G)).inv) ≫ (HasColimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor ⋙ G))).hom) ≫ (colimitIsoColimitCurryCompColim G).hom = colimit.ι ((curry.obj G).obj j) k ≫ colimit.ι (curry.obj G ⋙ colim) j"}}
+{"state": {"context": ["n : ℕ"], "goal": "(ack 0 n + 1) ^ 2 ≤ ack (0 + 3) n"}}
+{"state": {"context": ["q r : ℚ"], "goal": "q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : Type u_1", "CommSemiring R", "e : α ≃ β", "Semiring β", "Algebra R β", "b : β"], "goal": "(AlgEquiv.symm (Equiv.algEquiv R e)) b = e.symm b"}}
+{"state": {"context": ["z : ℂ", "n✝ n : ℕ", "hz : z ≠ 0"], "goal": "2 * ↑n + 2 = ↑(2 * n + 2)"}}
+{"state": {"context": ["α : Type u", "Fintype α", "p : α → Prop", "DecidablePred p", "h : {x | p x}.Finite"], "goal": "h.toFinset = Finset.filter p Finset.univ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData"], "goal": "(CategoryTheory.ShortComplex.HomologyMapData.zero h₁ h₂).left =\n  CategoryTheory.ShortComplex.LeftHomologyMapData.zero h₁.left h₂.left"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "f : E →L[ℝ] E", "hf : LinearMap.det ↑f ≠ 0", "s : Set E"], "goal": "μ (⇑f ⁻¹' s) = ENNReal.ofReal |(LinearMap.det ↑f)⁻¹| * μ s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_3", "M : Type u_2", "CommRing R", "Field S", "AddCommGroup M", "Algebra R S", "Module R M", "Module S M", "IsScalarTower R S M", "B : LinearMap.BilinForm S M", "N : Submodule R M", "x : M"], "goal": "x ∈ B.dualSubmodule N ↔ ∀ y ∈ N, (B x) y ∈ 1"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "inst✝²⁰ : CommRing A", "inst✝¹⁹ : Field K", "inst✝¹⁸ : CommRing B", "inst✝¹⁷ : Field L", "inst✝¹⁶ : Algebra A K", "inst✝¹⁵ : Algebra B L", "inst✝¹⁴ : Algebra A B", "inst✝¹³ : Algebra K L", "inst✝¹² : Algebra A L", "inst✝¹¹ : IsScalarTower A K L", "inst✝¹⁰ : IsScalarTower A B L", "inst✝⁹ : IsDomain A", "inst✝⁸ : IsFractionRing A K", "inst✝⁷ : IsIntegralClosure B A L", "inst✝⁶ : IsFractionRing B L", "inst✝⁵ : FiniteDimensional K L", "inst✝⁴ : Algebra.IsSeparable K L", "inst✝³ : IsDomain B", "inst✝² : IsIntegrallyClosed A", "I : Submodule B L", "ι : Type u_4", "inst✝¹ : DecidableEq ι", "inst✝ : Fintype ι", "b : Basis ι K L", "hb : ∀ (i : ι), IsIntegral A (b i)", "a x : L", "ha : a ∈ I", "hx : x ∈ Iᵛ", "hinv : IsUnit (traceMatrix K ⇑b).det", "H : (traceMatrix K ⇑b *ᵥ (traceMatrix K ⇑b).cramer fun i => (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • fun i => (Algebra.trace K L) (x * a * b i)", "this : Function.Injective (traceMatrix K ⇑b).mulVec"], "goal": "IsIntegral A (discr K ⇑b • a * x)"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "κ : ι → Sort u_7", "s t : (i : ι) → κ i → Set α", "h : ∀ (i : ι) (j : κ i), s i j ⊆ t i j"], "goal": "⋂ i, ⋂ j, s i j ⊆ ⋂ i, ⋂ j, t i j"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "Nontrivial E"], "goal": "Function.Surjective nnnorm"}}
+{"state": {"context": ["α : Type u", "inst✝ : DivisionRing α", "x : α", "hx1 : x ≠ 1", "hx0 : x ≠ 0", "n : ℕ", "h₁ : x⁻¹ ≠ 1", "h₂ : x⁻¹ - 1 ≠ 0", "h₃ : x - 1 ≠ 0"], "goal": "∑ i ∈ range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "M : Type u_3", "N : Type u_4", "ι : Type u_5", "κ : Type u_6", "CommSemiring R", "AddCommMonoid M", "Module R M", "AddCommMonoid N", "Module R N", "Semiring S", "Algebra R S", "Module S M", "IsScalarTower R S M", "f : ι →₀ M", "g : κ →₀ N", "i : ι", "k : κ"], "goal": "((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "x : α"], "goal": "IsGenericPoint x (closure {x})"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E"], "goal": "dist 1 = norm"}}
+{"state": {"context": ["z : ℂ"], "goal": "Complex.equivRealProd z = (z.re, z.im)"}}
+{"state": {"context": ["ι : Type u_1", "I J₁ J₂ : BoxIntegral.Box ι", "π : BoxIntegral.Prepartition I", "h₁ : J₁ ∈ π", "h₂ : J₂ ∈ π", "hle : J₁ ≤ J₂"], "goal": "J₁ = J₂"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "c : CategoryTheory.Limits.PullbackCone f g"], "goal": "c.op.inl = c.fst.op"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "K : Type u_4", "A : Type u_5", "A' : Type u_6", "A'' : Type u_7", "V : Type u", "V' : Type u_8", "x : ι → A", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing A'", "inst✝³ : CommRing A''", "inst✝² : Algebra R A", "inst✝¹ : Algebra R A'", "inst✝ : Algebra R A''", "a b : R", "h : Injective ⇑(algebraMap R A)", "s : Set A", "hs : AlgebraicIndependent R Subtype.val ∧ ∅ ⊆ s ∧ s ⊆ univ ∧ ∀ (x : Set A), AlgebraicIndependent R Subtype.val → s ⊆ x → x ⊆ univ → x = s", "t : Set A", "ht : AlgebraicIndependent R Subtype.val", "hr : s ≤ t"], "goal": "s = t"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "k : Set G", "p : G → Prop", "x : G", "h : x ∈ AddSubgroup.closure k", "mem : ∀ x ∈ k, p x", "one : p 0", "mul : ∀ (x y : G), p x → p y → p (x + y)", "inv : ∀ (x : G), p x → p (-x)"], "goal": "p x"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "LinearOrder α", "m n : ℕ", "s : Finset α", "hm : s.card = m", "hn : sᶜ.card = n", "i : Fin n"], "goal": "(finSumEquivOfFinset hm hn) (Sum.inr i) = (sᶜ.orderEmbOfFin hn) i"}}
+{"state": {"context": ["L : Language", "ι : Type v", "inst✝⁴ : Preorder ι", "G : ι → Type w", "inst✝³ : (i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "inst✝² : IsDirected ι fun x x_1 => x ≤ x_1", "inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)", "inst✝ : Nonempty ι", "S : L.Substructure (DirectLimit G f)", "S_fg : S.FG", "A : Finset (DirectLimit G f)", "A_closure : (Substructure.closure L).toFun ↑A = S", "i : ι", "y : { x // x ∈ A } → G i", "eq_y : (fun a => ↑a) = fun a => ⟦Structure.Sigma.mk f i (y a)⟧"], "goal": "∃ T, Substructure.map (of L ι G f i).toHom T = S"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "G : SimpleGraph α", "DecidableRel G.Adj", "ε : ℝ", "hG : G.FarFromTriangleFree ε"], "goal": "SimpleGraph.triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(G.cliqueFinset 3).card"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "h : s.encard = 1"], "goal": "∃ x, s = {x}"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝⁶ : AddCommMonoid R✝", "inst✝⁵ : Pow R✝ ℕ", "inst✝⁴ : BinomialRing R✝", "R : Type u_2", "inst✝³ : NonAssocSemiring R", "inst✝² : Pow R ℕ", "inst✝¹ : NatPowAssoc R", "inst✝ : BinomialRing R", "r : R", "k : ℕ"], "goal": "(r + 1 + (↑k).smeval (r + 1)) * (ascPochhammer ℕ k).smeval (r + 1) = k.succ • (ascPochhammer ℕ k).smeval (r + 1) + r * (ascPochhammer ℕ k).smeval (r + 1)"}}
+{"state": {"context": ["K : Type u_13", "K₁ : Type u_14", "K₂ : Type u_15", "V : Type u_16", "V₁ : Type u_17", "V₂ : Type u_18", "Field K", "AddCommGroup V", "Module K V", "Field K₁", "AddCommGroup V₁", "Module K₁ V₁", "Field K₂", "AddCommGroup V₂", "Module K₂ V₂", "I₁ : K₁ →+* K", "I₂ : K₂ →+* K", "B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V", "x : V₁", "y : V₂", "a : K₁", "ha : a ≠ 0"], "goal": "B.IsOrtho x y ↔ B.IsOrtho (a • x) y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "E' : Type u_3", "F : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝²⁰ : RCLike 𝕜", "inst✝¹⁹ : NormedAddCommGroup E", "inst✝¹⁸ : InnerProductSpace 𝕜 E", "inst✝¹⁷ : CompleteSpace E", "inst✝¹⁶ : NormedAddCommGroup E'", "inst✝¹⁵ : InnerProductSpace 𝕜 E'", "inst✝¹⁴ : CompleteSpace E'", "inst✝¹³ : NormedSpace ℝ E'", "inst✝¹² : NormedAddCommGroup F", "inst✝¹¹ : NormedSpace 𝕜 F", "inst✝¹⁰ : NormedAddCommGroup G", "inst✝⁹ : NormedAddCommGroup G'", "inst✝⁸ : NormedSpace ℝ G'", "inst✝⁷ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "E'' : Type u_8", "𝕜' : Type u_9", "inst✝⁶ : RCLike 𝕜'", "inst✝⁵ : NormedAddCommGroup E''", "inst✝⁴ : InnerProductSpace 𝕜' E''", "inst✝³ : CompleteSpace E''", "inst✝² : NormedSpace ℝ E''", "inst✝¹ : NormedSpace ℝ G", "hm✝ hm : m ≤ m0", "inst✝ : SigmaFinite (μ.trim hm)", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "x : G"], "goal": "AEStronglyMeasurable (↑↑(condexpIndSMul hm hs hμs x)) μ"}}
+{"state": {"context": ["n : ℕ", "c : Fin n → ℂ"], "goal": "torusMap c 0 = Function.const (Fin n → ℝ) c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Group α", "MulAction α β", "a : α", "x : β"], "goal": "(MulAction.toPerm a) x = a • x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "ι : Type u_4", "v : ι → E", "hv : Orthonormal 𝕜 v", "l : ι →₀ 𝕜", "i : ι"], "goal": "⟪(Finsupp.total ι E 𝕜 v) l, v i⟫_𝕜 = (starRingEnd 𝕜) (l i)"}}
+{"state": {"context": ["m n : ℕ", "h : m.Coprime n"], "goal": "m.lcm n = m * n"}}
+{"state": {"context": ["c : ℂ", "R : ℝ"], "goal": "Continuous (circleMap c R)"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : Preadditive C", "F : C ⥤ C", "inst✝ : F.Additive", "A₁ A₂ : Algebra F", "α β : A₁ ⟶ A₂"], "goal": "F.map (α.f + β.f) ≫ A₂.str = A₁.str ≫ (α.f + β.f)"}}
+{"state": {"context": ["motive : Nat → Sort u_1", "zero : motive 0", "succ : (n : Nat) → motive n → motive (n + 1)", "n : Nat"], "goal": "Nat.recAux zero succ (n + 1) = succ n (Nat.recAux zero succ n)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "K : Type u_5", "CommRing K", "Algebra R K", "IsFractionRing R K", "I J : Ideal R"], "goal": "IsLocalization.coeSubmodule K I ≤ IsLocalization.coeSubmodule K J ↔ I ≤ J"}}
+{"state": {"context": ["x : Cardinal.{v}", "m : ℕ∞"], "goal": "↑m ≤ Cardinal.lift.{u, v} x ↔ ↑m ≤ x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "s t : Set α", "x : α", "ht : IsLowerSet t"], "goal": "Disjoint (↑(upperClosure s)) t ↔ Disjoint s t"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroMorphisms D", "S : CategoryTheory.ShortComplex C", "h : S.ShortExact", "F : CategoryTheory.Functor C D", "F.PreservesZeroMorphisms", "F.PreservesLeftHomologyOf S", "F.PreservesRightHomologyOf S", "CategoryTheory.Mono (F.map S.f)", "CategoryTheory.Epi (F.map S.g)"], "goal": "(S.map F).ShortExact"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "β : Type u_3", "ι : Type u_4", "inst✝ : TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s s₁ s₂ : Set α", "m : τ → τ", "f₁ f₂ : Filter τ", "hf : Tendsto m f₁ f₂"], "goal": "ω f₁ (fun t x => ϕ (m t) x) s ⊆ ω f₂ ϕ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "β₂ : Type u_3", "γ : Type u_4", "ι : Sort u_5", "ι' : Sort u_6", "κ : ι → Sort u_7", "κ' : ι' → Sort u_8", "inst✝ : CompleteLattice α", "f✝ g s t : ι → α", "a✝ b : α", "p : ι → Prop", "f : (i : ι) → p i → α", "a : α", "h : ∃ i, p i"], "goal": "a ⊔ ⨆ i, ⨆ (h : p i), f i h = ⨆ i, ⨆ (h : p i), a ⊔ f i h"}}
+{"state": {"context": ["A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "L : CategoryTheory.Functor A T", "R₁ R₂ : CategoryTheory.Functor B T", "r r' : R₁ ⟶ R₂", "h : r = r'", "X : CategoryTheory.Comma L R₁"], "goal": "((CategoryTheory.Comma.mapRightEq L r r' h).inv.app X).left = CategoryTheory.CategoryStruct.id X.left"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α"], "goal": "Set.Ioo a b ⊆ Set.Ioi a"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "K₁ K₂ : Compacts G", "h : Disjoint K₁.carrier K₂.carrier", "h₂ : IsClosed K₂.carrier", "U₁ U₂ : Set G", "h1U₁ : IsOpen U₁", "h1U₂ : IsOpen U₂", "h2U₁ : K₁.carrier ⊆ U₁", "h2U₂ : K₂.carrier ⊆ U₂", "hU : Disjoint U₁ U₂", "L₁ : Set G", "h1L₁ : L₁ ∈ 𝓝 1", "V₁ : Set G", "h1V₁ : V₁ ⊆ L₁", "h2V₁ : IsOpen V₁", "h3V₁ : 1 ∈ V₁", "h2L₁ : K₁.carrier * V₁ ⊆ U₁", "L₂ : Set G", "h1L₂ : L₂ ∈ 𝓝 1", "V₂ : Set G", "h1V₂ : V₂ ⊆ L₂", "h2V₂ : IsOpen V₂", "h3V₂ : 1 ∈ V₂", "h2L₂ : K₂.carrier * V₂ ⊆ U₂", "eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)"], "goal": "chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : m → n → Type v", "(i : m) → (j : n) → Neg (α i j)", "M : DMatrix m n α", "i : m", "j : n"], "goal": "(-M) i j = -M i j"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "s : ι → Set α"], "goal": "⋂ i, s i = Set.univ ↔ ∀ (i : ι), s i = Set.univ"}}
+{"state": {"context": ["A : Type u_1", "inst✝ : SeminormedCommGroup A", "a : A", "m n : ℕ", "δ : ℝ", "hn : 0 < n", "han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n"], "goal": "⋃ i, ⋃ (_ : orderOf i = n), ball (a * i) δ = ⋃ x, ⋃ (_ : orderOf x = n), ball x δ"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "k : Type u_5", "S : Type u_6", "S₃ : Type u_7", "T : Type u_8", "M : Type u_9", "M₁ : Type u_10", "M₂ : Type u_11", "M₃ : Type u_12", "N₁ : Type u_13", "N₂ : Type u_14", "N₃ : Type u_15", "ι : Type u_16", "inst✝¹ : AddCommMonoid M", "inst✝ : AddCommMonoid M₂", "f g : M →+ M₂", "h : f.toNatLinearMap = g.toNatLinearMap", "x : M"], "goal": "f x = g x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type w", "inst✝² : MetricSpace X", "inst✝¹ : MetricSpace Y", "Φ✝ : Z → X", "Ψ✝ : Z → Y", "ε✝ : ℝ", "inst✝ : Nonempty Z", "Φ : Z → X", "Ψ : Z → Y", "ε : ℝ", "ε0 : 0 < ε", "x y : Y", "h : glueDist Φ Ψ ε (Sum.inr x) (Sum.inr y) = 0"], "goal": "Sum.inr x = Sum.inr y"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "E : ι → Type wE", "G : Type wG", "Fintype ι", "NontriviallyNormedField 𝕜", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "SeminormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ContinuousMultilinearMap 𝕜 E G", "𝕜' : Type u_1", "NormedField 𝕜'", "NormedSpace 𝕜' G", "SMulCommClass 𝕜 𝕜' G", "c : 𝕜'"], "goal": "‖c • f‖ ≤ ‖c‖ * ‖f‖"}}
+{"state": {"context": ["E : Type u_1", "SeminormedAddCommGroup E", "δ : ℝ", "x : E"], "goal": "-Metric.closedBall x δ = Metric.closedBall (-x) δ"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "a : α"], "goal": "0 ≤ a ^ (-2)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "P : CategoryTheory.MorphismProperty C", "F : C ⥤ D", "X Y : C", "f : X ⟶ Y", "hf : P f"], "goal": "P.map F (F.map f)"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : Abelian C", "inst✝ : HasExt C", "X Y Z T : C", "n : ℕ", "α : Ext X Y n", "m : ℕ", "β₁ β₂ : Ext Y Z m", "p : ℕ", "h : n + m = p", "this : HasDerivedCategory C := HasDerivedCategory.standard C"], "goal": "(α.comp (β₁ + β₂) h).hom = (α.comp β₁ h + α.comp β₂ h).hom"}}
+{"state": {"context": ["R : Type u", "CommRing R", "b c d : R"], "goal": "preNormEDS' b c d 4 = d"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f✝ f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x✝ : 𝕜", "s✝ t✝ : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "f : 𝕜 → F", "s t : Set 𝕜", "h : s ⊆ closure (s ∩ t)", "x : 𝕜", "H : (𝓝[s \\ {x}] x).NeBot", "H' : DifferentiableWithinAt 𝕜 f s x", "I : (𝓝[(s ∩ t) \\ {x}] x).NeBot", "this : Tendsto (slope f x) (𝓝[(s ∩ t) \\ {x}] x) (𝓝 (derivWithin f s x))", "y : 𝕜", "hy : y ∈ (s ∩ t) \\ {x}"], "goal": "(y - x)⁻¹ • (f y - f x) ∈ (Submodule.span 𝕜 (f '' t)).topologicalClosure"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq α", "Fintype β", "f : β → α", "Fintype ↑(Set.range f)"], "goal": "(Set.range f).toFinset = Finset.image f Finset.univ"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝⁴ : FunLike F α β", "inst✝³ : BiheytingAlgebra α", "inst✝² : BiheytingAlgebra β", "inst✝¹ : BiheytingAlgebra γ", "inst✝ : BiheytingAlgebra δ", "f f₁ f₂ : BiheytingHom α β", "g g₁ g₂ : BiheytingHom β γ", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "a : α"], "goal": "g (f₁ a) = g (f₂ a)"}}
+{"state": {"context": ["L : FirstOrder.Language", "ι : Type v", "Preorder ι", "Nonempty ι", "IsDirected ι fun x x_1 => x ≤ x_1", "M : Type u_1", "L.Structure M", "S : ι →o L.Substructure M", "i : ι", "x : ↥(S i)"], "goal": "(FirstOrder.Language.DirectLimit.Equiv_iSup S)\n    ((FirstOrder.Language.DirectLimit.of L ι (fun x => ↥(S x))\n        (fun x x_1 h => FirstOrder.Language.Substructure.inclusion ⋯) i)\n      x) =\n  (FirstOrder.Language.Substructure.inclusion ⋯) x"}}
+{"state": {"context": ["k : Type u_2", "Field k", "φ : PowerSeries k"], "goal": "φ⁻¹ = PowerSeries.inv.aux ((PowerSeries.constantCoeff k) φ)⁻¹ φ"}}
+{"state": {"context": ["μ ν : YoungDiagram"], "goal": "μ = ν.transpose.transpose ↔ μ = ν"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a✝ b : R", "n✝ : ℕ", "inst✝ : CommRing R", "p : R[X]", "a : R", "n : ℕ", "h : p ≠ 0", "h2 : p * (X - C a) ^ n ≠ 0"], "goal": "rootMultiplicity a (p * (X - C a) ^ n) = rootMultiplicity a p + n"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "DecidableEq ι", "Fintype ι", "NontriviallyNormedField 𝕜", "E : ι → Type u_3", "(i : ι) → NormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "x : (i : ι) → E i", "i : ι", "y : E i"], "goal": "HasFDerivAt (Function.update x i) (ContinuousLinearMap.pi (Pi.single i (ContinuousLinearMap.id 𝕜 (E i)))) y"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "CommMonoid M", "ψ : AddChar R M", "r x : R"], "goal": "(ψ.mulShift r) x = ψ (r * x)"}}
+{"state": {"context": ["k : Type u", "inst✝ : CommRing k", "G : Grp", "X Y : Rep k ↑G", "f : ↑X.V →ₗ[k] ↑Y.V", "g : ↑G"], "goal": "Y.ρ g ∘ₗ f ∘ₗ X.ρ g⁻¹ = f ↔ f ∘ₗ X.ρ g = Y.ρ g ∘ₗ f"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "Cardinal.mk ↑s ≤ Cardinal.aleph0 ↔ s.Countable"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "F : Type u_3", "K : Type u_4", "P : Cubic R", "inst✝³ : CommRing R", "inst✝² : CommRing S", "φ : R →+* S", "inst✝¹ : IsDomain R", "inst✝ : DecidableEq R"], "goal": "P.roots.toFinset.card ≤ 3"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "l : Filter α", "f₁ f₂ : α → E", "g : α → F", "hf : f₁ =ᶠ[l] f₂", "h : f₂ =O[l] g"], "goal": "f₁ =O[l] g"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "H : Type u_5", "inst✝⁸ : NontriviallyNormedField 𝕜", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "q : FormalMultilinearSeries 𝕜 F G", "p : FormalMultilinearSeries 𝕜 E F", "hq : 0 < q.radius", "hp : 0 < p.radius", "rq rp : ℝ≥0", "hrp : rp < 1 ∧ ↑rp < p.radius", "hrq : rq < 1 ∧ ↑rq < q.radius", "rp_pos : 0 < rp", "rq_pos : 0 < rq", "Cq : ℝ≥0", "_hCq0 : Cq > 0", "hCq : ∀ (n : ℕ), ‖q n‖₊ * rq ^ n ≤ Cq", "Cp : ℝ≥0", "hCp1 : Cp ≥ 1", "hCp : ∀ (n : ℕ), ‖p n‖₊ * rp ^ n ≤ Cp", "r0 : ℝ≥0 := (4 * Cp)⁻¹", "r0_pos : 0 < r0", "r : ℝ≥0 := rp * rq * r0", "r_pos : 0 < r"], "goal": "∃ r > 0, Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁶ : LinearOrder ι", "inst✝⁵ : MeasurableSpace ι", "inst✝⁴ : TopologicalSpace ι", "inst✝³ : OrderTopology ι", "inst✝² : SecondCountableTopology ι", "inst✝¹ : BorelSpace ι", "inst✝ : TopologicalSpace β", "u : ι → Ω → β", "τ : Ω → ι", "f : Filtration ι m", "h : ProgMeasurable f u", "hτ : IsStoppingTime f τ", "n : ι", "hτ_le : ∀ (ω : Ω), τ ω ≤ n"], "goal": "StronglyMeasurable (stoppedValue u τ)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "ContinuousSMul 𝕜 E", "x : E"], "goal": "Bornology.IsVonNBounded 𝕜 {x}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "G : Type u_4", "M : Type u_5", "N : Type u_6", "inst✝¹ : CommMonoid M", "inst✝ : CommMonoid N", "f g : α → M", "a b : α", "s t : Set α", "h : a ∉ s → f a = 1", "x : α", "hx : x ∈ mulSupport f", "hxs : x ∈ s"], "goal": "x ∈ s"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "f : β → β → γ", "g : α → β", "a b : α"], "goal": "(f on g) a b = f (g a) (g b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "fab : α → β", "fbc : β → γ", "ga : α → α", "gb : β → β", "gc : γ → γ", "hbc : Function.Semiconj fbc gb gc", "hab : Function.Semiconj fab ga gb"], "goal": "Function.Semiconj (fbc ∘ fab) ga gc"}}
+{"state": {"context": ["a x z✝ z : ℂ", "hz : z = 0"], "goal": "0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ z = 0) ↔ 0 < z.re ∨ z = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s s₁✝ s₂✝ t✝ t₁ t₂ u t s₁ s₂ : Set α", "h : s₁ ⊆ s₂", "x : α"], "goal": "x ∈ t.ite s₁ s₂ ↔ x ∈ s₁ ∪ s₂ \\ t"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {a : α} {l : List α}, (a :: l).get 0 = a"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "TopologicalSpace M", "m m' : M", "ContinuousMul M", "f : ℤ → M", "hf₁ : HasProd (fun n => f ↑n) m", "hf₂ : HasProd (fun n => f (-(↑n + 1))) m'"], "goal": "HasProd f (m * m')"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "H : Subgroup G", "R S : Set G", "inst✝ : Finite ↑(commutatorSet G)", "hG : ¬(center G).index = 0", "this✝ : (center G).FiniteIndex", "h1 : (center G).relindex (_root_.commutator G) ∣ (center G).index", "this : ((center G).subgroupOf (_root_.commutator G)).FiniteIndex", "h2 : Group.rank ↥((center G).subgroupOf (_root_.commutator G)) ≤ ((center G).subgroupOf (_root_.commutator G)).index * Group.rank ↥(_root_.commutator G)", "h3 : (center G).relindex (_root_.commutator G) * Group.rank ↥(_root_.commutator G) ≤ (center G).index * Nat.card ↑(commutatorSet G)"], "goal": "∀ (g : ↥((center G).subgroupOf (_root_.commutator G))), g ^ (center G).index = 1"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "Polynomial.derivative 1 = 0"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Add α", "s t : Finset α"], "goal": "Finset.image (fun x => x.1 + x.2) (s ×ˢ t) = s + t"}}
+{"state": {"context": ["p q y : ℝ", "r : ℚ", "m : ℤ", "n : ℕ", "hp : 1 < p", "M : ℤ", "h : LiouvilleWith p ↑M"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "w x y z : α", "inst✝ : GeneralizedBooleanAlgebra α", "s : z ⊔ x ⊓ y = x \\ y ⊔ x ⊓ y", "i : x ⊓ y ⊓ z = x \\ y ⊓ (x ⊓ y)"], "goal": "x \\ y = z"}}
+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝ : Monoid M", "a x y : M"], "goal": "y ∈ closure {x} ↔ ∃ n, x ^ n = y"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "s t : Set α", "h : Disjoint s t"], "goal": "⨆ a ∈ s, LinearMap.range (Finsupp.lsingle a) ≤ ⨅ a ∈ t, LinearMap.ker (Finsupp.lapply a)"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "E : Set α"], "goal": "Metric.cthickening 0 E = closure E"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "f : MeasureTheory.SimpleFunc α ℝ≥0∞", "μ : MeasureTheory.Measure α"], "goal": "∫⁻ (a : α), ↑f a ∂μ = f.lintegral μ"}}
+{"state": {"context": ["C : Type u_3", "CategoryTheory.Category.{u_1, u_3} C", "r : HomRel C", "D : Type u_4", "CategoryTheory.Category.{u_2, u_4} D", "F : CategoryTheory.Functor (CategoryTheory.Quotient r) D"], "goal": "CategoryTheory.Quotient.natTransLift r (CategoryTheory.CategoryStruct.id ((CategoryTheory.Quotient.functor r).comp F)) =\n  CategoryTheory.CategoryStruct.id F"}}
+{"state": {"context": ["N : Type u_1", "G : Type u_2", "Group N", "Group G", "φ : G →* MulAut N", "a : N ⋊[φ] G"], "goal": "a⁻¹.left = (φ a.right⁻¹) a.left⁻¹"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "E : EllipticCurve R", "C : WeierstrassCurve.VariableChange R", "hu : (↑C.u * ↑C.u⁻¹) ^ 12 = 1"], "goal": "↑C.u ^ 12 * ↑E.Δ'⁻¹ * (↑C.u⁻¹ ^ 4 * E.c₄) ^ 3 = E.j"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "inst✝² : Zero β✝", "inst✝¹ : Zero γ", "μ : Measure α", "f✝ : α →ₛ β✝", "β : Type u_5", "inst✝ : MonoidWithZero β", "f g : α →ₛ β", "hf : f.FinMeasSupp μ", "hg : g.FinMeasSupp μ"], "goal": "(map (fun p => p.1 * p.2) (f.pair g)).FinMeasSupp μ"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "f g : CategoryTheory.ComposableArrows C 2", "app₀ : f.obj' 0 ⋯ ≅ g.obj' 0 ⋯", "app₁ : f.obj' 1 ⋯ ≅ g.obj' 1 ⋯", "app₂ : f.obj' 2 ⋯ ≅ g.obj' 2 ⋯", "w₀ :\n  CategoryTheory.CategoryStruct.comp (f.map' 0 1 ⋯ ⋯) app₁.hom =\n    CategoryTheory.CategoryStruct.comp app₀.hom (g.map' 0 1 ⋯ ⋯)", "w₁ :\n  CategoryTheory.CategoryStruct.comp (f.map' 1 2 ⋯ ⋯) app₂.hom =\n    CategoryTheory.CategoryStruct.comp app₁.hom (g.map' 1 2 ⋯ ⋯)"], "goal": "(CategoryTheory.ComposableArrows.isoMk₂ app₀ app₁ app₂ w₀ w₁).hom =\n  CategoryTheory.ComposableArrows.homMk₂ app₀.hom app₁.hom app₂.hom w₀ w₁"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "s✝ t✝ r s t : Set α", "h : Disjoint s t"], "goal": "range ⇑(sumSet h) = s ∪ t"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℕ", "a b : α", "DecidableEq α", "h : a ≠ b", "h₁ : f a = 1"], "goal": "Nat.multinomial {a, b} f = (f b).succ"}}
+{"state": {"context": ["G : Type v", "Group G", "PseudoEMetricSpace G", "IsometricSMul G G", "IsometricSMul Gᵐᵒᵖ G"], "goal": "∀ (a : G), (IsometryEquiv.inv G) a = a⁻¹"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁷ : Category.{u_3, u_1} C", "inst✝⁶ : Category.{?u.2010, u_2} D", "inst✝⁵ : Preadditive C", "inst✝⁴ : HasZeroObject C", "inst✝³ : HasBinaryBiproducts C", "inst✝² : Preadditive D", "inst✝¹ : HasZeroObject D", "inst✝ : HasBinaryBiproducts D", "K L : CochainComplex C ℤ", "φ : K ⟶ L", "p q : ℤ", "hpq : p + -1 = q"], "goal": "(inl φ).v p q hpq ≫ (triangle φ).mor₃.f q = -(K.shiftFunctorObjXIso 1 q p ⋯).inv"}}
+{"state": {"context": ["n : ℕ", "R : Type u_1", "Semiring R"], "goal": "(↑n).coeff 0 = ↑n"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "Ring R", "Ring S", "f : R →+* S", "I : Type u_3", "e : I → R", "Fintype I", "he : CompleteOrthogonalIdempotents e"], "goal": "CompleteOrthogonalIdempotents (⇑f ∘ e)"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝¹ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "c c₁ c₂ : ℝ≥0", "ε ε₁ ε₂ : ℝ", "π₁ π₂ : TaggedPrepartition I", "inst✝ : CompleteSpace F", "h : Integrable I l f vol", "h0 : 0 < ε", "hπ : l.MemBaseSet I c (h.convergenceR ε c) π", "π₀ : Prepartition I", "hU : π.iUnion = π₀.iUnion", "δ : ℝ", "δ0 : 0 < δ", "δ' : ℝ := δ / (↑π₀.boxes.card + 1)"], "goal": "dist (integralSum f vol π) (∑ J ∈ π₀.boxes, integral J l f vol) ≤ ε + δ"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "E' : Type u_6", "F' : Type u_7", "Norm E", "SeminormedAddCommGroup E'", "SeminormedAddCommGroup F'", "f : α → E", "f' : α → E'", "g' : α → F'", "l : Filter α", "h : f =O[l] g'"], "goal": "f =O[l] fun x => (f' x, g' x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "LinearOrder α", "Preorder β", "OrderTop β", "f : α → β", "H : StrictMono f", "a : α", "h_top : f a = ⊤", "x : α"], "goal": "x ≤ a"}}
+{"state": {"context": ["X Y : Type u", "x : X", "y : Y"], "goal": "(β_ X Y).hom (x, y) = (y, x)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_3", "M : Type u_5", "M₂ : Type u_7", "Semiring R", "Semiring R₂", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R₂ M₂", "σ₁₂ : R →+* R₂", "F : Type u_9", "FunLike F M M₂", "SemilinearMapClass F σ₁₂ M M₂", "f : F", "q q' : Submodule R₂ M₂"], "goal": "q ≤ q' → Submodule.comap f q ≤ Submodule.comap f q'"}}
+{"state": {"context": ["α : Type u_1", "AddCommSemigroup α", "PartialOrder α", "ExistsAddOfLE α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Sub α", "OrderedSub α", "a b c : α", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "h₁ : a ≤ b", "h₂ : c ≤ b"], "goal": "a ≤ b - c ↔ c ≤ b - a"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a : α", "l : List α"], "goal": "Cycle.Chain r ↑(a :: l) ↔ List.Chain r a (l ++ [a])"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrder α", "inst✝¹ : LinearOrder β", "f : α → β", "𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)", "s t u : Finset α", "a b : α", "r : ℕ", "inst✝ : Fintype α", "t₁ t₂ : Finset α", "ht₁ : s.card = t₁.card ∧ { ofColex := t₁ } ≤ { ofColex := s }", "ht₂ : { ofColex := t₂ } < { ofColex := t₁ } ∧ t₂.card = s.card"], "goal": "{ ofColex := t₂ } ≤ { ofColex := s }"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : UniformSpace α", "inst✝² : Group α", "inst✝¹ : UniformGroup α", "inst✝ : UniformSpace β", "f : β → α", "hf : UniformContinuous f", "n : ℕ"], "goal": "UniformContinuous fun x => f x ^ Int.negSucc n"}}
+{"state": {"context": ["x : ℝ", "hx : x ≠ 0"], "goal": "Function.Injective fun y => y ^ x"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝⁸ : CommRing R✝", "S✝ : Submonoid R✝", "P✝ : Type u_2", "inst✝⁷ : CommRing P✝", "inst✝⁶ : Algebra R✝ P✝", "loc : IsLocalization S✝ P✝", "R : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : Nontrivial R", "S : Submonoid R", "P : Type u_4", "inst✝³ : Nontrivial P", "inst✝² : CommRing P", "inst✝¹ : Algebra R P", "inst✝ : NoZeroSMulDivisors R P", "hR : IsNoetherianRing R", "hS : S ≤ R⁰", "I : FractionalIdeal S P", "this : FG I.num"], "goal": "(↑I).FG"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M₁ : Type u_4", "M₂ : Type u_5", "M₃ : Type u_6", "AddCommGroup M₁", "AddCommGroup M₂", "AddCommGroup M₃", "Module R M₁", "Module R M₂", "Module R M₃", "Q₁ : QuadraticForm R M₁", "Q₂ : QuadraticForm R M₂", "Q₃ : QuadraticForm R M₃", "e₁₂ : QuadraticMap.IsometryEquiv Q₁ Q₂", "e₂₃ : QuadraticMap.IsometryEquiv Q₂ Q₃"], "goal": "(CliffordAlgebra.equivOfIsometry e₁₂).trans (CliffordAlgebra.equivOfIsometry e₂₃) =\n  CliffordAlgebra.equivOfIsometry (e₁₂.trans e₂₃)"}}
+{"state": {"context": ["k G : Type u", "Field k", "Monoid G", "V : Type u", "AddCommGroup V", "Module k V", "FiniteDimensional k V", "ρ : Representation k G V"], "goal": "(FDRep.of ρ).ρ = ρ"}}
+{"state": {"context": ["X : Type u_1", "inst✝² : NormedAddCommGroup X", "M : Type u_2", "inst✝¹ : Ring M", "inst✝ : Module M X", "P : { P // IsLprojection X P }"], "goal": "↑P * ↑Pᶜ = 0"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁵ : NormedRing A", "inst✝⁴ : StarRing A", "inst✝³ : CstarRing A", "inst✝² : CompleteSpace A", "inst✝¹ : NormedAlgebra ℂ A", "inst✝ : StarModule ℂ A", "a : A", "t : ℝ≥0", "ha : IsSelfAdjoint a", "ht : ‖a‖₊ ≤ t", "this : IsSelfAdjoint ((algebraMap ℝ A) ↑t - a)"], "goal": "SpectrumRestricts a ⇑ContinuousMap.realToNNReal ↔ ‖(algebraMap ℝ A) ↑t - a‖₊ ≤ t"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "AddGroup G", "AddAction G α", "g h : G", "comm : AddCommute g h", "j : ℤ"], "goal": "j • h +ᵥ AddAction.fixedBy α g = AddAction.fixedBy α g"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "gt_zero : 0 < b (min_bi b)"], "goal": "b (min_bi b) / 2 < 1"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "Module.Finite R M", "ι : Type u_1", "Fintype ι", "b : ι → M", "h : LinearIndependent R b"], "goal": "Fintype.card ι ≤ FiniteDimensional.finrank R M"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Preorder α", "Zero β", "Preorder β", "Zero γ", "Preorder γ", "SMul α β", "SMul α γ", "f : β → γ", "SMulPosMono α γ", "hf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂", "smul : ∀ (a : α) (b : β), f (a • b) = a • f b", "zero : f 0 = 0"], "goal": "SMulPosMono α β"}}
+{"state": {"context": ["ι : Type u_1", "π : ι → Sort u_2", "Fintype ι", "DecidableEq ι", "i : ι", "f g : (i : ι) → π i"], "goal": "(Finset.univ.erase i).piecewise f g = Function.update f i (g i)"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalNonAssocSemiring R", "r : R"], "goal": "⇑(AddMonoidHom.mulRight r) = fun x => x * r"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "m : MeasurableSpace α", "hm : m ≤ m0"], "goal": "AEMeasurable id μ"}}
+{"state": {"context": ["t : ℝ≥0∞", "t_ne_top : t ≠ ⊤"], "goal": "Monotone t.truncateToReal"}}
+{"state": {"context": ["m n✝ n : ℕ", "inst✝ : NeZero n", "p : ℕ × Fin n"], "goal": "(fun a => (a / n, ↑a)) ((fun p => p.1 * n + ↑p.2) p) = p"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "W : Submodule R M", "v : Module.Dual R M"], "goal": "(∀ (w : M), v w = 0) → v = 0"}}
+{"state": {"context": ["u : ℤˣ", "n : ℕ"], "goal": "u ^ n = u ^ (n % 2)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "LinearOrder α", "PartialOrder β", "f : α → β", "hf : Antitone f"], "goal": "StrictAnti f ↔ Function.Injective f"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "a : α"], "goal": "s \\ {a} ⩿ s"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : SeminormedAddCommGroup F", "f : ι → E", "g : ι → ℝ", "hg : Summable g", "h : ∀ᶠ (i : ι) in cofinite, ‖f i‖ ≤ g i"], "goal": "CauchySeq fun s => ∑ i ∈ s, f i"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "ι : Type u_4", "π : ι → Type u_5", "inst✝² : OrderedSemiring 𝕜", "inst✝¹ : AddCommMonoid E", "inst✝ : SMul 𝕜 E", "A B C : Set E", "x : E", "hAB : IsExtreme 𝕜 A B", "hBC : IsExtreme 𝕜 B C"], "goal": "IsExtreme 𝕜 A C"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "H : Type u_3", "inst✝³ : _root_.RCLike 𝕜", "inst✝² : NormedAddCommGroup H", "inst✝¹ : NormedSpace 𝕜 H", "inst✝ : Fintype ι", "f : H → EuclideanSpace 𝕜 ι", "f' : H →L[𝕜] EuclideanSpace 𝕜 ι", "t : Set H", "y : H"], "goal": "(∀ (i : ι), DifferentiableAt 𝕜 (fun x => (⇑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹¹ : TopologicalSpace M", "inst✝¹⁰ : ChartedSpace H M", "E' : Type u_5", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁶ : TopologicalSpace M'", "inst✝⁵ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝¹ : TopologicalSpace M''", "inst✝ : ChartedSpace H'' M''", "f f₀ f₁ : M → M'", "x : M", "s t : Set M", "g : M' → M''", "u : Set M'", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "I''s : SmoothManifoldWithCorners I'' M''", "f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)", "g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))", "h : MDifferentiableWithinAt I I' f s x"], "goal": "HasFDerivWithinAt (writtenInExtChartAt I I' x f) (mfderivWithin I I' f s x) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)"}}
+{"state": {"context": ["α : Type u", "CompleteLattice α", "f : α →o α", "x : α"], "goal": "∃ a < (Order.succ (Cardinal.mk α)).ord,\n  ∃ b < (Order.succ (Cardinal.mk α)).ord, a ≠ b ∧ OrdinalApprox.lfpApprox f x a = OrdinalApprox.lfpApprox f x b"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : SMulCommClass ℝ 𝕜 F", "A : nhds 0 = ⨅ i, Filter.comap (⇑(schwartzSeminormFamily ℝ E F i)) (nhds 0)"], "goal": "nhds 0 = ⨅ i, Filter.comap (⇑(schwartzSeminormFamily 𝕜 E F i)) (nhds 0)"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : FiniteDimensional ℝ E", "s : Set E", "x : E", "hs : s ∈ 𝓝 x"], "goal": "∃ f, tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝¹ : PseudoMetricSpace α", "x y z : α", "δ ε ε₁ ε₂ : ℝ", "s✝ : Set α", "inst✝ : PseudoMetricSpace β", "f : α → β", "a : α", "s : Set α"], "goal": "ContinuousWithinAt f s a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x : α}, x ∈ s → dist x a < δ → dist (f x) (f a) < ε"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "DecidableRel fun x x_1 => x ≤ x_1", "x : α", "t : Ordset α", "h_mem : x ∈ t"], "goal": "0 < t.size"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "f : ↥(lp E ⊤)"], "goal": "0 ≤ ‖f‖"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q : ℚ", "h : ↑p ^ (-padicValRat p q) = 0", "hq : ¬q = 0"], "goal": "↑p ≠ 0"}}
+{"state": {"context": ["G : Type u", "α : Type v", "β : Type w", "inst✝¹ : Group G", "inst✝ : Finite G", "p n : ℕ", "hp : Fact (Nat.Prime p)", "hdvd : p ^ (n + 1) ∣ Nat.card G", "H : Subgroup G", "hH : Nat.card ↥H = p ^ n", "s : ℕ", "hs : Nat.card G = s * p ^ (n + 1)", "hcard : Nat.card (G ⧸ H) = s * p", "hm : s * p % p = Nat.card (↥H.normalizer ⧸ H.subgroupOf H.normalizer) % p", "hm' : p ∣ Nat.card (↥H.normalizer ⧸ H.subgroupOf H.normalizer)", "x : ↥H.normalizer ⧸ H.subgroupOf H.normalizer", "hx : orderOf x = p", "hequiv : ↥H ≃ ↥(H.subgroupOf H.normalizer)", "y : G", "hy : y ∈ H"], "goal": "(fun x_1 => x ^ x_1) 0 = ↑⟨y, ⋯⟩"}}
+{"state": {"context": ["R : Type w", "inst✝² : Ring R", "J : Type u", "inst✝¹ : Category.{v, u} J", "F : J ⥤ ModuleCat R", "inst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrp)", "j : J", "r : R"], "goal": "colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j ≫ (mkOfSMul (coconePointSMul F)).smul r = (F.obj j).smul r ≫ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "s : Set E", "s_conv : Convex ℝ s", "f : E → F", "f' : E → E →L[ℝ] F", "f'' : E →L[ℝ] E →L[ℝ] F", "hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x", "x : E", "xs : x ∈ s", "hx : HasFDerivWithinAt f' f'' (interior s) x", "v w : E", "hv : x + v ∈ interior s", "hw : x + v + w ∈ interior s"], "goal": "(fun h => f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝⁴ : LinearOrderedCancelAddCommMonoid α", "inst✝³ : Nontrivial α", "inst✝² : DenselyOrdered α", "inst✝¹ : ExistsAddOfLE α", "inst✝ : Finite ι", "x : ι → α", "val✝ : Fintype ι", "h✝ : Nonempty ι", "ε : ι → α", "hε✝ : ∀ (i : ι), 0 < ε i", "h : ∀ (i : ι), x i < (x + ε) i", "hxε : ∀ (i : ι), x i + ε i = (x + ε) i", "hε : 0 < Finset.univ.inf' ⋯ ε", "δ : α", "hδ : 0 < δ", "hδε : δ < Finset.univ.inf' ⋯ ε"], "goal": "∃ ε_1, 0 < ε_1 ∧ ∀ (i : ι), x i + ε_1 < (x + ε) i"}}
+{"state": {"context": ["t t' : Type u → Type u", "eqv : (α : Type u) → t α ≃ t' α", "Traversable t", "m : Type u → Type u", "Applicative m", "α β : Type u", "f : α → m β", "x : t' α"], "goal": "Equiv.traverse eqv f x = ⇑(eqv β) <$> traverse f ((eqv α).symm x)"}}
+{"state": {"context": ["p : ℝ[X]"], "goal": "Tendsto (fun x => Polynomial.eval x⁻¹ p * rexp (-x⁻¹)) (𝓝 0 ⊓ 𝓟 {x | ¬x ≤ 0}) (𝓝 0)"}}
+{"state": {"context": ["C : Type u_1", "A : Type u_2", "inst✝⁶ : Category.{?u.45742, u_1} C", "inst✝⁵ : Category.{?u.45746, u_2} A", "F : C ⥤ A", "M : Type u_3", "inst✝⁴ : AddMonoid M", "inst✝³ : HasShift C M", "G : Type u_4", "inst✝² : AddGroup G", "inst✝¹ : HasShift C G", "inst✝ : F.ShiftSequence M", "n m a a' a'' : M", "ha' : n + a = a'", "ha'' : m + a' = a''", "X : C"], "goal": "m + n + a = a''"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : α → α → Prop", "f : β → α", "b : β", "l : List β"], "goal": "List.Chain R (f b) (List.map f l) ↔ List.Chain (fun a b => R (f a) (f b)) b l"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "s : Set X"], "goal": "IsCompact s ↔ IsCompact Set.univ"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "f : X → Y", "g : Y → Z", "TopologicalSpace X", "TopologicalSpace Y", "hf : OpenEmbedding f", "l : Filter Z", "x : X"], "goal": "Filter.Tendsto (g ∘ f) (𝓝 x) l ↔ Filter.Tendsto g (𝓝 (f x)) l"}}
+{"state": {"context": ["z✝ : ℂ", "E : Type u_1", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "x z : ℂ"], "goal": "z * x = x * z"}}
+{"state": {"context": ["x : ℝ", "hx : |x| ≤ 1"], "goal": "(cexp (-(↑x * I)) - cexp (↑x * I)) * I - (2 * ↑x - ↑1 * ↑x * ↑x * ↑x / 3) = (cexp (-(↑x * I)) - (1 / (↑0 + 1) + -(1 * (↑x * I)) / (↑0 + 1) + 1 * (↑x * I) * (↑x * I) / (↑0 + 1 + 1) + -(1 * (↑x * I) * (↑x * I) * (↑x * I)) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) - (cexp (↑x * I) - (1 / (↑0 + 1) + 1 * (↑x * I) / (↑0 + 1) + 1 * (↑x * I) * (↑x * I) / (↑0 + 1 + 1) + 1 * (↑x * I) * (↑x * I) * (↑x * I) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)))) * I"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f fa : α → α", "fb : β → β", "x y : α", "m n✝ : ℕ", "hx : x ∈ periodicPts f", "n : ℕ"], "goal": "((List.map (fun n => f^[n] x) (List.range (minimalPeriod f x))).rotate n).length = (List.map (fun n_1 => f^[n_1] (f^[n] x)) (List.range (minimalPeriod f (f^[n] x)))).length"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁵ : Group G", "inst✝⁴ : Group G'", "inst✝³ : Group G''", "A : Type u_4", "inst✝² : AddGroup A", "N : Type u_5", "inst✝¹ : Group N", "C : Type u_6", "inst✝ : CommGroup C", "s t : Subgroup C", "x✝ x y z : C"], "goal": "z ∈ closure {x, y} ↔ ∃ m n, x ^ m * y ^ n = z"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "Group G", "MulAction G α", "MeasurableSpace α", "s : Set α", "μ : MeasureTheory.Measure α", "G' : Type u_6", "Group G'", "MulAction G' α", "MeasurableSpace G'", "MeasurableSMul G' α", "MeasureTheory.SMulInvariantMeasure G' α μ", "SMulCommClass G' G α", "h : MeasureTheory.IsFundamentalDomain G s μ", "g : G'"], "goal": "MeasureTheory.IsFundamentalDomain G (g • s) μ"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "w₁✝ : InfinitePlace K", "B : ℕ", "hB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B", "w₁ : InfinitePlace K"], "goal": "∃ u, ∀ (w : InfinitePlace K), w ≠ w₁ �� Real.log (w ((algebraMap (𝓞 K) K) ↑u)) < 0"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝¹ : Semiring R✝", "r✝ : R✝", "f✝ : R✝[X]", "R : Type u_2", "inst✝ : CommRing R", "f : R[X]", "r : R", "h : ∀ (k : ℕ), eval r ((hasseDeriv k) f) = 0"], "goal": "(taylor r) f = (taylor r) 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "R : Cᵒᵖ ⥤ RingCat", "J : Type u₂", "CategoryTheory.Category.{v₂, u₂} J", "F : J ⥤ PresheafOfModules R", "{X Y : Cᵒᵖ} →\n    (f : X ⟶ Y) →\n      CategoryTheory.Limits.PreservesColimit (F ⋙ PresheafOfModules.evaluation R Y)\n        (ModuleCat.restrictScalars (R.map f))", "∀ (X : Cᵒᵖ), CategoryTheory.Limits.HasColimit (F ⋙ PresheafOfModules.evaluation R X)", "X : Cᵒᵖ"], "goal": "(PresheafOfModules.colimitBundledCore F).obj X = CategoryTheory.Limits.colimit (F ⋙ PresheafOfModules.evaluation R X)"}}
+{"state": {"context": ["a b : ℝ"], "goal": "∫ (x : ℝ) in a..b, rexp x = rexp b - rexp a"}}
+{"state": {"context": ["k : Turing.PartrecToTM2.Cont'"], "goal": "Turing.PartrecToTM2.codeSupp Turing.ToPartrec.Code.succ k =\n  Turing.PartrecToTM2.trStmts₁ (Turing.PartrecToTM2.trNormal Turing.ToPartrec.Code.succ k) ∪\n    Turing.PartrecToTM2.contSupp k"}}
+{"state": {"context": ["X : PartialFun"], "goal": "partialFunEquivPointed.unitIso.inv.app X =\n  pointedToPartialFun.map { toFun := Option.elim' none fun a => some ⟨some a, ⋯⟩, map_point := ⋯ } ≫\n    pointedToPartialFun.map (Pointed.Iso.mk (Equiv.optionSubtypeNe none) ⋯).hom ≫\n      (PartialFun.Iso.mk\n          { toFun := fun a => ⟨some a, ⋯⟩, invFun := fun a => (↑a).get ⋯, left_inv := ⋯, right_inv := ⋯ }).inv"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝ : ConditionallyCompleteLinearOrder β", "f : Filter α", "u v : α → β", "h₁ : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝", "h₂ : autoParam (IsCoboundedUnder (fun x x_1 => x ≤ x_1) f v) _auto✝", "h₃ : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝", "h₄ : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f v) _auto✝", "bddmax : IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun a => u a ⊔ v a", "cobddmax : IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun a => max (u a) (v a)"], "goal": "limsup (fun a => max (u a) (v a)) f ≤ max (limsup u f) (limsup v f)"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2", "hnsol : Polynomial.eval a F ≠ 0", "z : ℤ_[p]", "hev : Polynomial.eval z F = 0", "hnlt : ‖z - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖", "soln_dist : ‖z - soln‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖", "h : ℤ_[p] := z - soln", "q : ℤ_[p]", "hq : Polynomial.eval (soln + h) F = Polynomial.eval soln F + Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2"], "goal": "Polynomial.eval (soln + h) F = Polynomial.eval soln (Polynomial.derivative F) * h + q * h ^ 2"}}
+{"state": {"context": ["α : Type u_1", "s : Set αᵒᵖ", "a : αᵒᵖ"], "goal": "Opposite.unop a ∈ s.unop ↔ a ∈ s"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "inst✝ : TopologicalSpace G", "K₀ : Set G", "f : Compacts G → ℝ"], "goal": "f ∈ haarProduct K₀ ↔ ∀ (K : Compacts G), f K ∈ Icc 0 ↑(index (↑K) K₀)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F G : CategoryTheory.Functor C (Type w)"], "goal": "CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.coprod.inr\n    (CategoryTheory.FunctorToTypes.binaryCoproductIso F G).hom =\n  CategoryTheory.FunctorToTypes.coprod.inr"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "ι : Sort u_6", "κ : ι → Sort u_7", "inst✝³ : FunLike F α β", "inst✝² : SupSet α", "inst✝¹ : SupSet β", "inst✝ : sSupHomClass F α β", "f : F", "g : ι → α"], "goal": "f (⨆ i, g i) = ⨆ i, f (g i)"}}
+{"state": {"context": ["c : Prop", "Decidable c", "a b : Ordering"], "goal": "((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{?u.30268, u_1} C", "inst✝⁴ : Preadditive C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝ : Pretriangulated C", "X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C", "u₁₂ : X₁ ⟶ X₂", "u₂₃ : X₂ ⟶ X₃", "u₁₃ : X₁ ⟶ X₃", "comm : u₁₂ ≫ u₂₃ = u₁₃", "v₁₂ : X₂ ⟶ Z₁₂", "w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁", "h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles", "v₂₃ : X₃ ⟶ Z₂₃", "w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂", "h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles", "v₁₃ : X₃ ⟶ Z₁₃", "w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁", "h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles", "h : Octahedron comm h₁₂ h₂₃ h₁₃"], "goal": "u₁₃ ≫ 𝟙 X₃ = u₁₂ ≫ u₂₃"}}
+{"state": {"context": ["R : Type u", "A : Type u_1", "CommSemiring R", "CommRing A", "Algebra R A", "I J : Ideal A", "h : I ≤ J"], "goal": "(↑(DoubleQuot.quotQuotEquivQuotOfLEₐ R h).symm).comp (Ideal.Quotient.mkₐ R J) = DoubleQuot.quotQuotMkₐ R I J"}}
+{"state": {"context": ["R✝ : Type u", "S : Type u_1", "σ : Type v", "M : Type w", "inst✝⁴ : CommRing R✝", "inst✝³ : CommRing S", "inst✝² : AddCommGroup M", "inst✝¹ : Module R✝ M", "R : Type u_2", "inst✝ : CommSemiring R", "m n : ℕ", "F : MvPolynomial σ R", "hF : F.totalDegree ≤ m", "f : σ → R[X]", "hf : ∀ (i : σ), (f i).natDegree ≤ n", "d : σ →₀ ℕ", "hd : d ∈ F.support"], "goal": "(Polynomial.C (F d) * d.prod fun n e => f n ^ e).natDegree ≤ m * n"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "N : Type u_7", "TopologicalSpace N", "ChartedSpace H' N", "J : ModelWithCorners 𝕜 E' H'", "I.Boundaryless"], "goal": "(I.prod J).boundary (M × N) = Set.univ.prod (J.boundary N)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_4", "s : Set α", "t : ι → Set β", "f : α → β", "H : ∀ (i : ι), Set.SurjOn f s (t i)"], "goal": "Set.SurjOn f s (⋃ i, t i)"}}
+{"state": {"context": ["R : Type u", "inst✝⁴ : CommRing R", "n G : Type v", "inst✝³ : DecidableEq n", "inst✝² : Fintype n", "α β : Type v", "inst✝¹ : DecidableEq α", "M : Matrix n n R", "p : ℕ", "inst✝ : Fact (Nat.Prime p)", "k : ℕ", "hk : 1 - X • M.map ⇑C ∣ 1"], "goal": "IsUnit M.charpolyRev"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝ : BooleanAlgebra α", "s : Finset ι", "hs : s.Nonempty", "f : ι → α", "a : α"], "goal": "(s.inf fun b => a \\ f b) = a \\ s.sup f"}}
+{"state": {"context": ["G : Type u", "inst✝ : Group G"], "goal": "commutator G = Subgroup.normalClosure (commutatorSet G)"}}
+{"state": {"context": ["α : Type u_2", "Div α", "f g₁ g₂ : Filter α"], "goal": "g₁ ≤ g₂ → f / g₁ ≤ f / g₂"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "α : Type u'", "β : Type v'", "ft : (n : ℕ) → L.Term (α ⊕ Fin n) ≃ L'.Term (β ⊕ Fin n)", "fr : (n : ℕ) → L.Relations n ≃ L'.Relations n", "n : ℕ"], "goal": "∀ (a : L'.BoundedFormula β n),\n  (FirstOrder.Language.BoundedFormula.mapTermRelEquiv ft fr).symm a =\n    FirstOrder.Language.BoundedFormula.mapTermRel (fun n => ⇑(ft n).symm) (fun n => ⇑(fr n).symm) (fun x => id) a"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_4", "DecidableEq ι", "Fintype ι", "AddCommMonoid α", "s : Finset ι", "f : ι → α"], "goal": "(∑ i : ι, if i ∈ s then f i else 0) = ∑ i ∈ s, f i"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "inst✝¹ : Fintype F", "inst✝ : DecidableEq F", "a : F", "ha : a ≠ 0"], "goal": "(quadraticChar F) (a ^ 2) = 1"}}
+{"state": {"context": ["r s : ℝ", "hrs : r.IsConjExponent s"], "goal": "{x | ∃ k > 0, beattySeq r k = x} ∆ {x | ∃ k > 0, beattySeq' s k = x} ⊆ {n | 0 < n}"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "F : Type u_9", "Mul M", "Mul N", "FunLike F M N", "MulHomClass F M N", "f : F", "x y : M"], "goal": "f (x * y) = f x * f y"}}
+{"state": {"context": ["α : Type u", "l : List α", "n m : ℕ", "hml : m < l.length", "hm : (drop (n % l.length) l).length ≤ m"], "goal": "(drop (n % l.length) l ++ take (n % l.length) l)[m]? = l[(m + n) % l.length]?"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "V : Type u_3", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing A", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Algebra R A", "inst✝³ : Module R V", "inst✝² : Module A V", "inst✝¹ : IsScalarTower R A V", "inst✝ : Invertible 2", "Q : QuadraticForm R V", "v : V"], "goal": "(toBaseChange A Q) (involute ((ι (QuadraticForm.baseChange A Q)) (1 ⊗ₜ[R] v))) = (Algebra.TensorProduct.map (AlgHom.id A A) involute) ((toBaseChange A Q) ((ι (QuadraticForm.baseChange A Q)) (1 ⊗ₜ[R] v)))"}}
+{"state": {"context": ["x✝ y✝ x : ℝ", "hex : 1 ≤ x", "y : ℝ", "hey : 1 ≤ y", "hxy : x ≤ y"], "goal": "log x * x ≤ log y * y"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : LinearOrderedAddCommGroup α", "inst✝ : Archimedean α", "p : α", "hp : 0 < p", "a b : α", "hl : a ≤ b - toIcoDiv hp a b • p", "hr : b - toIcoDiv hp a b • p < a + p"], "goal": "a + toIcoDiv hp a b • p + p = a + p + toIcoDiv hp a b • p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝¹ : Countable α", "inst✝ : MeasurableSingletonClass α", "f : α → ℝ≥0∞"], "goal": "∑' (i : α), ∫⁻ (a : α), f a ∂μ {i} • dirac i = ∑' (a : α), f a * μ {a}"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a b : α"], "goal": "Finset.Icc a b = ∅ ↔ ¬a ≤ b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S"], "goal": "T R 2 = 2 * X ^ 2 - 1"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Lattice α", "Lattice β", "f : LatticeHom α β"], "goal": "Monotone (Sublattice.map f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "MeasurableSpace.CountablyGenerated γ", "κ : ProbabilityTheory.Kernel α (γ × β)", "ν : ProbabilityTheory.Kernel α γ", "hκν : κ.fst ≤ ν", "n : ℕ", "a : α", "x : γ", "s s' : Set β", "h : s ⊆ s'"], "goal": "κ.densityProcess ν n a x s ≤ κ.densityProcess ν n a x s'"}}
+{"state": {"context": ["γ : Type u_3", "α₁ : Type u_4", "α₂ : Type u_5", "Fintype α₁", "Fintype α₂", "AddCommMonoid γ", "f : α₁ × α₂ → γ"], "goal": "∑ x : α₁ × α₂, f x = ∑ y : α₂, ∑ x : α₁, f (x, y)"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝ : GroupWithZero G₀", "a x y x' y' : G₀", "h : SemiconjBy a x y", "h' : SemiconjBy a x' y'"], "goal": "SemiconjBy a (x / x') (y / y')"}}
+{"state": {"context": ["E : Type u_1", "AddCommGroup E", "Module ℝ E", "FiniteDimensional ℝ E", "mE : MeasurableSpace E", "tE : TopologicalSpace E", "TopologicalAddGroup E", "BorelSpace E", "T2Space E", "ContinuousSMul ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "g : E → ℝ", "h1 : g 0 = 0", "h2 : ∀ (x : E), g (-x) = g x", "h3 : ∀ (x y : E), g (x + y) ≤ g x + g y", "h4 : ∀ {x : E}, g x = 0 → x = 0", "h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x", "Nontrivial E", "r : ℝ"], "goal": "μ {x | g x ≤ r} = μ {x | g x < r}"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "p : F[X]", "sp : Polynomial.IsSplittingField F E p", "hp : p.Separable", "hFE : FiniteDimensional F E", "this✝¹ : DecidableEq E := Classical.decEq E", "s : Set E := p.rootSet E", "adjoin_root : IntermediateField.adjoin F s = ⊤", "P : IntermediateField F E → Prop := fun K => Fintype.card (↥K →ₐ[F] E) = finrank F ↥K", "K : IntermediateField F E", "x : E", "hx : x ∈ (p.aroots E).toFinset", "hK : Fintype.card (↥K →ₐ[F] E) = finrank F ↥K", "this✝ : Module ↥K ↥(↥K)⟮x⟯ := (↥K)⟮x⟯.module", "this : IsScalarTower F ↥K ↥(↥K)⟮x⟯ := IntermediateField.isScalarTower (↥K)⟮x⟯"], "goal": "finrank F ↥(IntermediateField.restrictScalars F (↥K)⟮x⟯) = finrank F ↥(↥K)⟮x⟯"}}
+{"state": {"context": ["α : Sort u_2", "β : Sort u_1", "p : α → β → Prop"], "goal": "(∀ (a : α) (b : β), p a b) ↔ ∀ (b : β) (a : α), p a b"}}
+{"state": {"context": ["x : ℝ", "n : ℤ", "h : -1 = 1", "hn : ↑(n / 2) * (2 * π) + π = x", "hn1 : n % 2 = 1"], "goal": "↑(n / 2) * (2 * π) = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "CommMonoid α", "TopologicalSpace α", "α' : Type u_5", "CommMonoid α'", "TopologicalSpace α'", "e : α' → α", "hes : Function.Surjective e", "f : β → α", "g : γ → α'", "he : ∀ {a : α'}, HasProd f (e a) ↔ HasProd g a"], "goal": "Multipliable f ↔ Multipliable g"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_3, u_2} D", "CategoryTheory.Preadditive C", "CategoryTheory.Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ", "H : C ⥤ D", "H.Additive", "HomologicalComplex.HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)"], "goal": "(H.mapHomologicalComplex (ComplexShape.up ℤ)).map (CochainComplex.mappingCone.inr φ) ≫\n    (CochainComplex.mappingCone.mapHomologicalComplexIso φ H).hom =\n  CochainComplex.mappingCone.inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)"}}
+{"state": {"context": ["α : Type u_1", "C : Set (Set α)", "s t : Set α", "I : Finset (Set α)", "hC : IsSetSemiring C", "hs : s ∈ C", "hI : ↑I ⊆ C", "h_dis : (↑I).PairwiseDisjoint id", "u : Set α", "hu : u ∈ ↑I", "v : Set α", "hv : v ∈ ↑(hC.diffFinset₀ hs hI)", "x✝ : u ≠ v"], "goal": "Disjoint u v"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Ring R", "x : R", "hx : IsUnit (1 + x)"], "goal": "↑hx.unit⁻¹ - 1 + x + x * (↑hx.unit⁻¹ - 1) = 0 ∧ x + (↑hx.unit⁻¹ - 1) + (↑hx.unit⁻¹ - 1) * x = 0"}}
+{"state": {"context": ["K : Type u_4", "L : Type u_5", "Field K", "Field L", "Algebra K L", "IsFractionRing K L", "I : FractionalIdeal K⁰ L"], "goal": "I = 0 ∨ I = 1"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Group (G i)", "inst✝ : Group H", "φ : (i : ι) → H →* G i", "hφ : ∀ (i : ι), Injective ⇑(φ i)", "i j : ι", "hij : i ≠ j"], "goal": "((of j).comp (φ j)).range ≤ (of j).range"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Setoid α"], "goal": "(∀ (x y : Quotient s), x = y) ↔ ∀ {x y : α}, ⊤ → s.Rel x y"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "f g : α → ℝ", "s : Set α", "inst✝ : SigmaFinite μ", "f_int : IntegrableOn f s μ", "g_int : IntegrableOn g s μ", "hs : MeasurableSet s", "hfg : f ≤ᶠ[ae (μ.restrict s)] g", "h : g - f =ᶠ[ae (μ.restrict s)] fun x => ↑(g x - f x).toNNReal"], "goal": "(μ.prod volume) (regionBetween f g s) = ENNReal.ofReal (∫ (y : α) in s, (g - f) y ∂μ)"}}
+{"state": {"context": ["n m : ℕ", "j : Fin n", "i : Fin (n + 1)", "hi : i ≠ 0"], "goal": "i.pred hi < j ↔ i < j.succ"}}
+{"state": {"context": ["α : Type u", "CommGroup α", "LT α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α"], "goal": "a * b⁻¹ < c ↔ a < b * c"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "s : ℕ", "hs : s ≤ k", "hodd : p ≠ 2"], "goal": "(Algebra.norm ℤ) (hζ.toInteger ^ ↑p ^ s - 1) = ↑↑p ^ ↑p ^ s"}}
+{"state": {"context": ["ι : Type u", "f✝ : ι → Ordinal.{max u v} → Ordinal.{max u v}", "f : ι → Ordinal.{max u u_1} → Ordinal.{max u u_1}", "o : Ordinal.{max u_1 u}"], "goal": "derivFamily f o < derivFamily f (succ o)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "a✝ b c d : ℝ≥0∞", "r p q : ℝ≥0", "x y z ε ε₁ ε₂ : ℝ≥0∞", "s : Set ℝ≥0∞", "ι : Sort u_4", "ι' : Sort u_5", "inst✝¹ : Nonempty ι", "inst✝ : Nonempty ι'", "f : ι → ℝ≥0∞", "g : ι' → ℝ≥0∞", "a : ℝ≥0∞", "h : ∀ (i : ι) (j : ι'), f i + g j ≤ a"], "goal": "iSup f + iSup g ≤ a"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "l : List α", "f : Equiv.Perm α"], "goal": "f ∈ permsOfList l → ∀ (x : α), f x ≠ x → x ∈ l"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : 𝕜 → E", "a b : 𝕜"], "goal": "(b - a) • dslope f a b = f b - f a"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "v : ℝ → E → E", "s : ℝ → Set E", "K : ℝ≥0", "f g f' g' : ℝ → E", "a b t₀ εf εg δ : ℝ", "hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)", "hf : ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt f (v t (f t)) t ∧ f t ∈ s t", "hg : ∀ᶠ (t : ℝ) in 𝓝 t₀, HasDerivAt g (v t (g t)) t ∧ g t ∈ s t", "heq : f t₀ = g t₀", "ε : ℝ", "hε : ε > 0", "h : ∀ y ∈ ball t₀ ε, (HasDerivAt f (v y (f y)) y ∧ f y ∈ s y) ∧ HasDerivAt g (v y (g y)) y ∧ g y ∈ s y"], "goal": "∃ s ∈ 𝓝 t₀, EqOn f g s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "CompleteLattice α", "CompleteLattice β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "s : Set β"], "goal": "u (sInf s) = ⨅ a ∈ s, u a"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "Algebra.IsIntegral R A", "h' : Algebra.FiniteType R A"], "goal": "Module.Finite R A"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "inst✝² : NormedField 𝕜", "inst✝¹ : AddCommGroup V", "inst✝ : Module 𝕜 V", "e : ENorm 𝕜 V", "x y : V"], "goal": "↑e (x - y) = ↑e (x + -y)"}}
+{"state": {"context": ["f : StieltjesFunction", "a b : ℝ", "c d : ℕ → ℝ", "ss : Icc a b ⊆ ⋃ i, Ioo (c i) (d i)"], "goal": "ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℝ E", "s t : Set E", "a : ℝ", "hs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s", "r : ℝ", "hr : r > 0", "b : ℝ", "hb : 0 < b", "x : E", "hx' : x ∈ s", "hs₂ : Absorbs ℝ s {(fun x => b • x) x}", "hx : (fun x => b • x) x ∉ a • s", "h : ∀ (c : ℝ), r ≤ ‖c‖ → {(fun x => b • x) x} ⊆ c • s"], "goal": "a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "f : α → β", "g : β → ��", "b : β", "s : Set β", "inst✝ : Decidable (b ∈ s)", "hb : b ∈ s"], "goal": "(fun x => b) ⁻¹' s = univ"}}
+{"state": {"context": ["α : Type u", "AddGroup α", "Preorder α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a : α", "h : a < 0"], "goal": "a < -a"}}
+{"state": {"context": ["b : Cardinal.{u}", "n m : ℕ", "h : IsUnit 0"], "goal": "0 = 1"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "E : ι → Type u_3", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)"], "goal": "PiTensorProduct.injectiveSeminorm =\n  sSup\n    {p |\n      ∃ G x x_1,\n        p =\n          (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp\n            (PiTensorProduct.toDualContinuousMultilinearMap G)}"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "F : Type u_3", "AddCommGroup F", "Module R F", "f : E →ₗ.[R] F"], "goal": "↑f 0 = 0"}}
+{"state": {"context": ["R✝ : Type u_1", "A✝ : Type u_2", "inst✝¹³ : CommSemiring R✝", "inst✝¹² : NonUnitalRing A✝", "inst✝¹¹ : Module R✝ A✝", "inst✝¹⁰ : IsScalarTower R✝ A✝ A✝", "inst✝⁹ : SMulCommClass R✝ A✝ A✝", "S : Type u_3", "R : Type u_4", "A : Type u_5", "inst✝⁸ : Semifield R", "inst✝⁷ : Field S", "inst✝⁶ : NonUnitalRing A", "inst✝⁵ : Algebra R S", "inst✝⁴ : Module S A", "inst✝³ : IsScalarTower S A A", "inst✝² : SMulCommClass S A A", "inst✝¹ : Module R A", "inst✝ : IsScalarTower R S A", "a : A", "r : R"], "goal": "(algebraMap R S) r ∈ quasispectrum S a ↔ r ∈ quasispectrum R a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "P : Algebra.Generators R S", "R' : Type u_1", "S' : Type u_2", "CommRing R'", "CommRing S'", "Algebra R' S'", "P' : Algebra.Generators R' S'", "Algebra R R'", "Algebra S S'", "Algebra R S'", "IsScalarTower R R' S'", "IsScalarTower R S S'", "f : P.Hom P'", "x : ↥P.ker"], "goal": "(Algebra.Generators.Cotangent.map f) (Algebra.Generators.Cotangent.mk x) =\n  Algebra.Generators.Cotangent.mk ⟨f.toAlgHom ↑x, ⋯⟩"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : EuclideanDomain R", "inst✝ : DecidableEq R", "a : R"], "goal": "(if a0 : a = 0 then 0 else let_fun x := ⋯; gcd (0 % a) a) = a"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "n : ℕ"], "goal": "(Polynomial.erase n p).toFinsupp = Finsupp.erase n p.toFinsupp"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "A : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "p : Submodule R M", "x : A ⊗[R] M"], "goal": "x ∈ span A ↑(LinearMap.range ((TensorProduct.mk R A M) 1))"}}
+{"state": {"context": ["R : Type uR", "M₂ : Type uM₂", "CommRing R", "AddCommGroup M₂", "Module R M₂", "Invertible 2", "Q₂ : QuadraticForm R M₂"], "goal": "QuadraticMap.comp Q₂ ↑(TensorProduct.lid R M₂) = QuadraticForm.tmul QuadraticMap.sq Q₂"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a", "i j n : ℕ", "ipos : 0 < i", "hin : i ≤ n", "j4n : j ≤ 4 * n", "h : xn a1 j ≡ xn a1 i [MOD xn a1 n]", "i2n : i ≤ 2 * n", "j2n : 2 * n < j"], "goal": "2 * n + 2 * n ≤ 2 * n + j"}}
+{"state": {"context": ["m n : ℕ+"], "goal": "m ∣ n ↔ m.gcd n = m"}}
+{"state": {"context": ["n : Nat"], "goal": "0 ≤ ↑n"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "NonAssocSemiring α", "Preorder α", "NonAssocSemiring β", "Preorder β", "NonAssocSemiring γ", "Preorder γ", "NonAssocSemiring δ", "Preorder δ", "f : γ →+*o δ", "g : β →+*o γ", "h : α →+*o β"], "goal": "(f.comp g).comp h = f.comp (g.comp h)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "c : R", "f : RingSeminorm R", "hf1 : f 1 ≤ 1", "hc : f c ≠ 0", "hpm : IsPowMul ⇑f", "x : R", "m n : ℕ", "hmn : m ≤ n"], "goal": "f (x * c ^ n) / f c ^ n ≤ f (x * c ^ m) / f c ^ m"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "d : G.Dart", "p : G.Walk u v"], "goal": "d ∈ p.reverse.darts ↔ d.symm ∈ p.darts"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E"], "goal": "Continuous fun a => ‖a‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "Countable ι", "μ : ι → MeasureTheory.Measure β"], "goal": "(ProbabilityTheory.Kernel.sum fun n => ProbabilityTheory.Kernel.const α (μ n)) =\n  ProbabilityTheory.Kernel.const α (MeasureTheory.Measure.sum μ)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : NormedAddCommGroup F'", "inst✝³ : NormedSpace ℝ F'", "inst✝² : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedField 𝕜", "inst✝ : NormedSpace 𝕜 E", "T : Set α → E →L[ℝ] F", "C : ℝ", "hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal", "f : ↥(simpleFunc E 1 μ)"], "goal": "‖SimpleFunc.setToSimpleFunc T (toSimpleFunc f)‖ ≤ C * ∑ x ∈ (toSimpleFunc f).range, (μ (↑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖"}}
+{"state": {"context": ["F : Type u_1", "R✝ : Type u_2", "S : Type u_3", "x y : R✝", "r : ℝ", "inst✝⁴ : NonAssocRing R✝", "inst✝³ : DecidableEq R✝", "inst✝² : NoZeroDivisors R✝", "inst✝¹ : Nontrivial R✝", "R : Type u_4", "inst✝ : Ring R", "f g : MulRingNorm R", "c : ℝ", "hcpos : 0 < c", "h : (fun x => f x ^ c) = ⇑g"], "goal": "0 < 1 / c ∧ (fun x => g x ^ (1 / c)) = ⇑f"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{u_2, u_1} C", "X Y : SimplicialObject C", "s : Splitting X", "Z : C", "Δ : SimplexCategoryᵒᵖ", "F : (A : IndexSet Δ) → s.N (unop A.fst).len ⟶ Z", "A : IndexSet Δ"], "goal": "(s.cofan Δ).inj A ≫ s.desc Δ F = F A"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : UniformSpace α", "inst✝² : Group α", "inst✝¹ : UniformGroup α", "inst✝ : UniformSpace β", "f : β → α", "hf : UniformContinuous f"], "goal": "UniformContinuous fun x => f x ^ 0"}}
+{"state": {"context": ["α : Type w₂", "C : Type u", "CategoryTheory.Category.{v, u} C", "f : α → C", "CategoryTheory.Limits.HasCoproduct f"], "goal": "(CategoryTheory.Limits.Sigma.map fun a => 𝟙 (f a)) = 𝟙 (∐ f)"}}
+{"state": {"context": ["L : Language", "L' : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'", "γ : Type u_3", "f : L.Functions 2", "t₁ t₂ : L.Term α", "v : α → M"], "goal": "(funMap f fun i => realize v (![t₁, t₂] i)) = funMap f ![realize v t₁, realize v t₂]"}}
+{"state": {"context": ["a b : UInt64", "h : a = b"], "goal": "a.val = b.val"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "f : X ⟶ Y", "g : X ⟶ Z", "t : CategoryTheory.Limits.PushoutCocone f g"], "goal": "t.flip.inr = t.inl"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "OrderedRing R", "AddCommGroup V", "Module R V", "AddTorsor V P", "NoZeroSMulDivisors R V", "w x y z : P", "h₁ : Sbtw R w x z", "h₂ : Sbtw R x y z"], "goal": "Sbtw R w y z"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedHeytingAlgebra α", "a b c : α"], "goal": "a ⇨ b ⇨ c = a ⊓ b ⇨ c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PartialOrder α", "PartialOrder β", "x y : α × β"], "goal": "x.swap ⩿ y.swap ↔ x ⩿ y"}}
+{"state": {"context": ["L : FirstOrder.Language", "ι : Type v", "Preorder ι", "G : ι → Type w", "(i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "IsDirected ι fun x x_1 => x ≤ x_1", "DirectedSystem G fun i j h => ⇑(f i j h)", "Nonempty ι", "P : Type u₁", "L.Structure P", "g : (i : ι) → G i ↪[L] P", "Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x"], "goal": "(FirstOrder.Language.DirectLimit.lift L ι G f g Hg).toHom.range = ⨆ i, (g i).toHom.range"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "mβ : MeasurableSpace β", "f : α → β", "hf : MeasurableEmbedding f", "hμν : μ ≪ ν", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "s : Set α", "x✝¹ : MeasurableSet s", "x✝ : ν s < ⊤", "hs_eq : s = f ⁻¹' (f '' s)"], "goal": "∫⁻ (a : β), (map f μ).rnDeriv (map f ν) a ∂map f (ν.restrict s) = μ s"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Mul α", "s t : Finset α"], "goal": "(s * t).Nonempty ↔ s.Nonempty ∧ t.Nonempty"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁴ : Category.{u_3, u_1} C", "B : C", "α : Type u_2", "X : α → C", "π : (a : α) → X a ⟶ B", "inst✝³ : HasCoproduct X", "inst✝² : ∀ {Z : C} (g : Z ⟶ ∐ X) (a : α), HasPullback g (Sigma.ι X a)", "inst✝¹ : ∀ {Z : C} (g : Z ⟶ ∐ X), HasCoproduct fun a => pullback g (Sigma.ι X a)", "inst✝ : ∀ {Z : C} (g : Z ⟶ ∐ X), Epi (Sigma.desc fun a => pullback.fst g (Sigma.ι X a))", "W : C", "e : (a : α) → X a ⟶ W", "h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂", "Z : C", "g₁ g₂ : Z ⟶ ∐ fun b => X b", "hg : g₁ ≫ Sigma.desc π = g₂ ≫ Sigma.desc π", "a b : α"], "goal": "pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.snd g₁ (Sigma.ι X a) ≫ e a = pullback.fst (pullback.fst g₁ (Sigma.ι X a) ≫ g₂) (Sigma.ι X b) ≫ pullback.fst g₁ (Sigma.ι X a) ≫ g₂ ≫ Sigma.desc e"}}
+{"state": {"context": ["r : ℝ"], "goal": "r⁻¹.sign = r.sign"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "CategoryTheory.Category.{v, u} C", "AddGroup A", "CategoryTheory.HasShift C A", "X Y : C", "f : X ⟶ Y", "i : A"], "goal": "(CategoryTheory.shiftFunctor C i).map ((CategoryTheory.shiftFunctor C (-i)).map f) =\n  CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctorCompIsoId C (-i) i ⋯).hom.app X)\n    (CategoryTheory.CategoryStruct.comp f ((CategoryTheory.shiftFunctorCompIsoId C (-i) i ⋯).inv.app Y))"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M"], "goal": "S = ⊥ ↔ ∀ x ∈ S, x = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type x", "UniformSpace β", "F : ι → α → β", "s : Set α", "p : Filter ι", "s' : Set α", "hf : UniformCauchySeqOn F p s", "hss' : s' ⊆ s"], "goal": "UniformCauchySeqOn F p s'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "a b c : Ordinal.{u_4}", "h : b.IsLimit", "H : ∀ b' < b, a * b' ≤ c"], "goal": "¬c < a * b"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "S : Type u_6", "H K : S", "SetLike S G", "AddSubgroupClass S G", "h : H ≤ K", "x : G", "hx : x ∈ H"], "goal": "(AddSubgroupClass.inclusion h) ⟨x, hx⟩ = ⟨x, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "inst✝ : LinearOrder α", "a b : α"], "goal": "compare a b = Ordering.lt ↔ a < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : MeasurableSpace α", "inst✝¹ : TopologicalSpace α", "μ : Measure α", "p q : Set α → Prop", "U s✝ : Set α", "ε r✝ : ℝ≥0∞", "inst✝ : SigmaFinite μ", "s : Set α", "hs : MeasurableSet s", "r : ℝ≥0∞", "hr : r < μ s", "B : ℕ → Set α := spanningSets μ"], "goal": "∃ K ⊆ s, (fun s => MeasurableSet s ∧ μ s ≠ ⊤) K ∧ r < μ K"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "DecidableEq (Associates α)", "(p : Associates α) → Decidable (Irreducible p)", "p : Associates α", "hp : ¬Irreducible p"], "goal": "p.count = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "h : S.HomologyData"], "goal": "CategoryTheory.ShortComplex.homologyMap' (CategoryTheory.CategoryStruct.id S) h h =\n  CategoryTheory.CategoryStruct.id h.left.H"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "inst✝ : SFinite ν", "ρ : Measure (α × β)"], "goal": "(map Prod.swap ρ).snd = ρ.fst"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X : SimplicialObject C", "Y : C", "n q : ℕ", "φ : Y ⟶ X _[n + 1]", "v : HigherFacesVanish q φ", "i : Fin (n + 1)", "hj₁ : i.succ ≠ 0", "hj₂ : n + 2 ≤ ↑i.succ + q"], "goal": "n + 1 ≤ ↑i + q"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "P Q : C", "S T : D", "f : (P, S) ⟶ (Q, T)"], "goal": "CategoryTheory.IsIso f ↔ CategoryTheory.IsIso f.1 ∧ CategoryTheory.IsIso f.2"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "MeasurableSpace X", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure X", "f g : ↥(MeasureTheory.Lp E p μ)", "s : Set X"], "goal": "MeasureTheory.Memℒp.toLp ↑↑(f + g) ⋯ = MeasureTheory.Memℒp.toLp ↑↑f ⋯ + MeasureTheory.Memℒp.toLp ↑↑g ⋯"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CommGroup α", "inst✝ : DecidableEq α", "A✝ B✝ C✝ A B C : Finset α", "hB : B.Nonempty", "hB' : B ∈ B.powerset.erase ∅", "U : Finset α", "hU : U.Nonempty ∧ U ⊆ B", "hUA : ∀ x' ∈ B.powerset.erase ∅, ↑(U * A).card / ↑U.card ≤ ↑(x' * A).card / ↑x'.card"], "goal": "↑((A * C).card * B.card) ≤ ↑((A * B).card * (B * C).card)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "f : Module.End R M", "h : IsUnit f", "x : M"], "goal": "h.unit.inv (f x) = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m✝ : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t✝ : Set α", "s : ℕ → Set α", "hs : ∑' (i : ℕ), μ (s i) ≠ ⊤", "t : ℕ → Set α := fun n => toMeasurable μ (s n)", "ht : ∑' (i : ℕ), μ (t i) ≠ ⊤", "n m : ℕ", "hnm : n ≤ m", "x : α"], "goal": "(∃ i, x ∈ t (i + m)) → ∃ i, x ∈ t (i + n)"}}
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+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Div α", "s : Finset α"], "goal": "s / ∅ = ∅"}}
+{"state": {"context": ["pqr : Multiset ℕ+", "H : E' 3 = pqr"], "goal": "1 < sumInv pqr"}}
+{"state": {"context": ["M : Type u_1", "inst✝¹ : OrderedCancelAddCommMonoid M", "inst✝ : ExistsAddOfLE M", "a b c d : M"], "goal": "BijOn (fun x => x + d) (Ioi a ∩ Iio b) (Ioi (a + d) ∩ Iio (b + d))"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{u_2, u_1} C", "A B X Y : C", "f : A ⟶ X", "i : A ⟶ B", "p : X ⟶ Y", "g : B ⟶ Y", "sq : CommSq f i p g"], "goal": "sq.HasLift → Nonempty sq.LiftStruct"}}
+{"state": {"context": ["n d : ℤ"], "goal": "(n /. d).den = if d = 0 then 1 else d.natAbs / n.gcd d"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "S : Set E", "x : E"], "goal": "x ∈ adjoin F S ↔ ∃ r s, x = (MvPolynomial.aeval Subtype.val) r / (MvPolynomial.aeval Subtype.val) s"}}
+{"state": {"context": ["κ : Cardinal.{w}", "T : FirstOrder.Language.empty.Theory"], "goal": "κ.Categorical T"}}
+{"state": {"context": ["α : Type u", "AddGroup α", "LT α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a b : α", "h : a < b", "c : α"], "goal": "a - c < b - c"}}
+{"state": {"context": ["c : ℂ", "R θ : ℝ"], "goal": "circleMap c R θ = c ↔ R = 0"}}
+{"state": {"context": ["k p : ℕ", "hk : k ≠ 0", "hp : Nat.Prime p"], "goal": "(p ^ k).primeFactors = {p}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a₁ a₂ : α", "b₁ b₂ : β", "SMul α β", "Preorder α", "Preorder β", "Zero α", "Zero β", "PosSMulStrictMono α β", "SMulPosStrictMono α β", "ha : a₁ < a₂", "hb : b₁ < b₂", "h₁ : 0 < a₁", "h₂ : 0 < b₂"], "goal": "a₁ • b₁ < a₂ • b₂"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "c : ℂ", "R : ℝ", "h0 : 0 < R", "f : ℂ → E", "y : E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c R \\ {c})", "hd : ∀ z ∈ (ball c R \\ {c}) \\ s, DifferentiableAt ℂ f z", "hy : Tendsto f (𝓝[≠] c) (𝓝 y)", "ε : ℝ", "ε0 : 0 < ε"], "goal": "‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ ε"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹¹ : AddCommGroup E", "inst✝¹⁰ : UniformSpace E", "inst✝⁹ : UniformAddGroup E", "inst✝⁸ : AddCommGroup F", "inst✝⁷ : UniformSpace F", "inst✝⁶ : FirstCountableTopology E", "inst✝⁵ : RCLike 𝕜", "inst✝⁴ : Module 𝕜 E", "inst✝³ : ContinuousSMul 𝕜 E", "inst✝² : RCLike 𝕜'", "inst✝¹ : Module 𝕜' F", "inst✝ : ContinuousSMul 𝕜' F", "σ : 𝕜 →+* 𝕜'", "f : E →ₛₗ[σ] F", "hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)", "b : ℕ → Set E", "bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)", "bE : (𝓝 0).HasAntitoneBasis fun i => b i", "bE' : (𝓝 0).HasBasis (fun x => x ≠ 0) fun n => (↑n)⁻¹ • b n", "V : Set F", "hV : V ∈ 𝓝 0", "u : ℕ → E", "hu : ∀ (ia : ℕ), ia ≠ 0 → u ia ∈ (↑ia)⁻¹ • b ia", "hu' : ∀ (ia : ℕ), ia ≠ 0 → f (u ia) ∉ V", "h_tendsto : Tendsto (fun n => ↑n • u n) atTop (𝓝 0)"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Encodable β", "f : β → Set α", "C : Set α → Prop", "H0 : C ∅", "H1 : ∀ (b : β), C (f b)", "n : ℕ"], "goal": "C (⋃ b ∈ Encodable.decode₂ β n, f b)"}}
+{"state": {"context": ["R : Type u", "A : Type z", "CommSemiring R", "Semiring A", "Algebra R A", "x : A"], "goal": "(Polynomial.aeval x) Polynomial.X = x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasBinaryBiproducts C", "K₁ L₁ K₂ L₂ : CochainComplex C ℤ", "φ₁ : K₁ ⟶ L₁", "φ₂ : K₂ ⟶ L₂", "a : K₁ ⟶ K₂", "b : L₁ ⟶ L₂", "comm : φ₁ ≫ b = a ≫ φ₂"], "goal": "(CochainComplex.mappingCone.triangleMap φ₁ φ₂ a b comm).hom₁ = a"}}
+{"state": {"context": ["J : Type u₁", "inst✝⁴ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝³ : Category.{v₂, u₂} K", "C : Type u", "inst✝² : Category.{v, u} C", "F✝ F : J ⥤ C", "inst✝¹ : HasColimit F", "G : K ⥤ C", "inst✝ : HasColimit G", "e : J ≌ K", "w : e.functor ⋙ G ≅ F", "j : J"], "goal": "colimit.ι F j ≫ (isoOfEquivalence e w).hom = F.map (e.unit.app j) ≫ w.inv.app ((e.functor ⋙ e.inverse).obj j) ≫ colimit.ι G (e.functor.obj ((e.functor ⋙ e.inverse).obj j))"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "𝒜 ℬ : Finset (Finset α)", "s t : Finset α", "a : α", "n : ℕ", "h : 𝒜.Shatters s"], "goal": "∃ a ∈ 𝒜, s ⊆ a"}}
+{"state": {"context": ["f : (a : Ordinal.{u_4}) → a < 0 → Ordinal.{max u_4 u_5}"], "goal": "Ordinal.bsup 0 f = 0"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : b ⟶ c"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.associator f (CategoryTheory.CategoryStruct.id b) g).hom\n    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.leftUnitor g).hom) =\n  CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.rightUnitor f).hom g"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "δ : ℝ", "hδ : δ ≤ 0", "s : Set α"], "goal": "Metric.thickening δ s = ∅"}}
+{"state": {"context": ["X : Type u_2", "PseudoEMetricSpace X", "δ : ℝ", "δ_pos : 0 < δ", "F : Set X", "x : X", "hxF : x ∈ F"], "goal": "1 ≤ (thickenedIndicator δ_pos F) x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "S : CategoryTheory.Subgroupoid C"], "goal": "S.disconnect.objs = S.objs"}}
+{"state": {"context": ["J : Type w", "C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f : J → (X ⟶ Y)"], "goal": "(CategoryTheory.Limits.parallelFamily f).obj CategoryTheory.Limits.WalkingParallelFamily.zero = X"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "K L : CochainComplex C ℤ", "n : ℤ", "γ : CochainComplex.HomComplex.Cochain K L n", "a n' m' : ℤ", "hn' : n' + a = n", "m : ℤ", "hm' : m' + a = m"], "goal": "CochainComplex.HomComplex.δ n' m' (γ.rightShift a n' hn') =\n  a.negOnePow • (CochainComplex.HomComplex.δ n m γ).rightShift a m' hm'"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Multiset α"], "goal": "t ≤ s ∪ t"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p : R[X]", "hroots : card p.roots = p.natDegree", "k : ℕ", "h : k ≤ p.natDegree"], "goal": "p.leadingCoeff * (map (fun a => X - C a) p.roots).prod.coeff k = p.leadingCoeff * ((-1) ^ (p.natDegree - k) * p.roots.esymm (p.natDegree - k))"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I J : Set α", "hI : M.Indep I", "hJ : M.Basis J I"], "goal": "J = I"}}
+{"state": {"context": [], "goal": "Set.MapsTo Real.sin (Set.Ioo (-(π / 2)) (π / 2)) (Set.Ioo (-1) 1)"}}
+{"state": {"context": ["R : Type u_1", "m : Type u_3", "n : Type u_4", "α : Type u_5", "Fintype m", "Fintype n", "SeminormedRing R", "SeminormedAddCommGroup α", "Module R α", "BoundedSMul R α"], "goal": "BoundedSMul R (Matrix m n α)"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R"], "goal": "W.φ 2 = C X * W.ψ₂ ^ 2 - C W.Ψ₃"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "e : PartialHomeomorph M M'", "he : PartialHomeomorph.MDifferentiable I I' e", "SmoothManifoldWithCorners I M", "SmoothManifoldWithCorners I' M'", "x : M", "hx : x ∈ e.source"], "goal": "Function.Surjective ⇑(mfderiv I I' (↑e) x)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a b : α"], "goal": "Relation.ReflTransGen r a b ↔ b = a ∨ Relation.TransGen r a b"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "β : Type u_3", "ι : Type u_4", "inst✝ : TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s s₁ s₂ : Set α", "u : Set τ", "hu : u ∈ f", "a✝ : β", "hx : ∀ (i : ↑f.sets), a✝ ∈ closure (image2 ϕ (↑i) s)"], "goal": "a✝ ∈ closure (image2 ϕ u s)"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type uX", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t u✝ : Set E", "f f₁ : E → F", "g : F → G", "x✝ x₀ : E", "c : F", "m n✝ : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "n : ℕ", "hs : UniqueDiffOn 𝕜 s", "H : ContDiffOn 𝕜 (↑(n + 1)) f s", "x : E", "hx : x ∈ s", "u : Set E", "hu : u ∈ 𝓝[insert x s] x", "f' : E → E →L[𝕜] F", "hff' : ∀ x ∈ u, HasFDerivWithinAt f (f' x) u x", "hf' : ContDiffWithinAt 𝕜 (↑n) f' u x", "o : Set E", "o_open : IsOpen o", "xo : x ∈ o", "ho : s ∩ o ⊆ u", "this : ContDiffWithinAt 𝕜 (↑n) f' s x"], "goal": "(fun y => fderivWithin 𝕜 f s y) =ᶠ[𝓝[s] x] f'"}}
+{"state": {"context": ["R✝ : Type u₁", "inst✝⁶ : CommSemiring R✝", "p : ℕ", "hp : Fact (Nat.Prime p)", "inst✝⁵ : CharP R✝ p", "R : Type u₁", "inst✝⁴ : CommSemiring R", "inst✝³ : CharP R p", "inst✝² : PerfectRing R p", "S : Type u₂", "inst✝¹ : CommSemiring S", "inst✝ : CharP S p", "f : R →+* S", "r : R", "n : ℕ"], "goal": "(fun n => f ((⇑↑(frobeniusEquiv R p).symm)^[n] r)) (n + 1) ^ p = (fun n => f ((⇑↑(frobeniusEquiv R p).symm)^[n] r)) n"}}
+{"state": {"context": ["ι : Type u_4", "ι' : Type u_5", "β : ι' → Type u_6", "(j : ι') → UniformSpace (β j)", "e : ι ≃ ι'"], "goal": "∀ (a : (b : ι) → β (e.symm.symm b)) (a_1 : ι'),\n  (UniformEquiv.piCongrLeft e) a a_1 = (Equiv.piCongrLeft' β e.symm).symm a a_1"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommRing R", "S : Type v", "inst✝¹ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝ : IsDedekindDomain S", "hp0 : map f p ≠ ⊥", "hP : P.IsPrime", "le : map f p ≤ P", "hP0 : P ≠ ⊥", "hPirr : Irreducible P", "P' : Ideal S", "hP' : P' ∈ normalizedFactors (map f p)", "P'_eq : Associated P P'"], "goal": "Multiset.count P (normalizedFactors (map f p)) ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ", "η : ProbabilityTheory.Kernel α γ", "ProbabilityTheory.IsSFiniteKernel η", "a : α", "s : Set (β × γ)", "hs : MeasurableSet s"], "goal": "((κ.prod η) a) s = ∫⁻ (b : β), (η a) {c | (b, c) ∈ s} ∂κ a"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁸ : TopologicalSpace α", "inst✝⁷ : NormalSpace α", "inst✝⁶ : R1Space α", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : BorelSpace α", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "μ : Measure α", "p : ℝ≥0∞", "inst✝² : NormedSpace ℝ E", "inst✝¹ : WeaklyLocallyCompactSpace α", "inst✝ : μ.Regular", "hp : p ≠ ⊤", "f : α → E", "hf : Memℒp f p μ", "ε✝ : ℝ≥0∞", "hε✝ : ε✝ ≠ 0", "c : E", "t : Set α", "ht : MeasurableSet t", "htμ : μ t < ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → eLpNorm f p μ ≤ δ → eLpNorm g p μ ≤ δ → eLpNorm (f + g) p μ < ε", "η : ℝ≥0", "ηpos : 0 < η", "hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x => c) p μ ≤ δ", "hη_pos' : 0 < ↑η", "s : Set α", "st : s ⊆ t", "s_compact : IsCompact s", "s_closed : IsClosed s", "μs : μ (t \\ s) < ↑η", "hsμ : μ s < ⊤", "I1 : eLpNorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ", "k : Set α", "k_compact : IsCompact k", "sk : s ⊆ interior k"], "goal": "∃ g, eLpNorm (g - t.indicator fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ HasCompactSupport g"}}
+{"state": {"context": ["X : Type u", "Preorder X", "x : Xᵒᵖ"], "goal": "(CategoryTheory.orderDualEquivalence X).inverse.obj x = OrderDual.toDual (Opposite.unop x)"}}
+{"state": {"context": ["B : Type u_1", "B' : Type u_2", "e : B ≃ B'", "W : Type u_3", "H : Type u_4", "inst✝¹ : Group W", "inst✝ : Group H", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "p : W → Prop", "w : W", "one : p 1", "mul_simple_left : ∀ (w : W) (i : B), p w → p (cs.simple i * w)", "p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop := fun w x => p w", "this : w ∈ Submonoid.closure (Set.range cs.simple)"], "goal": "p' 1 ⋯"}}
+{"state": {"context": ["J : Type v'", "inst✝⁷ : Category.{u', v'} J", "C : Type u", "inst✝⁶ : Category.{v, u} C", "W✝ X✝ Y✝ Z✝ : C", "f✝ : W✝ ⟶ X✝", "g✝ : W✝ ⟶ Y✝", "h✝ : X✝ ⟶ Z✝", "i✝ : Y✝ ⟶ Z✝", "D : Type u''", "inst✝⁵ : Category.{v'', u''} D", "F : C ⥤ D", "inst✝⁴ : Adhesive D", "H₁ : ∀ {X Y S : C} (f : X ⟶ S) (g : Y ⟶ S) [inst : Mono f], HasPullback f g", "H₂ : ∀ {X Y S : C} (f : S ⟶ X) (g : S ⟶ Y) [inst : Mono f], HasPushout f g", "inst✝³ : PreservesLimitsOfShape WalkingCospan F", "inst✝² : ReflectsLimitsOfShape WalkingCospan F", "inst✝¹ : PreservesColimitsOfShape WalkingSpan F", "inst✝ : ReflectsColimitsOfShape WalkingSpan F", "W X Y Z : C", "f : W ⟶ X", "g : W ⟶ Y", "h : X ⟶ Z", "i : Y ⟶ Z", "hf : Mono f", "H : IsPushout f g h i"], "goal": "IsVanKampenColimit ((Cocones.precompose (diagramIsoSpan (span f g ⋙ F)).inv).obj (F.mapCocone (PushoutCocone.mk h i ⋯)))"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y X₁ Y₁ : C", "α : X ≅ X₁", "β : Y ≅ Y₁", "f : X₁ ⟶ Y₁"], "goal": "α.inv ≫ (α.hom ≫ f ≫ β.inv) ≫ β.hom = f"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "ι : Type u_3", "α : Type u_4", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "x : ℕ"], "goal": "x ≤ ⌊max 0 (↑x + 1)⌋₊"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "n : ℕ", "F : CategoryTheory.ComposableArrows C n", "X : C", "f : X ⟶ F.left"], "goal": "CategoryTheory.ComposableArrows.Precomp.map F f 0 0 ⋯ = CategoryTheory.CategoryStruct.id X"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹⁰ : CommSemiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝⁹ : Fintype n", "inst✝⁸ : Finite m", "inst✝⁷ : DecidableEq n", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁶ : AddCommMonoid M₁", "inst✝⁵ : AddCommMonoid M₂", "inst✝⁴ : Module R M₁", "inst✝³ : Module R M₂", "v₁ : Basis n R M₁", "v₂ : Basis m R M₂", "G : Type u_7", "inst✝² : Group G", "inst✝¹ : DistribMulAction G M₁", "inst✝ : SMulCommClass G R M₁", "g : G", "f : M₁ →ₗ[R] M₂", "i✝ : m", "j✝ : n"], "goal": "(toMatrix (g • v₁) v₂) f i✝ j✝ = (toMatrix v₁ v₂) (f ∘ₗ DistribMulAction.toLinearMap R M₁ g) i✝ j✝"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "Algebra R A", "StarRing A", "StarModule R A", "S : Subalgebra R A"], "goal": "↑(star S) = star ↑S"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "s₁✝ s₂✝ : Finmap β", "s₁ s₂ : AList β", "h : ⟦s₁⟧.Disjoint ⟦s₂⟧"], "goal": "⟦s₁⟧ ∪ ⟦s₂⟧ = ⟦s₂⟧ ∪ ⟦s₁⟧"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "val : β"], "goal": "(RelEmbedding.sumLiftRelInr r s) val = Sum.inr val"}}
+{"state": {"context": ["f g : CircleDeg1Lift"], "goal": "dist (f 0 + g 0) (f (g 0)) < 1"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "β₂ : Type u_3", "γ : Type u_4", "ι : Sort u_5", "ι' : Sort u_6", "κ : ι → Sort u_7", "κ' : ι' → Sort u_8", "α : Type u_9", "s : Set (α → Prop)", "a : α"], "goal": "sInf s a ↔ ∀ r ∈ s, r a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_5", "ι : Type u_8", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "Nonempty ι", "TopologicalSpace E", "p : SeminormFamily 𝕜 E ι", "hp : WithSeminorms p", "x : E"], "goal": "(𝓝 x).HasBasis (fun sr => 0 < sr.2) fun sr => (sr.1.sup p).ball x sr.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "r : α → β → Prop", "b : Multiset β"], "goal": "Multiset.Rel r 0 b ↔ b = 0"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s s₁ s₂ t : Set X", "p p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X"], "goal": "closure (s ∪ t) = closure s ∪ closure t"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "s : Subfield K"], "goal": "↑0 = 0"}}
+{"state": {"context": ["X : Type u_1", "X' : Type u_2", "Y : Type u_3", "Y' : Type u_4", "Z : Type u_5", "Z' : Type u_6", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : TopologicalSpace X'", "inst✝³ : TopologicalSpace Y", "inst✝² : TopologicalSpace Y'", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "e : PartialHomeomorph X Y", "s : Set X", "hs : IsOpen s", "hse : s ⊆ e.source"], "goal": "IsOpen (↑e '' s)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y Z : C", "f g : X ⟶ Y", "h : Y ⟶ Z"], "goal": "(f - g) ≫ h = f ≫ h - g ≫ h"}}
+{"state": {"context": ["ι : Type u_2", "α : ι → Type u_3", "Fintype ι", "(i : ι) → Dist (α i)", "p : ℝ≥0∞", "hp : 0 < p.toReal", "f g : PiLp p α"], "goal": "dist f g = (∑ i : ι, dist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E ≃L[𝕜] F", "s : Set E", "c : ℝ≥0", "hf : ApproximatesLinearOn f (↑f') s c", "hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹"], "goal": "AntilipschitzWith (‖↑f'.symm‖₊⁻¹ - c)⁻¹ (s.restrict f)"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "inst✝¹ : Infinite F", "E : Type u_2", "inst✝ : Field E", "ϕ : F →+* E", "α β : E", "f g : F[X]", "sf : Multiset E := (Polynomial.map ϕ f).roots", "sg : Multiset E := (Polynomial.map ϕ g).roots", "s : Finset E := (sf.bind fun α' => Multiset.map (fun β' => -(α' - α) / (β' - β)) sg).toFinset", "s' : Finset F := s.preimage ⇑ϕ ⋯", "c : F", "hc : ¬∃ a ∈ sf, ∃ a_1 ∈ sg, -(a - α) / (a_1 - β) = ϕ c"], "goal": "∃ c, ∀ α' ∈ (Polynomial.map ϕ f).roots, ∀ β' ∈ (Polynomial.map ϕ g).roots, -(α' - α) / (β' - β) ≠ ϕ c"}}
+{"state": {"context": ["α : Type u", "AddCommMonoid α", "Mul α", "a₀ a₁ b₀ b₁ : α"], "goal": "Matrix.dotProduct ![a₀, a₁] ![b₀, b₁] = a₀ * b₀ + a₁ * b₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "A : Type u_3", "B : Type u_4", "F : Type u_5", "inst✝³ : AddMonoidWithOne B", "inst✝² : FunLike F ℕ A", "inst✝¹ : AddMonoidWithOne A", "inst✝ : AddMonoidHomClass F ℕ A", "f : F", "h1 : f 1 = 1"], "goal": "f 0 = ↑0"}}
+{"state": {"context": ["R : Type u", "inst✝⁴ : CommRing R", "n G : Type v", "inst✝³ : DecidableEq n", "inst✝² : Fintype n", "α β : Type v", "inst✝¹ : DecidableEq α", "M✝ : Matrix n n R", "p✝ : ℕ", "inst✝ : Fact (Nat.Prime p✝)", "M : Matrix n n R", "a✝ : Nontrivial R", "t : R[T;T⁻¹] := T 1", "t_inv : R[T;T⁻¹] := T (-1)", "p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det", "q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det", "ht : t_inv * t = 1", "hp : toLaurentAlg M.charpoly = p", "hq : toLaurentAlg M.charpolyRev = q"], "goal": "(1 - t_inv • M.map ⇑LaurentPolynomial.C).det = (invert.mapMatrix (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C)).det"}}
+{"state": {"context": ["α : Type u", "Semiring α", "M M' : Ideal α", "hM : M.IsMaximal", "hM' : M'.IsMaximal", "hne : M ≠ M'"], "goal": "M ⊔ M' = ⊤"}}
+{"state": {"context": ["X : Type u_5", "t : TopologicalSpace X", "self : TopologicalSpace.PseudoMetrizableSpace X"], "goal": "∃ m, UniformSpace.toTopologicalSpace = t"}}
+{"state": {"context": ["ι : Type u_2", "α : ι → Type u_3", "(i : ι) → AddGroup (α i)", "(i : ι) → Lattice (α i)", "f : (i : ι) → α i"], "goal": "|f| = fun i => |f i|"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Preorder α", "a : α", "inst✝ : SuccOrder α", "ha : ¬IsMax a"], "goal": "¬IsSuccLimit (succ a)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "AddCommMonoid M", "f g : α → M", "s t : Set α", "h₀ : s = t", "h₁ : ∀ x ∈ t, f x = g x"], "goal": "∑ᶠ (i : α) (_ : i ∈ s), f i = ∑ᶠ (i : α) (_ : i ∈ t), g i"}}
+{"state": {"context": ["F : PFunctor.{u}", "r : F.M → Sort u_2", "x : ↑F F.M", "f : (x : ↑F F.M) → r (PFunctor.M.mk x)"], "goal": "(PFunctor.M.mk x).casesOn f = f x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "X : C × D × E"], "goal": "(CategoryTheory.prod.inverseAssociator C D E).obj X = ((X.1, X.2.1), X.2.2)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "P : Set X → Prop", "c : Urysohns.CU P"], "goal": "c.left.C = c.C"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "κ' : Kernel α (γ × β)", "hκκ' : κ ≤ κ'", "hκ'ν : κ'.fst ≤ ν", "n : ℕ", "a : α", "x : γ", "s : Set β", "h0 : ¬(ν a) (countablePartitionSet n x) = 0", "h_le : (κ' a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)"], "goal": "((κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal ≤ ((κ' a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x)).toReal"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β i)", "inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)", "i : ι", "f : (i : ι) → β i", "s : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }"], "goal": "i ∈ { toFun := f, support' := Trunc.mk s }.support ↔ { toFun := f, support' := Trunc.mk s } i ≠ 0"}}
+{"state": {"context": ["G : Type u_1", "MeasurableSpace G", "Group G", "MeasurableMul₂ G", "μ ν : MeasureTheory.Measure G", "MeasureTheory.SFinite ν", "MeasureTheory.SFinite μ", "MeasurableInv G", "μ.IsMulRightInvariant"], "goal": "MeasureTheory.Measure.QuasiMeasurePreserving (fun p => p.1 / p.2) (μ.prod ν) μ"}}
+{"state": {"context": ["α : Type u_2"], "goal": "TopologicalSpace.IsTopologicalBasis (Set.range (Set.Iic ∘ 𝓟))"}}
+{"state": {"context": ["R : Type u", "ι : Type w", "s : Finset ι", "CommSemiring R", "f : ι → R[X]"], "goal": "(∏ i ∈ s, f i).coeff 0 = ∏ i ∈ s, (f i).coeff 0"}}
+{"state": {"context": ["X : Type u_1", "inst✝ : UniformSpace X", "i : (𝓤 X).IsCountablyGenerated", "s : Set X", "this : PseudoEMetricSpace X := PseudoMetricSpace.toPseudoEMetricSpace", "h : ∀ ε > 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, EMetric.ball y ε"], "goal": "TopologicalSpace.IsSeparable s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l✝ l₁ l₂ : List α", "a t : α", "ts is : List α", "IH1 : ∀ (l : List α), l ~ t :: is ++ ts → (∃ is', ∃ (_ : is' ~ t :: is), l = is' ++ ts) ∨ l ∈ ts.permutationsAux (t :: is)", "IH2 : ∀ (l : List α), l ~ [] ++ is → (∃ is', ∃ (_ : is' ~ []), l = is' ++ is) ∨ l ∈ is.permutationsAux []", "l : List α", "p : l ~ is ++ t :: ts"], "goal": "(∃ is', ∃ (_ : is' ~ is), l = is' ++ t :: ts) ∨ l ∈ ts.permutationsAux (t :: is) ∨ ∃ l₁ l₂, l₁ ++ l₂ ∈ is.permutations ∧ l₂ ≠ [] ∧ l = l₁ ++ t :: l₂ ++ ts"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "inst✝¹ : LieRing L'", "inst✝ : LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "I₁ I₂ : LieIdeal R L", "x y₁ : L", "hy₁ : y₁ ∈ I₁", "y₂ : L", "hy₂ : y₂ ∈ I₂", "hx : ⁅↑⟨y₁, hy₁⟩, ↑⟨y₂, hy₂⟩⁆ = x"], "goal": "⁅↑⟨y₁, hy₁⟩, ↑⟨y₂, hy₂⟩⁆ ∈ ↑(comap f ⁅map f I₁, map f I₂⁆)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "r : α → β → Prop"], "goal": "StrictAnti (Prod.snd ∘ Concept.toProd)"}}
+{"state": {"context": ["α : Type u_1", "β✝ : α → Type u", "δ : α → Sort v", "inst✝ : DecidableEq α", "s✝ : Finset α", "t : (a : α) → Finset (β✝ a)", "β : Type u_2", "s : Finset α", "f : α → β", "a : (a : α) → a ∈ s → β", "ha : a ∈ s.pi fun a => {f a}", "i : α", "hi : i ∈ s"], "goal": "a i hi = f i"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasBinaryBiproducts C", "P Q R : C"], "goal": "(CategoryTheory.Limits.biprod.associator P Q R).inv =\n  CategoryTheory.Limits.biprod.lift\n    (CategoryTheory.Limits.biprod.lift CategoryTheory.Limits.biprod.fst\n      (CategoryTheory.Limits.biprod.snd ≫ CategoryTheory.Limits.biprod.fst))\n    (CategoryTheory.Limits.biprod.snd ≫ CategoryTheory.Limits.biprod.snd)"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.SmallCategory J", "F : J ⥤ CommRingCat", "x : CommRingCat.Colimits.Prequotient F"], "goal": "Quot.mk Setoid.r x.neg = -Quot.mk Setoid.r x"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ ν κ : MeasureTheory.Measure α", "MeasureTheory.SigmaFinite μ", "MeasureTheory.SigmaFinite ν", "MeasureTheory.SigmaFinite κ", "hμν : μ ≪ ν"], "goal": "μ.rnDeriv ν * ν.rnDeriv κ =ᵐ[κ] μ.rnDeriv κ"}}
+{"state": {"context": ["α : Type u_2", "I : Type u_1", "inst✝² : CommMonoid α", "inst✝¹ : DecompositionMonoid α", "x y z : α", "s : I → α", "t : Finset I", "inst✝ : DecidableEq I"], "goal": "Pairwise (IsRelPrime on fun i => s ↑i) ↔ ∀ i ∈ t, IsRelPrime (s i) (∏ j ∈ t \\ {i}, s j)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "CommRing S", "Algebra R S", "P : Polynomial R", "x : S", "h : IsNilpotent ((Polynomial.aeval x) P)", "n : ℕ"], "goal": "IsNilpotent (P.newtonMap^[n] x - x)"}}
+{"state": {"context": ["α : Type u", "HasSSubset α", "IsIrrefl α fun x x_1 => x ⊂ x_1", "a : α"], "goal": "¬a ⊂ a"}}
+{"state": {"context": ["B : Type u_1", "B' : Type u_2", "e : B ≃ B'", "W : Type u_3", "H : Type u_4", "inst✝¹ : Group W", "inst✝ : Group H", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "this : Set.range cs.simple = Set.range cs.simple ∪ (Set.range cs.simple)⁻¹"], "goal": "Submonoid.closure (Set.range cs.simple) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f✝ g f : Perm α"], "goal": "f.support.card = 0 ↔ f = 1"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "δ : Type u_2", "π : ι → Type u_3", "r : α → α → Prop", "inst✝³ : Preorder α", "inst✝² : Nonempty α", "inst✝¹ : NoMinOrder α", "inst✝ : NoMaxOrder α", "inhabited_h : Inhabited α", "f : ℕ → α", "hf : StrictMono f", "hf₀ : f 0 = default"], "goal": "∃ f, StrictMono f"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "R I J✝ X Y B J : Set α", "e : α", "hI : M.Basis I X", "hJ : M.Basis J Y", "hJX : J ⊆ X", "hIY : I ⊆ Y"], "goal": "M.Basis I Y"}}
+{"state": {"context": ["G : Type u_1", "Group G", "Fintype G", "x : G"], "goal": "x ^ Fintype.card G = 1"}}
+{"state": {"context": ["α : Type u", "Preorder α", "IsDirected α fun x x_1 => x ≥ x_1", "s t : Set α"], "goal": "BddBelow (s ∪ t) ↔ BddBelow s ∧ BddBelow t"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "Z : D", "F : CategoryTheory.Functor C D", "M : (x : C) → Z ⟶ F.obj x", "hM : ∀ (x y : C) (f : x ⟶ y), CategoryTheory.CategoryStruct.comp (M x) (F.map f) = M y", "X : C"], "goal": "(CategoryTheory.WithInitial.inclLift F M hM).hom.app X =\n  CategoryTheory.CategoryStruct.id\n    (match CategoryTheory.WithInitial.incl.obj X with\n    | CategoryTheory.WithInitial.of x => F.obj x\n    | CategoryTheory.WithInitial.star => Z)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹³ : NormedAddCommGroup E", "inst✝¹² : NormedSpace ℝ E", "inst✝¹¹ : FiniteDimensional ℝ E", "F : Type u_2", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "inst✝⁸ : CompleteSpace F", "H : Type u_3", "inst✝⁷ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_4", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "inst✝⁴ : SmoothManifoldWithCorners I M", "inst✝³ : MeasurableSpace M", "inst✝² : BorelSpace M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "f f' : M → F", "μ : Measure M", "hf : LocallyIntegrable f μ", "h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → ∫ (x : M), g x • f x ∂μ = 0", "this✝³ : LocallyCompactSpace H", "this✝² : LocallyCompactSpace M", "this✝¹ : SecondCountableTopology H", "this✝ : SecondCountableTopology M", "this : MetrizableSpace M", "x✝ : MetricSpace M := metrizableSpaceMetric M", "s : Set M", "hs : IsCompact s", "δ : ℝ", "δpos : 0 < δ", "hδ : IsCompact (cthickening δ s)"], "goal": "0 = ∫ (x : M) in s, f x ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "p : α → Prop", "inst✝ : DecidablePred p", "s : Multiset α"], "goal": "filter p s + filter (fun a => ¬p a) s = s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "x y : R", "inst✝ : MonoidWithZero R", "hx : nilpotencyClass x = 1"], "goal": "x = 0"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : IsDomain B", "inst✝² : Algebra A B", "K : Type u_6", "inst✝¹ : Field K", "inst✝ : Algebra K S", "x : S", "h : IsIntegral K x", "g : Fin (minpoly K x).natDegree → K", "hg : ∑ i : Fin (minpoly K x).natDegree, g i • x ^ ↑i = 0", "i : Fin (minpoly K x).natDegree"], "goal": "g i = 0"}}
+{"state": {"context": ["β : Type u_2", "AddCommMonoid β", "f : Fin 8 → β"], "goal": "∑ i : Fin 8, f i = f 0 + f 1 + f 2 + f 3 + f 4 + f 5 + f 6 + f 7"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p : ℕ", "f : R[X]"], "goal": "((Polynomial.expand R p) f).natDegree = f.natDegree * p"}}
+{"state": {"context": ["n : ℤ"], "goal": "↑n = 1 ↔ n = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → StieltjesFunction", "inst✝ : IsFiniteKernel κ", "hf : IsCondKernelCDF f κ ν", "a : α", "x : ℝ", "s : Set β", "hs : MeasurableSet s"], "goal": "∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) (Iic x) ∂ν a = (κ a) (s ×ˢ Iic x)"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "m : ℤ", "n : ℕ"], "goal": "(WittVector.wittZSMul p m n).vars ⊆ Finset.univ ×ˢ Finset.range (n + 1)"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "W : Type v", "G : SimpleGraph V", "G₁ G₂ : G.Subgraph", "a b : V", "G' : G.Subgraph", "v : V", "inst✝ : Fintype ↑(G'.neighborSet v)"], "goal": "(∃! a, a ∈ (G'.neighborSet v).toFinset) ↔ ∃! w, G'.Adj v w"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "G : Type wG", "Fintype ι", "NontriviallyNormedField 𝕜", "SeminormedAddCommGroup G", "NormedSpace 𝕜 G", "z : G"], "goal": "‖ContinuousMultilinearMap.mkPiRing 𝕜 ι z‖ = ‖z‖"}}
+{"state": {"context": ["g✝ : SL(2, ℤ)", "z : ℍ", "p : Fin 2 → ℤ", "hp : IsCoprime (p 0) (p 1)", "this : (↑(p 0) ^ 2 + ↑(p 1) ^ 2)⁻¹ ≠ 0", "f : ℝ ≃ₜ ℝ := Homeomorph.mulRight₀ (↑(p 0) ^ 2 + ↑(p 1) ^ 2)⁻¹ this", "ff : ℝ ≃ₜ ℝ := Homeomorph.addRight ((↑(p 1) * ↑z - ↑(p 0)) / ((↑(p 0) ^ 2 + ↑(p 1) ^ 2) * (↑(p 0) * ↑z + ↑(p 1)))).re", "e_1✝ : ↑((fun g => ↑g 1) ⁻¹' {p}) = { g // ↑g 1 = p }", "g : ↑((fun g => ↑g 1) ⁻¹' {p})"], "goal": "(↑g • z).re = (lcRow0 p) ↑((SpecialLinearGroup.map (Int.castRingHom ℝ)) ↑g) / (↑(p 0) ^ 2 + ↑(p 1) ^ 2) + ((↑(p 1) * ↑z - ↑(p 0)) / ((↑(p 0) ^ 2 + ↑(p 1) ^ 2) * (↑(p 0) * ↑z + ↑(p 1)))).re"}}
+{"state": {"context": ["b✝ x y a b c : ℝ", "h₁ : b ≠ 0", "h₂ : b ≠ 1", "h₃ : b ≠ -1"], "goal": "log b / log a * (log c / log b) = log c / log a"}}
+{"state": {"context": ["G : Type u_1", "Group G", "N : Type u_5", "Group N", "K : Subgroup N", "f : G →* N"], "goal": "↑(Subgroup.comap f K) = ⇑f ⁻¹' ↑K"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "inst✝² : CommRing K", "inst✝¹ : IsDomain K", "inst✝ : Fintype Kˣ"], "goal": "∏ x : Kˣ, x = -1"}}
+{"state": {"context": ["α : Type u_2", "AddGroup α", "b : α"], "goal": "Finset.preimage 0 (fun x => x + -b) ⋯ = {b}"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "h : S.HomologyData"], "goal": "(CategoryTheory.ShortComplex.HomologyMapData.id h).right = CategoryTheory.ShortComplex.RightHomologyMapData.id h.right"}}
+{"state": {"context": ["R : Type u_1", "NormedCommRing R", "CompleteSpace R", "f : ℕ → R", "hsum : Summable f", "hf₀ : f 0 = 0", "ε : ℝ", "εpos : 0 < ε"], "goal": "∃ N, ∀ (s : Finset ℕ), N.primesBelow ≤ s → ‖∑' (m : ℕ), f m - ∑' (m : ↑(Nat.factoredNumbers s)), f ↑m‖ < ε"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "u v w x : V", "p : G.Walk u v", "h : G.Adj v w", "q : G.Walk w x"], "goal": "(p.concat h).append q = p.append (cons h q)"}}
+{"state": {"context": ["G : Type u_1", "a b : G", "Group G", "s : Set G", "hs : IsSubgroup s", "h : a ∈ s"], "goal": "a * b ∈ s ↔ b ∈ s"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "H : Subgroup G", "inst✝ : H.Normal"], "goal": "∀ (n : ℕ), upperCentralSeries G n ≤ upperCentralSeries G (n + 1)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "b : β a", "l₁ l₂ : List (Sigma β)", "nd₁ : l₁.NodupKeys", "p : l₁.Perm l₂"], "goal": "(List.kinsert a b l₁).Perm (List.kinsert a b l₂)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "self : X ≃ₕ Y"], "goal": "(self.invFun.comp self.toFun).Homotopic (ContinuousMap.id X)"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "a b : E", "r : ℝ"], "goal": "b ∈ Metric.ball a r ↔ ‖b - a‖ < r"}}
+{"state": {"context": ["r : ℝ≥0"], "goal": "↑r ≠ 0 ↔ r ≠ 0"}}
+{"state": {"context": ["a b : Cardinal.{v}"], "goal": "Cardinal.lift.{u, v} (max a b) = max (Cardinal.lift.{u, v} a) (Cardinal.lift.{u, v} b)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "S : Type u_8", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "AddCommMonoid S", "Module R M", "Module R N", "Module R P", "Module R Q", "Module R S", "φ ψ : TensorProduct R (TensorProduct R M N) (TensorProduct R P Q) →ₗ[R] S", "H : ∀ (w : M) (x : N) (y : P) (z : Q), φ ((w ⊗ₜ[R] x) ⊗ₜ[R] y ⊗ₜ[R] z) = ψ ((w ⊗ₜ[R] x) ⊗ₜ[R] y ⊗ₜ[R] z)"], "goal": "φ = ψ"}}
+{"state": {"context": ["u : XgcdType", "a b : ℕ+", "this : 0 < ↑a"], "goal": "(↑a - 1 + 1 + 0, 0 + (↑b - 1 + 1)) = (↑a, ↑b)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "NoZeroDivisors R", "p q : R[X]", "h1 : p ∣ q", "h2 : q ≠ 0"], "goal": "p.degree ≤ q.degree"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "s : AffineSubspace k P", "h : (↑s).Nonempty"], "goal": "s.direction = ⊤ ↔ s = ⊤"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "R✝ I J X Y R : Set α", "hR : R ⊆ M.E", "hX : (M ↾ R).Dep X"], "goal": "M.Dep X"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t : Finset α", "P : Finpartition s", "hP : P.IsEquipartition", "f : { x // x ∈ s } ≃ (t : { x // x ∈ P.parts }) × Fin (↑t).card", "hf : ∀ (a b : { x // x ∈ s }), P.part ↑a = P.part ↑b ↔ (f a).fst = (f b).fst", "g : { x // x ∈ P.parts } ≃ Fin P.parts.card", "hg : ∀ (t : { x // x ∈ P.parts }), (↑t).card = s.card / P.parts.card + 1 ↔ ↑(g t) < s.card % P.parts.card"], "goal": "∃ f, ∀ (a b : { x // x ∈ s }), P.part ↑a = P.part ↑b ↔ ↑(f a) % P.parts.card = ↑(f b) % P.parts.card"}}
+{"state": {"context": ["K : Type u₁", "inst✝² : Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "inst✝¹ : CommRing O", "inst✝ : Algebra O K", "hv : v.Integers O", "p : ℕ", "hp : Fact (Nat.Prime p)", "hvp : Fact (v ↑p ≠ 1)", "x y : ModP K v O hv p", "hxy0 : x * y ≠ 0"], "goal": "preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "H : Type u_4", "A : Type u_5", "B : Type u_6", "α : Type u_7", "β : Type u_8", "γ : Type u_9", "δ : Type u_10", "inst✝⁷ : Monoid M", "inst✝⁶ : MulAction M α", "inst✝⁵ : Group G", "inst✝⁴ : MulAction G α", "g✝ : G", "a✝ b : α", "inst✝³ : Group H", "inst✝² : MulAction G H", "inst✝¹ : SMulCommClass G H H", "inst✝ : IsScalarTower G H H", "g : G", "a : H", "n : ℤ"], "goal": "(g • a) ^ n = g ^ n • a ^ n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "MulOneClass α", "AddZeroClass β", "f : Additive α →+ β", "a : α"], "goal": "(AddMonoidHom.toMultiplicative' f) a = Multiplicative.ofAdd (f (Additive.ofMul a))"}}
+{"state": {"context": ["α : Type u_1", "Semiring α", "a b c : α", "m n : ℕ", "ha : c ^ m ∣ a", "hb : c ^ n ∣ b"], "goal": "c ^ min m n ∣ a + b"}}
+{"state": {"context": ["α : Type u", "DivisionCommMonoid α", "c : α", "h : IsUnit c", "a b : α"], "goal": "c * a / (c * b) = a / b"}}
+{"state": {"context": ["α : Type u_1", "A : Set α", "β : Type u_2", "Zero β", "TopologicalSpace β", "ι : Type u_3", "L : Filter ι", "As : ι → Set α", "b : β", "nhd_b : {0}ᶜ ∈ 𝓝 b", "nhd_o : {b}ᶜ ∈ 𝓝 0"], "goal": "Filter.Tendsto (fun i => (As i).indicator fun x => b) L (𝓝 (A.indicator fun x => b)) ↔\n  ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "u : ℕ → ℝ", "c : ℝ", "hc : 0 ≤ c", "n : ℕ", "hn : 0 < n", "h : ∀ k < n, c * u k < u (k + 1)"], "goal": "c ^ n * u 0 < u n"}}
+{"state": {"context": ["α : Type u_1", "Add α", "a x y : αᵃᵒᵖ"], "goal": "AddSemiconjBy (AddOpposite.unop a) (AddOpposite.unop y) (AddOpposite.unop x) ↔ AddSemiconjBy a x y"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "PartialOrder Γ", "AddCommMonoid R", "α : Type u_3", "β : Type u_4", "s : HahnSeries.SummableFamily Γ R α", "f : α ↪ β", "a : α"], "goal": "(s.embDomain f) (f a) = s a"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x y z : X", "ι : Type u_3", "γ✝ : Path x y", "a b : X", "γ : Path a b", "t₀ t₁ : ℝ"], "goal": "∀ (x : ℝ) (h : x ∈ I), (γ.truncate t₀ t₁) ⟨x, h⟩ ∈ range γ.extend"}}
+{"state": {"context": ["z w : ℍ", "r R : ℝ", "a b c : ℍ"], "goal": "0 < dist (↑b) ((starRingEnd ℂ) ↑b)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "I : Ideal R"], "goal": "↑(tensorQuotEquivQuotSMul M I).symm ∘ₗ (I • ⊤).mkQ = (TensorProduct.mk R M (R ⧸ I)).flip 1"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "R : Type u_3", "S : Type u_4", "A : Type u_5", "inst✝⁴ : CommRing K", "inst✝³ : Field L", "inst✝² : Algebra K L", "inst✝¹ : NoZeroSMulDivisors K L", "inst✝ : Algebra.IsAlgebraic K L", "f : L →ₐ[K] L"], "goal": "Function.Bijective ⇑f"}}
+{"state": {"context": [], "goal": "AkraBazziRecurrence.GrowsPolynomially Real.log"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "hp : p ≠ 0", "hp' : p ≠ ⊤", "f : α → F", "C : ℝ≥0", "s : Set α", "hs : MeasurableSet s", "hf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊"], "goal": "C • μ s ^ (1 / p.toReal) ≤ MeasureTheory.eLpNorm f p μ"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "ι : Type u_5", "inst✝¹ : Countable ι", "μ : Measure X", "inst✝ : NormedAddCommGroup E", "f : X → E", "s : ι → Set X", "hs : ∀ (i : ι), MeasurableSet (s i)", "hi : ∀ (i : ι), IntegrableOn f (s i) μ", "h : Summable fun i => ∫ (x : X) in s i, ‖f x‖ ∂μ", "B : ∀ (i : ι), ∫⁻ (a : X) in s i, ↑‖f a‖₊ ∂μ = ENNReal.ofReal (∫ (a : X) in s i, ↑‖f a‖₊ ∂μ)", "S' : Summable fun i => ⟨∫ (x : X) in s i, ↑‖f x‖₊ ∂μ, ⋯⟩", "S'' : ∑' (a : ι), ENNReal.ofReal (∫ (x : X) in s a, ‖f x‖ ∂μ) = ENNReal.ofReal ↑(∑' (b : ι), ⟨∫ (x : X) in s b, ‖f x‖ ∂μ, ⋯⟩)"], "goal": "∑' (b : ι), ENNReal.ofReal (∫ (a : X) in s b, ↑‖f a‖₊ ∂μ) < ⊤"}}
+{"state": {"context": ["ι : Sort u_1", "κ : ι → Sort u_2", "α : Type u_3", "β : Type u_4", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "inst✝ : AlexandrovDiscrete α", "S : Set (Set α)", "f✝ f : ι → Set α"], "goal": "(closure (⋃ i, f i))ᶜ = (⋃ i, closure (f i))ᶜ"}}
+{"state": {"context": ["F : Type u_1", "α✝ : Type u", "β : Type v", "γ : Type w", "α : Type u_2", "inst✝ : TopologicalSpace α", "f : α →ᵇ ℝ", "x : α"], "goal": "(fun f => f.toFun) 0 x ≤ (fun f => f.toFun) (const α ‖f‖ - f) x"}}
+{"state": {"context": ["a n m : Nat", "h : 1 < a", "w : n ≤ m"], "goal": "a ^ n ≤ a ^ m"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : TopologicalSpace E", "inst✝³ : AddCommGroup E", "inst✝² : TopologicalAddGroup E", "inst✝¹ : Module ℝ E", "inst✝ : ContinuousSMul ℝ E", "s t : Set E", "x y : E", "hs₁ : Convex ℝ s", "hs₂ : IsOpen s", "ht : Convex ℝ t", "disj : Disjoint s t", "a₀ : E", "ha₀ : a₀ ∈ s", "b₀ : E", "hb₀ : b₀ ∈ t", "x₀ : E := b₀ - a₀", "C : Set E := x₀ +ᵥ (s - t)", "this✝² : 0 ∈ C", "this✝¹ : Convex ℝ C", "this✝ : x₀ ∉ C", "f : E →L[ℝ] ℝ", "hf₁ : f x₀ = 1", "hf₂ : ∀ x ∈ C, f x < 1", "this : f b₀ = f a₀ + 1"], "goal": "∃ f u, (∀ a ∈ s, f a < u) ∧ ∀ b ∈ t, u ≤ f b"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "r : Ordnode α"], "goal": "(l.node' x r).dual = r.dual.node' x l.dual"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "f : α ≃ β", "o : ∀ ⦃a b : α⦄, s (f a) (f b) ↔ r a b"], "goal": "⇑{ toEquiv := f, map_rel_iff' := o } = ⇑f"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "S : Type u_2", "τ : Type u_3", "x : τ → S", "CommSemiring R", "CommSemiring S", "f : R →+* MvPolynomial τ S", "g : σ → MvPolynomial τ S", "p : MvPolynomial σ R"], "goal": "(MvPolynomial.eval x) (MvPolynomial.eval₂ f g p) =\n  MvPolynomial.eval₂ ((MvPolynomial.eval x).comp f) (fun s => (MvPolynomial.eval x) (g s)) p"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "W : C", "ι : W ⟶ X", "self : CategoryTheory.IsSplitEqualizer f g ι"], "goal": "CategoryTheory.CategoryStruct.comp f self.rightRetraction = CategoryTheory.CategoryStruct.comp self.leftRetraction ι"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Field K", "inst✝² : IsDedekindDomain A", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "I J : Ideal A", "x✝ : DvdNotUnit I J", "hI : I ≠ 0", "H : Ideal A", "hunit : ¬IsUnit H", "hmul : J = I * H", "h : J = I"], "goal": "I * H = I * 1"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m : ι → MeasurableSpace Ω"], "goal": "∀ {x : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω},\n  ProbabilityTheory.iIndep m μ → ProbabilityTheory.iIndepSets (fun x => {s | MeasurableSet s}) μ"}}
+{"state": {"context": ["a b : ℝ", "m n : ℕ"], "goal": "∫ (x : ℝ) in a..b, Real.sin x ^ (2 * m) * Real.cos x ^ (2 * n) =\n  ∫ (x : ℝ) in a..b, ((1 - Real.cos (2 * x)) / 2) ^ m * ((1 + Real.cos (2 * x)) / 2) ^ n"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "H : Type u_2", "TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "γ : ℝ → M", "v : (x : M) → TangentSpace I x", "s : Set ℝ", "hγ : IsIntegralCurveOn γ v s"], "goal": "ContinuousOn γ s"}}
+{"state": {"context": ["R : Type u₁", "L : Type u₂", "CommRing R", "LieRing L", "LieAlgebra R L", "A : Type u₃", "Ring A", "Algebra R A", "g₁ g₂ : UniversalEnvelopingAlgebra R L →ₐ[R] A", "h : g₁.toLieHom.comp (UniversalEnvelopingAlgebra.ι R) = g₂.toLieHom.comp (UniversalEnvelopingAlgebra.ι R)"], "goal": "g₁ = g₂"}}
+{"state": {"context": ["X : Type u_1", "X' : Type u_2", "Y : Type u_3", "Y' : Type u_4", "Z : Type u_5", "Z' : Type u_6", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : TopologicalSpace X'", "inst✝³ : TopologicalSpace Y", "inst✝² : TopologicalSpace Y'", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "e : PartialHomeomorph X Y", "f' : Y ≃ₜ Z", "f'' : Z ≃ₜ Z'"], "goal": "(e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "K : IntermediateField F E", "S : Set E"], "goal": "IntermediateField.restrictScalars F (IntermediateField.adjoin (↥K) S) = IntermediateField.adjoin F (↑K ∪ S)"}}
+{"state": {"context": ["R : Type u_1", "N₂ : Type u_11", "n : Type u_12", "m : Type u_13", "CommSemiring R", "AddCommMonoid N₂", "Module R N₂", "DecidableEq n", "Fintype n", "DecidableEq m", "Fintype m"], "goal": "Matrix.toLinearMap₂ (Pi.basisFun R n) (Pi.basisFun R m) = Matrix.toLinearMap₂' R"}}
+{"state": {"context": ["γ : Type u_3", "β : Type u_5", "tβ : TopologicalSpace β", "T2Space β", "MeasurableSpace β", "s : Set γ", "f : γ → β", "OpensMeasurableSpace β", "tγ : TopologicalSpace γ", "PolishSpace γ", "MeasurableSpace γ", "BorelSpace γ", "hs : MeasurableSet s", "f_cont : ContinuousOn f s", "f_inj : Set.InjOn f s"], "goal": "MeasurableSet (f '' s)"}}
+{"state": {"context": ["G : Type w", "TopologicalSpace G", "Group G", "TopologicalGroup G", "x y : G"], "goal": "Inseparable x y ↔ x / y ∈ closure 1"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "S : Type u_2", "inst✝¹ : CommRing S", "f : R →+* S", "I J : Ideal S", "inst✝ : I.IsPrime", "hIJ : I ≤ J", "r : S", "p : R[X]", "p_ne_zero : Polynomial.map (Quotient.mk (comap f I)) p ≠ 0", "hpI : eval₂ f r p ∈ I", "hrJ : r ∈ ↑J", "hrI : r ∉ ↑I", "rbar_ne_zero : (Quotient.mk I) r ≠ 0", "rbar_mem_J : (Quotient.mk I) r ∈ map (Quotient.mk I) J", "quotient_f : ∀ x ∈ comap f I, ((Quotient.mk I).comp f) x = 0", "rbar_root : eval₂ (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) ((Quotient.mk I) r) (Polynomial.map (Quotient.mk (comap f I)) p) = 0", "i : ℕ", "ne_zero : (Polynomial.map (Quotient.mk (comap f I)) p).coeff i ≠ 0", "mem : (Polynomial.map (Quotient.mk (comap f I)) p).coeff i ∈ comap (Quotient.lift (comap f I) ((Quotient.mk I).comp f) quotient_f) (map (Quotient.mk I) J)"], "goal": "∃ i, p.coeff i ∈ ↑(comap f J) \\ ↑(comap f I)"}}
+{"state": {"context": ["α : Type u", "p : α → Prop", "a : α", "l : List α", "h : p a"], "goal": "∃ x, x ∈ a :: l ∧ p x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s✝ t✝ u : Set α", "π : δ → Type u_6", "inst✝² : MeasurableSpace α", "inst✝¹ : (a : δ) → MeasurableSpace (π a)", "inst✝ : MeasurableSpace γ", "s : Set δ", "t : (i : δ) → Set (π i)", "hs : s.Countable", "ht : ∀ i ∈ s, MeasurableSet (t i)"], "goal": "MeasurableSet (s.pi t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : LinearOrder α", "inst✝² : ConditionallyCompleteLinearOrder β", "inst✝¹ : TopologicalSpace β", "inst✝ : OrderTopology β", "f : α → β", "hf : Monotone f", "x y : α", "h : x < y", "this✝ : TopologicalSpace α := Preorder.topology α", "this : OrderTopology α", "h' : 𝓝[<] y ≠ ⊥"], "goal": "f x ≤ leftLim f y"}}
+{"state": {"context": ["R : Type u", "self : EuclideanDomain R", "a : R", "b : R"], "goal": "b ≠ 0 → EuclideanDomain.r (EuclideanDomain.remainder a b) b"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s t : Set α"], "goal": "s ⊆ t → IsChain r t → IsChain r s"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_3", "Semiring A", "Algebra R A", "s : Set A"], "goal": "Algebra.adjoin R s = ((FreeAlgebra.lift R) Subtype.val).range"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "s : Set α"], "goal": "Filter.EventuallyConst s l ↔ (∀ᶠ (x : α) in l, x ∈ s) ∨ ∀ᶠ (x : α) in l, x ∉ s"}}
+{"state": {"context": ["α : Type u_1", "SubtractionCommMonoid α", "a b : α", "n : ℕ"], "goal": "n • (a - b) = n • a - n • b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹ : CommSemiring R", "σ : Type u_3", "inst✝ : AddCommMonoid M", "w✝ : σ → M", "n : M", "φ✝ ψ : MvPolynomial σ R", "w : σ → M", "φ : MvPolynomial σ R", "m : M"], "goal": "(weightedHomogeneousComponent w m) φ ∈ weightedHomogeneousSubmodule R w m"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : ℝ → F", "K : Set F", "ε : ℝ", "hε : 0 < ε", "x : ℝ", "hx : DifferentiableWithinAt ℝ f (Ici x) x"], "goal": "∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a : α", "x✝ : ℤ"], "goal": "x✝ ∈ Int.cast ⁻¹' Iic a ↔ x✝ ∈ Iic ⌊a⌋"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x : X"], "goal": "x ⤳ x"}}
+{"state": {"context": ["G : Type w", "Group G", "TopologicalSpace G", "TopologicalGroup G", "a : G"], "goal": "IsClosedMap fun x => x / a"}}
+{"state": {"context": ["K : Type u_1", "Field K", "NumberField K", "w : { w // w.IsReal }"], "goal": "(NumberField.mixedEmbedding.indexEquiv K) (Sum.inl w) = (↑w).embedding"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁴ : MeasurableSpace δ", "inst✝³ : NormedAddCommGroup β", "inst✝² : NormedAddCommGroup γ", "E : Type u_5", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : α → ℝ≥0∞", "hf : Measurable f", "hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤", "g : α → E"], "goal": "Integrable (fun x => (f x).toNNReal • g x) μ ↔ Integrable (fun x => (f x).toReal • g x) μ"}}
+{"state": {"context": [], "goal": "↑T⁻¹ = !![1, -1; 0, 1]"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0", "hf : Measurable f", "g : α → ℝ≥0∞"], "goal": "AEMeasurable g (μ.withDensity fun x => ↑(f x)) ↔ AEMeasurable (fun x => ↑(f x) * g x) μ"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "s : Set E", "A : Tendsto (fun r => ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall 0 1)) (𝓝[<] 1) (𝓝 (μ (closedBall 0 1)))"], "goal": "μ (closedBall 0 1) ≤ μ (ball 0 1)"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "F : Type u_2", "SeminormedAddCommGroup F", "NormedSpace 𝕜 F", "NormedSpace ℝ F", "IsScalarTower ℝ 𝕜 F", "fr : F →L[ℝ] ℝ", "x : F"], "goal": "fr.extendTo𝕜' x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))"}}
+{"state": {"context": ["J : Type u_1", "C : Type u_2", "CategoryTheory.Category.{u_3, u_1} J", "CategoryTheory.Category.{u_4, u_2} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)"], "goal": "∀ (X_1 : J),\n  (((CategoryTheory.ShortComplex.FunctorEquivalence.counitIso J C).hom.app X).app X_1).τ₂ =\n    CategoryTheory.CategoryStruct.id (X.obj X_1).X₂"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁵ : Field F", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Algebra F K", "inst✝¹ : Algebra F L", "inst✝ : Algebra.IsAlgebraic F K", "splits : ∀ (x : K), Splits (algebraMap F L) (minpoly F x)"], "goal": "normalClosure F K ↥(normalClosure F K L) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "a : α", "s : Multiset α"], "goal": "s = Multiset.replicate (Multiset.card s) a ↔ ∀ b ∈ s, b = a"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a b : Cardinal.{u_2}", "ha : ℵ₀ ≤ a", "b1 : 1 < b", "b0 : b ≠ 0"], "goal": "a < (b ^ a).ord.cof"}}
+{"state": {"context": [], "goal": "Real.tan (Real.pi / 2) = 0"}}
+{"state": {"context": ["ι : Sort u_1", "M : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝ : Mul M", "S T : Subsemigroup M", "this : T ≤ S ⊔ T"], "goal": "∀ {x : M}, x ∈ T → x ∈ S ⊔ T"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "inst✝⁵ : ConcreteCategory C", "inst✝⁴ : HasColimits C", "inst✝³ : HasLimits C", "inst✝² : PreservesLimits (CategoryTheory.forget C)", "inst✝¹ : PreservesFilteredColimits (CategoryTheory.forget C)", "inst✝ : (CategoryTheory.forget C).ReflectsIsomorphisms", "X Y : SheafedSpace C", "f : ↑X.toPresheafedSpace ⟶ ↑Y.toPresheafedSpace", "fc gc : Y.presheaf ⟶ (pushforward C f).obj X.presheaf", "h' : ∀ (x : ↑↑X.toPresheafedSpace), { base := f, c := fc }.stalkMap x = (Y.presheaf.stalkCongr ⋯).hom ≫ { base := f, c := gc }.stalkMap x", "U : Opens ↑↑Y.toPresheafedSpace", "s : (CategoryTheory.forget C).obj (Y.presheaf.obj (op U))"], "goal": "(fc.app (op U)) s = (gc.app (op U)) s"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "i j : ℕ"], "goal": "bernoulli' (i, j).1 / ↑(i, j).1! * ((↑(i, j).2 + 1) * ↑(i, j).2!)⁻¹ * ↑((i, j).1 + (i, j).2)! = ↑(((i, j).1 + (i, j).2).choose (i, j).2) / (↑(i, j).2 + 1) * bernoulli' (i, j).1"}}
+{"state": {"context": ["α : Type u_3", "DivisionSemiring α", "q r : ℚ≥0", "hq : ↑q.den ≠ 0", "hr : ↑r.den ≠ 0"], "goal": "↑(q * r) = ↑q * ↑r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : Fintype α", "s t : Finset α"], "goal": "↑s = Set.univ ↔ s = univ"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁶ : Category.{u_2, u_1} C", "inst✝⁵ : HasZeroObject C", "inst✝⁴ : HasShift C ℤ", "inst✝³ : Preadditive C", "inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝¹ : Pretriangulated C", "S : Subcategory C", "X Y : C", "f : X ⟶ Y", "inst✝ : IsIso f"], "goal": "S.W f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : CommGroup G", "a✝ b✝ c✝ d a b c : G"], "goal": "a / c * (b * c) = a * b"}}
+{"state": {"context": ["α : Sort u", "p : α → Prop", "b : Prop", "h₁ : Exists fun x => p x", "h₂ : ∀ (a : α), p a → b"], "goal": "b"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "DecidableRel fun x x_1 => x ∣ x_1", "p a : α", "hp : Prime p", "k : ℕ"], "goal": "multiplicity p (a ^ k) = k • multiplicity p a"}}
+{"state": {"context": ["m n : ℕ", "h : m ∣ n"], "goal": "fib m ∣ fib n"}}
+{"state": {"context": ["R : Type u", "a b : R", "Semiring R"], "goal": "(Polynomial.C a * Polynomial.X + Polynomial.C b).natDegree ≤ 1"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "NormedField 𝕜₁", "NormedField 𝕜₂", "σ : 𝕜₁ →+* 𝕜₂", "E : Type u_3", "F : Type u_4", "AddCommGroup E", "Module 𝕜₁ E", "TopologicalSpace E", "AddCommGroup F", "Module 𝕜₂ F", "TopologicalSpace F", "TopologicalAddGroup F", "𝔖 : Set (Set E)", "h𝔖₁ : 𝔖.Nonempty", "h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖"], "goal": "(𝓝 0).HasBasis (fun SV => SV.1 ∈ 𝔖 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | ∀ x ∈ SV.1, f x ∈ SV.2}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "mβ : MeasurableSpace β", "MeasurableSingletonClass α", "f : α → β", "s : Set α", "hs : s.Finite", "hf : Measurable (sᶜ.restrict f)"], "goal": "Measurable f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "PartialOrder α", "TopologicalSpace β", "a : α", "f : α → β"], "goal": "ContinuousWithinAt f (Set.Ioi a) a ↔ ContinuousWithinAt f (Set.Ici a) a"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "a b : A", "n x : ℕ", "hx : x ∈ Finset.range n.succ"], "goal": "(algebraMap ℚ A) (1 / ↑x !) * (algebraMap ℚ A) (1 / ↑(n - x)!) = ↑(n.choose x) * (algebraMap ℚ A) (1 / ↑n !)"}}
+{"state": {"context": ["R : Type u", "S : Type w", "Star R", "SetLike S R", "hS : StarMemClass S R", "s : S", "x : ↥s"], "goal": "↑(star x) = star ↑x"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X"], "goal": "IsClosed ∅"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "R X : Set α"], "goal": "(M.restrict R).Dep X ↔ ¬M.Indep X ∧ X ⊆ R"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : PseudoEMetricSpace α", "f : ℕ → α", "d : ℕ → ℝ≥0", "hf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ (fun i => ↑(d i)) n", "hd : ∑' (i : ℕ), ↑(d i) ≠ ⊤"], "goal": "CauchySeq f"}}
+{"state": {"context": ["α✝ : Sort u", "β : Sort v", "γ : Sort w", "α : Type u_1", "s : Set α", "inst✝ : DecidablePred fun x => x ∈ s", "x : α", "hx : x ∈ sᶜ"], "goal": "(Set.sumCompl s).symm ↑⟨x, hx⟩ = Sum.inr ⟨x, hx⟩"}}
+{"state": {"context": ["α : Type u_2", "I : Type u_1", "inst✝¹ : CommMonoid α", "inst✝ : DecompositionMonoid α", "x y z : α", "s : I → α", "t : Finset I"], "goal": "(∀ i ∈ t, IsRelPrime (s i) x) → IsRelPrime (∏ i ∈ t, s i) x"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "s : Set E", "f✝ : E → E", "f' : E → E →L[ℝ] E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "f : PartialHomeomorph E E", "hf' : ∀ x ∈ f.source, HasFDerivAt (↑f) (f' x) x", "g : E → F"], "goal": "∫ (x : E) in f.target, g x ∂μ = ∫ (x : E) in f.source, |(f' x).det| • g (↑f x) ∂μ"}}
+{"state": {"context": ["J : Type w", "K✝ : Type u_1", "C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "K : Type", "inst✝¹ : Finite K", "inst✝ : HasFiniteBiproducts C", "f : K → C", "p : Set K", "j : Subtype pᶜ", "k : Subtype p"], "goal": "0 ≫ biproduct.π (Subtype.restrict pᶜ f) j = 0"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Group (G i)", "inst✝ : Group H", "φ : (i : ι) → H →* G i", "hφ : ∀ (i : ι), Injective ⇑(φ i)", "i j : ι", "hij : i ≠ j", "x : PushoutI φ", "g₁ : G i", "hg₁ : (of i) g₁ = x", "g₂ : G j", "hg₂ : (of j) g₂ = x", "hx : ¬x ∈ (base φ).range", "hx1 : x ≠ 1", "hg₁1 : g₁ ≠ 1", "hg₂1 : g₂ ≠ 1"], "goal": "False"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : Set (AddSubmonoid M)", "Sne : S.Nonempty", "hS : DirectedOn (fun x x_1 => x ≤ x_1) S", "x : M"], "goal": "x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s"}}
+{"state": {"context": ["s n : Nat"], "goal": "range' s (n + 1) = range' s n ++ [s + n]"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝³ : LinearOrderedField R", "inst✝² : AddCommGroup V", "inst✝¹ : Module R V", "inst✝ : AddTorsor V P", "x y : P", "h : x ≠ y"], "goal": "(pointReflection R x) x ≠ (pointReflection R x) y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "NormedAddCommGroup β", "ι : Type u_5", "m : MeasurableSpace α", "f : α → β", "μ : ι → MeasureTheory.Measure α", "s : Finset ι"], "goal": "MeasureTheory.Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, MeasureTheory.Integrable f (μ i)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝⁷ : Preorder α", "inst✝⁶ : Preorder β", "inst✝⁵ : Preorder γ", "inst✝⁴ : Preorder δ", "inst✝³ : MulOneClass α", "inst✝² : MulOneClass β", "inst✝¹ : MulOneClass γ", "inst✝ : MulOneClass δ", "f g✝ : α →*o β", "g : β →*o γ", "f₁ f₂ : α →*o β", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "a : α"], "goal": "g (f₁ a) = g (f₂ a)"}}
+{"state": {"context": ["R' : Type u", "M : Type v", "CommSemiring R'", "AddCommMonoid M", "Module R' M", "Module R'ᵐᵒᵖ M", "IsCentralScalar R' M"], "goal": "TrivSqZeroExt.map LinearMap.id = AlgHom.id R' (tsze R' M)"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "self : AkraBazziRecurrence T g a b r"], "goal": "0 < self.n₀"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : GeneralizedCoheytingAlgebra α", "a b c d : α"], "goal": "a \\ (b ⊔ a ⊓ b) ⊔ b \\ (a ⊔ a ⊓ b) = a ∆ b"}}
+{"state": {"context": ["F : Type u_1", "α✝ : Type u", "β : Type v", "γ : Type w", "α : Type u_2", "inst✝ : TopologicalSpace α", "f : α →ᵇ ℝ", "x : α"], "goal": "(fun f => f.toFun) 0 x ≤ (fun f => f.toFun) (f + const α ‖f‖) x"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeSup α", "OrderBot α", "f : β → α"], "goal": "∅.sup f = ⊥"}}
+{"state": {"context": ["R : Type u", "ι : Type w", "s : Finset ι", "inst✝ : CommSemiring R", "f : ι → R[X]", "t : Multiset R[X]"], "goal": "(prod 0).coeff 0 = (Multiset.map (fun f => f.coeff 0) 0).prod"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "self : F.IsEquivalence"], "goal": "F.Full"}}
+{"state": {"context": ["α β : Type u", "s : Set α", "t : (a : α) → a ∈ s → Set β", "c : Cardinal.{u}", "hc : c.IsRegular", "hs : #↑s < c"], "goal": "#↑(⋃ a, ⋃ (h : a ∈ s), t a h) < c ↔ ∀ (a : α) (ha : a ∈ s), #↑(t a ha) < c"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : NormedField 𝕜", "α : Type u_2", "E : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "c : 𝕜", "s t : Set E", "x y : E", "r : ℝ≥0", "hc : 1 < ‖c‖", "h₀ : r ≠ 0 ∨ x ≠ 0"], "goal": "egauge 𝕜 (ball 0 ↑r) x ≤ ↑‖c‖₊ * ↑‖x‖₊ / ↑r"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "f✝ : α → β", "f : β → γ", "g : α → β", "hf : LocallyLipschitz f", "hg : LocallyLipschitz g", "x : α", "Kg : ℝ≥0", "t : Set α", "ht : t ∈ 𝓝 x", "hgL : LipschitzOnWith Kg g t", "Kf : ℝ≥0", "u : Set β", "hu : u ∈ 𝓝 (g x)", "hfL : LipschitzOnWith Kf f u"], "goal": "LipschitzOnWith (Kf * Kg) (f ∘ g) (t ∩ g ⁻¹' u)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "Z : Type u_6", "Semiring Z", "f : C(X, Y)", "g : LocallyConstant Y Z"], "goal": "∀ (a : X), ((LocallyConstant.comapRingHom f) g) a = g (f a)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "ι : Type u_1", "I : ι → Ideal R"], "goal": "Function.Injective ⇑(Ideal.quotientInfToPiQuotient I)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "ContinuousSMul 𝕜 E", "TopologicalAddGroup E", "s t : Set E", "hs : s.Nonempty", "ht : t.Nonempty"], "goal": "Bornology.IsVonNBounded 𝕜 (s - t) ↔ Bornology.IsVonNBounded 𝕜 s ∧ Bornology.IsVonNBounded 𝕜 t"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F✝ : Type u_3", "𝕜 : Type u_4", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : SMulCommClass ℝ 𝕜 E", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace ℝ F✝", "inst✝⁴ : CompleteSpace F✝", "G : Type u_5", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace ℝ G", "f✝ g : α → E", "m : MeasurableSpace α", "μ✝ : Measure α", "X : Type u_6", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X", "μ : Measure α", "f : ℕ → α → ℝ", "F : α → ℝ", "hf : ∀ (n : ℕ), Integrable (f n) μ", "hF : Integrable F μ", "h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x", "h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))", "f' : ℕ → α → ℝ := fun n x => f n x - f 0 x", "hf'_nonneg : ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, 0 ≤ f' i a", "hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ", "hF_ge : 0 ≤ᶠ[ae μ] fun x => (F - f 0) x"], "goal": "∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ ≠ ⊤"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_5", "DecidableEq n", "MulZeroClass α", "v w : n → α"], "goal": "(Matrix.diagonal v).hadamard (Matrix.diagonal w) = Matrix.diagonal (v * w)"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : Matrix.SpecialLinearGroup n R", "v : n → R"], "goal": "(Matrix.SpecialLinearGroup.toLin' A) v = (Matrix.toLin' ↑A) v"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "N : Type w'", "g : M ≃ N"], "goal": "(Equiv.bundledInduced L g).str = g.inducedStructure"}}
+{"state": {"context": ["R : Type u_4", "S : Type u_5", "ConditionallyCompleteLinearOrder R", "TopologicalSpace R", "OrderTopology R", "ConditionallyCompleteLinearOrder S", "TopologicalSpace S", "OrderTopology S", "F : Filter R", "F.NeBot", "f : R → S", "f_decr : Antitone f", "f_cont : ContinuousAt f F.limsSup", "bdd_above : Filter.IsBounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault", "cobdd : Filter.IsCobounded (fun x x_1 => x ≤ x_1) F := by isBoundedDefault"], "goal": "f F.limsSup = Filter.liminf f F"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝³⁷ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³⁶ : NormedAddCommGroup E", "inst✝³⁵ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝³⁴ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝³³ : TopologicalSpace M", "inst✝³² : ChartedSpace H M", "inst✝³¹ : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝³⁰ : NormedAddCommGroup E'", "inst✝²⁹ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝²⁸ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝²⁷ : TopologicalSpace M'", "inst✝²⁶ : ChartedSpace H' M'", "inst✝²⁵ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝²⁴ : NormedAddCommGroup E''", "inst✝²³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝²² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝²¹ : TopologicalSpace M''", "inst✝²⁰ : ChartedSpace H'' M''", "F : Type u_11", "inst✝¹⁹ : NormedAddCommGroup F", "inst✝¹⁸ : NormedSpace 𝕜 F", "G : Type u_12", "inst✝¹⁷ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N : Type u_13", "inst✝¹⁶ : TopologicalSpace N", "inst✝¹⁵ : ChartedSpace G N", "inst✝¹⁴ : SmoothManifoldWithCorners J N", "F' : Type u_14", "inst✝¹³ : NormedAddCommGroup F'", "inst✝¹² : NormedSpace 𝕜 F'", "G' : Type u_15", "inst✝¹¹ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_16", "inst✝¹⁰ : TopologicalSpace N'", "inst✝⁹ : ChartedSpace G' N'", "inst✝⁸ : SmoothManifoldWithCorners J' N'", "F₁ : Type u_17", "inst✝⁷ : NormedAddCommGroup F₁", "inst✝⁶ : NormedSpace 𝕜 F₁", "F₂ : Type u_18", "inst✝⁵ : NormedAddCommGroup F₂", "inst✝⁴ : NormedSpace 𝕜 F₂", "F₃ : Type u_19", "inst✝³ : NormedAddCommGroup F₃", "inst✝² : NormedSpace 𝕜 F₃", "F₄ : Type u_20", "inst✝¹ : NormedAddCommGroup F₄", "inst✝ : NormedSpace 𝕜 F₄", "e : PartialHomeomorph M H", "e' : PartialHomeomorph M' H'", "f f₁ : M → M'", "s s₁ t : Set M", "x✝ : M", "m n : ℕ∞", "x : M"], "goal": "ContMDiffAt I I' n f x ↔ ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (↑(extChartAt I' (f x)) ∘ f) x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_5", "OrderedSemiring 𝕜", "AddCommMonoid E", "OrderedAddCommMonoid β", "SMul 𝕜 E", "Module 𝕜 β", "OrderedSMul 𝕜 β", "s : Set E", "f : E → β", "hf : ConcaveOn 𝕜 s f", "r : β"], "goal": "Convex 𝕜 {x | x ∈ s ∧ r ≤ f x}"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq β", "inst✝ : Monoid α", "s t : Finset α", "a : α", "m n : ℕ", "hst : s ⊆ t"], "goal": "s ^ 0 ⊆ t ^ 0"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "N : Type u_3", "DecidableEq α", "DecidableEq N", "DecidableEq M", "Zero M", "Zero N", "f g : α →₀ N", "F : N → M", "F0 : F 0 = 0", "hF : Function.Injective F"], "goal": "(Finsupp.mapRange F F0 f).neLocus (Finsupp.mapRange F F0 g) = f.neLocus g"}}
+{"state": {"context": ["p : ℝ × ℝ", "hp_fst : 0 < p.1"], "goal": "∀ᶠ (x : ℝ × ℝ) in 𝓝 p, 0 < x.1"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝²⁶ : NormedAddCommGroup E", "inst✝²⁵ : NormedAddCommGroup E'", "inst✝²⁴ : NormedAddCommGroup E''", "inst✝²³ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x x' : G", "y y' : E", "inst✝²² : RCLike 𝕜", "inst✝²¹ : NormedSpace 𝕜 E", "inst✝²⁰ : NormedSpace 𝕜 E'", "inst✝¹⁹ : NormedSpace 𝕜 E''", "inst✝¹⁸ : NormedSpace ℝ F", "inst✝¹⁷ : NormedSpace 𝕜 F", "n : ℕ∞", "inst✝¹⁶ : CompleteSpace F", "inst✝¹⁵ : MeasurableSpace G", "μ ν : Measure G", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝¹⁴ : NormedAddCommGroup F'", "inst✝¹³ : NormedSpace ℝ F'", "inst✝¹² : NormedSpace 𝕜 F'", "inst✝¹¹ : CompleteSpace F'", "inst✝¹⁰ : NormedAddCommGroup F''", "inst✝⁹ : NormedSpace ℝ F''", "inst✝⁸ : NormedSpace 𝕜 F''", "inst✝⁷ : CompleteSpace F''", "k : G → E''", "L₂ : F →L[𝕜] E'' →L[𝕜] F'", "L₃ : E →L[𝕜] F'' →L[𝕜] F'", "L₄ : E' →L[𝕜] E'' →L[𝕜] F''", "inst✝⁶ : AddGroup G", "inst✝⁵ : SFinite μ", "inst✝⁴ : SFinite ν", "inst✝³ : μ.IsAddRightInvariant", "inst✝² : MeasurableAdd₂ G", "inst✝¹ : ν.IsAddRightInvariant", "inst✝ : MeasurableNeg G", "hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)", "x₀ : G", "hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν", "hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ", "hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)", "t : G", "ht : ConvolutionExistsAt g k (x₀ - t) L₄ μ"], "goal": "(fun s => ∫ (u : G), (L₃ (f s)) ((L₄ (g u)) (k (x₀ - s - u))) ∂μ) t = (fun s => (L₃ (f s)) (∫ (u : G), (L₄ (g u)) (k (x₀ - s - u)) ∂μ)) t"}}
+{"state": {"context": ["m n : ℕ"], "goal": "↑(m % n) = ↑m % ↑n"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "cut : α → Ordering", "f : α → α", "t : RBSet α cmp"], "goal": "(∀ {x : α}, RBNode.find? cut t.val = some x → cmpEq cmp (f x) x) → OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val).fst"}}
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+{"state": {"context": ["α : Type u_1", "PartialOrder α", "PredOrder α", "a : α", "OrderBot α"], "goal": "a ≤ Order.pred a ↔ a = ⊥"}}
+{"state": {"context": ["l m r : List Char", "s : Substring"], "goal": "Substring.ValidFor l m r s → s.next { byteIdx := String.utf8Len m } = { byteIdx := String.utf8Len m }"}}
+{"state": {"context": ["α✝ : Sort u_1", "β : Sort u_2", "α : Sort u_3", "r : α → α → Prop", "iseqv✝¹ : Equivalence r", "p : α → α → Prop", "iseqv✝ : Equivalence p", "Eq : ∀ (a b : α), Setoid.r a b ↔ Setoid.r a b", "this : r = p"], "goal": "{ r := r, iseqv := iseqv✝¹ } = { r := p, iseqv := iseqv✝ }"}}
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+{"state": {"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "𝒜 ℬ : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "h𝒜₁ : 𝒜.Nonempty", "h𝒜₂ : univ ∉ 𝒜", "m : ℕ", "hm : m = 𝒜.card"], "goal": "supSum 𝒜 = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹"}}
+{"state": {"context": ["x : ℝ", "h : 1 ≤ x"], "goal": "x.smoothTransition = 1"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]"], "goal": "X = X - C 0"}}
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+{"state": {"context": ["x✝ : ℝ"], "goal": "(↑(π / 2) - ↑x✝).sin = (↑x✝).cos"}}
+{"state": {"context": ["n : ℕ"], "goal": "↑n.totient = ↑n * ∏ p ∈ n.primeFactors, (1 - (↑p)⁻¹)"}}
+{"state": {"context": ["α β : Type u", "c : Cardinal.{?u.225151}", "S T : Set α", "h : T ⊆ S"], "goal": "Disjoint (S \\ T) T"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M"], "goal": "(FirstOrder.Language.Equiv.refl L M).toHom = FirstOrder.Language.Hom.id L M"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β → Option γ", "BEq α", "Hashable α", "m : Batteries.HashMap.Imp α β", "H : m.WF"], "goal": "(Batteries.HashMap.Imp.filterMap f m).WF"}}
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+{"state": {"context": ["o : Ordinal.{u_4}"], "goal": "Order.succ o.pred = o ↔ ∃ a, o = Order.succ a"}}
+{"state": {"context": ["X : Type u_5", "Y : Type u_6", "TopologicalSpace X", "TopologicalSpace Y", "s : Set Y", "DiscreteTopology ↑s", "f : X → Y", "hc : Continuous f", "hinj : Function.Injective f"], "goal": "DiscreteTopology ↑(f ⁻¹' s)"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁷ : Group G", "inst✝⁶ : Group G'", "inst✝⁵ : Group G''", "A : Type u_4", "inst✝⁴ : AddGroup A", "N : Type u_5", "P : Type u_6", "inst✝³ : Group N", "inst✝² : Group P", "K : Subgroup G", "M✝ : Type u_7", "inst✝¹ : MulOneClass M✝", "M : Type u_8", "inst✝ : Monoid M", "f : G →* M", "x : G"], "goal": "x ∈ f.toHomUnits.ker ↔ x ∈ f.ker"}}
+{"state": {"context": ["R : Type u", "inst₁ inst₂ : AddCommGroupWithOne R", "h_add : HAdd.hAdd = HAdd.hAdd", "h_one : One.one = One.one", "this : toAddCommGroup = toAddCommGroup"], "goal": "inst₁ = inst₂"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "rω : cs.IsReduced ω"], "goal": "(cs.leftInvSeq ω).Nodup"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a : α"], "goal": "upperBounds {a} = Set.Ici a"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : GroupWithZero G₀", "a b c d : G₀", "m n : ℕ", "hb : b ≠ 0", "h : b = a⁻¹ * c"], "goal": "a * b = c"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasInitial C", "ι : Type u_3", "F : CategoryTheory.Functor (CategoryTheory.Discrete ι) C", "c : CategoryTheory.Limits.Cocone F", "hc : CategoryTheory.IsVanKampenColimit c", "i j : CategoryTheory.Discrete ι", "hi : i ≠ j"], "goal": "CategoryTheory.IsPullback (CategoryTheory.Limits.initial.to (F.obj i)) (CategoryTheory.Limits.initial.to (F.obj j))\n  (c.ι.app i) (c.ι.app j)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : PartialEquiv α β"], "goal": "e.restr Set.univ = e"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝¹ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "inst✝ : IsFiniteKernel ν", "a : α", "seq : ℕ → Set β", "hseq : Monotone seq", "hseq_iUnion : ⋃ i, seq i = univ", "hseq_meas : ∀ (m : ℕ), MeasurableSet (seq m)", "this : IsFiniteKernel κ", "h_cont : ContinuousAt ENNReal.toReal ((κ a) univ)", "h : Tendsto (⇑(κ a) ∘ fun m => univ ×ˢ seq m) atTop (𝓝 ((κ a) (⋃ n, univ ×ˢ seq n)))"], "goal": "univ = univ ×ˢ univ"}}
+{"state": {"context": ["α : Type u_1", "r c✝ : ℝ", "l : Filter α", "f g : α → ℝ", "hr : 0 < r", "hg : 0 ≤ᶠ[l] g", "h : f =o[l] g", "c : ℝ", "hc : 0 < c"], "goal": "IsBigOWith c l (fun x => f x ^ r) fun x => g x ^ r"}}
+{"state": {"context": ["R : Type u", "σ : Type v", "CommRing R", "I : Ideal (MvPolynomial σ R)", "p : MvPolynomial σ R", "hcoe : ∀ (m : σ →₀ ℕ), MvPolynomial.coeff m p ∈ Ideal.comap MvPolynomial.C I"], "goal": "p ∈ I"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C"], "goal": "(CategoryTheory.Iso.refl X).symm = CategoryTheory.Iso.refl X"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "Ring R", "AddCommGroup V", "TopologicalSpace V", "TopologicalSpace R", "Module R V", "SeparatingDual R V", "x : V", "hx : x ≠ 0"], "goal": "∃ f, f x ≠ 0"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "α₂ : Type u_5", "Semiring β", "Mul α", "Mul α₂", "F : Type u_6", "FunLike F α α₂", "MulHomClass F α α₂", "f : F", "x y : MonoidAlgebra β α"], "goal": "MonoidAlgebra.mapDomain (⇑f) (x * y) = MonoidAlgebra.mapDomain (⇑f) x * MonoidAlgebra.mapDomain (⇑f) y"}}
+{"state": {"context": ["R : Type u", "inst₁ inst₂ : NonUnitalCommRing R"], "goal": "inst₁ = inst₂ ↔ HAdd.hAdd = HAdd.hAdd ∧ HMul.hMul = HMul.hMul"}}
+{"state": {"context": ["α✝ : Type u_1", "M : Type u_2", "α : Type u_3", "inst✝² : Monoid M", "inst✝¹ : MeasurableSpace M", "inst✝ : MeasurableMul₂ M", "m : MeasurableSpace α", "μ : Measure α", "f : α → M", "l : List (α → M)", "ihl : (∀ f ∈ l, AEMeasurable f μ) → AEMeasurable l.prod μ", "hl : AEMeasurable f μ ∧ ∀ x ∈ l, AEMeasurable x μ"], "goal": "AEMeasurable (f * l.prod) μ"}}
+{"state": {"context": ["b : Bool", "n : ℕ"], "goal": "(↑(Nat.bit b n)).testBit 0 = b"}}
+{"state": {"context": ["R : Type u", "M : Type v", "CommSemiring R", "AddCommMonoid M", "Module R M", "S : Set R", "T : Set M"], "goal": "Ideal.span S • Submodule.span R T = Submodule.span R (⋃ s ∈ S, ⋃ t ∈ T, {s • t})"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "p : ι → Prop", "s : ι → Set α", "𝓕 : Filter ι", "a x : α"], "goal": "x ∈ ⋂₀ {a | {x | p x ∧ ¬s x ⊆ a}.Finite} ↔ x ∈ {x | {n | p n ∧ x ∈ s n}.Infinite}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : TopologicalSpace β", "inst✝³ : PseudoMetrizableSpace β", "inst✝² : MeasurableSpace β", "inst✝¹ : BorelSpace β", "ι : Type u_3", "μ : Measure α", "f : ι → α → β", "g : α → β", "u : Filter ι", "hu : u.NeBot", "inst✝ : u.IsCountablyGenerated", "hf : ∀ (n : ι), AEMeasurable (f n) μ", "h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))", "v : ℕ → ι", "hv : Tendsto v atTop u", "h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ", "p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))", "hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x", "aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else ⋯.some"], "goal": "g =ᶠ[ae μ] aeSeqLim"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "CommSemiring ι", "DecidableEq ι", "CommRing R", "Ring A", "Ring B", "Algebra R A", "Algebra R B", "𝒜 : ι → Submodule R A", "ℬ : ι → Submodule R B", "GradedAlgebra 𝒜", "GradedAlgebra ℬ", "M : Type u_5", "AddCommMonoid M", "Module R M", "f g : 𝒜 ᵍ⊗[R] ℬ →ₗ[R] M", "h : f ∘ₗ ↑(GradedTensorProduct.of R 𝒜 ℬ) = g ∘ₗ ↑(GradedTensorProduct.of R 𝒜 ℬ)"], "goal": "f = g"}}
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+{"state": {"context": ["R : Type u_1", "F : Type u", "inst✝ : Field F", "p q : F[X]", "f✝ g : RatFunc F", "f : F[X]", "hf : f ≠ 0"], "goal": "↑f ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "μ : MeasureTheory.Measure β", "MeasureTheory.SFinite μ", "ν : MeasureTheory.Measure γ", "MeasureTheory.SFinite ν"], "goal": "(ProbabilityTheory.Kernel.const α μ).prod (ProbabilityTheory.Kernel.const α ν) =\n  ProbabilityTheory.Kernel.const α (μ.prod ν)"}}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field K", "inst✝² inst✝¹ : NumberField K", "inst✝ : IsCyclotomicExtension {n} ℚ K", "hno : Odd ↑n", "ζ x : K", "hζ : IsPrimitiveRoot ζ ↑n", "hx : IsOfFinOrder x", "hnegζ : IsPrimitiveRoot (-ζ) (2 * ↑n)", "k : ℕ", "hkpos : 0 < k", "hkn : x ^ k = 1", "l : ℕ", "hl : l ∈ k.divisors", "hlzero : NeZero l", "this : NeZero ↑l", "a : ℕ", "ha : 2 * ↑n = l * a", "hlroot : x ^ (2 * ↑n) = 1"], "goal": "∃ r, x = (-ζ) ^ r"}}
+{"state": {"context": ["α : Type u", "Top α"], "goal": "⊤.down = ⊤"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "s₁ s₂ : Set P"], "goal": "affineSpan k s₁ ∥ affineSpan k s₂ ↔ vectorSpan k s₁ = vectorSpan k s₂ ∧ (s₁ = ∅ ↔ s₂ = ∅)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "I : Type u_3", "J : Type u_4", "CategoryTheory.Category.{u_6, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "Zero I", "DecidableEq I", "CategoryTheory.Limits.HasInitial C", "F : CategoryTheory.Functor C (CategoryTheory.Functor D D)", "X : C", "e : F.obj X ≅ CategoryTheory.Functor.id D", "(Y : D) → CategoryTheory.Limits.PreservesColimit (CategoryTheory.Functor.empty C) (F.flip.obj Y)", "p : I × J → J", "hp : ∀ (j : J), p (0, j) = j", "Y : CategoryTheory.GradedObject J D", "j : J"], "goal": "CategoryTheory.Limits.Cofan.inj (CategoryTheory.GradedObject.mapBifunctorLeftUnitorCofan F X e p hp Y j) ⟨(0, j), ⋯⟩ =\n  CategoryTheory.CategoryStruct.comp ((F.map (CategoryTheory.GradedObject.singleObjApplyIso 0 X).hom).app (Y j))\n    (e.hom.app (Y j))"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : LieRing L", "inst✝⁶ : LieAlgebra R L", "inst✝⁵ : Field K", "inst✝⁴ : LieAlgebra K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "inst✝ : IsKilling K L", "x : L", "hx : x ∈ H", "hx' : _root_.IsNilpotent ((ad K L) x)", "y : ↥H"], "goal": "((traceForm K (↥H) L) ⟨x, hx⟩) y = 0 y"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_4", "v : ι → E", "hv : Orthonormal 𝕜 v", "l₁ l₂ : ι →₀ 𝕜"], "goal": "⟪(Finsupp.total ι E 𝕜 v) l₁, (Finsupp.total ι E 𝕜 v) l₂⟫_𝕜 = l₂.sum fun i y => (starRingEnd 𝕜) (l₁ i) * y"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B : Set α", "hB : M.Base B"], "goal": "B ⊆ M.E"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "Sup α", "Sup β", "Sup γ", "f : SupHom β γ", "g : SupHom α β", "a : α"], "goal": "(f.comp g) a = f (g a)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "e : S₁ ≅ S₂"], "goal": "S₁.Exact ∧ CategoryTheory.Mono S₁.f ↔ S₂.Exact ∧ CategoryTheory.Mono S₂.f"}}
+{"state": {"context": ["C : Type u_1", "A : Type u_3", "CategoryTheory.Category.{u_5, u_1} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "CategoryTheory.Pretriangulated C", "CategoryTheory.Category.{u_4, u_3} A", "CategoryTheory.Abelian A", "F : CategoryTheory.Functor C A", "F.IsHomological", "F.ShiftSequence ℤ", "T : CategoryTheory.Pretriangulated.Triangle C", "hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles", "n₀ : ℤ"], "goal": "(CategoryTheory.ShortComplex.mk ((F.shift n₀).map T.mor₁) ((F.shift n₀).map T.mor₂) ⋯).Exact"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "ι : Type u_4", "w : ι → k", "p : ι → P"], "goal": "(∅.weightedVSub p) w = 0"}}
+{"state": {"context": ["z x✝ y✝ : ℝ", "n : ℕ", "α : Type u_1", "inst✝¹ : LinearOrderedField α", "inst✝ : Archimedean α", "hn : n ≠ 0", "x y : α", "h : x < y", "hy : 0 < y", "q₂ : ℚ", "hx₂ : max x 0 < ↑q₂", "hy₂ : ↑q₂ < y", "q₁ : ℚ", "hx₁ : max x 0 < ↑q₁", "hq₁₂ : ↑q₁ < ↑q₂", "this : 0 < ↑q₂"], "goal": "∃ q, 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "s : E", "p q : FormalMultilinearSeries 𝕜 𝕜 E", "f g : 𝕜 → E", "n : ℕ", "z z₀ : 𝕜", "a : ℕ → E", "hs : HasSum (fun m => z ^ m • a m) s", "ha : ∀ k < n, a k = 0"], "goal": "∃ t, z ^ n • t = s ∧ HasSum (fun m => z ^ m • a (m + n)) t"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "R : Type u_5", "Fintype n", "CommRing R", "Fintype m", "DecidableEq m", "A : Matrix m m R", "B : Matrix m n R", "hA : IsUnit A.det"], "goal": "(A * B).rank = B.rank"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "inst✝⁵ : Ring R", "inst✝⁴ : Ring S", "M : Type u_4", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "m : Submodule R M", "N : Type u_5", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "s : Set ι", "p : ι → Submodule R M", "hp : ∀ i ∈ s, IsSemisimpleModule R ↥(p i)", "hp' : ⨆ i ∈ s, p i = ⊤"], "goal": "IsSemisimpleModule R M"}}
+{"state": {"context": ["a : ℤ", "p : ℕ", "inst✝ : Fact (Nat.Prime p)", "h : J(a | p) = 1"], "goal": "¬1 = -1"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "h : ∀ (u : ℕ → E), (Summable fun x => ‖u x‖) → ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a)", "u : ℕ → E", "hu : CauchySeq u", "f : ℕ → ℕ", "hf₁ : StrictMono f", "hf₂ : Summable fun i => dist (u (f (i + 1))) (u (f i))"], "goal": "∃ a, Tendsto u atTop (𝓝 a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "ht : t.Finite", "hst : s ⊆ t", "hts : (t \\ s).encard + s.encard ≤ 0 + s.encard", "hdiff : t ⊆ s"], "goal": "s = t"}}
+{"state": {"context": ["ι : Type u_1", "I : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : I → Type u_6", "x y : (i : I) → f✝ i", "i : I", "inst✝¹ : One γ", "inst✝ : LE γ", "f : α → β", "g : α → γ", "e : β → γ", "hg : g ≤ 1", "he : e ≤ 1", "_b : β"], "goal": "extend f g e _b ≤ 1 _b"}}
+{"state": {"context": [], "goal": "Measurable Complex.sin"}}
+{"state": {"context": ["A : Type u_1", "A' : Type u_2", "B : Type u_3", "B' : Type u_4", "inst✝³ : Category.{u_5, u_1} A", "inst✝² : Category.{u_7, u_2} A'", "inst✝¹ : Category.{u_6, u_3} B", "inst✝ : Category.{u_8, u_4} B'", "eA : A ≌ A'", "eB : B ≌ B'", "e' : A' ≌ B'", "F : A ⥤ B'", "hF : eA.functor ⋙ e'.functor ≅ F", "G : B ⥤ A", "hG : eB.functor ⋙ e'.inverse ≅ G ⋙ eA.functor", "X : A"], "goal": "(equivalence₁ hF).unitIso.hom.app X ≫ (equivalence₁ hF).inverse.map (eB.counitIso.inv.app (F.obj X)) = (equivalence₂UnitIso eB hF).hom.app X"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "FunLike F G H", "a : G", "b : H", "AddGroup G", "AddGroup H", "AddConstMapClass F G H a b", "f : F", "x : G", "n : ℤ"], "goal": "f (x + n • a) = f x + n • b"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "DecidableEq α", "A B : Finset α", "C : Finset α", "hA : ∀ A' ⊆ A, (A + B).card * A'.card ≤ (A' + B).card * A.card"], "goal": "(A + B + C).card * A.card ≤ (A + B).card * (A + C).card"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v', u'} D", "F : C ⥤ D", "r : HomRel C", "A : Type w", "AddMonoid A", "CategoryTheory.HasShift C A", "CategoryTheory.HasShift D A", "r.IsCompatibleWithShift A", "F.CommShift A", "hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂", "a : A", "X : C"], "goal": "(CategoryTheory.Quotient.LiftCommShift.iso F r hF a).inv.app ((CategoryTheory.Quotient.functor r).obj X) =\n  (F.commShiftIso a).inv.app X ≫\n    (CategoryTheory.Quotient.lift r F hF).map (((CategoryTheory.Quotient.functor r).commShiftIso a).hom.app X)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "N : Type u_5", "AddGroup N", "H : AddSubgroup N"], "goal": "⊤.prod H = AddSubgroup.comap (AddMonoidHom.snd G N) H"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "T : E →ₗ[𝕜] E", "FiniteDimensional 𝕜 E", "_hT : Fact T.IsSymmetric", "x : E", "μ : Module.End.Eigenvalues T"], "goal": "((DirectSum.decompose fun μ => Module.End.eigenspace T (↑T μ)) x) μ =\n  (orthogonalProjection (Module.End.eigenspace T (↑T μ))) x"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : TopologicalSpace α", "β : Type u_2", "inst✝ : Preorder β", "f✝ g : α → β", "x : α", "s t : Set α", "y z : β", "ι : Type u_3", "f : ι → α → ℝ≥0∞", "h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x", "b : Finset ι"], "goal": "LowerSemicontinuousWithinAt (fun x' => ∑ i ∈ b, f i x') s x"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G"], "goal": "↑⊤ = Set.univ"}}
+{"state": {"context": ["F : Type u_1", "K : Type u_2", "E : Type u_3", "inst✝¹³ : Field F", "inst✝¹² : Field K", "inst✝¹¹ : Field E", "inst✝¹⁰ : Algebra F K", "inst✝⁹ : Algebra F E", "inst✝⁸ : Algebra K E", "inst✝⁷ : IsScalarTower F K E", "A₁ : Type u_4", "B₁ : Type u_5", "A₂ : Type u_6", "B₂ : Type u_7", "inst✝⁶ : Field A₁", "inst✝⁵ : Field B₁", "inst✝⁴ : Field A₂", "inst✝³ : Field B₂", "inst✝² : Algebra A₁ B₁", "inst✝¹ : Algebra A₂ B₂", "e₁ : A₁ ≃+* A₂", "e₂ : B₁ ≃+* B₂", "he : (algebraMap A₂ B₂).comp ↑e₁ = (↑e₂).comp (algebraMap A₁ B₁)", "inst✝ : Algebra.IsSeparable A₁ B₁", "this✝¹ : Algebra A₁ A₂ := e₁.toRingHom.toAlgebra", "this✝ : Algebra A₂ B₁ := ((algebraMap A₁ B₁).comp e₁.symm.toRingHom).toAlgebra", "this : IsScalarTower A₁ A₂ B₁", "e : B₁ ≃ₐ[A₂] B₂ := { toEquiv := e₂.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }"], "goal": "Algebra.IsSeparable A₂ B₂"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁶ : Semiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : AddCommMonoid M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a b : R", "x y : M", "n : ℕ", "H : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → s.card ≤ n", "s : Set M", "li : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val"], "goal": "∀ (s_1 : Finset ↑s), s_1.card ≤ n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g : X ⟶ Y"], "goal": "(CategoryTheory.Limits.parallelPair f g).obj CategoryTheory.Limits.WalkingParallelPair.zero = X"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s'✝ : Set α", "x : α", "p : Filter ι", "p'✝ : Filter α", "g : ι → α", "ι' : Type u_1", "α' : Type u_2", "β' : Type u_3", "inst✝ : UniformSpace β'", "F' : ι' → α' → β'", "f' : α' → β'", "p' : Filter ι'", "s' : Set α'", "h : TendstoUniformlyOnFilter F f p (𝓟 s)", "h' : TendstoUniformlyOnFilter F' f' p' (𝓟 s')"], "goal": "TendstoUniformlyOnFilter (fun i => Prod.map (F i.1) (F' i.2)) (Prod.map f f') (p ×ˢ p') (𝓟 (s ×ˢ s'))"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝ : Field K", "P : K[X]", "hP : ¬P = 0"], "goal": "(Polynomial.idealX K).intValuation P = (idealX K).intValuation ↑P"}}
+{"state": {"context": ["α : Sort u_1", "a b c : List Nat", "ctx : Lean.Data.AC.Context α", "h : Std.Commutative ctx.op", "h₂ : Lean.Data.AC.evalList α ctx a = Lean.Data.AC.evalList α ctx b", "h₃ : a ≠ []", "h₄ : b ≠ []"], "goal": "Lean.Data.AC.evalList α ctx (Lean.Data.AC.sort.loop a c) = Lean.Data.AC.evalList α ctx (Lean.Data.AC.sort.loop b c)"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝ : PseudoEMetricSpace α", "δ✝ ε✝ : ℝ", "s✝ t : Set α", "x : α", "ε δ : ℝ", "s : Set α", "hε : 0 ≤ ε", "hδ : δ ≤ 0"], "goal": "thickening ε (thickening δ s) ⊆ thickening (ε + δ) s"}}
+{"state": {"context": ["θ : Angle"], "goal": "↑θ.toReal = θ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "TopologicalSpace H", "TopologicalSpace M", "I : ModelWithCorners 𝕜 E H", "ChartedSpace H M", "x : M"], "goal": "(extChartAt I x).source ∈ 𝓝 x"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hε₁ : ε ≤ 1", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "hUV : U ≠ V", "hunif : ¬G.IsUniform ε U V"], "goal": "4 / 5 * ε ≤ ↑(star hP G ε hU V).card / 4 ^ P.parts.card"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Group G", "inst✝¹ : MulAction G α", "inst✝ : MulAction G β", "g : G", "x : Quotient G α"], "goal": "Set.MapsTo (fun x => g • x) (MulAction.orbit G (Quotient.out' x)) (MulAction.orbit G (Quotient.out' x))"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "inst✝² : LinearOrderedSemifield α", "inst✝¹ : Archimedean α", "inst✝ : ExistsAddOfLE α", "x y ε : α", "hx : 0 < x", "hy : 1 < y"], "goal": "∃ n, x ∈ Ico (y ^ n) (y ^ (n + 1))"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Semiring R", "AddCommMonoid M", "Module R M", "m : M"], "goal": "(TrivSqZeroExt.inrHom R M) m = TrivSqZeroExt.inr m"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "m' : o → Type u_5", "n' : o → Type u_6", "R : Type u_7", "S : Type u_8", "α : Type v", "β : Type w", "γ : Type u_9", "inst✝¹ : NonUnitalNonAssocSemiring α", "inst✝ : Fintype m", "A B : Matrix m n α", "x : m → α", "x✝ : n"], "goal": "(x ᵥ* (A + B)) x✝ = (x ᵥ* A + x ᵥ* B) x✝"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "X' : Type u_4", "Y : Type u_5", "Z : Type u_6", "α : Type u_7", "α' : Type u_8", "β : Type u_9", "β' : Type u_10", "γ : Type u_11", "𝓕 : Type u_12", "tX : TopologicalSpace X", "tY : TopologicalSpace Y", "tZ : TopologicalSpace Z", "uα : UniformSpace α", "uβ : UniformSpace β", "uγ : UniformSpace γ", "inst✝ : Finite ι", "F : ι → X → α", "S : Set X", "x₀ : X"], "goal": "EquicontinuousWithinAt F S x₀ ↔ ∀ (i : ι), ContinuousWithinAt (F i) S x₀"}}
+{"state": {"context": ["X : RingedSpace", "U V : (Opens ↑↑X.toPresheafedSpace)ᵒᵖ", "i : U ⟶ V", "inst✝ : IsIso i", "f : ↑(X.presheaf.obj U)"], "goal": "X.basicOpen ((X.presheaf.map i) f) = X.basicOpen f"}}
+{"state": {"context": ["α : Type u_1", "f : α → α"], "goal": "Function.Commute f id"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D"], "goal": "(CategoryTheory.Monoidal.transportedMonoidalCounitIso e).inv =\n  CategoryTheory.monoidalUnit (CategoryTheory.Monoidal.fromTransported e) e.symm.toAdjunction"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : Category.{v₁, u₂} D", "i : D ⥤ C", "inst✝² : HasFiniteProducts C", "inst✝¹ : CartesianClosed C", "inst✝ : Reflective i", "h : (A : C) → i ⋙ exp A ⋙ reflector i ⋙ i ≅ i ⋙ exp A"], "goal": "ExponentialIdeal i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "g : γ → α", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ", "hg : Measurable g"], "goal": "(ProbabilityTheory.Kernel.sum fun n => (κ.seq n).comap g hg) = κ.comap g hg"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : Abelian C", "S : ShortComplex C"], "goal": "S.Exact ↔ Epi (imageToKernel S.f S.g ⋯)"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : Monoid G", "a b x✝ y : G", "n m : ℕ", "M : Type ?u.50075", "inst✝ : Monoid M", "x : M", "hx : IsOfFinOrder x"], "goal": "x * x ^ (orderOf x - 1) = 1"}}
+{"state": {"context": ["R : Type u", "inst✝ : Semiring R", "q : ℕ", "h : Nat.Prime q", "hchar✝ : CharP R q"], "goal": "max q 1 = q"}}
+{"state": {"context": ["X Y : Scheme", "f : X ⟶ Y"], "goal": "MorphismProperty.StableUnderBaseChange @QuasiCompact"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "u : ℕ → E", "l : E", "h : Tendsto u atTop (𝓝 l)"], "goal": "Tendsto (fun n => (↑n)⁻¹ • ∑ i ∈ range n, u i) atTop (𝓝 l)"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "u v w : V"], "goal": "G.dist u v = G.dist v u"}}
+{"state": {"context": ["a b : Nat", "h : a ≤ b"], "goal": "min a b = a"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.SmallCategory J", "F : CategoryTheory.Functor J CategoryTheory.Cat", "X : CategoryTheory.Limits.limit (F.comp CategoryTheory.Cat.objects)"], "goal": "CategoryTheory.CategoryStruct.id X =\n  CategoryTheory.Limits.Types.Limit.mk (CategoryTheory.Cat.HasLimits.homDiagram X X)\n    (fun j => CategoryTheory.CategoryStruct.id (CategoryTheory.Limits.limit.π (F.comp CategoryTheory.Cat.objects) j X))\n    ⋯"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "F G K L : CochainComplex C ℤ", "n m : ℤ", "hnm : ¬n + 1 = m", "z : Cochain F G n", "p q : ℤ", "hpq : p + m = q"], "goal": "(Cochain.mk fun p q hpq => z.v p (p + n) ⋯ ≫ G.d (p + n) q + m.negOnePow • F.d p (p + m - n) ≫ z.v (p + m - n) q ⋯).v p q hpq = Cochain.v 0 p q hpq"}}
+{"state": {"context": ["G : Type u_3", "AddCommSemigroup G", "a b c : G"], "goal": "a + b + c = a + c + b"}}
+{"state": {"context": ["Ω : Type u_1", "Nonempty Ω", "m0 : MeasurableSpace Ω", "μ : MeasureTheory.ProbabilityMeasure Ω"], "goal": "μ.toFiniteMeasure.normalize = μ"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p✝ q✝ r : R[X]", "inst✝ : Semiring S", "f : R →+* S", "p q : R[X]"], "goal": "∀ (a : R), map f ((C a).comp q) = (map f (C a)).comp (map f q)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "TopologicalSpace β", "TotallyDisconnectedSpace β", "f : α → β", "h : Continuous f", "g : ConnectedComponents α → β", "hg : g ∘ ConnectedComponents.mk = f"], "goal": "g = h.connectedComponentsLift"}}
+{"state": {"context": ["α : Type u", "UniformSpace α"], "goal": "UniformInducing id"}}
+{"state": {"context": ["z : ℂ", "hz : ‖z‖ < 1", "this : HasSum (fun x => -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2))) (-I / 2 * log ((1 + z * I) / (1 - z * I)))", "k : ℕ"], "goal": "HasSum (fun c => -I / 2 * (((-1) ^ (2 * (k, c).1 + ↑(k, c).2 + 1) + 1) * (z * I) ^ (2 * (k, c).1 + ↑(k, c).2) / ↑(2 * (k, c).1 + ↑(k, c).2))) ((-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1))"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ✝ : Measure α", "f✝ g✝ : α → ℝ", "s✝ : Set α", "μ : Measure α", "f g : α → ℝ", "f_mble : AEMeasurable f μ", "g_mble : AEMeasurable g μ", "s : Set α", "s_mble : NullMeasurableSet s μ"], "goal": "NullMeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ico (f p.1) (g p.1)} (μ.prod volume)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K✝ : ℝ≥0", "f✝ : α → β", "x y : α", "r : ℝ≥0∞", "f : Function.End α", "K : ℝ≥0", "h : LipschitzWith K f", "n : ℕ"], "goal": "LipschitzWith (K ^ n * K) (f ^ n * f)"}}
+{"state": {"context": ["M : Type u_4", "AddMonoid M", "n : ℕ", "a : M"], "goal": "(List.replicate n a).sum = n • a"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "s : Set X", "x : { x // x ∈ s }"], "goal": "𝓝 x = Filter.comap Subtype.val (𝓝[s] ↑x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α", "M : Type u_3", "inst✝² : AddCommMonoid M", "inst✝¹ : TopologicalSpace M", "inst✝ : OrderedAddCommMonoid M", "s✝ : SignedMeasure α", "i j : Set α", "s : SignedMeasure α", "f : ℕ → ℝ", "left✝ : Antitone f", "hf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))", "B : ℕ → Set α", "hB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ restrict s B ≤ restrict 0 B} ∧ ↑s (B n) = f n", "hB₁ : ∀ (n : ℕ), MeasurableSet (B n)", "hB₂ : ∀ (n : ℕ), restrict s (B n) ≤ restrict 0 (B n)"], "goal": "∃ i, MeasurableSet i ∧ restrict 0 i ≤ restrict s i ∧ restrict s iᶜ ≤ restrict 0 iᶜ"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "M : Type u_3", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : LieAlgebra.IsNilpotent R L", "inst✝² : NoZeroSMulDivisors ℤ R", "inst✝¹ : NoZeroSMulDivisors R M", "inst✝ : IsNoetherian R M", "α : L → R", "β : Weight R L M", "hα : α ≠ 0"], "goal": "⇑(chainTop α β) = chainTopCoeff α β • α + ⇑β"}}
+{"state": {"context": ["R : Type u1", "inst✝² : CommRing R", "M : Type u2", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "d d' : Module.Dual R M", "m : CliffordAlgebra Q", "x : M", "hx : (contractLeft d) ((contractLeft d') m) = -(contractLeft d') ((contractLeft d) m)"], "goal": "(contractLeft d) ((contractLeft d') ((ι Q) x * m)) = -(contractLeft d') ((contractLeft d) ((ι Q) x * m))"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace ℂ F", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners ℂ E H", "I.Boundaryless", "M : Type u_4", "TopologicalSpace M", "CompactSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "f : M → F", "hf : MDifferentiable I 𝓘(ℂ, F) f"], "goal": "IsLocallyConstant f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝² : ConditionallyCompleteLinearOrder α", "f✝ : Filter α", "b : α", "v : Filter ι", "p : ι' → Prop", "s : ι' → Set ι", "inst✝¹ : Countable (Subtype p)", "inst✝ : Nonempty (Subtype p)", "hv : v.HasBasis p s", "f : ι → α", "H : ¬∃ j, s ↑j = ∅", "H' : ¬∀ (j : Subtype p), ¬BddBelow (range fun i => f ↑i)"], "goal": "∀ (j : Subtype p), (s ↑j).Nonempty"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x✝ y : α", "s t : Set α", "Φ : α → β", "x : α", "E : Set α"], "goal": "0 < infEdist x (closure E) ↔ x ∉ closure E"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "I J : Ideal R"], "goal": "zeroLocus ↑I ⊆ zeroLocus ↑J ↔ J ≤ I.radical"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "C₃ : Type u_3", "CategoryTheory.Category.{u_4, u_1} C₁", "CategoryTheory.Category.{u_5, u_2} C₂", "CategoryTheory.Category.{u_6, u_3} C₃", "W : CategoryTheory.MorphismProperty C₁", "F : CategoryTheory.Functor C₁ C₂", "G : CategoryTheory.Functor C₂ C₃"], "goal": "(W.map F).IsInvertedBy G ↔ W.IsInvertedBy (F.comp G)"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "m1 : MeasurableSpace β", "f : α → β", "hf : MeasurableEmbedding f", "μ : Measure α", "s t : Set β", "ht : MeasurableSet t"], "goal": "((Measure.map f μ).restrict s) t = (Measure.map f (μ.restrict (f ⁻¹' s))) t"}}
+{"state": {"context": ["R : Type u_1", "S : Type v₃", "CommSemiring R", "CommSemiring S", "Algebra R S", "R' : Type u_6", "S' : Type u_7", "CommSemiring R'", "CommSemiring S'", "Algebra R R'", "Algebra S S'", "Algebra R' S'", "Algebra R S'", "IsScalarTower R R' S'", "IsScalarTower R S S'", "H : Algebra.IsPushout R S R' S'", "A : Type u_8", "Semiring A", "Algebra R A", "f : S →ₐ[R] A", "g : R' →ₐ[R] A", "hf : ∀ (x : S) (y : R'), f x * g y = g y * f x", "a : S'"], "goal": "(Algebra.pushoutDesc S' f g hf) a = (⋯.lift g.toLinearMap) a"}}
+{"state": {"context": ["n : Num"], "goal": "↑n.succ' = ↑n + 1"}}
+{"state": {"context": ["α : Type u_3", "n : ℕ", "β : Type u_2", "γ : Type u_1", "σ₂ : Type", "xs : Mathlib.Vector α n", "f₂ : α → σ₂ → σ₂ × β", "s : σ₂", "f₁ : β → γ"], "goal": "Mathlib.Vector.map f₁ (Mathlib.Vector.mapAccumr f₂ xs s).2 =\n  (Mathlib.Vector.mapAccumr\n      (fun x s =>\n        let r := f₂ x s;\n        (r.1, f₁ r.2))\n      xs s).2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "δ : Type u_7", "δ' : Type u_8", "ε : Type u_9", "DecidableEq δ", "DecidableEq δ'", "DecidableEq ε", "s : Finset α", "t : Finset β", "γ : Type u_14", "u : Finset γ", "f : δ → γ → ε", "g : α → β → δ", "f' : α → γ → δ'", "g' : δ' → β → ε", "h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b"], "goal": "Finset.image₂ f (Finset.image₂ g s t) u = Finset.image₂ g' (Finset.image₂ f' s u) t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "Zero M", "l : α →₀ M", "f : α ↪ β", "a : β", "b : M", "hb : b ≠ 0", "h : Finsupp.embDomain f l = Finsupp.single a b"], "goal": "∃ x, l = Finsupp.single x b ∧ f x = a"}}
+{"state": {"context": ["K : Type u_4", "L : Type u_5", "Field K", "Field L", "Algebra K L", "FiniteDimensional K L", "Algebra.IsSeparable K L", "pb : PowerBasis K L", "i : Fin pb.dim"], "goal": "((Algebra.traceForm K L).dualBasis ⋯ pb.basis) i =\n  (minpolyDiv K pb.gen).coeff ↑i / (Polynomial.aeval pb.gen) (Polynomial.derivative (minpoly K pb.gen))"}}
+{"state": {"context": ["α : Type u_1", "x : α", "as bs : List α"], "goal": "x ∈ as.reverseAux bs ↔ x ∈ as ∨ x ∈ bs"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a✝ : α", "f✝ g : α → β", "inst✝ : CommMonoid β", "s : Finset α", "f : α → β", "b₁ b₂ : β", "a : α", "ha : a ∈ s", "h : f a * b₁ = f a * b₂"], "goal": "(∏ a ∈ s, f a) * b₁ = (∏ a ∈ s, f a) * b₂"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "u v : V"], "goal": "G.dist u v = G.dist v u"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "CategoryTheory.Limits.HasColimit F", "c : CategoryTheory.Limits.Cocone F"], "goal": "(CategoryTheory.Limits.colimit.isColimit F).desc c = CategoryTheory.Limits.colimit.desc F c"}}
+{"state": {"context": ["R : Type r", "CommRing R", "W : WeierstrassCurve R", "n : ℤ"], "goal": "(WeierstrassCurve.Affine.CoordinateRing.mk W) (W.ψ n) = (WeierstrassCurve.Affine.CoordinateRing.mk W) (W.Ψ n)"}}
+{"state": {"context": ["a b c : ℝ≥0∞"], "goal": "a ≠ ⊤ → a + b ≤ a + c → b ≤ c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_4", "κ : ι → Sort u_7", "s : (i : ι) → κ i → Set α", "t : Set β", "f : α → β"], "goal": "Set.MapsTo f (⋃ i, ⋃ j, s i j) t ↔ ∀ (i : ι) (j : κ i), Set.MapsTo f (s i j) t"}}
+{"state": {"context": ["a : Bool", "m : ℕ", "b : Bool", "n : ℕ"], "goal": "(Nat.bit a m).ldiff (Nat.bit b n) = Nat.bit (a && !b) (m.ldiff n)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoMetricSpace α", "PseudoMetricSpace β", "f : α → β", "s : Set α"], "goal": "UniformContinuousOn f s ↔ ∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_3", "CommSemiring R"], "goal": "SetLike.GradedMonoid (MvPolynomial.homogeneousSubmodule σ R)"}}
+{"state": {"context": ["α : Type u_1", "f : Perm α", "n : ℤ", "x : α"], "goal": "f ((f ^ n) x) = (f ^ n) x ↔ f x = x"}}
+{"state": {"context": ["α : Type u_2", "One α", "h : Set.Finite 1 := ⋯"], "goal": "Set.Finite.toFinset h = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "F : Type u_2", "RCLike 𝕜", "NormedAddCommGroup F", "InnerProductSpace 𝕜 F", "CompleteSpace F", "f : F → 𝕜", "f' x : F", "s : Set F"], "goal": "HasGradientWithinAt f f' s x ↔ (fun x' => f x' - f x - ⟪f', x' - x⟫_𝕜) =o[𝓝[s] x] fun x' => x' - x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : WSeq α"], "goal": "Computation.IsBisimulation fun c1 c2 => ∃ l s, c1 = Computation.corec (fun x => match x with | (n, s) => match Seq.destruct s with | none => Sum.inl n | some (none, s') => Sum.inr (n, s') | some (some val, s') => Sum.inr (n + 1, s')) (l.length, s) ∧ c2 = Computation.map List.length (Computation.corec (fun x => match x with | (l, s) => match Seq.destruct s with | none => Sum.inl l.reverse | some (none, s') => Sum.inr (l, s') | some (some a, s') => Sum.inr (a :: l, s')) (l, s))"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : UniformSpace α", "pkg : AbstractCompletion α", "β : Type u_2", "inst✝² : UniformSpace β", "pkg' : AbstractCompletion α", "γ : Type u_3", "inst✝¹ : TopologicalSpace γ", "inst✝ : T3Space γ", "f : α → γ", "cont_f : Continuous f", "x✝¹ : UniformSpace pkg'.space := pkg'.uniformStruct", "x✝ : UniformSpace pkg.space := pkg.uniformStruct", "h : ∀ (a : pkg.space), Tendsto f (comap pkg.coe (𝓝 a)) (𝓝 (⋯.extend f a))"], "goal": "⋯.extend f ∘ pkg'.compare pkg = ⋯.extend f"}}
+{"state": {"context": ["f : ℕ → ℂ"], "goal": "f * LSeries.delta = f 1 • LSeries.delta"}}
+{"state": {"context": ["G : Type u_1", "CommGroup G", "s : Set G"], "goal": "s * ↑(MulAction.stabilizer G s) = s"}}
+{"state": {"context": ["a : ℕ"], "goal": "Nat.ceilRoot 0 a = 0"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "hx : ¬x = 0", "hy : ¬y = 0"], "goal": "(o.oangle (-x) y).sign = -(o.oangle x y).sign"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "AddCommMonoid β", "DecidableEq α", "s t : Finset α", "f g : α → β"], "goal": "∑ x ∈ s, t.piecewise f g x = ∑ x ∈ s ∩ t, f x + ∑ x ∈ s \\ t, g x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_7", "Semiring R", "Semiring S", "f : R ≃+* S"], "goal": "∀ (a : S), f.toSemilinearEquiv.symm a = f.invFun a"}}
+{"state": {"context": ["R : Type u_2", "Ring R", "M : Type u_4", "AddCommGroup M", "Module R M", "N : Type u_5", "AddCommGroup N", "Module R N", "IsSimpleModule R M", "IsSimpleModule R N", "f : M →ₗ[R] N"], "goal": "Function.Bijective ⇑f ∨ f = 0"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "s : Set P", "p : P", "hp : p ∈ s"], "goal": "Submodule.span k (s -ᵥ s) ≤ Submodule.span k ((fun x => x -ᵥ p) '' s)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "x : 𝕜", "R : Type u_1", "CommSemiring R", "Algebra R 𝕜", "q : R[X]"], "goal": "DifferentiableAt 𝕜 (fun x => (Polynomial.aeval x) q) x"}}
+{"state": {"context": ["k : Type u_1", "G : Type u_2", "inst✝¹ : Semiring k", "inst✝ : AddCancelCommMonoid G", "x : k[G]", "a b x✝ : G"], "goal": "(x /ᵒᶠ (a + b)) x✝ = (x /ᵒᶠ a /ᵒᶠ b) x✝"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n✝ : ℕ", "inst✝ : Semiring R", "p q r : R[X]", "x : R", "n : ℕ"], "goal": "eval x (↑n * p) = ↑n * eval x p"}}
+{"state": {"context": ["p : ℕ", "inst✝⁶ : Fact (Nat.Prime p)", "k✝ : Type u_1", "inst✝⁵ : CommRing k✝", "k : Type u_2", "inst✝⁴ : Field k", "inst✝³ : IsAlgClosed k", "inst✝² : CharP k p", "V : Type u_3", "inst✝¹ : AddCommGroup V", "inst✝ : Isocrystal p k V", "h_dim : finrank K(p, k) V = 1", "this✝ : Nontrivial V", "x : V", "hx : x ≠ 0", "this : Φ(p, k) x ≠ 0", "a : K(p, k)", "ha : a ≠ 0", "hax : Φ(p, k) x = a • x", "b : K(p, k)", "hb : b ≠ 0", "m : ℤ", "hmb : φ(p, k) b * a = ↑p ^ m * b"], "goal": "∃ m, Nonempty (StandardOneDimIsocrystal p k m ≃ᶠⁱ[p, k] V)"}}
+{"state": {"context": ["R : Type u_1", "k : Type u_2", "inst✝ : Field k", "f : k⟦X⟧", "hf : f ≠ 0", "h : IsUnit f", "this : f.order.get ⋯ = 0"], "goal": "divided_by_X_pow_order hf = X ^ f.order.get ⋯ * divided_by_X_pow_order hf"}}
+{"state": {"context": ["α : Type u_3", "Nonempty α", "SemilatticeInf α", "Countable α"], "goal": "Filter.atBot.HasCountableBasis (fun x => True) Set.Iic"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "s : AffineSubspace ℝ P", "Nonempty ↥s", "HasOrthogonalProjection s.direction", "p : P"], "goal": "EuclideanGeometry.orthogonalProjectionFn s p ∈ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "p : α → Prop", "DecidablePred p", "f : α → β", "hf : Function.Injective f", "s : Multiset α", "DecidablePred fun b => ∃ a, p a ∧ f a = b"], "goal": "Multiset.map f (Multiset.filter p s) = Multiset.filter (fun b => ∃ a, p a ∧ f a = b) (Multiset.map f s)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "↑(NonemptyInterval.pure a) = Interval.pure a"}}
+{"state": {"context": ["R : Type u", "L : Type v", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "L₂ : Type w", "inst✝¹ : LieRing L₂", "inst✝ : LieAlgebra R L₂", "f : L →ₗ⁅R⁆ L₂", "K K' : LieSubalgebra R L", "K₂ : LieSubalgebra R L₂"], "goal": "K.incl.range = K"}}
+{"state": {"context": ["f : ℕ → ℝ≥0∞"], "goal": "Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, f i) Filter.atTop (𝓝 (∑' (n : ℕ), f n))"}}
+{"state": {"context": ["f f' g : ℝ → ℝ", "a : ℝ", "hf : ContinuousOn f (Set.Ici a)", "hft : Filter.Tendsto f Filter.atTop Filter.atTop", "hff' : ∀ x ∈ Set.Ioi a, HasDerivWithinAt f (f' x) (Set.Ioi x) x", "hg_cont : ContinuousOn g (f '' Set.Ioi a)", "hg1 : MeasureTheory.IntegrableOn g (f '' Set.Ici a) MeasureTheory.volume", "hg2 : MeasureTheory.IntegrableOn (fun x => (g ∘ f) x * f' x) (Set.Ici a) MeasureTheory.volume"], "goal": "∫ (x : ℝ) in Set.Ioi a, (g ∘ f) x * f' x = ∫ (u : ℝ) in Set.Ioi (f a), g u"}}
+{"state": {"context": ["N : ℕ", "j : ℝ", "hj : 0 < j", "c : ℝ", "hc : 1 < c"], "goal": "∑ i ∈ Finset.filter (fun x => j < ↑⌊c ^ x⌋₊) (Finset.range N), 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "B' : Type u_3", "inst✝³ : CommRing A", "inst✝² : Ring B", "inst✝¹ : Algebra A B", "inst✝ : Nontrivial B", "x : A"], "goal": "(minpoly A ((algebraMap A B) x)).degree = 1"}}
+{"state": {"context": ["C : Type u", "𝒞 : CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X Y : C × C × C", "f : X ⟶ Y"], "goal": "(CategoryTheory.MonoidalCategory.leftAssocTensor C).map f =\n  CategoryTheory.MonoidalCategory.tensorHom (CategoryTheory.MonoidalCategory.tensorHom f.1 f.2.1) f.2.2"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : AddCommMonoid M'", "inst✝ : Module R M'", "b b₁ : Basis ι R M", "i : ι", "c : R", "x : M"], "goal": "b.repr.symm (Finsupp.single i c) = b.repr.symm (c • Finsupp.single i 1)"}}
+{"state": {"context": ["a b : Ordinal.{u}"], "goal": "a * b ≤ a ⨳ b"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "p : Polynomial R", "S : Type u_2", "NonAssocSemiring S", "Module R S", "IsScalarTower R S S", "Pow S ℕ", "NatPowAssoc S", "x : S", "n : ℕ"], "goal": "(p * Polynomial.X ^ n).smeval x = p.smeval x * x ^ n"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m✝ m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "hp : Fact (1 ≤ p)", "H : ∀ (f : ℕ → α → E), (∀ (n : ℕ), Memℒp (f n) p μ) → ∀ (B : ℕ → ℝ≥0∞), ∑' (i : ℕ), B i < ⊤ → (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) → ∃ f_lim, Memℒp f_lim p μ ∧ Tendsto (fun n => eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)", "B : ℕ → ℝ := fun n => (1 / 2) ^ n", "hB_pos : ∀ (n : ℕ), 0 < B n", "f : ℕ → ↥(Lp E p μ)", "hf : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) < B N", "M : ℝ", "hB : HasSum B M", "B1 : ℕ → ℝ≥0∞ := fun n => ENNReal.ofReal (B n)", "hB1_has : HasSum B1 (ENNReal.ofReal M)", "hB1 : ∑' (i : ℕ), B1 i < ⊤", "f1 : ℕ → α → E := fun n => ↑↑(f n)", "N n m : ℕ", "hn : N ≤ n", "hm : N ≤ m"], "goal": "eLpNorm (f1 n - f1 m) p μ < B1 N"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W : WeierstrassCurve R", "inst✝ : Invertible 2"], "goal": "IsUnit (2 ^ 4) ∧ IsUnit W.Δ ↔ IsUnit W.Δ"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁵ : NormedRing A", "inst✝⁴ : NormedAlgebra ℂ A", "inst✝³ : CompleteSpace A", "inst✝² : StarRing A", "inst✝¹ : CstarRing A", "inst✝ : StarModule ℂ A", "a : A", "ha : IsSelfAdjoint a", "z : ℂ", "hz : z ∈ spectrum ℂ a"], "goal": "z = ↑z.re"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedSemifield α", "a b c d e : α", "m n : ℤ", "ha : 1 ≤ a"], "goal": "a⁻¹ ≤ 1"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "A : Set X", "x : X"], "goal": "x ∈ derivedSet A ↔ AccPt x (𝓟 A)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁵ : Category.{u_3, u_1} C", "inst✝⁴ : Category.{?u.12624, u_2} D", "inst✝³ : Preadditive C", "inst✝² : Balanced C", "S₁ S₂ : ShortComplex C", "φ : S₁ ⟶ S₂", "h₁ : S₁.ShortExact", "h₂ : S₂.ShortExact", "inst✝¹ : IsIso φ.τ₁", "inst✝ : IsIso φ.τ₃", "this✝¹ : Mono S₁.f", "this✝ : Mono S₂.f", "this : Epi S₁.g"], "goal": "IsIso φ.τ₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G : SimpleGraph α", "n : ℕ", "f : α ↪ β", "Nonempty α"], "goal": "(SimpleGraph.map f G).CliqueFree n ↔ G.CliqueFree n"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝⁷ : CommSemiring R✝", "M : Submonoid R✝", "S✝ : Type u_2", "inst✝⁶ : CommSemiring S✝", "inst✝⁵ : Algebra R✝ S✝", "P : Type u_3", "inst✝⁴ : CommSemiring P", "inst✝³ : IsLocalization M S✝", "R : Type u_4", "S : Type u_5", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "r : R", "hr : IsUnit r", "H : Function.Bijective ⇑(algebraMap R S)", "n : ℕ"], "goal": "IsUnit ((algebraMap R S) ↑⟨(fun x => r ^ x) n, ⋯⟩)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "PseudoMetricSpace β", "f g : BoundedContinuousFunction α β"], "goal": "nndist f g = sInf {C | ∀ (x : α), nndist (f x) (g x) ≤ C}"}}
+{"state": {"context": ["A : Type u_1", "SeminormedCommGroup A", "m : ℕ", "δ : ℝ", "n : ℕ", "hm : 0 < m"], "goal": "(fun y => y ^ m) '' approxOrderOf A (n * m) δ ⊆ approxOrderOf A n (↑m * δ)"}}
+{"state": {"context": ["m n a✝ b✝ c✝ d a b c : ℕ", "hca : c ∣ a", "h : ¬c = 0"], "goal": "b % c < c"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommSemiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝⁸ : Fintype n", "inst✝⁷ : Fintype m", "inst✝⁶ : DecidableEq n", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁵ : AddCommMonoid M₁", "inst✝⁴ : AddCommMonoid M₂", "inst✝³ : Module R M₁", "inst✝² : Module R M₂", "v₁ : Basis n R M₁", "v₂ : Basis m R M₂", "M₃ : Type u_7", "inst✝¹ : AddCommMonoid M₃", "inst✝ : Module R M₃", "v₃ : Basis l R M₃", "x : R", "i✝ j✝ : n"], "goal": "(toMatrix v₁ v₁) (DistribMulAction.toLinearMap R M₁ x) i✝ j✝ = Matrix.diagonal (fun x_1 => x) i✝ j✝"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "H : Type u_2", "inst✝⁴ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝³ : TopologicalSpace M", "inst✝² : ChartedSpace H M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "this✝¹ : LocallyCompactSpace H", "this✝ : LocallyCompactSpace M", "this : SecondCountableTopology H"], "goal": "MetrizableSpace M"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "s : Set α", "a✝ b✝ c d a b : α"], "goal": "min a b = a ∨ min a b = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2"], "goal": "Sum.inr ⁻¹' Set.range Sum.inl = ∅"}}
+{"state": {"context": ["σ : Type u", "R : Type v", "inst✝ : CommSemiring R", "p✝ m : ℕ", "p : MvPolynomial σ R", "n : ℕ"], "goal": "p ∈ restrictDegree σ R n ↔ ∀ s ∈ p.support, ∀ (i : σ), s i ≤ n"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "σ : Type u_3", "AddCommMonoid M", "w : σ → M", "m : M", "p : MvPolynomial σ R", "hp : MvPolynomial.IsWeightedHomogeneous w p m"], "goal": "(MvPolynomial.weightedHomogeneousComponent w m) p = p"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "p q P : α → Prop"], "goal": "(∃ a, P a) ∨ ¬∃ x, P x"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "l : Filter ι", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "OpensMeasurableSpace α", "b : ι → α", "hb : Filter.Tendsto b l Filter.atTop", "NoMaxOrder α"], "goal": "MeasureTheory.AECover μ l fun i => Set.Iio (b i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_4", "EMetricSpace α", "FunLike F α β", "PseudoEMetricSpace β", "DilationClass F α β", "f : F"], "goal": "Embedding ⇑f"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝² : DecidableEq α", "inst✝¹ : CommSemiring R", "f : α → R", "s : Finset α", "inst✝ : CommSemiring R", "x : α → R", "n : ℕ"], "goal": "s.sum x ^ n = ∑ i : { x // x ∈ s.sym n }, ↑(↑↑i).multinomial * (Multiset.map x ↑↑i).prod"}}
+{"state": {"context": ["R : Type u_2", "CommSemiring R", "n m : ℕ", "f : PowerSeries R", "h : n < m"], "goal": "(PowerSeries.coeff R n) ↑(PowerSeries.trunc m f) = (PowerSeries.coeff R n) f"}}
+{"state": {"context": ["B : Type u", "Quiver B", "a b c : CategoryTheory.FreeBicategory B", "f : a ⟶ b", "g : b ⟶ c"], "goal": "CategoryTheory.FreeBicategory.Hom.comp f g = CategoryTheory.CategoryStruct.comp f g"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : _root_.RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : InnerProductSpace ℝ F", "K U V : Submodule 𝕜 E", "inst✝ : HasOrthogonalProjection U", "h : (orthogonalProjection U).comp V.subtypeL = 0", "u : E", "hu : u ∈ U", "v : E", "hv : v ∈ V"], "goal": "v = v - ↑((orthogonalProjection U) v)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "inst✝ : DecidableEq α", "l : List ι", "i : ℕ", "h : i < l.length"], "goal": "(take i l).length + (drop (i + 1) l).length + 1 = i + (drop (i + 1) l).length + 1"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝¹ : AddCommGroup R", "inst✝ : StarAddMonoid R", "x : R"], "goal": "x ∈ (skewAdjoint R).carrier ↔ star x = -x"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "I : Type u_3", "J : Type u_4", "inst✝⁷ : Category.{u_6, u_1} C", "inst✝⁶ : Category.{u_5, u_2} D", "inst✝⁵ : Zero I", "inst✝⁴ : DecidableEq I", "inst✝³ : HasInitial C", "F : D ⥤ C ⥤ D", "Y : C", "e : F.flip.obj Y ≅ 𝟭 D", "inst✝² : (X : D) → PreservesColimit (Functor.empty C) (F.obj X)", "p : J × I → J", "hp : ∀ (j : J), p (j, 0) = j", "X X' : GradedObject J D", "φ : X ⟶ X'", "inst✝¹ : (((mapBifunctor F J I).obj X).obj ((single₀ I).obj Y)).HasMap p", "inst✝ : (((mapBifunctor F J I).obj X').obj ((single₀ I).obj Y)).HasMap p", "j : J"], "goal": "Cofan.inj (mapBifunctorRightUnitorCofan F Y e p hp X j) ⟨(j, 0), ⋯⟩ = (F.obj (X j)).map (singleObjApplyIso 0 Y).hom ≫ e.hom.app (X j)"}}
+{"state": {"context": ["xl xr : Type u", "x y : PGame"], "goal": "x.IsOption (-y) ↔ (-x).IsOption y"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y : C", "CategoryTheory.Limits.HasBinaryProduct X Y", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C"], "goal": "CategoryTheory.IsPullback CategoryTheory.Limits.prod.fst CategoryTheory.Limits.prod.snd 0 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R"], "goal": "degrees 0 = 0"}}
+{"state": {"context": ["Ω : Type u_1", "MeasurableSpace Ω", "TopologicalSpace Ω", "HasOuterApproxClosed Ω", "BorelSpace Ω"], "goal": "Function.Injective MeasureTheory.FiniteMeasure.toWeakDualBCNN"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l : List α", "a : α"], "goal": "True ∧ ∀ (b : α), b ∈ l ∧ b = a → b = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "h : ∀ (x' : E), ‖x' - x‖ = 0 → ‖f x' - f x - f' (x' - x)‖ = 0"], "goal": "Tendsto (fun x_1 => ‖f x_1 - f x - f' (x_1 - x)‖ / ‖x_1 - x‖) L (𝓝 0) ↔ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0)"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁵ : DivisionRing K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "V₂ : Type v'", "inst✝² : AddCommGroup V₂", "inst✝¹ : Module K V₂", "ι : Type u_1", "inst✝ : Unique ι", "h : finrank K V = 1", "v : V", "hv : v ≠ 0"], "goal": "(basisSingleton ι h v hv) default = v"}}
+{"state": {"context": ["x✝ y x : ℝ"], "goal": "-x < √(1 + x ^ 2)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b c : α", "hab : a ⩿ b", "hbc : AntisymmRel (fun x x_1 => x ≤ x_1) b c"], "goal": "a ⩿ c"}}
+{"state": {"context": ["K : Type u", "L : Type v", "M : Type w", "inst✝² : DivisionRing K", "inst✝¹ : DivisionRing L", "inst✝ : DivisionRing M", "ι : Sort u_1", "hι : Nonempty ι", "S : ι → Subfield K", "hS : Directed (fun x x_1 => x ≤ x_1) S", "x : K"], "goal": "x ∈ ↑(⨆ i, S i) ↔ x ∈ ⋃ i, ↑(S i)"}}
+{"state": {"context": ["n : Int"], "goal": "n ∣ 0"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "μ : Measure ℝ", "l : Filter ι", "inst✝² : l.NeBot", "inst✝¹ : l.IsCountablyGenerated", "inst✝ : NormedAddCommGroup E", "a b : ι → ℝ", "f : ℝ → E", "I a₀ b₀ : ℝ", "hfi : ∀ (i : ι), IntegrableOn f (Ioc (a i) (b i)) volume", "ha : Tendsto a l (𝓝 a₀)", "hb : Tendsto b l (𝓝 b₀)", "h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I", "i : ι", "hi : ∫ (x : ℝ) in Ioc (a i) (b i), ‖f x‖ ≤ I"], "goal": "∀ (a : ℝ), 0 a ≤ (fun x => ‖f x‖) a"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "γ : Sort u_3", "t : γ → Set (ProjectiveSpectrum 𝒜)"], "goal": "ProjectiveSpectrum.vanishingIdeal (⋃ i, t i) = ⨅ i, ProjectiveSpectrum.vanishingIdeal (t i)"}}
+{"state": {"context": ["x y : SetTheory.PGame"], "goal": "-y ≤ x ↔ -x ≤ y"}}
+{"state": {"context": ["R : Type u_1", "OrderedRing R", "x : R"], "goal": "Sbtw R 1 x 0 ↔ x ∈ Set.Ioo 0 1"}}
+{"state": {"context": ["A : Set ℕ", "inst✝ : DecidablePred fun x => x ∈ A", "k : ℕ", "hk : k ∉ A", "hk' : k > 0"], "goal": "filter (fun x => x ∈ A) (Ioo 0 k) ⊆ Ioo 0 k"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "N : LieSubmodule R L M"], "goal": "Function.Injective ⇑N.incl"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : Sym2 α → Sym2 β → Prop", "i : Sym2 α", "j : Sym2 β", "hf : ∀ (a₁ a₂ : α) (b₁ b₂ : β), f s(a₁, a₂) s(b₁, b₂)"], "goal": "f i j"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : SeminormedCommGroup E", "ε δ : ℝ", "s t : Set E", "x y x✝¹ : E", "x✝ : x✝¹ ∈ s"], "goal": "ball x✝¹ δ = {x✝¹} * ball 1 δ"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝⁸ : CommSemiring R", "inst✝⁷ : CommSemiring S", "inst✝⁶ : Semiring A", "inst✝⁵ : Semiring B", "inst✝⁴ : Algebra R S", "inst✝³ : Algebra R A", "inst✝² : Algebra S A", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R S A", "s✝ t : Set A", "f : A →ₐ[R] B", "s : Set A"], "goal": "s ⊆ ⇑f ⁻¹' ↑(adjoin R (⇑f '' s))"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝⁹ : EMetricSpace X", "inst✝⁸ : EMetricSpace Y", "inst✝⁷ : MeasurableSpace X", "inst✝⁶ : BorelSpace X", "inst✝⁵ : MeasurableSpace Y", "inst✝⁴ : BorelSpace Y", "𝕜 : Type u_4", "E : Type u_5", "P : Type u_6", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : MeasurableSpace E", "inst✝ : BorelSpace E", "s : Set ℝ"], "goal": "μH[1] ((fun r => r • 0) '' s) = ‖0‖₊ • μH[1] s"}}
+{"state": {"context": ["xl xr yl yr : Type u_1", "xL : xl → SetTheory.PGame", "xR : xr → SetTheory.PGame", "yL : yl → SetTheory.PGame", "yR : yr → SetTheory.PGame", "i : xr", "j : yl"], "goal": "(SetTheory.PGame.mk xl xr xL xR * SetTheory.PGame.mk yl yr yL yR).moveRight (Sum.inr (i, j)) =\n  xR i * SetTheory.PGame.mk yl yr yL yR + SetTheory.PGame.mk xl xr xL xR * yL j - xR i * yL j"}}
+{"state": {"context": ["q : ℚ", "q_pos : 0 < q", "q_num_pos : 0 < q.num", "q_num_abs_eq_q_num : ↑q.num.natAbs = q.num", "q_inv : ℚ := ↑q.den / ↑q.num", "q_inv_def : q_inv = ↑q.den / ↑q.num", "q_inv_eq : q⁻¹ = q_inv", "q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den"], "goal": "↑q.den - q.num * ⌊q_inv⌋ < q.num"}}
+{"state": {"context": ["η : Type u_14", "Fintype η", "α : Type u_16", "ιs : η → Type u_17", "AddMonoid α", "f : (j : η) × ιs j →₀ α", "j : η", "i : ιs j"], "goal": "(Finsupp.sigmaFinsuppAddEquivPiFinsupp f j) i = f ⟨j, i⟩"}}
+{"state": {"context": ["G : Type u_1", "inst✝ : Group G", "H K : Subgroup G", "S T : Set G", "hST : IsComplement S T", "hHT : IsComplement (↑H) T", "hSK : IsComplement S ↑K", "g : G", "h1 : 1 ∈ T"], "goal": "↑(hST.equiv g).2 = 1 ↔ g ∈ S"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_2", "β : Type u_3", "Γ : Type u_4", "inst✝⁶ : Group Γ", "T : Type u_5", "inst✝⁵ : TopologicalSpace T", "inst✝⁴ : MulAction Γ T", "G₀ : Type u_6", "inst✝³ : GroupWithZero G₀", "inst✝² : MulAction G₀ α", "inst✝¹ : TopologicalSpace α", "inst✝ : ContinuousConstSMul G₀ α", "c : G₀", "s : Set α", "x : α", "hs : ∃ t ⊆ s, IsOpen t ∧ x ∈ t", "hc : c ≠ 0"], "goal": "∃ t ⊆ c • s, IsOpen t ∧ c • x ∈ t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder α", "LinearOrder β", "Countable α", "DenselyOrdered β", "Nontrivial β"], "goal": "Nonempty (α ↪o β)"}}
+{"state": {"context": ["α : Sort u_1", "IsEmpty α", "f : α → ℝ"], "goal": "⨅ i, f i = 0"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "AddGroup G", "AddGroup G'", "f : G →+ G'"], "goal": "f.ker.index = Nat.card ↑(Set.range ⇑f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "s : Multiset α", "n : ℕ"], "goal": "card (n • s) = n * card s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝² : CommMonoid α", "inst✝¹ : TopologicalSpace α", "f g : β → α", "a✝ b : α", "s✝ : Finset β", "inst✝ : ContinuousMul α", "ι : Type u_5", "s : Finset ι", "t : ι → Set β", "a : ι → α", "hs : (↑s).Pairwise (Disjoint on t)", "hf : ∀ i ∈ s, HasProd (f ∘ Subtype.val) (a i)"], "goal": "HasProd (f ∘ Subtype.val) (∏ i ∈ s, a i)"}}
+{"state": {"context": ["Γ : Type u_1", "Γ' : Type u_2", "inst✝¹ : Inhabited Γ", "inst✝ : Inhabited Γ'", "f : PointedMap Γ Γ'", "l : ListBlank Γ"], "goal": "(map f l).head = f.f l.head"}}
+{"state": {"context": ["n : ℕ+", "K : Type u_1", "inst✝⁴ : Field K", "L : Type u_2", "μ : L", "inst✝³ : CommRing L", "inst✝² : IsDomain L", "hμ : IsPrimitiveRoot μ ↑n", "inst✝¹ : Algebra K L", "inst✝ : IsCyclotomicExtension {n} K L", "f g : L ≃ₐ[K] L", "hf' : ∃ m, f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m", "hg' : ∃ m, g ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ m", "hfg : hf'.choose ≡ hg'.choose [MOD ↑n]", "hf : f ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ hf'.choose", "hg : g ↑↑hμ.toRootsOfUnity = ↑↑hμ.toRootsOfUnity ^ hg'.choose"], "goal": "hμ.toRootsOfUnity ^ hf'.choose = hμ.toRootsOfUnity ^ hg'.choose"}}
+{"state": {"context": ["x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame", "hy : y.Numeric", "ihyx : IH1 y x", "i : x.LeftMoves", "k : y.LeftMoves", "l : (-y).LeftMoves"], "goal": "y.moveLeft k < -(-y).moveLeft l"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "m : ℕ"], "goal": "Localization.mk (↑m) 1 = ↑m"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "x y : R", "m n : ℕ", "H : IsCoprime x y"], "goal": "IsCoprime (x ^ m) (y ^ n)"}}
+{"state": {"context": ["a b : ℕ", "hb₀ : (b / 2).beq 0 = false", "ha : a % 2 = 0", "hb₁ : b % 2 = 0", "hf : (↑a).gcd ↑b = 1", "h : 2 ∣ (↑a).gcd ↑b"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "R : Type u_3", "α : Type u_4", "inst✝³ : DivisionSemiring α", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSemiring α", "f g : ι → α", "a a₁ a₂ : α", "inst✝ : T2Space α", "hf : ¬Summable f", "ha : ¬a = 0"], "goal": "∑' (x : ι), f x * a = (∑' (x : ι), f x) * a"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "X₁ X₂ : Type u", "X Y : AlgebraCat R", "i : X ≅ Y", "x : ↑X"], "goal": "(𝟙 X) x = x"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "B' : Type u_3", "inst✝⁴ : CommRing A", "inst✝³ : Ring B", "inst✝² : Algebra A B", "x : B", "inst✝¹ : IsDomain A", "inst✝ : IsDomain B", "hx : IsIntegral A x", "f g : A[X]", "hf : f.Monic", "hg : g.Monic", "he : f * g = minpoly A x", "h : ¬IsUnit f ∧ ¬IsUnit g"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Ring R", "AddCommGroup M", "Module R M", "v : Basis ι R M", "w : ι → Rˣ", "i : ι"], "goal": "(v.unitsSMul w) i = w i • v i"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ₁ φ₂ : S₁ ⟶ S₂", "h : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂"], "goal": "h.symm.h₁ = -h.h₁"}}
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+{"state": {"context": ["c : Type u → Type u", "hom : ⦃α β : Type u⦄ → c α → c β → Type u", "self : CategoryTheory.BundledHom hom", "α β : Type u", "Iα : c α", "Iβ : c β"], "goal": "Function.Injective (self.toFun Iα Iβ)"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp : p ≠ 2", "this : -2 = ↑(-2)"], "goal": "legendreSym p (-2) = χ₈' ↑p"}}
+{"state": {"context": ["x : ℝ", "h : 0 ≤ x"], "goal": "arcsin (x / √(1 + x ^ 2)) = arcsin √(1 - (√(1 + x ^ 2))⁻¹ ^ 2)"}}
+{"state": {"context": ["c : ℝ", "f✝ g : ℕ → Bool", "n : ℕ", "f : ℕ → Bool", "h1 : 0 ≤ c", "h2 : c < 1"], "goal": "cantorFunctionAux c f 0 + ∑' (b : ℕ), cantorFunctionAux c f (b + 1) = (bif f 0 then 1 else 0) + c * cantorFunction c fun n => f (n + 1)"}}
+{"state": {"context": ["α : Type u", "γ : Type u_1", "inst✝² : TopologicalSpace γ", "inst✝¹ : T2Space γ", "inst✝ : CompactSpace γ", "f : α → γ", "b : Ultrafilter α", "c : γ", "h : Ultrafilter.extend f b = c", "b' : Ultrafilter (Ultrafilter α) := Ultrafilter.map pure b", "t : ↑b' ≤ 𝓝 b"], "goal": "↑(Ultrafilter.map f b) ≤ 𝓝 c"}}
+{"state": {"context": ["R : Type u_1", "LinearOrderedCommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "NoZeroSMulDivisors R M", "v : M", "r : R"], "goal": "SameRay R (r • v) v ↔ 0 ≤ r ∨ v = 0"}}
+{"state": {"context": ["p q : ℤ[X]", "k m m' n : ℕ", "hkm : k < m", "hmn : m < n", "hkm' : k < m'", "hmn' : m' < n", "u v w x z : ℤˣ", "hp : p = trinomial k m n ↑u ↑v ↑w", "hq : q = trinomial k m' n ↑x ↑v ↑z", "h : p * p.mirror = q * q.mirror", "hmul : ↑w * ↑u = ↑z * ↑x", "hadd : eval 1 (trinomial k m n ↑u ↑v ↑w) ^ 2 = eval 1 (trinomial k m' n ↑x ↑v ↑z) ^ 2"], "goal": "q = p ∨ q = p.mirror"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹¹ : TopologicalSpace M", "inst✝¹⁰ : ChartedSpace H M", "E' : Type u_5", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁶ : TopologicalSpace M'", "inst✝⁵ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝¹ : TopologicalSpace M''", "inst✝ : ChartedSpace H'' M''", "f f₀ f₁ : M → M'", "x : M", "s t : Set M", "g : M' → M''", "u : Set M'", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "I''s : SmoothManifoldWithCorners I'' M''", "f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)", "g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))", "n : ℕ∞", "h : ∀ x ∈ s, f x = f₁ x", "p : TangentBundle I M", "hp : p.proj ∈ s", "hs : UniqueMDiffWithinAt I s p.proj"], "goal": "HEq (tangentMapWithin I I' f s p).snd (tangentMapWithin I I' f₁ s p).snd"}}
+{"state": {"context": [], "goal": "deriv Real.cos = fun x => -Real.sin x"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "hS : S.Exact", "hS' : S.ShortExact", "v : ι → ↑S.X₁", "β : Type u_4", "u : ι ⊕ β → ↑S.X₂", "huv : u ∘ Sum.inl = ⇑S.f ∘ v", "hv : ⊤ ≤ span R (range v)", "hw : ⊤ ≤ span R (range (⇑S.g ∘ u ∘ Sum.inr))", "m : ↑S.X₂", "a✝ : m ∈ ⊤", "hgm : S.g m ∈ span R (range (⇑S.g ∘ u ∘ Sum.inr))"], "goal": "m ∈ span R (range u)"}}
+{"state": {"context": ["c f₁ f₂ : Type u", "r₁ r₂ : Type v"], "goal": "∀ (a : ℕ), (FirstOrder.Language.mk₂ c f₁ f₂ r₁ r₂).Relations a = FirstOrder.Sequence₂ PEmpty.{v + 1} r₁ r₂ a"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)"], "goal": "IsIntegralClosure (CyclotomicRing (p ^ k) ℤ ℚ) ℤ (CyclotomicField (p ^ k) ℚ)"}}
+{"state": {"context": ["α : Type u", "x : α"], "goal": "default = ⟨x, ⋯⟩"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "inst✝¹ : HasZeroObject C", "ι : Type u_1", "inst✝ : DecidableEq ι", "c : ComplexShape ι", "j : ι", "A✝ A : C", "i : ι", "hi : i ≠ j"], "goal": "IsZero (((single C c j).obj A).homology i)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "R I J✝ X Y B J : Set α", "e : α", "hI : M.Basis I X", "hJ : M.Basis J Y", "hJX : J ⊆ X", "hIY : I ⊆ Y", "hI' : M.Basis I (X ∩ Y)", "hJ' : M.Basis J (X ∩ Y)"], "goal": "M.Basis I Y"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "A : Type u₂", "inst✝² : Category.{v₂, u₂} A", "D : Type u_1", "inst✝¹ : Category.{u_2, u_1} D", "inst✝ : HasWeakSheafify J D", "P Q R : Cᵒᵖ ⥤ D", "η : P ⟶ Q", "γ : Q ⟶ R", "hR : Presheaf.IsSheaf J R"], "goal": "toSheafify J P ≫ sheafifyMap J η ≫ sheafifyLift J γ hR = η ≫ γ"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "One M", "One N", "f : OneHom M N"], "goal": "(OneHom.id N).comp f = f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "K : ℝ≥0", "s t✝ : Set α", "f : α → β", "t : Set ↑s"], "goal": "LipschitzOnWith K (s.restrict f) t ↔ LipschitzOnWith K f (s ∩ Subtype.val '' t)"}}
+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "s.ncard = s.encard.toNat"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "s✝ t✝ u : Finset α", "f : α → β", "n : ℕ", "inst✝ : DecidableEq α", "s t : Finset α"], "goal": "(t \\ (s ∩ t)).card ≤ (t \\ s).card"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ x y : R"], "goal": "↑x = ↑y ↔ x = y"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "f : M ≃+ N"], "goal": "(Finsupp.mapRange.addEquiv f).symm = Finsupp.mapRange.addEquiv f.symm"}}
+{"state": {"context": ["α : Type u", "σ : Type v", "s : σ", "a : Option α"], "goal": "εNFA.step 0 s a = ∅"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Zero α", "Preorder α", "a : α", "ha : 0 ≤ a"], "goal": "0 ≤ Function.const β a"}}
+{"state": {"context": ["f✝ g f g₁ g₂ : CircleDeg1Lift", "h : Semiconj ⇑f ⇑g₁ ⇑g₂"], "goal": "dist (g₁ 0) (f (g₁ 0) - f 0) + dist (f (g₁ 0) - f 0) (g₂ 0) = dist (f 0 + g₁ 0) (f (g₁ 0)) + dist (g₂ 0 + f 0) (g₂ (f 0))"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "V : Type u_18", "V₂ : Type u_17"], "goal": "⇑(LinearEquiv.curry R V V₂) = Function.curry"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ : Measure α", "p q : ℝ", "hpq : p.IsConjExponent q", "f g : α → ℝ≥0∞", "hf : AEMeasurable f μ", "hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤", "hg : AEMeasurable g μ", "hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤", "h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0", "h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤", "hp_not_nonpos : ¬p ≤ 0", "htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤"], "goal": "(∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "v : Fin n → α"], "goal": "(Matrix.circulant v).IsSymm ↔ ∀ (i : Fin n), v (-i) = v i"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t : Set X", "inst✝ : LindelofSpace X", "f : X → Y", "hf : Continuous f"], "goal": "IsLindelof (range f)"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedBooleanAlgebra α", "a : α"], "goal": "Function.Surjective fun x => a ∆ x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x₀ : E", "h : DifferentiableAt 𝕜 f x₀"], "goal": "(fun x => f x - f x₀) =O[𝓝 x₀] fun x => x - x₀"}}
+{"state": {"context": ["α : Type u_1", "R✝ : Type u_2", "k : Type u_3", "S : Type u_4", "M✝ : Type u_5", "M₂ : Type u_6", "M₃ : Type u_7", "ι : Type u_8", "R : Type u_9", "M : Type u_10", "inst✝⁴ : Semiring R", "inst✝³ : CharZero R", "inst✝² : AddCommMonoid M", "inst✝¹ : Module R M", "inst✝ : NoZeroSMulDivisors R M", "c : ℕ", "x : M"], "goal": "↑c = 0 ∨ x = 0 → c = 0 ∨ x = 0"}}
+{"state": {"context": ["M₀ : Type u_2", "G₀ : Type u_3", "F : Type u_6", "MonoidWithZero M₀", "GroupWithZero G₀", "Nontrivial M₀", "FunLike F G₀ M₀", "MonoidWithZeroHomClass F G₀ M₀", "f : F", "a : G₀"], "goal": "f a = 0 ↔ a = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "S : Set α", "hS : BooleanGenerators S", "T₁ T₂ : Set α", "hT₁ : T₁ ⊆ S", "hT₂ : T₂ ⊆ S", "X : Set α", "hX : X ⊆ S", "hX' : sSup T₁ ⊓ sSup T₂ = sSup X"], "goal": "∀ b ∈ X, b ≤ sSup (T₁ ∩ T₂)"}}
+{"state": {"context": ["α : Type u_2"], "goal": "Function.Injective (@LinearOrder.toPartialOrder α)"}}
+{"state": {"context": ["α : Type u", "R : Type v", "CommRing R", "f : FreeCommRing α →+* R"], "goal": "FreeCommRing.lift (⇑f ∘ FreeCommRing.of) = f"}}
+{"state": {"context": ["G : PGame", "inst✝ : G.Impartial"], "goal": "(∃ j, G.moveRight j ≈ 0) ↔ G ‖ 0"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "inst✝² : Group G", "inst✝¹ : Group H", "K : Type u_3", "inst✝ : Group K", "f : G →* K", "g : H →* K"], "goal": "(lift f g).range = f.range ⊔ g.range"}}
+{"state": {"context": ["n : ℕ", "α : TypeVec.{u} n", "F : TypeVec.{u} n → Type v", "MvFunctor F", "LawfulMvFunctor F", "x : F α"], "goal": "TypeVec.id <$$> x = x"}}
+{"state": {"context": ["X : Type u", "inst✝⁵ : MetricSpace X", "inst✝⁴ : CompactSpace X", "inst✝³ : Nonempty X", "Y : Type v", "inst✝² : MetricSpace Y", "inst✝¹ : CompactSpace Y", "inst✝ : Nonempty Y", "e : X ≃ᵢ Y", "f : ↑(range (kuratowskiEmbedding X)) ≃ᵢ X", "g : Y ≃ᵢ ↑(range (kuratowskiEmbedding Y))"], "goal": "⟦NonemptyCompacts.kuratowskiEmbedding X⟧ = ⟦NonemptyCompacts.kuratowskiEmbedding Y⟧"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ"], "goal": "HurwitzZeta.hurwitzZetaOdd (-a) s = -HurwitzZeta.hurwitzZetaOdd a s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "f✝ : α → β", "s✝ t✝ : Set α", "f : α → β", "s : Set α", "t : Set β"], "goal": "f '' (s \\ f ⁻¹' t) = f '' s \\ t"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "FloorRing α", "TopologicalSpace α", "OrderClosedTopology α", "n : ℤ"], "goal": "Filter.Tendsto (fun x => ↑⌈x⌉) (𝓝[≤] ↑n) (𝓝 ↑n)"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "C D : ℝ≥0", "f g : E → ℝ", "s : Set E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "h'f : HasCompactSupport f", "hg : Continuous g", "v : E", "K : Set E := cthickening ‖v‖ (tsupport f)", "K_compact : IsCompact K"], "goal": "Integrable (K.indicator fun x => ↑C * ‖v‖ * ‖g x‖) μ"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝¹¹ : NontriviallyNormedField 𝕜", "E : Type v", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "F : Type w", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace 𝕜 F", "F' : Type x", "inst✝⁶ : AddCommGroup F'", "inst✝⁵ : Module 𝕜 F'", "inst✝⁴ : TopologicalSpace F'", "inst✝³ : TopologicalAddGroup F'", "inst✝² : ContinuousSMul 𝕜 F'", "inst✝¹ : CompleteSpace 𝕜", "ι : Type u_1", "inst✝ : Finite ι", "f : ι → E", "val✝ : Fintype ι", "hf : LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i)) = ⊥", "K : ℝ≥0", "K0 : K > 0", "hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (f i))", "this : Tendsto (fun g => ∑ i : ι, ‖g i - f i‖) (𝓝 f) (𝓝 0)"], "goal": "∀ᶠ (g : ι → E) in 𝓝 f, LinearMap.ker ((LinearMap.lsum 𝕜 (fun x => 𝕜) ℕ) fun i => LinearMap.id.smulRight (g i)) = ⊥"}}
+{"state": {"context": ["E : Type u", "NormedAddCommGroup E", "NormedSpace ℂ E", "CompleteSpace E", "f : ℂ → E", "c : ℂ", "hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z", "ho : (fun z => f z - f c) =o[𝓝[≠] c] fun z => (z - c)⁻¹"], "goal": "Filter.Tendsto f (𝓝[≠] c) (𝓝 (limUnder (𝓝[≠] c) f))"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f g : α → E", "hf : MeasureTheory.Memℒp f p μ", "hg : MeasureTheory.Memℒp g p μ"], "goal": "MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp g hg ↔ f =ᵐ[μ] g"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F : CategoryTheory.Functor Cᵒᵖ (Type w)", "α : Type", "X : α → C", "c : CategoryTheory.Limits.Cofan X", "hc : CategoryTheory.Limits.IsColimit c"], "goal": "let_fun this := ⋯;\n(CategoryTheory.Limits.piComparison F fun x => Opposite.op (X x)) =\n  CategoryTheory.CategoryStruct.comp\n    (F.map\n      (CategoryTheory.Limits.opCoproductIsoProduct' hc\n          (CategoryTheory.Limits.productIsProduct fun x => Opposite.op (X x))).inv)\n    (CategoryTheory.Equalizer.Presieve.Arrows.forkMap F X c.inj)"}}
+{"state": {"context": ["α : Type u_1", "n : PosNum"], "goal": "0 < n.natSize"}}
+{"state": {"context": ["β : Type u_2", "α : Type u_6", "ConditionallyCompleteLattice β", "f : Filter α", "f.NeBot", "b : β"], "goal": "Filter.limsup (fun x => b) f = b"}}
+{"state": {"context": ["ι : Type u_1", "ι₂ : Type u_2", "ι₃ : Type u_3", "R : Type u_4", "inst✝¹¹ : CommSemiring R", "R₁ : Type u_5", "R₂ : Type u_6", "s : ι → Type u_7", "inst✝¹⁰ : (i : ι) → AddCommMonoid (s i)", "inst✝⁹ : (i : ι) → Module R (s i)", "M : Type u_8", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : Module R M", "E : Type u_9", "inst✝⁶ : AddCommMonoid E", "inst✝⁵ : Module R E", "F : Type u_10", "inst✝⁴ : AddCommMonoid F", "t : ι → Type u_11", "t' : ι → Type u_12", "inst✝³ : (i : ι) → AddCommMonoid (t i)", "inst✝² : (i : ι) → Module R (t i)", "inst✝¹ : (i : ι) → AddCommMonoid (t' i)", "inst✝ : (i : ι) → Module R (t' i)", "g : (i : ι) → t i →ₗ[R] t' i", "f : (i : ι) → s i →ₗ[R] t i", "x✝ : (i : ι) → s i"], "goal": "((map fun i => g i ∘ₗ f i).compMultilinearMap (tprod R)) x✝ = ((map g ∘ₗ map f).compMultilinearMap (tprod R)) x✝"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "V : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "FiniteDimensional ℝ V", "MeasurableSpace V", "BorelSpace V", "f : V → E", "N : ℕ∞", "hf : ∀ (n : ℕ), ↑n ≤ N → MeasureTheory.Integrable (fun v => ‖v‖ ^ n * ‖f v‖) MeasureTheory.volume", "h'f : MeasureTheory.AEStronglyMeasurable f MeasureTheory.volume", "n : ℕ", "hn : ↑n ≤ N"], "goal": "iteratedFDeriv ℝ n (𝓕 f) = 𝓕 fun v => VectorFourier.fourierPowSMulRight (innerSL ℝ) f v n"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "s : AffineSubspace ℝ P", "Nonempty ↥s", "HasOrthogonalProjection s.direction", "p : P"], "goal": "↑s ∩ ↑(AffineSubspace.mk' p s.directionᗮ) = {EuclideanGeometry.orthogonalProjectionFn s p}"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "x y : α"], "goal": "EMetric.diam {x, y} = edist x y"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁵ : Category.{v₁, u₁} C", "D✝ : Type u₂", "inst✝⁴ : Category.{v₂, u₂} D✝", "J✝ : Type u", "inst✝³ : Category.{v, u} J✝", "T : Comonad C", "D : J✝ ⥤ T.Coalgebra", "c : Cone (D ⋙ T.forget)", "t : IsLimit c", "inst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor", "inst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor", "inst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor", "s : Cone D", "m : s.pt ⟶ (liftedCone c t).pt", "J : ∀ (j : J✝), m ≫ (liftedCone c t).π.app j = s.π.app j"], "goal": "m.f = ((fun s => { f := t.lift (T.forget.mapCone s), h := ⋯ }) s).f"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P' Q' : Fin 3 → R", "u v : Rˣ"], "goal": "W'.add ((fun m => m • P') u) ((fun m => m • Q') v) ≈ W'.add P' Q'"}}
+{"state": {"context": ["n₁ n₂ : ℤ"], "goal": "(n₁ - n₂).negOnePow = n₁.negOnePow * n₂.negOnePow"}}
+{"state": {"context": ["α : Type u_1", "AddGroup α", "s : AddSubgroup α", "β : Sort u_2", "f : α → β", "h : ∀ (a b : α), Setoid.r a b → f a = f b", "x : α"], "goal": "Quotient.liftOn' (↑x) f h = f x"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "a✝ a : α", "s : Finset α", "hs : a ∉ s", "ih : ∀ {𝒜 ℬ : Finset (Finset α)}, IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → 𝒜.card * ℬ.card ≤ 2 ^ s.card * (𝒜 ∩ ℬ).card", "𝒜 ℬ : Finset (Finset α)", "h𝒜 : IsLowerSet ↑𝒜", "hℬ : IsLowerSet ↑ℬ", "h𝒜s : ∀ t ∈ 𝒜, t ⊆ insert a s", "hℬs : ∀ t ∈ ℬ, t ⊆ insert a s"], "goal": "(Finset.memberSubfamily a 𝒜).card * (Finset.nonMemberSubfamily a ℬ).card + (Finset.nonMemberSubfamily a 𝒜).card * (Finset.memberSubfamily a ℬ).card + ((Finset.memberSubfamily a 𝒜).card * (Finset.memberSubfamily a ℬ).card + (Finset.nonMemberSubfamily a 𝒜).card * (Finset.nonMemberSubfamily a ℬ).card) ≤ 2 ^ (s.card + 1) * (𝒜 ∩ ℬ).card"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "(i : ι) → Preorder (α i)", "x : (i : ι) → α i", "Nonempty ι"], "goal": "(Set.univ.pi fun i => Set.Ioi (x i)) ⊆ Set.Ioi x"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝² : NontriviallyNormedField 𝕜", "inst✝¹ : NontriviallyNormedField 𝕜'", "inst✝ : NormedAlgebra 𝕜 𝕜'", "f : 𝕜 → 𝕜'", "x : 𝕜", "a : 𝕜'", "ha : a ≠ 0"], "goal": "logDeriv (fun z => a * f z) x = logDeriv f x"}}
+{"state": {"context": ["a : ℤ", "b : ℕ", "ha2 : a % 2 = 0", "hb2 : b % 2 = 1"], "goal": "(if b % 8 = 3 ∨ b % 8 = 5 then -jacobiSym (a / 2) b else jacobiSym (a / 2) b) = jacobiSym a b"}}
+{"state": {"context": ["K R : Type u", "V V₁ V₂ V₃ : Type v", "V' V'₁ : Type v'", "V'' : Type v''", "ι : Type w", "ι' : Type w'", "η : Type u₁'", "φ : η → Type u_1", "inst✝ : DivisionRing K", "aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)"], "goal": "max ℵ₀ #K ≤ Module.rank K (ℕ → K)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "M : Type u_3", "N : Type u_4", "X : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace M", "inst✝¹ : Mul M", "inst✝ : ContinuousMul M", "a b : M"], "goal": "𝓝 a * 𝓝 b ≤ 𝓝 (a * b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f g : α → β", "Add β", "ContinuousAdd β", "hf : MeasureTheory.AEStronglyMeasurable f μ", "hg : MeasureTheory.AEStronglyMeasurable g μ"], "goal": "MeasureTheory.AEStronglyMeasurable (f + g) μ"}}
+{"state": {"context": ["α : Type u_1", "a b : α"], "goal": "(bif false then a else b) = b"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "κ : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : LinearOrderedField 𝕜", "r : α → β → Prop", "inst✝ : (a : α) → DecidablePred (r a)", "s s₁ s₂ : Finset α", "t t₁ t₂ : Finset β", "a : α", "b : β", "δ : 𝕜", "hs : s₂ ⊆ s₁", "ht : t₂ ⊆ t₁", "hδ : 0 ≤ δ", "hscard : (1 - δ) * ↑s₁.card ≤ ↑s₂.card", "htcard : (1 - δ) * ↑t₁.card ≤ ↑t₂.card", "h : 1 ≤ δ"], "goal": "|↑(edgeDensity r s₂ t₂) - ↑(edgeDensity r s₁ t₁)| ≤ 2 * δ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "mβ : MeasurableSpace β", "f : α → β", "hf : MeasurableEmbedding f", "hμν : μ ≪ ν", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν"], "goal": "Measurable fun x => (map f μ).rnDeriv (map f ν) (f x)"}}
+{"state": {"context": ["q r : ℚ", "s : ℚ := (q.num * ↑r.den + r.num * ↑q.den) /. (↑q.den * ↑r.den)", "hs : ↑q.den * ↑r.den ≠ 0", "c : ℤ", "c_mul_num : q.num * ↑r.den + r.num * ↑q.den = c * (↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).num", "c_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).den", "a✝ : ¬c = 0", "h : q + r = s"], "goal": "↑(↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).den * (q + r).num = (↑(q.num * ↑r.den + r.num * ↑q.den) / ↑(↑q.den * ↑r.den)).num * ↑(q + r).den"}}
+{"state": {"context": ["K : Type u_1", "v : K", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "n : ℕ"], "goal": "(of v).convs (n + 1) = ↑⌊v⌋ + 1 / (of (Int.fract v)⁻¹).convs n"}}
+{"state": {"context": ["α : Type u_2", "CompleteLattice α"], "goal": "(WellFounded fun x x_1 => x > x_1) ↔ CompleteLattice.IsSupFiniteCompact α"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "MeasureTheory.IsProbabilityMeasure μ", "hs : MeasurableSet s"], "goal": "s ∈ MeasureTheory.ae μ ↔ μ s = 1"}}
+{"state": {"context": ["n✝ : ℕ", "i✝ : Fin (n✝ + 1).succ", "i : Fin (n✝ + 1)"], "goal": "(if i = Fin.last n✝ then 0 else ↑i.succ + 1) = ↑((swap 0 1) ((finRotate (n✝ + 1)) i).succ)"}}
+{"state": {"context": ["𝕜 : Type u_1", "LinearOrderedField 𝕜", "x y : 𝕜", "hxy : x ≠ y"], "goal": "openSegment 𝕜 x y = Set.Ioo (min x y) (max x y)"}}
+{"state": {"context": ["α : Type u", "x y : α", "BooleanAlgebra α"], "goal": "x = yᶜ ↔ IsCompl x y"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "s : Finmap β"], "goal": "a ∉ Finmap.erase a s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasBinaryBiproducts C", "X₁ X₂ X₃ : CochainComplex C ℤ", "f : X₁ ⟶ X₂", "g : X₂ ⟶ X₃"], "goal": "(CochainComplex.mappingConeCompTriangle f g).obj₃ = CochainComplex.mappingCone g"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "inst✝² : Category.{v, u} V", "inst✝¹ : HasZeroMorphisms V", "c : ComplexShape ι", "C : HomologicalComplex V c", "inst✝ : HasKernels V", "i j : ι", "r : c.Rel i j"], "goal": "kernelSubobject (C.d i j ≫ (C.xNextIso r).inv) = kernelSubobject (C.d i j)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝ : LinearOrderedField 𝕜", "n : ℕ"], "goal": "ConvexOn 𝕜 (Ioi 0) fun x => x ^ ↑n"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹⁵ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹⁴ : TopologicalSpace M", "inst✝¹³ : ChartedSpace H M", "inst✝¹² : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹¹ : NormedAddCommGroup E'", "inst✝¹⁰ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁹ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁸ : TopologicalSpace M'", "inst✝⁷ : ChartedSpace H' M'", "inst✝⁶ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝⁵ : NormedAddCommGroup E''", "inst✝⁴ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝³ : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝² : TopologicalSpace M''", "inst✝¹ : ChartedSpace H'' M''", "inst✝ : SmoothManifoldWithCorners I'' M''", "s : Set M", "x✝ : M", "x : M × M'", "this : ∀ᶠ (y : E × E') in 𝓝[range ↑(I.prod I')] ↑(extChartAt (I.prod I') x) x, (↑(extChartAt I' x.2) ∘ Prod.snd ∘ ↑(extChartAt (I.prod I') x).symm) y = y.2"], "goal": "(↑(extChartAt I' x.2) ∘ Prod.snd ∘ ↑(extChartAt (I.prod I') x).symm) (↑(extChartAt (I.prod I') x) x) = (↑(extChartAt (I.prod I') x) x).2"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "n : ℕ", "h : 0 ≠ 0", "this : DecidableEq R := Classical.decEq R", "hp0 : ¬p = 0", "hpn0 : p ^ n = 0", "h1 : p.leadingCoeff ^ n ≠ 0"], "goal": "False"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "κ : Type u_7", "inst✝ : TopologicalSpace X", "T : (i : ι) → TopologicalSpace (π i)", "f : X → (i : ι) → π i", "i : Set ι", "s : (a : ι) → Set (π a)", "x : (a : ι) → π a", "hi : i.Finite", "hs : ∀ a ∈ i, s a ∈ 𝓝 (x a)"], "goal": "∀ i_1 ∈ i, eval i_1 ⁻¹' s i_1 ∈ 𝓝 x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : SuccOrder α", "a b : α", "n m : ℕ", "h_eq : succ^[n] a = succ^[m] a", "h_lt : n < m", "this : succ (succ^[n] a) = succ^[n + 1] a"], "goal": "succ (succ^[n] a) ≤ succ^[m] a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "r' : β → β → Prop", "t : Set β", "hs : IsAntichain r' t", "φ : r ≃r r'"], "goal": "IsAntichain r (⇑φ ⁻¹' t)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Semiring R", "AddCommMonoid M", "Module R M", "n : ℕ", "b : Basis (Fin n) R M", "k : Fin n"], "goal": "b.flag k.succ = Submodule.span R {b k} ⊔ b.flag k.castSucc"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I : Ideal R"], "goal": "I.cotangentToQuotientSquare ∘ₗ I.toCotangent = Submodule.mkQ (I ^ 2) ∘ₗ Submodule.subtype I"}}
+{"state": {"context": ["p : PosNum", "n : ℕ"], "goal": "p <<< n.succ = (p <<< n).bit0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α"], "goal": "toIocDiv hp a (p + b) = toIocDiv hp a b + 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "f : Filter α", "hf : Cauchy f", "U : ℕ → Set (α × α)", "U_mem : ∀ (n : ℕ), U n ∈ 𝓤 α", "U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s", "N m n : ℕ", "hm : N ≤ m", "hn : N ≤ n", "p : α × α", "hp : p ∈ setSeq hf U_mem m ×ˢ setSeq hf U_mem n"], "goal": "p ∈ U N"}}
+{"state": {"context": [], "goal": "⇑NNReal.toRealHom = NNReal.toReal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β", "r : Set α", "hf : Set.BijOn f s t", "hrs : r ⊆ s"], "goal": "Set.BijOn f r (f '' r)"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "R : Type u_11", "CommRing R", "Fintype m", "DecidableEq m", "Fintype n", "DecidableEq n", "e : m ≃ n", "M : Matrix m m R"], "goal": "((Matrix.reindexLinearEquiv R R e e) M).det = M.det"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "a : α"], "goal": "IsClosed {a}"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "inst✝ : CommRing R", "c₁ c₂ r x y z : R", "a b c : ℍ[R,c₁,c₂]", "x✝¹ x✝ : R"], "goal": "(algebraMap R ℍ[R,c₁,c₂]) x✝¹ = (algebraMap R ℍ[R,c₁,c₂]) x✝ → x✝¹ = x✝"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : IsDomain B", "inst✝² : Algebra A B", "K : Type u_6", "inst✝¹ : Field K", "inst✝ : Algebra A S", "pb : PowerBasis A S", "h✝ : Nontrivial A", "k : Fin pb.dim"], "goal": "pb.gen ^ (↑k + 1) = ∑ x : Fin pb.dim, (if ↑k + 1 = pb.dim then -pb.minpolyGen.coeff ↑x else if ↑x = ↑k + 1 then 1 else 0) • pb.gen ^ ↑x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "MeasurableSpace β", "e : α ≃ᵐ β", "s : Set β"], "goal": "MeasurableSet (⇑e ⁻¹' s) ↔ MeasurableSet s"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "x : ℚ_[p]", "f : CauSeq ℚ (padicNorm p)", "hf : Quotient.mk'' f ≠ 0"], "goal": "‖Quotient.mk'' f‖ = ↑p ^ (-valuation (Quotient.mk'' f))"}}
+{"state": {"context": ["ι : Type u_2", "β : ι → Type u_3", "Fintype ι", "(i : ι) → DecidableEq (β i)", "x y z : (i : ι) → β i"], "goal": "hammingDist x z ≤ hammingDist x y + hammingDist y z"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "h✝ : Nonempty α"], "goal": "atTop ×ˢ atTop = atTop"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "S : Type u_3", "inst✝¹⁵ : Semiring R", "inst✝¹⁴ : Semiring S", "inst✝¹³ : AddCommMonoid M", "inst✝¹² : AddCommMonoid M₂", "inst✝¹¹ : AddCommMonoid M₃", "inst✝¹⁰ : AddCommMonoid M₄", "inst✝⁹ : AddCommMonoid M₅", "inst✝⁸ : AddCommMonoid M₆", "inst✝⁷ : Module R M", "inst✝⁶ : Module R M₂", "inst✝⁵ : Module R M₃", "inst✝⁴ : Module R M₄", "inst✝³ : Module R M₅", "inst✝² : Module R M₆", "f✝ : M →ₗ[R] M₂", "inst✝¹ : Module S M₃", "inst✝ : SMulCommClass R S M₃", "f : M × M₂ →ₗ[R] M₃"], "goal": "{ toFun := fun f => f.1.coprod f.2, map_add' := ⋯, map_smul' := ⋯ }.toFun ((fun f => (f ∘ₗ inl R M M���, f ∘ₗ inr R M M₂)) f) = f"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "b : β a", "s : Finmap β", "h : Finmap.lookup a s = some b"], "goal": "Finmap.erase a s ∪ Finmap.singleton a b = s"}}
+{"state": {"context": ["T : Type u", "CategoryTheory.Category.{v, u} T", "f : CategoryTheory.Arrow T"], "goal": "(CategoryTheory.CategoryStruct.id f).right = CategoryTheory.CategoryStruct.id f.right"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve R", "A : Type v", "CommRing A", "φ : R →+* A"], "goal": "(W.map φ).c₄ = φ W.c₄"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : C", "𝒯 : LimitCone (Functor.empty C)", "ℬ : (X Y : C) → LimitCone (pair X Y)", "X₁ X₂ : C"], "goal": "∀ (j : Discrete WalkingPair), tensorHom ℬ (𝟙 X₁) (𝟙 X₂) ≫ (ℬ X₁ X₂).cone.π.app j = 𝟙 (tensorObj ℬ X₁ X₂) ≫ (ℬ X₁ X₂).cone.π.app j"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "a : α", "n : ℕ", "hn : n ≠ 0"], "goal": "n ≤ ⌊a⌋₊ ↔ ↑n ≤ a"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s✝ t s : Set M"], "goal": "span ℕ s ≤ AddSubmonoid.toNatSubmodule.symm.symm (AddSubmonoid.closure s)"}}
+{"state": {"context": ["ι : Type v", "Preorder ι", "G : ι → Type w", "f : (i j : ι) → i ≤ j → G i → G j", "self : DirectedSystem G f", "i j k : ι", "hij : i ≤ j", "hjk : j ≤ k", "x : G i"], "goal": "f j k hjk (f i j hij x) = f i k ⋯ x"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_4", "M : Type u_5", "E : B → Type u_6", "NontriviallyNormedField 𝕜", "EB : Type u_7", "NormedAddCommGroup EB", "NormedSpace 𝕜 EB", "HB : Type u_8", "TopologicalSpace HB", "IB : ModelWithCorners 𝕜 EB HB", "TopologicalSpace B", "ChartedSpace HB B", "EM : Type u_9", "NormedAddCommGroup EM", "NormedSpace 𝕜 EM", "HM : Type u_10", "TopologicalSpace HM", "IM : ModelWithCorners 𝕜 EM HM", "TopologicalSpace M", "ChartedSpace HM M", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module 𝕜 (E x)", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "TopologicalSpace (Bundle.TotalSpace F E)", "(x : B) → TopologicalSpace (E x)", "FiberBundle F E", "VectorBundle 𝕜 F E", "SmoothVectorBundle F E IB", "e : Trivialization F Bundle.TotalSpace.proj", "MemTrivializationAtlas e", "f : M → Bundle.TotalSpace F E", "s : Set M", "he : Set.MapsTo f s e.source"], "goal": "SmoothOn IM (IB.prod 𝓘(𝕜, F)) f s ↔\n  SmoothOn IM IB (fun x => (f x).proj) s ∧ SmoothOn IM 𝓘(𝕜, F) (fun x => (↑e (f x)).2) s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoEMetricSpace α", "Fintype β", "Nonempty β", "a b : α"], "goal": "(edist (fun x => a) fun x => b) = edist a b"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "Group G", "Fact (Nat.Prime p)", "Finite (Sylow p G)"], "goal": "Nat.card (Sylow p G) ≡ 1 [MOD p]"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "G : K ⥤ C", "e : K ≌ J", "α : e.functor ⋙ F ≅ G", "X : CategoryTheory.Limits.Cocone F"], "goal": "(CategoryTheory.Limits.Cocones.equivalenceOfReindexing e α).functor.obj X =\n  (CategoryTheory.Limits.Cocones.precompose α.inv).obj (CategoryTheory.Limits.Cocone.whisker e.functor X)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "M₂ : Type w₁", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "N N' : LieSubmodule R L M", "I J : LieIdeal R L", "N₂ : LieSubmodule R L M₂"], "goal": "⁅I, N ⊓ N'⁆ ≤ ⁅I, N⁆ ∧ ⁅I, N ⊓ N'⁆ ≤ ⁅I, N'⁆"}}
+{"state": {"context": ["L : Language", "T : L.Theory", "α : Type w", "S : L[[α]].Theory"], "goal": "{p | S ⊆ ↑p} = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ S).IsSatisfiable"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "ι : Sort u_1", "s : ι → Set α", "hs : ∀ (i : ι), IsComplete (s i)", "U : Set (α × α)", "hU : U ∈ 𝓤 α", "hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j", "S : Set α := ⋃ i, s i", "l : Filter α", "hl : Cauchy l", "hls : S ∈ l", "hl_ne : l.NeBot", "hl' : ��� s ∈ 𝓤 α, ∃ t ∈ l, t ×ˢ t ⊆ s", "t : Set α", "htS : t ⊆ S", "htl : t ∈ l", "htU : t ×ˢ t ⊆ U", "i : ι", "hi : t ⊆ s i"], "goal": "∃ x ∈ S, l ≤ 𝓝 x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s s₁ s₂ : Finset α", "t t₁ t₂ : α → Finset β", "inst✝ : DecidableEq β", "hs : s₁ = s₂", "ht : ∀ a ∈ s₁, t₁ a = t₂ a", "x : β"], "goal": "∀ (a : α), a ∈ s₁ ∧ x ∈ t₁ a ↔ a ∈ s₂ ∧ x ∈ t₂ a"}}
+{"state": {"context": ["k : Type u", "inst✝ : Field k", "n : ℕ", "x : Step k n", "hx : AlgebraicClosure.toStepOfLE' k n (n + 1) ⋯ x = (toStepSucc k n) x"], "goal": "(Ring.DirectLimit.of (Step k) (AlgebraicClosure.toStepOfLE' k) (n + 1)) (AlgebraicClosure.toStepOfLE' k n (n + 1) ⋯ x) = (Ring.DirectLimit.of (Step k) (AlgebraicClosure.toStepOfLE' k) n) x"}}
+{"state": {"context": ["S : Type uS", "CommSemiring S", "A : Type uA", "Semiring A", "Algebra S A", "c : RingCon A", "s : S"], "goal": "↑((algebraMap S A) s) = (algebraMap S c.Quotient) s"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "InnerProductSpaceable E", "r : ℚ"], "goal": "InnerProductSpaceable.innerProp' E ↑r"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "U : Set X", "hU : IsClosed U"], "goal": "IsProperMap Subtype.val"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "c g' : Perm (Fin 5)", "h : 3 ∈ (c * g').cycleType", "hd : c.Disjoint g'", "left✝ : c.IsCycle", "h3 : c.support.card = 3", "n : ℕ", "hn : n ∈ g'.cycleType"], "goal": "(c * g').support.card ≤ 3 + 2"}}
+{"state": {"context": ["α₁ : Type u_2", "β₁ : Type u_1", "β₂ : Type u_3", "e : α₁ → β₁ ≃ β₂", "a : α₁", "b : β₁"], "goal": "((sigmaEquivProd α₁ β₁).symm.trans (sigmaCongrRight e)) (a, b) = ((prodCongrRight e).trans (sigmaEquivProd α₁ β₂).symm) (a, b)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "Infinite R", "σ : Type u_2", "p q : MvPolynomial σ R"], "goal": "p = q ↔ ∀ (x : σ → R), (MvPolynomial.eval x) p = (MvPolynomial.eval x) q"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝ : TopologicalSpace α", "s t u v : Set α", "hu : IsOpen u", "hv : IsOpen v", "huv : Disjoint u v", "hsuv : s ⊆ u ∪ v", "hsu : (s ∩ u).Nonempty", "hs : IsPreconnected s", "hsv : (s ∩ v).Nonempty", "x : α", "left✝ : x ∈ s", "hx : x ∈ u ∩ v"], "goal": "False"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : WeierstrassCurve R", "dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} => natDegree_pow_le_of_le n", "h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ WeierstrassCurve.expDegree n ∧ (W.preΨ' n).coeff (WeierstrassCurve.expDegree n) = ↑(WeierstrassCurve.expCoeff n) := fun {n} => WeierstrassCurve.natDegree_coeff_preΨ' W n", "n : ℕ", "hd : (n + 1) ^ 2 - 1 = 2 * WeierstrassCurve.expDegree (n + 1) + if Even (n + 1) then 3 else 0"], "goal": "(W.ΨSq ↑(n + 1)).natDegree ≤ (n + 1) ^ 2 - 1 ∧ (W.ΨSq ↑(n + 1)).coeff ((n + 1) ^ 2 - 1) = ↑(↑(n + 1) ^ 2)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a✝ b✝ : R", "n : ℕ", "inst✝³ : CommRing R", "inst✝² : IsDomain R", "inst✝¹ : CommRing S", "inst✝ : IsDomain S", "φ : R →+* S", "f : R[X]", "h_mon : f.leadingCoeff = 1", "h_irr : Irreducible (Polynomial.map φ f)", "a b : R[X]", "h : f = a * b", "q : a.leadingCoeff * b.leadingCoeff = 1"], "goal": "IsUnit a ∨ IsUnit b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι✝ : Type u_2", "inst✝ : PseudoMetricSpace α", "s : Set α", "ι : Sort u_3", "c : ι → Set α", "hs : IsCompact s", "hc₁ : ∀ (i : ι), IsOpen (c i)", "hc₂ : s ⊆ ⋃ i, c i"], "goal": "∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "p : R[X]"], "goal": "p.IsPrimitive ↔ p.content = 1"}}
+{"state": {"context": ["a b c : ℤ", "h : Minimal a b c", "ha2 : a % 2 = 1", "hc : 0 < c", "ht : PythagoreanTriple (a ^ 2) (b ^ 2) c", "h2 : (a ^ 2).gcd (b ^ 2) = 1", "ha22 : a ^ 2 % 2 = 1", "m n : ℤ", "ht1 : a ^ 2 = m ^ 2 - n ^ 2", "ht2 : b ^ 2 = 2 * m * n", "ht3 : c = m ^ 2 + n ^ 2", "ht4 : m.gcd n = 1", "ht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0", "ht6 : 0 ≤ m", "htt : PythagoreanTriple a n m", "h3 : a.gcd n = 1", "hb20 : b ^ 2 ≠ 0", "h4 : 0 < m", "r s : ℤ", "left✝ : a = r ^ 2 - s ^ 2", "htt2 : n = 2 * r * s", "htt3 : m = r ^ 2 + s ^ 2", "htt4 : r.gcd s = 1", "htt5 : r % 2 = 0 ∧ s % 2 = 1 ∨ r % 2 = 1 ∧ s % 2 = 0", "htt6 : 0 ≤ r", "hcp : m.gcd (r * s) = 1", "b' : ℤ", "hb2' : b = 2 * b'"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "m0 : MeasurableSpace α", "μ : Measure α", "v : VitaliFamily μ", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : SecondCountableTopology α", "inst✝² : BorelSpace α", "inst✝¹ : IsLocallyFiniteMeasure μ", "ρ : Measure α", "inst✝ : IsLocallyFiniteMeasure ρ", "hρ : ρ ≪ μ", "x : α", "x✝ : x ∈ {x | v.limRatioMeas hρ x = ⊤}", "o : Set α", "xo : x ∈ o", "o_open : IsOpen o", "μo : ρ o < ⊤"], "goal": "∃ u ∈ 𝓝[{x | v.limRatioMeas hρ x = ⊤}] x, μ u = 0"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Submonoid R", "S : Type u_2", "CommRing S", "Algebra R S", "IsLocalization M S", "p : S[X]"], "goal": "∃ b, ∀ (i : ℕ), (algebraMap R S) ((IsLocalization.integerNormalization M p).coeff i) = ↑b • p.coeff i"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X", "f : X → Y", "hf : Inducing f", "hf' : IsClosed (range f)", "K : Set Y", "hK : IsCompact K"], "goal": "IsCompact (f ⁻¹' K)"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s : LowerSet α"], "goal": "(↑s).Nonempty ↔ s ≠ ⊥"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α"], "goal": "l ⊆ a :: l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e e' : PartialEquiv α β", "hs : Disjoint e.source e'.source", "ht : Disjoint e.target e'.target", "(x : α) → Decidable (x ∈ e.source)", "(y : β) → Decidable (y ∈ e.target)"], "goal": "↑(e.disjointUnion e' hs ht) = e.source.piecewise ↑e ↑e'"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "x : M", "h : I.IsInteriorPoint x"], "goal": "Set.range ↑I ∈ 𝓝 (↑(extChartAt I x) x)"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "W : Type u_3", "P : Type u_4", "Ring R", "AddCommGroup V", "Module R V", "TopologicalSpace P", "AddTorsor V P", "AddCommGroup W", "Module R W", "S : Type u_8", "TopologicalSpace W", "Monoid S", "DistribMulAction S W", "SMulCommClass R S W", "ContinuousConstSMul S W", "t : S", "f : P →ᴬ[R] W", "x : P"], "goal": "(t • f) x = t • f x"}}
+{"state": {"context": ["a o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{u}", "hf : a.IsFundamentalSequence o f", "i j : Ordinal.{u}", "hi : i < o", "hj : j < o", "hij : i ≤ j"], "goal": "f i hi ≤ f j hj"}}
+{"state": {"context": ["a b r : ℕ", "hr : 0 < r", "v : ℕ", "hr' : 0 < ↑r"], "goal": "count (fun x => x ≡ v [MOD r]) b = b / r + if v % r < b % r then 1 else 0"}}
+{"state": {"context": ["n : ℕ+", "K : Type u_1", "inst✝⁴ : Field K", "L : Type u_2", "μ : L", "inst✝³ : CommRing L", "inst✝² : IsDomain L", "hμ✝ : IsPrimitiveRoot μ ↑n", "inst✝¹ : Algebra K L", "inst✝ : IsCyclotomicExtension {n} K L", "h : Irreducible (cyclotomic (↑n) K)", "hζ : IsPrimitiveRoot (zeta n K L) ↑n := zeta_spec n K L", "hμ : ∀ (t : (ZMod ↑n)ˣ), IsPrimitiveRoot (zeta n K L ^ (↑t).val) ↑n := fun t => IsPrimitiveRoot.pow_of_coprime hζ (↑t).val (ZMod.val_coe_unit_coprime t)", "f : L ≃ₐ[K] L"], "goal": "(fun t => (IsPrimitiveRoot.powerBasis K hζ).equivOfMinpoly (IsPrimitiveRoot.powerBasis K ⋯) ⋯) ((↑__src✝).toFun f) = f"}}
+{"state": {"context": ["α✝ : Type u", "β : Type v", "inst✝ : SemilatticeSup α✝", "a b c✝ d : α✝", "α : Type u_1", "A B : SemilatticeSup α", "H : ∀ (x y : α), x ≤ y ↔ x ≤ y", "x y c : α"], "goal": "x ⊔ y ≤ c ↔ x ≤ c ∧ y ≤ c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁴ : MeasurableSpace α", "inst✝³ : MeasurableSpace β", "inst✝² : MeasurableSpace γ", "inst✝¹ : MeasurableSpace δ", "μa : Measure α", "μb : Measure β", "μc : Measure γ", "μd : Measure δ", "μ : Measure α", "f : α → α", "s : Set α", "inst✝ : IsFiniteMeasure μ", "hf : MeasurePreserving f μ μ", "hs : NullMeasurableSet s μ", "hs' : μ s ≠ 0"], "goal": "∃ x ∈ s, ∃ m, m ≠ 0 ∧ f^[m] x ∈ s"}}
+{"state": {"context": ["s : (n : ℕ) × Finset (Fin n)"], "goal": "(FormalMultilinearSeries.changeOriginIndexEquiv.symm s).fst = s.fst - s.snd.card"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type uG", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "n : ℕ∞", "E₁ : Type u_3", "E₂ : Type u_4", "NormedAddCommGroup E₁", "NormedAddCommGroup E₂", "NormedSpace 𝕜 E₁", "NormedSpace 𝕜 E₂", "g : E₁ × E₂ → G", "f₁ : F → E₁", "f₂ : F → E₂", "s : Set F", "hg : ContDiff 𝕜 n g", "hf₁ : ContDiffOn 𝕜 n f₁ s", "hf₂ : ContDiffOn 𝕜 n f₂ s"], "goal": "ContDiffOn 𝕜 n (fun x => g (f₁ x, f₂ x)) s"}}
+{"state": {"context": ["α : Type u", "Group α"], "goal": "⊤.toTopologicalSpace = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "a : α", "ha : a ∉ s"], "goal": "Prod.mk a ⁻¹' s ×ˢ t = ∅"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "Group G", "MulAction G α", "S : Set G"], "goal": "(∀ g ∈ Subgroup.closure S, (MulAction.fixedBy α g)ᶜ.Finite) ↔ ∀ g ∈ S, (MulAction.fixedBy α g)ᶜ.Finite"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "n : ℕ"], "goal": "(Matrix.vandermonde fun i => ↑↑i).det = ↑n.superFactorial"}}
+{"state": {"context": ["R : Type u", "a b c : R", "Semiring R"], "goal": "(Polynomial.C a * Polynomial.X ^ 2 + Polynomial.C b * Polynomial.X + Polynomial.C c).degree ≤ 2"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹² : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝⁹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝⁸ : TopologicalSpace M", "inst✝⁷ : ChartedSpace H M", "inst✝⁶ : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝⁵ : NormedAddCommGroup E'", "inst✝⁴ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝³ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : SmoothManifoldWithCorners I' M'", "s : Set M", "x : M", "hs : UniqueMDiffWithinAt I s x", "f : M → M'", "f' : E →L[𝕜] E'", "hf : HasMFDerivWithinAt I I' f s x f'", "hd : DenseRange ⇑f'", "this : UniqueMDiffWithinAt I (s ∩ f ⁻¹' (extChartAt I' (f x)).source) x"], "goal": "UniqueMDiffWithinAt I' (f '' s) (f x)"}}
+{"state": {"context": [], "goal": "GaussianInt.toComplex 0 = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "AddCommMonoid α", "LT α", "s : Finset ι", "f : ι → WithTop α"], "goal": "∑ i ∈ s, f i < ⊤ ↔ ∀ i ∈ s, f i < ⊤"}}
+{"state": {"context": [], "goal": "borel ℝ = MeasurableSpace.generateFrom (⋃ a, {Set.Ioi ↑a})"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : Ring R", "E : Type u_2", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module R E", "F : Type u_3", "inst✝³ : AddCommGroup F", "inst✝² : Module R F", "G : Type u_4", "inst✝¹ : AddCommGroup G", "inst✝ : Module R G", "f g : E →ₗ.[R] F", "h : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y"], "goal": "f ≤ f.sup g h"}}
+{"state": {"context": ["n : Type u_3", "α : Type v", "Add α", "A B : Matrix n n α"], "goal": "(A + B).diag = A.diag + B.diag"}}
+{"state": {"context": ["R : Type u_1", "M : Type v₁", "N : Type v₂", "S : Type v₃", "inst✝¹² : AddCommMonoid M", "inst✝¹¹ : AddCommMonoid N", "inst✝¹⁰ : CommSemiring R", "inst✝⁹ : CommSemiring S", "inst✝⁸ : Algebra R S", "inst✝⁷ : Module R M", "inst✝⁶ : Module R N", "inst✝⁵ : Module S N", "inst✝⁴ : IsScalarTower R S N", "f : M →ₗ[R] N", "h : IsBaseChange S f", "P : Type u_2", "Q : Type u_3", "inst✝³ : AddCommMonoid P", "inst✝² : Module R P", "inst✝¹ : AddCommMonoid Q", "inst✝ : Module S Q", "g₁ g₂ : N →ₗ[S] Q", "e : ∀ (x : M), g₁ (f x) = g₂ (f x)", "x : N"], "goal": "∀ (m : M), g₁ (f m) = g₂ (f m)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : TopologicalSpace α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : μ.WeaklyRegular", "inst✝ : SigmaFinite μ", "f : α → ℝ≥0", "fint : Integrable (fun x => ↑(f x)) μ", "fmeas : AEMeasurable f μ", "ε : ℝ≥0", "εpos : 0 < ↑ε", "δ : ℝ≥0", "δpos : 0 < δ", "hδε : δ < ε"], "goal": "∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ Integrable (fun x => (g x).toReal) μ ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), ↑(f x) ∂μ + ↑ε"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M M' : Bimod W X", "f : M ⟶ M'", "N : Bimod X Y", "P : Bimod Y Z"], "goal": "M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ colimit.ι (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)) ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫ M'.X ◁ TensorBimod.actLeft N P)) WalkingParallelPair.one = (α_ M.X N.X P.X).inv ≫ (((α_ M.X N.X P.X).hom ≫ f.hom ▷ (N.X ⊗ P.X)) ≫ M'.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)) ≫ coequalizer.π (M'.actRight ▷ TensorBimod.X N P) ((α_ M'.X X.X (TensorBimod.X N P)).hom ≫ M'.X ◁ TensorBimod.actLeft N P)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹ : TopologicalSpace M", "inst✝ : ChartedSpace H M"], "goal": "I.interior M ⊓ I.boundary M = ⊥"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "x : ℚ_[p]", "n : ℤ"], "goal": "‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : 𝕜", "e : 𝕜 →ₗ[𝕜] F"], "goal": "deriv (⇑e) x = e 1"}}
+{"state": {"context": ["α : Type u_1", "a : Array α", "v : α", "i : USize", "h : i.toNat < a.size"], "goal": "(a.uset i v h).size = a.size"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "c c' : CategoryTheory.Limits.Cone F", "f g : c ⟶ c'", "w : f.hom = g.hom"], "goal": "f = g"}}
+{"state": {"context": ["r : ℝ"], "goal": "↑r⁻¹ = (↑r)⁻¹"}}
+{"state": {"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ≤ 0"], "goal": "HurwitzZeta.evenKernel a x = 0"}}
+{"state": {"context": ["α : Type u", "AddGroup α", "f g : AddGroupTopology α", "h : TopologicalSpace.IsOpen = TopologicalSpace.IsOpen"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "SuccOrder α", "IsSuccArchimedean α", "a b : α"], "goal": "(∃ n, Order.succ^[n] a = b) ∨ ∃ n, Order.succ^[n] b = a"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "𝒜 : Finset (Finset α)", "s : Finset α", "a : α", "r k : ℕ", "ih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ u, u.card = k ∧ u ⊆ t ∧ t \\ u ∈ 𝒜", "t : Finset α"], "goal": "t ∈ ∂⁺ ^[k + 1] 𝒜 ↔ ∃ u, u.card = k + 1 ∧ u ⊆ t ∧ t \\ u ∈ 𝒜"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "pf : FormalMultilinearSeries 𝕜 E F", "x : E", "hf : HasFPowerSeriesAt f pf x", "v : Fin 0 → E"], "goal": "(pf 0) v = f x"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝¹ : Ring R", "inst✝ : Semiring S", "f : R →+* S", "x : S", "T : Subring R", "p : (↥T)[X]"], "goal": "↑(ofSubring T p).coeffs ⊆ ↑T"}}
+{"state": {"context": ["α : Type u_1", "Add α", "x y : αᵃᵒᵖ"], "goal": "AddOpposite.unop (x + y) = AddOpposite.unop y + AddOpposite.unop x"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_4, u_1} C", "D : Type u_2", "inst✝ : Category.{u_3, u_2} D", "ε : C ≌ D", "h : IsIdempotentComplete C", "X' : D", "p : X' ⟶ X'", "hp : p ≫ p = p", "φ : (𝟭 D).obj X' ≅ (ε.inverse ⋙ ε.functor).obj X' := ε.counitIso.symm.app X'"], "goal": "∃ Y i e, i ≫ e = 𝟙 Y ∧ e ≫ i = p"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ : R", "a : ℍ[R,c₁,c₂]"], "goal": "star a = ↑(2 * a.re) - a"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C"], "goal": "(Comon_.Comon_EquivMon_OpOp C).functor = Comon_.Comon_ToMon_OpOp C"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Group α", "x : Finset α × Finset α"], "goal": "Finset.mulETransformLeft 1 x = x"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : TopologicalSpace X", "a✝ b✝ b₀✝ b₁ b₂ : ℝ", "μ : Measure ℝ", "f : ℝ → E", "inst✝ : FirstCountableTopology X", "F : X → ℝ → E", "bound : ℝ → ℝ", "a b a₀ b₀ : ℝ", "x₀ : X", "hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))", "h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t", "bound_integrable : IntervalIntegrable bound μ a b", "h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x => F x t) x₀", "ha₀ : a₀ ∈ Ioo a b", "hb₀ : b₀ ∈ Ioo a b", "hμb₀ : μ {b₀} = 0", "hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b", "Ioo_nhds : Ioo a b ∈ 𝓝 b₀", "Icc_nhds : Icc a b ∈ 𝓝 b₀", "hx₀ : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x₀ t‖ ≤ bound t", "this : ∀ᶠ (p : X × ℝ) in 𝓝 (x₀, b₀), ∫ (s : ℝ) in a₀..p.2, F p.1 s ∂μ = ∫ (s : ℝ) in a₀..b₀, F p.1 s ∂μ + ∫ (s : ℝ) in b₀..p.2, F x₀ s ∂μ + ∫ (s : ℝ) in b₀..p.2, F p.1 s - F x₀ s ∂μ"], "goal": "ContinuousAt (fun x => ∫ (s : ℝ) in a₀..b₀, F x.1 s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x₀ s ∂μ + ∫ (s : ℝ) in b₀..x.2, F x.1 s - F x₀ s ∂μ) (x₀, b₀)"}}
+{"state": {"context": ["F : Type u_1", "E : Type u_2", "K : Type u_3", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Field K", "inst✝¹ : Algebra F E", "inst✝ : Algebra F K", "S : Set E", "c : Set (Lifts F E K)", "hc✝ : IsChain (fun x x_1 => x ≤ x_1) c", "t : ↑(insert ⊥ c) → IntermediateField F E := fun i => (↑i).carrier", "t' : ↑(insert ⊥ c) → Subalgebra F E := fun i => (t i).toSubalgebra", "hc : IsChain (fun x x_1 => x ≤ x_1) (insert ⊥ c)", "dir : Directed (fun x x_1 => x ≤ x_1) t"], "goal": "∃ ub, ∀ a ∈ c, a ≤ ub"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "K : Submodule 𝕜 E", "x : E", "hK : Dense ↑K", "h : x ∈ Kᗮ"], "goal": "x = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "F.IsEquivalence", "X Y : C", "f : X ⟶ Y"], "goal": "F.inv.map (F.map f) =\n  CategoryTheory.CategoryStruct.comp (F.asEquivalence.unitInv.app X)\n    (CategoryTheory.CategoryStruct.comp f (F.asEquivalence.unit.app Y))"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "n : ℕ", "h : p.coeff n ≠ 0"], "goal": "p.coeff n ∈ p.coeffs"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : C", "S✝ R✝ : Sieve X", "J : GrothendieckTopology C", "f : Y ⟶ X", "S R : Sieve X", "h : J.Covers S f", "k : ∀ {Z : C} (g : Z ⟶ X), S.arrows g → J.Covers R g"], "goal": "∀ ⦃Y_1 : C⦄ ⦃f_1 : Y_1 ⟶ Y⦄, (Sieve.pullback f S).arrows f_1 → Sieve.pullback f_1 (Sieve.pullback f R) ∈ J.sieves Y_1"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "n✝ : ℕ", "α : Type u_1", "β : Type u_2", "C : G.Coloring α", "n : ℕ", "hc : G.Colorable n", "s : Finset V", "h : G.IsClique ↑s", "hm : n < s.card"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι✝ : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s : Set E", "i j : ι✝", "c : R", "t✝ : Finset ι✝", "w✝² : ι✝ → R", "z : ι✝ → E", "t : Set F", "x : E", "y : F", "ι : Type", "w✝¹ : Fintype ι", "w : ι → R", "f : ι → E", "hw₀ : ∀ (i : ι), 0 ≤ w i", "hw₁ : ∑ i : ι, w i = 1", "hfs : ∀ (i : ι), f i ∈ s", "hf : ∑ i : ι, w i • f i = x", "κ : Type", "w✝ : Fintype κ", "v : κ → R", "g : κ → F", "hv₀ : ∀ (i : κ), 0 ≤ v i", "hv₁ : ∑ i : κ, v i = 1", "hgt : ∀ (i : κ), g i ∈ t", "hg : ∑ i : κ, v i • g i = y"], "goal": "∃ ι x_1 w z, (∀ (i : ι), 0 ≤ w i) ∧ ∑ i : ι, w i = 1 ∧ (∀ (i : ι), z i ∈ s ×ˢ t) ∧ ∑ i : ι, w i • z i = (x, y)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "k : Type u_3", "V : Type u_4", "P : Type u_5", "inst✝⁴ : AddCommGroup V", "inst✝³ : AffineSpace V P", "inst✝² : Ring k", "inst✝¹ : Module k V", "b : AffineBasis ι k P", "s : Finset ι", "i j : ι", "e : ι ≃ ι'", "inst✝ : Fintype ι", "w : ι → k", "hw : ∑ i : ι, w i = 1", "hq : (Finset.affineCombination k Finset.univ ⇑b) w ∈ affineSpan k (range ⇑b)", "x✝ : ι", "a✝ : x✝ ∈ Finset.univ"], "goal": "(b.coord x✝) ((Finset.affineCombination k Finset.univ ⇑b) w) = w x✝"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "s t : CategoryTheory.Bicategory.LeftExtension f g", "i : s ⟶ t", "x : B", "h : c ⟶ x"], "goal": "(CategoryTheory.Bicategory.LeftExtension.whiskerHom i h).right = CategoryTheory.Bicategory.whiskerRight i.right h"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq N", "inst✝² : Zero N", "inst✝¹ : DecidableEq M", "inst✝ : Zero M", "f g : α →₀ N", "F : N → M", "F0 : F 0 = 0", "x : α"], "goal": "x ∈ (mapRange F F0 f).neLocus (mapRange F F0 g) → x ∈ f.neLocus g"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁵ : LinearOrder ι", "f : Filtration ι m", "τ π : Ω → ι", "inst✝⁴ : TopologicalSpace ι", "inst✝³ : SecondCountableTopology ι", "inst✝² : OrderTopology ι", "inst✝¹ : MeasurableSpace ι", "inst✝ : BorelSpace ι", "hτ : IsStoppingTime f τ", "hπ : IsStoppingTime f π", "j : ι", "this : {ω | τ ω ≤ π ω} ∩ {ω | τ ω ≤ j} = {ω | min (τ ω) j ≤ min (π ω) j} ∩ {ω | τ ω ≤ j}"], "goal": "Measurable fun a => min (τ a) j"}}
+{"state": {"context": ["α : Type u", "Subsingleton α", "r : α → α → Prop", "hr : IsIrrefl α r"], "goal": "IsWellOrder α r"}}
+{"state": {"context": ["α : Type u", "a : α", "inst✝ : Group α", "h : IsCyclic α", "g : α", "hg : ∀ (x : α), x ∈ zpowers g"], "goal": "Nat.card ↥⊤ = Nat.card α"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_4", "Semiring R", "AddCommMonoid M", "Module R M", "x : M", "s : Set M", "h : x ∈ Submodule.span R s"], "goal": "Submodule.span R (insert x s) = Submodule.span R s"}}
+{"state": {"context": ["Y : Type u_1", "inst✝ : TopologicalSpace Y", "a✝¹ a✝ : Y"], "goal": "(∀ (p₁ p₂ : Path a✝¹ a✝), p₁.Homotopic p₂) ↔ Subsingleton (Path.Homotopic.Quotient a✝¹ a✝)"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "inst✝⁶ : Category.{v₁, u₁} C", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : Preadditive C", "inst✝³ : Preadditive D", "F : C ⥤ D", "inst✝² : HasBinaryProducts C", "inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) F", "inst✝ : F.PreservesZeroMorphisms", "this✝ : HasBinaryBiproducts C", "this : PreservesBinaryBiproducts F"], "goal": "F.Additive"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s s₁ s₂ : Finset α", "t✝ t₁ t₂ : α → Finset β", "inst✝ : DecidableEq β", "t : α → Finset β", "h : s₁ ⊆ s₂", "x : β"], "goal": "x ∈ s₁.biUnion t → x ∈ s₂.biUnion t"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : Module 𝕜 E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace 𝕜 F", "inst✝ : CompleteSpace F", "α : Type u_4", "l : Filter α", "f : α → E →WOT[𝕜]F", "A : E →WOT[𝕜]F"], "goal": "Tendsto f l (𝓝 A) ↔ ∀ (x : E) (y : F), Tendsto (fun a => ((InnerProductSpace.toDual 𝕜 F) y) ((f a) x)) l (𝓝 (((InnerProductSpace.toDual 𝕜 F) y) (A x)))"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝ : TopologicalSpace α", "s t u v : Set α", "x : α", "F : Set α", "hs : IsPreconnected s", "hxs : x ∈ s", "hsF : s ⊆ F"], "goal": "s ⊆ connectedComponentIn F x"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "n✝ : ℕ", "α : Type u_1", "β : Type u_2", "C : G.Coloring α", "n : ℕ", "h : sInf {n' | G.Colorable n'} ≤ n", "m : ℕ", "hm : G.Colorable m", "this : G.Colorable (sInf {n' | G.Colorable n'})"], "goal": "G.Colorable n"}}
+{"state": {"context": ["α : Type u_2", "ι : Type u_5", "DistribLattice α", "OrderBot α", "s : Finset ι", "f : ι → α", "a : α"], "goal": "Disjoint (s.sup f) a ↔ ∀ ⦃i : ι⦄, i ∈ s → Disjoint (f i) a"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : OrderedAddCommMonoid M", "f : ℕ → M", "u : ℕ → ℕ", "C : ℕ", "hf : ∀ ⦃m n : ℕ⦄, 1 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "h_nonneg : ∀ (n : ℕ), 0 ≤ f n", "hu : Monotone u", "h_succ_diff : SuccDiffBounded C u", "n : ℕ"], "goal": "∑ k ∈ range (n + 1), (u (k + 1) - u k) • f (u k) ≤ (u 1 - u 0) • f (u 0) + C • ∑ k ∈ Ico (u 0 + 1) (u n + 1), f k"}}
+{"state": {"context": ["M : Type u_4", "MulOneClass M", "a : M"], "goal": "[a].prod = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F₁ : Type u_3", "F₂ : Type u_4", "M : Type u_5", "E₁ : B → Type u_6", "E₂ : B → Type u_7", "inst✝³⁵ : NontriviallyNormedField 𝕜", "inst✝³⁴ : (x : B) → AddCommGroup (E₁ x)", "inst✝³³ : (x : B) → Module 𝕜 (E₁ x)", "inst✝³² : NormedAddCommGroup F₁", "inst✝³¹ : NormedSpace 𝕜 F₁", "inst✝³⁰ : TopologicalSpace (TotalSpace F₁ E₁)", "inst✝²⁹ : (x : B) → TopologicalSpace (E₁ x)", "inst✝²⁸ : (x : B) → AddCommGroup (E₂ x)", "inst✝²⁷ : (x : B) → Module 𝕜 (E₂ x)", "inst✝²⁶ : NormedAddCommGroup F₂", "inst✝²⁵ : NormedSpace 𝕜 F₂", "inst✝²⁴ : TopologicalSpace (TotalSpace F₂ E₂)", "inst✝²³ : (x : B) → TopologicalSpace (E₂ x)", "inst✝²² : ∀ (x : B), TopologicalAddGroup (E₂ x)", "inst✝²¹ : ∀ (x : B), ContinuousSMul 𝕜 (E₂ x)", "EB : Type u_8", "inst✝²⁰ : NormedAddCommGroup EB", "inst✝¹⁹ : NormedSpace 𝕜 EB", "HB : Type u_9", "inst✝¹⁸ : TopologicalSpace HB", "IB : ModelWithCorners 𝕜 EB HB", "inst✝¹⁷ : TopologicalSpace B", "inst✝¹⁶ : ChartedSpace HB B", "EM : Type u_10", "inst✝¹⁵ : NormedAddCommGroup EM", "inst✝¹⁴ : NormedSpace 𝕜 EM", "HM : Type u_11", "inst✝¹³ : TopologicalSpace HM", "IM : ModelWithCorners 𝕜 EM HM", "inst✝¹² : TopologicalSpace M", "inst✝¹¹ : ChartedSpace HM M", "inst✝¹⁰ : SmoothManifoldWithCorners IM M", "n : ℕ∞", "inst✝⁹ : FiberBundle F₁ E₁", "inst✝⁸ : VectorBundle 𝕜 F₁ E₁", "inst✝⁷ : FiberBundle F₂ E₂", "inst✝⁶ : VectorBundle 𝕜 F₂ E₂", "e₁ e₁' : Trivialization F₁ TotalSpace.proj", "e₂ e₂' : Trivialization F₂ TotalSpace.proj", "inst✝⁵ : SmoothVectorBundle F₁ E₁ IB", "inst✝⁴ : SmoothVectorBundle F₂ E₂ IB", "inst✝³ : MemTrivializationAtlas e₁", "inst✝² : MemTrivializationAtlas e₁'", "inst✝¹ : MemTrivializationAtlas e₂", "inst✝ : MemTrivializationAtlas e₂'", "h₁ : SmoothOn IB 𝓘(𝕜, F₁ →L[𝕜] F₁) (fun b => ↑(Trivialization.coordChangeL 𝕜 e₁' e₁ b)) (e₁'.baseSet ∩ e₁.baseSet)"], "goal": "SmoothOn IB 𝓘(𝕜, (F₁ →L[𝕜] F₂) →L[𝕜] F₁ →L[𝕜] F₂) (continuousLinearMapCoordChange (RingHom.id 𝕜) e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "R : Type u_2", "Semiring R", "MulSemiringAction M R", "m : M"], "goal": "HSMul.hSMul m = Polynomial.map (MulSemiringAction.toRingHom M R m)"}}
+{"state": {"context": ["M : Type u_1", "S : Set M", "AddMonoid M", "z : M"], "goal": "z ∈ AddSubmonoid.centralizer S ↔ ∀ g ∈ S, g + z = z + g"}}
+{"state": {"context": ["α : Type u", "SemilatticeSup α", "a b : α"], "goal": "b < a ⊔ b ↔ ¬a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "hμν : μ ≪ ν", "x : α", "hx : (μ.rnDeriv ν)⁻¹ x = ν.rnDeriv μ x"], "goal": "(ν.rnDeriv μ)⁻¹ x = μ.rnDeriv ν x"}}
+{"state": {"context": ["d : ℤ", "z w : ℤ√d"], "goal": "(z * w).re = z.re * w.re + d * z.im * w.im"}}
+{"state": {"context": ["R : Type u", "M : Type v", "SMul R M", "p : SubMulAction R M", "x : M"], "goal": "x ∈ p.carrier ↔ x ∈ ↑p"}}
+{"state": {"context": ["S : Profinite"], "goal": "ClosedEmbedding (ι S)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝² : Category.{?u.20465, u_1} C", "inst✝¹ : Category.{?u.20469, u_2} D", "inst✝ : Category.{?u.20473, u_3} E", "F G H : C ⥤ Karoubi D", "φ : F ⟶ G", "φ' : G ⟶ H", "P : Karoubi C"], "goal": "(F.map P.p).f ≫ (φ.app P.X).f ≫ (φ'.app P.X).f = (F.map P.p).f ≫ (φ.app P.X).f ≫ (G.map P.p).f ≫ (φ'.app P.X).f"}}
+{"state": {"context": ["X : Type u_2", "x : FreeAbelianGroup X"], "goal": "Finsupp.toFreeAbelianGroup (FreeAbelianGroup.toFinsupp x) = x"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]"], "goal": "Commute Polynomial.X p"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝² : MeasurableSpace Ω", "ι : Type u_2", "L : Filter ι", "μ : Measure Ω", "μs : ι → Measure Ω", "inst✝¹ : IsProbabilityMeasure μ", "inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)", "E : Set Ω", "E_mble : MeasurableSet E", "h : limsup (fun i => (μs i) E) L ≤ μ E", "hne : L.NeBot", "meas_Ec : μ Eᶜ = 1 - μ E", "meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E", "obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L", "this : 1 - limsup (fun i => (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L"], "goal": "1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "a b : Part α"], "goal": "a * b = do\n  let y ← a\n  Part.map (fun x => y * x) b"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type w'", "L.Structure M", "L.Structure N", "f : L.Equiv M N"], "goal": "f.symm.toHom.comp f.toHom = FirstOrder.Language.Hom.id L M"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Type u_3", "f : α → β", "m : ι → MeasureTheory.OuterMeasure α"], "goal": "(MeasureTheory.OuterMeasure.map f) (⨅ i, m i) ≤ ⨅ i, (MeasureTheory.OuterMeasure.map f) (m i)"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : Y ⟶ X", "AlgebraicGeometry.IsOpenImmersion f", "x : ↑↑X.toPresheafedSpace"], "goal": "(𝒰.add f).f x = some (𝒰.f x)"}}
+{"state": {"context": ["α : Type u_1", "AddZeroClass α", "Preorder α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "a b : α", "ha : 0 ≤ a", "hb : 0 < b"], "goal": "0 < a + b"}}
+{"state": {"context": ["β : Type u_1", "G : Type u_2", "α : Type u_3", "γ : Type u_4", "inst✝³ : Group G", "inst✝² : AddGroup α", "inst✝¹ : SMul γ α", "inst✝ : SlashAction β G α γ", "k : β", "g : G", "a : α"], "goal": "(-a) ∣[k;γ] g + a ∣[k;γ] g = 0"}}
+{"state": {"context": ["C : Type u_1", "H : Type u_3", "D : Type u_5", "D' : Type u_6", "CategoryTheory.Category.{u_7, u_1} C", "CategoryTheory.Category.{u_8, u_3} H", "CategoryTheory.Category.{u_9, u_5} D", "CategoryTheory.Category.{u_10, u_6} D'", "L : CategoryTheory.Functor C D", "L' : CategoryTheory.Functor C D'", "G : CategoryTheory.Functor D D'", "f : L' ⟶ L.comp G", "F : CategoryTheory.Functor C H", "X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit F) ((CategoryTheory.whiskeringLeft C D' H).obj L')"], "goal": "∀ {X_1 Y : D} (f_1 : X_1 ⟶ Y),\n  ((CategoryTheory.Functor.LeftExtension.postcomp₁ G f F).obj X).right.map f_1 = X.right.map (G.map f_1)"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "self : TopologicalSpace.Opens α"], "goal": "IsOpen self.carrier"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "n : ℕ", "α : Type u_1", "β : Type u_2", "C : G.Coloring α", "inst✝ : Fintype V"], "goal": "∀ (C : ⊤.Coloring V), Surjective ⇑C"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Bool", "l : List α", "h : ∀ (x : α), x ∈ l → p x = true → q x = true"], "goal": "List.countP p l ≤ List.countP q l"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "G : C ⥤ D", "inst✝¹ : HasCoequalizer f g", "inst✝ : HasCoequalizer (G.map f) (G.map g)", "Z : C", "h : Y ⟶ Z", "w : f ≫ h = g ≫ h"], "goal": "G.map f ≫ G.map h = G.map g ≫ G.map h"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "DecidableEq α", "DecidableEq β", "SemilatticeInf α", "SemilatticeInf β", "FunLike F α β", "InfHomClass F α β", "f : F", "s t : Finset α"], "goal": "Finset.image (⇑f) (s ⊼ t) = Finset.image (⇑f) s ⊼ Finset.image (⇑f) t"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "S : Set X", "f g h : C(X, Y)", "h₀ : f.HomotopicRel g S", "h₁ : g.HomotopicRel h S"], "goal": "f.HomotopicRel h S"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "a b : α", "s : Seq α"], "goal": "a = b ∨ a ∈ s → a ∈ cons b s"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁷ : LinearOrder ι", "f : Filtration ι m", "τ π : Ω → ι", "inst✝⁶ : Countable ι", "inst✝⁵ : TopologicalSpace ι", "inst✝⁴ : MeasurableSpace ι", "inst✝³ : BorelSpace ι", "inst✝² : OrderTopology ι", "inst✝¹ : MeasurableSingletonClass ι", "inst✝ : SecondCountableTopology ι", "hτ : IsStoppingTime f τ", "hπ : IsStoppingTime f π"], "goal": "∀ (i : ι), MeasurableSet ({ω | τ ω = π ω} ∩ {ω | τ ω ≤ i})"}}
+{"state": {"context": ["M : Type u_1", "inst✝⁵ : CommMonoid M", "inst✝⁴ : TopologicalSpace M", "m✝ m' : M", "G : Type u_2", "inst✝³ : CommGroup G", "g g' : G", "inst✝² : T2Space M", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : Encodable β", "inst✝ : CompleteLattice α", "m : α → M", "m0 : m ⊥ = 1", "s : β → α"], "goal": "∏' (i : ℕ), m (⨆ b ∈ decode₂ β i, s b) = ∏' (b : β), m (s b)"}}
+{"state": {"context": ["n j : ℕ", "left✝ : Prime j", "a1 : 1 ≠ 1", "hp : Prime (1 * j)"], "goal": "1 * j = 1"}}
+{"state": {"context": ["x : ℂ"], "goal": "(log x).im ≤ π"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "inst✝¹³ : NontriviallyNormedField 𝕜'", "σ : 𝕜 →+* 𝕜'", "σ' : 𝕜' →+* 𝕜", "inst✝¹² : RingHomInvPair σ σ'", "inst✝¹¹ : RingHomInvPair σ' σ", "inst✝¹⁰ : RingHomIsometric σ", "inst✝⁹ : RingHomIsometric σ'", "E : Type u_3", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F✝ : Type u_4", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace 𝕜' F✝", "f : E →SL[σ] F✝", "inst✝⁴ : CompleteSpace F✝", "inst✝³ : CompleteSpace E", "F : Type u_5", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "g : E →ₗ[𝕜] F", "hg : IsClosed ↑g.graph", "this✝ : CompleteSpace ↥g.graph := completeSpace_coe_iff_isComplete.mpr (IsClosed.isComplete hg)", "φ₀ : E →ₗ[𝕜] E × F := id.prod g", "this : LeftInverse Prod.fst ⇑φ₀", "φ : E ≃ₗ[𝕜] ↥g.graph := LinearEquiv.ofLeftInverse this ≪≫ₗ LinearEquiv.ofEq (range φ₀) g.graph ⋯"], "goal": "Continuous ⇑g"}}
+{"state": {"context": ["R : Type u", "CommRing R", "g : R[X]", "hg : g.Monic"], "goal": "IsIntegral R (AdjoinRoot.root g)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "X Y : Set α", "hX : X ⊆ M.E"], "goal": "X \\ Y ⊆ M.E"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type u_4", "Fintype m", "Fintype n", "DecidableEq m", "DecidableEq n", "CommRing α", "A : Matrix m n α", "B : Matrix n m α"], "goal": "(1 - A * B).det = (1 - B * A).det"}}
+{"state": {"context": ["a b : ℍ", "h : ↑a = ↑b"], "goal": "a = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "s s₁ s₂ : LowerSet α", "t t₁ t₂ : LowerSet β", "x : α × β"], "goal": "Disjoint (s₁ ×ˢ t₁) (s₂ ×ˢ t₂) ↔ Disjoint s₁ s₂ ∨ Disjoint t₁ t₂"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "Preorder α", "DecidableEq α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (Function.swap HMul.hMul) LE.le", "LocallyFiniteOrder α", "a b c d : α"], "goal": "Finset.Icc a b * Finset.Icc c d ⊆ Finset.Icc (a * c) (b * d)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s : Set (Set α)", "h : DirectedOn (fun x x_1 => x ⊆ x_1) s"], "goal": "(⋃₀ s).Pairwise r ↔ ∀ a ∈ s, a.Pairwise r"}}
+{"state": {"context": ["A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "L : CategoryTheory.Functor A T", "R : CategoryTheory.Functor B T", "X Y : CategoryTheory.Comma L R", "self : CategoryTheory.CommaMorphism X Y"], "goal": "CategoryTheory.CategoryStruct.comp (L.map self.left) Y.hom = CategoryTheory.CategoryStruct.comp X.hom (R.map self.right)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s s₁ s₂ t t₁ t₂ u v : Finset α", "a b : α"], "goal": "a ∉ s ∪ t ↔ a ∉ s ∧ a ∉ t"}}
+{"state": {"context": ["α : Type u_3", "SemilatticeSup α", "a : α"], "goal": "Filter.atTop = Filter.comap Subtype.val Filter.atTop"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "a : ℍ[R]"], "goal": "(-a).imI = -a.imI"}}
+{"state": {"context": ["z : ℍ"], "goal": "0 < (-↑z)⁻¹.im"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "Zero M", "g : α →₀ M"], "goal": "Finsupp.mapRange id ⋯ g = g"}}
+{"state": {"context": ["G : Type u", "α : Type v", "β : Type w", "inst✝² : Group G", "inst✝¹ : Finite G", "p : ℕ", "inst✝ : Fact (Nat.Prime p)", "P : Sylow p G", "hdvd : p ^ 1 ∣ Nat.card G"], "goal": "p ∣ Nat.card ↥↑P"}}
+{"state": {"context": ["a o o' : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{u}", "hf : a.IsFundamentalSequence o f", "g : (b : Ordinal.{u}) → b < o' → Ordinal.{u}", "hg : o.IsFundamentalSequence o' g"], "goal": "a.IsFundamentalSequence o' fun i hi => f (g i hi) ⋯"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "P : CategoryTheory.Idempotents.Karoubi C"], "goal": "P.decompId_i.f = P.p"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "P₂ : Type u_3", "V₁ : Type u_6", "V₂ : Type u_7", "Ring k", "AddCommGroup V₁", "Module k V₁", "AddTorsor V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AddTorsor V₂ P₂", "TopologicalSpace P₁", "TopologicalSpace P₂", "e : P₁ ≃ᵃL[k] P₂", "s : Set P₂"], "goal": "⇑e '' (⇑e ⁻¹' s) = s"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p✝ : FormalMultilinearSeries 𝕜 E F", "r✝ : ℝ≥0", "p : FormalMultilinearSeries 𝕜 E F", "h : 0 < p.radius", "r : ℝ≥0", "r0 : 0 < ↑r", "rlt : ↑r < p.radius", "rpos : 0 < ↑r", "C : ℝ", "Cpos : C > 0", "hCp : ∀ (n : ℕ), ‖p n‖ ≤ C / ↑r ^ n", "n : ℕ"], "goal": "‖p n‖ ≤ C / ↑r ^ n"}}
+{"state": {"context": ["α : Type u", "s : Computation α"], "goal": "s.think.Equiv s"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "s : Set (ι → ℝ)", "hs : IsUpperSet s"], "goal": "MeasureTheory.volume (frontier s) = 0"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "I : C", "F : Cᵒᵖ ⥤ Type w", "hF : IsSheafFor F (ofArrows Empty.elim fun a => instIsEmptyEmpty.elim a)", "hI : IsInitial I", "α : Type", "X : α → C", "c : Cofan X", "hc : IsColimit c", "inst✝³ : (ofArrows X c.inj).hasPullbacks", "inst✝² : HasInitial C", "inst✝¹ : ∀ (i : α), Mono (c.inj i)", "hd : Pairwise fun i j => IsPullback (initial.to (X i)) (initial.to (X j)) (c.inj i) (c.inj j)", "inst✝ : PreservesLimit (Discrete.functor fun x => op (X x)) F", "this : HasCoproduct X", "hi : IsIso (piComparison F fun x => op (X x))"], "goal": "∀ (y : Equalizer.Presieve.Arrows.FirstObj F X), Equalizer.Presieve.Arrows.firstMap F X c.inj y = Equalizer.Presieve.Arrows.secondMap F X c.inj y → ∃! x, Equalizer.Presieve.Arrows.forkMap F X c.inj x = y"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "inst✝³ : Field A", "inst✝² : Ring B", "inst✝¹ : Algebra A B", "x : B", "inst✝ : Nontrivial B", "p : A[X]", "hp1 : Irreducible p", "hp2 : (Polynomial.aeval x) p = 0"], "goal": "p * C p.leadingCoeff⁻¹ = minpoly A x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x✝ x : Fin n → ℕ", "hx : x ∈ sphere n d k"], "goal": "‖(WithLp.equiv 2 (Fin n → ℝ)).symm (Nat.cast ∘ x)‖ = √↑k"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X"], "goal": "T1Space X ↔ ∀ ⦃x y : X⦄, x ≠ y → Disjoint (𝓝 x) (pure y)"}}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re", "n : ℕ", "hn : n ≠ 0", "this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n", "this : ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) = ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)", "x : ℝ", "hx : x ∈ Ioc 0 1", "hn' : ↑n ≠ 0", "A : ↑n ^ s = ↑n ^ (s - 1) * ↑n"], "goal": "↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) = ↑n * ((1 - ↑x) ^ n * (↑x * ↑n) ^ (s - 1))"}}
+{"state": {"context": ["α : Type u_1", "Add α", "LE α", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "a b c : α", "bc : a + b ≤ a + c"], "goal": "b ≤ c"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α"], "goal": "LipschitzWith 1 fun x => Metric.infDist x s"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "I : Ideal R", "M : Type u_2", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "N : Type u_3", "inst✝² : AddCommGroup N", "inst✝¹ : Module R N", "inst✝ : Module.Finite R M", "n : ℕ", "p : (Fin n → R) →ₗ[R] M", "hp : Function.Surjective ⇑p", "f : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M := ofTensorProduct I M ∘ₗ LinearMap.baseChange (AdicCompletion I R) p", "g : AdicCompletion I R ⊗[R] (Fin n → R) →ₗ[AdicCompletion I R] AdicCompletion I M := map I p ∘ₗ ofTensorProduct I (Fin n → R)", "hfg : f = g", "hf : Function.Surjective ⇑f"], "goal": "Function.Surjective ⇑(ofTensorProduct I M)"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝¹ : Ring R", "inst✝ : Semiring S", "f : R →+* S", "x : S", "p : R[X]", "T : Subring R", "hp : ↑p.coeffs ⊆ ↑T"], "goal": "(p.toSubring T hp).Monic ↔ p.Monic"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "LinearOrderedRing α", "LinearOrderedAddCommGroup β", "Module α β", "OrderedSMul α β", "σ : Equiv.Perm ι", "f : ι → α", "g : ι → β", "Fintype ι", "hfg : Monovary f g"], "goal": "∑ i : ι, f i • g (σ i) < ∑ i : ι, f i • g i ↔ ¬Monovary f (g ∘ ⇑σ)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "n : ℕ", "hn : Even n", "a : α"], "goal": "|a| ^ n = a ^ n"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "β : Sort u_1", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "X Y : C", "f : X ⟶ Y"], "goal": "f = eqToHom ⋯ ≫ f ≫ eqToHom ⋯"}}
+{"state": {"context": ["ι : Type u_1", "LinearOrder ι", "SuccOrder ι", "IsSuccArchimedean ι", "PredOrder ι", "i0 i j : ι", "h_le : toZ i0 i ≤ toZ i0 j"], "goal": "i ≤ j"}}
+{"state": {"context": ["n k k' : ℕ", "e : k + 2 = k'", "h : MinFacHelper n k", "np : n.minFac ≠ k", "h2 : k < n.minFac"], "goal": "k + 2 ≤ n.minFac"}}
+{"state": {"context": ["R : Type u", "A : Type z", "CommSemiring R", "Semiring A", "Algebra R A"], "goal": "∀ (a : A), Polynomial.CAlgHom a = Polynomial.C a"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x y : X", "γ : Path x y"], "goal": "γ 1 = y"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "z w : K"], "goal": "z < w ↔ RCLike.re z < RCLike.re w ∧ RCLike.im z = RCLike.im w"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "MeasurableSpace β", "self : ProbabilityTheory.Kernel α β"], "goal": "Measurable self.toFun"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "DivisionRing k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : Set P", "h : Collinear k s", "p₁ p₂ : P", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "hp₁p₂ : p₁ ≠ p₂"], "goal": "affineSpan k {p₁, p₂} = affineSpan k s"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "N : Type u_5", "AddGroup N", "H : AddSubgroup G", "f : G →+ N", "hf : Function.Bijective ⇑f"], "goal": "AddSubgroup.map f H.normalizer = (AddSubgroup.map f H).normalizer"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝⁹ : NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "G : Type u_5", "H : Type u_6", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedSpace 𝕜 G", "inst✝² : NormedAddCommGroup H", "inst✝¹ : NormedSpace 𝕜 H", "inst✝ : CompleteSpace F", "f g : E → F", "s : Set E", "x : E", "hf : AnalyticWithinAt 𝕜 f s x", "hs : f =ᶠ[𝓝[s] x] g", "hx : f x = g x"], "goal": "AnalyticWithinAt 𝕜 g s x"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "α : Type u_2", "MulAction M α", "m : M", "hm : IsUnit m"], "goal": "Function.Bijective fun a => m • a"}}
+{"state": {"context": ["α✝ : Type u_1", "inst✝¹ : MeasurableSpace α✝", "μ✝ : Measure α✝", "α : Type u_2", "inst✝ : MeasurableSpace α", "μ : Measure α", "f g : α → ℝ≥0∞", "hf : AEMeasurable f μ", "hg : AEMeasurable g μ", "p q : ℝ", "hp✝ : 0 ≤ p", "hq✝ : 0 ≤ q", "hpq : p + q = 1", "hp : 0 < p", "hq : 0 < q", "h2p : 1 < 1 / p"], "goal": "∫⁻ (a : α), f a ^ p * g a ^ q ∂μ ≤ (∫⁻ (a : α), f a ∂μ) ^ p * (∫⁻ (a : α), g a ∂μ) ^ q"}}
+{"state": {"context": ["h : Summable ({p | Nat.Prime p}.indicator fun n => 1 / ↑n)", "k : ℕ", "hk : ∑' (x : ℕ), ({p | Nat.Prime p} ∩ {p | k ≤ p}).indicator (fun n => 1 / ↑n) x < 1 / 2"], "goal": "False"}}
+{"state": {"context": ["α : Type u_2", "Div α", "s t u : Set α"], "goal": "s / t ⊆ u ↔ ∀ x ∈ s, ∀ y ∈ t, x / y ∈ u"}}
+{"state": {"context": ["p : ℕ", "hp : Prime p", "n : ℕ"], "goal": "∀ x < p ^ (n + 1), ¬p ∣ x ↔ ∀ (x_1 : ℕ), ¬(x_1 < p ^ n ∧ x_1 * p = x)"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Preadditive C", "CategoryTheory.Preadditive D", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroObject D", "CategoryTheory.Limits.HasBinaryBiproducts C", "F : C ⥤ₗ D"], "goal": "((CategoryTheory.AdditiveFunctor.ofLeftExact C D).obj F).obj = F.obj"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "M : Type w", "M' : Type w₁", "inst✝¹² : CommRing R", "inst✝¹¹ : LieRing L", "inst✝¹⁰ : LieAlgebra R L", "inst✝⁹ : LieRing L'", "inst✝⁸ : LieAlgebra R L'", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M'", "inst✝² : Module R M'", "inst✝¹ : LieRingModule L M'", "inst✝ : LieModule R L M'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "h₁ : Function.Surjective ⇑f", "x : L", "hx : x ∈ ↑I"], "goal": "f ↑⟨x, hx⟩ = ↑f x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "s : Set α", "t : Set β", "f : Filter α", "g g₁ g₂ : Filter β"], "goal": "f ×ˢ (g₁ ⊓ g₂) = f ×ˢ g₁ ⊓ f ×ˢ g₂"}}
+{"state": {"context": ["α : Type u_1"], "goal": "ofInt' ↑0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Multiset α", "t : Multiset β", "a : α"], "goal": "Sum.inl a ∈ s.disjSum t ↔ a ∈ s"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "Polynomial.Monic 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁷ : TopologicalSpace α", "inst✝⁶ : NormalSpace α", "inst✝⁵ : R1Space α", "inst✝⁴ : MeasurableSpace α", "inst✝³ : BorelSpace α", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "μ : Measure α", "p : ℝ≥0∞", "inst✝¹ : NormedSpace ℝ E", "inst✝ : μ.WeaklyRegular", "hp : p ≠ ⊤", "f : α → E", "hf : Memℒp f p μ", "ε : ℝ≥0∞", "hε : ε ≠ 0"], "goal": "∀ (f g : α → E), Continuous f ∧ Memℒp f p μ ∧ IsBounded (range f) → Continuous g ∧ Memℒp g p μ ∧ IsBounded (range g) → Continuous (f + g) ∧ Memℒp (f + g) p μ ∧ IsBounded (range (f + g))"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "H : IsOpenImmersion f", "x : (Opens ↑↑Y)ᵒᵖ"], "goal": "(((isoRestrict f).hom ≫ Y.ofRestrict ⋯).c ≫ whiskerRight (eqToHom ⋯) X.presheaf).app x = f.c.app x"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "s : Set α", "ι : Type u_3", "f : ι → α → ℝ≥0∞", "h : ∀ (i : ι), LowerSemicontinuousOn (f i) s"], "goal": "LowerSemicontinuousOn (fun x' => ∑' (i : ι), f i x') s"}}
+{"state": {"context": ["K : Type u", "CommRing K", "IsDomain K", "p : K[X]"], "goal": "RatFunc.mk p 0 = { toFractionRing := 0 }"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Preadditive C", "ι : Type u_2", "c : ComplexShape ι", "F G K : HomologicalComplex C c", "φ : F ⟶ G", "inst✝¹ : HasHomotopyCofiber φ", "inst✝ : DecidableRel c.Rel", "i j : ι", "hij : c.Rel i j", "A : C", "f g : A ⟶ X φ i", "h₁ : f ≫ fstX φ i j hij = g ≫ fstX φ i j hij", "h₂ : f ≫ sndX φ i = g ≫ sndX φ i"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α", "C : ℝ", "h : ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ C"], "goal": "EMetric.diam s ≤ ENNReal.ofReal C"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{?u.121680, u_1} C", "inst✝² : HasZeroMorphisms C", "S S₁ S₂ S₃ : ShortComplex C", "e : S₁ ≅ S₂", "inst✝¹ : S₁.HasLeftHomology", "inst✝ : S₂.HasLeftHomology"], "goal": "cyclesMap e.hom ≫ cyclesMap e.inv = 𝟙 S₁.cycles"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{?u.117978, u_1} C", "inst✝² : HasZeroMorphisms C", "S S₁ S₂ S₃ : ShortComplex C", "e : S₁ ≅ S₂", "inst✝¹ : S₁.HasLeftHomology", "inst✝ : S₂.HasLeftHomology"], "goal": "leftHomologyMap e.hom ≫ leftHomologyMap e.inv = 𝟙 S₁.leftHomology"}}
+{"state": {"context": ["ι : Type u", "X : Type v", "TopologicalSpace X", "s : Set X", "NormalSpace X", "ParacompactSpace X", "p : (X → ℝ) → Prop", "h01 :\n  ∀ (s t : Set X),\n    IsClosed s →\n      IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1", "hs : IsClosed s", "U : ι → Set X", "ho : ∀ (i : ι), IsOpen (U i)", "hU : s ⊆ ⋃ i, U i"], "goal": "∃ f, (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "s : MeasureTheory.SignedMeasure α", "μ : MeasureTheory.Measure α", "MeasureTheory.SigmaFinite μ"], "goal": "s ≪ᵥ μ.toENNRealVectorMeasure ↔ μ.withDensityᵥ (s.rnDeriv μ) = s"}}
+{"state": {"context": ["n : ℕ"], "goal": "↑(n + 1) = ↑n + 1"}}
+{"state": {"context": ["G : Type u_3", "Mul G", "IsRightCancelMul G", "a : G", "b c : G"], "goal": "b * a ≠ c * a ↔ b ≠ c"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "ι : Type u_1", "f : ι → R[X]", "s✝ : Finset ι", "a : ι", "s : Finset ι", "has : a ∉ s", "ih : (∀ x ∈ s, ∀ y ∈ s, x ≠ y → IsCoprime (f x) (f y)) → (∀ x ∈ s, (f x).Separable) → (∏ x ∈ s, f x).Separable", "h1 : ∀ x ∈ insert a s, ∀ y ∈ insert a s, x ≠ y → IsCoprime (f x) (f y)", "h2 : ∀ x ∈ insert a s, (f x).Separable"], "goal": "(∏ x ∈ insert a s, f x).Separable"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M", "x : M"], "goal": "Filter.map (⇑DomAddAct.mk) (𝓝 x) = 𝓝 (DomAddAct.mk x)"}}
+{"state": {"context": ["R : Type u", "inst✝⁵ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "T : Type u_1", "inst✝² : CommSemiring T", "inst✝¹ : Algebra R T", "inst✝ : IsLocalization S T", "m : M", "s : ↥S"], "goal": "IsLocalization.mk' T 1 1 • mk m s = mk m s"}}
+{"state": {"context": ["α : Type u", "a : α", "l m : List α", "ainm : a ∈ m", "lsubm : l ⊆ m"], "goal": "a :: l ⊆ m"}}
+{"state": {"context": ["M : Type u_1", "B : Type u_3", "AddMonoid M", "SetLike B M", "AddSubmonoidClass B M", "S : B", "l : List M", "hl : ∀ x ∈ l, x ∈ S"], "goal": "l.sum ∈ S"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "f : ι → α", "s t : Set α"], "goal": "Sum.inr ⁻¹' range Sum.inl = ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "x : E", "ι : Sort u_4", "s : ι → Set E", "hs : ∀ (i : ι), StarConvex 𝕜 x (s i)"], "goal": "StarConvex 𝕜 x (⋃ i, s i)"}}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁶ : CommRing A", "inst✝⁵ : CommRing B", "inst✝⁴ : Algebra A B", "inst✝³ : IsCyclotomicExtension {n} A B", "inst✝² : Field L", "ζ : L", "hζ✝ : IsPrimitiveRoot ζ ↑n", "inst✝¹ : Field K", "inst✝ : Algebra K L", "k : ℕ", "hζ : IsPrimitiveRoot ζ (2 ^ k)", "hk : 2 ≤ k", "H : IsCyclotomicExtension {2 ^ k} K L", "hirr : Irreducible (cyclotomic (2 ^ k) K)", "this : 2 < 2 ^ k"], "goal": "(Algebra.norm K) (ζ - 1) = 2"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_3", "CommRing R", "Invertible 2", "AddCommGroup V", "Module R V", "AddTorsor V P", "a b : P"], "goal": "(AffineMap.homothety a ⅟2) b = midpoint R a b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "n : ℕ", "hn : 0 < n"], "goal": "↑n ! / ↑n ^ n ≤ 1 / ↑n"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : LocalRing R", "inst✝ : IsNoetherianRing R", "s : Set ↥(maximalIdeal R)"], "goal": "Submodule.span (ResidueField R) (⇑(maximalIdeal R).toCotangent '' s) = ⊤ ↔ Submodule.span R s = ⊤"}}
+{"state": {"context": ["α : Type u_2", "CoheytingAlgebra α", "a b : α", "h : a ≤ b"], "goal": "Codisjoint (￢a) b"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : ℕ → α → ℝ≥0∞", "hf : ∀ (n : ℕ), Measurable (f n)", "h_mono : Monotone f"], "goal": "∫⁻ (a : α), ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ"}}
+{"state": {"context": ["α : Type u_1", "l : List α"], "goal": "l.dropLast.Sublist l"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "r : ℝ"], "goal": "↑r⁻¹ = (↑r)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "F : ℕ → α → β", "f : α → β", "bound : α → ℝ", "F_measurable : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (F n) μ", "bound_hasFiniteIntegral : MeasureTheory.HasFiniteIntegral bound μ", "h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a", "h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun n => F n a) Filter.atTop (𝓝 (f a))"], "goal": "Filter.Tendsto (fun n => ∫⁻ (a : α), ENNReal.ofReal ‖F n a - f a‖ ∂μ) Filter.atTop (𝓝 0)"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝²⁶ : NormedAddCommGroup E", "inst✝²⁵ : NormedAddCommGroup E'", "inst✝²⁴ : NormedAddCommGroup E''", "inst✝²³ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x x' : G", "y y' : E", "inst✝²² : RCLike 𝕜", "inst✝²¹ : NormedSpace 𝕜 E", "inst✝²⁰ : NormedSpace 𝕜 E'", "inst✝¹⁹ : NormedSpace 𝕜 E''", "inst✝¹⁸ : NormedSpace ℝ F", "inst✝¹⁷ : NormedSpace 𝕜 F", "n : ℕ∞", "inst✝¹⁶ : CompleteSpace F", "inst✝¹⁵ : MeasurableSpace G", "μ ν : Measure G", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝¹⁴ : NormedAddCommGroup F'", "inst✝¹³ : NormedSpace ℝ F'", "inst✝¹² : NormedSpace 𝕜 F'", "inst✝¹¹ : CompleteSpace F'", "inst✝¹⁰ : NormedAddCommGroup F''", "inst✝⁹ : NormedSpace ℝ F''", "inst✝⁸ : NormedSpace 𝕜 F''", "inst✝⁷ : CompleteSpace F''", "k : G → E''", "L₂ : F →L[𝕜] E'' →L[𝕜] F'", "L₃ : E →L[𝕜] F'' →L[𝕜] F'", "L₄ : E' →L[𝕜] E'' →L[𝕜] F''", "inst✝⁶ : AddGroup G", "inst✝⁵ : SFinite μ", "inst✝⁴ : SFinite ν", "inst✝³ : μ.IsAddRightInvariant", "inst✝² : MeasurableAdd₂ G", "inst✝¹ : ν.IsAddRightInvariant", "inst✝ : MeasurableNeg G", "hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)", "x₀ : G", "hf : AEStronglyMeasurable f ν", "hg : AEStronglyMeasurable g μ", "hk : AEStronglyMeasurable k μ", "hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν", "hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (mul ℝ ℝ) μ", "hfgk : ConvolutionExistsAt (fun x => ‖f x‖) ((fun x => ‖g x‖) ⋆[mul ℝ ℝ, μ] fun x => ‖k x‖) x₀ (mul ℝ ℝ) ν"], "goal": "((f ⋆[L, ν] g) ⋆[L₂, μ] k) x₀ = (f ⋆[L₃, ν] g ⋆[L₄, μ] k) x₀"}}
+{"state": {"context": ["M : Type u_1", "P : Type u_3", "AddZeroClass M", "AddZeroClass P", "c : AddCon M", "f : M →+ P", "H : c ≤ AddCon.ker f"], "goal": "AddMonoidHom.mrange (c.lift f H) = AddMonoidHom.mrange f"}}
+{"state": {"context": ["n k k' : ℕ", "e : k + 2 = k'", "h : Mathlib.Meta.NormNum.MinFacHelper n k", "np : n.minFac ≠ k"], "goal": "Mathlib.Meta.NormNum.MinFacHelper n k'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "a b c d : ℝ≥0∞", "r p q : ℝ≥0", "x y z ε ε₁ ε₂ : ℝ≥0∞", "s : Set ℝ≥0∞", "ι : Sort u_4", "f g : ι → ℝ≥0∞", "h : ∀ (i j : ι), ∃ k, f i + g j ≤ f k + g k", "h✝ : IsEmpty ι"], "goal": "iSup f + iSup g = ⨆ a, f a + g a"}}
+{"state": {"context": [], "goal": "Nat.Primrec Nat.pred"}}
+{"state": {"context": ["Ω : Type u_1", "Ω' : Type u_2", "inst✝⁵ : MeasurableSpace Ω", "inst✝⁴ : MeasurableSpace Ω'", "inst✝³ : TopologicalSpace Ω", "inst✝² : OpensMeasurableSpace Ω", "inst✝¹ : TopologicalSpace Ω'", "inst✝ : BorelSpace Ω'", "f : Ω → Ω'", "f_cont : Continuous f"], "goal": "Continuous fun ν => ν.map ⋯"}}
+{"state": {"context": ["a : ℤ"], "goal": "Ideal.span {↑a.natAbs} = Ideal.span {a}"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "Module R (Additive ℤˣ)", "s : ℤˣ", "x y : R"], "goal": "s ^ (x - y) = s ^ x / s ^ y"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Balanced C", "G : C"], "goal": "CategoryTheory.IsSeparator G → CategoryTheory.IsDetector G"}}
+{"state": {"context": ["F : Type u_1", "α✝ : Type u_2", "β✝ : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝² : SemilatticeSup α✝", "inst✝¹ : OrderBot α✝", "s✝ s₁ s₂ : Finset β✝", "f✝ g : β✝ → α✝", "a✝ : α✝", "α : Type u_7", "β : Type u_8", "inst✝ : GeneralizedBooleanAlgebra α", "s : Finset β", "f : β → α", "a : α"], "goal": "(s.sup fun b => f b \\ a) = s.sup f \\ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_4", "PseudoEMetricSpace α", "PseudoEMetricSpace β", "FunLike F α β", "DilationClass F α β", "f : F"], "goal": "UniformInducing ⇑f"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "ι' : Type v'", "E : ι → Type wE", "E₁ : ι → Type wE₁", "E' : ι' → Type wE'", "G : Type wG", "G' : Type wG'", "inst✝¹⁷ : Fintype ι", "inst✝¹⁶ : Fintype ι'", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "inst✝¹⁴ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝¹³ : (i : ι) → NormedSpace 𝕜 (E i)", "inst✝¹² : (i : ι) → SeminormedAddCommGroup (E₁ i)", "inst✝¹¹ : (i : ι) → NormedSpace 𝕜 (E₁ i)", "inst✝¹⁰ : (i : ι') → SeminormedAddCommGroup (E' i)", "inst✝⁹ : (i : ι') → NormedSpace 𝕜 (E' i)", "inst✝⁸ : SeminormedAddCommGroup G", "inst✝⁷ : NormedSpace 𝕜 G", "inst✝⁶ : SeminormedAddCommGroup G'", "inst✝⁵ : NormedSpace 𝕜 G'", "c : 𝕜", "f✝ g : ContinuousMultilinearMap 𝕜 E G", "m : (i : ι) → E i", "𝕜' : Type u_1", "inst✝⁴ : NormedField 𝕜'", "inst✝³ : NormedSpace 𝕜' G", "inst✝² : SMulCommClass 𝕜 𝕜' G", "inst✝¹ : (i' : ι') → SeminormedAddCommGroup (E' i')", "inst✝ : (i' : ι') → NormedSpace 𝕜 (E' i')", "f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')", "x✝ : ℝ≥0"], "goal": "‖pi f‖₊ ≤ x✝ ↔ ‖f‖₊ ≤ x✝"}}
+{"state": {"context": ["R₁ : Type u_1", "Semiring R₁", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "Module R₁ M₁", "ContinuousAdd M₁", "self : M₁ →L[R₁] M₁"], "goal": "ContinuousLinearMap.toLinearMapRingHom self = ↑self"}}
+{"state": {"context": ["ι : Type u", "α : ι → Type v", "i✝ j : ι", "l : List ι", "f : (i : ι) → α i", "inst✝ : DecidableEq ι", "i : ι", "is : List ι", "hl : (i :: is).Nodup", "v w : TProd α (i :: is)", "hvw : ∀ (i_1 : ι) (hi : i_1 ∈ i :: is), v.elim hi = w.elim hi"], "goal": "v.2 = w.2"}}
+{"state": {"context": ["V : Type (u + 1)", "inst✝² : LargeCategory V", "G : MonCat", "inst✝¹ : MonoidalCategory V", "H : Grp", "X : Action V ((forget₂ Grp MonCat).obj H)", "inst✝ : LeftRigidCategory V", "h : ↑H"], "goal": "(ᘁX).ρ h = ᘁX.ρ h⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Finite β", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "σ τ : Perm α", "h0 : Nat.Prime (Fintype.card α)", "h1 : σ.IsCycle", "h2 : σ.support = univ", "x y : α", "h4 : x ≠ y", "h5 : τ = swap x y", "i : ℕ", "hi : (σ ^ i) x = y"], "goal": "closure {σ, swap x ((σ ^ i) x)} = ⊤"}}
+{"state": {"context": ["m✝ n✝ m : ℕ∞", "n : ℕ", "h : m ≤ ↑n"], "goal": "m.toNat ≤ n"}}
+{"state": {"context": ["F : Type u_3", "R : Type u_4", "S : Type u_5", "Star R", "Star S", "FunLike F R S", "StarHomClass F R S", "x : R", "hx : IsSelfAdjoint x", "f : F"], "goal": "IsSelfAdjoint (f x)"}}
+{"state": {"context": ["n d : ℤ", "h : n ≠ 0", "hd : d ≠ 0"], "goal": "Rat.divInt n d ≠ 0"}}
+{"state": {"context": ["a : Int", "b : Int", "hb : 0 < b"], "goal": "0 ≤ a.fmod b"}}
+{"state": {"context": ["α : Type u_7", "β : Type u_8", "t₁ t₂ : TopologicalSpace α", "t₃ : TopologicalSpace β", "h₁ : t₂ ≤ t₁", "s : Set α", "f : α → β", "h₂ : ContinuousOn f s"], "goal": "ContinuousOn f s"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "CategoryTheory.Bicategory.HasLeftKanExtension f g", "x : B", "h : c ⟶ x", "CategoryTheory.Bicategory.HasLeftKanExtension f (CategoryTheory.CategoryStruct.comp g h)", "i :\n  CategoryTheory.Bicategory.lan f (CategoryTheory.CategoryStruct.comp g h) ≅\n    CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.lan f g) h", "w :\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.lanUnit f (CategoryTheory.CategoryStruct.comp g h))\n      (CategoryTheory.Bicategory.whiskerLeft f i.hom) =\n    CategoryTheory.CategoryStruct.comp\n      (CategoryTheory.Bicategory.whiskerRight (CategoryTheory.Bicategory.lanUnit f g) h)\n      (CategoryTheory.Bicategory.associator f (CategoryTheory.Bicategory.lan f g) h).hom"], "goal": "CategoryTheory.Bicategory.Lan.CommuteWith f g h"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommSemiring R", "σ : Type u_3", "inst✝¹ : AddCommMonoid M", "w : σ → M", "n✝ : M", "φ ψ : MvPolynomial σ R", "inst✝ : DecidableEq M", "m n : M", "p : MvPolynomial σ R", "h : IsWeightedHomogeneous w p n", "x : σ →₀ ℕ"], "goal": "coeff x ((weightedHomogeneousComponent w m) p) = coeff x (if m = n then p else 0)"}}
+{"state": {"context": ["C : Type u", "inst✝¹² : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w₁", "inst✝¹¹ : Category.{max v u, w₁} D", "E : Type w₂", "inst✝¹⁰ : Category.{max v u, w₂} E", "F✝ : D ⥤ E", "inst✝⁹ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) D", "inst✝⁸ : ∀ (α β : Type (max v u)) (fst snd : β → α), HasLimitsOfShape (WalkingMulticospan fst snd) E", "inst✝⁷ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F✝", "P✝ : Cᵒᵖ ⥤ D", "inst✝⁶ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝⁵ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ E", "inst✝⁴ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F✝", "F G : D ⥤ E", "η : F ⟶ G", "P : Cᵒᵖ ⥤ D", "inst✝³ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ F", "inst✝² : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan F", "inst✝¹ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ G", "inst✝ : (X : C) → (W : J.Cover X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (W.index P).multicospan G", "X : Cᵒᵖ", "W : (J.Cover (unop X))ᵒᵖ"], "goal": "η.app (multiequalizer ((unop W).index P)) ≫ (J.diagramCompIso G P (unop X)).hom.app W ≫ colimit.ι (J.diagram (P ⋙ G) (unop X)) W = (J.diagramCompIso F P (unop X)).hom.app W ≫ Multiequalizer.lift ((unop W).index (P ⋙ G)) ((J.diagram (P ⋙ F) (unop X)).obj W) (fun i => Multiequalizer.ι ((unop W).index (P ⋙ F)) i ≫ η.app (P.obj (op i.Y))) ⋯ ≫ colimit.ι (J.diagram (P ⋙ G) (unop X)) W"}}
+{"state": {"context": ["n : ℕ", "c : Composition n", "j : Fin n", "i : Fin c.length := c.index j", "H : ↑j < c.sizeUpTo ↑i", "i_pos : 0 < ↑i", "i₁ : ℕ := (↑i).pred"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "x y : α", "n : ℕ", "hx : Function.IsPeriodicPt f n x", "hy : Function.IsPeriodicPt f n y", "hn : 0 < n", "h : f x = f y"], "goal": "x = y"}}
+{"state": {"context": ["n k : ℕ", "hk : k ≠ 0"], "goal": "(1 ^ k).minFac = minFac 1"}}
+{"state": {"context": ["ι : Type u", "α : ι → Type v", "i j : ι", "l : List ι", "DecidableEq ι", "hl : (i :: l).Nodup", "hj : j ∈ l", "v : List.TProd α (i :: l)"], "goal": "v.elim ⋯ = List.TProd.elim v.2 hj"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α"], "goal": "Fintype.card { a // a.IsDiag } = Fintype.card α"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "R : Type x", "inst✝ : NonUnitalNonAssocRing α", "a b c : α"], "goal": "(a - b) * c = a * c - b * c"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C"], "goal": "(CategoryTheory.MonoidalOpposite.mopEquiv C).counitIso =\n  CategoryTheory.Iso.refl (CategoryTheory.unmopFunctor C ⋙ CategoryTheory.mopFunctor C)"}}
+{"state": {"context": ["β : Type u_1", "G : Type u_2", "M : Type u_3", "inst✝² : Monoid M", "inst✝¹ : Preorder M", "inst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "x a : M", "ha : 1 < a", "l : ℕ", "hk : l.succ ≠ 0"], "goal": "1 < a ^ l.succ"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "v : Mathlib.Vector α n.succ"], "goal": "↑v.tail = (↑v).tail"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "E : Type u_6", "MeasurableSpace α", "MeasurableSpace β", "ν : MeasureTheory.Measure β", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasureTheory.SFinite ν", "f : α → β → E", "hf : MeasureTheory.StronglyMeasurable (Function.uncurry f)"], "goal": "MeasureTheory.StronglyMeasurable fun x => ∫ (y : β), f x y ∂ν"}}
+{"state": {"context": ["b : ℝ", "hb : 0 < b", "s : ℝ", "hs : -1 < s"], "goal": "IntegrableOn (fun x => x ^ s * rexp (-b * x ^ 2)) (Ioi 0) volume"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝⁴ : TopologicalSpace X", "inst✝³ : TopologicalSpace Y", "inst✝² : TopologicalSpace Z", "inst✝¹ : (i : ι) → TopologicalSpace (π i)", "x y z : X", "s : Set X", "f g : X → Y", "inst✝ : DecidablePred fun x => x ∈ s", "hs : IsOpen s", "hf : Continuous f", "hg : Continuous g", "hspec : ∀ (x : X), f x ⤳ g x", "this : ∀ (U : Set Y), IsOpen U → g ⁻¹' U ⊆ f ⁻¹' U", "U : Set Y", "hU : IsOpen U"], "goal": "IsOpen (s.piecewise f g ⁻¹' U)"}}
+{"state": {"context": ["α : Sort u_1", "h✝ : Nonempty α"], "goal": "⨆ x, 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝² : CommSemiring S", "inst✝¹ : Algebra R S", "P : Type u_3", "inst✝ : CommSemiring P", "a : R", "b : ↥M", "c : R"], "goal": "1 * (↑b * ↑b * (a + c)) = 1 * (↑b * (↑b * c + ↑b * a))"}}
+{"state": {"context": ["b : Int", "e : Nat"], "goal": "b ^ (e + 1) = b ^ e * b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "S : Set (Set α)", "h : S ⊆ {∅, univ}"], "goal": "(∃ x, ∃ (_ : x ∈ S), ¬x = ∅) → ⋃₀ S = univ"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedField F", "inst✝ : CompleteSpace F", "f : ℕ →*₀ F", "p : ℕ", "hp : Nat.Prime p", "hsum : Summable fun x => ‖f x‖"], "goal": "(1 - f p)⁻¹ = ∑' (e : ℕ), f (p ^ e)"}}
+{"state": {"context": ["α : Type u", "inst✝² : DecidableEq α", "β : Type v", "inst✝¹ : Fintype α", "f : Perm α", "p : α → Prop", "inst✝ : DecidablePred p", "h₁ : ∀ (x : α), p x ↔ p (f x)", "h₂ : ∀ (x : α), f x ≠ x → p x", "l : { l // l.prod = f.subtypePerm h₁ ∧ ∀ g ∈ l, g.IsSwap } := (f.subtypePerm h₁).truncSwapFactors.out", "hl' : ∀ g' ∈ List.map ⇑ofSubtype ↑l, g'.IsSwap"], "goal": "sign (f.subtypePerm h₁) = sign f"}}
+{"state": {"context": ["x : ℝ", "hx : 0 ≤ x"], "goal": "(↑x).arg = 0"}}
+{"state": {"context": ["f₁ f₂ : CircleDeg1Lift", "h₁ : Function.Bijective ⇑f₁", "h₂ : Function.Bijective ⇑f₂", "h : f₁.translationNumber = f₂.translationNumber"], "goal": "∃ F, Function.Semiconj ⇑F ⇑f₁ ⇑f₂"}}
+{"state": {"context": ["E : Type u_4", "F : Type u_5", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F", "k n : ℕ", "f : SchwartzMap E F", "x : E"], "goal": "‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β", "r : ℝ", "hr : 0 ≤ r", "H1 : ∀ x ∈ s, infDist x t ≤ r", "H2 : ∀ x ∈ t, infDist x s ≤ r", "h1 : ¬hausdorffEdist s t = ⊤", "hs : s = ∅"], "goal": "hausdorffDist s t ≤ r"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_4", "OrderedSemiring 𝕜", "AddCommMonoid E", "LE β", "SMul 𝕜 E", "s : Set E", "f : E → β", "hs : Convex 𝕜 s", "h : ∀ (r : β), Convex 𝕜 {x | f x ≤ r}"], "goal": "QuasiconvexOn 𝕜 s f"}}
+{"state": {"context": ["ι : Type uι", "E : Type uE", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : FiniteDimensional ℝ E", "F : Type uF", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ F", "H : Type uH", "inst✝⁵ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝⁴ : TopologicalSpace M", "inst✝³ : ChartedSpace H M", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "t : M → Set F", "n : ℕ∞", "s : Set H", "hs : IsOpen s", "h's : IsOpen (↑I.symm ⁻¹' s)"], "goal": "∃ f, support f = s ∧ Smooth I 𝓘(ℝ, ℝ) f ∧ range f ⊆ Icc 0 1"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "H : Type u_3", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "CategoryTheory.Category.{u_6, u_3} H", "L : CategoryTheory.Functor C D", "F : CategoryTheory.Functor C H", "E : L.LeftExtension F", "h : E.IsPointwiseLeftKanExtension", "G : L.LeftExtension F", "f₁ f₂ : E ⟶ G"], "goal": "f₁ = f₂"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : Set α", "OrderBot α", "hs : IsLowerSet s"], "goal": "⊥ ∉ s ↔ s = ∅"}}
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+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "SemilatticeSup α", "OrderBot α", "SemilatticeSup γ", "OrderBot γ", "s : Finset β", "f : β → α", "g : α → γ", "g_sup : ∀ (x y : α), g (x ⊔ y) = g x ⊔ g y", "bot : g ⊥ = ⊥"], "goal": "g (s.sup f) = s.sup (g ∘ f)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "Semiring A", "Algebra R A", "S : Subalgebra R A"], "goal": "(Subalgebra.toSubmodule S).toSubalgebra ⋯ ⋯ = S"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type w", "Nonempty Z", "MetricSpace Z", "MetricSpace X", "MetricSpace Y", "Φ : Z → X", "Ψ : Z → Y", "hΦ : Isometry Φ", "hΨ : Isometry Ψ"], "goal": "Isometry (Metric.toGlueL hΦ hΨ)"}}
+{"state": {"context": ["α : Type u_2", "OrderedCommMonoid α", "s : Multiset α"], "goal": "(∀ x ∈ s, 1 ≤ x) → ∀ x ∈ s, x ≤ s.prod"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "E : Type u_3", "E' : Type u_4", "F : Type u_5", "ι : Type u_6", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : Module 𝕜 E", "inst✝² : TopologicalSpace E", "inst✝¹ : ContinuousSMul 𝕜 E", "inst✝ : ContinuousAdd E", "s t : Set E", "hs : s.Nonempty", "ht : t.Nonempty"], "goal": "IsVonNBounded 𝕜 (s + t) ↔ s = ∅ ∨ t = ∅ ∨ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t"}}
+{"state": {"context": ["M : Type u_2", "Monoid M", "σ : M →* M", "self : MonoidHom.CompTriple.IsId σ"], "goal": "σ = MonoidHom.id M"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "F : J ⥤ C", "H H' : C ⥤ D", "α : H ≅ H'", "c : CategoryTheory.Limits.Cone F"], "goal": "(CategoryTheory.Functor.postcomposeWhiskerLeftMapCone α c).inv.hom = α.inv.app c.pt"}}
+{"state": {"context": ["α : Type u_1", "Group α", "H : Type u_2", "Group H", "K : Subgroup H", "f : α →* H", "hf : Function.Injective ⇑f"], "goal": "Nat.card ↥(Subgroup.comap f K) ∣ Nat.card ↥K"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → β → Prop"], "goal": "Relation.Comp (fun x x_1 => x = x_1) r = r"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : T2Space X", "inst✝ : CompactSpace X", "x : X"], "goal": "⋂ Z, ↑Z ⊆ connectedComponent x"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α", "PredOrder α", "IsPredArchimedean α", "NoMinOrder α", "h : Order.IsPredLimit a"], "goal": "IsMax a"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_3", "m : MeasurableSpace Ω", "Preorder ι", "f : MeasureTheory.Filtration ι m", "τ : Ω → ι", "hτ : MeasureTheory.IsStoppingTime f τ", "i : ι", "hτ_le : ∀ (ω : Ω), i ≤ τ ω"], "goal": "↑f i ≤ hτ.measurableSpace"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "A : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "p : Submodule R M", "s : Set M"], "goal": "⇑((TensorProduct.mk R A M) 1) '' ↑(span R s) ⊆ ↑(span A (⇑((TensorProduct.mk R A M) 1) '' s))"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "l✝ : Filter α", "f : α → β", "l : Filter β", "inst✝ : l.IsCountablyGenerated", "h : Tendsto f cofinite l"], "goal": "(f ⁻¹' l.ker)ᶜ.Countable"}}
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+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq β", "inst✝ : Monoid α", "s✝ t : Finset α", "a✝ : α", "m n : ℕ", "a : α", "s : Fin n → Finset α"], "goal": "(∃ f, (List.ofFn fun i => ↑(f i)).prod = a) ↔ ∃ f, (List.ofFn fun i => ↑(f i)).prod = a"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_3", "f : Filter α", "g : Filter γ", "p : α × γ × γ → Prop"], "goal": "(∀ᶠ (x : α × γ × γ) in f ×ˢ g ×ˢ g, p x) → ∀ᶠ (x : α × γ) in f ×ˢ g, p (x.1, x.2, x.2)"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : E → ℝ", "s : Set E", "x y : E", "f' : E → E →L[ℝ] ℝ", "hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x", "hs : Convex ℝ s", "xs : x ∈ s", "ys : y ∈ s", "g : ℝ → E := fun t => (AffineMap.lineMap x y) t", "I : Set ℝ := Icc 0 1", "hsub : Ioo 0 1 ⊆ I", "hmaps : MapsTo g I s", "hfg : ∀ t ∈ I, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) I t", "t : ℝ", "Ht : t ∈ Ioo 0 1", "hMVT' : (f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)"], "goal": "f y - f x = (f' ((AffineMap.lineMap x y) t)) (y - x)"}}
+{"state": {"context": ["R : Type u", "M : Type v", "AddGroupWithOne R", "AddGroup M", "z : ℤ"], "goal": "(↑z).snd = 0"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_7", "M₂ : Type u_10", "AddCommMonoid M₂", "Module R M₂", "M₃ : Type u_11", "AddCommMonoid M₃", "Module R M₃", "S : Type u_12", "Semiring S", "Module S N", "SMulCommClass R S N", "e : M ≃ₗ[R] M₂", "f : M₂ ≃ₗ[R] M₃"], "goal": "AlternatingMap.domLCongr R N ι S e ≪≫ₗ AlternatingMap.domLCongr R N ι S f = AlternatingMap.domLCongr R N ι S (e ≪≫ₗ f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f g : α → β", "s : Set α", "x : α", "h : f =ᶠ[𝓝[s] x] g", "hx : f x = g x"], "goal": "ContinuousWithinAt f s x ↔ ContinuousWithinAt g s x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x y z : E", "δ ε : ℝ", "hε : 0 ≤ ε", "hδ : 0 < δ", "a b : E"], "goal": "Metric.closedBall a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)"}}
+{"state": {"context": ["R : Type u_3", "Γ : Type u_5", "LinearOrderedCancelAddCommMonoid Γ", "NonUnitalNonAssocSemiring R", "NoZeroDivisors R", "x y : HahnSeries Γ R", "hx : x ≠ 0", "hy : y ≠ 0"], "goal": "(x * y).order = x.order + y.order"}}
+{"state": {"context": ["V₁ : Type u_2", "V₂ : Type u_3", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂"], "goal": "⇑0 = 0"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "N : Type w₁", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup N", "inst✝² : Module R N", "inst✝¹ : LieRingModule L N", "inst✝ : LieModule R L N", "x : L"], "goal": "x ∈ LieModule.ker R L M ↔ ∀ (m : M), ⁅x, m⁆ = 0"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "X Y : C", "p : ExactPairing X Y"], "goal": "(leftDualIso p p).hom = (Iso.refl X).hom"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Algebra F E", "K : Type w", "inst✝¹ : Field K", "inst✝ : Algebra F K", "x : F"], "goal": "(minpoly F x).Separable"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "Interval.dual (Interval.pure a) = Interval.pure (OrderDual.toDual a)"}}
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+{"state": {"context": ["α : Type u_1", "UniformSpace α", "β : Type u_2", "UniformSpace β", "f : α → β", "g : UniformSpace.Completion α → UniformSpace.Completion β", "hg : UniformContinuous g", "h : ∀ (a : α), ↑β (f a) = g (↑α a)"], "goal": "UniformSpace.Completion.map f = g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : PartialOrder α", "a : α", "inst✝ : IsStronglyAtomic α", "ha : (Set.Ici a).Infinite", "hfin : {x | a ⋖ x}.Finite", "h : ∀ (b : α), a ⋖ b → ¬(Set.Ici b).Infinite", "b : α", "hb : b ∈ Set.Ici a \\ {a}", "x : α", "hax : a ⋖ x", "hxb : x ≤ b"], "goal": "b ∈ ⋃ i ∈ {x | a ⋖ x}, Set.Ici i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : α → Set β", "Preorder α"], "goal": "Monotone (Set.Accumulate s)"}}
+{"state": {"context": ["𝕜 : Type u_1", "β : Type u_5", "OrderedSemiring 𝕜", "TopologicalSpace β", "LinearOrderedCancelAddCommMonoid β", "OrderTopology β", "Module 𝕜 β", "OrderedSMul 𝕜 β", "s : Set β", "hs : s.OrdConnected"], "goal": "StrictConvex 𝕜 s"}}
+{"state": {"context": ["D : Type u_1", "C : Type u_2", "inst✝³ : Groupoid D", "inst✝² : Category.{?u.11305, u_2} C", "inst✝¹ : MonoidalCategory C", "inst✝ : MonoidalClosed C", "F G : D ⥤ C", "X Y : D", "f : X ⟶ Y"], "goal": "(F.map f ⊗ (ihom (F.obj X)).map (G.map f) ≫ (pre (CategoryTheory.inv (F.map f))).app (G.obj Y)) ≫ (ihom.ev (F.obj Y)).app (G.obj Y) = (ihom.ev (F.obj X)).app (G.obj X) ≫ G.map f"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "X' : Type u_4", "Y : Type u_5", "Z : Type u_6", "α : Type u_7", "α' : Type u_8", "β : Type u_9", "β' : Type u_10", "γ : Type u_11", "𝓕 : Type u_12", "tX : TopologicalSpace X", "tY : TopologicalSpace Y", "tZ : TopologicalSpace Z", "uα : UniformSpace α", "uβ : UniformSpace β", "uγ : UniformSpace γ", "p : κ → Prop", "s : κ → Set (β × β)", "F : ι → β → α", "hβ : (𝓤 β).HasBasis p s"], "goal": "(∀ ib ∈ 𝓤 α, ∃ ia, p ia ∧ ∀ (a b : β), (a, b) ∈ s ia → ((⇑UniformFun.ofFun ∘ swap F) a, (⇑UniformFun.ofFun ∘ swap F) b) ∈ UniformFun.gen ι α ib) ↔ ∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ (x y : β), (x, y) ∈ s k → ∀ (i : ι), (F i x, F i y) ∈ U"}}
+{"state": {"context": ["r : ℝ", "h : r.sign = 0"], "goal": "r = 0"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "NormalizedGCDMonoid R", "p : Polynomial R"], "goal": "p = Polynomial.C p.content * p.primPart"}}
+{"state": {"context": ["F : Type u_1", "A : Type u_2", "B : Type u_3", "C : Type u_4", "D : Type u_5", "E : Type u_6", "inst✝¹² : Monoid A", "inst✝¹¹ : Monoid B", "inst✝¹⁰ : Monoid C", "inst✝⁹ : Monoid D", "inst✝⁸ : CommGroup E", "inst✝⁷ : TopologicalSpace A", "inst✝⁶ : TopologicalSpace B", "inst✝⁵ : TopologicalSpace C", "inst✝⁴ : TopologicalSpace D", "inst✝³ : TopologicalSpace E", "inst✝² : TopologicalGroup E", "inst✝¹ : ContinuousMul B", "inst✝ : T2Space B", "x y : A"], "goal": "IsClosed {x_1 | x_1 (x * y) = x_1 x * x_1 y}"}}
+{"state": {"context": ["R : Type u_1", "G : Type u_2", "CommSemiring R", "Group G", "MulAction G R", "SMulCommClass G R R", "IsScalarTower G R R", "x : G", "y z : R"], "goal": "IsCoprime (x • y) z ↔ IsCoprime y z"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "inst✝ : CommRing R", "x : 𝕎 R", "hx : x ≠ 0", "hex : ∃ k, x.coeff k ≠ 0", "n : ℕ := Nat.find hex"], "goal": "∃ n x', x'.coeff 0 ≠ 0 ∧ x = (⇑verschiebung)^[n] x'"}}
+{"state": {"context": ["α✝ : Type u_1", "inst✝¹ : DecidableEq α✝", "s✝ s'✝ : Cycle α✝", "α : Type u_2", "inst✝ : DecidableEq α", "s s' : Cycle α", "l l' : List α"], "goal": "∀ {hs : Nodup (Quotient.mk'' l)} {hs' : Nodup (Quotient.mk'' l')}, formPerm (Quotient.mk'' l) hs = formPerm (Quotient.mk'' l') hs' ↔ Quotient.mk'' l = Quotient.mk'' l' ∨ length (Quotient.mk'' l) ≤ 1 ∧ length (Quotient.mk'' l') ≤ 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "LinearOrder α", "Preorder β", "f : α → β", "s : Set α", "hf : StrictMonoOn f s", "a b : α", "ha : a ∈ s", "hb : b ∈ s", "o : Ordering"], "goal": "o.Compares (f a) (f b) ↔ o.Compares a b"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : Measure α", "p : α → Prop", "f : α → ℝ≥0∞", "hf : AEMeasurable f μ"], "goal": "NullMeasurableSet {x | f x ≠ 0} μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : NormedLinearOrderedField 𝕜", "α : Type u_2", "u v : α → 𝕜", "l : Filter α", "huv : u =O[l] v", "hu : Tendsto (fun x => ‖u x‖) l atTop"], "goal": "Tendsto (fun x => ‖v x‖) l atTop"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "LocallyFiniteOrder α", "a b : αᵒᵈ"], "goal": "Finset.Ioo (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ioo b a)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "α : Type u_1", "C : G.Coloring α", "v : V"], "goal": "C.colorClass (C v) ∈ C.colorClasses"}}
+{"state": {"context": ["ι : Type u_5", "σ : ι → Type u_7", "(i : ι) → TopologicalSpace (σ i)", "i : ι"], "goal": "ClosedEmbedding (Sigma.mk i)"}}
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+{"state": {"context": ["o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}", "H : ∀ (i : Ordinal.{u}) (hi : i < o), Ordinal.IsNormal (f i hi)", "a : Ordinal.{max u v}"], "goal": "(∀ (i : Ordinal.{u}) (hi : i < o), f i hi a = a) ↔ ∃ b, o.derivBFamily f b = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s✝ s₁ t u✝ : Set E", "f f₁ : E → F", "g : F → G", "x✝ x₀ : E", "c✝ : F", "b : E × F → G", "m✝ n✝ : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "s : Set E", "hs : UniqueDiffOn 𝕜 s", "n : ℕ∞", "c : E → F →L[𝕜] G", "hc : ContDiffOn 𝕜 n c s", "i : ℕ", "hi : ↑i ≤ n", "x : E", "hx : x ∈ s", "u : F", "m : Fin i → E"], "goal": "(iteratedFDerivWithin 𝕜 i (fun y => (c y) u) s x) m = ((iteratedFDerivWithin 𝕜 i c s x) m) u"}}
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+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V", "h : ⟪x, -y⟫_ℝ = 0", "h0 : x ≠ 0"], "goal": "angle x (x - y) < π / 2"}}
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+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "f : α → β", "s t : Set α", "Hf : AntitoneOn f t", "Hst : s ⊆ t"], "goal": "f '' (lowerBounds s ∩ t) ⊆ upperBounds (f '' s)"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "z1 z2 : ℚ_[p]", "h : ‖z1 + z2‖ < ‖z1‖", "hne : ¬‖z1‖ = ‖z2‖"], "goal": "‖z1 + z2‖ ≥ ‖z1‖"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "s : Set G"], "goal": "AddSubgroup.centralizer s = ⊤ ↔ s ⊆ ↑(AddSubgroup.center G)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹ : NontriviallyNormedField 𝕜", "inst✝ : CompleteSpace 𝕜", "x : 𝕜", "x✝ : x ∈ Set.univ", "U : Set 𝕜", "hU : U ∈ 𝓝 x", "c : 𝕜", "c_pos : 0 < ‖c‖", "hc : ‖c‖ < 1", "A : Tendsto (fun n => x + c ^ n) atTop (𝓝 x)", "B : ∀ᶠ (n : ℕ) in atTop, x + c ^ n ∈ U", "n : ℕ", "hn : x + c ^ n ∈ U"], "goal": "¬c ^ n = 0"}}
+{"state": {"context": ["R : Type u", "Γ₀ : Type v", "CommRing R", "LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "O : Type w", "CommRing O", "Algebra O R", "self : v.Integers O", "r : R"], "goal": "v r ≤ 1 → ∃ x, (algebraMap O R) x = r"}}
+{"state": {"context": ["N : Type u_2", "r : N → N → Prop", "Group N"], "goal": "Covariant N N (fun x x_1 => x * x_1) r ↔ Contravariant N N (fun x x_1 => x * x_1) r"}}
+{"state": {"context": ["M : Type u_1", "SeminormedAddCommGroup M", "R : Type u_3", "Ring R", "Module R M", "S : Submodule R M", "m : M"], "goal": "‖Submodule.Quotient.mk m‖ ≤ ‖m‖"}}
+{"state": {"context": ["n : ℕ", "a b : ℤ", "_hb : 1 < b"], "goal": "IsOpen (ball (↑a / ↑b) (1 / ↑b ^ n) \\ {↑a / ↑b})"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "CategoryTheory.Limits.HasBinaryBiproduct X Y"], "goal": "CategoryTheory.Limits.cokernelBiprodInlIso.hom =\n  CategoryTheory.Limits.colimit.desc (CategoryTheory.Limits.parallelPair CategoryTheory.Limits.biprod.inl 0)\n    (CategoryTheory.Limits.biprod.inlCokernelCofork X Y)"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f₁ f₂ : End R M", "μ₁ μ₂ : R", "h : Commute f₁ f₂", "m : M", "hm : m ∈ (⨆ k, (f₁.genEigenspace μ₁) k) ⊓ ⨆ k, (f₂.genEigenspace μ₂) k"], "goal": "m ∈ ⨆ k, ((f₁ + f₂).genEigenspace (μ₁ + μ₂)) k"}}
+{"state": {"context": ["α : Sort u", "β : α → Sort v", "DecidableEq α", "f : (a : α) → β a", "a : α", "b : β a"], "goal": "Function.update f a b = f ↔ b = f a"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁶ : TopologicalSpace α", "inst✝⁵ : TopologicalSpace β", "inst✝⁴ : Group α", "inst✝³ : MulAction α β", "inst✝² : ContinuousInv α", "inst✝¹ : ContinuousSMul α β", "s : Set α", "t✝ : Set β", "inst✝ : CompactSpace α", "t : Set β", "ht : IsClosed t"], "goal": "⋃ g, (fun x => g • x) '' t = univ • t"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "IsUpperModularLattice α", "a b : α"], "goal": "a ⊓ b ⋖ a → b ⋖ a ⊔ b"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : Affine R", "x y L : R"], "goal": "eval (C L * (X - C x) + C y) (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)) = -(C { a := 1, b := -L ^ 2 - W.a₁ * L + W.a₂, c := 2 * x * L ^ 2 + (W.a₁ * x - 2 * y - W.a₃) * L + (-W.a₁ * y + W.a₄), d := -x ^ 2 * L ^ 2 + (2 * x * y + W.a₃ * x) * L - (y ^ 2 + W.a₃ * y - W.a₆) }.a * X ^ 3 + C { a := 1, b := -L ^ 2 - W.a₁ * L + W.a₂, c := 2 * x * L ^ 2 + (W.a₁ * x - 2 * y - W.a₃) * L + (-W.a₁ * y + W.a₄), d := -x ^ 2 * L ^ 2 + (2 * x * y + W.a₃ * x) * L - (y ^ 2 + W.a₃ * y - W.a₆) }.b * X ^ 2 + C { a := 1, b := -L ^ 2 - W.a₁ * L + W.a₂, c := 2 * x * L ^ 2 + (W.a₁ * x - 2 * y - W.a₃) * L + (-W.a₁ * y + W.a₄), d := -x ^ 2 * L ^ 2 + (2 * x * y + W.a₃ * x) * L - (y ^ 2 + W.a₃ * y - W.a₆) }.c * X + C { a := 1, b := -L ^ 2 - W.a₁ * L + W.a₂, c := 2 * x * L ^ 2 + (W.a₁ * x - 2 * y - W.a₃) * L + (-W.a₁ * y + W.a₄), d := -x ^ 2 * L ^ 2 + (2 * x * y + W.a₃ * x) * L - (y ^ 2 + W.a₃ * y - W.a₆) }.d)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "R I J X Y : Set α", "hR : autoParam (R ⊆ M.E) _auto✝"], "goal": "(M ↾ R).Basis I X ↔ M.Basis I X ∧ X ⊆ R"}}
+{"state": {"context": ["P : Type u_1", "L : Type u_2", "inst✝³ : Membership P L", "inst✝² : ProjectivePlane P L", "inst✝¹ : Finite P", "inst✝ : Finite L", "p₁ p₂ p₃ : P", "l₁ l₂ l₃ : L", "h₂₁ : p₂ ∉ l₁", "h₂₂ : p₂ ∈ l₂", "h₂₃ : p₂ ∈ l₃", "h₃₁ : p₃ ∉ l₁", "h₃₂ : p₃ ∈ l₂", "h₃₃ : p₃ ∉ l₃"], "goal": "1 < order P L"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "hC : CategoryTheory.Pretriangulated C", "T T' : CategoryTheory.Pretriangulated.Triangle C", "e : T ≅ T'", "hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles"], "goal": "(CategoryTheory.Pretriangulated.shortComplexOfDistTriangleIsoOfIso e hT).inv.τ₁ = e.inv.hom₁"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_4", "(i : ι) → Group (G i)", "i j : ι", "w : Monoid.CoprodI.NeWord G i j"], "goal": "w.inv.prod = w.prod⁻¹"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p✝ q p : R[X]", "hp : p ≠ 0", "s : Multiset R"], "goal": "(Multiset.map (fun a => X - C a) s).prod ∣ p ↔ s ≤ p.roots"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroMorphisms D", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "F : CategoryTheory.Functor C D", "F.PreservesZeroMorphisms", "h₁.left.IsPreservedBy F", "h₁.right.IsPreservedBy F", "h₂.left.IsPreservedBy F", "h₂.right.IsPreservedBy F"], "goal": "F.map (CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂) =\n  CategoryTheory.ShortComplex.homologyMap' (F.mapShortComplex.map φ) (h₁.map F) (h₂.map F)"}}
+{"state": {"context": ["ι : Type uι", "M₁ : ι → Type v₁", "M₂ : Type v₂", "(i : ι) → AddCommMonoid (M₁ i)", "AddCommMonoid M₂", "R' : Type u_1", "A : Type u_2", "Monoid R'", "Semiring A", "(i : ι) → Module A (M₁ i)", "DistribMulAction R' M₂", "Module A M₂", "SMulCommClass A R' M₂", "c : R'", "f : MultilinearMap A M₁ M₂"], "goal": "⇑(c • f) = c • ⇑f"}}
+{"state": {"context": ["α β : Type u", "n : ℕ", "a : Cardinal.{u}"], "goal": "↑n < lift.{v, u} a ↔ ↑n < a"}}
+{"state": {"context": ["ι : Type u_1", "A : Type u_3", "σ : Type u_4", "Semiring A", "DecidableEq ι", "CanonicallyOrderedAddCommMonoid ι", "SetLike σ A", "AddSubmonoidClass σ A", "𝒜 : ι → σ", "GradedRing 𝒜", "a b : A", "n i : ι", "Sub ι", "OrderedSub ι", "ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "a_mem : a ∈ 𝒜 i", "h : i ≤ n"], "goal": "↑(((DirectSum.decompose 𝒜) (a * b)) n) = a * ↑(((DirectSum.decompose 𝒜) b) (n - i))"}}
+{"state": {"context": ["R : Type u_2", "StrictOrderedSemiring R", "Archimedean R"], "goal": "Filter.atTop.HasCountableBasis (fun x => True) fun n => Set.Ici ↑n"}}
+{"state": {"context": ["F : Type u → Type u", "Applicative F", "α' β' : Type u", "f : α' → F β'", "a : α'", "l : List α'"], "goal": "traverse f (a :: l) = Seq.seq ((fun x x_1 => x :: x_1) <$> f a) fun x => traverse f l"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "inst✝² : Preorder α✝", "inst✝¹ : Preorder β", "s✝ t : Set α✝", "α : Type u_3", "inst✝ : PartialOrder α", "s : Set α", "hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Ioo x y ⊆ s", "x : α", "hx hy : x ∈ s", "hxy : x ≤ x"], "goal": "Icc x x ⊆ s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v', u'} D", "f : CategoryTheory.Functor C D", "hf : f.FullyFaithful", "X : C"], "goal": "∀ (a : X ⟶ X), (hf.mulEquivEnd X) a = f.map a"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝³ : Category.{v₂, u₂} D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "F F' : C ⥤ D", "G : D ⥤ E", "obj : C → D", "map : {X Y : C} → (X ⟶ Y) → (obj X ⟶ obj Y)", "G_obj : D → E", "map✝¹ : {X Y : D} → (X ⟶ Y) → (G_obj X ⟶ G_obj Y)", "map_id✝¹ : ∀ (X : D), { obj := G_obj, map := map✝¹ }.map (𝟙 X) = 𝟙 ({ obj := G_obj, map := map✝¹ }.obj X)", "map_comp✝¹ : ∀ {X Y Z : D} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj, map := map✝¹ }.map (f ≫ g) = { obj := G_obj, map := map✝¹ }.map f ≫ { obj := G_obj, map := map✝¹ }.map g", "inst✝¹ : { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.Faithful", "map✝ : {X Y : C} → (X ⟶ Y) → ((G_obj ∘ obj) X ⟶ (G_obj ∘ obj) Y)", "map_id✝ : ∀ (X : C), { obj := G_obj ∘ obj, map := map✝ }.map (𝟙 X) = 𝟙 ({ obj := G_obj ∘ obj, map := map✝ }.obj X)", "map_comp✝ : ∀ {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z), { obj := G_obj ∘ obj, map := map✝ }.map (f ≫ g) = { obj := G_obj ∘ obj, map := map✝ }.map f ≫ { obj := G_obj ∘ obj, map := map✝ }.map g", "inst✝ : { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.Faithful", "h_obj : ∀ (X : C), { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.obj (obj X) = { obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.obj X", "h_map✝ : ∀ {X Y : C} {f : X ⟶ Y}, HEq ({ obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map (map f)) ({ obj := G_obj ∘ obj, map := map✝, map_id := map_id✝, map_comp := map_comp✝ }.map f)", "h_map : ∀ {X Y : C} {f : X ⟶ Y}, map✝¹ (map f) = map✝ f"], "goal": "(fun {X Y} f => { obj := G_obj, map := map✝¹, map_id := map_id✝¹, map_comp := map_comp✝¹ }.map ({ obj := obj, map := map, map_id := ⋯, map_comp := ⋯ }.map f)) = map✝"}}
+{"state": {"context": ["n m : ℕ", "h : m ∈ n.properDivisors"], "goal": "0 < m"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M", "AddMonoid M"], "goal": "Inducing ⇑(AddUnits.embedProduct M)"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "B : ℝ", "x✝ : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)"], "goal": "x✝ ∈ convexBodySum K B ↔ x✝ ∈ convexBodySumFun ⁻¹' Set.Icc 0 B"}}
+{"state": {"context": ["C : FloatCfg"], "goal": "emin = max (emin + ↑(Nat.size 0) - ↑prec) emin"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "X : C"], "goal": "(α_ (𝟙_ C) (𝟙_ C) X).inv ≫ (β_ (𝟙_ C ⊗ 𝟙_ C) X).hom ≫ (α_ X (𝟙_ C) (𝟙_ C)).inv ≫ (β_ (𝟙_ C) X).inv ▷ 𝟙_ C ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ 𝟙_ C ◁ (ρ_ X).hom = (α_ (𝟙_ C) (𝟙_ C) X).inv ≫ (β_ (𝟙_ C ⊗ 𝟙_ C) X).hom ≫ X ◁ (ρ_ (𝟙_ C)).hom ≫ (β_ (𝟙_ C) X).inv"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "K : Submodule 𝕜 E", "v : E"], "goal": "v ∈ Kᗮ ↔ ∀ u ∈ K, ⟪u, v⟫_𝕜 = 0"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁴ : MeasurableSpace X", "inst✝³ : TopologicalSpace X", "inst✝² : OpensMeasurableSpace X", "inst✝¹ : T0Space X", "inst✝ : CompletelyRegularSpace X", "μ : ↑(range diracProba)", "F : Filter ↑(range diracProba)", "key : Tendsto diracProba (map (⇑diracProbaEquiv.symm) F) (𝓝 (diracProba (diracProbaEquiv.symm μ))) ↔ Tendsto id (map (⇑diracProbaEquiv.symm) F) (𝓝 (diracProbaEquiv.symm μ))"], "goal": "Tendsto (⇑diracProbaEquiv.symm) F (𝓝 (diracProbaEquiv.symm μ)) ↔ Tendsto id F (𝓝 μ)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝ : ConditionallyCompleteLinearOrder R", "s : Finset S", "f : S → WithTop R"], "goal": "trop (sInf (f '' ↑s)) = ∑ i ∈ s, trop (f i)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Ring α", "m n : ZNum"], "goal": "↑(m * n) = ↑m * ↑n"}}
+{"state": {"context": ["α : Type u_1", "acc : Array α", "as : List (Option α)", "as' : List α"], "goal": "List.fillNonesTR.go as as' acc = acc.data ++ as.fillNones as'"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : CommSemiring α", "s✝ : Multiset ι", "f g : ι → α", "a : ι", "s : Multiset ι", "ih : (map (fun i => f i + g i) s).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) s.antidiagonal).sum"], "goal": "(map (fun i => f i + g i) (a ::ₘ s)).prod = (map (fun p => (map f p.1).prod * (map g p.2).prod) (a ::ₘ s).antidiagonal).sum"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → MeasurableSpace (α i)", "P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)", "μ ν : Measure ((i : ι) → α i)", "inst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)", "hμ : IsProjectiveLimit μ P", "hν : IsProjectiveLimit ν P", "this : IsFiniteMeasure μ", "I : Finset ι", "S : Set ((i : { x // x ∈ I }) → α ↑i)", "hS : MeasurableSet S", "hs : cylinder I S ∈ measurableCylinders α"], "goal": "μ (cylinder I S) = ν (cylinder I S)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝³ : Countable ι", "m✝ : MeasurableSpace α", "μ✝ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "f g : α → β", "inst✝ : PseudoMetrizableSpace β", "m : MeasurableSpace α", "μ : ι → Measure α", "h : ∀ (i : ι), AEStronglyMeasurable f (μ i)", "this✝¹ : MeasurableSpace β := borel β", "this✝ : BorelSpace β", "A : ∀ (i : ι), ∃ t, IsSeparable t ∧ f ⁻¹' t ∈ ae (μ i)"], "goal": "∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t"}}
+{"state": {"context": ["X : Type u_2", "EMetricSpace X", "s t : Set X"], "goal": "dimH (s ∪ t) = max (dimH s) (dimH t)"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f g : E → ℂ", "hf : Differentiable ℂ f", "hg : Differentiable ℂ g", "h0 : ∀ (x : E), f x ∈ Complex.slitPlane"], "goal": "Differentiable ℂ fun x => f x ^ g x"}}
+{"state": {"context": ["α : Type u", "s : Set α", "a : α", "h : a ∉ s"], "goal": "s ⊂ insert a s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "f : ↑s ↪ β", "h✝ : Nonempty (↑sᶜ ≃ ↑(range ⇑f)ᶜ)", "g : ↑sᶜ ≃ ↑(range ⇑f)ᶜ", "h : α ≃ β := (Set.sumCompl s).symm.trans (((ofInjective ⇑f ⋯).sumCongr g).trans (Set.sumCompl (range ⇑f)))"], "goal": "∃ g, ∀ (x : ↑s), g ↑x = f x"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "StrictMono Submodule.toSubMulAction"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "f : ↥(MeasureTheory.Lp.simpleFunc E p μ)"], "goal": "MeasureTheory.Lp.simpleFunc.toLp (MeasureTheory.Lp.simpleFunc.toSimpleFunc f) ⋯ = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f g : α → ℝ≥0∞", "hf : Measurable f", "hg : Measurable g"], "goal": "∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), (⨆ n, ↑(eapprox f n) a) + ⨆ n, ↑(eapprox g n) a ∂μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "p : FormalMultilinearSeries 𝕜 E F", "r : ℝ≥0∞", "n : ℕ", "x y : E", "hf : HasFiniteFPowerSeriesOnBall f p x n r", "h : y ∈ EMetric.ball x r"], "goal": "CPolynomialAt 𝕜 f y"}}
+{"state": {"context": ["α : Type u_2", "One α"], "goal": "1 ∈ 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : Finite α", "s : Set α", "f : ↑s ↪ β", "h : lift.{max u v, u} #α = lift.{max u v, v} #β", "g : α ≃ β"], "goal": "lift.{max u v, u} #↑s = lift.{max u v, v} #↑(range ⇑f)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.HasShift C ���", "X₁ X₂ : C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasBinaryBiproduct X₁ X₂"], "goal": "(CategoryTheory.Pretriangulated.binaryBiproductTriangle X₁ X₂).obj₃ = X₂"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β", "r : ℝ", "h : x ∈ s", "H : hausdorffDist s t < r", "fin : hausdorffEdist s t ≠ ⊤", "r0 : 0 < r"], "goal": "∃ y ∈ t, dist x y < r"}}
+{"state": {"context": ["G : Type w", "AddGroup G", "t₁ t₂ : TopologicalSpace G", "h₁ : TopologicalAddGroup G", "h₂ : TopologicalAddGroup G"], "goal": "TopologicalAddGroup G"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "f : 𝕜 → 𝕜", "x : 𝕜", "hf : MeromorphicAt f x", "n : ℕ"], "goal": "MeromorphicAt (f ^ n) x"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "e : Perm α"], "goal": "Perm.sign (optionCongr 1) = Perm.sign 1"}}
+{"state": {"context": ["C : Type u", "D : Type u'", "inst✝⁴ : Category.{v, u} C", "inst✝³ : ConcreteCategory C", "inst✝² : Category.{v', u'} D", "inst✝¹ : ConcreteCategory D", "inst✝ : HasForget₂ C D", "X Y Z : C", "f : X ⟶ Y", "g : Y ⟶ Z", "x : (forget D).obj ((forget₂ C D).obj X)"], "goal": "((forget₂ C D).map (f ≫ g)) x = ((forget₂ C D).map g) (((forget₂ C D).map f) x)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "a b c : α", "l : List α"], "goal": "count a ((c :: l).erase b) = count a (c :: l) - if a = b then 1 else 0"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "Preorder α", "Preorder β", "EquivLike F α β", "OrderIsoClass F α β", "f : F", "a b : α"], "goal": "f a < f b ↔ a < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "inst✝ : MeasurableSpace δ", "μa : Measure α", "μb : Measure β", "μc : Measure γ", "μd : Measure δ", "f : α → β", "hf : MeasurePreserving f μa μb", "h₂ : MeasurableEmbedding f", "s : Set β"], "goal": "map f (μa.restrict (f ⁻¹' s)) = μb.restrict s"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "inst✝² : Field A", "inst✝¹ : Ring B", "inst✝ : Algebra A B", "x : B", "p : A[X]", "pmonic : p.Monic", "hp : (Polynomial.aeval x) p = 0", "pmin : ∀ (q : A[X]), q.Monic → (Polynomial.aeval x) q = 0 → p.degree ≤ q.degree", "hx : IsIntegral A x"], "goal": "minpoly A x - p = 0"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "n : ℕ", "u : Rˣ", "hn : IsUnit ↑n", "a✝ : Nontrivial R", "hpos : n > 0", "n' : R", "hn' : n' * ↑n = 1"], "goal": "∃ a b, a * (X ^ n - C ↑u) + b * derivative (X ^ n - C ↑u) = 1"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a✝ b : R", "n : ℕ", "inst✝ : Ring R", "p q : R[X]", "a : R", "h0 : p ≠ 0", "this : Nontrivial R"], "goal": "0 < 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t✝ s : Set α", "t : Set β", "f : α → β", "hf : MapsTo f s t", "f_inj : InjOn f s"], "goal": "f '' s ⊆ t"}}
+{"state": {"context": ["X : Type u_1", "Y : Sort u_2", "A : Type ?u.10594", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace A", "f : X → Y"], "goal": "IsSeparatedMap f ↔ ClosedEmbedding (toPullbackDiag f)"}}
+{"state": {"context": ["E : Type u_2", "SeminormedAddCommGroup E", "NormedSpace ℝ E", "δ ε : ℝ", "hε : 0 < ε", "hδ : 0 < δ", "a b : E"], "goal": "Metric.ball a ε - Metric.ball b δ = Metric.ball (a - b) (ε + δ)"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_3", "S : Type u_4", "CommSemiring R", "CommSemiring S", "f : R →+* S"], "goal": "(MvPolynomial.map f).comp MvPolynomial.C = MvPolynomial.C.comp f"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "p p₀ p₁ p₂ p1 p2 p3 : P", "h : dist p3 p1 = dist p3 p2", "m : P := midpoint ℝ p1 p2", "h1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m)", "h2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m)"], "goal": "∠ p3 (midpoint ℝ p1 p2) p1 = π / 2"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "r r' : α → α → Prop", "f : β → α", "s✝ t : Set α", "x y : α", "s : Set β"], "goal": "(f '' s).WellFoundedOn r ↔ s.WellFoundedOn (r on f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "f : α → β", "s✝ : Set β", "ι : Type u_3", "U : ι → Opens β", "hU : iSup U = ⊤", "H : IsClosedMap f", "s : Set β", "t : Set ↑(f ⁻¹' s)"], "goal": "IsClosed t → IsClosed (s.restrictPreimage f '' t)"}}
+{"state": {"context": ["F : Type v'", "R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝¹⁶ : CommSemiring R", "inst✝¹⁵ : StarRing R", "inst✝¹⁴ : NonUnitalSemiring A", "inst✝¹³ : StarRing A", "inst✝¹² : Module R A", "inst✝¹¹ : IsScalarTower R A A", "inst✝¹⁰ : SMulCommClass R A A", "inst✝⁹ : StarModule R A", "inst✝⁸ : NonUnitalSemiring B", "inst✝⁷ : StarRing B", "inst✝⁶ : Module R B", "inst✝⁵ : IsScalarTower R B B", "inst✝⁴ : SMulCommClass R B B", "inst✝³ : StarModule R B", "inst✝² : FunLike F A B", "inst✝¹ : NonUnitalAlgHomClass F R A B", "inst✝ : NonUnitalStarAlgHomClass F R A B", "S : NonUnitalStarSubalgebra R A", "h : ∀ (x : A), x ∈ S", "x : A"], "goal": "x ∈ S ↔ x ∈ ⊤"}}
+{"state": {"context": ["z : ℂ"], "goal": "abs z ≤ |z.re| + |z.im|"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "D : Set ℝ", "hD : Convex ℝ D", "f : ℝ → ℝ", "hf : ContinuousOn f D", "hf' : DifferentiableOn ℝ f (interior D)", "C : ℝ", "hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x", "x : ℝ", "hx : x ∈ D", "y : ℝ", "hy : y ∈ D", "hxy : x ≤ y", "hxy' : x < y", "hxyD : Icc x y ⊆ D", "hxyD' : Ioo x y ⊆ interior D", "a : ℝ", "a_mem : a ∈ Ioo x y", "ha : deriv f a = (f y - f x) / (y - x)", "this : C ≤ (f y - f x) / (y - x)"], "goal": "C * (y - x) ≤ f y - f x"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "h : Fact (∀ (I : Ideal R), I ≠ ⊤ → CharZero (R ⧸ I))", "n : ℕ+"], "goal": "¬CharZero (R ⧸ Ideal.span {↑↑n})"}}
+{"state": {"context": ["α β : Type u_2", "f : α → β", "l : List α"], "goal": "f <$> l = List.map f l"}}
+{"state": {"context": ["F : Type u_1", "Field F", "ι : Type u_2", "DecidableEq ι", "s : Finset ι", "v : ι → F", "r : ι → F", "x : F", "hvs : Set.InjOn v ↑s", "hs : s.Nonempty", "hx : ∀ i ∈ s, x ≠ v i"], "goal": "Polynomial.eval x ((Lagrange.interpolate s v) r) =\n  (∑ i ∈ s, Lagrange.nodalWeight s v i * (x - v i)⁻¹ * r i) / ∑ i ∈ s, Lagrange.nodalWeight s v i * (x - v i)⁻¹"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "s : Set E", "f : E → E", "f' : E → E →L[ℝ] E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hs : MeasurableSet s", "h's : μ s ≠ ⊤", "hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x", "this : Tendsto (fun ε => ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ + 2 * ↑ε * μ s) (𝓝[>] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ))"], "goal": "∀ a ∈ Ioi 0, μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ + 2 * ↑a * μ s"}}
+{"state": {"context": ["τ : Type u_1", "TopologicalSpace τ", "AddMonoid τ", "ContinuousAdd τ", "α : Type u_2", "TopologicalSpace α", "f : Filter τ", "ϕ : Flow τ α", "s : Set α", "hf : ∀ (t : τ), Filter.Tendsto (fun x => t + x) f f"], "goal": "IsInvariant ϕ.toFun (omegaLimit f ϕ.toFun s)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : Measure α", "f✝ g : α → F'", "s : Set α", "hμ : NeZero μ", "f : α → F'", "hμ_finite : IsFiniteMeasure μ", "h_meas : StronglyMeasurable (μ[f|⊥])", "c : F'", "h_eq : μ[f|⊥] = fun x => c", "h_integral : (μ Set.univ).toReal • c = ∫ (x : α), f x ∂μ"], "goal": "¬μ Set.univ = 0 ∧ ¬μ Set.univ = ⊤"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝⁴ : CommMonoid M", "inst✝³ : CommMonoid N", "inst✝² : DivisionCommMonoid G", "k l : ℕ", "ζ : R", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "n : ℕ+", "hζ : IsPrimitiveRoot ζ ↑n", "p : ℕ := ringChar R", "hfin : multiplicity.Finite p ↑n"], "goal": "NeZero ↑↑n"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "𝕜 : Type u_2", "RCLike 𝕜", "G : Type u_3", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ι → 𝕜 → G", "g : 𝕜 → G", "f' : ι → 𝕜 → G", "g' : 𝕜 → G", "x : 𝕜", "l.NeBot", "hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)", "hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2", "hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Filter.Tendsto (fun n => f n y) l (𝓝 (g y))"], "goal": "HasDerivAt g (g' x) x"}}
+{"state": {"context": ["α✝ β α : Type u", "n : ℕ", "H : ↑n < #α"], "goal": "↑(n + 1) ≤ #α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s t u : Set α", "x : Set ℝ"], "goal": "x ∈ ⋃ a, ⋃ b, ⋃ (_ : a < b), {Ioo ↑a ↑b} ↔ x ∈ {S | ∃ l u, ↑l < ↑u ∧ Ioo ↑l ↑u = S}"}}
+{"state": {"context": ["p q : ℚ", "K : Type u_5", "LinearOrderedField K"], "goal": "↑p ≤ ↑q ↔ p ≤ q"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "μ : M → N → N", "PartialOrder N"], "goal": "(Covariant M N μ fun x x_1 => x < x_1) → Covariant M N μ fun x x_1 => x ≤ x_1"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "∀ (i : 𝒰.J), CategoryTheory.Limits.HasPullback (𝒰.map i ≫ f) g", "i j : 𝒰.J"], "goal": "AlgebraicGeometry.Scheme.Pullback.pullbackFstιToV 𝒰 f g i j ≫\n    CategoryTheory.Limits.pullback.snd (CategoryTheory.Limits.pullback.fst (𝒰.map j ≫ f) g ≫ 𝒰.map j) (𝒰.map i) =\n  CategoryTheory.Limits.pullback.fst\n      (CategoryTheory.Limits.pullback.fst (AlgebraicGeometry.Scheme.Pullback.p1 𝒰 f g) (𝒰.map i))\n      ((AlgebraicGeometry.Scheme.Pullback.gluing 𝒰 f g).ι j) ≫\n    CategoryTheory.Limits.pullback.snd (AlgebraicGeometry.Scheme.Pullback.p1 𝒰 f g) (𝒰.map i)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "P : Type u_3", "CommSemiring P", "IsLocalization M S", "h : R ≃+* P"], "goal": "IsLocalization (Submonoid.map h.toMonoidHom M) S"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝⁹ : Countable ι", "f✝ g : α → β", "X : Type u_5", "Y : Type u_6", "inst✝⁸ : Zero α", "inst✝⁷ : TopologicalSpace X", "inst✝⁶ : TopologicalSpace Y", "inst✝⁵ : MeasurableSpace X", "inst✝⁴ : MeasurableSpace Y", "inst✝³ : OpensMeasurableSpace X", "inst✝² : OpensMeasurableSpace Y", "inst✝¹ : TopologicalSpace α", "inst✝ : PseudoMetrizableSpace α", "f : X × Y → α", "hf : Continuous f", "h'f : HasCompactSupport f", "this✝¹ : MeasurableSpace α := borel α", "this✝ : BorelSpace α"], "goal": "IsSeparable (range f)"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "inst✝⁵ : FunLike F G H", "a : G", "b : H", "inst✝⁴ : LinearOrderedAddCommGroup G", "inst✝³ : Archimedean G", "inst✝² : AddGroup H", "inst✝¹ : AddConstMapClass F G H a b", "f : F", "R : H → H → Prop", "inst✝ : IsTrans H R", "hR : CovariantClass H H (fun x y => y + x) R", "ha : 0 < a", "l : G", "hf : ∀ x ∈ Icc l (l + a), ∀ y ∈ Icc l (l + a), x < y → R (f x) (f y)", "x y : G", "hxy : (fun x x_1 => x < x_1) x y"], "goal": "R (f x) (f y)"}}
+{"state": {"context": ["α : Type u", "Preorder α", "f : ℕ → α", "hf : Antitone f", "n : ℕ", "x : α", "h1 : f (n + 1) < x", "h2 : x < f n", "a : ℕ"], "goal": "f a ≠ x"}}
+{"state": {"context": ["R : Type u_1", "c₁ c₂ : R", "Infinite R"], "goal": "Cardinal.mk ℍ[R,c₁,c₂] = Cardinal.mk R"}}
+{"state": {"context": ["G : Type u", "α : Type v", "β : Type w", "inst✝¹ : Group G", "inst✝ : Finite G", "p n : ℕ", "hp : Fact (Nat.Prime p)", "hdvd : p ^ (n + 1) ∣ Nat.card G", "H : Subgroup G", "hH : Nat.card ↥H = p ^ n", "s : ℕ", "hs : Nat.card G = s * p ^ (n + 1)"], "goal": "Nat.card (G ⧸ H) * Nat.card ↥H = s * p * Nat.card ↥H"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "inst✝⁴ : DivisionRing K", "inst✝³ : AddCommGroup V", "inst✝² : AddCommGroup V'", "inst✝¹ : Module K V", "inst✝ : Module K V'", "v : ι → V", "s t : Set V", "x✝ y✝ z x y : V", "hx : x ≠ 0"], "goal": "LinearIndependent K ![x, y] ↔ ∀ (a : K), a • x ≠ y"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VAdd α β", "t : Set β", "a : α", "s : Set α"], "goal": "a ∈ s → a +ᵥ t ⊆ s +ᵥ t"}}
+{"state": {"context": ["x : ℝ", "hx : Liouville x", "n : ℕ"], "goal": "∃ᶠ (b : ℕ) in atTop, ∃ a, x ≠ ↑a / ↑b ∧ |x - ↑a / ↑b| < 1 / ↑b ^ n"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M : Type u_5", "M₁ : Type u_6", "M₂ : Type u_7", "M₃ : Type u_8", "Mₗ₁ : Type u_9", "Mₗ₁' : Type u_10", "Mₗ₂ : Type u_11", "Mₗ₂' : Type u_12", "K : Type u_13", "K₁ : Type u_14", "K₂ : Type u_15", "V : Type u_16", "V₁ : Type u_17", "V₂ : Type u_18", "n : Type u_19", "inst✝⁷ : Field K", "inst✝⁶ : AddCommGroup V", "inst✝⁵ : Module K V", "inst✝⁴ : Field K₁", "inst✝³ : AddCommGroup V₁", "inst✝² : Module K₁ V₁", "inst✝¹ : AddCommGroup V₂", "inst✝ : Module K V₂", "J : K →+* K", "J₁ J₁' : K₁ →+* K", "B : V₁ →ₛₗ[J₁] V₁ →ₛₗ[J₁'] V₂", "x : V₁", "hx : ¬B.IsOrtho x x", "μ : V₁ → K₁", "h : B.IsOrtho x (μ x • x)"], "goal": "μ x = 0"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f : α → β → β", "op : α → α → α", "inst✝¹ : Monoid α", "inst✝ : Monoid β", "s : Multiset α", "comm : {x | x ∈ s}.Pairwise Commute", "p : α → Prop", "hom : ∀ (a b : α), p a → p b → p (a * b)", "unit : p 1", "base : ∀ x ∈ s, p x"], "goal": "p (s.noncommProd comm)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁵ : Semiring R", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "inst✝² : AddCommMonoid M'", "inst✝¹ : Module R M'", "b b₁ : Basis ι R M", "i : ι", "c : R", "x : M", "inst✝ : Fintype ι", "m : M"], "goal": "b.sumCoords m = (∑ i : ι, b.coord i) m"}}
+{"state": {"context": ["n p : ℕ", "hp : p ∈ n.primeFactors"], "goal": "0 < p"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "e : M₁ ≃SL[σ₁₂] M₂"], "goal": "⇑e.symm ∘ ⇑e = id"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_6", "M : Type u_9", "M₃ : Type u_12", "Semiring R", "Semiring S", "AddCommMonoid M", "AddCommMonoid M₃", "Module R M", "Module S M₃", "σ : R →+* S", "f g : M →ₛₗ[σ] M₃", "h : f = g", "x : M"], "goal": "f x = g x"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "LocallyFiniteOrder α", "a b c d : α", "hcb : c ≤ b", "had : a ≤ d"], "goal": "Finset.Ico a b ∪ Finset.Ico c d = Finset.Ico (min a c) (max b d)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "A : Type u", "CommRing A", "Algebra R A", "r : R", "IsLocalization.Away r A"], "goal": "Algebra.Smooth R A"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M", "x y : ↥S"], "goal": "↑(x + y) = ↑x + ↑y"}}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "T : Type w", "inst✝² : NonAssocSemiring R", "M : Submonoid R", "inst✝¹ : NonAssocSemiring S✝", "inst✝ : NonAssocSemiring T", "S : Set (Subsemiring R)", "Sne : S.Nonempty", "hS : DirectedOn (fun x x_1 => x ≤ x_1) S", "x : R"], "goal": "x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m' mΩ : MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "hm' : m' ≤ mΩ", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "β : ι → Type u_3", "m : (x : ι) → MeasurableSpace (β x)", "f : (i : ι) → Ω → β i", "hf_Indep : ProbabilityTheory.iCondIndepFun m' hm' m f μ", "i j : ι", "hij : i ≠ j"], "goal": "ProbabilityTheory.CondIndepFun m' hm' (f i) (f j) μ"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁴ : DecidableEq l", "inst✝³ : DecidableEq m", "inst✝² : DecidableEq n", "inst✝¹ : Semiring α", "i j : n", "c : α", "i' j' : n", "inst✝ : Fintype n"], "goal": "(stdBasisMatrix i i c).trace = c"}}
+{"state": {"context": ["f : ℝ → ℝ", "hf : Differentiable ℝ f", "hf'_mono : Monotone (deriv f)"], "goal": "ConvexOn ℝ Set.univ f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f g : α → β → α", "a : α", "l : List β", "H : ∀ (a : α) (b : β), b ∈ l → f a b = g a b"], "goal": "List.foldl f a l = List.foldl g a l"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "V2 : Type u_4", "P2 : Type u_5", "Ring k", "AddCommGroup V1", "Module k V1", "AffineSpace V1 P1", "AddCommGroup V2", "Module k V2", "AffineSpace V2 P2", "f : P1 →ᵃ[k] P2"], "goal": "Function.Surjective ⇑f.linear ↔ Function.Surjective ⇑f"}}
+{"state": {"context": ["α : Type u", "a : α", "BEq α"], "goal": "List.elem a List.nil = false"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : (i : ι) → TopologicalSpace (π i)", "x y z : X", "s : Set X", "f g : X → Y", "t : Set (SeparationQuotient X)", "hs : IsClosed s"], "goal": "mk ⁻¹' (mk '' s) = s"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "a : α", "l₁ l₂ : List α", "p : l₁ ~ l₂", "h₁ : ¬a ∈ l₁", "h₂ : ¬a ∈ l₂"], "goal": "l₁ ~ l₂"}}
+{"state": {"context": ["w : Nat", "x : BitVec w"], "goal": "BitVec.allOnes w - x = ~~~x"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "Field K", "AddCommGroup V", "Module K V", "W : Subspace K (Module.Dual K V)", "FiniteDimensional K ↥W"], "goal": "Function.Bijective ⇑(Submodule.quotDualCoannihilatorToDual W)"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E", "inst✝⁵ : Algebra F E", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "inst✝² : Algebra E K", "inst✝¹ : IsScalarTower F E K", "inst✝ : IsPurelyInseparable F E", "ι : Type u_1", "v : ι → K", "hsep : ∀ (i : ι), IsSeparable F (v i)", "h : LinearIndependent F v", "q : ℕ", "h✝ : ExpChar F q", "this : ExpChar K q"], "goal": "LinearIndependent E v"}}
+{"state": {"context": ["M : Type u_3", "AddMonoid M", "LinearOrder M", "CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "x : M", "n : ℕ", "hn : n ≠ 0"], "goal": "0 ≤ n • x ↔ 0 ≤ x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f g h : C", "α : f ⟶ g", "β : g ⟶ h"], "goal": "CategoryTheory.CategoryStruct.comp α β = CategoryTheory.CategoryStruct.comp α β"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s t u : Set α", "x : Set ℝ"], "goal": "x ∈ ⋃ a, {Ici ↑a} ↔ x ∈ Ici '' (Rat.cast '' univ)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "U : Opens ↑(PrimeSpectrum.Top R)", "s : ↑((structureSheaf R).val.obj (op U))", "ι : Type u_1", "t : Finset ι", "a h : ι → R", "iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ U", "h_cover : U ≤ ⨆ i ∈ t, PrimeSpectrum.basicOpen (h i)", "hs : ∀ (i : ι), const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯ = ((structureSheaf R).val.map (iDh i).op) s", "fractions_eq : ∀ (i j : ι), IsLocalization.mk' (Localization.Away (h i * h j)) (a i * h j) ⟨h i * h j, ⋯⟩ = IsLocalization.mk' (Localization.Away (h i * h j)) (h i * a j) ⟨h i * h j, ⋯⟩", "exists_power : ∀ (i j : ι), ∃ n, a i * h j * (h i * h j) ^ n = h i * a j * (h i * h j) ^ n", "n : ι × ι → ℕ := fun p => ⋯.choose", "n_spec : ∀ (p : ι × ι), a p.1 * h p.2 * (h p.1 * h p.2) ^ ⋯.choose = h p.1 * a p.2 * (h p.1 * h p.2) ^ ⋯.choose", "N : ℕ := (t ×ˢ t).sup n", "basic_opens_eq : ∀ (i : ι), PrimeSpectrum.basicOpen (h i ^ (N + 1)) = PrimeSpectrum.basicOpen (h i)", "i : ι", "a✝ : i ∈ t"], "goal": "((structureSheaf R).val.map ((fun i => eqToHom ⋯ ≫ iDh i) i).op) s = const R ((fun i => a i * h i ^ N) i) ((fun i => h i ^ (N + 1)) i) (PrimeSpectrum.basicOpen ((fun i => h i ^ (N + 1)) i)) ⋯"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "s✝ : Set E", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : Module 𝕜 E", "inst✝⁴ : Module ℝ E", "inst✝³ : SMulCommClass ℝ 𝕜 E", "inst✝² : TopologicalSpace E", "inst✝¹ : LocallyConvexSpace ℝ E", "inst✝ : ContinuousSMul 𝕜 E", "s : Set E", "hs : s ∈ 𝓝 0 ∧ Convex ℝ s"], "goal": "Balanced 𝕜 ((convexHull ℝ) (balancedCore 𝕜 s)) ∧ Convex ℝ ((convexHull ℝ) (balancedCore 𝕜 s))"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p : ℕ", "ExpChar R p", "ι : Type u_2", "s : Finset ι", "f : ι → R"], "goal": "(∑ i ∈ s, f i) ^ p = ∑ i ∈ s, f i ^ p"}}
+{"state": {"context": ["β : Type v", "γ : Type w", "δ : Type u_2", "TopologicalSpace β", "TopologicalSpace γ", "TopologicalSpace δ", "Zero δ", "g : CocompactMap β γ"], "goal": "ZeroAtInftyContinuousMap.comp 0 g = 0"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : Semigroup M", "inst✝¹ : TopologicalSpace M", "inst✝ : T2Space M", "continuous_mul_left : ∀ (r : M), Continuous fun x => x * r", "s : Set M", "snemp : s.Nonempty", "s_compact : IsCompact s", "s_add : ∀ x ∈ s, ∀ y ∈ s, x * y ∈ s", "M' : Type u_1 := { m // m ∈ s }", "this✝² : Semigroup M' := Semigroup.mk ⋯", "this✝¹ : CompactSpace M'", "this✝ : Nonempty M'", "this : ∀ (p : M'), Continuous fun x => x * p", "m : M", "hm : m ∈ s", "idem : ⟨m, hm⟩ * ⟨m, hm⟩ = ⟨m, hm⟩"], "goal": "∃ m ∈ s, m * m = m"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "inst✝ : AddGroup G", "T : AddTorsor G P", "p₁ p₂ p : P", "h : p -ᵥ p₁ = p -ᵥ p₂"], "goal": "p -ᵥ p₂ +ᵥ p₁ = p -ᵥ p₂ +ᵥ p₂"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop", "hn : n % 2 = 1"], "goal": "2 * (n / 2) = n - 1"}}
+{"state": {"context": ["α : Type u_1", "One α"], "goal": "AddOpposite.unop 1 = 1"}}
+{"state": {"context": ["Ω : Type u_1", "MeasurableSpace Ω", "MeasurableSingletonClass Ω", "s t u : Set Ω", "hs : s.Finite"], "goal": "(ProbabilityTheory.condCount s) (t ∩ u) = (ProbabilityTheory.condCount (s ∩ u)) t * (ProbabilityTheory.condCount s) u"}}
+{"state": {"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝⁴ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝³ : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝² : CommRing R", "inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀", "inst✝ : LinearOrderedCommMonoidWithZero Γ'₀", "v : Valuation R Γ₀", "a s : R", "h : s ∈ v.supp"], "goal": "v (a + s) = v a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p : α → Prop", "s : Finset α", "H : ∀ (x : α), x ∈ s ↔ p x", "inst✝ : Fintype { x // p x }"], "goal": "card { x // p x } = s.card"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X"], "goal": "(Homeomorph.refl X).toPartialHomeomorph = PartialHomeomorph.refl X"}}
+{"state": {"context": ["m n : ℕ"], "goal": "2 * n ≠ 2 * m + 1"}}
+{"state": {"context": ["m✝ n✝¹ n✝ a n : ℕ", "x : ℤ", "hl : x.natAbs ≤ n / 2", "m : ℤ", "hm : m ≠ 0"], "goal": "n / 2 + n / 2 ≤ (↑n * m + x - x).natAbs"}}
+{"state": {"context": [], "goal": "Tendsto eulerMascheroniSeq atTop (𝓝 eulerMascheroniConstant)"}}
+{"state": {"context": ["R✝ : Type u₁", "inst✝⁶ : CommSemiring R✝", "p : ℕ", "hp : Fact (Nat.Prime p)", "inst✝⁵ : CharP R✝ p", "R : Type u₁", "inst✝⁴ : CommSemiring R", "inst✝³ : CharP R p", "inst✝² : PerfectRing R p", "S : Type u₂", "inst✝¹ : CommSemiring S", "inst✝ : CharP S p", "f : R →+* Ring.Perfection S p", "r : R", "n : ℕ"], "goal": "(coeff S p 0) (f ((⇑(frobeniusEquiv R p).symm)^[n] r)) = (coeff S p n) (f r)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "x y : List α", "h : x <+: y", "hl : x.length < y.length"], "goal": "take x.length y ++ [y.get ⟨x.length, hl⟩] = take (x.length + 1) y"}}
+{"state": {"context": ["ι : Type u_4", "F : Type u_5", "Fintype ι", "NormedAddCommGroup F", "InnerProductSpace ℝ F", "FiniteDimensional ℝ F", "MeasurableSpace F", "BorelSpace F", "b : OrthonormalBasis ι ℝ F"], "goal": "b.toBasis.addHaar = MeasureTheory.volume"}}
+{"state": {"context": ["α : Type u_1", "x : α"], "goal": "¬List.Duplicate x []"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "D : LieDerivation R L M"], "goal": "(↑D).toFun = ⇑D"}}
+{"state": {"context": ["m n : ℕ", "h : m ≤ n"], "goal": "(n - m).gcd n = m.gcd n"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : LinearOrderedCommRing R", "a b : R", "h : IsCoprime a b", "h' : a ^ 2 + b ^ 2 = 0", "ha : a ^ 2 = 0", "hb : b ^ 2 = 0"], "goal": "False"}}
+{"state": {"context": ["M : Type u_1", "Add M", "s t : Set M", "h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s", "h_mul' : ∀ {a b : M}, a ∈ t → b ∈ t → a + b ∈ t"], "goal": "{ carrier := s, add_mem' := h_mul } ≤ { carrier := t, add_mem' := h_mul' } ↔ s ⊆ t"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "NontriviallyNormedField 𝕜", "NonUnitalNormedRing A", "NormedSpace 𝕜 A", "SMulCommClass 𝕜 A A", "IsScalarTower 𝕜 A A", "a : 𝓜(𝕜, A)"], "goal": "‖a‖₊ = ‖DoubleCentralizer.toProdHom a‖₊"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹⁵ : CommRing R", "inst✝¹⁴ : CommRing S", "inst✝¹³ : Algebra R S", "M : Type u_1", "inst✝¹² : AddCommGroup M", "inst✝¹¹ : Module R M", "inst✝¹⁰ : Module S M", "inst✝⁹ : IsScalarTower R S M", "A : Type u_2", "B : Type u_3", "inst✝⁸ : CommRing A", "inst✝⁷ : CommRing B", "inst✝⁶ : Algebra R A", "inst✝⁵ : Algebra R B", "inst✝⁴ : Algebra A B", "inst✝³ : Algebra S B", "inst✝² : IsScalarTower R A B", "inst✝¹ : IsScalarTower R S B", "inst✝ : SMulCommClass S A B", "h : Function.Surjective ⇑(algebraMap A B)", "y : A"], "goal": "(map R S A B) ((D R A) y) ∈ ↑(LinearMap.range (map R S A B))"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : E", "iso : E ≃ₗᵢ[𝕜] F"], "goal": "HasStrictFDerivAt (⇑iso) (↑{ toLinearEquiv := iso.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }) x"}}
+{"state": {"context": ["F : Type u_1", "R : Type u_2", "S : Type u_3", "A : Type u_4", "inst✝⁵ : Field R", "inst✝⁴ : Ring A", "inst✝³ : Algebra R A", "inst✝² : SetLike S A", "hSA : NonUnitalSubringClass S A", "hSRA : SMulMemClass S R A", "s : S", "h1 : 1 ∉ s", "inst✝¹ : FunLike F (Unitization R ↥s) A", "inst✝ : AlgHomClass F R (Unitization R ↥s) A", "f : F", "hf : ∀ (x : ↥s), f ↑x = ↑x", "r : R", "hr : r ≠ 0", "hr' : (algebraMap R A) r ∈ s"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "f : α → ℝ≥0∞", "hf : Measurable f", "hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤", "g : α → ℝ", "this : (fun x => g x * (f x).toReal) = fun x => (f x).toReal • g x"], "goal": "Integrable g (μ.withDensity f) ↔ Integrable (fun x => (f x).toReal • g x) μ"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "ι : Type u_2", "s : Finset ι", "v : ι → R", "DecidableEq ι"], "goal": "Polynomial.derivative (Lagrange.nodal s v) = ∑ i ∈ s, Lagrange.nodal (s.erase i) v"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E✝ : Type u", "inst✝⁷ : NormedAddCommGroup E✝", "inst✝⁶ : NormedSpace 𝕜 E✝", "F : Type v", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "ι : Type u_2", "E : ι → Type u_3", "inst✝³ : (i : ι) → NormedAddCommGroup (E i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E i)", "inst✝¹ : Fintype ι", "f : ContinuousMultilinearMap 𝕜 E F", "n : ℕ∞", "x : (i : ι) → E i", "inst✝ : DecidableEq ι", "y : (i : ι) → E i", "h✝ : Nonempty ι"], "goal": "(∑ a : { s // s.card = Fintype.card ι - 1 }, ((f.toFormalMultilinearSeries.changeOriginSeriesTerm 1 (Fintype.card ι - 1) ↑a ⋯) fun x_1 => x) fun x => y) = ∑ i : ι, f (Function.update x i (y i))"}}
+{"state": {"context": ["V : Type w'", "FirstOrder.Language.graph.Structure V", "V ⊨ FirstOrder.Language.Theory.simpleGraph", "x y : V"], "goal": "(FirstOrder.Language.simpleGraphOfStructure V).Adj x y =\n  FirstOrder.Language.Structure.RelMap FirstOrder.Language.adj ![x, y]"}}
+{"state": {"context": ["p : ℕ+", "K : Type u", "Field K", "CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p} ℚ K", "hζ : IsPrimitiveRoot ζ ↑p"], "goal": "hζ.integralPowerBasis'.dim = φ ↑p"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹² : Category.{u_4, u_1} C", "inst✝¹¹ : Category.{u_6, u_2} D", "F : C ⥤ D", "ι : Type u_3", "c : ComplexShape ι", "inst✝¹⁰ : Preadditive C", "inst✝⁹ : Preadditive D", "inst✝⁸ : CategoryWithHomology C", "inst✝⁷ : CategoryWithHomology D", "inst✝⁶ : (HomologicalComplex.quasiIso C c).HasLocalization", "inst✝⁵ : (HomologicalComplex.quasiIso D c).HasLocalization", "inst✝⁴ : F.Additive", "inst✝³ : F.PreservesHomology", "inst✝² : c.QFactorsThroughHomotopy C", "inst✝¹ : c.QFactorsThroughHomotopy D", "inst✝ : (HomotopyCategory.quotient C c).IsLocalization (HomologicalComplex.homotopyEquivalences C c)", "K : HomologicalComplex C c"], "goal": "(Localization.liftNatTrans (HomotopyCategory.quotient C c) (HomologicalComplex.homotopyEquivalences C c) (HomotopyCategory.quotient C c ⋙ HomologicalComplexUpToQuasiIso.Qh ⋙ F.mapHomologicalComplexUpToQuasiIso c) (HomotopyCategory.quotient C c ⋙ F.mapHomotopyCategory c ⋙ HomologicalComplexUpToQuasiIso.Qh) (HomologicalComplexUpToQuasiIso.Qh ⋙ F.mapHomologicalComplexUpToQuasiIso c) (F.mapHomotopyCategory c ⋙ HomologicalComplexUpToQuasiIso.Qh) (F.mapHomologicalComplexUpToQuasiIsoFactors c).hom).app ((HomotopyCategory.quotient C c).obj K) = (F.mapHomologicalComplexUpToQuasiIso c).map ((HomologicalComplexUpToQuasiIso.quotientCompQhIso C c).hom.app K) ≫ (F.mapHomologicalComplexUpToQuasiIsoFactors c).hom.app K ≫ (HomologicalComplexUpToQuasiIso.quotientCompQhIso D c).inv.app ((F.mapHomologicalComplex c).obj K) ≫ HomologicalComplexUpToQuasiIso.Qh.map ((F.mapHomotopyCategoryFactors c).inv.app K)"}}
+{"state": {"context": ["l : Type u_1", "R : Type u_2", "DecidableEq l", "Fintype l", "CommRing R", "A : ↥(Matrix.symplecticGroup l R)"], "goal": "↑A⁻¹ = (↑A)⁻¹"}}
+{"state": {"context": ["R : Type u", "M : Type v", "CommRing R", "AddCommGroup M", "Module R M"], "goal": "Module.Flat R M ↔ Module.Baer R (CharacterModule M)"}}
+{"state": {"context": [], "goal": "Nat.fib 2 = 1"}}
+{"state": {"context": ["M : Type u_3", "CommMonoid M", "n : ℕ", "i : Fin n", "v : Fin n.succ → M"], "goal": "∏ j ∈ Finset.Ioi i.succ, v j = ∏ j ∈ Finset.Ioi i, v j.succ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "T : CategoryTheory.Monad C", "J : Type u", "CategoryTheory.Category.{v, u} J", "D : CategoryTheory.Functor J T.Algebra", "c : CategoryTheory.Limits.Cocone (D.comp T.forget)", "t : CategoryTheory.Limits.IsColimit c", "CategoryTheory.Limits.PreservesColimit (D.comp T.forget) T.toFunctor", "CategoryTheory.Limits.PreservesColimit ((D.comp T.forget).comp T.toFunctor) T.toFunctor"], "goal": "(CategoryTheory.Monad.ForgetCreatesColimits.coconePoint c t).a = CategoryTheory.Monad.ForgetCreatesColimits.lambda c t"}}
+{"state": {"context": ["K : Type u_1", "NontriviallyNormedField K", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace K F", "m : ℕ", "f : K → F", "hf : ContDiff K (↑m + 1) f"], "goal": "Differentiable K (iteratedDeriv m f)"}}
+{"state": {"context": ["R : Type u", "M : Type v", "CommSemiring R", "AddCommMonoid M", "Module R M", "I : Ideal R", "M₁ M₂ : Submodule R M"], "goal": "I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "n : ℕ", "f : 𝕜 → F", "s : Set 𝕜", "x✝ x : 𝕜", "hxs : UniqueDiffWithinAt 𝕜 s x"], "goal": "iteratedDerivWithin (n + 1) f s x = derivWithin (iteratedDerivWithin n f s) s x"}}
+{"state": {"context": ["C : Type u_1", "ι : Type u_2", "inst✝⁵ : Category.{?u.55275, u_1} C", "inst✝⁴ : Abelian C", "c : ComplexShape ι", "S : ShortComplex (HomologicalComplex C c)", "hS : S.Exact", "inst✝³ : Epi S.g", "i : ι", "inst✝² : S.X₁.HasHomology i", "inst✝¹ : S.X₂.HasHomology i", "inst✝ : S.X₃.HasHomology i"], "goal": "opcyclesMap S.f i ≫ opcyclesMap S.g i = 0"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "z : K"], "goal": "RCLike.re (RCLike.I * z) = -RCLike.im z"}}
+{"state": {"context": ["R : Type u_2", "M₁ : Type u_4", "M₂ : Type u_5", "N : Type u_8", "CommSemiring R", "AddCommMonoid M₁", "AddCommMonoid M₂", "AddCommMonoid N", "Module R M₁", "Module R M₂", "Module R N", "Q₁ : QuadraticMap R M₁ N", "Q₂ : QuadraticMap R M₂ N", "f : Q₁ →qᵢ Q₂"], "goal": "f.comp (QuadraticMap.Isometry.id Q₁) = f"}}
+{"state": {"context": ["α : Type u_2", "Monoid α", "s t : Set α", "hst : s ⊆ t", "n : ℕ"], "goal": "s ^ n ⊆ t ^ n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : FirstCountableTopology X", "inst✝ : T1Space X", "x : X", "U : ℕ → Set X", "hU : ∀ (i : ℕ), x ∈ U i ∧ IsOpen (U i)", "h_basis : (𝓝 x).HasAntitoneBasis fun i => U i"], "goal": "IsGδ {x}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : UniformSpace α", "inst✝¹ : UniformSpace β", "inst✝ : CompactSpace α"], "goal": "𝓤 α ≤ 𝓝ˢ (diagonal α)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Finite β", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "σ τ : Perm α", "h0 : Nat.Prime (Fintype.card α)", "h1 : σ.IsCycle", "h2 : σ.support = univ", "x y : α", "h4 : x ≠ y", "h5 : τ = swap x y", "i : ℕ", "hi : (σ ^ i) x = y", "m : ℕ", "hm : i = Fintype.card α * m"], "goal": "x = y"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝¹⁴ : Field K", "inst✝¹³ : CommRing R", "inst✝¹² : Nontrivial R", "inst✝¹¹ : LieRing L", "inst✝¹⁰ : LieAlgebra K L", "inst✝⁹ : LieAlgebra R L", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "inst✝⁶ : LieRingModule L M", "inst✝⁵ : LieModule R L M", "inst✝⁴ : Module.Finite K L", "inst✝³ : Module.Finite R L", "inst✝² : Module.Free R L", "inst✝¹ : Module.Finite R M", "inst✝ : Module.Free R M", "x y : L", "r : R", "b : Basis (ChooseBasisIndex R L) R L := chooseBasis R L"], "goal": "Polynomial.map ((evalRingHom r).comp ↑(MvPolynomial.aeval fun i => C ((b.repr y) i) * X + C ((b.repr x) i))) ((↑φ).polyCharpoly b) = charpoly (φ (r • y + x))"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁰ : MeasurableSpace G", "inst✝⁹ : Group G", "inst✝⁸ : MeasurableMul₂ G", "μ ν : Measure G", "inst✝⁷ : SFinite ν", "inst✝⁶ : SFinite μ", "s : Set G", "inst✝⁵ : MeasurableInv G", "inst✝⁴ : μ.IsMulLeftInvariant", "μ' ν' : Measure G", "inst✝³ : SigmaFinite μ'", "inst✝² : SigmaFinite ν'", "inst✝¹ : μ'.IsMulLeftInvariant", "inst✝ : ν'.IsMulLeftInvariant", "sm : MeasurableSet s", "h2s : ν' s ≠ 0", "h3s : ν' s ≠ ⊤", "f : G → ℝ≥0∞", "hf : Measurable f", "g : G → ℝ≥0∞ := fun y => f y⁻¹ / ν' ((fun x => x * y⁻¹) ⁻¹' s)"], "goal": "μ' s * lintegral ν' g = ∫⁻ (x : G), f x ∂μ'"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "G : SimpleGraph V", "G' : SimpleGraph V'", "f : G →g G'", "hinj : Function.Injective ⇑f", "u v : V", "p : G.Path u v"], "goal": "↑(SimpleGraph.Path.map f hinj p) = SimpleGraph.Walk.map f ↑p"}}
+{"state": {"context": ["θ₂ : Angle", "x✝ : ℝ"], "goal": "(↑x✝ + θ₂).expMapCircle = (↑x✝).expMapCircle * θ₂.expMapCircle"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : ProbabilityTheory.Kernel α (β × ℝ)", "ν : ProbabilityTheory.Kernel α β", "f : α × β → ℚ → ℝ", "ProbabilityTheory.IsFiniteKernel κ", "hf : ProbabilityTheory.IsRatCondKernelCDF f κ ν", "a : α", "x : ℝ"], "goal": "∫ (b : β), ↑(ProbabilityTheory.stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (Set.univ ×ˢ Set.Iic x)).toReal"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "h : MeasureTheory.NullMeasurableSet s μ"], "goal": "MeasureTheory.toMeasurable μ s =ᵐ[μ] s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β", "r : Set β", "hf : Set.BijOn f s t", "hrt : r ⊆ t"], "goal": "Set.BijOn f (s ∩ f ⁻¹' r) r"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "I✝ : Ideal R", "a b : R", "S : Type v", "x y : R", "I : Ideal R"], "goal": "IsDomain (R ⧸ I) ↔ I.IsPrime"}}
+{"state": {"context": ["ι : Type u_1", "β : Type u_4", "CommMonoid β", "κ : Type u_7", "s : Finset ι", "t : ι → Set κ", "f : κ → β"], "goal": "(↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "f : α → β", "s : Set α"], "goal": "StrictAntiOn (f ∘ ⇑OrderDual.ofDual) s ↔ StrictMonoOn f s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "a b : α", "ha : a ∉ s", "hb : b ∈ s"], "goal": "a ∉ s \\ {b}"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "ι : Type u_5", "LinearOrder α", "OrderTop α", "s : Finset ι", "f : ι → α", "SemilatticeInf β", "OrderTop β", "g : α → β", "mono_g : Monotone g", "top : g ⊤ = ⊤"], "goal": "g (s.inf f) = s.inf (g ∘ f)"}}
+{"state": {"context": ["θ : Real.Angle"], "goal": "θ.sin ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ ↑π"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "E : ι → Type wE", "Fintype ι", "NontriviallyNormedField 𝕜", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "ι' : Type v'", "Fintype ι'", "E' : ι' → Type wE'", "(i' : ι') → SeminormedAddCommGroup (E' i')", "(i' : ι') → NormedSpace 𝕜 (E' i')", "f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')"], "goal": "‖ContinuousMultilinearMap.pi f‖ = ‖f‖"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "htwo : p ^ (k + 1) ≠ 2", "n : ℤ", "x : 𝓞 K", "pB : PowerBasis ℤ (𝓞 K) := hζ.integralPowerBasis", "hdim : 1 < pB.dim", "h : (pB.basis.repr hζ.toInteger) ⟨1, hdim⟩ = (pB.basis.repr (↑↑p * x + ↑n)) ⟨1, hdim⟩", "this : pB.gen = hζ.toInteger"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Multiset α", "t : Multiset β", "s₁ s₂ : Multiset α", "t₁ t₂ : Multiset β", "a✝ : α", "b : β", "x : α ⊕ β", "a : α", "left✝ : a ∈ s", "ha : inl a = inr b"], "goal": "False"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type u_1", "L.Structure M", "L.Structure N", "φ : L.Hom M N", "S : L.Substructure N"], "goal": "↑(FirstOrder.Language.Substructure.comap φ S) = ⇑φ ⁻¹' ↑S"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁶ : MeasurableSpace α", "μ : Measure α", "𝕜 : Type u_2", "inst✝⁵ : RCLike 𝕜", "E : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedSpace 𝕜 E", "H : Type u_4", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "F✝ : H → α → E", "x₀✝ : H", "bound : α → ℝ", "ε : ℝ", "F : 𝕜 → α → E", "x₀ : 𝕜", "F' : α → E", "ε_pos : 0 < ε", "hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ", "hF_int : Integrable (F x₀) μ", "hF'_meas : AEStronglyMeasurable F' μ", "h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (ball x₀ ε)", "bound_integrable : Integrable bound μ", "L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (L (F' a)) x₀", "hm : AEStronglyMeasurable (⇑L ∘ F') μ", "key : HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀", "hF'_int : Integrable F' μ", "hE : CompleteSpace E"], "goal": "HasDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasKernels C", "CategoryTheory.Limits.HasCokernels C"], "goal": "∀ {X Y : CategoryTheory.ShortComplex C} (φ : X ⟶ Y),\n  (CategoryTheory.ShortComplex.rightHomologyFunctor C).map φ = CategoryTheory.ShortComplex.rightHomologyMap φ"}}
+{"state": {"context": ["n : ℕ", "x y : ℤ", "he : ↑x = ↑y", "hl : x.natAbs ≤ n / 2"], "goal": "x.natAbs ≤ y.natAbs"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "p : ι → Prop", "s : ι → Set α", "𝓕 : Filter ι", "a : α"], "goal": "blimsup s cofinite p = {x | {n | p n ∧ x ∈ s n}.Infinite}"}}
+{"state": {"context": ["G : Type u_1", "MeasurableSpace G", "Group G", "MeasurableMul₂ G", "μ : MeasureTheory.Measure G", "MeasureTheory.SFinite μ", "s : Set G", "MeasurableInv G", "μ.IsMulLeftInvariant", "y : G"], "goal": "μ ((fun x => x * y) ⁻¹' s) = 0 ↔ μ s = 0"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "m m0 : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "μ : Measure α", "f : α → E", "s : Set α", "hs : MeasurableSet s", "hf : f =ᶠ[ae (μ.restrict s)] 0", "hm : m ≤ m0", "hμm this : SigmaFinite (μ.trim hm)"], "goal": "μ[f|m] =ᶠ[ae (μ.restrict s)] 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "h :\n  ∀ (u : ℕ → E),\n    (Summable fun x => ‖u x‖) → ∃ a, Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, u i) Filter.atTop (𝓝 a)"], "goal": "CompleteSpace E"}}
+{"state": {"context": ["α : Type u_1", "DivisionMonoid α", "a : α", "n : ℤ"], "goal": "a⁻¹ ^ n = (a ^ n)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : MeasurableSpace α'", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace β'", "inst✝³ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝² : NormedAddCommGroup E", "inst✝¹ : SFinite ν", "inst✝ : SFinite μ", "s : Set α", "t : Set β"], "goal": "(μ.restrict s).prod (ν.restrict t) = (μ.prod ν).restrict (s ×ˢ t)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "DivisionRing k", "AddCommGroup V", "Module k V", "AffineSpace V P", "DecidableEq P", "p : ι → P", "hi : AffineIndependent k p", "s : Finset ι", "sp : AffineSubspace k P", "FiniteDimensional k ↥sp.direction", "hle : affineSpan k ↑(Finset.image p s) ≤ sp", "hc : s.card = FiniteDimensional.finrank k ↥sp.direction + 1"], "goal": "affineSpan k ↑(Finset.image p s) = sp"}}
+{"state": {"context": ["R : Type u", "inst✝¹⁸ : CommRing R", "S : Type v", "inst✝¹⁷ : CommRing S", "f✝ : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝¹⁶ : Algebra R S", "K : Type u_1", "inst✝¹⁵ : Field K", "inst✝¹⁴ : Algebra R K", "hRK : IsFractionRing R K", "L : Type u_2", "inst✝¹³ : Field L", "inst✝¹² : Algebra S L", "inst✝¹¹ : IsFractionRing S L", "V : Type u_3", "V' : Type u_4", "V'' : Type u_5", "inst✝¹⁰ : AddCommGroup V", "inst✝⁹ : Module R V", "inst✝⁸ : Module K V", "inst✝⁷ : IsScalarTower R K V", "inst✝⁶ : AddCommGroup V'", "inst✝⁵ : Module R V'", "inst✝⁴ : Module S V'", "inst✝³ : IsScalarTower R S V'", "inst✝² : AddCommGroup V''", "inst✝¹ : Module R V''", "inst✝ : IsDedekindDomain R", "hRS : RingHom.ker (algebraMap R S) ≠ ⊤", "f : V'' →ₗ[R] V", "hf : Function.Injective ⇑f", "f' : V'' →ₗ[R] V'", "ι : Type u_6", "b : ι → V''", "s : Finset ι", "g : ι → K", "eq✝ : ∑ i ∈ s, g i • (⇑f ∘ b) i = 0", "j' : ι", "hj's : j' ∈ s", "hj'g : ¬g j' = 0", "a : K", "j : ι", "hjs : j ∈ s", "hgI✝ : a * g j ∉ ↑(RingHom.ker (algebraMap R S))", "g'' : (i : ι) → i ∈ s → R", "hg'' : ∀ (i : ι) (a_1 : i ∈ s), (algebraMap R K) (g'' i a_1) = a * g i", "this : (a : Prop) → Decidable a := Classical.propDecidable", "g' : ι → R := fun i => if h : i ∈ s then g'' i h else 0", "hg' : ∀ i ∈ s, (algebraMap R K) (g' i) = a * g i", "hgI : (algebraMap R S) (g' j) ≠ 0", "eq : f (∑ i ∈ s, g' i • b i) = 0"], "goal": "∑ i ∈ s, (fun i => (algebraMap R S) (g' i)) i • (⇑f' ∘ b) i = 0"}}
+{"state": {"context": ["E : Type u", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "s : Set E", "a x y : E", "h : IsLocalMinOn f s a", "hf : HasFDerivWithinAt f f' s a", "hy : y ∈ posTangentConeAt s a", "hy' : -y ∈ posTangentConeAt s a"], "goal": "f' y = 0"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "x : MvPolynomial σ R", "s : Set σ", "this : x ∈ Ideal.span ((fun x => (monomial (Finsupp.single x 1)) 1) '' s) ↔ ∀ xi ∈ x.support, ∃ si ∈ (fun i => Finsupp.single i 1) '' s, si ≤ xi"], "goal": "x ∈ Ideal.span (X '' s) ↔ ∀ m ∈ x.support, ∃ i ∈ s, m i ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "s : Set H", "x : H"], "goal": "ContinuousWithinAt (↑I) s x"}}
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+{"state": {"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₃ : Type u_3", "J : Type u_4", "Zero I₂", "r : I₁ × I₂ × I₃ → J", "π : I₁ × I₃ → J", "self : CategoryTheory.GradedObject.TriangleIndexData r π", "i : I₁ × I₂ × I₃"], "goal": "π (i.1, self.p₂₃ i.2) = r i"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "h01 : 0 = 1"], "goal": "IsArtinianRing R"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "S : Set E", "α : E", "ι : Type u_3", "f : ι → IntermediateField F E", "x : E", "hx : x ∈ ⨆ i, f i", "i : ι"], "goal": "f i ≤ ⨆ i, F⟮↑i.snd⟯"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "G : SimpleGraph V", "a b c u v✝ w : V", "e✝ : Sym2 V", "inst✝ : DecidableEq V", "v : V", "e : Sym2 V", "he : e ∈ G.edgeSet", "hv : v ∈ e"], "goal": "G.otherVertexOfIncident ⋯ ∈ G.neighborSet v"}}
+{"state": {"context": ["r : ℝ", "hr : 0 < r", "x : ℂ", "hx : x ≠ 0"], "goal": "Complex.log (x * ↑r) = ↑(Real.log r) + Complex.log x"}}
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+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "n : ℕ"], "goal": "↑↑n = ↑n"}}
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+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p₁ p₂ p₃ : P", "h : Sbtw ℝ p₁ p₂ p₃"], "goal": "EuclideanGeometry.angle p₁ p₃ p₂ = 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1"], "goal": "p.leadingCoeff = 0 ↔ p.degree = ⊥"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : Monoid α", "s t : Set α", "a : α", "m n✝ : ℕ", "ha : a ∈ s", "n : ℕ"], "goal": "a ^ (n + 1) ∈ s ^ (n + 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PartialOrder α", "PartialOrder β", "TopologicalSpace α", "TopologicalSpace β", "OrderTopology α", "OrderTopology β", "e : α ≃o β"], "goal": "Continuous ⇑e"}}
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+{"state": {"context": ["α : Type u_1", "inst✝ : AddGroupWithOne α", "a b : PosNum"], "goal": "↑(pos a + neg b) = ↑(pos a) + ↑(neg b)"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p q : ℝ≥0", "h : 0 < b → b < a → c ≠ ⊤"], "goal": "(a - b) * c = a * c - b * c"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort uι", "MeasurableSpace α", "Countable ι", "p : ι → α → Prop", "hp : ∀ (i : ι), Measurable (p i)"], "goal": "Measurable fun a => ∃ i, p i a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝¹ : GroupWithZero α", "inst✝ : MulAction α β", "a✝ a : α", "ha : a ≠ 0", "x y : β", "h : a⁻¹ • x = y"], "goal": "x = a • y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : Filter α", "g : Filter β", "h : Filter γ"], "goal": "Filter.map (⇑(Equiv.prodAssoc α β γ)) ((f ×ˢ g) ×ˢ h) = f ×ˢ g ×ˢ h"}}
+{"state": {"context": ["ι : Type u_1", "α✝ : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "α : ι → Type u_6", "inst✝² : (i : ι) → LinearOrder (α i)", "inst✝¹ : ∀ (i : ι), IsWellOrder (α i) fun x x_1 => x < x_1", "inst✝ : Finite ι", "s : Set ((i : ι) → α i)", "val✝ : Fintype ι"], "goal": "s.IsPWO"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝⁴ : CommMonoid M", "inst✝³ : CommMonoid N", "inst✝² : DivisionCommMonoid G", "k l✝ : ℕ", "ζ : M", "f : F", "h✝ : IsPrimitiveRoot ζ k", "inst✝¹ : FunLike F M N", "inst✝ : MonoidHomClass F M N", "h : IsPrimitiveRoot (f ζ) k", "hf : Injective ⇑f", "l : ℕ", "hl : f ζ ^ l = 1"], "goal": "orderOf (f ζ) ∣ l"}}
+{"state": {"context": ["α : Type u_2", "Ring α", "self : Ring.PositiveCone α", "a b : ��"], "goal": "self.pos a → self.pos b → self.pos (a * b)"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R", "s : Set R", "p : R → R → Prop", "a b : R", "ha : a ∈ NonUnitalSubring.closure s", "hb : b ∈ NonUnitalSubring.closure s", "Hs : ∀ x ∈ s, ∀ y ∈ s, p x y", "H0_left : ∀ (x : R), p 0 x", "H0_right : ∀ (x : R), p x 0", "Hneg_left : ∀ (x y : R), p x y → p (-x) y", "Hneg_right : ∀ (x y : R), p x y → p x (-y)", "Hadd_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ + x₂) y", "Hadd_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ + y₂)", "Hmul_left : ∀ (x₁ x₂ y : R), p x₁ y → p x₂ y → p (x₁ * x₂) y", "Hmul_right : ∀ (x y₁ y₂ : R), p x y₁ → p x y₂ → p x (y₁ * y₂)"], "goal": "p a b"}}
+{"state": {"context": ["Ω : Type u_1", "f g : Ω → ℝ≥0∞", "Mf Mg mΩ : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "hMf : Mf ≤ mΩ", "hMg : Mg ≤ mΩ", "h_ind : ProbabilityTheory.Indep Mf Mg μ", "h_meas_f : Measurable f", "h_meas_g : Measurable g"], "goal": "∫⁻ (ω : Ω), f ω * g ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), g ω ∂μ"}}
+{"state": {"context": ["x : ℕ", "r : ℝ"], "goal": "Nat.cast ⁻¹' Metric.closedBall (↑x) r = Metric.closedBall x r"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "a✝ b✝ c✝ : Cat", "f✝ g✝ : a✝ ⟶ b✝", "η✝ : f✝ ⟶ g✝", "h✝ : b✝ ⟶ c✝"], "goal": "(prelaxfunctor.map₂ (Bicategory.whiskerRight η✝ h✝)).app = (((fun {a b c} => mapComp) f✝ h✝).hom ≫ Bicategory.whiskerRight (prelaxfunctor.map₂ η✝) (prelaxfunctor.map h✝) ≫ ((fun {a b c} => mapComp) g✝ h✝).inv).app"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "β : ι → Type u_3", "γ : ι → Type u_4", "inst✝ : DecidableEq ι", "f g : ⦃i : ι⦄ → α i → β i → Finset (γ i)", "a✝ : (i : ι) × α i", "b✝ : (i : ι) × β i", "h : ∀ ⦃i : ι⦄ ⦃a : α i⦄ ⦃b : β i⦄, f a b ⊆ g a b", "a : (i : ι) × α i", "b : (i : ι) × β i", "x : Sigma γ", "hx : x ∈ sigmaLift f a b"], "goal": "x ∈ sigmaLift g a b"}}
+{"state": {"context": ["V : Type u_1", "W : Type u_2", "G : SimpleGraph V", "G' : SimpleGraph W", "f : G ↪g G'", "e : ↑G.edgeSet"], "goal": "f.mapEdgeSet e = SimpleGraph.Hom.mapEdgeSet (RelEmbedding.toRelHom f) e"}}
+{"state": {"context": ["m n a b : ℕ", "hn : a ≡ b [MOD n]", "hm : a ≡ b [MOD m]"], "goal": "a ≡ b [MOD n.lcm m]"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop"], "goal": "Set.range Subtype.val = {x | p x}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : PseudoMetrizableSpace β", "inst✝⁴ : MeasurableSpace β", "inst✝³ : BorelSpace β", "ι : Type u_3", "inst✝² : Countable ι", "inst✝¹ : Nonempty ι", "μ : Measure α", "f : ι → α → β", "L : Filter ι", "inst✝ : L.IsCountablyGenerated", "hf : ∀ (n : ι), AEMeasurable (f n) μ", "h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)", "inhabited_h : Inhabited ι", "hL : L.NeBot", "p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)", "hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x"], "goal": "∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f g : E → F", "p pf pg : FormalMultilinearSeries 𝕜 E F", "x : E", "r r' : ℝ≥0∞", "hf : HasFPowerSeriesOnBall f p x r", "A : ContinuousOn (fun y => y - x) (EMetric.ball x r)"], "goal": "MapsTo (fun y => y - x) (EMetric.ball x r) (EMetric.ball 0 r)"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "Group G", "Group H", "f : G → H", "hf : IsGroupHom f", "a b : G"], "goal": "f a = f b ↔ f (a⁻¹ * b) = 1"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "CountableInterFilter l", "p : Set α → Prop", "HasCountableSeparatingOn α p Set.univ", "hl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l"], "goal": "∃ s, s.Subsingleton ∧ s ∈ l"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "inst✝¹ : CompleteSpace F", "μ ν : Measure α", "s t : Set α", "f✝ g : α → ℝ≥0∞", "inst✝ : IsProbabilityMeasure μ", "f : α → ℝ≥0∞"], "goal": "⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂μ"}}
+{"state": {"context": ["k : Type u_1", "Field k", "K : Type u_2", "Field K", "F : Type u_3", "Field F", "w : NumberField.InfinitePlace K", "f : F →+* K", "g : k →+* F"], "goal": "w.comap (f.comp g) = (w.comap f).comap g"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "P : Algebra.Generators R S", "R' : Type u'", "S' : Type v'", "CommRing R'", "CommRing S'", "Algebra R' S'", "P' : Algebra.Generators R' S'", "Algebra R R'", "Algebra S S'", "Algebra R S'", "IsScalarTower R R' S'", "IsScalarTower R S S'", "f : P.Hom P'", "x : P.Cotangent"], "goal": "(Algebra.Generators.CotangentSpace.map f) (P.cotangentComplex x) =\n  P'.cotangentComplex ((Algebra.Generators.Cotangent.map f) x)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S : SimplicialObject.Split C"], "goal": "SimplicialObject.Split.nondegComplexFunctor.obj S = S.s.nondegComplex"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Monoid (G i)", "inst✝ : Monoid H", "φ : (i : ι) → H →* G i", "i : ι", "x : H"], "goal": "((of i).comp (φ i)) x = (base φ) x"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddCommMonoid A", "inst✝ : HasShift C A", "X Y : C", "f : X ⟶ Y", "i j : A"], "goal": "(shiftFunctor C j).map ((shiftFunctor C i).map f) = (shiftComm X i j).hom ≫ (shiftFunctor C i).map ((shiftFunctor C j).map f) ≫ (shiftComm Y j i).hom"}}
+{"state": {"context": ["R : Type u_5", "S : Type u_6", "Semiring R", "OrderedSemiring S", "f : R →ₙ* S", "h₁ : ∀ (x : R), 0 ≤ f.toFun x", "h₂ : ∀ (x : R), f.toFun x = 0 ↔ x = 0", "h₃ : ∀ (x y : R), f.toFun (x + y) ≤ f.toFun x + f.toFun y"], "goal": "⇑{ toMulHom := f, nonneg' := h₁, eq_zero' := h₂, add_le' := h₃ } = ⇑f"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra R L", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : LieRingModule L M", "inst✝ : LieModule R L M", "N N' : LieSubmodule R L M", "I J : LieIdeal R L", "s t : Set M", "h : s ⊆ t"], "goal": "s ⊆ ↑(lieSpan R L t)"}}
+{"state": {"context": ["x✝¹ x✝ : ℝ", "h : ∃ k, 2 * (x✝¹ + x✝) - π = 2 * π * ↑k"], "goal": "(↑x✝).tan = (↑x✝¹).tan⁻¹"}}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "F : Type w", "inst✝⁷ : CommRing R", "inst✝⁶ : Semiring S✝", "R₁ : Type u_1", "R₂ : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝⁵ : CommSemiring R₁", "inst✝⁴ : CommSemiring R₂", "inst✝³ : CommRing A", "inst✝² : Algebra R₁ A", "inst✝¹ : Algebra R₂ A", "S : Type v", "inst✝ : CommRing S", "I : Ideal R", "J : Ideal S", "f : R ≃+* S", "hIJ : J = map (↑f) I"], "goal": "map (↑f) I ≤ comap (↑f.symm) I"}}
+{"state": {"context": ["n : ℕ", "R : Type u_1", "inst✝³ : CommRing R", "inst✝² : IsJacobson R", "σ : Type u_2", "inst✝¹ : _root_.Finite σ", "S : Type u_3", "inst✝ : Field S", "f : MvPolynomial σ R →+* S", "hf : Function.Surjective ⇑f"], "goal": "(f.comp MvPolynomial.C).IsIntegral"}}
+{"state": {"context": ["ctx : Nat.Linear.Context", "e : Nat.SOM.Expr"], "goal": "Nat.SOM.Poly.denote ctx e.toPoly = Nat.SOM.Expr.denote ctx e"}}
+{"state": {"context": ["α : Type u", "n : Nat"], "goal": "List.nil.rotateLeft n = List.nil"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β₁ : Type u_3", "β₂ : Type u_4", "γ₁ : Type u_5", "γ₂ : Type u_6", "f : α₁ → β₁ → Finset γ₁", "g : α₂ → β₂ → Finset γ₂", "a : α₁ ⊕ α₂", "b : β₁ ⊕ β₂", "c₁ : γ₁"], "goal": "Sum.inl c₁ ∈ Finset.sumLift₂ f g a b ↔ ∃ a₁ b₁, a = Sum.inl a₁ ∧ b = Sum.inl b₁ ∧ c₁ ∈ f a₁ b₁"}}
+{"state": {"context": [], "goal": "|rexp 1 - expNear 0 1 (2244083 / 825552)| ≤ |1| ^ 0 / ↑(Nat.factorial 0) * (1 / 10 ^ 10)"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommRing R", "p : MvPolynomial σ R"], "goal": "(-p).support = p.support"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : DecidableEq α", "inst✝¹ : Fintype α", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "s✝ : Finset α", "P : Finpartition univ", "ε : ℝ", "hε : 0 < ε", "hP₁ : P.IsEquipartition", "hP₃ : P.parts.card ≤ bound (ε / 8) ⌈4 / ε⌉₊", "x y z : α", "s : Finset α", "hX : s ∈ P.parts", "Y : Finset α", "hY : Y ∈ P.parts", "xX : x ∈ s", "yY : y ∈ Y", "nXY : s ≠ Y", "uXY : G.IsUniform (ε / 8) s Y", "dXY : ε / 4 ≤ ↑(G.edgeDensity s Y)", "Z : Finset α", "hZ : Z ∈ P.parts", "zZ : z ∈ Z", "hX' : s ∈ P.parts", "xX' : x ∈ s", "nXZ : s ≠ Z", "uXZ : G.IsUniform (ε / 8) s Z", "dXZ : ε / 4 ≤ ↑(G.edgeDensity s Z)", "hY' : Y ∈ P.parts", "yY' : y ∈ Y", "hZ' : Z ∈ P.parts", "zZ' : z ∈ Z", "nYZ : Y ≠ Z", "uYZ : G.IsUniform (ε / 8) Y Z", "dYZ : ε / 4 ≤ ↑(G.edgeDensity Y Z)"], "goal": "triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(G.cliqueFinset 3).card"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "M : Submonoid R", "B : Type w", "CommRing B", "Algebra R B", "f : A →ₐ[R] B"], "goal": "Submonoid.map f (Algebra.algebraMapSubmonoid A M) = Algebra.algebraMapSubmonoid B M"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι"], "goal": "IsCompact (BoxIntegral.Box.Icc I)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "W✝ : Affine R", "f : R →+* S", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "h₁ : W.Equation x₁ y₁", "h₂ : W.Equation x₂ y₂", "hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂", "sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c"], "goal": "∃ x, (∃ a a_1 a_2, a_2 * (Y - W.negPolynomial) = x + a * C (X - C x₁) + a_1 * C (X - C x₂)) ∧ (AdjoinRoot.mk W.polynomial) x = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq β", "f : α → β", "s : Finset α", "t : Finset β", "h : ∀ x ∈ s, f x ∈ t"], "goal": "Finset.filter (fun y => y ∈ Finset.image f s) t = Finset.image f s"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝¹ : (i : ι) → Zero (α i)", "r : ι → ι → Prop", "s : (i : ι) → α i → α i → Prop", "hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0", "hs : ∀ (i : ι), WellFounded (s i)", "inst✝ : DecidableEq ι", "i✝ i : ι", "h✝ : ∀ (y : ι), (rᶜ ⊓ fun x x_1 => x ≠ x_1) y i → Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) y", "ih : ∀ (y : ι), (rᶜ ⊓ fun x x_1 => x ≠ x_1) y i → ∀ (a : α y), Acc (DFinsupp.Lex r s) (single y a)", "a✝ a : α i", "ha : ∀ (y : α i), s i y a → (fun x => Acc (DFinsupp.Lex r s) (single i x)) y"], "goal": "(fun x => Acc (DFinsupp.Lex r s) (single i x)) a"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : PseudoMetricSpace α", "b : ℝ"], "goal": "{b} = ⋂ r, ⋂ (_ : r > 0), Icc (b - r) (b + r)"}}
+{"state": {"context": [], "goal": "Finset.univ = ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : InnerProductSpace 𝕜 E", "inst✝⁴ : InnerProductSpace 𝕜 F", "inst✝³ : InnerProductSpace 𝕜 G", "inst✝² : CompleteSpace E", "inst✝¹ : CompleteSpace G", "inst✝ : CompleteSpace F", "A : F →L[𝕜] G", "B : E →L[𝕜] F", "v : G"], "goal": "(adjoint (A.comp B)) v = ((adjoint B).comp (adjoint A)) v"}}
+{"state": {"context": ["X : TopCat", "p₀ : ↑X", "inst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)", "C : Type v", "inst✝² : Category.{u, v} C", "A : C", "inst✝¹ : HasTerminal C", "inst✝ : HasColimits C", "𝓕 : Presheaf C X", "c : C", "f : 𝓕.stalk p₀ ⟶ c", "U : (OpenNhds p₀)ᵒᵖ"], "goal": "(𝓕.germ ⟨p₀, ⋯⟩ ≫ f ≫ eqToHom ⋯) ≫ eqToHom ⋯ = colimit.ι ((OpenNhds.inclusion p₀).op ⋙ 𝓕) U ≫ f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "NormedAddCommGroup β", "f : α → β", "x : α"], "goal": "Filter.Tendsto (fun M => {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) Filter.atTop (𝓝 0)"}}
+{"state": {"context": ["σ : Type u_1", "Fintype σ", "DecidableEq σ", "R : Type u_2", "CommRing R", "k : ℕ", "f : Finset σ × σ → MvPolynomial σ R"], "goal": "∑ t ∈ Finset.filter (fun t => t.1.card < k) (MvPolynomial.NewtonIdentities.pairs σ k), f t =\n  ∑ a ∈ Finset.filter (fun a => a.1 < k) (Finset.antidiagonal k),\n    ∑ A ∈ Finset.powersetCard a.1 Finset.univ, ∑ j : σ, f (A, j)"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : Distrib M", "a b : M", "ha : a ∈ center M", "hb : b ∈ center M", "x✝¹ x✝ : M"], "goal": "(a + b) * (x✝¹ * x✝) = (a + b) * x✝¹ * x✝"}}
+{"state": {"context": [], "goal": "@AlgebraicGeometry.UniversallyClosed = (AlgebraicGeometry.topologically @IsClosedMap).universally"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "hp : 0 < ↑p.natTrailingDegree"], "goal": "p.coeff (p.natTrailingDegree - 1) = 0"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℝ≥0∞"], "goal": "∑' (a : α), f a = ⨆ s, ∑ a ∈ s, f a"}}
+{"state": {"context": ["K : Type u", "A : Type v", "inst✝² : Field K", "inst✝¹ : Ring A", "inst✝ : Algebra K A", "x : A", "p : K[X]", "hp : p ≠ 0", "hpx : (aeval x) p = 0"], "goal": "eval₂ (algebraMap K A) x (p * C p.leadingCoeff⁻¹) = 0"}}
+{"state": {"context": ["R : Type uR", "ι : Type uι", "M₁ : ι → Type v₁", "M₂ : Type v₂", "CommSemiring R", "(i : ι) → AddCommMonoid (M₁ i)", "AddCommMonoid M₂", "(i : ι) → Module R (M₁ i)", "Module R M₂", "M₁' : ι → Type u_1", "(i : ι) → AddCommMonoid (M₁' i)", "(i : ι) → Module R (M₁' i)", "g : MultilinearMap R M₁' M₂"], "goal": "∀ (a : (i : ι) → (fun i => M₁ i →ₗ[R] M₁' i) i) (m : (i : ι) ��� M₁ i),\n  ((MultilinearMap.piLinearMap g) a) m = g fun i => (a i) (m i)"}}
+{"state": {"context": ["α : Type u", "Nonempty α", "l : List α"], "goal": "l.rotate 1 = l ↔ ∃ a, l = List.replicate l.length a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "W : Affine R", "f : R →+* S", "inst✝ : IsDomain R", "p q : R[X]", "hdp : (p ^ 2).degree = 2 • p.degree", "hdpq : (p * q * (C W.a₁ * X + C W.a₃)).degree ≤ p.degree + q.degree + 1", "hdq : (q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)).degree = 2 • q.degree + 3"], "goal": "(p ^ 2 - p * q * (C W.a₁ * X + C W.a₃) - q ^ 2 * (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)).degree = max (2 • p.degree) (2 • q.degree + 3)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "N : LieSubmodule R L M"], "goal": "N = ⊥ ↔ ∀ m ∈ N, m = 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C"], "goal": "∀ {X Y : CategoryTheory.CosimplicialObject C} (f : X ⟶ Y),\n  (AlgebraicTopology.alternatingCofaceMapComplex C).map f = AlgebraicTopology.AlternatingCofaceMapComplex.map f"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "f : R[X]", "T : Type u_1", "inst✝³ : CommRing T", "inst✝² : Algebra R T", "U : Type u_2", "inst✝¹ : CommRing U", "inst✝ : Algebra R U", "h : IsAdjoinRoot S f", "h' : IsAdjoinRoot T f", "h'' : IsAdjoinRoot U f", "z : S"], "goal": "((h.aequiv h').trans (h'.aequiv h'')) z = (h.aequiv h'') z"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "B : Type u_3", "Semiring B", "TopologicalSpace B", "Algebra R A", "Algebra R B", "T2Space B", "s : Set A", "f g : A →A[R] B", "h : Set.EqOn (⇑f) (⇑g) s"], "goal": "Set.EqOn (⇑f) (⇑g) (closure ↑(Algebra.adjoin R s))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : InnerProductSpace 𝕜 E", "inst✝⁴ : InnerProductSpace 𝕜 F", "inst✝³ : InnerProductSpace 𝕜 G", "inst✝² : CompleteSpace E", "inst✝¹ : CompleteSpace G", "inst✝ : CompleteSpace F", "A : E →L[𝕜] F", "v : E"], "goal": "(adjointAux (adjointAux A)) v = A v"}}
+{"state": {"context": ["f : ℝ → ℝ", "s : Set ℝ", "hf : MonotoneOn f s"], "goal": "∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X : SimplicialObject C", "Y : C", "n a q : ℕ", "φ : Y ⟶ X _[n + 1]", "v : HigherFacesVanish q φ", "hnaq : n = a + q", "hnaq_shift : ∀ (d : ℕ), n + d = a + d + q", "simplif : ∀ (a b c d e f : Y ⟶ X _[n + 1]), b = f → d + e = 0 → c + a = 0 → a + b + (c + d + e) = f"], "goal": "∑ i : Fin (a + 1), ((-1) ^ a * (-1) ^ ↑i) • φ ≫ X.δ ⟨↑i, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ + ((-1) ^ a * (-1) ^ (a + 1)) • φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ + (∑ i : Fin (a + 1), ((-1) ^ ↑i * (-1) ^ (a + 1)) • φ ≫ X.σ ⟨a + 1, ⋯⟩ ≫ X.δ ⟨↑i, ⋯⟩ + ((-1) ^ (a + 1) * (-1) ^ (a + 1)) • φ ≫ X.σ ⟨a + 1, ⋯⟩ ≫ X.δ ⟨a + 1, ⋯⟩ + ((-1) ^ (a + 2) * (-1) ^ (a + 1)) • φ ≫ X.σ ⟨a + 1, ⋯⟩ ≫ X.δ ⟨a + 2, ⋯⟩) = -φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩"}}
+{"state": {"context": [], "goal": "-1 < 1"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "X Y : C", "φ : W.RightFraction X Y"], "goal": "φ.op.f = φ.f.op"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "s : Set M", "self : IsSubmonoid s"], "goal": "1 ∈ s"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "G : AlgebraCat R", "H : Type u", "Ring H", "Algebra R H", "f : ↑G →ₐ[R] H"], "goal": "f.comp (𝟙 G) = f"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "n : ℕ∞", "f : M → M'", "x : M"], "goal": "ContMDiffAt I I' n f x ↔\n  ContinuousAt f x ∧\n    ContDiffWithinAt 𝕜 n (↑(extChartAt I' (f x)) ∘ f ∘ ↑(extChartAt I x).symm) (Set.range ↑I) (↑(extChartAt I x) x)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_5", "MulOneClass M", "MulOneClass N", "MulOneClass P", "f : Monoid.Coprod M N →* P"], "goal": "MonoidHom.mrange f = MonoidHom.mrange (f.comp Monoid.Coprod.inl) ⊔ MonoidHom.mrange (f.comp Monoid.Coprod.inr)"}}
+{"state": {"context": ["p n : ℕ", "hp : Fact (Nat.Prime p)", "hn : 0 < n", "div : p ∣ n"], "goal": "1 ≤ padicValNat p n"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝⁶ : CommRing 𝕜", "inst✝⁵ : NoZeroDivisors 𝕜", "inst✝⁴ : TopologicalSpace 𝕜", "inst✝³ : TopologicalRing 𝕜", "inst✝² : TopologicalSpace A", "inst✝¹ : Semiring A", "inst✝ : Algebra 𝕜 A", "a : ↑(characterSpace 𝕜 A)"], "goal": "((fun a => { toFun := fun φ => φ a, continuous_toFun := ⋯ }) 1) a = 1 a"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "P : Type u_3", "N : Type u_4", "inst✝⁸ : Ring R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup P", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R M", "inst✝³ : Module R P", "inst✝² : Module R N", "inst✝¹ : IsArtinian R M", "inst✝ : IsArtinian R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "hf : Function.Injective ⇑f", "hg : Function.Surjective ⇑g", "h : LinearMap.range f = LinearMap.ker g"], "goal": "∀ (a : Submodule R N), Submodule.comap g (Submodule.map g a) = a ⊔ LinearMap.range f"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t u✝ : Set E", "f✝ f₁ : E → F", "g : F → G", "x✝ x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "ι : Type u_3", "f : ι → E → F", "u : Finset ι", "i : ℕ", "x : E", "hs : UniqueDiffOn 𝕜 s", "hx : x ∈ s", "h : ∀ j ∈ u, ContDiffOn 𝕜 (↑i) (f j) s"], "goal": "iteratedFDerivWithin 𝕜 i (fun x => ∑ j ∈ u, f j x) s x = ∑ j ∈ u, iteratedFDerivWithin 𝕜 i (f j) s x"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "m : Type u_4", "n : Type u_5", "Fintype n", "M : Matrix m n R"], "goal": "LinearMap.ker M.mulVecLin = ⊥ ↔ ∀ (v : n → R), M *ᵥ v = 0 → v = 0"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "AddGroup G", "AddTorsor G P"], "goal": "Subsingleton G ↔ Subsingleton P"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α", "inst✝² : CountablyGenerated α", "inst✝¹ : CountablyGenerated α", "inst✝ : SeparatesPoints α", "x y : α", "hxy : mapNatBool α x = mapNatBool α y", "n : ℕ"], "goal": "Decidable (y ∈ natGeneratingSequence α n)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "I✝ : Ideal R", "a b : R", "S : Type v", "x y : R", "I : Ideal R"], "goal": "Subsingleton (R ⧸ I) ↔ I = ⊤"}}
+{"state": {"context": ["α : Type u", "Nonempty α"], "goal": "#(List α) = max #α ℵ₀"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddCommMonoid α", "TopologicalSpace α", "Fintype β", "f : β → α"], "goal": "HasSum f (∑ b : β, f b)"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : AddGroup α", "inst✝² : Preorder α", "inst✝¹ : DecidableRel fun x x_1 => x < x_1", "inst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a : α"], "goal": "(if a < 0 then 1 else if 0 < a then -1 else 0) = -if 0 < a then 1 else if a < 0 then -1 else 0"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "f : R[X]", "𝓟 : Ideal R", "self : f.IsEisensteinAt 𝓟"], "goal": "f.leadingCoeff ∉ 𝓟"}}
+{"state": {"context": ["M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "α : Type u_7", "β : Type u_8"], "goal": "∀ (a : α ⊕ β →₀ M), (Finsupp.sumFinsuppLEquivProdFinsupp R) a = Finsupp.sumFinsuppAddEquivProdFinsupp.toFun a"}}
+{"state": {"context": ["A : Type u_2", "B : Type u_3", "AddMonoid A", "AddMonoid B", "TopologicalSpace A", "TopologicalSpace B", "ContinuousAdd B", "T2Space B"], "goal": "ClosedEmbedding ContinuousAddMonoidHom.toContinuousMap"}}
+{"state": {"context": ["b c a : Prop", "h : b ↔ c"], "goal": "a ∨ b ↔ a ∨ c"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "Nontrivial E"], "goal": "(Bornology.cobounded E).NeBot"}}
+{"state": {"context": ["x : ℝ"], "goal": "↑x ≠ 1 ↔ x ≠ 1"}}
+{"state": {"context": ["α β : Sort u_1", "h : α = β", "x : α"], "goal": "(Equiv.cast h) x = cast h x"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "H : SimpleGraph V", "h : H ≤ G"], "goal": "∀ (a a_1 : V), (SimpleGraph.toSubgraph H h).Adj a a_1 = H.Adj a a_1"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p q : R[X]", "n : ℕ", "m : ℕ", "dvd : q ^ n ∣ p"], "goal": "q ^ (n - m) ∣ (⇑Polynomial.derivative)^[m] p"}}
+{"state": {"context": ["A : Set ℕ", "DecidablePred fun x => x ∈ A", "DecidablePred fun x => x ∈ insert 0 A"], "goal": "schnirelmannDensity (insert 0 A) = schnirelmannDensity A"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "X : Type u_2", "A : Type u_3", "Semiring A", "Algebra R A", "f : X → A", "x : X"], "goal": "((FreeAlgebra.lift R) f) (FreeAlgebra.ι R x) = f x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : Part α", "g : α → Part β"], "goal": "(f.bind g).Dom ↔ ∃ (h : f.Dom), (g (f.get h)).Dom"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "Preorder ι", "LinearOrder α", "u : ι → α", "h : Monotone u", "H : ¬BddAbove (Set.range u)"], "goal": "Filter.Tendsto u Filter.atTop Filter.atTop"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : ProbabilityTheory.Kernel α γ", "ProbabilityTheory.IsFiniteKernel κ", "ProbabilityTheory.IsFiniteKernel η", "a : α"], "goal": "((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a) = 0"}}
+{"state": {"context": ["X Y : Scheme", "U✝ : X.Opens", "hU : IsAffineOpen U✝", "f✝ : ↑Γ(X, U✝)", "f : X ⟶ Y", "H : IsOpenImmersion f", "U : { U // ↑U ≤ Scheme.Hom.opensRange f }"], "goal": "IsAffineOpen ↑↑U"}}
+{"state": {"context": ["β : Type u_2", "AddCommMonoid β", "x : β", "n : ℕ"], "goal": "((multiplesAddHom β) x) n = n • x"}}
+{"state": {"context": ["L : Language", "α : Type w", "V : Type w'", "n : ℕ", "S : Language.graph.Structure V", "inst✝ : V ⊨ Theory.simpleGraph", "x✝² : ℕ", "x✝¹ : Language.graph.Relations x✝²", "x✝ : Fin x✝² → V"], "goal": "RelMap x✝¹ x✝ ↔ RelMap x✝¹ x✝"}}
+{"state": {"context": ["p : ℕ", "R S : Type u", "σ : Type u_1", "idx : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "hp : Fact (Nat.Prime p)", "n : ℕ"], "goal": "∀ b ∈ Finset.range (n + 1), b ≠ 0 → (bind₁ onePoly) (C (↑p ^ b) * X b ^ p ^ (n - b)) = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f fa : α → α", "fb : β → β", "x y : α", "m n✝ : ℕ", "hx : x ∈ periodicPts f", "n : ℕ"], "goal": "f^[n] x ∈ periodicPts f"}}
+{"state": {"context": ["α : Type u", "Inf α", "a b : α"], "goal": "{ down := a ⊓ b } = { down := a } ⊓ { down := b }"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NontriviallyNormedField 𝕜'", "σ : 𝕜 →+* 𝕜'", "σ' : 𝕜' →+* 𝕜", "inst✝⁹ : RingHomInvPair σ σ'", "inst✝⁸ : RingHomInvPair σ' σ", "inst✝⁷ : RingHomIsometric σ", "inst✝⁶ : RingHomIsometric σ'", "E : Type u_3", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type u_4", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜' F", "f : E →SL[σ] F", "inst✝¹ : CompleteSpace F", "inst✝ : CompleteSpace E", "surj : Surjective ⇑f", "s : Set E", "hs : IsOpen s"], "goal": "IsOpen (⇑f '' s)"}}
+{"state": {"context": ["F : Type u → Type u", "q : QPF F", "α : Type u", "x : F α"], "goal": "id <$> x = x"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_7, u_1} C", "D : Type u_2", "inst✝⁴ : Category.{u_5, u_2} D", "E : Type u_3", "inst✝³ : Category.{?u.128478, u_3} E", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "L : GrothendieckTopology E", "A : Type u_4", "inst✝² : Category.{u_6, u_4} A", "G : C ⥤ D", "inst✝¹ : G.IsCoverDense K", "inst✝ : G.IsLocallyFull K", "ℱ✝ : Dᵒᵖ ⥤ A", "ℱ'✝ ℱ ℱ' : Sheaf K A", "α : ℱ ⟶ ℱ'", "i : IsIso (whiskerLeft G.op α.val)"], "goal": "α = (sheafIso (asIso (whiskerLeft G.op α.val))).hom"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : α → β → Prop", "x : List α", "y : List β", "h : List.Forall₂ R x y", "i : ℕ", "hx : i < x.length", "hy : i < y.length"], "goal": "R (x.nthLe i hx) (y.nthLe i hy)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "s₁ s₂ : Finset α", "f g : α → β", "AddCommMonoid β", "h : s₁ = s₂"], "goal": "(∀ x ∈ s₂, f x = g x) → s₁.sum f = s₂.sum g"}}
+{"state": {"context": ["U : Type u_1", "V : Type u_2", "W : Type u_3", "inst✝⁴ : Quiver U", "inst✝³ : Quiver V", "inst✝² : Quiver W", "V' : Type u_4", "inst✝¹ : Quiver V'", "inst✝ : HasReverse V'", "φ : V ⥤q V'"], "goal": "∀ (X : V), (of ⋙q lift φ).obj X = φ.obj X"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : CancelCommMonoidWithZero α", "inst✝¹ : NormalizedGCDMonoid α", "s s₁ s₂ : Finset β", "f : β → α", "inst✝ : Nontrivial α"], "goal": "s.lcm f = 0 ↔ 0 ∈ f '' ↑s"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "U : TopologicalSpace.Opens α"], "goal": "(↑U).Nonempty ↔ ∃ x, x ∈ U"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "f✝¹ : ℝ → E", "g' g✝ φ : ℝ → ℝ", "a b : ℝ", "inst✝² : CompleteSpace E", "f✝ f'✝ : ℝ → E", "G : Type u_6", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "f f' : ℝ → ℝ", "g : ℝ → G", "hf : ContinuousOn f [[a, b]]", "hff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x", "hf' : ContinuousOn f' [[a, b]]", "hg : ContinuousOn g (f '' [[a, b]])"], "goal": "IntegrableOn g (f '' [[a, b]]) volume"}}
+{"state": {"context": ["Γ : Subgroup SL(2, ℤ)", "k : ℤ", "α : Type u_2", "SMul α ℂ", "IsScalarTower α ℂ ℂ", "f : CuspForm Γ k", "n : α"], "goal": "⇑(n • f) = n • ⇑f"}}
+{"state": {"context": ["n m l : ℕ", "hlm : l ≤ m"], "goal": "filter (fun x => decide (x < l)) (Ico n m) = Ico n l"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "ConditionallyCompleteLinearOrderBot ι", "a b : ℝ", "f : ι → Ω → ℝ", "n : ℕ", "ω : Ω"], "goal": "MeasureTheory.upperCrossingTime a b f ⊥ n ω = ⊥"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "r : Ordnode α", "hl : l.Sized", "hr : r.Sized"], "goal": "(l.balance' x r).size = l.size + r.size + 1"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α"], "goal": "0 ⟂ₘ μ"}}
+{"state": {"context": ["R : Type u_1", "Γ₀ : Type u_2", "inst✝¹ : CommRing R", "inst✝ : LinearOrderedCommMonoidWithZero Γ₀", "v : Valuation R Γ₀"], "goal": "Ideal.map (Ideal.Quotient.mk v.supp) v.supp = 0"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "A B : Subgroup G", "φ : ↥A ≃* ↥B", "H : Type u_2", "inst✝¹ : Group H", "M : Type u_3", "inst✝ : Monoid M", "a : ↥A"], "goal": "of ↑(φ a) = of ↑(φ a) * t * t⁻¹"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Prop", "t : RBNode α"], "goal": "All (fun a => p a ∧ q a) t ↔ All p t ∧ All q t"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "H : LieSubalgebra R L", "self : H.IsCartanSubalgebra"], "goal": "LieAlgebra.IsNilpotent R ↥H"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "A : Type u'", "CategoryTheory.Category.{v', u'} A", "CategoryTheory.ConcreteCategory A", "F₁ F₂ : CategoryTheory.Sheaf J A", "φ : F₁ ⟶ F₂"], "goal": "CategoryTheory.Presheaf.IsLocallySurjective J ((CategoryTheory.sheafToPresheaf J A).map φ) ↔\n  CategoryTheory.Sheaf.IsLocallySurjective φ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "SemilatticeInf β", "f g : α → β", "a : α", "l : Filter α", "hf : IsMaxFilter f l a", "hg : IsMaxFilter g l a"], "goal": "IsMaxFilter (fun x => f x ⊓ g x) l a"}}
+{"state": {"context": ["q : ℤ[X]", "k m n : ℕ", "hkm : k < m", "hmn : m < n", "x y z : ℤ", "hp : {k, m, n}.card = 3 ∧ ∀ k_1 ∈ {k, m, n}, IsUnit ((C x * X ^ k + C y * X ^ m + C z * X ^ n).coeff k_1)", "hx : IsUnit ((x + y * if k = m then 1 else 0) + z * if k = n then 1 else 0)", "hy : IsUnit ((x * if m = k then 1 else 0) + y + z * if m = n then 1 else 0)", "hz : IsUnit (((x * if n = k then 1 else 0) + y * if n = m then 1 else 0) + z)"], "goal": "(C x * X ^ k + C y * X ^ m + C z * X ^ n).IsUnitTrinomial"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s✝ t✝ : WSeq α", "c1 c2 : Computation (Option (α × WSeq α))", "s t : WSeq α", "h : c1 = (s.append t).destruct ∧ c2 = s.destruct.bind (destruct_append.aux t)"], "goal": "Computation.BisimO (fun c1 c2 => ∃ s t, c1 = (s.append t).destruct ∧ c2 = s.destruct.bind (destruct_append.aux t)) (s.append t).destruct.destruct (s.destruct.bind (destruct_append.aux t)).destruct"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrderedAddCommGroup α", "inst✝¹ : OrderTopology α", "l : Filter β", "f g : β → α", "inst✝ : NoMaxOrder α", "a : α"], "goal": "(𝓝 a).HasBasis (fun ε => 0 < ε) fun ε => {b | |b - a| < ε}"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p q r : R[X]", "inst✝ : Nontrivial R", "i j : ℕ"], "goal": "(monomial i) 1 = (monomial j) 1 ↔ i = j"}}
+{"state": {"context": ["T : ℝ", "hT : 0 < T", "t : ℝ", "μ : autoParam (Measure ℝ) _auto✝", "x : ℝ"], "goal": "∃! g, g +ᵥ x ∈ Ioc t (t + T)"}}
+{"state": {"context": ["a x✝ z x y : ℂ", "hx : x ≠ 0", "hy : y ≠ 0"], "goal": "x * y = ↑(abs x) * ↑(abs y) * (cexp (↑x.arg * I) * cexp (↑y.arg * I))"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "B : Type u_4", "CommSemiring R", "StarRing R", "Semiring A", "Algebra R A", "StarRing A", "Semiring B", "Algebra R B", "StarRing B", "StarModule R B", "f : A →⋆ₐ[R] B", "hf : Function.Injective ⇑f", "a : A"], "goal": "(StarAlgEquiv.ofInjective f hf) a = f.rangeRestrict a"}}
+{"state": {"context": ["S X₁ X₂ : Type u", "f : S ⟶ X₁", "g : S ⟶ X₂", "inst✝ : Mono f", "a b : X₁ ⊕ X₂"], "goal": "Quot.mk (Rel f g) a = Quot.mk (Rel f g) b ↔ Rel' f g a b"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "s : Finset α", "a b : α", "h₁ : a < b", "h₂ : s.min = ↑b"], "goal": "a ∉ s"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "M✝ : Matroid α", "F X✝ Y : Set α", "e : α", "M : Matroid α", "X : Set α"], "goal": "∀ (x : α), (∀ (t : Set α), M.Flat t → X ∩ M.E ⊆ t → x ∈ t) ↔ x ∈ M.E ∧ ∀ a ⊆ M.E, X ∩ M.E ⊆ a → M.Flat a → x ∈ a"}}
+{"state": {"context": ["α : Type u_1", "is l : List α", "H : l.Perm ([] ++ is) → (∃ ts', ∃ (_ : ts'.Perm []), l = ts' ++ is) ∨ l ∈ is.permutationsAux []"], "goal": "l.Perm is → l ∈ is.permutations"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁶ : NormedRing A", "inst✝⁵ : NormedAlgebra ℂ A", "inst✝⁴ : StarRing A", "inst✝³ : CstarRing A", "inst✝² : StarModule ℂ A", "inst✝¹ : CompleteSpace A", "a : A", "inst✝ : IsStarNormal a", "S✝ S : StarSubalgebra ℂ A", "hS : IsClosed ↑S", "x : A", "h : IsUnit x", "hxS : x ∈ S", "hx : IsUnit (star x * x)", "this : IsStarNormal (star x * x)", "hx' : IsUnit ⟨star x * x, ⋯⟩"], "goal": "↑hx.unit⁻¹ ∈ elementalStarAlgebra ℂ (star x * x)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁴ : MeasurableSpace X", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "f g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "inst✝¹ : PartialOrder X", "x y : X", "inst✝ : NoAtoms μ"], "goal": "∫ (t : X) in Icc x y, f t ∂μ = ∫ (t : X) in Ioo x y, f t ∂μ"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "f : Perm α", "e : α ≃ Fin n", "h : ∀ (x : α), f x ≠ x → x ∈ []"], "goal": "signAux ((e.symm.trans f).trans e) = signAux2 [] f"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "n : ℕ", "s : Affine.Simplex ℝ P (n + 1)"], "goal": "0 < s.circumradius"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : PseudoMetricSpace X", "inst✝ : IsUltrametricDist X", "x y z : X", "r✝ s r : ℝ", "hr : r ≠ 0"], "goal": "IsOpen (closedBall x r ⊓ (ball x r)ᶜ)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : RegularSpace X", "x✝ : X", "s✝ : Set X", "B : Set (Set X)", "hB : IsTopologicalBasis B", "x : X", "s : Set X", "h : s ∈ 𝓝 x"], "goal": "∃ t ∈ B, x ∈ t ∧ closure t ⊆ s"}}
+{"state": {"context": ["R : Type u", "Semiring R", "Nontrivial R", "p : R[X]"], "goal": "(p * Polynomial.X).degree = p.degree + 1"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "f : α → E", "p q : ℝ", "hp0_lt : 0 < p", "hpq : p ≤ q", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.eLpNorm' f p μ ≤ MeasureTheory.eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "f : X ⟶ Y", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Limits.HasZeroMorphisms D", "G : C ⥤ D", "G.PreservesZeroMorphisms", "CategoryTheory.Limits.HasKernel f", "CategoryTheory.Limits.HasKernel (G.map f)", "Z : C", "h : Z ⟶ X", "w : h ≫ f = 0"], "goal": "G.map (CategoryTheory.Limits.kernel.lift f h w) ≫ CategoryTheory.Limits.kernelComparison f G =\n  CategoryTheory.Limits.kernel.lift (G.map f) (G.map h) ⋯"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : CategoryTheory.Functor J C", "CategoryTheory.Limits.HasLimit F", "j : J"], "goal": "(CategoryTheory.Limits.limit.toStructuredArrow F).obj j =\n  CategoryTheory.StructuredArrow.mk (CategoryTheory.Limits.limit.π F j)"}}
+{"state": {"context": ["α : Type u", "s : Set (Set α)"], "goal": "#↑{t | GenerateMeasurable s t} ≤ max (#↑s) 2 ^ ℵ₀"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "I : Ideal R", "a b : R", "S : Type v", "x✝ y✝ x y : R"], "goal": "(mk (span {x})) y = 0 ↔ x ∣ y"}}
+{"state": {"context": ["R : Type u", "K : Type v", "L : Type z", "p : R", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : Algebra K L", "inst✝⁶ : Algebra R L", "inst✝⁵ : Algebra R K", "inst✝⁴ : IsScalarTower R K L", "inst✝³ : Algebra.IsSeparable K L", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "B : PowerBasis K L", "hp : _root_.Prime p", "hBint : IsIntegral R B.gen", "z : L", "Q : R[X]", "hQ : (aeval B.gen) Q = p • z", "hzint : IsIntegral R z", "hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})", "this : Module.Finite K L := finite B", "P : R[X] := minpoly R B.gen", "n : ℕ", "hn : B.dim = n.succ", "finrank_K_L : FiniteDimensional.finrank K L = B.dim", "deg_K_P : (minpoly K B.gen).natDegree = B.dim", "deg_R_P : P.natDegree = B.dim", "f : ℕ → L", "hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i", "aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n", "hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))", "r : R", "hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))"], "goal": "Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n) = p ^ n.succ * r"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "f : ι → α", "p : ↑(Set.range f) → Prop"], "goal": "(∀ (a : ↑(Set.range f)), p a) ↔ ∀ (i : ι), p ⟨f i, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "i✝ : Nat", "a b : Array α", "i j : Nat", "h : j ≤ a.size"], "goal": "((a ++ b).extract i j).size = (a.extract i j).size"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m✝ : MeasurableSpace α", "μ : Measure α", "m : MeasurableSpace α", "T T' : Set α → F →L[ℝ] F'", "f : α →ₛ F"], "goal": "setToSimpleFunc (T + T') f = setToSimpleFunc T f + setToSimpleFunc T' f"}}
+{"state": {"context": ["x : ℝ", "hx : 0 < x"], "goal": "Real.log x ≤ 0 ↔ x ≤ 1"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_5", "M : Type u_6", "CommSemiring R", "AddCommMonoid M", "Module R M", "e : Basis ι R M", "v : ι' → M", "j : ι'"], "goal": "(e.toMatrix v)ᵀ j = ⇑(e.repr (v j))"}}
+{"state": {"context": ["s step m n : Nat", "h : m + 1 < n + 1"], "goal": "(range' s (n + 1) step)[m + 1]? = some (s + step * (m + 1))"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "SuccOrder α", "a b : α", "h : a < Order.succ b"], "goal": "Set.Ioc a (Order.succ b) = insert (Order.succ b) (Set.Ioc a b)"}}
+{"state": {"context": ["F : Type v", "Field F", "W : WeierstrassCurve.Jacobian F", "P : Fin 3 → F", "hP : W.Nonsingular P", "hPz : P z = 0"], "goal": "IsUnit (P y)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X", "p : X → Prop"], "goal": "(∀ᶠ (x : X) in 𝓝ˢ s, p x) ↔ ∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, p x"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "G : SimpleGraph V", "a b c u v w : V", "e : Sym2 V", "H : SimpleGraph V", "s✝ s₁ s₂ s : Set (Sym2 V)", "x✝¹ x✝ : V"], "goal": "(G.deleteEdges s).Adj x✝¹ x✝ ↔ (G.deleteEdges (s ∩ G.edgeSet)).Adj x✝¹ x✝"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝¹ : SeminormedRing 𝕜", "inst✝ : SMul 𝕜 E", "s✝ t✝ : Set E", "x : E", "s : Set E", "a : 𝕜", "ha : ‖a‖ ≤ 1", "y : E", "t : Set E", "ht1 : Balanced 𝕜 t", "ht2 : t ⊆ s", "hy : y ∈ t"], "goal": "(fun x => a • x) y ∈ balancedCore 𝕜 s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "M : Type u_4", "AddMonoid M", "CategoryTheory.HasShift C M", "X Y : C", "CategoryTheory.Preadditive C", "m₀ : M", "hm₀ : m₀ = 0"], "goal": "CategoryTheory.ShiftedHom.mk₀ m₀ hm₀ 0 = 0"}}
+{"state": {"context": ["b x : ℝ", "hb : 1 < b", "hx_pos : 0 < x", "hx : x ≠ 1"], "goal": "Real.logb b x ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "S : Submonoid R", "A : Type u_2", "CommRing A", "Algebra R A", "IsLocalization S A", "M : Type u_3", "AddCommMonoid M", "Module R M", "M' : Type u_4", "AddCommMonoid M'", "Module R M'", "Module A M'", "IsScalarTower R A M'", "f : M →ₗ[R] M'"], "goal": "IsLocalizedModule S f ↔ IsBaseChange A f"}}
+{"state": {"context": ["α : Type u_4", "p : Finset α → Prop", "DecidableEq α", "S : Finset α", "h₁ : p ∅", "h₂ : ∀ {a : α} {s : Finset α}, a ∈ S → s ⊆ S → a ∉ s → p s → p (insert a s)"], "goal": "p S"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "m : Type u_5", "n : Type u_6", "Fintype m", "DecidableEq m", "Fintype n", "DecidableEq n", "A : Matrix m n 𝕜"], "goal": "Matrix.toEuclideanLin A.conjTranspose = LinearMap.adjoint (Matrix.toEuclideanLin A)"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "j : ℕ"], "goal": "cs.wordProd ω * (cs.rightInvSeq ω).getD j 1 = cs.wordProd (ω.eraseIdx j)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "hF : F.FullyFaithful", "X Y : C", "e : F.obj X ≅ F.obj Y"], "goal": "(hF.preimageIso e).hom = hF.preimage e.hom"}}
+{"state": {"context": ["α✝ : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝⁵ : LinearOrder α", "inst✝⁴ : TopologicalSpace α", "inst✝³ : TopologicalSpace β", "inst✝² : TopologicalSpace γ", "inst✝¹ : OrderTop α", "inst✝ : CompactIccSpace α", "x✝ : Set α", "hs : x✝ ∈ atBot", "h✝ : Nonempty α"], "goal": "∃ t, IsCompact t ∧ tᶜ ⊆ x✝"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "F : C → D", "CategoryTheory.Functorial F", "self : CategoryTheory.LaxMonoidal F", "X Y Z : C"], "goal": "CategoryTheory.MonoidalCategory.whiskerRight (CategoryTheory.LaxMonoidal.μ X Y) (F Z) ≫\n    CategoryTheory.LaxMonoidal.μ (CategoryTheory.MonoidalCategory.tensorObj X Y) Z ≫\n      CategoryTheory.map F (CategoryTheory.MonoidalCategory.associator X Y Z).hom =\n  (CategoryTheory.MonoidalCategory.associator (F X) (F Y) (F Z)).hom ≫\n    CategoryTheory.MonoidalCategory.whiskerLeft (F X) (CategoryTheory.LaxMonoidal.μ Y Z) ≫\n      CategoryTheory.LaxMonoidal.μ X (CategoryTheory.MonoidalCategory.tensorObj Y Z)"}}
+{"state": {"context": ["m : Type u", "n : Type v", "α : Type w", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "inst✝ : CommRing α", "A : Matrix n n α", "h : Fintype.card n ≠ 1", "n✝ : ℕ", "h_card : Fintype.card n = n✝ + 1 + 1", "A' : Matrix n n (MvPolynomial (n × n) ℤ) := mvPolynomialX n n ℤ"], "goal": "A.adjugate.adjugate = A.det ^ (Fintype.card n - 2) • A"}}
+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝ : MulOneClass M", "ι : Sort u_4", "hι : Nonempty ι", "S : ι → Submonoid M", "hS : Directed (fun x x_1 => x ≤ x_1) S", "x : M", "hx : x ∈ closure (⋃ i, ↑(S i))"], "goal": "∀ (x y : M), (∃ i, x ∈ S i) → (∃ i, y ∈ S i) → ∃ i, x * y ∈ S i"}}
+{"state": {"context": ["R : Type u", "M : Type v", "M₂ : Type w", "Semiring R", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂", "q : Submodule R M₂"], "goal": "Submodule.map (LinearMap.inr R M M₂) q = ⊥.prod q"}}
+{"state": {"context": ["R : Type u_2", "AddZeroClass R", "Mul R", "c : RingCon R"], "goal": "↑0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ✝ : Measure α", "inst✝ : NormedAddCommGroup β", "μ : Measure α", "p : ℝ≥0∞", "f✝ : ℕ → α → β", "g✝ : α → β", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "f : ℕ → α → β", "g : α → β", "haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ", "hg' : Memℒp g p μ", "hui : UnifIntegrable f p μ", "hut : UnifTight f p μ", "hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))", "hf : ∀ (n : ℕ), StronglyMeasurable (AEStronglyMeasurable.mk (f n) ⋯)", "hff' : ∀ (n : ℕ), f n =ᶠ[ae μ] AEStronglyMeasurable.mk (f n) ⋯"], "goal": "Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (𝓝 0)"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ : NumberField K", "inst✝ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "u : (𝓞 K)ˣ", "x : 𝓞 K"], "goal": "-↑η - 1 + ↑η + 1 = 0"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁹ : CommRing R", "inst✝⁸ : IsDomain R", "inst✝⁷ : CommRing S", "L : Type u_3", "inst✝⁶ : Field L", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra S L", "inst✝³ : Algebra R L", "inst✝² : IsScalarTower R S L", "inst✝¹ : IsIntegralClosure S R L", "inst✝ : Algebra.IsAlgebraic R L", "inj : Function.Injective ⇑(algebraMap R L)", "a b : S", "hb : b ≠ 0"], "goal": "∃ c d, d ≠ 0 ∧ d • a = b * c"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "ι : Type u_3", "hm : MeasurableSpace α", "μ✝ : Measure α", "inst✝¹⁰ : TopologicalSpace α", "inst✝⁹ : BorelSpace α", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "g✝ : α → E", "l : Filter ι", "x₀ : α", "s t : Set α", "φ✝ : ι → α → ℝ", "a : E", "inst✝⁶ : CompleteSpace E", "F : Type u_4", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : FiniteDimensional ℝ F", "inst✝² : MeasurableSpace F", "inst✝¹ : BorelSpace F", "μ : Measure F", "inst✝ : μ.IsAddHaarMeasure", "φ : F → ℝ", "hφ : ∀ (x : F), 0 ≤ φ x", "h'φ : ∫ (x : F), φ x ∂μ = 1", "h : Tendsto (fun x => ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)", "g : F → E", "hg : Integrable g μ", "h'g : ContinuousAt g 0", "I : Integrable φ μ"], "goal": "∀ᶠ (i : ℝ) in atTop, AEStronglyMeasurable (fun x => i ^ finrank ℝ F * φ (i • x)) μ"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "IsSimpleAddGroup G", "H : AddSubgroup G", "Hn : H.Normal"], "goal": "H = ⊥ ∨ H = ⊤"}}
+{"state": {"context": ["b x : ℕ", "hx : x ≠ 0", "hxb : x < b"], "goal": "b.digits x = [x]"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "h : DifferentiableOn 𝕜 (fun y => -f y) s"], "goal": "DifferentiableOn 𝕜 f s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ComposableArrows C 0"], "goal": "S.IsComplex"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_4", "SemilatticeSup α", "Preorder β", "u : β → α", "l : LowerAdjoint u", "x y : α"], "goal": "u (l.toFun x) ⊔ u (l.toFun y) ≤ u (l.toFun (x ⊔ y))"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂", "p₁ : Submodule R M", "p₂ : Submodule R M₂"], "goal": "p₁.prod p₂ = ⊤ ↔ p₁ = ⊤ ∧ p₂ = ⊤"}}
+{"state": {"context": ["α : Type u_1", "X : Type u_2", "Y : Type u_3", "Z : Type u_4", "T : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace T", "K : Set X", "U : Set Y", "f g : C(X, Y)", "h : ⇑f ⤳ ⇑g"], "goal": "f ⤳ g"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α"], "goal": "IsPreconnected ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : _root_.RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : InnerProductSpace 𝕜 E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace ℝ F", "ι : Type u_4", "inst✝ : DecidableEq E", "s : Set E"], "goal": "(∀ (i j : { x // x ∈ s }), ⟪↑i, ↑j⟫_𝕜 = if i = j then 1 else 0) ↔ ∀ v ∈ s, ∀ w ∈ s, ⟪v, w⟫_𝕜 = if v = w then 1 else 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommSemiring R", "CommSemiring S", "S' : Type u_1", "CommSemiring S'", "f : R →+* S", "g : S →+* S'", "x : PrimeSpectrum S'"], "goal": "(PrimeSpectrum.comap (g.comp f)) x = (PrimeSpectrum.comap f) ((PrimeSpectrum.comap g) x)"}}
+{"state": {"context": ["α : Type u_1", "a : α", "o : Option α", "h : o.isSome = true"], "goal": "a ∈ o → o.get h = a"}}
+{"state": {"context": ["n : Nat", "x y : Nat"], "goal": "BitVec.ofNat n x - BitVec.ofNat n y = BitVec.ofNat n (2 ^ n - y % 2 ^ n + x)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X : C"], "goal": "∀ (X_1 : Cᴹᵒᵖ),\n  ((CategoryTheory.MonoidalOpposite.tensorRightMopIso X).hom.app X_1).unmop =\n    𝟙 (CategoryTheory.MonoidalCategory.tensorObj X X_1.unmop)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "CommRing R", "CommRing A", "CommRing B", "Algebra R A", "Algebra R B", "f : A →ₐ[R] B", "hf : f.FinitePresentation"], "goal": "f.FiniteType"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁵ : TopologicalSpace α", "inst✝⁴ : TopologicalSpace β", "inst✝³ : Group α", "inst✝² : MulAction α β", "inst✝¹ : ContinuousInv α", "inst✝ : ContinuousSMul α β", "s : Set α", "t : Set β", "ht : IsClosed t", "hs : IsCompact s", "this : ∀ x ∈ s • t, ∃ g ∈ s, g⁻¹ • x ∈ t"], "goal": "IsClosed (s • t)"}}
+{"state": {"context": ["α : Type u", "σ : Type v", "M : DFA α σ", "α' : Type u_1", "f : α' → α", "x : List α'"], "goal": "(DFA.comap f M).eval x = M.eval (List.map f x)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "Zero α", "i : ι", "b : α"], "goal": "Finsupp.single i b = Finsupp.indicator {i} fun x x => b"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "TopologicalSpace α", "UniformSpace β", "X : Set (C(α, β) × C(α, β))"], "goal": "X ∈ 𝓤 C(α, β) ↔ ∃ K V, IsCompact K ∧ V ∈ 𝓤 β ∧ {fg | ∀ x ∈ K, (fg.1 x, fg.2 x) ∈ V} ⊆ X"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "g : β → γ", "x : FreeGroup α"], "goal": "(FreeGroup.map g) ((FreeGroup.map f) x) = (FreeGroup.map (g ∘ f)) x"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "a b : α"], "goal": "⌊a⌋ + ⌊b⌋ ≤ ⌊a + b⌋"}}
+{"state": {"context": ["l : Filter ℂ", "hl : IsExpCmpFilter l", "n : ℕ"], "goal": "∀ᶠ (x : ℂ) in l, ↑n * ‖Real.log (max x.re |x.im|)‖ ≤ ‖x.re‖"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type uX", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t✝ u✝ : Set E", "f f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "m✝ n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "m : ℕ", "h : ContDiffWithinAt 𝕜 n f s x", "hmn : ↑m < n", "hs : UniqueDiffOn 𝕜 (insert x s)", "u : Set E", "uo : IsOpen u", "xu : x ∈ u", "t : Set E := insert x s ∩ u", "hu : ContDiffOn 𝕜 (↑(Add.add (↑m) 1)) f t", "A : t =ᶠ[𝓝[≠] x] s"], "goal": "DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x"}}
+{"state": {"context": ["R : Type u_4", "CommRing R", "Ring.DimensionLEOne R", "p : Ideal R", "h : p.IsPrime", "hp : p ≠ ⊥"], "goal": "p.IsMaximal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : Filter α", "k f g g' : α → β", "inst✝³ : TopologicalSpace β", "inst✝² : CommSemiring β", "inst✝¹ : ContinuousAdd β", "inst✝ : ContinuousMul β", "hf : SuperpolynomialDecay l k f", "p : β[X]", "n : ℕ", "c : β"], "goal": "SuperpolynomialDecay l k fun x => eval (k x) ((monomial n) c) * f x"}}
+{"state": {"context": ["α : Type u_2", "β : α → Type u_1", "l₁ : List α", "l₂ : (a : α) → List (β a)"], "goal": "∀ (acc : Array ((a : α) × β a)) (a : α), (foldl (fun acc b => acc.push ⟨a, b⟩) acc (l₂ a)).data = acc.data ++ ?h.h.h.h.G a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "A : Type u'", "CategoryTheory.Category.{v', u'} A", "CategoryTheory.ConcreteCategory A", "F₁ F₂ F₃ : Cᵒᵖ ⥤ A", "f₁ : F₁ ⟶ F₂", "f₂ : F₂ ⟶ F₃", "CategoryTheory.Presheaf.IsLocallyInjective J f₁", "CategoryTheory.Presheaf.IsLocallySurjective J f₁"], "goal": "CategoryTheory.Presheaf.IsLocallyInjective J (f₁ ≫ f₂) ↔ CategoryTheory.Presheaf.IsLocallyInjective J f₂"}}
+{"state": {"context": ["a : ℝ", "ha : a ∈ Ico 0 1", "p : ℝ", "hp : 0 < p", "hp' : (fun t => F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)"], "goal": "∃ p, 0 < p ∧ (fun t => F_int 0 (↑a) t - if ↑a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)"}}
+{"state": {"context": [], "goal": "StrictConvexOn ℝ (Set.Ici 0) fun x => x * Real.log x"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Semiring α", "AddCommMonoid β", "Module α β", "a b : α", "s : Set β"], "goal": "(a + b) • s ⊆ a • s + b • s"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Submonoid R", "S : Type u_2", "CommRing S", "Algebra R S", "Rₘ : Type u_4", "Sₘ : Type u_5", "CommRing Rₘ", "CommRing Sₘ", "Algebra R Rₘ", "IsLocalization M Rₘ", "Algebra S Sₘ", "i : IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ", "Algebra Rₘ Sₘ", "Algebra R Sₘ", "IsScalarTower R Rₘ Sₘ", "IsScalarTower R S Sₘ", "x : Rₘ"], "goal": "(algebraMap Rₘ Sₘ) x = (IsLocalization.map Sₘ (algebraMap R S) ⋯) x"}}
+{"state": {"context": [], "goal": "((fun x x_1 => x ↔ x_1) ⇒ (fun x x_1 => x ↔ x_1) ⇒ fun x x_1 => x ↔ x_1) (fun x x_1 => x ∨ x_1) fun x x_1 => x ∨ x_1"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "Nontrivial M"], "goal": "¬LinearMap.BilinForm.Nondegenerate 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "self : CategoryTheory.Limits.HasImages C", "X Y : C", "f : X ⟶ Y"], "goal": "CategoryTheory.Limits.HasImage f"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "Field K", "Field L", "Field L'", "Algebra K L", "Algebra K L'", "f : L →ₐ[K] L'"], "goal": "↑f.fieldRange = Set.range ⇑f"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "i : ι", "t : (i : ι) → Set (α i)"], "goal": "{i}.pi t = Function.eval i ⁻¹' t i"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "Nonempty α", "s : Set α", "MeasurableSpace α", "hs : MeasurableSet s"], "goal": "(PMF.uniformOfFintype α).toMeasure s = ↑(Fintype.card ↑s) / ↑(Fintype.card α)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t u : Finset α", "h : s ∪ t ⊆ u"], "goal": "t ⊆ u"}}
+{"state": {"context": ["n : ℕ"], "goal": "image (Prod.snd ∘ ⇑(Equiv.prodComm ℕ ℕ).toEmbedding) n.divisorsAntidiagonal = n.divisors"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E", "inst✝⁵ : Algebra F E", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "inst✝² : IsPurelyInseparable F E", "L : Type w", "inst✝¹ : CommRing L", "inst✝ : IsReduced L", "f g : E ��+* L", "x : E", "q : ℕ := ringExpChar F", "n : ℕ", "y : F", "h : (algebraMap F E) y = x ^ q ^ n", "heq : f x ^ q ^ n = g x ^ q ^ n", "a✝ : Nontrivial L", "this : ExpChar L q"], "goal": "f x = g x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝² : Ring R", "inst✝¹ : Ring S", "f : R →+* S", "I : Type u_3", "e : I → R", "he✝ : OrthogonalIdempotents e", "inst✝ : Fintype I", "he : CompleteOrthogonalIdempotents e", "e₁ e₂ : R", "he₁ : IsIdempotentElem e₁", "he₂ : IsIdempotentElem e₂", "H' : Commute e₁ e₂", "this : (e₁ - e₂) ^ 3 = e₁ - e₂", "n : ℕ", "hn : (e₁ - e₂) ^ n = 0"], "goal": "e₁ = e₂"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝⁶ : CommRing R", "inst✝⁵ : IsDomain R", "p✝ q✝ : R[X]", "inst✝⁴ : CommRing T", "inst✝³ : CommRing S", "inst✝² : IsDomain S", "inst✝¹ : Algebra T S", "inst✝ : NoZeroSMulDivisors T S", "p q : T[X]", "hpq : p * q ≠ 0"], "goal": "(p * q).aroots S = p.aroots S + q.aroots S"}}
+{"state": {"context": ["a b : ℝ≥0∞", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "0 < a * b"}}
+{"state": {"context": ["G : Type u", "H : Type v", "Add G", "Add H", "f : G ≃+ H"], "goal": "UniqueSums G ↔ UniqueSums H"}}
+{"state": {"context": ["b✝ x y : ℝ", "b : ℕ", "r : ℝ", "hb : 1 < b", "hr✝ : 0 ≤ r", "hr : 0 < r", "hb1' : 1 < ↑b"], "goal": "Int.log b r ≤ ⌊logb (↑b) r⌋"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "p : G.ConnectedComponent → Prop"], "goal": "(∀ (c : G.ConnectedComponent), p c) ↔ ∀ (v : V), p (G.connectedComponentMk v)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝² : MeasurableSpace α", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : OpensMeasurableSpace α", "e : ℕ → α", "x : α", "hx : x ∈ closure (Set.range e)", "ε : ℝ≥0∞", "hε : 0 < ε", "N : ℕ", "hN : edist x (e N) < ε"], "goal": "∃ ia, True ∧ ∀ x_1 ∈ Set.Ici ia, ↑(nearestPt e x_1) x ∈ ball x ε"}}
+{"state": {"context": ["self : CategoryTheory.FullSubcategory AlgebraicGeometry.Scheme.Spec.essImage"], "goal": "AlgebraicGeometry.AffineScheme.forgetToScheme.obj self = self.obj"}}
+{"state": {"context": ["k k' : Turing.ToPartrec.Cont", "v : List ℕ"], "goal": "(k.then k').eval v = k.eval v >>= k'.eval"}}
+{"state": {"context": ["L : FirstOrder.Language", "T : L.Theory", "α : Type w", "n : ℕ", "φ : L.BoundedFormula α n"], "goal": "T.SemanticallyEquivalent φ ∼∼φ"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ"], "goal": "(CochainComplex.HomComplex.Cochain.ofHom (CochainComplex.mappingCone.inr φ)).comp ↑(CochainComplex.mappingCone.fst φ)\n    ⋯ =\n  0"}}
+{"state": {"context": ["𝕜 : Type u_4", "F : Type u_5", "RCLike 𝕜", "AddCommGroup F", "Module 𝕜 F", "self : InnerProductSpace.Core 𝕜 F", "x : F"], "goal": "0 ≤ RCLike.re ⟪x, x⟫_𝕜"}}
+{"state": {"context": ["A : Type u_4", "M₁ : Type u_5", "SetLike A M₁", "Zero M₁", "hA : ZeroMemClass A M₁", "S' : A"], "goal": "↑0 = 0"}}
+{"state": {"context": ["R : Type u", "M : Type w", "CommRing R", "AddCommGroup M", "Module R M", "f : M →ₗ[R] M", "p q : R[X]", "hpq : IsCoprime p q"], "goal": "Disjoint (LinearMap.ker ((Polynomial.aeval f) p)) (LinearMap.ker ((Polynomial.aeval f) q))"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "a b : α", "h : ¬b ≤ a"], "goal": "a < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p : Set γ", "f f₁ f₂ f₃ : α → β", "g g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a : α", "b : β", "h₁ : SurjOn f s₁ t₁", "h₂ : SurjOn f s₂ t₂", "h : InjOn f (s₁ ∪ s₂)", "x₁ : α", "hx₁ : x₁ ∈ s₁", "hy : f x₁ ∈ t₁ ∩ t₂"], "goal": "f x₁ ∈ f '' (s₁ ∩ s₂)"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : α → β → β", "op : α → α → α", "inst✝¹ : Monoid β", "inst✝ : Monoid γ", "s : Finset α", "f : α → β", "comm : (↑s).Pairwise fun a b => Commute (f a) (f b)"], "goal": "{x | ∃ a ∈ s.val, f a = x}.Pairwise Commute"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π : BoxIntegral.TaggedPrepartition I"], "goal": "π.iUnion = π.iUnion"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "inst✝³ : Group G", "inst✝² : MulAction G α", "M : Type u_3", "inst✝¹ : Monoid M", "inst✝ : MulAction M α", "s : Set α", "g : G"], "goal": "s ∈ fixedBy (Set α) g ↔ ∀ (x : α), g • x ∈ s ↔ x ∈ s"}}
+{"state": {"context": ["G : Type u_1", "Group G", "s : Set G"], "goal": "↑(MulAction.stabilizer G s) * s = s"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_3", "CommSemiring R", "DecidableEq σ", "f : MvPowerSeries σ R", "n : ℕ", "d : σ →₀ ℕ"], "goal": "(MvPowerSeries.coeff R d) (f ^ n) =\n  ∑ l ∈ (Finset.range n).finsuppAntidiag d, ∏ i ∈ Finset.range n, (MvPowerSeries.coeff R (l i)) f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : TopologicalSpace β", "inst✝² : Preorder β", "inst✝¹ : OrderTop β", "inst✝ : OrderTopology β", "l : Filter α", "f g : α → β", "hg : f ≤ᶠ[l] g", "hf : ∀ i < ⊤, ∀ᶠ (a : α) in l, f a ∈ Ioi i", "x : β", "hx : x < ⊤"], "goal": "∀ᶠ (a : α) in l, g a ∈ Ioi x"}}
+{"state": {"context": ["R : Type u_1", "LinearOrderedSemifield R", "FloorSemiring R", "b : ℕ", "hb : 1 < b", "z : ℤ"], "goal": "Int.clog b (↑b ^ z) = z"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : AddCommSemigroup α", "inst✝⁴ : PartialOrder α", "inst✝³ : ExistsAddOfLE α", "inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a b c d : α", "h : c ≤ b"], "goal": "a - c - (b - c) = a - b"}}
+{"state": {"context": ["α : Type u_1", "AddCommSemigroup α", "PartialOrder α", "ExistsAddOfLE α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Sub α", "OrderedSub α", "a b c : α", "hc : AddLECancellable c", "h : c ≤ b"], "goal": "a < b - c ↔ a + c < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β → γ", "x : Option (α × β)"], "goal": "Option.map (Function.uncurry f) x = Option.map₂ f (Option.map Prod.fst x) (Option.map Prod.snd x)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasPullbacks C", "𝒢 : Set C", "Small.{v₁, u₁} ↑𝒢", "h𝒢 : CategoryTheory.IsDetecting 𝒢"], "goal": "CategoryTheory.WellPowered C"}}
+{"state": {"context": ["R : Type u", "L₁ : Type v", "L₂ : Type w", "L₃ : Type w₁", "CommRing R", "LieRing L₁", "LieRing L₂", "LieRing L₃", "LieAlgebra R L₁", "LieAlgebra R L₂", "LieAlgebra R L₃", "e₁ : L₁ ≃ₗ⁅R⁆ L₂", "e₂ : L₂ ≃ₗ⁅R⁆ L₃"], "goal": "(e₁.trans e₂).symm = e₂.symm.trans e₁.symm"}}
+{"state": {"context": ["G₁ : Type u_5", "G₂ : Type u_6", "G₃ : Type u_7", "Group G₁", "Group G₂", "Group G₃", "f : G₁ →* G₂", "f_inv : G₂ → G₁", "hf : Function.RightInverse f_inv ⇑f", "g : G₁ →* G₃", "hg : f.ker ≤ g.ker", "h : G₂ →* G₃", "hh : h.comp f = g"], "goal": "h = (f.liftOfRightInverse f_inv hf) ⟨g, hg⟩"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : Zero α", "inst✝³ : TopologicalSpace α", "inst✝² : PartialOrder α", "inst✝¹ : DecidableRel fun x x_1 => x < x_1", "inst✝ : OrderTopology α", "a : α", "h : a < 0"], "goal": "(fun x => -1) =ᶠ[nhds a] ⇑SignType.sign"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "IsDomain R", "CommSemiring S", "Algebra R S", "f : R[X]", "z : S", "hz : (Polynomial.aeval z) f = 0", "inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0"], "goal": "(Polynomial.aeval (z * (algebraMap R S) f.leadingCoeff)) f.integralNormalization = 0"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_4", "AddGroup α", "StarAddMonoid α", "A : Matrix n n α", "h : A.IsHermitian"], "goal": "(-A).IsHermitian"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "RegularSpace X", "x : X", "s : Set X"], "goal": "Disjoint (𝓝 x) (𝓝ˢ s) ↔ x ∉ closure s"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁴ : Group G", "inst✝³ : Group G'", "inst✝² : Group G''", "A : Type u_4", "inst✝¹ : AddGroup A", "N : Type u_5", "inst✝ : Group N", "f✝ : G →* N", "H : Subgroup G", "f : G ≃* N", "x : N", "a✝ : G"], "goal": "(a✝ ∈ H ↔ f.symm x * a✝ * (f.symm x)⁻¹ ∈ H) ↔ (f.symm (f.toEquiv a✝) ∈ H ↔ f.symm (x * f.toEquiv a✝ * x⁻¹) ∈ H)"}}
+{"state": {"context": ["E : Sort u_1", "α : Sort u_3", "β : Sort u_4", "γ : Sort u_5", "iE : EquivLike E α β", "e : E", "f : β → γ"], "goal": "Function.Bijective (f ∘ ⇑e) ↔ Function.Bijective f"}}
+{"state": {"context": [], "goal": "⊤ = true"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type uX", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F"], "goal": "ContDiffAt 𝕜 ⊤ f x ↔ ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) f x"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "p a : α"], "goal": "a ≡ a [PMOD p]"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : α → ℝ", "μ : MeasureTheory.Measure α", "hf : AEMeasurable f μ"], "goal": "AEMeasurable (fun x => ENNReal.ofReal (f x)) μ"}}
+{"state": {"context": ["𝕜 : Type u_8", "RCLike 𝕜", "n : Type u_9", "Fintype n", "x y : EuclideanSpace 𝕜 n"], "goal": "dist x y = √(∑ i : n, dist (x i) (y i) ^ 2)"}}
+{"state": {"context": ["α : Type u_1", "AddGroup α", "s : AddSubgroup α", "x y : α"], "goal": "Setoid.r x y ↔ -x + y ∈ s"}}
+{"state": {"context": ["α : Type u", "inst✝ : LinearOrder α", "a b : α", "h : b ≤ a"], "goal": "∀ {d : α}, a ≤ d → b ≤ d → a ≤ d"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X : Cᴹᵒᵖᴹᵒᵖ"], "goal": "((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).unitIso.inv.app X).unmop.unmop = (𝟙 X.unmop).unmop"}}
+{"state": {"context": ["w : Nat", "x y : BitVec w"], "goal": "x.add y = x + y"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝ : Semiring R", "p q✝ r : R[X]", "hm : p.Monic", "hpu : IsUnit p", "a✝ : Nontrivial R", "q : R[X]", "h : p * q = 1"], "goal": "p = 1"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B✝ I J X Y : Set α", "e : α", "hI : M.Basis I X", "B : Set α", "hB : M.Base B", "hIB : I ⊆ B"], "goal": "B ∩ X ⊆ I"}}
+{"state": {"context": ["n : ℕ"], "goal": "∫ (x : ℝ) in 0 ..π / 2, Real.cos x ^ n = 1 / 2 * ∫ (x : ℝ) in 0 ..π, Real.sin x ^ n"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "CommSemiring R", "Semiring A", "Algebra R A", "Semiring B", "Algebra R B", "ι : Type u_4", "Nonempty ι", "K : ι → Subalgebra R A", "dir : Directed (fun x x_1 => x ≤ x_1) K", "f : (i : ι) → ↥(K i) →ₐ[R] B", "hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (Subalgebra.inclusion h)", "T : Subalgebra R A", "hT : T = iSup K", "i : ι", "h : K i ≤ T"], "goal": "(Subalgebra.iSupLift K dir f hf T hT).comp (Subalgebra.inclusion h) = f i"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "CategoryTheory.Pretriangulated C", "A : Type u_2", "CategoryTheory.Category.{u_4, u_2} A", "CategoryTheory.Abelian A", "F : CategoryTheory.Functor Cᵒᵖ A", "F.IsHomological", "T : CategoryTheory.Pretriangulated.Triangle C", "hT : T ∈ CategoryTheory.Pretriangulated.distinguishedTriangles"], "goal": "((CategoryTheory.Pretriangulated.shortComplexOfDistTriangle T hT).op.map F).Exact"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝¹ : HasCoequalizers C", "R S T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)"], "goal": "(α_ P.X Q.X (T.X ⊗ T.X)).hom ≫ (P.X ◁ (α_ Q.X T.X T.X).inv ≫ P.X ◁ Q.actRight ▷ T.X ≫ P.X ◁ Q.actRight) ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) = (α_ (P.X ⊗ Q.X) T.X T.X).inv ≫ (α_ P.X Q.X T.X).hom ▷ T.X ≫ (P.X ◁ Q.actRight) ▷ T.X ≫ (α_ P.X Q.X T.X).hom ≫ P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}}
+{"state": {"context": ["x y : UInt16"], "goal": "x = y ↔ x ≤ y ∧ y ≤ x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "t u : Finset α", "f : α → β", "n : ℕ"], "goal": "∅.card ≤ 1 ↔ ∀ a ∈ ∅, ∀ b ∈ ∅, a = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ≥0∞", "s : Set α", "g : α →ₛ ℝ≥0∞", "hg : ↑g ≤ fun a => s.indicator f a", "this : ↑g ≤ f", "t : ℝ≥0∞", "H : ¬t = 0", "x : α"], "goal": "x ∈ ↑g ⁻¹' {t} ↔ x ∈ ↑g ⁻¹' {t} ∩ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝¹ : MeasurableSpace α", "K : Type u_5", "inst✝ : Zero β", "f : α →ₛ β", "s : Set α", "hs : MeasurableSet s", "a : α"], "goal": "↑(f.restrict s) a = s.indicator (↑f) a"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t : Set X", "f : X → Y", "hs : IsLindelof s", "hf : ContinuousOn f s", "l : Filter Y", "lne : l.NeBot", "inst✝ : CountableInterFilter l", "ls : l ≤ 𝓟 (f '' s)"], "goal": "∃ x ∈ f '' s, ClusterPt x l"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst���¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f✝ f₀ f₁ g : E → F", "f'✝ f₀' f₁' g' e : E →L[𝕜] F", "x✝ : E", "s✝ t : Set E", "L L₁ L₂ : Filter E", "iso : E ≃L[𝕜] F", "f : G → E", "s : Set G", "x : G", "f' : G →L[𝕜] E", "H : HasFDerivWithinAt (⇑iso ∘ f) ((↑iso).comp f') s x", "A : f = ⇑iso.symm ∘ ⇑iso ∘ f", "B : f' = (↑iso.symm).comp ((↑iso).comp f')"], "goal": "HasFDerivWithinAt (⇑iso.symm ∘ ⇑iso ∘ f) ((↑iso.symm).comp ((↑iso).comp f')) s x"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁸ : AddCommGroup E", "inst✝⁷ : Module ℝ E", "inst✝⁶ : FiniteDimensional ℝ E", "mE : MeasurableSpace E", "tE : TopologicalSpace E", "inst✝⁵ : TopologicalAddGroup E", "inst✝⁴ : BorelSpace E", "inst✝³ : T2Space E", "inst✝² : ContinuousSMul ℝ E", "μ : Measure E", "inst✝¹ : μ.IsAddHaarMeasure", "g : E → ℝ", "h1 : g 0 = 0", "h2 : ∀ (x : E), g (-x) = g x", "h3 : ∀ (x y : E), g (x + y) ≤ g x + g y", "h4 : ∀ {x : E}, g x = 0 → x = 0", "h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x", "inst✝ : Nontrivial E", "r : ℝ", "F : Type u_1 := E", "this✝⁴ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯", "this✝³ : NormedSpace ℝ F := NormedSpace.mk ⋯", "this✝² : TopologicalSpace F := UniformSpace.toTopologicalSpace", "this✝¹ : MeasurableSpace F := borel F", "this✝ : BorelSpace F", "φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv", "ν : Measure F := map (⇑φ) μ", "this : ν.IsAddHaarMeasure"], "goal": "μ {x | g x ≤ r} = μ {x | g x < r}"}}
+{"state": {"context": ["n : ℕ", "f✝ : Vector ℕ n → ℕ", "f : ℕ → ℕ", "hf : Nat.Primrec f"], "goal": "Primrec' fun v => f v.head"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝³ : LinearOrderedAddCommGroup 𝕜", "inst✝² : TopologicalSpace 𝕜", "inst✝¹ : OrderTopology 𝕜", "p a : 𝕜", "hp : Fact (0 < p)", "inst✝ : Archimedean 𝕜", "f : 𝕜 → B", "x : 𝕜", "hx : x ∈ Ico 0 p"], "goal": "x ∈ Ico 0 (0 + p)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "μ' : Measure α", "c : ℝ≥0∞", "hc : c ≠ ⊤", "hμ'_le : μ' ≤ c • μ", "f : α → β", "hf : Integrable f μ"], "goal": "Integrable f μ'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "g : β → α", "h₁ : Function.LeftInverse g f", "h₂ : Function.RightInverse g f"], "goal": "Set.image f = Set.preimage g"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "x✝ y✝ z x y : E", "a : 𝕜", "h₀ : 0 < a", "h₁ : a < 1"], "goal": "(lineMap x y) a ∈ insert ((lineMap x y) a) (⇑(lineMap x ((lineMap x y) a)) '' Ioo 0 1 ∪ ⇑(lineMap ((lineMap x y) a) y) '' Ioo 0 1)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : LinearOrderedRing R", "a b n✝ : ℤ", "x : R", "n : ℤ", "hx : |x| ≤ 1"], "goal": "0 ≤ ↑n * x + ↑n * ↑n"}}
+{"state": {"context": ["M₀ : Type u_2", "CancelMonoidWithZero M₀", "a b : M₀"], "goal": "a * b = a ↔ b = 1 ∨ a = 0"}}
+{"state": {"context": ["α β : Type u", "f : Cardinal.{u} → Cardinal.{max u v}", "s : Set Cardinal.{u}", "hs : BddAbove s"], "goal": "BddAbove (f '' s)"}}
+{"state": {"context": ["E : Type u_5", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "p : ℝ"], "goal": "MeasureTheory.eLpNormLESNormFDerivOneConst μ p = MeasureTheory.lintegralPowLePowLIntegralFDerivConst μ p ^ p⁻¹"}}
+{"state": {"context": ["p : ℕ", "g : ⦃R : Type u_3⦄ → [inst : CommRing R] → WittVector p R → WittVector p R → WittVector p R", "f : ⦃R : Type u_3⦄ → [inst : CommRing R] → WittVector p R → WittVector p R", "WittVector.IsPoly₂ p g", "WittVector.IsPoly p f"], "goal": "WittVector.IsPoly₂ p fun _R _Rcr x y => g x (f y)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "M : Type u_2", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M"], "goal": "Submodule.comap involute.toLinearMap (LinearMap.range (ι Q)) = LinearMap.range (ι Q)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G"], "goal": "H.relindex H = 1"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q r : ℚ", "hqr : q + r ≠ 0", "hq : q ≠ 0", "hr : r ≠ 0", "hval : padicValRat p q < padicValRat p r"], "goal": "padicValRat p (q + r) = padicValRat p q"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "f : { f // ∀ (i : α), (f.2 i).isSome = true ↔ i ∈ f.1 }", "i : α", "x : β i", "left✝¹ : ⟨i, x⟩.fst ∈ (↑f).1", "hx : (↑f).2 ⟨i, x⟩.fst = some ⟨i, x⟩.snd", "y : β i", "left✝ : ⟨i, y⟩.fst ∈ (↑f).1", "hy : (↑f).2 ⟨i, y⟩.fst = some ⟨i, y⟩.snd"], "goal": "⟨i, x⟩ = ⟨i, y⟩"}}
+{"state": {"context": ["a b c d : Int", "h₁ : a < c", "h₂ : b ≤ d", "h₃ : 0 < b", "h₄ : 0 ≤ c"], "goal": "a * b < c * d"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : QuotientMap f"], "goal": "Continuous f"}}
+{"state": {"context": ["b : EReal", "h : b ≤ 0"], "goal": "Antitone fun a => a / b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : TopologicalSpace α", "inst✝⁴ : LinearOrder α", "inst✝³ : DenselyOrdered α", "inst✝² : OrderTopology α", "inst✝¹ : TopologicalSpace β", "inst✝ : RegularSpace β", "f : α → β", "a b : α", "la lb : β", "hab : a ≠ b", "hf : ContinuousOn f (Ioo a b)", "ha : Tendsto f (𝓝[>] a) (𝓝 la)", "hb : Tendsto f (𝓝[<] b) (𝓝 lb)"], "goal": "ContinuousOn (extendFrom (Ioo a b) f) (Icc a b)"}}
+{"state": {"context": ["α✝ : Sort u", "β✝ : Sort v", "γ : Sort w", "α : Type u_1", "β : Type u_2", "e : α ≃ β", "s : Set α", "t : Set β"], "goal": "⇑e.symm '' t ⊆ s ↔ t ⊆ ⇑e '' s"}}
+{"state": {"context": ["α : Type u_1", "μ₁ μ₂ : MeasureTheory.OuterMeasure α", "h : ∀ (s : Set α), s.Nonempty → μ₁ s = μ₂ s"], "goal": "μ₁ = μ₂"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "G : C ⥤ D", "f : X ⟶ Z", "g : Y ⟶ Z", "CategoryTheory.Limits.HasPullback f g", "CategoryTheory.Limits.HasPullback (G.map f) (G.map g)"], "goal": "CategoryTheory.Limits.pullbackComparison G f g ≫ CategoryTheory.Limits.pullback.fst (G.map f) (G.map g) =\n  G.map (CategoryTheory.Limits.pullback.fst f g)"}}
+{"state": {"context": ["this : -(π / 2) < π / 2"], "goal": "Tendsto tan (𝓝[>] (-(π / 2))) atBot"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "T0Space E", "a : E"], "goal": "‖a‖ = 0 ↔ a = 0"}}
+{"state": {"context": ["n : ℕ", "hn : Squarefree n"], "goal": "IsSquare (-1) ↔ ∀ {q : ℕ}, Nat.Prime q → q ∣ n → q % 4 ≠ 3"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l✝ l₁ l₂ : List α", "r : α → α → Prop", "a b : α", "p : α → Bool", "l : List α"], "goal": "l.Nodup → (List.filter p l).Nodup"}}
+{"state": {"context": ["ι : Type u", "α : ι → Type v"], "goal": "Cardinal.mk ((i : ι) → α i) = Cardinal.prod fun i => Cardinal.mk (α i)"}}
+{"state": {"context": ["R : Type u_5", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "s : Multiset R", "t : Multiset M"], "goal": "s.sum • t.sum = (Multiset.map (fun p => p.1 • p.2) (s ×ˢ t)).sum"}}
+{"state": {"context": ["p : Prop"], "goal": "(p ∧ True) = p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁸ : TopologicalSpace α", "inst✝⁷ : TopologicalSpace β", "inst✝⁶ : LinearOrder α", "inst✝⁵ : LinearOrder β", "inst✝⁴ : OrderTopology α", "inst✝³ : OrderTopology β", "inst✝² : DenselyOrdered α", "inst✝¹ : NoMinOrder α", "inst✝ : FirstCountableTopology α", "x y : α", "hy : y < x", "u : ℕ → α", "hu_mono : StrictMono u", "hu_mem : ∀ (n : ℕ), u n ∈ Ioo y x", "hux : Tendsto u atTop (𝓝 x)"], "goal": "∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x)"}}
+{"state": {"context": ["x✝ y : ℂ", "x : ℝ"], "goal": "(starRingEnd ℂ) (tanh ↑x) = tanh ↑x"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_3", "m : MeasurableSpace Ω", "Preorder ι", "s : Set (MeasureTheory.Filtration ι m)", "i : ι"], "goal": "↑(sSup s) i = sSup ((fun f => ↑f i) '' s)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "inst✝² : TopologicalSpace H'", "inst✝¹ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝ : I.Boundaryless"], "goal": "IsOpen (f.extend I).target"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "x : S", "h : adjoin R {x} = ⊤"], "goal": "conductor R x = ⊤"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "TopologicalSpace α", "CompactIccSpace α", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.IsLocallyFiniteMeasure μ", "a b : α"], "goal": "μ (Set.Ioc a b) < ⊤"}}
+{"state": {"context": ["α : Type u_1", "a b : α", "as bs : Multiset α"], "goal": "a ::ₘ as = b ::ₘ bs ↔ a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "ρ : MeasureTheory.Measure (α × ℝ)", "MeasureTheory.IsFiniteMeasure ρ", "r : ℝ"], "goal": "MeasureTheory.IsFiniteMeasure (ρ.IicSnd r)"}}
+{"state": {"context": ["Δ Δ' Δ'' : SimplexCategory", "φ : Δ ⟶ Δ''", "e : Δ ⟶ Δ'", "inst✝¹ : Epi e", "i : Δ' ⟶ Δ''", "inst✝ : Mono i", "fac : e ≫ i = φ"], "goal": "image φ = Δ'"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ : MeasurableSpace Ω", "μ : Measure Ω", "f g : Ω → ℝ≥0∞", "X✝ Y✝ : Ω → ℝ", "β : Type u_2", "inst✝² : MeasurableSpace β", "X Y : Ω → β", "inst✝¹ : NormedDivisionRing β", "inst✝ : BorelSpace β", "hXY : IndepFun X Y μ", "h'XY : Integrable (X * Y) μ", "hX : AEStronglyMeasurable X μ", "hY : AEStronglyMeasurable Y μ", "h'X : ¬X =ᶠ[ae μ] 0", "I : ∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ ≠ 0", "H : ∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ = ⊤", "J : IndepFun (fun ω => ↑‖X ω‖₊) (fun ω => ↑‖Y ω‖₊) μ", "A : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) * ⊤ < ⊤"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s' : Set α", "x : α", "p : Filter ι", "p' : Filter α", "g : ι → α", "inst✝ : p.NeBot", "hF : UniformCauchySeqOnFilter F p p'", "hF' : ∀ᶠ (x : α) in p', Tendsto (fun n => F n x) p (𝓝 (f x))", "u : Set (β × β)", "hu : u ∈ 𝓤 β", "t : Set (β × β)", "ht : t ∈ 𝓤 β", "htsymm : ∀ {a b : β}, (a, b) ∈ t → (b, a) ∈ t", "htmem : t ○ t ⊆ u", "hmc : ∀ᶠ (x : (ι × ι) × α) in (p ×ˢ p) ×ˢ p', Tendsto (fun n => F n x.2) p (𝓝 (f x.2))", "this : ∀ᶠ (x : (α × ι) × ι) in (p' ×ˢ p) ×ˢ p, (F ((Equiv.prodAssoc α ι ι) x).swap.1.1 ((Equiv.prodAssoc α ι ι) x).swap.2, F ((Equiv.prodAssoc α ι ι) x).swap.1.2 ((Equiv.prodAssoc α ι ι) x).swap.2) ∈ t ∧ Tendsto (fun n => F n ((Equiv.prodAssoc α ι ι) x).swap.2) p (𝓝 (f ((Equiv.prodAssoc α ι ι) x).swap.2))"], "goal": "∀ᶠ (y : α × ι) in p' ×ˢ p, (f y.swap.2, F y.swap.1 y.swap.2) ∈ u"}}
+{"state": {"context": ["n : Nat", "i : Nat", "h : i < n"], "goal": "0 < ⟨i.succ, ⋯⟩"}}
+{"state": {"context": ["ι : Type w", "s : Finset ι", "S : Type u_1", "Semiring S", "f : ι → S[X]"], "goal": "(∑ i ∈ s, f i).natDegree ≤ Finset.fold max 0 (Polynomial.natDegree ∘ f) s"}}
+{"state": {"context": [], "goal": "USize.size ≤ 2 ^ 64"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "NonUnitalNonAssocSemiring α", "Fintype n", "A : Matrix m n α"], "goal": "A *ᵥ 0 = 0"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝³ : DecidableEq ι", "inst✝² : (i : ι) → AddZeroClass (β i)", "inst✝¹ : (i : ι) → AddZeroClass (β₁ i)", "inst✝ : (i : ι) → AddZeroClass (β₂ i)", "f : (i : ι) → β i ≃+ β₁ i", "f₂ : (i : ι) → β₁ i ≃+ β₂ i"], "goal": "(addEquiv fun i => (f i).trans (f₂ i)) = (addEquiv f).trans (addEquiv f₂)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "C : Type u_3", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : StarRing R", "inst✝⁹ : NonUnitalSemiring A", "inst✝⁸ : StarRing A", "inst✝⁷ : Module R A", "inst✝⁶ : SMulCommClass R A A", "inst✝⁵ : IsScalarTower R A A", "inst✝⁴ : StarModule R A", "inst✝³ : Semiring C", "inst✝² : Algebra R C", "inst✝¹ : StarRing C", "inst✝ : StarModule R C", "φ : A →⋆ₙₐ[R] C", "r✝ : R", "a✝ : A"], "goal": "(↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (star (inl r✝ + ↑a✝)) = star ((↑↑(lift φ.toNonUnitalAlgHom).toRingHom).toFun (inl r✝ + ↑a✝))"}}
+{"state": {"context": ["𝕜✝ : Type u", "inst✝¹⁴ : NontriviallyNormedField 𝕜✝", "E✝ : Type v", "inst✝¹³ : NormedAddCommGroup E✝", "inst✝¹² : NormedSpace 𝕜✝ E✝", "F : Type w", "inst✝¹¹ : NormedAddCommGroup F", "inst✝¹⁰ : NormedSpace 𝕜✝ F", "F' : Type x", "inst✝⁹ : AddCommGroup F'", "inst✝⁸ : Module 𝕜✝ F'", "inst✝⁷ : TopologicalSpace F'", "inst✝⁶ : TopologicalAddGroup F'", "inst✝⁵ : ContinuousSMul 𝕜✝ F'", "inst✝⁴ : CompleteSpace 𝕜✝", "𝕜 : Type u_1", "inst✝³ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : LocallyCompactSpace E", "r : ℝ", "rpos : 0 < r", "hr : IsCompact (closedBall 0 r)", "c : 𝕜", "hc : 1 < ‖c‖", "hC : ∀ (n : ℕ), IsCompact (closedBall 0 (‖c‖ ^ n * r))"], "goal": "ProperSpace E"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : Measure α", "f✝ g✝ : α → F'", "s : Set α", "fs gs : ℕ → α → F'", "f g : α → F'", "hfs_int : ∀ (n : ℕ), Integrable (fs n) μ", "hgs_int : ∀ (n : ℕ), Integrable (gs n) μ", "hfs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => fs n x) atTop (𝓝 (f x))", "hgs : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => gs n x) atTop (𝓝 (g x))", "bound_fs : α → ℝ", "h_int_bound_fs : Integrable bound_fs μ", "bound_gs : α → ℝ", "h_int_bound_gs : Integrable bound_gs μ", "hfs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖fs n x‖ ≤ bound_fs x", "hgs_bound : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, ‖gs n x‖ ≤ bound_gs x", "hfg : ∀ (n : ℕ), μ[fs n|m] =ᶠ[ae μ] μ[gs n|m]", "hm : m ≤ m0", "hμm this : SigmaFinite (μ.trim hm)"], "goal": "μ[f|m] =ᶠ[ae μ] μ[g|m]"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_4", "l : Type u_8", "m : Type u_9", "n : Type u_10", "p : Type u_11", "CommSemiring R", "AddCommMonoid α", "AddCommMonoid β", "Module R α", "Module R β", "A : Matrix l m α", "B : Matrix n p β", "i₁ : l", "i₂ : n", "j₁ : m", "j₂ : p"], "goal": "Matrix.kroneckerMap (fun x x_1 => x ⊗ₜ[R] x_1) A B (i₁, i₂) (j₁, j₂) = A i₁ j₁ ⊗ₜ[R] B i₂ j₂"}}
+{"state": {"context": ["r : ℚ", "p' : ℕ", "hp_prime : Fact (Nat.Prime (0 + p' + 1))", "x : ℤ_[0 + p' + 1]"], "goal": "x.zmodRepr % (p' + 1) = x.zmodRepr"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "DecidableEq β", "SMul α β", "s : Finset α", "t : Finset β"], "goal": "s • t = ∅ ↔ s = ∅ ∨ t = ∅"}}
+{"state": {"context": ["K : Type u_2", "L : Type u_3", "LieRing L", "Field K", "LieAlgebra K L", "FiniteDimensional K L", "H : LieSubalgebra K L", "H.IsCartanSubalgebra", "LieModule.IsTriangularizable K (↥H) L", "LieAlgebra.IsKilling K L", "CharZero K", "α : LieModule.Weight K (↥H) L", "hα : α.IsNonZero"], "goal": "α ((LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)) ≠ 0"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "K K' : ChainComplex C ℕ", "f : K ⟶ K'", "Δ Δ'' : SimplexCategory", "i : Δ ⟶ Δ", "inst✝ : Mono i", "hi : Isδ₀ i"], "goal": "False"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v u' v' : V", "p : G.Walk u v", "hu : u = u'", "hv : v = v'"], "goal": "(p.copy hu hv).edges = p.edges"}}
+{"state": {"context": ["n : ℕ", "hn0 : ¬n = 0", "hn : 1 ≤ n"], "goal": "∑ x ∈ Finset.Ioc 1 n, (↑x)⁻¹ ≤ Real.log ↑n"}}
+{"state": {"context": ["x : ℤ"], "goal": "(∃ n, n * n = x) ↔ Int.sqrt x * Int.sqrt x = x"}}
+{"state": {"context": ["V₁ : Type u_2", "V₂ : Type u_3", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂", "f : NormedAddGroupHom V₁ V₂"], "goal": "LipschitzWith ⟨‖f‖, ⋯⟩ ⇑f"}}
+{"state": {"context": ["𝕜 : Type u", "A : Type v", "inst✝³ : Field 𝕜", "inst✝² : Ring A", "inst✝¹ : Algebra 𝕜 A", "inst✝ : Nontrivial A", "k : 𝕜"], "goal": "σ (↑ₐ k) = {k}"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "v : V"], "goal": "G.incidenceSet v ⊆ G.edgeSet"}}
+{"state": {"context": ["J : Type v", "inst✝¹ : SmallCategory J", "F : J ⥤ MonCatMax", "inst✝ : IsFiltered J", "j₁ : J", "x : ↑(F.obj j₁)", "j₂ : J", "y : ↑(F.obj j₂)", "j₃ : J", "x' : ↑(F.obj j₃)", "l : J", "f : ⟨j₁, x⟩.fst ⟶ l", "g : ⟨j₃, x'⟩.fst ⟶ l", "hfg : (F.map f) x = (F.map g) x'", "s : J", "α : IsFiltered.max j₁ j₂ ⟶ s", "β : l ⟶ s", "γ : IsFiltered.max j₃ j₂ ⟶ s", "h₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β", "h₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = IsFiltered.rightToMax j₃ j₂ ≫ γ", "h₃ : IsFiltered.leftToMax j₃ j₂ ≫ γ = g ≫ β"], "goal": "(F.map (IsFiltered.leftToMax j₁ j₂) ≫ F.map α) x * (F.map (IsFiltered.rightToMax j₁ j₂) ≫ F.map α) y = (F.map (IsFiltered.leftToMax j₃ j₂) ≫ F.map γ) x' * (F.map (IsFiltered.rightToMax j₃ j₂) ≫ F.map γ) y"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : Type u_5", "x : ι → A", "CommRing R", "CommRing A", "Algebra R A"], "goal": "AlgebraicIndependent R x ↔ RingHom.ker (MvPolynomial.aeval x).toRingHom = ⊥"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f f₁ : E → F", "x : E", "s : Set E", "h : f₁ =ᶠ[𝓝 x] f"], "goal": "fderivWithin 𝕜 f₁ s x = fderivWithin 𝕜 f s x"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "k : ℕ+", "hc : (↑k).Coprime ↑(torsionOrder K)", "ζ : (𝓞 K)ˣ", "h : ζ ^ ↑k = 1"], "goal": "orderOf ζ ∣ ↑(torsionOrder K)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "x : G", "h : IsOfFinAddOrder x", "i : ℤ"], "goal": "IsOfFinAddOrder (i • x)"}}
+{"state": {"context": ["n : ℕ"], "goal": "n ∈ properDivisors 0 ↔ n ∣ 0 ∧ n < 0"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : Semiring R", "S : Type v", "inst✝ : Semiring S", "f : R →+* S", "s : S", "n : ℕ", "hl : (monomial n) s ∈ Set.range (map f)", "hzero : ¬s = 0"], "goal": "∃ q, map f q = (monomial n) s ∧ q.degree = ((monomial n) s).degree"}}
+{"state": {"context": ["n : ℕ"], "goal": "↑n = 0"}}
+{"state": {"context": ["f : ℝ → ℝ", "hfc : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.volume.restrict (Set.Ioi 0))", "a s : ℝ", "hf : f =O[Filter.atTop] fun x => x ^ (-a)", "hs : s < a"], "goal": "∃ c, 0 < c ∧ MeasureTheory.IntegrableOn (fun t => t ^ (s - 1) * f t) (Set.Ioi c) MeasureTheory.volume"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "D : Type w", "CategoryTheory.Category.{max v u, w} D", "CategoryTheory.ConcreteCategory D", "X : C", "P : CategoryTheory.Functor Cᵒᵖ D", "S : J.Cover X", "x : CategoryTheory.Meq P S", "I : S.Relation"], "goal": "(P.map I.r.g₁.op) (↑x ((S.index P).fstTo I)) = (P.map I.r.g₂.op) (↑x ((S.index P).sndTo I))"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁶ : Semiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid M'", "inst✝² : Module R M'", "b b₁ : Basis ι R M", "i : ι", "c : R", "x : M", "b' : Basis ι' R M'", "e : ι ≃ ι'", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq M", "bi : M", "hbi : bi ∈ Finset.image (⇑b) Finset.univ"], "goal": "b.reindexFinsetRange ⟨bi, hbi⟩ = ↑⟨bi, hbi⟩"}}
+{"state": {"context": ["ι : Type u_1", "Ω : Type u_2", "inst✝³ : PseudoMetricSpace Ω", "inst✝² : SeparableSpace Ω", "inst✝¹ : MeasurableSpace Ω", "inst✝ : OpensMeasurableSpace Ω", "ε : ℝ", "ε_pos : 0 < ε", "X_emp : ¬IsEmpty Ω"], "goal": "∃ As, (∀ (n : ℕ), MeasurableSet (As n)) ∧ (∀ (n : ℕ), Bornology.IsBounded (As n)) ∧ (∀ (n : ℕ), diam (As n) ≤ ε) ∧ ⋃ n, As n = univ ∧ Pairwise fun n m => Disjoint (As n) (As m)"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "R : Cᵒᵖ ⥤ RingCat", "F : D ⥤ PresheafOfModules R", "inst✝¹ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections", "c : Cone F", "hc : IsLimit c", "hF : ∀ (j : D), Presheaf.IsSheaf J (F.obj j).presheaf", "inst✝ : HasLimitsOfShape D AddCommGrp", "G : D ⥤ Sheaf J AddCommGrp := { obj := fun j => { val := (F.obj j).presheaf, cond := ⋯ }, map := fun {X Y} φ => { val := (toPresheaf R).map (F.map φ) }, map_id := ⋯, map_comp := ⋯ }"], "goal": "Presheaf.IsSheaf J c.pt.presheaf"}}
+{"state": {"context": ["m n₁ n₂ : ℕ"], "goal": "ack m n₁ = ack m n₂ ↔ n₁ = n₂"}}
+{"state": {"context": ["s : ℂ"], "goal": "LSeriesSummable 0 s"}}
+{"state": {"context": ["α : Type u", "n : ℕ"], "goal": "map Fin.val (finRange n.succ) = map Fin.val ((map Fin.castSucc (finRange n)).concat (Fin.last n))"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁵ : NormedAddCommGroup E", "inst✝¹⁴ : NormedAddCommGroup E'", "inst✝¹³ : NormedAddCommGroup E''", "inst✝¹² : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NormedSpace 𝕜 E", "inst✝⁹ : NormedSpace 𝕜 E'", "inst✝⁸ : NormedSpace 𝕜 E''", "inst✝⁷ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝⁶ : MeasurableSpace G", "μ ν : Measure G", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : AddCommGroup G", "inst✝³ : μ.IsAddLeftInvariant", "inst✝² : μ.IsNegInvariant", "inst✝¹ : MeasurableNeg G", "inst✝ : MeasurableAdd G", "x : G"], "goal": "∫ (t : G), (L.flip (g t)) (f (x - t)) ∂μ = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ"}}
+{"state": {"context": ["K : Type u", "Field K", "s : Subfield K", "ι : Type u_1", "t : Finset ι", "f : ι → K", "h : ∀ c ∈ t, f c ∈ s"], "goal": "∏ i ∈ t, f i ∈ s"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "inst✝ : T0Space α", "s : Closeds α", "hs : IsAtom s"], "goal": "∃ a, ↑s = {a}"}}
+{"state": {"context": ["α : Type u", "CommSemiring α", "x : α"], "goal": "Ideal.span {x} = ⊤ ↔ IsUnit x"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝³ : DecidableEq ι", "κ : Type u_1", "inst✝² : DecidableEq ι", "inst✝¹ : DecidableEq κ", "inst✝ : (i : ι) → Zero (β i)", "h : κ → ι", "h' : ι → κ", "hh' : Function.LeftInverse h' h", "k : κ", "x : β (h k)", "i : κ"], "goal": "(single (h k) x) (h i) = (single k x) i"}}
+{"state": {"context": ["V W : SemiNormedGrp"], "goal": "SemiNormedGrp.completion.map 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "p n : ℕ", "CommSemiring R", "ExpChar R p", "PerfectRing R p"], "goal": "(iterateFrobeniusEquiv R p n).symm = (frobeniusEquiv R p).symm ^ n"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "F : Type u_3", "π : ι → Type u_4", "inst✝² : Fintype ι", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → NormedAddCommGroup (π i)", "i : ι", "y : π i"], "goal": "‖single i y‖₊ = ‖y‖₊"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d x : α", "h : Ici x = {x}", "y : α", "H : x < y", "this : y = x"], "goal": "False"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : E", "s : Set E", "iso : E ≃L[𝕜] F", "hxs : UniqueDiffWithinAt 𝕜 s x"], "goal": "fderivWithin 𝕜 (⇑iso) s x = ↑iso"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "CanonicallyOrderedCommMonoid α", "TopologicalSpace α", "OrderClosedTopology α", "f : ι → α", "a : α", "hf : HasProd f a"], "goal": "IsLUB (Set.range fun s => ∏ i ∈ s, f i) a"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "inst✝² : CommRing R", "inst✝¹ : CommRing R'", "inst✝ : Finite Rˣ", "χ : MulChar R ℂ", "a : R"], "goal": "star (χ a) = χ⁻¹ a"}}
+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "n : ℕ", "inst✝ : Group G", "g : G →₀ k", "f : Fin n → G", "r : k"], "goal": "(diagonalSucc k G n).inv.hom (0 ⊗ₜ[k] single f r) = ((Finsupp.lift ((Fin (n + 1) → G) →₀ k) k G) fun a => single (a • partialProd f) r) 0"}}
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+{"state": {"context": ["f : ℕ → ℝ≥0", "hf : Summable f"], "goal": "Tendsto f cofinite (𝓝 0)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]"], "goal": "p.trailingDegree ≠ 0 ↔ p.coeff 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "M' : Type u_3", "AddCommGroup M'", "Module R M'", "ι : Type u_4", "DecidableEq ι", "Fintype ι", "b : Basis ι R M", "f : M ≃ₗ[R] M'", "v : ι → M'"], "goal": "(b.map f).det v = b.det (⇑f.symm ∘ v)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "C : Levenshtein.Cost α β δ", "inst✝ : CanonicallyLinearOrderedAddCommMonoid δ", "xs : List α", "y : β", "ys : List β", "a : List α", "suff : a <:+ xs", "a' : List α", "suff' : a' <:+ a"], "goal": "(List.map (fun xs' => levenshtein C xs' ys) xs.tails).minimum ≤ ↑(levenshtein C a' ys)"}}
+{"state": {"context": ["S : Finset Λ'", "W : ∀ {q : Λ'}, q ∈ trStmts₁ q", "q : Option Γ' → Λ'", "q_ih : ∀ (a : Option Γ'), Λ'.Supports S (q a) → trStmts₁ (q a) ⊆ S → Supports (trStmts₁ (q a)) S", "H₁ : Λ'.Supports S (Λ'.read q)", "HS₁ : insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s)) ⊆ S", "h₁ : Λ'.read q ∈ S", "h₂ : ∀ ⦃x : Λ'⦄ (x_1 : Option Γ'), x ∈ trStmts₁ (q x_1) → x ∈ S"], "goal": "Supports (insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s))) S"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "x : E", "s : Set E", "f₂ : E → F × G", "h : DifferentiableWithinAt 𝕜 f₂ s x"], "goal": "DifferentiableWithinAt 𝕜 (fun x => (f₂ x).1) s x"}}
+{"state": {"context": ["x : ZFSet"], "goal": "⋃₀ {x} = x"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "p : ℝ≥0∞", "q : ℝ", "μ : Measure α", "f✝ g✝ f g : α → E", "hf : AEStronglyMeasurable f μ", "hg : AEStronglyMeasurable g μ", "hp1 : 1 ≤ p", "hp0 : ¬p = 0", "hp_top : ¬p = ⊤", "hp1_real : 1 ≤ p.toReal"], "goal": "eLpNorm' (f + g) p.toReal μ ≤ eLpNorm' f p.toReal μ + eLpNorm' g p.toReal μ"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F G : J ⥤ C", "s : CategoryTheory.Limits.Cocone F", "t : CategoryTheory.Limits.Cocone G", "P : CategoryTheory.Limits.IsColimit s", "Q : CategoryTheory.Limits.IsColimit t", "w : F ≅ G", "j : J"], "goal": "s.ι.app j ≫ (P.coconePointsIsoOfNatIso Q w).hom = w.hom.app j ≫ t.ι.app j"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝ : TopologicalSpace X", "r : Rel Y X", "l : Filter Y", "x : X"], "goal": "∀ (s t : Set X), s ⊆ t → r.preimage s ⊆ r.preimage t"}}
+{"state": {"context": ["α : Type u_4", "Finite α"], "goal": "PartENat.card α = ↑(Nat.card α)"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "H : Subgroup G", "inst✝ : H.Normal", "h : Group.IsNilpotent (G ⧸ center G)", "n : ℕ", "hn : upperCentralSeries (G ⧸ center G) n = ⊤"], "goal": "Group.IsNilpotent G"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝³ : Field K", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : Algebra.IsIntegral K L", "f g : L ≃ₐ[K] L", "hfg : f ≠ g", "φ : L ≃ₐ[K] L := f⁻¹ * g", "x : L", "hx : f x ≠ g x", "hφx : φ x ≠ x", "E : IntermediateField K L := K⟮x⟯", "h_findim : FiniteDimensional K ↥E := IntermediateField.adjoin.finiteDimensional (Algebra.IsIntegral.isIntegral x)", "H : Subgroup (L ≃ₐ[K] L) := E.fixingSubgroup", "h_basis : ↑H ∈ galGroupBasis K L", "W : Set (L ≃ₐ[K] L)", "hWH : W ⊆ ↑H", "hW_open : IsOpen W", "hW_1 : 1 ∈ W", "w1 : L ≃ₐ[K] L", "hw1 : w1 ∈ W", "w2 : L ≃ₐ[K] L", "hw2 : w2 ∈ W", "h : w1 * w2⁻¹ = f⁻¹ * g", "h_in_H : x ∈ E → φ x = x"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing S", "I : Ideal R", "y : R", "inst✝¹ : Algebra R S", "inst✝ : Away y S", "H : IsJacobson R"], "goal": "∀ (P : Ideal S), P.IsPrime → P.jacobson = P"}}
+{"state": {"context": ["α : Type u", "inst✝⁵ : Primcodable α", "inst✝⁴ : Inhabited α", "β : Type v", "inst✝³ : Primcodable β", "inst✝² : Inhabited β", "γ : Type w", "inst✝¹ : Primcodable γ", "inst✝ : Inhabited γ", "d₁ d₂ : ManyOneDegree"], "goal": "d₁ ≤ d₂ → d₂ ≤ d₁ → d₁ = d₂"}}
+{"state": {"context": ["x x₁ x₂ x₃ x' y y₁✝ y₂✝ y₃ y' y₁ y₂ : PGame", "h : ∀ (i : x₂.LeftMoves), IH3 x₁ (x₂.moveLeft i) x₂ y₁ y₂", "hs : ∀ (i : (-x₁).LeftMoves), IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂", "hl : x₁ < x₂", "i : x₁.RightMoves", "hi : x₁.moveRight i ≤ x₂"], "goal": "P3 x₁ x₂ y₁ y₂"}}
+{"state": {"context": ["α : Type u_2", "Monoid α"], "goal": "ThreeGPFree ∅"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "ι : Type u_4", "f : X → Y", "g : Y → Z", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "hf : Inducing f", "s : Set X", "x : X"], "goal": "x ∈ closure s ↔ x ∈ f ⁻¹' closure (f '' s)"}}
+{"state": {"context": ["a b c : ℤ", "h : a.gcd b = 1", "heq : a * b = c ^ 2"], "goal": "∃ a0, a = a0 ^ 2 ∨ a = -a0 ^ 2"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "T₁ T₂ : CategoryTheory.Monad C", "f g : T₁ ⟶ T₂", "h : f.app = g.app"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝⁴ : (i : ι) → Zero (α i)", "inst✝³ : (i : ι) → PartialOrder (α i)", "inst✝² : (i : ι) → LocallyFiniteOrder (α i)", "f g : Π₀ (i : ι), α i", "i : ι", "a : α i", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → DecidableEq (α i)", "x : ι", "hx : x ∈ (f.rangeIcc g).support", "h : x ∉ f.support ∪ g.support"], "goal": "Icc 0 0 = 0"}}
+{"state": {"context": ["α : Type u", "a : α", "inst✝³ : Group α", "inst✝² : DecidableEq α", "inst✝¹ : Fintype α", "inst✝ : IsCyclic α", "n : ℕ", "hn0 : 0 < n", "g : α", "hg : ∀ (x : α), x ∈ zpowers g"], "goal": "(↑(zpowers (g ^ (Fintype.card α / n.gcd (Fintype.card α))))).toFinset.card ≤ n"}}
+{"state": {"context": ["K : Type u", "inst✝³ : Field K", "n : ℕ", "hζ✝ : (primitiveRoots n K).Nonempty", "hn✝ : 0 < n", "a : K", "H : Irreducible (X ^ n - C a)", "L : Type u_1", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : IsSplittingField K L (X ^ n - C a)", "α : L", "hα : α ^ n = (algebraMap K L) a", "ζ : K", "hζ : IsPrimitiveRoot ζ n", "m : ℤ", "hn : 0 < n"], "goal": "((autEquivZmod H L hζ).symm (Multiplicative.ofAdd ↑m)) α = ζ ^ m • α"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {l₁ l l₂ : List α}, l₁ ⊆ l → l.Disjoint l₂ → l₁.Disjoint l₂"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "n : ℕ", "s : Affine.Simplex ℝ P n", "i : Fin (n + 1)"], "goal": "dist s.circumcenter (s.points i) = s.circumradius"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M", "c : Con M"], "goal": "↑1 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "f g : ι → α → β", "p : ℝ≥0∞", "hf : UnifIntegrable f p μ", "hg : UnifIntegrable g p μ", "hp : 1 ≤ p", "hf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ", "hg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ", "ε : ℝ", "hε : 0 < ε", "hε2 : 0 < ε / 2"], "goal": "∃ δ, ∃ (_ : 0 < δ), ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator ((f + g) i)) p μ ≤ ENNReal.ofReal ε"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Add α", "s t : Set α", "Fintype ↑s", "Fintype ↑t", "Fintype ↑(s + t)"], "goal": "(s + t).toFinset = s.toFinset + t.toFinset"}}
+{"state": {"context": ["u v m n : ℤ", "h : n % 2 = 0"], "goal": "n % 2 + 2 * (n / 2) = n / 2 + n / 2"}}
+{"state": {"context": ["M : Type u_1", "G : Type u_2", "α : Type u_3", "inst✝⁵ : Monoid M", "inst✝⁴ : Group G", "inst✝³ : TopologicalSpace α", "inst✝² : MulAction M α", "inst✝¹ : MulAction G α", "inst✝ : ContinuousConstSMul M α", "H : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ", "x✝ : α"], "goal": "¬closure (orbit M x✝) = ∅"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "β : Type u_7", "inst✝ : AddCommMonoid β", "T✝ T' T : Set α → β", "T_empty : T ∅ = 0", "h_add : FinMeasAdditive μ T", "ι : Type u_8", "S : ι → Set α", "sι : Finset ι", "hS_meas : ∀ (i : ι), MeasurableSet (S i)"], "goal": "(∀ i ∈ sι, μ (S i) ≠ ⊤) → (∀ i ∈ sι, ∀ j ∈ sι, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i)"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "FunLike F G H", "AddGroupWithOne G", "AddGroupWithOne H", "AddConstMapClass F G H 1 1", "f : F", "x : G", "n : ℤ"], "goal": "f (x - ↑n) = f x - ↑n"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "NonUnitalRing A", "StarRing A", "TopologicalSpace A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "self : UniqueNonUnitalContinuousFunctionalCalculus R A", "a : A"], "goal": "CompactSpace ↑(σₙ R a)"}}
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+{"state": {"context": ["x : ℝ"], "goal": "Real.cos (-x) = Real.cos x"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "R : Type u_5", "S : Type u_6", "inst✝⁷ : Semiring R", "inst✝⁶ : Semiring S", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid N", "inst✝² : Module R N", "inst✝¹ : AddCommMonoid P", "inst✝ : Module R P", "s t : Set α", "h : Disjoint s t", "a₁ : α", "h₁ : a₁ ∈ s"], "goal": "∀ ⦃x : M⦄, x ∈ ⊤ → ∀ i ∈ t, (lapply i ∘ₗ lsingle a₁) x = 0"}}
+{"state": {"context": ["R : Type u", "EuclideanDomain R", "a : R", "b : R"], "goal": "b ≠ 0 → EuclideanDomain.r (a % b) b"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : OrderedAddCommMonoid M", "f : ℕ → M", "u : ℕ → ℕ", "hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "hu : Monotone u", "n : ℕ", "ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)", "this : ∀ k ∈ Ico (u n) (u (n + 1)), f k ≤ f (u n)"], "goal": "∑ k ∈ Ico (u n) (u (n + 1)), f k ≤ (u (n + 1) - u n) • f (u n)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹² : NontriviallyNormedField 𝕜", "inst✝¹¹ : CompleteSpace 𝕜", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : MeasurableSpace E", "inst✝⁸ : BorelSpace E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : MeasurableSpace F", "inst✝⁴ : BorelSpace F", "inst✝³ : NormedSpace 𝕜 F", "L : E →ₗ[𝕜] F", "μ : Measure E", "ν : Measure F", "inst✝² : μ.IsAddHaarMeasure", "inst✝¹ : ν.IsAddHaarMeasure", "inst✝ : LocallyCompactSpace E", "h : Function.Surjective ⇑L", "s : Set F", "hs : MeasurableSet s", "this✝ : FiniteDimensional 𝕜 E", "this : AEMeasurable (⇑L) μ", "c : ℝ≥0∞", "c_pos : 0 < c", "hc : map (⇑L) μ = c • ν"], "goal": "(∀ᵐ (y : F) ∂map (⇑L) μ, s y) ↔ ∀ᵐ (y : F) ∂ν, y ∈ s"}}
+{"state": {"context": ["T : Type u₁", "inst✝ : Category.{v₁, u₁} T", "X✝ X Y Z : T", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "autoParam (∀ (X_1 Y_1 : Over X) (f_1 : X_1 ⟶ Y_1), (map (f ≫ g)).map f_1 = eqToHom ⋯ ≫ (map f ⋙ map g).map f_1 ≫ eqToHom ⋯) _auto✝"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s✝ s₁ s₂ t : Set X", "p✝ p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X", "p : ι → Prop", "s : ι → Set X", "h : (𝓝 x).HasBasis p s"], "goal": "(∀ (i : ι), p i → (s i ∩ t).Nonempty) ↔ ∀ (i : ι), p i → ∃ y ∈ t, y ∈ s i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "CommMonoid α", "TopologicalSpace α", "f : β → α", "a : α", "g : γ → α", "i : ↑(Function.mulSupport g) → β", "hi : Function.Injective i", "hf : Function.mulSupport f ⊆ Set.range i", "hfg : ∀ (x : ↑(Function.mulSupport g)), f (i x) = g ↑x"], "goal": "HasProd f a ↔ HasProd g a"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "m : M"], "goal": "m ∈ LieModule.maxTrivSubmodule R L M ↔ ∀ (x : L), ⁅x, m⁆ = 0"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : PartialOrder α", "inst✝² : PartialOrder β", "inst✝¹ : OrderBot α", "inst✝ : OrderBot β", "f : β ↪o α", "hbot : f ⊥ = ⊥", "b : β", "hb : ⊥ ⋖ f b"], "goal": "⊥ ⋖ b"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "LocallyFiniteOrder α", "a b : αᵒᵈ"], "goal": "Finset.Ioc (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ico b a)"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "s✝ : Set α", "ν : Measure α", "hμν : μ ≪ ν", "hm : m ≤ m0", "s : Set α", "hs : MeasurableSet s", "hsν : (ν.trim hm) s = 0"], "goal": "(μ.trim hm) s = 0"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "inst✝¹ : HasShift C ℤ", "inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "n : ℤ", "X✝ Y✝ : Triangle C", "f : X✝ ⟶ Y✝"], "goal": "n.negOnePow • (CategoryTheory.shiftFunctor C n).map X✝.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app X✝.obj₁ ≫ (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map f.hom₁) = n.negOnePow • (CategoryTheory.shiftFunctor C n).map X✝.mor₃ ≫ (CategoryTheory.shiftFunctor C n).map ((CategoryTheory.shiftFunctor C 1).map f.hom₁) ≫ (shiftFunctorComm C 1 n).hom.app Y✝.obj₁"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "η₁ η₂ : R", "hη₁ : η₁ ∈ nthRootsFinset n R", "hη₂ : η₂ ∈ nthRootsFinset n R"], "goal": "η₁ * η₂ ∈ nthRootsFinset n R"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "N : Type u_4", "CommRing R", "AddCommGroup M", "Module R M", "AddCommGroup N", "Module R N", "self : RootPairing ι R M N", "i j : ι"], "goal": "self.root j - (self.toLin (self.root j)) (self.coroot i) • self.root i = self.root ((self.reflection_perm i) j)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹¹ : TopologicalSpace M", "inst✝¹⁰ : ChartedSpace H M", "E' : Type u_5", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁶ : TopologicalSpace M'", "inst✝⁵ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝¹ : TopologicalSpace M''", "inst✝ : ChartedSpace H'' M''", "f f₀ f₁ : M → M'", "x : M", "s t : Set M", "g : M' → M''", "u : Set M'", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "I''s : SmoothManifoldWithCorners I'' M''", "f' f₀' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)", "g' : TangentSpace I' (f x) →L[𝕜] TangentSpace I'' (g (f x))", "h : MDifferentiableWithinAt I I' f s x"], "goal": "mfderivWithin I I' f s x = fderivWithin 𝕜 (writtenInExtChartAt I I' x f) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I) (↑(extChartAt I x) x)"}}
+{"state": {"context": ["f : StieltjesFunction", "l : ℝ", "hf : Filter.Tendsto (↑f) Filter.atTop (𝓝 l)", "x : ℝ"], "goal": "f.measure (Set.Ici x) = ENNReal.ofReal (l - Function.leftLim (↑f) x)"}}
+{"state": {"context": ["r✝ : ℝ≥0", "r : ℝ := ↑r✝"], "goal": "HasSum (fun n => rexp (-↑r✝) * ↑r✝ ^ n / ↑n !) 1"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : AddMonoid R", "x : HahnSeries Γ Rᵃᵒᵖ"], "goal": "(if h : ∀ (a : Γ), AddOpposite.unop (x.coeff a) = 0 a then 0 else AddOpposite.unop (x.coeff (⋯.min ⋯))) = AddOpposite.unop (if h : ∀ (a : Γ), x.coeff a = coeff 0 a then 0 else x.coeff (⋯.min ⋯))"}}
+{"state": {"context": ["G : Type u_1", "inst✝⁹ : TopologicalSpace G", "inst✝⁸ : Group G", "inst✝⁷ : TopologicalGroup G", "inst✝⁶ : MeasurableSpace G", "inst✝⁵ : BorelSpace G", "inst✝⁴ : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝³ : μ.IsHaarMeasure", "inst✝² : IsFiniteMeasureOnCompacts μ'", "inst✝¹ : μ'.IsMulLeftInvariant", "inst✝ : μ.InnerRegularCompactLTTop", "s : Set G", "hs : MeasurableSet s", "h's : IsCompact (closure s)", "ν : Measure G := μ'.haarScalarFactor μ • μ", "f : G → ℝ", "f_cont : Continuous f", "hf : EqOn (⇑{ toFun := f, continuous_toFun := f_cont }) 1 (closure s)", "f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }"], "goal": "μ' s ≤ ν s"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "𝒢 : Filtration ℕ m0", "f : ℕ → Ω → ℝ", "τ π : Ω → ℕ", "inst✝ : IsFiniteMeasure μ", "h : Submartingale f 𝒢 μ", "hτ : IsStoppingTime 𝒢 τ"], "goal": "∀ (i : ℕ), Integrable (stoppedProcess f τ i) μ"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "inst✝⁶ : CommRing A", "S : Submonoid A", "hS : S ≤ A⁰", "inst✝⁵ : Field K", "inst✝⁴ : Algebra A K", "inst✝³ : IsFractionRing A K", "B : Type u_3", "inst✝² : CommRing B", "inst✝¹ : Algebra A B", "inst✝ : IsLocalization S B", "x : K", "x✝² x✝¹ : A", "x✝ : x✝¹ ∈ S"], "goal": "(algebraMap A K) x✝² * ((algebraMap A K) x✝¹)⁻¹ = IsLocalization.mk' K x✝² ⟨x✝¹, ⋯⟩"}}
+{"state": {"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1"], "goal": "(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H K : AddSubgroup G", "hK : H ≤ K", "hN : (H.addSubgroupOf K).Normal", "a b : G", "hb : b ∈ K", "h : a + b ∈ H"], "goal": "b + a ∈ H"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l₁ l₂ : List α", "r : α → α → Prop", "a b : α", "h : a ∉ l", "h' : l.Nodup"], "goal": "(l.concat a).Nodup"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α", "m : ℤ"], "goal": "toIocDiv hp a (m • p + b) = m + toIocDiv hp a b"}}
+{"state": {"context": ["a : Int"], "goal": "a.sign = 0 → a = 0"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x✝ y✝ x y : V", "hx : x ≠ 0", "hy : y ≠ 0", "h : ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖", "h₁ : ‖x‖ * ‖y‖ ≠ 0"], "goal": "angle x y = 0"}}
+{"state": {"context": ["R : Type u1", "inst✝⁴ : CommRing R", "M : Type u2", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "A : Type u_1", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "n : ℕ", "f : Fin n → ↑↑(LinearMap.range (ι R))", "hu : (List.ofFn fun i => ↑(f i)).prod ∈ ↑(LinearMap.range (ι R)) ^ n", "i : Fin n", "v : M", "hv : (ι R) v = ↑(f i)"], "goal": "(ι R) (ιInv ↑(f i)) = ↑(f i)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "a b : G", "h : Commute a b", "m : ℤ"], "goal": "Commute (a ^ m) b"}}
+{"state": {"context": ["n : ℕ"], "goal": "n.boddDiv2 = (n.bodd, n.div2)"}}
+{"state": {"context": ["M : Type u_2", "AddMonoid M", "a : M", "m n : ℕ"], "goal": "(m * n) • a = m • n • a"}}
+{"state": {"context": ["u v : ZFSet"], "goal": "(insert u v).Nonempty"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "a✝ : α", "n : ℕ", "a : α"], "goal": "⌊a⌋₊ ≤ ⌈a⌉₊"}}
+{"state": {"context": ["m k : ℕ+", "h₀ : ↑m % ↑k + ↑k * (↑m / ↑k) = ↑m := Nat.mod_add_div ↑m ↑k", "this✝ : ¬(↑m % ↑k = 0 ∧ ↑m / ↑k = 0)", "this : ↑(k.modDivAux (↑m % ↑k) (↑m / ↑k)).1 + ↑k * (k.modDivAux (↑m % ↑k) (↑m / ↑k)).2 = ↑m % ↑k + ↑k * (↑m / ↑k)"], "goal": "↑(m.mod k) + ↑k * m.div k = ↑m"}}
+{"state": {"context": ["n : ℤ", "d : ℤ", "d_ne_zero : d ≠ 0"], "goal": "∃ c, n = c * (↑n / ↑d).num ∧ d = c * ↑(↑n / ↑d).den"}}
+{"state": {"context": ["α : Type u", "β : Type v", "a : α", "f : Filter α", "x✝ : TopologicalSpace α := nhdsAdjoint a f"], "goal": "pure a ⊔ f ≤ 𝓝 a"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "s : AffineSubspace ℝ P", "inst✝ : HasOrthogonalProjection s.direction", "ps : Set P", "hnps : ps.Nonempty", "p : P", "hps : ps ⊆ ↑s", "hp : p ∉ s", "this : Nonempty ↥s", "cc : P", "cr : ℝ", "hcccru : ∀ (y : Sphere P), y.center ∈ s ∧ ps ⊆ Metric.sphere y.center y.radius → y = { center := cc, radius := cr }", "hcc : cc ∈ s", "hcr : ps ⊆ Metric.sphere cc cr", "x : ℝ := dist cc ↑((orthogonalProjection s) p)", "y : ℝ := dist p ↑((orthogonalProjection s) p)", "hy0 : y ≠ 0"], "goal": "∃! cs₂, cs₂.center ∈ affineSpan ℝ (insert p ↑s) ∧ insert p ps ⊆ Metric.sphere cs₂.center cs₂.radius"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "Ring k", "AddCommGroup V1", "Module k V1", "AffineSpace V1 P1", "NoZeroSMulDivisors k V1", "p₀ p₁ : P1", "c₁ c₂ : k"], "goal": "(AffineMap.lineMap p₀ p₁) c₁ = (AffineMap.lineMap p₀ p₁) c₂ ↔ p₀ = p₁ ∨ c₁ = c₂"}}
+{"state": {"context": ["α : Type u_2", "Add α", "DecidableEq α", "s t u : Finset α"], "goal": "s + t ⊆ u ↔ ∀ b ∈ t, AddOpposite.op b +ᵥ s ⊆ u"}}
+{"state": {"context": ["R : Type u", "M : Type v", "M' : Type v'", "Ring R", "AddCommGroup M", "AddCommGroup M'", "Module R M", "Module R M'", "f : M →ₗ[R] M'"], "goal": "Cardinal.lift.{v, v'} (Module.rank R ↥(LinearMap.range f)) ≤ Cardinal.lift.{v', v} (Module.rank R M)"}}
+{"state": {"context": ["E : Type u_1", "AddCommGroup E", "Module ℝ E", "TopologicalSpace E", "ContinuousSMul ℝ E", "F : Type u_2", "AddCommGroup F", "Module ℝ F", "TopologicalSpace F", "ContinuousSMul ℝ F", "T2Space F", "f : E →+ F", "hf : Continuous ⇑f"], "goal": "⇑(f.toRealLinearMap hf) = ⇑f"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "UniqueFactorizationMonoid α", "DecidableEq (Associates α)", "(p : Associates α) → Decidable (Irreducible p)", "m p : Associates α", "h₁ : m ≠ 0", "h₂ : Irreducible p", "k : ℕ"], "goal": "p ^ k ≤ m ↔ k ≤ p.count m.factors"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G", "l : List ↥H"], "goal": "↑l.sum = (List.map Subtype.val l).sum"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "t : W", "ht : cs.IsReflection t"], "goal": "Odd (cs.length t)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B I J X Y : Set α", "e : α", "hIX : M.Basis I X", "hIY : M.Basis I Y"], "goal": "M.Indep I"}}
+{"state": {"context": ["α : Type u_1", "A : Type u_5", "AddMonoid A", "Monoid α", "DistribMulAction α A", "a : α", "S T : AddSubmonoid A"], "goal": "a • (S ⊔ T) = a • S ⊔ a • T"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "inst✝⁵ : HasColimits C", "X Y Z : TopCat", "inst✝⁴ : ConcreteCategory C", "inst✝³ : PreservesFilteredColimits (forget C)", "inst✝² : HasLimits C", "inst✝¹ : PreservesLimits (forget C)", "inst✝ : (forget C).ReflectsIsomorphisms", "F : Sheaf C X", "U : Opens ↑X", "s t : (forget C).obj (F.val.obj (op U))", "h : ∀ (x : ↥U), (F.presheaf.germ x) s = (F.presheaf.germ x) t", "V : ↥U → Opens ↑X", "m : ∀ (x : ↥U), ↑x ∈ V x", "i₁ i₂ : (x : ↥U) → V x ⟶ U", "heq : ∀ (x : ↥U), (F.presheaf.map (i₁ x).op) s = (F.presheaf.map (i₂ x).op) t"], "goal": "U ≤ iSup V"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι : Type u_12", "inst✝¹⁸ : SeminormedRing 𝕜", "inst✝¹⁷ : SeminormedRing 𝕜₂", "inst✝¹⁶ : SeminormedRing 𝕜₃", "σ₁₂ : 𝕜 →+* 𝕜₂", "inst✝¹⁵ : RingHomIsometric σ₁₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "inst✝¹⁴ : RingHomIsometric σ₂₃", "σ₁₃ : 𝕜 →+* 𝕜₃", "inst✝¹³ : RingHomIsometric σ₁₃", "inst✝¹² : AddCommGroup E", "inst✝¹¹ : AddCommGroup E₂", "inst✝¹⁰ : AddCommGroup E₃", "inst✝⁹ : AddCommGroup F", "inst✝⁸ : AddCommGroup G", "inst✝⁷ : Module 𝕜 E", "inst✝⁶ : Module 𝕜₂ E₂", "inst✝⁵ : Module 𝕜₃ E₃", "inst✝⁴ : Module 𝕜 F", "inst✝³ : Module 𝕜 G", "inst✝² : SMul R ℝ", "inst✝¹ : SMul R ℝ≥0", "inst✝ : IsScalarTower R ℝ≥0 ℝ", "p q : Seminorm 𝕜 E", "a b : ℝ≥0", "hpq : p ≤ q", "hab : a ≤ b"], "goal": "∀ (x : E), (a • p) x ≤ (b • q) x"}}
+{"state": {"context": ["α : Type u_1", "Finite α", "DecidableEq α", "f : Equiv.Perm α", "hf : f.IsCycle"], "goal": "∃! s, (↑s).formPerm ⋯ = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m m0 : MeasurableSpace α", "E : Type u_5", "inst✝³ : NormedAddCommGroup E", "inst✝² : MeasurableSpace E", "inst✝¹ : OpensMeasurableSpace E", "μ : Measure α", "hm : m ≤ m0", "inst✝ : SigmaFinite (μ.trim hm)", "C : ℝ≥0∞", "f : α → ℝ≥0∞", "hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C", "this : ∫⁻ (x : α) in univ, f x ∂μ = ∫⁻ (x : α), f x ∂μ", "S : ℕ → Set α", "x✝ : ∀ (n : ℕ), MeasurableSet (S n)", "hS_mono : Monotone S"], "goal": "Directed (fun x x_1 => x ⊆ x_1) S"}}
+{"state": {"context": ["J : Type w", "f : J → AddCommGrp", "s : CategoryTheory.Limits.Fan f"], "goal": "(AddCommGrp.HasLimit.productLimitCone f).isLimit.lift s = AddCommGrp.HasLimit.lift f s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "D : Type u_2", "CategoryTheory.Category.{u_6, u_2} D", "K : CategoryTheory.GrothendieckTopology D", "A : Type u_4", "CategoryTheory.Category.{u_7, u_4} A", "G : CategoryTheory.Functor C D", "G.IsCoverDense K", "G.IsLocallyFull K", "ℱ ℱ' : CategoryTheory.Sheaf K A", "i : G.op.comp ℱ.val ≅ G.op.comp ℱ'.val"], "goal": "(CategoryTheory.Functor.IsCoverDense.sheafIso i).hom.val = (CategoryTheory.Functor.IsCoverDense.presheafIso i).hom"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasFiniteCoproducts C", "K : ChainComplex C ℕ"], "goal": "AlgebraicTopology.DoldKan.N₂Γ₂.hom.app ((CategoryTheory.Idempotents.toKaroubi (ChainComplex C ℕ)).obj K) =\n  AlgebraicTopology.DoldKan.N₂Γ₂ToKaroubiIso.hom.app K ≫ AlgebraicTopology.DoldKan.N₁Γ₀.hom.app K"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α β : TypeVec.{u} n", "g : α.Arrow β", "x y : (MvQPF.P F).W α"], "goal": "MvQPF.WEquiv x y → MvQPF.WEquiv (MvFunctor.map g x) (MvFunctor.map g y)"}}
+{"state": {"context": ["x y z✝ : ℝ", "n : ℕ", "hx : 0 ≤ x", "hy : 0 ≤ y", "z : ℝ"], "goal": "(x / y) ^ z = x ^ z / y ^ z"}}
+{"state": {"context": ["b : ℕ", "hb : 1 < b", "x : ℕ"], "goal": "Nat.log b (b ^ x) = x"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Function.Surjective Language.reverse"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁶ : CommSemiring R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid N", "inst✝² : Module R N", "inst✝¹ : AddCommMonoid P", "inst✝ : Module R P", "f : N →ₗ[R] P", "Q : QuadraticMap R M N", "b : R", "x : M"], "goal": "(fun x => f (Q x)) (b • x) = (b * b) • (fun x => f (Q x)) x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F G : CategoryTheory.Functor (CategoryTheory.Discrete PEmpty.{w + 1}) C"], "goal": "F = G"}}
+{"state": {"context": ["α : Type u", "s : Set α", "hs : s.Infinite"], "goal": "s.Nontrivial"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x y : α", "s t : Set α", "Φ : α → β", "U : Set α", "hU : IsOpen U", "a : ℝ≥0∞", "a_pos : 0 < a", "a_lt_one : a < 1", "F : ℕ → Set α := fun n => (fun x => infEdist x Uᶜ) ⁻¹' Ici (a ^ n)", "F_subset : ∀ (n : ℕ), F n ⊆ U"], "goal": "∃ F, (∀ (n : ℕ), IsClosed (F n)) ∧ (∀ (n : ℕ), F n ⊆ U) ∧ ⋃ n, F n = U ∧ Monotone F"}}
+{"state": {"context": ["a b c : ℕ", "a1 : 1 < a", "b1 : 1 < b", "h : a ≡ b [MOD c]", "n : ℕ"], "goal": "yn a1 (n + 2) + yn ⋯ n ≡ yn b1 (n + 2) + yn ⋯ n [MOD c]"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "M : Type u_5", "AddMonoid M", "TopologicalSpace M", "ContinuousAdd M", "l : List (α → M)", "hl : ∀ f ∈ l, MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.AEStronglyMeasurable l.sum μ"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "IsTrans α r"], "goal": "IsTrans α (Function.swap r)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s✝ t✝ u v : Finset α", "a b : α", "s t : Finset α", "h : s ⊂ t"], "goal": "∃ a ∈ t, s ⊆ t.erase a"}}
+{"state": {"context": [], "goal": "∀ {α : Sort u_1} {a' : α}, ∃ a, a = a'"}}
+{"state": {"context": ["o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{max u v} → Ordinal.{max u v}", "H : ∀ (i : Ordinal.{u}) (hi : i < o), IsNormal (f i hi)", "a : Ordinal.{max u v}", "h : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi a ≤ a", "i : Ordinal.{u}", "hi : i < o"], "goal": "f i hi a ≤ a"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "CommRing R", "Ms : ι → Type u_3", "(i : ι) → AddCommGroup (Ms i)", "(i : ι) → Module R (Ms i)", "Fintype ι", "DecidableEq ι", "p : (i : ι) → Submodule R (Ms i)"], "goal": "∀ (a : (i : ι) → Ms i ⧸ p i),\n  (Submodule.quotientPi p).symm a = (Submodule.piQuotientLift p (Submodule.pi Set.univ p) LinearMap.single ⋯) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : PartialOrder α", "a : α", "inst✝ : IsStronglyAtomic α", "ha : (Set.Ici a).Infinite", "hfin : {x | a ⋖ x}.Finite"], "goal": "∃ b, a ⋖ b ∧ (Set.Ici b).Infinite"}}
+{"state": {"context": ["p n : ℕ", "hp : 1 < p", "hn : 0 < n", "this : 0 < p"], "goal": "go 0 p (p * n) = go 0 p n + 1"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : ι → Type u_4", "CommSemiring R", "(i : ι) → Semiring (A i)", "(i : ι) → Algebra R (A i)", "κ : Type u_5", "s : Finset κ", "x : κ → (i : ι) → A i", "hx : (↑s).Pairwise fun a b => Commute (x a) (x b)"], "goal": "(PiTensorProduct.tprod R) (s.noncommProd x hx) = s.noncommProd (fun k => (PiTensorProduct.tprod R) (x k)) ⋯"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalNonAssocSemiring R"], "goal": "⇑AddMonoidHom.mul = AddMonoidHom.mulLeft"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "n : ℕ", "n.AtLeastTwo"], "goal": "(OfNat.ofNat n).comp p = ↑n"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Groupoid C", "S : Subgroupoid C", "D : Type u_1", "inst✝ : Groupoid D", "φ : C ⥤ D", "hφ : Function.Injective φ.obj", "hφ' : im φ hφ = ⊤", "Sn : S.IsNormal", "d' : D", "c : C", "g : φ.obj c ⟶ d'", "γ : c ⟶ c", "γS : γ ∈ S.arrows c c", "cd' : φ.obj c = φ.obj c", "this : d' ∈ (im φ hφ).objs"], "goal": "Groupoid.inv g ≫ (eqToHom ⋯ ≫ φ.map γ ≫ eqToHom cd') ≫ g ∈ (map φ hφ S).arrows d' d'"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "ι : Type u_2", "DecidableEq ι", "Fintype ι", "N : ι → Type u_3", "(i : ι) → Monoid (N i)", "ϕ : (i : ι) → N i →* M", "hcomm : Pairwise fun i j => ∀ (x : N i) (y : N j), Commute ((ϕ i) x) ((ϕ j) y)", "i : ι", "y : N i"], "goal": "(MonoidHom.noncommPiCoprod ϕ hcomm) (Pi.mulSingle i y) = (ϕ i) y"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X Y Z : Cᵒᵖ", "f : X ⟶ Y", "g : X ⟶ Z", "c : CategoryTheory.Limits.PushoutCocone f g"], "goal": "c.unop.pt = Opposite.unop c.pt"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "R₃ : Type u_3", "inst✝⁷ : Semiring R₁", "inst✝⁶ : Semiring R₂", "inst✝⁵ : Semiring R₃", "σ₁₂ : R₁ →+* R₂", "σ₂₃ : R₂ →+* R₃", "M₁ : Type u_4", "M₂ : Type u_5", "M₃ : Type u_6", "inst✝⁴ : TopologicalSpace M₁", "inst✝³ : TopologicalSpace M₂", "inst✝² : TopologicalSpace M₃", "inst✝¹ : AddCommMonoid M₁", "inst✝ : Module R₁ M₁", "f : M₁ → M₂", "g : M₂ → M₃", "hg : Continuous g", "K : Set M₂", "hK : IsCompact K", "hKf : f ⁻¹' K ∈ 𝓝 0"], "goal": "f ⁻¹' K ⊆ f ⁻¹' (g ⁻¹' (g '' K))"}}
+{"state": {"context": ["K : Type u_3", "Field K", "R₁ : Type u_4", "CommRing R₁", "IsDomain R₁", "Algebra R₁ K", "IsFractionRing R₁ K", "x : K"], "goal": "(FractionalIdeal.spanSingleton R₁⁰ x)⁻¹ = FractionalIdeal.spanSingleton R₁⁰ x⁻¹"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "β : Type u_7", "Group β", "f g : α → β"], "goal": "f / g =ᵐ[μ] 1 ↔ f =ᵐ[μ] g"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "p : X → Prop", "s : Set { x // p x }"], "goal": "IsCompact s ↔ IsCompact (Subtype.val '' s)"}}
+{"state": {"context": ["α : Type u_1", "RatCast α", "q : ℚ"], "goal": "AddOpposite.op ↑q = ↑q"}}
+{"state": {"context": ["E : Type u_3", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "f : ℝ → E", "c : E", "a b : ℝ", "hf : IntervalIntegrable f MeasureTheory.volume a b", "hmeas : StronglyMeasurableAtFilter f (𝓝 a) MeasureTheory.volume", "hb : Filter.Tendsto f (𝓝 a ⊓ MeasureTheory.ae MeasureTheory.volume) (𝓝 c)"], "goal": "deriv (fun u => ∫ (x : ℝ) in u..b, f x) a = -c"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝² : EMetricSpace X", "inst✝¹ : EMetricSpace Y", "C✝ K r✝ : ℝ≥0", "f✝ : X → Y", "s✝ t✝ : Set X", "inst✝ : SecondCountableTopology X", "r : ℝ≥0", "f : X → Y", "hr : 0 < r", "s : Set X", "C : X → ℝ≥0", "t : X → Set X", "htn : ∀ x ∈ s, t x ∈ 𝓝[s] x", "hC : ∀ x ∈ s, HolderOnWith (C x) r f (t x)"], "goal": "dimH (f '' s) ≤ dimH s / ↑r"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "β✝ : Type u_4", "β' : Type u_5", "γ : Type u_6", "γ' : Type u_7", "_mα : MeasurableSpace α", "_mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "f✝ : Ω → β✝", "g : Ω → β'", "β : ι → Type u_8", "m : (i : ι) → MeasurableSpace (β i)", "f : (i : ι) → Ω → β i", "inst✝¹ : IsMarkovKernel κ", "inst✝ : Subsingleton ι", "s : Finset ι", "f' : (i : ι) → Set (β i)", "hf' : ∀ i ∈ s, MeasurableSet (f' i)"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ s, f i ⁻¹' f' i) = ∏ i ∈ s, (κ a) (f i ⁻¹' f' i)"}}
+{"state": {"context": ["M : Type u_11", "N : Type u_12", "AddMonoid N", "VAdd M N", "h : ∀ (x : M) (y : N), x +ᵥ 0 + y = x +ᵥ y"], "goal": "VAddAssocClass M N N"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M"], "goal": "Subsingleton (AddSubmonoid M) ↔ Subsingleton M"}}
+{"state": {"context": ["α : Type u_1", "s t : Set α", "h : s ⊆ t", "x y : ↑s"], "goal": "Set.inclusion h x = Set.inclusion h y ↔ x = y"}}
+{"state": {"context": ["a b : Cardinal.{u_1}", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "a * b < Cardinal.aleph0 ↔ a < Cardinal.aleph0 ∧ b < Cardinal.aleph0"}}
+{"state": {"context": ["n : Nat", "x : BitVec n", "y : Fin (2 ^ n)"], "goal": "x < { toFin := y } ↔ x.toFin < y"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "CommRing R", "AddCommGroup M", "AddCommGroup N", "AddCommGroup P", "Module R M", "Module R N", "Module R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "Q : Type u_5", "AddCommGroup Q", "Module R Q", "hfg : Function.Exact ⇑f ⇑g", "h : P → N", "hgh : Function.RightInverse h ⇑g"], "goal": "lTensor.inverse_of_rightInverse Q hfg hgh ∘ₗ LinearMap.lTensor Q g = (LinearMap.range (LinearMap.lTensor Q f)).mkQ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : ℝ → E", "x : ℝ", "n : ℕ", "s : Set ℝ", "hs : UniqueDiffOn ℝ s", "hf : ContDiffOn ℝ (↑n) f s"], "goal": "ContinuousOn (fun t => ∑ k ∈ Finset.range (n + 1), ((↑k !)⁻¹ * (x - t) ^ k) • iteratedDerivWithin k f s t) s"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_5", "Zero R", "Zero M", "SMulWithZero R M", "NoZeroSMulDivisors R M", "c : R", "x : M", "hc : c ≠ 0"], "goal": "c • x = 0 ↔ x = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι✝ → Sort u_7", "κ₁ : ι✝ → Sort u_8", "κ₂ : ι✝ → Sort u_9", "κ' : ι' → Sort u_10", "ι : Type u_11", "t : Set ι", "s : ι → Set β", "w : ⋃ i ∈ t, s i = ⊤", "x : β", "p : ∃ i, x ∈ ⋃ (_ : i ∈ t), s i"], "goal": "∃ i ∈ t, x ∈ s i"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t✝ : Set X", "t : Set Y", "hs : ∀ (f : Ultrafilter X), s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x", "ht : ∀ (f : Ultrafilter Y), t ∈ f → ∃ x ∈ t, ↑f ≤ 𝓝 x", "f : Ultrafilter (X × Y)", "hfs : s ×ˢ t ∈ f"], "goal": "∃ x ∈ s ×ˢ t, ↑f ≤ 𝓝 x"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "f : PadicSeq p", "hf : ¬f ≈ 0", "v1 v3 : ℕ"], "goal": "stationaryPoint hf ≤ stationaryPoint hf"}}
+{"state": {"context": ["α : Type u_1", "LT α", "x : WithBot α"], "goal": "⊥ < x ↔ x ≠ ⊥"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "K : HomologicalComplex₂ C (ComplexShape.up ℤ) (ComplexShape.up ℤ)", "x y : ℤ", "K.HasTotal (ComplexShape.up ℤ)"], "goal": "((HomologicalComplex₂.shiftFunctor₂ C y).obj K).totalShift₁Iso x ≪≫\n    (CategoryTheory.shiftFunctor (CochainComplex C ℤ) x).mapIso (K.totalShift₂Iso y) =\n  (x * y).negOnePow •\n    HomologicalComplex₂.total.mapIso ((HomologicalComplex₂.shiftFunctor₁₂CommIso C x y).app K) (ComplexShape.up ℤ) ≪≫\n      ((HomologicalComplex₂.shiftFunctor₁ C x).obj K).totalShift₂Iso y ≪≫\n        (CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) y).mapIso (K.totalShift₁Iso x) ≪≫\n          (CategoryTheory.shiftFunctorComm (CochainComplex C ℤ) x y).app (K.total (ComplexShape.up ℤ))"}}
+{"state": {"context": ["m n : ℕ"], "goal": "(∀ (a : ℕ), m ∣ a ↔ n ∣ a) ↔ m = n"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "m : M"], "goal": "-((v Q) m * e0 Q) = e0 Q * (v Q) m"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "x : E", "s : Set E", "n : ℕ∞", "hf : ContDiffWithinAt ℝ n f s x"], "goal": "ContDiffWithinAt ℝ n (fun x => Real.sin (f x)) s x"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "W X Y Z X₁ X₂ X₃ Z₁ Z₂ : C", "g₁ : Z₁ ⟶ X₁", "g₂ : Z₁ ⟶ X₂", "g₃ : Z₂ ⟶ X₂", "g₄ : Z₂ ⟶ X₃", "inst✝³ : HasPushout g₁ g₂", "inst✝² : HasPushout g₃ g₄", "inst✝¹ : HasPushout (g₃ ≫ pushout.inr g₁ g₂) g₄", "inst✝ : HasPushout g₁ (g₂ ≫ pushout.inl g₃ g₄)"], "goal": "pushout.inr g₃ g₄ ≫ pushout.inr g₁ (g₂ ≫ pushout.inl g₃ g₄) ≫ (pushoutAssoc g₁ g₂ g₃ g₄).inv = pushout.inr (g₃ ≫ pushout.inr g₁ g₂) g₄"}}
+{"state": {"context": ["α β : Type u", "f g : FreeAddMagma (α → β)", "x : FreeAddMagma α"], "goal": "(Seq.seq (f + g) fun x_1 => x) = (Seq.seq f fun x_1 => x) + Seq.seq g fun x_1 => x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "p q : R[X]", "CommSemiring S", "f : R →+* S", "k : ℕ", "t : S"], "goal": "Polynomial.eval₂ f t (p.comp^[k] q) = (fun x => Polynomial.eval₂ f x p)^[k] (Polynomial.eval₂ f t q)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "f : α → ℂ", "s : Set α", "x : α", "h : ContinuousWithinAt f s x"], "goal": "ContinuousWithinAt (fun y => Complex.exp (f y)) s x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "β₂ : Type u_3", "γ : Type u_4", "ι : Sort u_5", "ι' : Sort u_6", "κ : ι → Sort u_7", "κ' : ι' → Sort u_8", "inst✝¹ : CompleteLattice α", "f✝ g s✝ t : ι → α", "a b : α", "inst✝ : CompleteLattice β", "f : α ≃o β", "s : Set α"], "goal": "f (sInf s) = sInf (⇑f.symm ⁻¹' s)"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : Cᵒᵖ", "s : Cocone (coyoneda.obj X)", "Y : C"], "goal": "(colimitCocone X).ι.app Y ≫ (fun s x => s.ι.app (unop X) (𝟙 (unop X))) s = s.ι.app Y"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Countable ι", "a : ι → ℂ", "p : ι → ℝ", "F : ℝ → ℂ", "s : ℂ", "hp : ∀ (i : ι), 0 ≤ p i", "hs : 0 < s.re", "hF : ∀ t ∈ Ioi 0, HasSum (fun i => if p i = 0 then 0 else a i * ↑(rexp (-π * p i * t))) (F t)", "h_sum : Summable fun i => ‖a i‖ / p i ^ s.re", "hs' : s ≠ 0", "a' : ι → ℂ := fun i => if p i = 0 then 0 else a i", "hp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i"], "goal": "HasSum (fun i => ↑π ^ (-s) * Complex.Gamma s * a i / ↑(p i) ^ s) (mellin F s)"}}
+{"state": {"context": ["n k : ℕ"], "goal": "(ascPochhammer ℕ (n + k + 1)).smeval (-↑n) = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : RCLike 𝕜", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : SMulCommClass ℝ 𝕜 F", "f : 𝓢(E, F)", "n : ℕ", "IH : ∀ (m : Fin (n + 1) → E), (iteratedPDeriv 𝕜 m) f = (iteratedPDeriv 𝕜 (Fin.init m)) ((pderivCLM 𝕜 (m (Fin.last n))) f)", "m : Fin (n + 1 + 1) → E", "hmzero : Fin.init m 0 = m 0", "hmtail : Fin.tail m (Fin.last n) = m (Fin.last n.succ)"], "goal": "(iteratedPDeriv 𝕜 m) f = (iteratedPDeriv 𝕜 (Fin.init m)) ((pderivCLM 𝕜 (m (Fin.last (n + 1)))) f)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "AddCommMonoid β", "DecidableEq α", "a : α", "x : β", "s : Finset α"], "goal": "∑ a' ∈ s, Pi.single a x a' = if a ∈ s then x else 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasPullbacks C", "X Y S X' Y' S' : C", "f : X ⟶ S", "g : Y ⟶ S", "f' : X' ⟶ S'", "g' : Y' ⟶ S'", "i₁ : X ⟶ X'", "i₂ : Y ⟶ Y'", "i₃ : S ⟶ S'", "e₁ : f ≫ i₃ = i₁ ≫ f'", "e₂ : g ≫ i₃ = i₂ ≫ g'", "CategoryTheory.Mono i₃"], "goal": "(CategoryTheory.Limits.pullbackFstFstIso f g f' g' i₁ i₂ i₃ e₁ e₂).inv =\n  CategoryTheory.Limits.pullback.lift\n    (CategoryTheory.Limits.pullback.lift (CategoryTheory.Limits.pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂)\n      (CategoryTheory.Limits.pullback.fst f g) ⋯)\n    (CategoryTheory.Limits.pullback.lift (CategoryTheory.Limits.pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂)\n      (CategoryTheory.Limits.pullback.snd f g) ⋯)\n    ⋯"}}
+{"state": {"context": ["a b : ℤ", "b0 : b ≠ 0", "n : ℤ", "d : ℕ", "h : d ≠ 0", "c : n.natAbs.Coprime d", "e : a * ↑d = n * b"], "goal": "n.natAbs ∣ a.natAbs * d"}}
+{"state": {"context": ["n : ℕ"], "goal": "IsSquare ↑n ↔ IsSquare n"}}
+{"state": {"context": ["p : ℕ", "hp : Prime p", "this : p = p - 2 + 2"], "goal": "p.minFac :: (p / p.minFac).primeFactorsList = [p]"}}
+{"state": {"context": ["K : Type u_3", "Field K", "R₁ : Type u_6", "CommRing R₁", "Algebra R₁ K", "IsLocalization R₁⁰ K", "I : FractionalIdeal R₁⁰ K", "(↑I).IsPrincipal", "h : I ≠ 0"], "goal": "I * FractionalIdeal.spanSingleton R₁⁰ (Submodule.IsPrincipal.generator ↑I)⁻¹ = 1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "X Y : C", "CategoryTheory.Simple X", "f : X ⟶ Y", "CategoryTheory.Epi f", "w : f ≠ 0"], "goal": "CategoryTheory.IsIso f"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "X : CategoryTheory.SimplicialObject C", "Y : C", "n q : ℕ", "φ : Y ⟶ X _[n + 1]", "v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ", "hqn : n < q"], "goal": "φ ≫ (AlgebraicTopology.DoldKan.Hσ q).f (n + 1) = 0"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝²² : Category.{u_4, u_1} C", "inst✝²¹ : Category.{u_6, u_2} D", "inst✝²⁰ : Category.{u_5, u_3} E", "inst✝¹⁹ : HasShift C ℤ", "inst✝¹⁸ : HasShift D ℤ", "inst✝¹⁷ : HasShift E ℤ", "F : C ⥤ D", "inst✝¹⁶ : F.CommShift ℤ", "G : D ⥤ E", "inst✝¹⁵ : G.CommShift ℤ", "inst✝¹⁴ : HasZeroObject C", "inst✝¹³ : HasZeroObject D", "inst✝¹² : HasZeroObject E", "inst✝¹¹ : Preadditive C", "inst✝¹⁰ : Preadditive D", "inst✝⁹ : Preadditive E", "inst✝⁸ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝⁷ : ∀ (n : ℤ), (shiftFunctor D n).Additive", "inst✝⁶ : ∀ (n : ℤ), (shiftFunctor E n).Additive", "inst✝⁵ : Pretriangulated C", "inst✝⁴ : Pretriangulated D", "inst✝³ : Pretriangulated E", "inst✝² : (F ⋙ G).IsTriangulated", "inst✝¹ : F.IsTriangulated", "inst✝ : F.mapArrow.EssSurj", "T : Triangle D", "hT : T ∈ distinguishedTriangles", "T' : Triangle C", "hT' : T' ∈ distinguishedTriangles", "e : F.mapTriangle.obj T' ≅ T"], "goal": "G.mapTriangle.obj T ∈ distinguishedTriangles"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "k : ℕ", "a : R", "m n : ℕ", "h : 1 = (Int.castRingHom R) 1"], "goal": "(2 * Chebyshev.T ℚ ↑m).comp (Chebyshev.T ℚ ↑n) = (2 * Chebyshev.T ℚ ↑m).comp ((C ⅟2 * X).comp (2 * Chebyshev.T ℚ ↑n))"}}
+{"state": {"context": ["α : Type u_1"], "goal": "0 = ∅"}}
+{"state": {"context": ["D : GlueData", "U : Set ↑↑D.glued.toPresheafedSpace"], "goal": "∀ (a : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J), IsOpen (⇑(D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι a) ⁻¹' (⇑(TopCat.homeoOfIso D.isoCarrier.symm) ⁻¹' U)) ↔ IsOpen (⇑(D.ι a).val.base ⁻¹' U)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "h₁ : S.LeftHomologyData", "h₂ : S.RightHomologyData", "CategoryTheory.IsIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)"], "goal": "(CategoryTheory.ShortComplex.HomologyData.ofIsIsoLeftRightHomologyComparison' h₁ h₂).iso =\n  CategoryTheory.asIso (CategoryTheory.ShortComplex.leftRightHomologyComparison' h₁ h₂)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "K : Type w", "inst✝ : CommRing R", "f : R[X]"], "goal": "Algebra.adjoin R {root f} = ⊤"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ ν : MeasureTheory.Measure α", "MeasureTheory.SigmaFinite μ", "μ.HaveLebesgueDecomposition ν", "hμν : μ.AbsolutelyContinuous ν"], "goal": "(fun x => rexp (MeasureTheory.llr μ ν x)) =ᵐ[μ] fun x => (μ.rnDeriv ν x).toReal"}}
+{"state": {"context": ["α : Type u_1", "p : PosNum"], "goal": "↑(pos p).pred = (↑(pos p)).pred"}}
+{"state": {"context": ["a✝ b n✝ n a : ℕ"], "goal": "(n.floorRoot a).factorization = a.factorization ⌊/⌋ n"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "Group G", "A : Rep k G", "f : ↥(groupCohomology.twoCocycles A)", "g : G"], "goal": "↑f (g, 1) = (A.ρ g) (↑f (1, 1))"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDedekindDomain R", "K : Type v", "Field K", "Algebra R K", "IsFractionRing R K", "n : ℕ", "hn : Fact (0 < n)"], "goal": "IsDedekindDomain.selmerGroup.fromUnit.ker = (powMonoidHom n).range"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F✝ : Type u_3", "𝕜 : Type u_4", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : SMulCommClass ℝ 𝕜 E", "inst✝⁴ : NormedAddCommGroup F✝", "inst✝³ : NormedSpace ℝ F✝", "inst✝² : CompleteSpace F✝", "G : Type u_5", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "f✝ g : α → E", "m : MeasurableSpace α", "μ : Measure α", "ι : Type u_6", "f : α → G", "hfi : Integrable f μ", "F : ι → α → G", "l : Filter ι", "hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ", "hF : Tendsto (fun i => ∫⁻ (x : α), ↑‖F i x - f x‖₊ ∂μ) l (𝓝 0)", "s : Set α"], "goal": "∀ᶠ (i : ι) in l, Integrable (F i) (μ.restrict s)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "𝔖 : Set (Set α)", "hs : s ∈ 𝔖", "V : Set (β × β)", "hV : V ∈ 𝓤 β"], "goal": "UniformOnFun.gen 𝔖 s V ∈ ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 (UniformOnFun.gen 𝔖 s V)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : List (Option α)"], "goal": "(¬∀ (x : Option α), x ∈ l → x.isSome = true) ↔ none ∈ l"}}
+{"state": {"context": ["l : List ℕ", "k : ℕ"], "goal": "l.prod.Coprime k ↔ ∀ n ∈ l, n.Coprime k"}}
+{"state": {"context": ["α : Type u", "Inhabited α", "l₁ l₂ : List α"], "goal": "List.takeI l₁.length (l₁ ++ l₂) = l₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "h₁ : min a b ≤ max c d", "h₂ : min c d ≤ max a b"], "goal": "Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d)"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "w x y z : α", "inst✝ : GeneralizedBooleanAlgebra α"], "goal": "(x \\ y) \\ z = x \\ y ⊓ x \\ z"}}
+{"state": {"context": ["ι : Type u_1", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι", "t : TransvectionStruct ι ℝ"], "goal": "MeasurePreserving (⇑(toLin' t.toMatrix)) volume volume"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w"], "goal": "∀ {s t : WSeq α}, s ~ʷ t → s.head ~ t.head"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "P : Type u_3", "inst✝¹ : CommSemiring P", "inst✝ : IsLocalization M S", "x y : R", "x✝ : ∃ c, ↑c * x = ↑c * y", "c : ↥M", "h : (algebraMap R S) ↑c * (algebraMap R S) x = (algebraMap R S) ↑c * (algebraMap R S) y"], "goal": "(algebraMap R S) x = (algebraMap R S) y"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "S T : ConvexCone 𝕜 E"], "goal": "↑(S ⊓ T) = ↑S ∩ ↑T"}}
+{"state": {"context": ["A B : Grp", "f : A ⟶ B", "x : ↑B"], "goal": "((Grp.SurjectiveOfEpiAuxs.h f) x) (Grp.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset ⟨↑(MonoidHom.range f), ⋯⟩) =\n  Grp.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset ⟨↑(MonoidHom.range f), ⋯⟩"}}
+{"state": {"context": ["c : Cardinal.{u_1}"], "goal": "Cardinal.aleph0 ≤ c ↔ ∀ (n : ℕ), ↑n ≤ c"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "M : Type w", "M' : Type w₁", "inst✝¹² : CommRing R", "inst✝¹¹ : LieRing L", "inst✝¹⁰ : LieAlgebra R L", "inst✝⁹ : LieRing L'", "inst✝⁸ : LieAlgebra R L'", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M'", "inst✝² : Module R M'", "inst✝¹ : LieRingModule L M'", "inst✝ : LieModule R L M'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "x : L'", "y : L", "hy₂ : ↑f y = x", "z₁ : L", "hz₁ : z₁ ∈ I", "z₂ : L", "hz₂ : z₂ ∈ f.ker", "hy : z₁ + z₂ = y"], "goal": "↑f (z₁ + z₂) ∈ ↑(Submodule.map (↑f) (lieIdealSubalgebra R L I).toSubmodule)"}}
+{"state": {"context": ["α β : Type u", "f : α → β", "x : FreeAddMagma α"], "goal": "(Seq.seq (pure f) fun x_1 => x) = f <$> x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁸ : AddCommGroup E", "inst✝⁷ : Module ℝ E", "s✝ t : Set E", "x : E", "a✝ : ℝ", "α : Type u_4", "inst✝⁶ : LinearOrderedField α", "inst✝⁵ : MulActionWithZero α ℝ", "inst✝⁴ : OrderedSMul α ℝ", "inst✝³ : Module α E", "inst✝² : SMulCommClass α ℝ ℝ", "inst✝¹ : IsScalarTower α ℝ ℝ", "inst✝ : IsScalarTower α ℝ E", "s : Set E", "symmetric : ∀ x ∈ s, -x ∈ s", "a : α"], "goal": "gauge (a • s) = gauge (|a| • s)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "CommSemiring R", "AddCommMonoid A", "Module R A", "AddCommMonoid B", "Module R B", "CoalgebraStruct R A", "CoalgebraStruct R B", "f : A → B", "h₁ : ∀ (x y : A), f (x + y) = f x + f y", "h₂ :\n  ∀ (m : R) (x : A),\n    { toFun := f, map_add' := h₁ }.toFun (m • x) = (RingHom.id R) m • { toFun := f, map_add' := h₁ }.toFun x", "h₃ : Coalgebra.counit ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ } = Coalgebra.counit", "h₄ :\n  TensorProduct.map { toFun := f, map_add' := h₁, map_smul' := h₂ } { toFun := f, map_add' := h₁, map_smul' := h₂ } ∘ₗ\n      Coalgebra.comul =\n    Coalgebra.comul ∘ₗ { toFun := f, map_add' := h₁, map_smul' := h₂ }"], "goal": "⇑{ toFun := f, map_add' := h₁, map_smul' := h₂, counit_comp := h₃, map_comp_comul := h₄ } = f"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "B : ℝ", "h : minkowskiBound K I ≤ volume (convexBodySum K B)"], "goal": "∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I J : Ideal R", "hIJ : I ≤ J", "x : R"], "goal": "(Ideal.Quotient.mk I) x ∈ Ideal.map (Ideal.Quotient.mk I) J ↔ x ∈ J"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "s : Set α"], "goal": "SigmaFinite (μ.trim ⋯) ↔ IsFiniteMeasure μ"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R"], "goal": "(MvPowerSeries.coeff R 0) 1 = 1"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "N : Type u_4", "CommRing R", "AddCommGroup M", "Module R M", "AddCommGroup N", "Module R N", "P : RootPairing ι R M N", "i : ι"], "goal": "Set.MapsTo (⇑(P.coreflection i)) (Set.range ⇑P.coroot) (Set.range ⇑P.coroot)"}}
+{"state": {"context": ["α : Type u", "Infinite α"], "goal": "#(List α) = #α"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace 𝕜 E", "inst✝² : InnerProductSpace 𝕜 F", "inst✝¹ : CompleteSpace E", "inst✝ : CompleteSpace F", "T : E →L[𝕜] E", "hT : T.IsPositive", "S : F →L[𝕜] E"], "goal": "((adjoint S).comp (T.comp S)).IsPositive"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E✝ : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace α", "inst✝¹ : NormedAddCommGroup E✝", "f✝ g : α → E✝", "s✝ t : Set α", "μ ν : Measure α", "E : Type u_5", "inst✝ : NormedAddCommGroup E", "p : ℝ≥0∞", "s : Set α", "f : ↥(Lp E p μ)", "hp : 1 ≤ p", "hμs : μ s ≠ ⊤", "hμ_restrict_univ : (μ.restrict s) univ < ⊤", "hμ_finite : IsFiniteMeasure (μ.restrict s)"], "goal": "Memℒp (↑↑f) 1 (μ.restrict s)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t : Finset α", "m n a b : ℕ", "P : Finpartition s", "hs : a * m + b * (m + 1) = s.card", "m_pos : m > 0"], "goal": "∃ Q, (∀ x ∈ Q.parts, x.card = m ∨ x.card = m + 1) ∧ (∀ x ∈ P.parts, (x \\ (filter (fun y => y ⊆ x) Q.parts).biUnion id).card ≤ m) ∧ (filter (fun i => i.card = m + 1) Q.parts).card = b"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ : Measure α", "f : α → ℝ", "f_intble : Integrable f μ", "f_nn : 0 ≤ᶠ[ae μ] f", "key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}", "lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ⊤", "rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ⊤", "rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ⊤"], "goal": "∫ (ω : α), f ω ∂μ = (∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ).toReal"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "K : ChainComplex C ℕ", "Δ Δ' : SimplexCategory", "i : Δ' ⟶ Δ", "CategoryTheory.Mono i", "hi : AlgebraicTopology.DoldKan.Isδ₀ i"], "goal": "AlgebraicTopology.DoldKan.Γ₀.Obj.Termwise.mapMono K i = K.d Δ.len Δ'.len"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "p : ℝ", "hp : 1 < p"], "goal": "ContDiff ℝ 1 fun x => ‖x‖ ^ p"}}
+{"state": {"context": ["R : Type u_2", "S : Type u_1", "CommRing R", "CommRing S", "Algebra R S", "x : S", "hx : IsIntegral R x", "i : ℕ", "hi : i ≤ (minpolyDiv R x).natDegree"], "goal": "(minpolyDiv R x).coeff ((minpolyDiv R x).natDegree - i) - x ^ i ∈ Submodule.span R ((fun x_1 => x ^ x_1) '' Set.Iio i)"}}
+{"state": {"context": ["J : Type u", "CategoryTheory.Category.{u_1, u} J", "F : J ⥤ Type v", "j : J", "x : F.obj j"], "goal": "x ∈ F.eventualRange j ↔ ∀ ⦃i : J⦄ (f : i ⟶ j), x ∈ Set.range (F.map f)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F G : C ⥤ D", "app' : (X : C) → F.obj X ≅ G.obj X", "naturality : ∀ {X Y : C} (f : X ⟶ Y), F.map f ≫ (app' Y).hom = (app' X).hom ≫ G.map f", "X : C"], "goal": "(CategoryTheory.NatIso.ofComponents app' naturality).app X = app' X"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{u_2, u_1} C", "inst✝¹ : Preadditive C", "inst✝ : HasFiniteCoproducts C", "P : Karoubi (SimplicialObject C)"], "goal": "(((whiskeringLeft (SimplicialObject C) (Karoubi (SimplicialObject C)) (Karoubi (SimplicialObject C))).obj (toKaroubi (SimplicialObject C))).preimage (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans)).app P = (N₂ ⋙ Γ₂).map P.decompId_i ≫ (Γ₂N₂ToKaroubiIso.hom ≫ Γ₂N₁.natTrans).app P.X ≫ P.decompId_p"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "Preorder β", "f : α → β", "s : Set α", "a : α", "TopologicalSpace β", "hf : IsLocalMinOn f s a", "(𝓝[<] f a).NeBot"], "goal": "¬𝓝 (f a) ≤ Filter.map f (𝓝[s] a)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "ι : Sort u_3", "δ' : Type u_5", "ConditionallyCompleteLinearOrder δ'", "f : ι → α → δ'", "bdd : ∀ (x : α), BddBelow (Set.range fun i => f i x)", "h : ∀ (i : ι), UpperSemicontinuous (f i)"], "goal": "UpperSemicontinuous fun x' => ⨅ i, f i x'"}}
+{"state": {"context": ["n m : ℕ"], "goal": "m ≤ n / 2 ↔ ↑m * 2 ≤ ↑n"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : MeasurableSpace α'", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace β'", "inst✝³ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝² : NormedAddCommGroup E", "inst✝¹ : SFinite ν", "inst✝ : SFinite ν'", "h1 : μ ≪ μ'", "h2 : ν ≪ ν'", "s : Set (α × β)", "hs : MeasurableSet s", "h2s : (fun x => ν' (Prod.mk x ⁻¹' s)) =ᶠ[ae μ'] 0"], "goal": "(fun x => ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F", "I J : BoxIntegral.Box ι", "Fintype ι", "l : BoxIntegral.IntegrationParams", "f : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "CompleteSpace F", "h : BoxIntegral.Integrable I l f vol", "hJ : J ≤ I"], "goal": "BoxIntegral.Integrable J l f vol"}}
+{"state": {"context": ["p k n : ℕ", "hp : Fact (Nat.Prime p)", "h : k ≤ n + k"], "goal": "(p - 1) * padicValNat p (n + k)! - ((p - 1) * padicValNat p k ! + (p - 1) * padicValNat p (n + k - k)!) = (p.digits k).sum + (p.digits n).sum - (p.digits (n + k)).sum"}}
+{"state": {"context": ["R : Type u_1", "r : R", "inst✝ : CommRing R", "P : R[X]", "hunit : IsUnit (P.coeff 0)", "hnil : ∀ (i : ℕ), i ≠ 0 → IsNilpotent (P.coeff i)"], "goal": "IsUnit P"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "s : AffineSubspace ℝ P", "ps : Set P", "h : ps ⊆ ↑s", "Nonempty ↥s", "HasOrthogonalProjection s.direction"], "goal": "EuclideanGeometry.Cospherical ps ↔ ∃ center ∈ s, ∃ radius, ∀ p ∈ ps, dist p center = radius"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "R : Type u_2", "inst✝⁶ : CommSemiring k", "inst✝⁵ : Monoid G", "inst✝⁴ : Monoid H", "A : Type u₃", "inst✝³ : Semiring A", "inst✝² : Algebra k A", "B : Type u_3", "inst✝¹ : Semiring B", "inst✝ : Algebra k B", "F : G →* A", "a : G", "b : k"], "goal": "((lift k G A) F) (single a b) = b • F a"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "p : MvPolynomial σ R"], "goal": "p ≠ 0 ↔ ∃ d, MvPolynomial.coeff d p ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "OrderedSemiring α", "x y : ↑(Set.Icc 0 1)"], "goal": "x * y ≤ x"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "p q : R[X]", "n : ℕ", "h2 : n ≤ p.natDegree", "h3 : ¬p.natTrailingDegree ≤ n"], "goal": "p.mirror.coeff n = p.coeff (p.natDegree + p.natTrailingDegree - n)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "X Y : C", "f : Y ⟶ X", "self : CategoryTheory.EffectiveEpi f"], "goal": "Nonempty (CategoryTheory.EffectiveEpiStruct f)"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "M : Type u_1", "CommSemiring R", "Semiring A", "a : A", "Algebra R A", "AddCommMonoid M", "Module A M", "Module R M", "IsScalarTower R A M", "p : Submodule R M", "hp : p ≤ Submodule.comap ((Algebra.lsmul R R M) a) p"], "goal": "(Module.AEval.comapSubmodule R M a) (Module.AEval.mapSubmodule a hp) = p"}}
+{"state": {"context": ["α : Type u", "γ : Type u_1", "inst✝² : TopologicalSpace γ", "inst✝¹ : T2Space γ", "inst✝ : CompactSpace γ", "f : α → γ", "b : Ultrafilter α", "c : γ", "h : Ultrafilter.extend f b = c", "b' : Ultrafilter (Ultrafilter α) := Ultrafilter.map pure b"], "goal": "↑(Ultrafilter.map f b) ≤ 𝓝 c"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "𝕜' : Type u_3", "NonUnitalSeminormedRing 𝕜'", "NormedSpace 𝕜 𝕜'", "IsScalarTower 𝕜 𝕜' 𝕜'", "SMulCommClass 𝕜 𝕜' 𝕜'", "x : 𝕜'"], "goal": "‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖ ≤ ‖x‖"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁵ : TopologicalSpace E", "inst✝⁴ : AddCommGroup E", "inst✝³ : TopologicalAddGroup E", "inst✝² : Module ℝ E", "inst✝¹ : ContinuousSMul ℝ E", "s t : Set E", "x y : E", "inst✝ : LocallyConvexSpace ℝ E", "hs₁ : Convex ℝ s", "hs₂ : IsCompact s", "ht₁ : Convex ℝ t", "ht₂ : IsClosed t", "disj : Disjoint s t", "hs : s.Nonempty", "_ht : t.Nonempty", "U V : Set E", "hU : IsOpen U", "hV : IsOpen V", "hU₁ : Convex ℝ U", "hV₁ : Convex ℝ V", "sU : s ⊆ U", "tV : t ⊆ V", "disj' : Disjoint U V"], "goal": "∃ f u v, (∀ a ∈ s, f a < u) ∧ u < v ∧ ∀ b ∈ t, v < f b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "Preorder α", "t : OrderClosedTopology α", "TopologicalSpace β", "f g : β → α", "hf : Continuous f", "hg : Continuous g"], "goal": "IsClosed {b | f b ≤ g b}"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Preadditive V", "c : ComplexShape ι", "C D : HomologicalComplex V c", "f : (i j : ι) → C.X i ⟶ D.X j", "j : ι"], "goal": "(prevD j) f = (toPrev j) f ≫ D.dTo j"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "ι : Sort u_1", "S : ι → Subalgebra R A"], "goal": "Subalgebra.toSubmodule (⨅ i, S i) = ⨅ i, Subalgebra.toSubmodule (S i)"}}
+{"state": {"context": ["Γ : Type u_1", "inst✝ : Inhabited Γ", "l₁ l₂ : List Γ", "i j : ℕ", "e : l₁ ++ List.replicate i default = l₂ ++ List.replicate j default", "h : l₁.length ≤ l₂.length"], "goal": "BlankExtends l₁ l₂"}}
+{"state": {"context": ["α : Type u_2", "R : Type u_7", "CommRing R", "f : α → ℤ", "s : Finset α"], "goal": "↑(∏ i ∈ s, f i) = ∏ i ∈ s, ↑(f i)"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : FiniteDimensional ℝ E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "r : ��", "hnr : ↑(finrank ℝ E) < r", "hr : 0 < r", "h_meas : Measurable fun ω => (1 + ‖ω‖) ^ (-r)", "h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)", "h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))"], "goal": "∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ (1 + ‖a‖) ^ (-r)} < ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝ : LinearOrderedField 𝕜", "x y z : 𝕜", "hxy : x ≠ y"], "goal": "openSegment 𝕜 x y = Ioo (min x y) (max x y)"}}
+{"state": {"context": ["x : PartENat"], "goal": "x + ⊤ = ⊤"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "DivisionRing K", "AddCommGroup V", "Module K V", "Fintype K", "Fintype V", "k : ℕ", "hk : k ≤ FiniteDimensional.finrank K V"], "goal": "Nat.card { s // LinearIndependent K s } =\n  ∏ i : Fin k, (Fintype.card K ^ FiniteDimensional.finrank K V - Fintype.card K ^ ↑i)"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Type u_3", "inst✝ : Nonempty ι", "f : α → β", "m : ι → OuterMeasure β", "s : Set β"], "goal": "(⨅ i, (map f) ((comap f) (m i))) s ≤ ((map f) (⨅ i, (comap f) (m i))) s"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "CommMonoid M", "f : α → M", "s : Finset (Option α)"], "goal": "∏ x ∈ Finset.eraseNone s, f x = ∏ x ∈ s, Option.elim' 1 f x"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝¹ : Semiring R✝", "r✝ : R✝", "f✝ : R✝[X]", "R : Type u_2", "inst✝ : CommSemiring R", "f : R[X]", "r s : R"], "goal": "(taylor r) ((taylor s) f) = (taylor (r + s)) f"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "FiniteDimensional F E", "h : IntermediateField.fixedField ⊤ = ⊥"], "goal": "IsGalois F E"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "ι : Type u_3", "CommRing R", "AddCommGroup M", "Module R M", "e : ι → M", "ε : ι → Module.Dual R M", "DecidableEq ι", "h : Module.DualBases e ε", "m : M", "i : ι"], "goal": "(h.coeffs m) i = (ε i) m"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "x : X"], "goal": "IsCompact {x}"}}
+{"state": {"context": ["K : Type u_1", "Field K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3"], "goal": "↑⋯.unit ^ 2 + ↑⋯.unit + 1 = 0"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "s : MeasureTheory.SignedMeasure α", "i : Set α"], "goal": "Pairwise (Disjoint on MeasureTheory.SignedMeasure.restrictNonposSeq s i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : DecidableEq α", "inst✝¹ : LinearOrderedCommSemiring β", "inst✝ : ExistsAddOfLE β", "𝒜✝ ℬ✝ : Finset (Finset α)", "a : α", "f f₁ f₂ f₃ f₄ g μ : Finset α → β", "s✝ t✝ u : Finset α", "hu : a ∉ u", "h₁ : 0 ≤ f₁", "h₂ : 0 ≤ f₂", "h₃ : 0 ≤ f₃", "h₄ : 0 ≤ f₄", "h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)", "𝒜 ℬ : Finset (Finset α)", "s : Finset α", "hsu : s ⊆ u", "t : Finset α", "htu : t ⊆ u", "this✝¹ : s ⊆ insert a u", "this✝ : t ⊆ insert a u", "this : insert a s ⊆ insert a u"], "goal": "collapse 𝒜 a f₁ s * collapse ℬ a f₂ t ≤ collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)"}}
+{"state": {"context": ["α : Type u_2", "DistribLattice α", "a : α"], "goal": "InfPrime a ↔ InfIrred a"}}
+{"state": {"context": ["α β : Type u", "β' : Type v", "inst✝ : Infinite α", "leq : lift.{v, u} #α = lift.{u, v} #β'"], "goal": "#(α ≃ β') = 2 ^ lift.{v, u} #α"}}
+{"state": {"context": ["t : ℝ", "ht : -1 ≤ t ∧ t ≤ 1", "x : ℝ"], "goal": "rexp (t * x) ≤ cosh x + t * sinh x"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α"], "goal": "Set.Iic a ∩ Set.Ioi b = Set.Ioc b a"}}
+{"state": {"context": ["f : ℝ → ℝ", "hf' : Differentiable ℝ f", "hf'' : Differentiable ℝ (deriv f)", "hf''_nonneg : ∀ (x : ℝ), 0 ≤ deriv^[2] f x"], "goal": "ConvexOn ℝ Set.univ f"}}
+{"state": {"context": ["α : Type v", "M : Type u", "Monoid M", "MulAction M α", "exp_pos : 0 < Monoid.exponent M", "m : M", "a : α"], "goal": "MulAction.period m a ≤ Monoid.exponent M"}}
+{"state": {"context": ["M : Type u_1", "Semigroup M", "V : Ultrafilter M"], "goal": "Continuous fun x => x * V"}}
+{"state": {"context": ["θ : Angle"], "goal": "(-θ).sin = -θ.sin"}}
+{"state": {"context": ["α : Type u_1", "G H : SimpleGraph α", "n : ℕ", "h : G ≤ H"], "goal": "G.cliqueSet n ⊆ H.cliqueSet n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s✝ t✝ : Multiset α", "a✝ : α", "inst✝ : DecidableEq α", "a : α", "s t : Multiset α", "d : s.Nodup", "h : a ∈ s - t", "h' : a ∈ t"], "goal": "count a s ≤ count a t"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V"], "goal": "o.oangle x y + o.oangle y x = 0"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "B : Type u_3", "Semiring B", "TopologicalSpace B", "Algebra R A", "Algebra R B", "C : Type u_4", "Semiring C", "Algebra R C", "TopologicalSpace C", "D : Type u_5", "Semiring D", "Algebra R D", "TopologicalSpace D", "h : C →A[R] D", "g : B →A[R] C", "f : A →A[R] B"], "goal": "(h.comp g).comp f = h.comp (g.comp f)"}}
+{"state": {"context": ["ι : Type u_1", "Ω : Type u_2", "inst✝¹ : MeasurableSpace Ω", "inst✝ : PseudoEMetricSpace Ω", "μ ν : Measure Ω", "δ : ℝ≥0∞", "h : ∀ (ε : ℝ≥0∞) (B : Set Ω), 0 < ε → ε < ⊤ → MeasurableSet B → μ B ≤ ν (thickening (δ + ε).toReal B) + δ + ε ∧ ν B ≤ μ (thickening (δ + ε).toReal B) + δ + ε", "ε : ℝ≥0", "hε : 0 < ε", "a✝ : δ < ⊤", "ε_top : ¬↑ε = ⊤"], "goal": "levyProkhorovEDist μ ν ≤ δ + ↑ε"}}
+{"state": {"context": ["α : Type u_1", "m : Set α → ℝ≥0∞", "m_empty : m ∅ = 0", "s t : Set α", "h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ⊤", "f : ℕ → Set α", "hf : s ∪ t ⊆ ⋃ i, f i", "μ : OuterMeasure α := OuterMeasure.ofFunction m m_empty", "i : ℕ", "hs : (s ∩ f i).Nonempty", "ht : (t ∩ f i).Nonempty"], "goal": "μ s + μ t ≤ ∑' (i : ℕ), m (f i)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "k : ℕ", "a : R", "n : ℕ", "h : 2 ≤ n"], "goal": "dickson k a n = X * dickson k a (n - 1) - C a * dickson k a (n - 2)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "M : Type u_1", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : Module S M", "inst✝ : IsScalarTower R S M", "x : Ω[S⁄R]"], "goal": "x ∈ Submodule.span S (Set.range ⇑(D R S))"}}
+{"state": {"context": ["Γ : Subgroup SL(2, ℤ)", "k : ℤ", "α : Type u_2", "SMul α ℂ", "IsScalarTower α ℂ ℂ", "f : SlashInvariantForm Γ k", "n : α"], "goal": "⇑(n • f) = n • ⇑f"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝¹ : HasCoequalizers C", "R S T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)"], "goal": "(α_ P.X Q.X (T.X ⊗ T.X)).hom ≫ (P.X ◁ (α_ Q.X T.X T.X).inv ≫ P.X ◁ Q.actRight ▷ T.X ≫ P.X ◁ Q.actRight) ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) = coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ (T.X ⊗ T.X) ≫ (α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X T.X).inv ≫ actRight P Q ▷ T.X ≫ actRight P Q"}}
+{"state": {"context": ["R : Type u_1", "m : Type u_2", "n₁ : Type u_6", "n₂ : Type u_7", "A₁ : Matrix m n₁ R", "A₂ : Matrix m n₂ R", "B₁ : Matrix m n₁ R", "B₂ : Matrix m n₂ R"], "goal": "A₁.fromColumns A₂ = B₁.fromColumns B₂ ↔ A₁ = B₁ ∧ A₂ = B₂"}}
+{"state": {"context": ["X : Type u_2", "EMetricSpace X", "m : Set X → ℝ≥0∞"], "goal": "(MeasureTheory.OuterMeasure.mkMetric' m).IsMetric"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp : p ≠ 2"], "goal": "IsSquare (-2) ↔ p % 8 = 1 ∨ p % 8 = 3"}}
+{"state": {"context": ["M : Type u_1", "X : Type u_2", "TopologicalSpace M", "AddMonoid M", "TopologicalSpace X", "f : X → AddUnits M"], "goal": "Continuous f ↔ Continuous (AddUnits.val ∘ f) ∧ Continuous fun x => ↑(-f x)"}}
+{"state": {"context": ["m : Type u → Type u", "inst✝¹ : _root_.Monad m", "inst✝ : LawfulMonad m", "X✝ Y✝ Z✝ : Kleisli (ofTypeMonad m)", "f : X✝ ⟶ Y✝", "g : Y✝ ⟶ Z✝", "this✝ : _root_.Monad (ofTypeMonad m).obj := let_fun this := inferInstance; this", "this : LawfulMonad (ofTypeMonad m).obj := let_fun this := inferInstance; this"], "goal": "{ obj := fun X => X, map := fun {X Y} f => f }.map (f ≫ g) = { obj := fun X => X, map := fun {X Y} f => f }.map f ≫ { obj := fun X => X, map := fun {X Y} f => f }.map g"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "LinearOrderedSemifield α", "l : Filter β", "f : β → α", "r : α", "hr : 0 < r"], "goal": "Filter.Tendsto (fun x => f x / r) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop"}}
+{"state": {"context": ["k : ℕ"], "goal": "∫ (x : ℝ) in 0 ..π, sin x ^ (2 * k + 1) ≤ ∫ (x : ℝ) in 0 ..π, sin x ^ (2 * k)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "f g : CauSeq α abs", "hf : f.LimZero", "hg : g.LimZero"], "goal": "(f ⊓ g).LimZero"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Groupoid C", "S : Subgroupoid C", "D : Type u_1", "inst✝ : Groupoid D", "φ : C ⥤ D", "hφ : Function.Injective φ.obj", "hφ' : im φ hφ = ⊤", "Sn : S.IsNormal", "c : C"], "goal": "Map.Arrows φ hφ S (φ.obj c) (φ.obj c) (𝟙 (φ.obj c))"}}
+{"state": {"context": ["X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "s : Set ↑Γ(X, ⊤)", "hs : Ideal.span s = ⊤", "hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)", "U V : ↑X.affineOpens", "s' : Finset ↑Γ(X, ⊤)", "hs' : ↑s' ⊆ s", "e : Ideal.span ↑s' = ⊤", "i : ↑↑s'"], "goal": "IsCompact (↑↑U ∩ ↑(X.basicOpen ↑i))"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "K : Type u_3", "Field K", "Algebra R K", "IsFractionRing R K", "I : Ideal R"], "goal": "↑I ≠ 1 ↔ I ≠ 1"}}
+{"state": {"context": ["α : Type u_9", "β : Type u_10", "s : Set (α → β → Prop)", "a : α", "b : β"], "goal": "sInf s a b ↔ ∀ r ∈ s, r a b"}}
+{"state": {"context": ["X Y Z : SemiNormedGrp", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0", "c : NNReal", "h : ‖g‖ ≤ ↑c"], "goal": "‖SemiNormedGrp.explicitCokernelDesc w‖ ≤ ↑c"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "PartialOrder G", "PartialOrder P", "SMul G P", "IsOrderedCancelSMul G P", "s : Set G", "t : Set P", "u : Set G", "hu : u.IsPWO", "a : P", "hs : s.IsPWO", "ht : t.IsPWO", "h : u ⊆ s"], "goal": "Finset.SMulAntidiagonal hu ht a ⊆ Finset.SMulAntidiagonal hs ht a"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "X Y Z : AlgebraicGeometry.PresheafedSpace C", "α : X ⟶ Y", "β : Y ⟶ Z", "U : (TopologicalSpace.Opens ↑↑Z)ᵒᵖ"], "goal": "(α ≫ β).c.app U = β.c.app U ≫ α.c.app (Opposite.op ((TopologicalSpace.Opens.map β.base).obj (Opposite.unop U)))"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "ι : Sort u_3", "S : ι → IntermediateField F E"], "goal": "(iInf S).toSubfield = ⨅ i, (S i).toSubfield"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "LinearOrderedField α", "DivisionRing β", "a : α", "b : β"], "goal": "b ∈ LinearOrderedField.cutMap β a ↔ ∃ q, ↑q < a ∧ ↑q = b"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommRing R", "Ring A", "Algebra R A", "a : A", "r : R"], "goal": "{r} - spectrum R a = spectrum R ((algebraMap R A) r - a)"}}
+{"state": {"context": ["R : Type u", "AddCommGroup R"], "goal": "↑(AddCommGrp.of R) = R"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "F G H : CategoryTheory.OplaxFunctor B C", "η θ : F ⟶ G", "Γ : η ⟶ θ", "ι : G ⟶ H", "a : B"], "goal": "(CategoryTheory.OplaxNatTrans.whiskerRight Γ ι).app a = CategoryTheory.Bicategory.whiskerRight (Γ.app a) (ι.app a)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "b : β", "s : Computation α", "h : b ∈ Computation.map f s"], "goal": "∃ a, a ∈ s ∧ f a = b"}}
+{"state": {"context": ["α : Type u_1", "η : Type u_14", "Fintype η", "ιs : η → Type u_15", "Zero α", "f : (j : η) × ιs j →₀ α", "j : η", "i : ιs j"], "goal": "(Finsupp.sigmaFinsuppEquivPiFinsupp f j) i = f ⟨j, i⟩"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "G : C ⥤ D", "X Y : C", "f g : X ⟶ Y", "W : C", "ι : W ⟶ X", "q : IsSplitEqualizer f g ι", "F : C ⥤ D"], "goal": "F.map f ≫ F.map q.rightRetraction = F.map q.leftRetraction ≫ F.map ι"}}
+{"state": {"context": ["x y : ℝ"], "goal": "sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x"}}
+{"state": {"context": ["α : Type u_3", "TopologicalSpace α", "PolishSpace α"], "goal": "T2Space α"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "N : Type w₁", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "AddCommGroup N", "Module R N", "LieRingModule L N", "f : M →ₗ⁅R,L⁆ N"], "goal": "f.ker = ⊥ ↔ Function.Injective ⇑f"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "s : Set α", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "c c' : E"], "goal": "MeasureTheory.indicatorConstLp p hs hμs c - MeasureTheory.indicatorConstLp p hs hμs c' =\n  MeasureTheory.indicatorConstLp p hs hμs (c - c')"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹¹ : CommRing R", "M : Submonoid R", "S : Type u_2", "inst✝¹⁰ : CommRing S", "inst✝⁹ : Algebra R S", "P : Type u_3", "inst✝⁸ : CommRing P", "A : Type u_4", "inst✝⁷ : CommRing A", "inst✝⁶ : IsDomain A", "K : Type u_5", "B : Type u_6", "inst✝⁵ : CommRing B", "inst✝⁴ : IsDomain B", "inst✝³ : Field K", "L : Type u_7", "inst✝² : Field L", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "g : A →+* L", "h : R ≃+* P", "x : P"], "goal": "x ∈ nonZeroDivisors P ↔ x ∈ Submonoid.map h.toMonoidHom (nonZeroDivisors R)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : Preorder α", "s : LowerSet α", "t : Set α", "a : α"], "goal": "s.sdiff t = s ↔ Disjoint (↑s) t"}}
+{"state": {"context": ["f : ℝ → ℝ", "b : ℝ", "hb : b ∈ Set.Ioo 0 1", "hb₀ : 0 < b", "h_tendsto : Tendsto (fun x => 1 / 2 * log x) atTop atTop"], "goal": "∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, log u ∈ Set.Icc (1 / 2 * log x) (1 * log x)"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "IsAscendingCentralSeries (upperCentralSeries G)"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "MeasurableSpace E", "m : MeasurableSpace Ω", "X : Ω → E", "ℙ : MeasureTheory.Measure Ω", "μ : MeasureTheory.Measure E := by volume_tac", "hX : MeasureTheory.HasPDF X ℙ μ"], "goal": "MeasureTheory.Measure.map X ℙ = MeasureTheory.Measure.withDensity μ (MeasureTheory.pdf X ℙ μ)"}}
+{"state": {"context": ["X : RingedSpace", "U : Opens ↑↑X.toPresheafedSpace", "f : ↑(X.presheaf.obj (op U))", "h : ∀ (x : ↥U), IsUnit ((X.presheaf.germ x) f)", "V : ↥U → Opens ↑↑X.toPresheafedSpace", "iVU : (x : ↥U) → V x ⟶ U", "m : ∀ (x : ↥U), ↑x ∈ V x", "h_unit : ∀ (x : ↥U), IsUnit ((X.presheaf.map (iVU x).op) f)", "hcover : U ≤ iSup V", "g : (x : ↥U) → ↑(X.presheaf.obj (op (V x)))", "hg : ∀ (x : ↥U), (X.presheaf.map (iVU x).op) f * g x = 1", "ic : IsCompatible (sheaf X).val V g", "gl : (CategoryTheory.forget CommRingCat).obj ((sheaf X).val.obj (op U))", "gl_spec : ∀ (i : ↥U), ((sheaf X).val.map (iVU i).op) gl = g i", "i : ↥U"], "goal": "((sheaf X).val.map (iVU i).op) f * g i = 1"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "a b : ℤ", "h : IsCoprime a b"], "goal": "IsCoprime ↑a ↑b"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : a ⟶ c", "t : CategoryTheory.Bicategory.LeftExtension f g", "H : t.IsKan", "s : CategoryTheory.Bicategory.LeftExtension f g"], "goal": "CategoryTheory.CategoryStruct.comp t.unit (CategoryTheory.Bicategory.whiskerLeft f (H.desc s)) = s.unit"}}
+{"state": {"context": ["F : Type u_3", "NormedAddCommGroup F", "InnerProductSpace ℝ F", "x y : F"], "goal": "⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ"}}
+{"state": {"context": ["h : LSeriesSummable (fun n => ↑(μ n)) 1"], "goal": "Summable (-fun i => ↑({p | Nat.Prime p}.indicator (fun n => 1 / ↑n) i))"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "N : Type u_5", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "N' : Type u_8", "AddCommGroup N'", "Module R N'", "f : N →ₗ[R] N'", "Q : QuadraticMap R M N"], "goal": "(f.compQuadraticMap' Q).polarBilin = LinearMap.compr₂ Q.polarBilin f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "hab : a ≤ b", "h : c ≤ b"], "goal": "Ioc a b ∪ Ioi c = Ioi (min a c)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "NonUnitalRing A", "Module R A", "Star A", "S : NonUnitalStarSubalgebra R A"], "goal": "NonUnitalSubringClass.subtype S = ↑(NonUnitalStarSubalgebraClass.subtype S)"}}
+{"state": {"context": ["q : ℚ", "f : ℚ → ℚ → ℚ", "f₁ f₂ : ℤ → ℤ → ℤ → ℤ → ℤ", "fv : ∀ {n₁ : ℤ} {d₁ : ℕ} {h₁ : d₁ ≠ 0} {c₁ : n₁.natAbs.Coprime d₁} {n₂ : ℤ} {d₂ : ℕ} {h₂ : d₂ ≠ 0} {c₂ : n₂.natAbs.Coprime d₂}, f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } { num := n₂, den := d₂, den_nz := h₂, reduced := c₂ } = f₁ n₁ (↑d₁) n₂ ↑d₂ /. f₂ n₁ (↑d₁) n₂ ↑d₂", "f0 : ∀ {n₁ d₁ n₂ d₂ : ℤ}, d₁ ≠ 0 → d₂ ≠ 0 → f₂ n₁ d₁ n₂ d₂ ≠ 0", "a b c d : ℤ", "b0 : b ≠ 0", "d0 : d ≠ 0", "H : ∀ {n₁ d₁ n₂ d₂ : ℤ}, a * d₁ = n₁ * b → c * d₂ = n₂ * d → f₁ n₁ d₁ n₂ d₂ * f₂ a b c d = f₁ a b c d * f₂ n₁ d₁ n₂ d₂", "n₁ : ℤ", "d₁ : ℕ", "h₁ : d₁ ≠ 0", "c₁ : n₁.natAbs.Coprime d₁", "ha : a /. b = n₁ /. ↑d₁"], "goal": "f { num := n₁, den := d₁, den_nz := h₁, reduced := c₁ } (c /. d) = f₁ a b c d /. f₂ a b c d"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "J✝ : Type v", "inst✝ : HasFiniteLimits C", "J : Type", "x✝ : Finite J", "val✝ : Fintype J"], "goal": "HasLimitsOfShape (WidePullbackShape J) C"}}
+{"state": {"context": ["K : Type u", "inst✝⁵ : Field K", "n : ℕ", "hζ✝ : (primitiveRoots n K).Nonempty", "hn : 0 < n", "a : K", "H : Irreducible (X ^ n - C a)", "L : Type u_1", "inst✝⁴ : Field L", "inst✝³ : Algebra K L", "inst✝² : IsGalois K L", "inst✝¹ : FiniteDimensional K L", "inst✝ : IsCyclic (L ≃ₐ[K] L)", "hK : (primitiveRoots (finrank K L) K).Nonempty", "ζ : K", "hζ : IsPrimitiveRoot ζ (finrank K L)", "σ : L ≃ₐ[K] L", "hσ : ∀ (x : L ≃ₐ[K] L), x ∈ Subgroup.zpowers σ", "hσ' : orderOf σ = Fintype.card (L ≃ₐ[K] L)", "this : (minpoly K σ.toLinearMap).IsRoot ζ", "v : L", "hv : Module.End.HasEigenvector σ.toLinearMap ζ v", "hv' : ∀ (n : ℕ), (σ.toLinearMap ^ n) v = ζ ^ n • v"], "goal": "∃ α, α ^ finrank K L ∈ Set.range ⇑(algebraMap K L) ∧ K⟮α⟯ = ⊤"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝⁷ : Preorder ι", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "f✝ g : ι → Ω → E", "ℱ : Filtration ι m0", "F : Type u_4", "inst✝³ : NormedLatticeAddCommGroup F", "inst✝² : NormedSpace ℝ F", "inst✝¹ : CompleteSpace F", "inst✝ : OrderedSMul ℝ F", "f : ι → Ω → F", "c : ℝ", "hc : 0 ≤ c", "hf : Supermartingale f ℱ μ", "i j : ι", "hij : i ≤ j"], "goal": "μ[(c • f) j|↑ℱ i] ≤ᶠ[ae μ] (c • f) i"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_7", "M₂ : Type u_8", "M₁' : Type u_9", "M₂' : Type u_10", "n : Type u_12", "m : Type u_13", "n' : Type u_14", "m' : Type u_15", "CommSemiring R", "AddCommMonoid M₁", "Module R M₁", "AddCommMonoid M₂", "Module R M₂", "DecidableEq n", "Fintype n", "DecidableEq m", "Fintype m", "b₁ : Basis n R M₁", "b₂ : Basis m R M₂", "AddCommMonoid M₁'", "Module R M₁'", "AddCommMonoid M₂'", "Module R M₂'", "b₁' : Basis n' R M₁'", "b₂' : Basis m' R M₂'", "Fintype n'", "Fintype m'", "DecidableEq n'", "DecidableEq m'", "B : M₁ →ₗ[R] M₂ →ₗ[R] R", "l : M₁' →ₗ[R] M₁", "r : M₂' →ₗ[R] M₂"], "goal": "(LinearMap.toMatrix₂ b₁' b₂') (B.compl₁₂ l r) =\n  ((LinearMap.toMatrix b₁' b₁) l)ᵀ * (LinearMap.toMatrix₂ b₁ b₂) B * (LinearMap.toMatrix b₂' b₂) r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "CompleteLattice β", "f : CompleteLatticeHom α β", "L : CompleteSublattice α"], "goal": "(CompleteSublattice.map f L).carrier = ⇑f '' ↑L"}}
+{"state": {"context": ["M : Type u_3", "f : ℕ → M", "m n : ℕ", "AddCommGroup M", "hmn : m < n"], "goal": "∑ i ∈ Finset.Ico m n, f i - f m = ∑ i ∈ Finset.Ico (m + 1) n, f i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "g : β → γ", "s : Set β", "t : Set γ", "hg : Set.MapsTo g s t", "f : α → β"], "goal": "Set.MapsTo (g ∘ f) (f ⁻¹' s) t"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "l : List α", "d : (List.map f l).Nodup", "x : α"], "goal": "x ∈ l → ∀ ⦃y : α⦄, y ∈ l → f x = f y → x = y"}}
+{"state": {"context": ["α : Type u", "β : Type v", "e : α ≃ β", "R : Type u_1", "SMul R β", "r : R", "x : α"], "goal": "r • x = e.symm (r • e x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "e : E →L[𝕜] F", "x : E", "L : Filter E"], "goal": "HasFDerivAtFilter (⇑e) e x L"}}
+{"state": {"context": ["α : Type u", "β : Type v", "δ : Type w", "s₁ s₂ : Stream' α"], "goal": "s₁.tail ⋈ s₂.tail = (s₁.head :: s₂.head :: (s₁.tail ⋈ s₂.tail)).tail.tail"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "Zero M", "p : α → Prop", "AddCommMonoid N", "v : α →₀ M", "h : α → M → N", "hp : ∀ x ∈ v.support, p x"], "goal": "((Finsupp.subtypeDomain p v).sum fun a b => h (↑a) b) = v.sum h"}}
+{"state": {"context": ["n : Type u_3", "α : Type v", "ι : Type u_6", "Mul α", "AddCommMonoid α", "Unique ι", "a b : n → α"], "goal": "(Matrix.col ι a * Matrix.row ι b).diag = a * b"}}
+{"state": {"context": ["α : Type u_1", "s : Multiset α", "hs : s ≠ 0", "t : Set α"], "goal": "(PMF.ofMultiset s hs).toOuterMeasure t =\n  (∑' (x : α), ↑(Multiset.count x (Multiset.filter (fun x => x ∈ t) s))) / ↑(Multiset.card s)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M"], "goal": "(∀ (a : Set (Submodule R M)), a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬I < M') ↔ IsArtinian R M"}}
+{"state": {"context": ["x : PGame"], "goal": "x.birthday = 0 ↔ IsEmpty x.LeftMoves ∧ IsEmpty x.RightMoves"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "A : Type u_3", "inst✝¹ : Mul M", "s : Set M", "inst✝ : Add A", "t : Set A", "S✝ : Subsemigroup M", "ι : Sort u_4", "S : ι → Subsemigroup M", "x : M"], "goal": "x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i"}}
+{"state": {"context": ["G : Type u_1", "Group G", "N : Type u_5", "Group N", "f : G →* N", "H K : Subgroup N", "hf : Function.Surjective ⇑f"], "goal": "Subgroup.comap f H ⊔ Subgroup.comap f K = Subgroup.comap f (H ⊔ K)"}}
+{"state": {"context": ["L : Language", "M : Type u_1", "N : Type u_2", "P : Type u_3", "Q : Type u_4", "inst✝³ : L.Structure M", "inst✝² : L.Structure N", "inst✝¹ : L.Structure P", "inst✝ : L.Structure Q", "f : M ↪ₑ[L] N", "φ : L.Sentence"], "goal": "M ⊨ φ ↔ N �� φ"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "ho : o.IsLimit", "hsC : contained C o", "l : Products I", "h : isGood (π C fun x => ord I x < o) l"], "goal": "((↑l).toFinset.sup fun a => Order.succ (ord I a)) < o ∧ isGood (π C fun x => ord I x < (↑l).toFinset.sup fun a => Order.succ (ord I a)) l"}}
+{"state": {"context": ["α : Type u", "s : Set α", "h : s.Countable", "default : α"], "goal": "Set.range (Set.enumerateCountable h default) ⊆ insert default s"}}
+{"state": {"context": ["a b n : ℕ", "hn : 0 < n.succ"], "goal": "(a ^ n.succ).Coprime b ↔ a.Coprime b"}}
+{"state": {"context": ["K : Type u_1", "Field K", "w : NumberField.InfinitePlace K", "x : ({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)"], "goal": "(NumberField.mixedEmbedding.normAtPlace w) (-x) = (NumberField.mixedEmbedding.normAtPlace w) x"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "J : Type u₂", "inst✝ : Category.{v₂, u₂} J", "X : Type v₂", "α : Type u_1", "Z : α → C", "f f' : Fan Z", "c : Cofan fun x => op (Z x)", "hf : IsLimit f", "hf' : IsLimit f'", "hc : IsColimit c", "j : α"], "goal": "((opProductIsoCoproduct' hf' hc).inv.unop ≫ (opProductIsoCoproduct' hf hc).hom.unop ≫ f.proj j).op = (f'.π.app { as := j }).op"}}
+{"state": {"context": ["α : Type u_1", "i : Nat", "as bs : Array α", "h : i < (as ++ bs).size", "hle : as.size ≤ i", "hlt : i - as.size < bs.size := ⋯"], "goal": "(as ++ bs)[i] = bs[i - as.size]"}}
+{"state": {"context": ["V : Type u_1", "W : Type u_2", "Quiver V", "Quiver W", "obj : V → W", "map : {X Y : V} → (X ⟶ Y) → (obj X ⟶ obj Y)", "X : V"], "goal": "{ obj := obj, map := map }.obj X = obj X"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E", "inst✝⁵ : Algebra F E", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "inst✝² : Algebra E K", "inst✝¹ : IsScalarTower F E K", "inst✝ : IsPurelyInseparable F E", "ι : Type u_1", "v : ι → K", "hsep : ∀ (i : ι), IsSeparable F (v i)", "h : LinearIndependent F v", "q : ℕ", "h✝ : ExpChar F q"], "goal": "LinearIndependent E v"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "MeasurableSpace M", "MeasurableMul₂ M", "μ ν ρ : MeasureTheory.Measure M", "MeasureTheory.SFinite μ", "MeasureTheory.SFinite ν", "MeasureTheory.SFinite ρ"], "goal": "(μ + ν).mconv ρ = μ.mconv ρ + ν.mconv ρ"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "x : α", "inst✝ : LinearOrderedRing α", "k : ℕ", "h : ¬Even (k + 1 + 1)", "hx : x + 1 < 0"], "goal": "0 < ∑ i ∈ range (k + 1 + 1), x ^ i"}}
+{"state": {"context": ["k : Type u_1", "inst✝³ : Field k", "K : Type u_2", "inst✝² : Field K", "F : Type u_3", "inst✝¹ : Field F", "inst✝ : NumberField K", "w : InfinitePlace K", "a : 𝓞 K", "ha : a ≠ 0", "h : ∀ ⦃z : InfinitePlace K⦄, z ≠ w → z ↑a < 1"], "goal": "1 ≤ ↑|(Algebra.norm ℚ) ↑a|"}}
+{"state": {"context": ["k n : Nat"], "goal": "0 < k → n < k + n"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : CategoryTheory.Center C"], "goal": "(X.tensorObj Y).fst = X.fst ⊗ Y.fst"}}
+{"state": {"context": ["V₁ : Type u_2", "V₂ : Type u_3", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂", "f : NormedAddGroupHom V₁ V₂", "K : AddSubgroup V₂", "C : ℝ", "h : f.SurjectiveOnWith K C"], "goal": "Set.SurjOn (⇑f) Set.univ ↑K"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "Nonempty α", "s : Set α"], "goal": "(PMF.uniformOfFintype α).toOuterMeasure s = ↑(Fintype.card ↑s) / ↑(Fintype.card α)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R", "M : Type u_2", "AddCommGroup M", "Module R M", "N : Type u_3", "AddCommGroup N", "Module R N", "f : M →ₗ[R] N", "n : ℕ", "x : AdicCompletion I M"], "goal": "↑((AdicCompletion.map I f) x) n = (LinearMap.reduceModIdeal (I ^ n) f) (↑x n)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "D' : Type u_3", "inst✝³ : Category.{u_4, u_1} C", "inst✝² : Category.{u_5, u_2} D", "inst✝¹ : Category.{?u.28361, u_3} D'", "L : C ⥤ D", "W : MorphismProperty C", "inst✝ : L.IsLocalization W", "Y : C", "P : CostructuredArrow L (L.obj Y) → Prop", "hP₀ : P (CostructuredArrow.mk (𝟙 (L.obj Y)))", "hP₁ : ∀ ⦃X₁ X₂ : C⦄ (f : X₁ ⟶ X₂) (φ : L.obj X₂ ⟶ L.obj Y), P (CostructuredArrow.mk φ) → P (CostructuredArrow.mk (L.map f ≫ φ))", "hP₂ : ∀ ⦃X₁ X₂ : C⦄ (w : X₁ ⟶ X₂) (hw : W w) (φ : L.obj X₁ ⟶ L.obj Y), P (CostructuredArrow.mk φ) → P (CostructuredArrow.mk ((isoOfHom L W w hw).inv ≫ φ))", "g : CostructuredArrow L (L.obj Y)", "g' : StructuredArrow (op ((Functor.fromPUnit (L.obj Y)).obj g.right)) L.op := StructuredArrow.mk g.hom.op"], "goal": "P (CostructuredArrow.mk g'.hom.unop)"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : Measure α", "p q : Set α → Prop", "U✝ : Set α", "ε : ℝ≥0∞", "H : μ.InnerRegularWRT p q", "c : ℝ≥0∞", "U : Set α", "hU : q U", "r : ℝ≥0∞", "hr : r < c * ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K"], "goal": "∃ K ⊆ U, p K ∧ r < (c • μ) K"}}
+{"state": {"context": ["α : Type u", "inst✝³ : UniformSpace α", "β : Type v", "γ : Type w", "inst✝² : UniformSpace β", "inst✝¹ : UniformSpace γ", "inst✝ : T0Space β", "f : α → β", "hf : UniformContinuous f", "a : α"], "goal": "⋯.extend f (pureCauchy a) = f a"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "I : Ideal R", "hI : I ≠ 0", "h_ne_zero : Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors) ≠ 0", "hv : Irreducible (Associates.mk v.asIdeal)", "h_dvd : v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors ∣ ∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors", "h_not_dvd : ¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣ ∏ᶠ (v : HeightOneSpectrum R), v.asIdeal ^ (Associates.mk v.asIdeal).count (Associates.mk I).factors"], "goal": "(Associates.mk v.asIdeal).count (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)).factors = (Associates.mk v.asIdeal).count (Associates.mk I).factors"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : WeierstrassCurve R", "C : VariableChange R"], "goal": "(↑C.u⁻¹ ^ 3 * (W.a₃ + C.r * W.a₁ + 2 * C.t)) ^ 2 + 4 * (↑C.u⁻¹ ^ 6 * (W.a₆ + C.r * W.a₄ + C.r ^ 2 * W.a₂ + C.r ^ 3 - C.t * W.a₃ - C.t ^ 2 - C.r * C.t * W.a₁)) = ↑C.u⁻¹ ^ 6 * (W.a₃ ^ 2 + 4 * W.a₆ + 2 * C.r * (2 * W.a₄ + W.a₁ * W.a₃) + C.r ^ 2 * (W.a₁ ^ 2 + 4 * W.a₂) + 4 * C.r ^ 3)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : T1Space X", "s t : Set X", "x : X", "hx : x ∈ s", "h : {x}ᶜ ∈ 𝓝ˢ t → {x}ᶜ ∈ 𝓝ˢ s"], "goal": "x ∈ t"}}
+{"state": {"context": ["α : Type u_1", "One α", "AddZeroClass α", "PartialOrder α", "ZeroLEOneClass α", "NeZero 1", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a : α"], "goal": "a < a + 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hε₁ : ε ≤ 1", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "A B : Finset (Finset α)", "hA : A ⊆ (chunk hP G ε hU).parts", "hB : B ⊆ (chunk hP G ε hV).parts"], "goal": "(∑ ab ∈ A.product B, ↑(G.edgeDensity ab.1 ab.2)) / (↑A.card * ↑B.card) ≤ ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : (o.oangle x y).sign ≠ 0"], "goal": "x ≠ y"}}
+{"state": {"context": ["p : Prop → Prop"], "goal": "(∃ h, p h) ↔ p False ∨ p True"}}
+{"state": {"context": ["cs cs' : List Char"], "goal": "{ data := cs ++ cs' }.prev { byteIdx := String.utf8Len cs } = { byteIdx := String.utf8Len cs.dropLast }"}}
+{"state": {"context": ["ι : Type u_1", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "c : ComplexShape ι", "K L M : HomologicalComplex C c", "φ : K ⟶ L", "φ' : L ⟶ M", "∀ (i : ι), K.HasHomology i", "∀ (i : ι), L.HasHomology i", "∀ (i : ι), M.HasHomology i", "hφ' : QuasiIso φ'"], "goal": "QuasiIso (φ ≫ φ') ↔ QuasiIso φ"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : Seminorm 𝕜 E", "x : E", "r : ℝ", "hr : r ≤ 0"], "goal": "p.ball x r = ∅"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "I J : Ideal R", "f : R →+* S", "hf : Function.Injective ⇑f", "p : Ideal R", "H : p ∈ minimalPrimes R", "this✝ : p.IsPrime", "this : Nontrivial (Localization (Submonoid.map f p.primeCompl))", "M : Ideal (Localization (Submonoid.map f p.primeCompl))", "hM : M.IsMaximal"], "goal": "comap (algebraMap R (Localization p.primeCompl)) (comap (IsLocalization.map (Localization (Submonoid.map f p.primeCompl)) f ⋯) M) = p"}}
+{"state": {"context": ["x y : SetTheory.PGame", "i : x.RightMoves", "j : y.RightMoves"], "goal": "(x * y).moveLeft (SetTheory.PGame.toLeftMovesMul (Sum.inr (i, j))) =\n  x.moveRight i * y + x * y.moveRight j - x.moveRight i * y.moveRight j"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommSemiring R", "p : ℕ", "x : R", "hp : ¬1 < p"], "goal": "x = 0 ↔ ∃ n, x ^ p ^ n = 0"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_3", "m : MeasurableSpace Ω", "LinearOrder ι", "μ : MeasureTheory.Measure Ω", "ℱ : MeasureTheory.Filtration ι m", "τ : Ω → ι", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "f : Ω → E", "Filter.atTop.IsCountablyGenerated", "hτ : MeasureTheory.IsStoppingTime ℱ τ", "i : ι", "MeasureTheory.SigmaFinite (μ.trim ⋯)"], "goal": "μ[f|⋯.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ i}] μ[f|hτ.measurableSpace]"}}
+{"state": {"context": ["X : Type u", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : NormalSpace X", "s : Set X", "hs : IsClosed s", "𝕜 : Type v", "inst✝⁴ : RCLike 𝕜", "inst✝³ : TietzeExtension 𝕜", "E : Type w", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "f : ↑s →ᵇ E", "hf : ¬‖f‖ = 0", "this : TietzeExtension ↑(Metric.closedBall 0 ‖f‖)", "hf' : ∀ (x : ↑s), f x ∈ Metric.closedBall 0 ‖f‖", "g : C(X, E)", "hg : ContinuousMap.restrict s g = ↑f", "hg_mem : ∀ (x : X), ‖g x‖ ≤ ‖f‖"], "goal": "∃ g, ‖g‖ = ‖f‖ ∧ g.restrict s = f"}}
+{"state": {"context": ["X : RingedSpace", "U : Opens ↑↑X.toPresheafedSpace", "f : ↑(X.presheaf.obj (op U))"], "goal": "IsUnit ((X.presheaf.map (homOfLE ⋯).op) f)"}}
+{"state": {"context": ["S : Set ℝ", "h₁ : S.Nonempty", "h₂ : BddAbove S"], "goal": "IsLUB S (Classical.choose ⋯)"}}
+{"state": {"context": ["α : Type u_2", "Zero α", "f : Filter α", "h : f.NeBot"], "goal": "f ≤ 0 ↔ f = 0"}}
+{"state": {"context": ["X : Type u_1", "X' : Type u_2", "Y : Type u_3", "Y' : Type u_4", "Z : Type u_5", "Z' : Type u_6", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : TopologicalSpace X'", "inst✝³ : TopologicalSpace Y", "inst✝² : TopologicalSpace Y'", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "e : PartialHomeomorph X Y", "f : Z → X", "x : Z", "h : f ⁻¹' e.source ∈ 𝓝 x", "hx : f x ∈ e.source", "h' : f ⁻¹' e.source ∈ 𝓝[univ] x"], "goal": "ContinuousAt f x ↔ ContinuousAt (↑e ∘ f) x"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : DecidableEq R", "hp : Fact (Nat.Prime p)", "inst✝ : Invertible ↑p", "n : ℕ", "IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0"], "goal": "∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝⁴ : Ring k", "inst✝³ : AddCommGroup V", "inst✝² : Module k V", "inst✝¹ : AffineSpace V P", "ι : Type u_4", "inst✝ : Nontrivial k", "p : ι → P", "ha : ∀ (s1 s2 : Finset ι) (w1 w2 : ι → k), ∑ i ∈ s1, w1 i = 1 → ∑ i ∈ s2, w2 i = 1 → (affineCombination k s1 p) w1 = (affineCombination k s2 p) w2 → (↑s1).indicator w1 = (↑s2).indicator w2", "s1 s2 : Set ι", "p0 : P", "fs1 : Finset ι", "hfs1 : ↑fs1 ⊆ s1", "w1 : ι → k", "hw1 : ∑ i ∈ fs1, w1 i = 1", "hp0s1 : p0 = (affineCombination k fs1 p) w1", "fs2 : Finset ι", "hfs2 : ↑fs2 ⊆ s2", "w2 : ι → k", "hw2 : ∑ i ∈ fs2, w2 i = 1", "hp0s2 : p0 = (affineCombination k fs2 p) w2"], "goal": "∃ i, i ∈ s1 ∩ s2"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "m0 : MeasurableSpace α", "μ : Measure α", "s✝ t s : Set α", "ht : NullMeasurableSet t μ"], "goal": "μ (s ∪ t) + μ (s ∩ t) = μ s + μ t"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "k : Nat"], "goal": "List.drop k l = [] ↔ l.length ≤ k"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "Preorder β", "f : α → β", "s : Set α", "a : α", "hf : IsMaxOn f s a", "hs : s ∈ 𝓝 a"], "goal": "IsLocalMax f a"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommRing R", "inst✝² : IsDomain R", "p n : ℕ", "inst✝¹ : ExpChar R p", "f : R[X]", "inst✝ : PerfectRing R p", "y : R", "m : ℕ"], "goal": "((X ^ p ^ n - C y) ^ m).roots = (m * p ^ n) • {(iterateFrobeniusEquiv R p n).symm y}"}}
+{"state": {"context": ["α : Type u_2", "Sub α", "f₁ f₂ g : Filter α"], "goal": "f₁ ≤ f₂ → f₁ - g ≤ f₂ - g"}}
+{"state": {"context": ["k✝ G✝ : Type u", "inst✝⁷ : CommRing k✝", "inst✝⁶ : Monoid G✝", "k : Type u_1", "G : Type u_2", "inst✝⁵ : CommRing k", "inst✝⁴ : Monoid G", "V : Type u_3", "W : Type u_4", "inst✝³ : AddCommGroup V", "inst✝² : AddCommGroup W", "inst✝¹ : Module k V", "inst✝ : Module k W", "ρ : G →* V →ₗ[k] V", "σ : G →* W →ₗ[k] W", "f : V →ₗ[k] W", "w : ∀ (g : G), f ∘ₗ ρ g = σ g ∘ₗ f", "r : MonoidAlgebra k G", "x : V"], "goal": "∀ (r : k) (f_1 : MonoidAlgebra k G), f ((((MonoidAlgebra.lift k G (V →ₗ[k] V)) ρ) f_1) x) = (((MonoidAlgebra.lift k G (W →ₗ[k] W)) σ) f_1) (f x) → f ((((MonoidAlgebra.lift k G (V →ₗ[k] V)) ρ) (r • f_1)) x) = (((MonoidAlgebra.lift k G (W →ₗ[k] W)) σ) (r • f_1)) (f x)"}}
+{"state": {"context": ["X✝ : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X✝", "inst✝¹ : FirstCountableTopology X✝", "inst✝ : R1Space X✝", "x y : X✝", "X : Type u_3", "t₁ t₂ : TopologicalSpace X", "h₁ : R1Space X", "h₂ : R1Space X"], "goal": "R1Space X"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "M : Type w", "L.Structure M", "α : Type u'", "L'.Structure M", "φ : L →ᴸ L'", "φ.IsExpansionOn M", "ψ : L.Formula α", "v : α → M"], "goal": "(φ.onFormula ψ).Realize v ↔ ψ.Realize v"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "inst✝¹ : HasZeroMorphisms C", "β : Type w", "f : β → C", "inst✝ : HasProduct f", "h : IsCoseparator (∏ᶜ f)", "X Y : C", "u v : X ⟶ Y", "huv : ∀ G ∈ Set.range f, ∀ (h : Y ⟶ G), u ≫ h = v ≫ h"], "goal": "u = v"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s✝ t u : Set α✝", "inst✝⁹ : TopologicalSpace α✝", "inst✝⁸ : MeasurableSpace α✝", "inst✝⁷ : OpensMeasurableSpace α✝", "inst✝⁶ : MeasurableSpace δ", "inst✝⁵ : LinearOrder α✝", "inst✝⁴ : OrderClosedTopology α✝", "a b x✝ : α✝", "α : Type u_5", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : OrderTopology α", "inst✝ : SecondCountableTopology α", "s : Set α", "hd : Dense s", "hbot : ∀ (x : α), IsTop x → x ∈ s", "hIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → x ∈ s", "x y : αᵒᵈ", "hlt : x < y", "he : Ioo x y = ∅"], "goal": "Ioo y x = ∅"}}
+{"state": {"context": ["n : ℕ", "s : ℝ", "hs : s ∈ Ici 1"], "goal": "‖term (n + 1) s‖ ≤ term (n + 1) 1"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "P : Cᵒᵖ ⥤ Type (max v u)", "X : C", "R : Presieve X", "S : Sieve X", "x : FirstObj P S.arrows"], "goal": "((firstObjEqFamily P S.arrows).hom x).SieveCompatible ↔ firstMap P S x = secondMap P S x"}}
+{"state": {"context": ["M : Type u_6", "AddMonoid M", "TopologicalSpace M", "ContinuousAdd M", "a b : M", "h : Inseparable a b", "n : ℕ"], "goal": "Inseparable (n • a) (n • b)"}}
+{"state": {"context": ["X : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : RegularSpace X", "inst✝ : LocallyCompactSpace X", "s t : Set X", "hs : IsCompact s", "h's : IsGδ s", "ht : IsClosed t", "hd : Disjoint s t", "U : ℕ → Set X", "U_open : ∀ (n : ℕ), IsOpen (U n)", "hU : s = ⋂ n, U n", "m : Set X", "m_comp : IsCompact m", "sm : s ⊆ interior m", "mt : m ⊆ tᶜ", "f : ℕ → C(X, ℝ)", "fs : ∀ (n : ℕ), EqOn (⇑(f n)) 1 s", "fm : ∀ (n : ℕ), EqOn (⇑(f n)) 0 (U n ∩ interior m)ᶜ", "_hf : ∀ (n : ℕ), HasCompactSupport ⇑(f n)", "f_range : ∀ (n : ℕ) (x : X), (f n) x ∈ Icc 0 1", "u : ℕ → ℝ", "u_pos : ∀ (i : ℕ), 0 < u i", "u_sum : Summable u", "hu : ∑' (i : ℕ), u i = 1", "g : X → ℝ := fun x => ∑' (n : ℕ), u n * (f n) x", "hgmc : EqOn g 0 mᶜ", "I : ∀ (n : ℕ) (x : X), u n * (f n) x ≤ u n", "S : ∀ (x : X), Summable fun n => u n * (f n) x", "x : X"], "goal": "{ toFun := g, continuous_toFun := ?intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1 } x ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "Archimedean α", "x : α"], "goal": "∃ q, ↑q < x"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "NormedAddCommGroup F", "f : α → F", "μ ν : MeasureTheory.Measure α", "p : ℝ≥0∞"], "goal": "MeasureTheory.eLpNorm f p μ ≤ MeasureTheory.eLpNorm f p (μ + ν)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℝ E", "s : ConvexCone ℝ E", "f : E →ₗ.[ℝ] ℝ", "nonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x", "dense : ∀ (y : E), ∃ x, ↑x + y ∈ s", "hdom : f.domain ≠ ⊤"], "goal": "∃ g, f < g ∧ ∀ (x : ↥g.domain), ↑x ∈ s → 0 ≤ ↑g x"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "M' : Type u_4", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M'", "inst✝² : Module R M'", "inst✝¹ : LieRingModule L M'", "inst✝ : LieModule R L M'", "N N₁ N₂ : LieSubmodule R L M", "m : M", "hm : m ∈ N"], "goal": "m ∈ N.normalizer"}}
+{"state": {"context": ["R : Type u", "M M₁ : Type v", "M' : Type v'", "ι : Type w", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M₁", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M₁", "n : ℕ", "H : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → s.card ≤ n", "s : Set M", "li : LinearIndependent (ι := { x // x ∈ s }) R Subtype.val"], "goal": "#↑↑⟨s, li⟩ ≤ ↑n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommGroup α", "TopologicalSpace α", "TopologicalGroup α", "f : β → α", "T2Space α", "s : Finset β", "hf : Multipliable f"], "goal": "(∏ x ∈ s, f x) * ∏' (x : ↑(↑s)ᶜ), f ↑x = ∏' (x : β), f x"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a✝ b✝ : ℝ", "f : ℕ → Ω → ℝ", "N✝ n m : ℕ", "ω : Ω", "ℱ : Filtration ℕ m0", "inst✝ : IsFiniteMeasure μ", "hf : Submartingale f ℱ μ", "a b : ℝ", "N : ℕ"], "goal": "Submartingale (fun n => ∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) ℱ μ"}}
+{"state": {"context": ["a k : ℕ", "h0 : 0 < a", "h1 : a < k", "n : ℕ"], "goal": "∀ ⦃x : ℕ⦄, k ≤ x → x < k + n → Prime x → (k ≤ x ∧ x < k + n) ∧ a.Coprime x"}}
+{"state": {"context": ["ι : Type u_5", "σ : ι → Type u_7", "(i : ι) → TopologicalSpace (σ i)", "i : ι"], "goal": "OpenEmbedding (Sigma.mk i)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n✝ : ℕ", "inst✝ : Semiring R", "p✝ q✝ r : R[X]", "n : ℕ", "p q : R[X]", "qn : q.natDegree ≤ n", "h : (p + q).natDegree ≤ n"], "goal": "p.natDegree ≤ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝ : CompleteLattice α", "s : Set α", "hs : SetIndependent s", "x : α", "y : Set α", "hx : x ∈ s", "hy : y ⊆ s", "hxy : x ∉ y", "this : Disjoint x (sSup (insert x y \\ {x}))"], "goal": "Disjoint x (sSup y)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "N : Type u_3", "AddCommMonoid N", "Module R N", "ι : Type u_4", "DecidableEq ι", "i : ι", "m : M", "n : N"], "goal": "(TensorProduct.finsuppLeft R M N ι).symm (Finsupp.single i (m ⊗ₜ[R] n)) = Finsupp.single i m ⊗ₜ[R] n"}}
+{"state": {"context": ["n : ℕ", "f : Vector ℕ n →. ℕ", "pf : Partrec' f"], "goal": "_root_.Partrec f"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "t : Affine.Triangle ℝ P", "i₁ i₂ i₃ : Fin 3", "h₁₂ : i₁ ≠ i₂", "h₁₃ : i₁ ≠ i₃", "h₂₃ : i₂ ≠ i₃"], "goal": "Affine.Simplex.altitude t i₁ = Affine.Simplex.mongePlane t i₂ i₃"}}
+{"state": {"context": ["x y : SetTheory.PGame", "h : x ≤ y", "j : y.RightMoves"], "goal": "x.LF (y.moveRight j)"}}
+{"state": {"context": ["R : CommRingCat", "U V : (TopologicalSpace.Opens ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj R).toPresheafedSpace)ᵒᵖ", "i : U ⟶ V"], "goal": "(AlgebraicGeometry.Spec.locallyRingedSpaceObj R).presheaf.map i = (AlgebraicGeometry.Spec.structureSheaf ↑R).val.map i"}}
+{"state": {"context": ["α : Type u", "BEq α", "self : LawfulBEq α", "a b : α"], "goal": "(a == b) = true → a = b"}}
+{"state": {"context": ["R : Type u₁", "inst✝¹ : CommSemiring R", "p : ℕ", "hp : Fact (Nat.Prime p)", "inst✝ : CharP R p", "f : Ring.Perfection R p", "n m✝ m : ℕ", "ih : (coeff R p (n + m)) ((⇑(frobenius (Ring.Perfection R p) p))^[m] f) = (coeff R p n) f"], "goal": "(coeff R p (n + m.succ)) ((⇑(frobenius (Ring.Perfection R p) p))^[m.succ] f) = (coeff R p n) f"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_6", "AddGroup G", "s : Set α", "f : α → G"], "goal": "sᶜ.indicator f = f - s.indicator f"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x y : V"], "goal": "cos (angle x y) = cos π ↔ angle x y = π"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "μ : Measure ℝ", "l : Filter ι", "inst✝³ : l.IsCountablyGenerated", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "a b : ι → ℝ", "f : ℝ → E", "hfi : Integrable f μ", "ha : Tendsto a l atBot", "hb : Tendsto b l atTop", "φ : ι → Set ℝ := fun i => Ioc (a i) (b i)", "hφ : AECover μ l φ"], "goal": "Tendsto (fun i => ∫ (x : ℝ) in a i..b i, f x ∂μ) l (𝓝 (∫ (x : ℝ), f x ∂μ))"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℂ E", "V : Type u_2", "W : Type u_3", "inst✝⁸ : NormedAddCommGroup V", "inst✝⁷ : NormedSpace ℝ V", "inst✝⁶ : NormedAddCommGroup W", "inst✝⁵ : NormedSpace ℝ W", "L : V →L[ℝ] W →L[ℝ] ℝ", "f : V → E", "inst✝⁴ : SecondCountableTopology V", "inst✝³ : MeasurableSpace V", "inst✝² : BorelSpace V", "μ✝ : Measure V", "inst✝¹ : FiniteDimensional ℝ V", "μ : Measure V", "inst✝ : μ.IsAddHaarMeasure", "K N : ℕ∞", "hf : ContDiff ℝ N f", "h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ", "k n : ℕ", "hk : ↑k ≤ K", "hn : ↑n ≤ N", "w : W"], "goal": "∫ (v : V), ‖iteratedFDeriv ℝ n (fun v => fourierPowSMulRight L f v k) v‖ ∂μ ≤ (2 * π) ^ k * (2 * ↑k + 2) ^ n * ‖L‖ ^ k * ∑ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), ∫ (v : V), ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖ ∂μ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "l : List α"], "goal": "(foldr (fun a arr => Array.foldl (fun r l => r.push (a :: l)) arr arr 0) #[[]] l).toList = foldr (fun a r => (Array.foldl (fun r l => r.push (a :: l)) (toArray r) (toArray r) 0).toList) [[]] l"}}
+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "s.encard = 2 ↔ ∃ x y, x ≠ y ∧ s = {x, y}"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = 0 ∨ o.oangle x y = ↑π", "hx : ¬x = 0", "hy : ¬y = 0"], "goal": "x = 0 ∨ y = 0 ∨ |(o.oangle x y).toReal| = 0 ∨ |(o.oangle x y).toReal| = π"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "CategoryTheory.Epi φ.τ₁", "CategoryTheory.IsIso φ.τ₂", "CategoryTheory.Mono φ.τ₃"], "goal": "S₁.Exact ↔ S₂.Exact"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → Zero (β i)", "s : Finset ι", "x : (i : ↑↑s) → β ↑i", "i✝ i : ι", "xi : β i"], "goal": "i = i ∧ HEq xi 0 ∨ xi = 0 ∧ 0 = 0 ↔ xi = 0"}}
+{"state": {"context": ["f✝ f g : StieltjesFunction"], "goal": "(f + g).measure = f.measure + g.measure"}}
+{"state": {"context": [], "goal": "WellFounded fun x x_1 => x < x_1"}}
+{"state": {"context": ["I : Type w₁", "C : I → Type u₁", "(i : I) → CategoryTheory.Category.{v₁, u₁} (C i)", "J : Type w₂", "g : J → I", "j : J", "X Y : C (g j)", "f : X ⟶ Y"], "goal": "(CategoryTheory.Sigma.map C g).map (CategoryTheory.Sigma.SigmaHom.mk f) = CategoryTheory.Sigma.SigmaHom.mk f"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace ℝ F", "f : F → E", "hf : Differentiable ℝ f", "x : F", "p : ℝ", "hp : 1 < p"], "goal": "‖fderiv ℝ (fun x => ‖f x‖ ^ p) x‖ ≤ p * ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖"}}
+{"state": {"context": ["a b c : Ordinal.{u_4}"], "goal": "a < b / c → c * a < b"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v}", "c : Ordinal.{max u v}", "hι : Cardinal.lift.{v, u} #ι < c.cof", "hf : ∀ (i : ι), f i < c"], "goal": "Ordinal.lsub f < c"}}
+{"state": {"context": ["R : Type u", "S : Type u₁", "Monoid R", "Monoid S", "φ : R →* S", "A : Type v", "B : Type w", "NonUnitalNonAssocSemiring A", "DistribMulAction R A", "NonUnitalNonAssocSemiring B", "DistribMulAction S B", "f : A →ₛₙₐ[φ] B", "x y : A"], "goal": "f (x + y) = f x + f y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "CommMonoid M", "DecidableEq α", "DecidableEq β", "s : Finset (α × β)", "f : α × β → M"], "goal": "∏ᶠ (ab : α × β) (_ : ab ∈ s), f ab =\n  ∏ᶠ (a : α) (b : β) (_ : b ∈ Finset.image Prod.snd (Finset.filter (fun ab => ab.1 = a) s)), f (a, b)"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : GroupWithZero G₀", "a✝ b c d : G₀", "m n : ℕ", "a : G₀", "ha : a ≠ 0", "h : n ≤ m"], "goal": "a ^ (m - n) = a ^ m * (a ^ n)⁻¹"}}
+{"state": {"context": ["X : Type u", "ι : Sort w", "TopologicalSpace X", "Finite ι", "f : ι → Set X"], "goal": "closure (⋃ i, f i) = ⋃ i, closure (f i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → ℚ → ℝ", "hf : IsRatCondKernelCDFAux f κ ν", "inst✝ : IsFiniteKernel ν", "a : α", "q : ℚ", "t : β", "hbdd_below : ∀ (q : ℚ), BddBelow (range fun r => f (a, t) ↑r)", "h_nonneg : ∀ (q : ℚ), 0 ≤ f (a, t) q", "h_le_one : ∀ (q : ℚ), f (a, t) q ≤ 1"], "goal": "‖⨅ r, f (a, t) ↑r‖ ≤ 1"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁷ : OrderedRing R", "inst✝⁶ : AddCommGroup V", "inst✝⁵ : Module R V", "inst✝⁴ : AddTorsor V P", "inst✝³ : AddCommGroup V'", "inst✝² : Module R V'", "inst✝¹ : AddTorsor V' P'", "inst✝ : NoZeroSMulDivisors R V", "x y z : P", "h : Wbtw R x y z"], "goal": "Wbtw R x z y ↔ y = z"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : Stream' (Option α)", "al : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a"], "goal": "Stream'.map (fun o => Option.rec none (fun val => some val) o) f = f"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "G : D ⥤ C", "η : 𝟭 C ≅ F ⋙ G", "ε : G ⋙ F ≅ 𝟭 D", "X : C", "this : (ε.app (F.obj X)).hom ≫ (ε.app (F.obj X)).inv = 𝟙 ((G ⋙ F).obj (F.obj X))"], "goal": "F.map (η.hom.app X) ≫ (ε.hom.app (F.obj X) ≫ ε.inv.app (F.obj X)) ≫ F.map (η.inv.app X) = 𝟙 (F.obj X)"}}
+{"state": {"context": ["A D : Type u", "TopologicalSpace A", "TopologicalSpace D", "T1Space A", "CompactSpace D", "π : D → A", "π_cont : Continuous π", "π_surj : Function.Surjective π"], "goal": "∃ E, CompactSpace ↑E ∧ π '' E = Set.univ ∧ ∀ (E₀ : Set ↑E), E₀ ≠ Set.univ → IsClosed E₀ → E.restrict π '' E₀ ≠ Set.univ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasTerminal C"], "goal": "(CategoryTheory.subterminalsEquivMonoOverTerminal C).counitIso =\n  CategoryTheory.NatIso.ofComponents\n    (fun X =>\n      CategoryTheory.MonoOver.isoMk\n        (CategoryTheory.Iso.refl\n          (({ obj := fun X => { obj := X.obj.left, property := ⋯ }, map := fun {X Y} f => f.left, map_id := ⋯,\n                        map_comp := ⋯ }.comp\n                    {\n                      obj := fun X =>\n                        { obj := CategoryTheory.Over.mk (CategoryTheory.Limits.terminal.from X.obj), property := ⋯ },\n                      map := fun {X Y} f => CategoryTheory.MonoOver.homMk f ⋯, map_id := ⋯, map_comp := ⋯ }).obj\n                X).obj.left)\n        ⋯)\n    ⋯"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝¹ : Ring R", "inst✝ : Semiring S", "f : R →+* S", "x : S", "p : R[X]", "T : Subring R", "hp : ↑p.coeffs ⊆ ↑T", "i : ℕ"], "goal": "(p.toSubring T hp).coeff i = 0 ↔ ↑((p.toSubring T hp).coeff i) = 0"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l₁ l₂ r₁ r₂ : List α"], "goal": "l₁ ++ a :: r₁ ~ l₂ ++ a :: r₂ → l₁ ++ r₁ ~ l₂ ++ r₂"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "φ✝ φ ψ : R⟦X⟧", "n : ℕ", "hn : ↑n < φ.order + ψ.order"], "goal": "∀ x ∈ antidiagonal n, (coeff R x.1) φ * (coeff R x.2) ψ = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G H : SimpleGraph α", "s : Finset α", "a b c : α"], "goal": "(↑s).Subsingleton ↔ s.card ≤ 1"}}
+{"state": {"context": ["L : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "S✝ S : L.Substructure M", "x : M"], "goal": "x ∈ S.subtype.toHom.range ↔ x ∈ S"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α"], "goal": "Computable₂ fun l a => l ++ [a]"}}
+{"state": {"context": ["F : Type u_1", "NormedAddCommGroup F", "g : ℝ → F", "R : ℝ", "hR : R ≠ 0"], "goal": "MeasureTheory.Integrable (fun x => g (x * R)) MeasureTheory.volume ↔ MeasureTheory.Integrable g MeasureTheory.volume"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "U : Set X", "hf : IsProperMap f", "hU : IsClosed U"], "goal": "IsProperMap fun x => f ↑x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_5", "OrderedSemiring 𝕜", "AddCommMonoid E", "OrderedCancelAddCommMonoid β", "Module 𝕜 E", "Module 𝕜 β", "OrderedSMul 𝕜 β", "s : Set E", "f : E → β", "hf : ConvexOn 𝕜 s f"], "goal": "Convex 𝕜 {p | p.1 ∈ s ∧ f p.1 < p.2}"}}
+{"state": {"context": ["M : Type u_5", "S : Type u_6", "DivInvMonoid M", "SetLike S M", "hSM : SubgroupClass S M", "H : S", "x y : M", "hx : x ∈ H", "hy : y ∈ H"], "goal": "x / y ∈ H"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s✝ : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "s : Multiset (MvPolynomial σ R)", "l : List (MvPolynomial σ R)"], "goal": "l.prod.totalDegree ≤ (List.map totalDegree l).sum"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N₁ N₂ : Bimod Y Z", "f : N₁ ⟶ N₂"], "goal": "(α_ X.X M.X N₁.X).inv ≫ M.actLeft ▷ N₁.X ≫ M.X ◁ f.hom ≫ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one = (X.X ◁ M.X ◁ f.hom ≫ X.X ◁ colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one) ≫ TensorBimod.actLeft M N₂"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "f g : ι → ℝ≥0", "p : ℝ", "hp : 1 ≤ p"], "goal": "(∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)"}}
+{"state": {"context": ["M : Type u_1", "AddCommMonoid M", "TopologicalSpace M", "m m' : M", "ContinuousAdd M", "f : ℤ → M", "hf₁ : HasSum (fun n => f ↑n) m", "hf₂ : HasSum (fun n => f (-(↑n + 1))) m'"], "goal": "HasSum f (m + m')"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a : α", "b₁ b₂ : β", "SMul α β", "Preorder α", "Preorder β", "Zero α", "PosSMulReflectLE α β", "h : a • b₁ ≤ a • b₂", "ha : 0 < a"], "goal": "b₁ ≤ b₂"}}
+{"state": {"context": ["x : ℝ≥0∞", "y z : ℝ", "hx : x ≠ 0", "h'x : x ≠ ⊤"], "goal": "x ^ (y - z) = x ^ y / x ^ z"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : Group G", "inst✝¹ : MulAction G α", "inst✝ : MulAction G β", "H : Subgroup G", "x : Quotient G α", "a b : ↑x.orbit", "c : α", "h : ⟦a⟧ = ⟦b⟧"], "goal": "b ∈ orbit ⟦a⟧"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁶ : NontriviallyNormedField 𝕜", "H : Type u_2", "inst✝¹⁵ : TopologicalSpace H", "E : Type u_3", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G✝ : Type u_4", "inst✝¹² : Mul G✝", "inst✝¹¹ : TopologicalSpace G✝", "inst✝¹⁰ : ChartedSpace H G✝", "inst✝⁹ : SmoothMul I G✝", "E' : Type u_5", "inst✝⁸ : NormedAddCommGroup E'", "inst✝⁷ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁶ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_7", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H' M", "g✝ h✝ : G✝", "G : Type u_8", "inst✝³ : Semigroup G", "inst✝² : TopologicalSpace G", "inst✝¹ : ChartedSpace H G", "inst✝ : SmoothMul I G", "g h x✝ : G"], "goal": "(𝑳 I (g * h)) x✝ = ((𝑳 I g).comp (𝑳 I h)) x✝"}}
+{"state": {"context": ["x y : ℝ"], "goal": "√x ≠ 0 ↔ 0 < x"}}
+{"state": {"context": ["R✝ : Type u", "inst✝¹ : CommSemiring R✝", "x y z : R✝", "R : Type u_1", "inst✝ : CommRing R", "a b u v : ℤ", "H : u * a + v * b = 1"], "goal": "↑u * ↑a + ↑v * ↑b = 1"}}
+{"state": {"context": [], "goal": "Nat.zero = 0"}}
+{"state": {"context": ["f : StieltjesFunction", "l : ℝ", "hf : Tendsto (↑f) atBot (𝓝 l)", "x : ℝ"], "goal": "f.measure (Iic x) = ofReal (↑f x - l)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "c x : α", "Ring α", "h : Function.Periodic f c", "n : ℕ"], "goal": "f (↑n * c - x) = f (-x)"}}
+{"state": {"context": ["R : Type u", "AddGroupWithOne R", "m n : ℤ"], "goal": "↑(m + n) = ↑m + ↑n"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "h : CategoryTheory.IsIdempotentComplete Cᵒᵖ"], "goal": "CategoryTheory.IsIdempotentComplete C"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "f : C(X, Y)", "g : LocallyConstant Y Z", "x : X"], "goal": "(LocallyConstant.comap f g) x = g (f x)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_3", "One M", "s t : Set α", "f g : α → M", "l : Filter α", "hf : f =ᶠ[l ⊓ 𝓟 s] g", "hs : s =ᶠ[l] t"], "goal": "s.mulIndicator f =ᶠ[l] t.mulIndicator g"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝¹ : DecidableEq α", "s✝ t✝ u : Multiset α", "a b : α", "inst✝ : DecidableEq β", "f : α → β", "finj : Injective f", "s t : Multiset α", "l₁ l₂ : List α"], "goal": "List.map f (l₁.diff l₂ ++ l₂) = (List.map f l₁).diff (List.map f l₂) ++ List.map f l₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "p : ℝ≥0∞", "f✝ : α → β", "hp_one : 1 ≤ p", "hp_top : p ≠ ⊤", "n : ℕ", "h : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), Memℒp (f i) p μ) → UnifIntegrable f p μ", "f : Fin (n + 1) → α → β", "hfLp : ∀ (i : Fin (n + 1)), Memℒp (f i) p μ", "ε : ℝ", "hε : 0 < ε", "g : Fin n → α → β := fun k => f ↑↑k", "hgLp : ∀ (i : Fin n), Memℒp (g i) p μ", "δ₁ : ℝ", "hδ₁pos : 0 < δ₁", "hδ₁ : ∀ (i : Fin n) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ₁ → eLpNorm (s.indicator (g i)) p μ ≤ ENNReal.ofReal ε"], "goal": "∃ δ, ∃ (_ : 0 < δ), ∀ (i : Fin (n + 1)) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε"}}
+{"state": {"context": ["m : ℤ", "hm : 0 < m", "p : ℕ", "hp : Fact (Nat.Prime p)", "Hpv : (multiplicity (↑p) m).get ⋯ % 2 = 1"], "goal": "1 ≠ 0"}}
+{"state": {"context": ["ι : Sort u_1", "f : ι ��� ℝ≥0", "a : ℝ≥0"], "goal": "iInf f * a = ⨅ i, f i * a"}}
+{"state": {"context": ["g : SL(2, ℤ)", "z : ℍ", "h : z ∈ 𝒟ᵒ"], "goal": "3 < 4 * z.im ^ 2"}}
+{"state": {"context": ["s : ℝ", "hs : -1 ≤ s", "p : ℝ", "hp : 1 ≤ p"], "goal": "1 + p * s ≤ (1 + s) ^ p"}}
+{"state": {"context": ["a x✝ z x : ℂ", "hi : x.im = 0"], "goal": "(-x).arg = x.arg - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "n : ℕ∞", "f' : E → FormalMultilinearSeries 𝕜 E F", "hf : HasFTaylorSeriesUpTo n f f'"], "goal": "ContDiff 𝕜 n f"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "n : ℤ", "p q : ℤ", "hpq : p + n = q"], "goal": "CochainComplex.HomComplex.Cochain.v 0 p q hpq = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → ℚ → ℝ", "hf : IsRatCondKernelCDFAux f κ ν", "inst✝ : IsFiniteKernel ν", "a : α", "seq : ℕ → ℚ", "hseq : Monotone seq", "hseq_tendsto : Tendsto seq atTop atTop"], "goal": "∀ᵐ (c : β) ∂ν a, Tendsto (fun m => f (a, c) (seq m)) atTop (𝓝 1)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "s : Set α", "hs : Dense s"], "goal": "DenseEmbedding Subtype.val"}}
+{"state": {"context": ["x y : ℂ", "hj : ∀ (j : ℕ), ∑ m ∈ range j, (x + y) ^ m / ↑m.factorial = ∑ i ∈ range j, ∑ k ∈ range (i + 1), x ^ k / ↑k.factorial * (y ^ (i - k) / ↑(i - k).factorial)"], "goal": "cexp (x + y) = cexp x * cexp y"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "f : S → R[X]", "s : Finset S", "h : {i | i ∈ s ∧ f i ≠ 0}.Pairwise (Ne on Polynomial.natDegree ∘ f)"], "goal": "(s.sum f).natDegree = s.sup fun i => (f i).natDegree"}}
+{"state": {"context": ["M : Type u_2", "AddMonoid M", "a : M"], "goal": "1 • a = a"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "IsLocalization M S", "z : S", "h : M ≤ Submonoid.comap (RingHom.id R) M := ⋯"], "goal": "(IsLocalization.map S (RingHom.id R) h) z = z"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j : ι", "hj : c.next i = j", "h : K.d i j = 0", "K.HasHomology i"], "goal": "CategoryTheory.IsIso (K.homologyι i)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝³ : Category.{u_4, u_1} C", "inst✝² : Category.{u_5, u_2} D", "inst✝¹ : Category.{?u.101120, u_3} E", "inst✝ : IsIdempotentComplete D", "F G : Karoubi C ⥤ D", "τ : toKaroubi C ⋙ F ⟶ toKaroubi C ⋙ G", "P : Karoubi C"], "goal": "(((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage τ).app { X := P.X, p := 𝟙 P.X, idem := ⋯ } = (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).map (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage τ)).app P.X"}}
+{"state": {"context": ["J : Type v", "inst✝ : SmallCategory J", "X✝ Y✝ Z✝ X Y Z : TopCat", "f : X ⟶ Z", "g : Y ⟶ Z", "homeo : ↑(pullback f g) ≃ₜ ↑(of { p // f p.1 = g p.2 }) := homeoOfIso (pullbackIsoProdSubtype f g)"], "goal": "(pullback f g).str = induced (⇑(pullback.fst f g)) X.str ⊓ induced (⇑(pullback.snd f g)) Y.str"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "D : Type w", "CategoryTheory.Category.{max v u, w} D", "CategoryTheory.ConcreteCategory D", "CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget D)", "X : C", "P : CategoryTheory.Functor Cᵒᵖ D", "S : J.Cover X", "CategoryTheory.Limits.HasMultiequalizer (S.index P)", "x : (CategoryTheory.forget D).obj (CategoryTheory.Limits.multiequalizer (S.index P))", "I : S.Arrow"], "goal": "↑((CategoryTheory.Meq.equiv P S) x) I = (CategoryTheory.Limits.Multiequalizer.ι (S.index P) I) x"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁶ : Field F", "inst✝⁵ : Field E", "inst✝⁴ : Algebra F E", "K : Type w", "inst✝³ : Field K", "inst✝² : Algebra F K", "L : IntermediateField F E", "inst✝¹ : IsPurelyInseparable F ↥L", "inst✝ : Algebra.IsSeparable F ↥L", "x : E", "hx : x ∈ L", "y : F", "hy : (algebraMap F ↥L) y = ⟨x, hx⟩"], "goal": "x ∈ ⊥"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "S : Type u_8", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "AddCommMonoid S", "Module R M", "Module R N", "Module R P", "Module R Q", "Module R S", "f : M →ₗ[R] P", "g : N →ₗ[R] Q", "f' : S →ₗ[R] M"], "goal": "TensorProduct.map f g ���ₗ LinearMap.rTensor N f' = TensorProduct.map (f ∘ₗ f') g"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "φ : MvPolynomial σ R", "A : Type u_3", "CommSemiring A", "Algebra R A"], "goal": "(MvPolynomial.coeToMvPowerSeries.algHom A) φ = (MvPowerSeries.map σ (algebraMap R A)) ↑φ"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : α →ₘ[μ] E", "hf : MeasureTheory.eLpNorm (↑f) p μ < ⊤"], "goal": "↑⟨f, hf⟩ = f"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : AddCommMonoid M", "inst✝⁹ : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝⁸ : CommRing R₁", "inst✝⁷ : AddCommGroup M₁", "inst✝⁶ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝⁵ : Field K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M₂' : Type u_7", "inst✝² : AddCommMonoid M₂'", "inst✝¹ : Module R M₂'", "inst✝ : FiniteDimensional K V", "B : BilinForm K V", "W : Submodule K V", "b₁ : B.IsRefl", "b₂ : (B.restrict W).Nondegenerate", "b₃ : B.Nondegenerate", "h : W = ⊤"], "goal": "B.orthogonal W = ⊥"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : CompleteSpace F", "T T' T'' : Set α → E →L[ℝ] F", "C C' C'' : ℝ", "f✝ g : α → E", "hT : DominatedFinMeasAdditive μ T C", "fs : ℕ → α → E", "f : α → E", "bound : α → ℝ", "fs_measurable : ∀ (n : ℕ), AEStronglyMeasurable (fs n) μ", "bound_integrable : Integrable bound μ", "h_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖fs n a‖ ≤ bound a", "h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => fs n a) atTop (𝓝 (f a))", "f_measurable : AEStronglyMeasurable f μ", "fs_int : ∀ (n : ℕ), Integrable (fs n) μ"], "goal": "Tendsto (fun n => setToFun μ T hT (fs n)) atTop (𝓝 (setToFun μ T hT f))"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : Semiring R", "S : Type v", "inst✝ : Semiring S", "f : R →+* S", "s : S", "n : ℕ", "hzero : ¬s = 0", "q : R[X]", "hq : (map f q).coeff n = ((monomial n) s).coeff n", "hcoeff : f (q.coeff n) = s", "hqzero : q.coeff n ≠ 0"], "goal": "(C (q.coeff n) * X ^ n).degree = ((monomial n) s).degree"}}
+{"state": {"context": ["V : Type u_1", "R : Type u_2", "inst✝³ : Fintype V", "inst✝² : DecidableEq V", "G : SimpleGraph V", "inst✝¹ : DecidableRel G.Adj", "inst✝ : DecidableEq G.ConnectedComponent", "g : G.ConnectedComponent → ℝ", "h0 : ↑(∑ i : G.ConnectedComponent, g i • G.lapMatrix_ker_basis_aux i) = ↑0"], "goal": "∀ (i : G.ConnectedComponent), g i = 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedCommMonoidWithZero α", "a b : α", "h : b < a"], "goal": "a ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "InnerProductSpaceable E", "n : ℕ", "x y : E"], "goal": "inner_ 𝕜 (↑n • x) y = ↑n * inner_ 𝕜 x y"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LocallyFiniteOrder α", "a a₁ a₂ b b₁ b₂ c x : α"], "goal": "a ∈ [[b, c]] ↔ b ≤ a ∧ a ≤ c ∨ c ≤ a ∧ a ≤ b"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "D : LieDerivation R L M"], "goal": "⇑(-D) = -⇑D"}}
+{"state": {"context": ["E : Type u", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "F : Type v", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℂ F", "f : E → F", "hf : Differentiable ℂ f", "hb : Bornology.IsBounded (range f)", "z w : E", "g : ℂ → F := f ∘ fun t => t • (w - z) + z"], "goal": "Bornology.IsBounded (range g)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedCommGroup α", "a b : α", "hab : a ≤ b", "ha : a < 1", "hb : b < 1"], "goal": "mabs (a * b) = mabs a * mabs b ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1"}}
+{"state": {"context": ["m : ℕ", "hm : m ≠ 1", "hm' : m > 0", "n : ℕ", "hn : 0 < n", "this : image (fun x => x * m + 1) (Icc 0 ((n - 1) / m)) ⊆ filter (fun x => x % m = 1) (Ioc 0 n)", "a b : ℕ"], "goal": "(fun x => x * m + 1) a = (fun x => x * m + 1) b → a = b"}}
+{"state": {"context": ["V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P", "𝕜 : Type u_6", "NormedField 𝕜", "NormedSpace 𝕜 V", "p₁ p₂ : P"], "goal": "LipschitzWith (nndist p₁ p₂) ⇑(AffineMap.lineMap p₁ p₂)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : ProbabilityTheory.Kernel α β", "ProbabilityTheory.IsSFiniteKernel κ", "η : ProbabilityTheory.Kernel (α × β) γ", "ProbabilityTheory.IsSFiniteKernel η", "a : α", "h : ((κ.compProd η) a) s = 0"], "goal": "(fun b => (η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "⟪x, o.rightAngleRotationAux₁ y⟫_ℝ = -(o.areaForm x) y"}}
+{"state": {"context": ["b x y : ℝ", "b_pos : 0 < b", "b_lt_one : b < 1", "h : 0 < x"], "goal": "logb b x < 0 ↔ 1 < x"}}
+{"state": {"context": ["R : Type u", "M : Type v", "N : Type w", "inst✝⁹ : Ring R", "inst✝⁸ : TopologicalSpace R", "inst✝⁷ : TopologicalSpace M", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R M", "inst✝³ : ContinuousSMul R M", "inst✝² : Module R N", "inst✝¹ : ContinuousAdd M", "inst✝ : IsSimpleModule R N"], "goal": "IsClosed ↑(ker 0) ∨ Dense ↑(ker 0)"}}
+{"state": {"context": ["m n : ZNum"], "goal": "↑(m * n) = ↑m * ↑n"}}
+{"state": {"context": ["R : Type u_2", "L : Type u_3", "M : Type u_4", "CommRing R", "LieRing L", "LieAlgebra R L", "LieAlgebra.IsNilpotent R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "M₂ : Type u_5", "AddCommGroup M₂", "Module R M₂", "LieRingModule L M₂", "LieModule R L M₂", "χ : L → R", "f : M →ₗ⁅R,L⁆ M₂", "hf : Function.Injective ⇑f"], "goal": "LieSubmodule.comap f (LieModule.weightSpace M₂ χ) = LieModule.weightSpace M χ"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y f' : ℝ", "hfc : ConvexOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hf' : HasDerivWithinAt f f' S y"], "goal": "∃ l ∈ Iio y, Ico l y ⊆ S"}}
+{"state": {"context": ["R : Type u_3", "LinearOrderedSemiring R", "a b : R", "ha : 0 ≤ a", "hb : 0 ≤ b"], "goal": "a ^ 2 = b ^ 2 ↔ a = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "𝕜 : Type u_3", "E : Type u_4", "F : Type u_5", "inst✝⁵ : RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : CompleteSpace F", "u : α → ℝ", "inst✝ : NormedSpace 𝕜 F", "f : α → E → F", "f' : α → E → E →L[𝕜] F", "g g' : α → 𝕜 → F", "v : ℕ → α → ℝ", "s : Set E", "t : Set 𝕜", "x₀ x : E", "y₀ y : 𝕜", "N : ℕ∞", "hf : ∀ (i : α), ContDiff 𝕜 N (f i)", "hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)", "h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i"], "goal": "ContDiff 𝕜 N fun x => ∑' (i : α), f i x"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {cut : α → Ordering} {t : Batteries.RBNode α} {path : Batteries.RBNode.Path α}\n  {t' : Batteries.RBNode α} {path' : Batteries.RBNode.Path α} {c₀ : Batteries.RBColor} {n₀ : Nat}\n  {c : Batteries.RBColor} {n : Nat},\n  t.Balanced c n →\n    Batteries.RBNode.Path.Balanced c₀ n₀ path c n →\n      Batteries.RBNode.zoom cut t path = (t', path') →\n        ∃ c n, t'.Balanced c n ∧ Batteries.RBNode.Path.Balanced c₀ n₀ path' c n"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "α : Type u_1", "Fintype α", "hG : G.Colorable (Fintype.card α)"], "goal": "G.chromaticNumber = ↑(Fintype.card α) ↔ ∀ (C : G.Coloring α), Function.Surjective ⇑C"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : StrictConvexSpace ℝ V", "inst✝¹ : PseudoMetricSpace P", "inst✝ : NormedAddTorsor V P", "p p₁ p₂ p₃ : P", "h : Sbtw ℝ p₁ p₂ p₃"], "goal": "dist p₂ p < max (dist p₁ p) (dist p₃ p)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "ConditionallyCompleteLinearOrder β", "TopologicalSpace β", "OrderClosedTopology β", "f : α → β", "s : Set α", "Cf : ContinuousAt f (sInf s)", "Mf : Monotone f", "ne : s.Nonempty", "H : BddBelow s"], "goal": "f (sInf s) = sInf (f '' s)"}}
+{"state": {"context": ["α : Type u_1", "f : List α → List α", "n : Nat", "l : List α", "h : n ≤ l.length"], "goal": "∃ l₁ l₂, l = l₁ ++ l₂ ∧ l₁.length = n ∧ List.modifyNthTail f n l = l₁ ++ f l₂"}}
+{"state": {"context": ["R : Type u_1", "NonAssocSemiring R", "Nontrivial R"], "goal": "ringChar R ≠ 1"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "w x y z : R"], "goal": "span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z}"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {x : ��} {p : α → Bool} {as : Array α}, x ∈ Array.filter p as 0 ↔ x ∈ as ∧ p x = true"}}
+{"state": {"context": ["M : Type u", "Monoid M", "α : Type v", "MulAction M α", "m : M", "a : α"], "goal": "m • MulAction.orbit M a ⊆ MulAction.orbit M a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type uG", "NormedAddCommGroup G", "NormedSpace 𝕜 G"], "goal": "ContDiff 𝕜 ⊤ ⇑(Equiv.prodAssoc E F G).symm"}}
+{"state": {"context": ["M : Type u", "inst✝⁵ : Monoid M", "G : Type u", "inst✝⁴ : Group G", "F : Type v", "inst✝³ : Field F", "inst✝² : MulSemiringAction M F", "inst✝¹ : MulSemiringAction G F", "m : M", "inst✝ : Fintype G", "x : F", "f g : Polynomial ↥(subfield G F)", "hf : f.Monic", "hg : g.Monic", "hfg : f * g = minpoly G F x", "hf2 : f ∣ minpoly G F x", "hg2 : g ∣ minpoly G F x", "this : Polynomial.eval₂ (subfield G F).subtype x f = 0 ∨ Polynomial.eval₂ (subfield G F).subtype x g = 0"], "goal": "f = 1 ∨ g = 1"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "f : R →+* S", "x : S", "n : ℕ", "r : R"], "goal": "Polynomial.eval₂ f x ((Polynomial.monomial n) r) = f r * x ^ n"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "a : α"], "goal": "↑(Sym.replicate n a) = Multiset.replicate n a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝³ : ConditionallyCompleteLinearOrder β", "x : β", "f : α → β", "inst✝² : TopologicalSpace β", "inst✝¹ : FirstCountableTopology β", "inst✝ : OrderTopology β", "hf : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) (ae μ) f) _auto✝"], "goal": "μ {y | ¬f y ≤ essSup f μ} = 0"}}
+{"state": {"context": ["R : Type u_1", "F : Type u", "K : Type v", "L : Type w", "inst✝³ : CommRing R", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Field F", "i : K →+* L", "p : K[X]", "m : p.Monic", "hsplit : Splits (RingHom.id K) p"], "goal": "(Multiset.map (fun a => X - C a) p.roots).prod = C p.leadingCoeff * (Multiset.map (fun a => X - C a) p.roots).prod"}}
+{"state": {"context": ["p✝ x : ℝ", "hp✝ : p✝ ≠ 0", "p : ℝ", "hp : 0 < p", "hx : |x| ≤ p / 2", "hx' : x ≠ p / 2"], "goal": "‖↑x‖ = |x|"}}
+{"state": {"context": [], "goal": "Complex.log 1 = 0"}}
+{"state": {"context": ["α : Type u_1", "x : Quiver.SingleObj α", "p : Quiver.Path (Quiver.SingleObj.star α) x"], "goal": "Quiver.SingleObj.listToPath (Quiver.SingleObj.pathToList p) = Quiver.Path.cast ⋯ ⋯ p"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s s₁ s₂ t : Set X", "p p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X"], "goal": "closure s ⊆ s ↔ ∀ (x : X), (∃ᶠ (y : X) in 𝓝 x, y ∈ s) → x ∈ s"}}
+{"state": {"context": ["F : Type u_1", "E : Type u_2", "inst✝⁵ : CommRing F", "inst✝⁴ : StrongRankCondition F", "inst✝³ : Ring E", "inst✝² : Algebra F E", "S : Subalgebra F E", "inst✝¹ : Nontrivial E", "inst✝ : Module.Free F E"], "goal": "⊥ = ⊤ ↔ finrank F E = 1"}}
+{"state": {"context": ["α : Type v", "M : Type u", "Monoid M", "MulAction M α", "m : M", "a : α", "n : ℕ", "n_pos : 0 < n", "fixed : m ^ n • a = a"], "goal": "0 < MulAction.period m a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LT α", "inst✝ : LT β", "s t : Set α", "l : List α", "a : α"], "goal": "(∃ l ∈ s.subchain, l.length = 1) ↔ s.Nonempty"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.MonoidalPreadditive C", "CategoryTheory.Limits.HasFiniteBiproducts C", "J : Type", "Fintype J", "X Y : C", "f : J → C", "g h : X ⟶ CategoryTheory.MonoidalCategory.tensorObj Y (⨁ f)", "w :\n  ∀ (j : J),\n    CategoryTheory.CategoryStruct.comp g\n        (CategoryTheory.MonoidalCategory.whiskerLeft Y (CategoryTheory.Limits.biproduct.π f j)) =\n      CategoryTheory.CategoryStruct.comp h\n        (CategoryTheory.MonoidalCategory.whiskerLeft Y (CategoryTheory.Limits.biproduct.π f j))"], "goal": "g = h"}}
+{"state": {"context": ["ι : Type u_1", "B : Type u_2", "F : Type u_3", "X : Type u_4", "inst✝³ : TopologicalSpace X", "E : B → Type u_5", "inst✝² : TopologicalSpace B", "inst✝¹ : TopologicalSpace F", "inst✝ : (x : B) → TopologicalSpace (E x)", "a : FiberPrebundle F E", "e✝ e e' : Pretrivialization F TotalSpace.proj", "x✝ : e' ∈ a.pretrivializationAtlas"], "goal": "IsOpen e.source"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ≥0∞", "h : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0"], "goal": "∃ δ > 0, ∀ (s : Set α), μ s < δ → ∫⁻ (x : α) in s, f x ∂μ < ε"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "p q : R[X]"], "goal": "(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n x => x ^ 2"}}
+{"state": {"context": ["b c : Ordinal.{u}", "h : b ≤ c", "a : Ordinal.{u}"], "goal": "b ♯ a ≤ c ♯ a"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁷ : Category.{u_3, u_1} C", "inst✝⁶ : Category.{u_4, u_2} D", "inst✝⁵ : HasZeroMorphisms C", "inst✝⁴ : HasZeroMorphisms D", "S : ShortComplex C", "h₁✝ : S.LeftHomologyData", "h₂✝ : S.RightHomologyData", "F✝ : C ⥤ D", "inst✝³ : F✝.PreservesZeroMorphisms", "hl : S.LeftHomologyData", "hr : S.RightHomologyData", "S₁ S₂ : ShortComplex C", "φ : S₁ ⟶ S₂", "hl₁ : S₁.LeftHomologyData", "hr₁ : S₁.RightHomologyData", "hl₂ : S₂.LeftHomologyData", "hr₂ : S₂.RightHomologyData", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "F : C ⥤ D", "inst✝² : F.PreservesZeroMorphisms", "inst✝¹ : S.HasRightHomology", "inst✝ : F.PreservesRightHomologyOf S"], "goal": "rightHomologyMap' (𝟙 (S.map F)) (S.map F).rightHomologyData (S.rightHomologyData.map F) = rightHomologyMap' (𝟙 (S.map F)) (S.map F).rightHomologyData (hr.map F) ≫ F.map (rightHomologyMap' (𝟙 S) hr S.rightHomologyData)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "S : Type u_2", "Semiring S", "R₂ : Type u_3", "Semiring R₂", "S₂ : Type u_4", "Semiring S₂", "M : Type u_5", "N : Type u_6", "P : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "Module R M", "Module S N", "Module R₂ P", "Module S₂ P", "SMulCommClass S₂ R₂ P", "ρ₁₂ : R →+* R₂", "σ₁₂ : S →+* S₂", "f : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P", "m : M", "n : N"], "goal": "(f.flip n) m = (f m) n"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : DecidableEq L", "inst✝⁶ : Algebra R K", "inst✝⁵ : IsFractionRing R K", "inst✝⁴ : Algebra K L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra R L", "inst✝¹ : IsScalarTower R K L", "inst✝ : IsDomain R", "P : ClassGroup R → Prop", "h : ∀ (I : (FractionalIdeal R⁰ K)ˣ), P (mk I)", "x : ClassGroup R", "I : (FractionalIdeal R⁰ (FractionRing R))ˣ"], "goal": "P ↑I"}}
+{"state": {"context": ["V W : SemiNormedGrp", "CompleteSpace ↑W", "T0Space ↑W", "f : V ⟶ W"], "goal": "SemiNormedGrp.completion.incl ≫ SemiNormedGrp.completion.lift f = f"}}
+{"state": {"context": ["K : Type u_1", "NormedLinearOrderedField K", "HasSolidNorm K", "FloorRing K", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace K E", "FiniteDimensional K E", "ProperSpace E", "L : AddSubgroup E", "DiscreteTopology ↥L", "IsZlattice K L"], "goal": "Module.Free ℤ ↥L"}}
+{"state": {"context": ["X : CategoryTheory.RelCat"], "goal": "CategoryTheory.CategoryStruct.id X = fun x x_1 => x = x_1"}}
+{"state": {"context": ["ξ : ℝ", "n : ℕ"], "goal": "ξ.convergent (n + 1) = ↑⌊ξ⌋ + ((Int.fract ξ)⁻¹.convergent n)⁻¹"}}
+{"state": {"context": ["α✝ : Sort u", "β✝ : Sort v", "γ : Sort w", "α : Type u_1", "β : Type u_2", "f : α → β", "s : Set ↑(range f)", "y : ↑(range f)", "m : y ∈ s"], "goal": "⟨f ↑⟨rangeSplitting f y, ⋯⟩, ⋯⟩ ∈ s"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "D : Type u'", "inst✝² : Category.{v', u'} D", "inst✝¹ : HasZeroObject C", "inst✝ : HasZeroMorphisms C", "X Y : C", "f : X ⟶ Y", "i : X ≅ 0", "Z : C", "g h : Z ⟶ X", "x✝ : g ≫ f = h ≫ f"], "goal": "g = h"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "X : C"], "goal": "(α_ X (𝟙_ C) (𝟙_ C)).hom ≫ X ◁ (λ_ (𝟙_ C)).hom = (ρ_ X).hom ▷ 𝟙_ C"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "TopologicalSpace β", "s : Set β", "hs : IsConnected s", "f : α → β", "hinj : Function.Injective f", "hf : IsOpenMap f", "hsf : s ⊆ Set.range f"], "goal": "IsConnected (f ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "Preorder α", "DecidableEq α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (Function.swap HMul.hMul) LE.le", "LocallyFiniteOrderTop α", "a b : α"], "goal": "Finset.Ici a * Finset.Ici b ⊆ Finset.Ici (a * b)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_3", "B : Type u_4", "CommSemiring R", "Semiring A", "Semiring B", "Algebra R A", "Algebra R B", "f : A ≃ₐ[R] B"], "goal": "∀ (a : Aᵐᵒᵖ), (AlgEquiv.op f) a = MulOpposite.op (f (MulOpposite.unop a))"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "r✝ : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "r : α → α → Prop", "inst✝ : IsWellOrder α r", "o o' : Ordinal.{u}", "ho : o < type r", "ho' : o' < type r"], "goal": "¬r (enum r o' ho') (enum r o ho) ↔ o ≤ o'"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "a₁ a₂ : α", "b : α → β", "h : a₁ = a₂"], "goal": "(let x := a₁;\n  b x) =\n  let x := a₂;\n  b x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F G : CategoryTheory.Functor Cᵒᵖ D", "α : F ⟶ G", "X : C"], "goal": "(CategoryTheory.NatTrans.rightOp α).app X = (α.app (Opposite.op X)).op"}}
+{"state": {"context": ["β : Type u_3", "γ : Type u_4", "ConditionallyCompleteLinearOrderedField β", "ConditionallyCompleteLinearOrderedField γ"], "goal": "(LinearOrderedField.inducedOrderRingIso β γ).symm = LinearOrderedField.inducedOrderRingIso γ β"}}
+{"state": {"context": [], "goal": "UniqueDiffOn ℝ (Set.Icc 0 1)"}}
+{"state": {"context": ["N : ℕ", "a : Fin 2 → ZMod N", "v : Fin 2 → ℤ", "hv : v ∈ gammaSet N a", "γ : SL(2, ℤ)"], "goal": "v ᵥ* ↑γ ∈ gammaSet N (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ))"}}
+{"state": {"context": ["α : Type u_1", "One α", "a : α"], "goal": "AddOpposite.op a = 1 ↔ a = 1"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "inst✝ : AddGroup G", "T : AddTorsor G P", "p : P"], "goal": "{p} -ᵥ {p} = {0}"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C", "S : CategoryTheory.ShortComplex.SnakeInput C"], "goal": "CategoryTheory.CategoryStruct.comp S.L₀.g S.δ = 0"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H₁ H₂ : Subgroup G"], "goal": "⁅H₁, H₂⁆ = ⁅H₂, H₁⁆"}}
+{"state": {"context": ["n p : ℕ", "pp : Prime p", "i : ℕ"], "goal": "p.Coprime i ∨ ¬p.Coprime i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ✝ : Measure α", "inst✝ : NormedAddCommGroup β", "p✝ : ℝ≥0∞", "μ : Measure α", "p : ℝ≥0∞", "hp' : p ≠ ⊤", "s : Set α", "hs : MeasurableSet s", "f g : α → β", "c : ℝ", "hc : 0 ≤ c", "hf : ∀ x ∈ s, dist (f x) (g x) ≤ c", "hp : ¬p = 0", "this : ∀ (x : α), ‖s.indicator (f - g) x‖ ≤ ‖s.indicator (fun x => c) x‖"], "goal": "eLpNorm (s.indicator (f - g)) p μ ≤ ENNReal.ofReal c * μ s ^ (1 / p.toReal)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α", "h : a ≤ b"], "goal": "Nonempty ↑(Set.Icc a b)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : Semiring R", "inst✝ : Semiring S", "I✝ I' : Ideal R", "J J' : Ideal S", "I : Ideal (R × S)"], "goal": "I = (map (RingHom.fst R S) I).prod (map (RingHom.snd R S) I)"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "LinearOrder α", "a : α"], "goal": "Set.Unbounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a ≤ b}) ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s"}}
+{"state": {"context": ["α : Type u", "i : Nat", "a : Array α"], "goal": "a.extract i i = #[]"}}
+{"state": {"context": ["M : Type u_1", "inst✝¹ : CancelCommMonoidWithZero M", "inst✝ : Unique Mˣ", "a : M", "l₁ : List M", "b : M", "l₂ : List M", "h : (a :: l₁).prod = (b :: l₂).prod", "hl₁ : ∀ p ∈ a :: l₁, Prime p", "hl₂ : ∀ p ∈ b :: l₂, Prime p"], "goal": "a :: l₁ ~ b :: l₂"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x✝ y : α", "s t u : Set α", "Φ : α → β", "x : α", "xs : x ∈ s", "this : infEdist x ∅ ≤ hausdorffEdist s ∅"], "goal": "hausdorffEdist s ∅ = ⊤"}}
+{"state": {"context": ["J : Type u", "inst✝ : Category.{?u.10050, u} J", "F : J ⥤ Type v", "i j k : J", "s : Set (F.obj i)", "X✝ Y✝ : J", "g : X✝ ⟶ Y✝", "x : F.obj X✝", "h : ∀ (i_1 : X✝ ⟶ i), x ∈ F.map i_1 ⁻¹' s", "f : Y✝ ⟶ i"], "goal": "F.map f (F.map g x) ∈ s ↔ x ∈ F.map (g ≫ f) ⁻¹' s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : Finite α", "this : Fintype α"], "goal": "Nat.card (α → β) = Nat.card β ^ Nat.card α"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "X : Type u"], "goal": "(ρ_ ((free R).obj X)).hom = (𝟙 ((free R).obj X) ⊗ ε R) ≫ (μ R X (𝟙_ (Type u))).hom ≫ map (free R).obj (ρ_ X).hom"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : ClosedIicTopology α", "inst✝ : TopologicalSpace β", "a b c : α", "f : α → β"], "goal": "IsOpen (Ioi a)"}}
+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "⋃ (_ : s.Nonempty), s = s"}}
+{"state": {"context": ["α : Type u_1", "s t : Finset α"], "goal": "s ⊆ t ↔ s.val ⊆ t.val"}}
+{"state": {"context": ["n : ℕ", "h : n ≠ 0"], "goal": "Finset.filter (fun x => x ∣ n) (Finset.range n.succ) = n.divisors"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Lattice α", "inst✝ : Lattice β", "a✝ a₁ a₂ b✝ b₁ b₂ c x a b : α"], "goal": "[[a, b]] = [[b, a]]"}}
+{"state": {"context": ["R : Type u", "M : Type v", "M₂ : Type w", "M₃ : Type y", "CommSemiring R", "AddCommMonoid M", "AddCommMonoid M₂", "AddCommMonoid M₃", "Module R M", "Module R M₂", "Module R M₃"], "goal": "∀ (x : M), LinearMap.vecEmpty₂ x = LinearMap.vecEmpty"}}
+{"state": {"context": ["p : ℕ", "hp : Nat.blt 1 p = true", "h : sMod' (2 ^ p - 1) (p - 2) = 0"], "goal": "LucasLehmerTest p"}}
+{"state": {"context": ["X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "inst✝¹ : IsPreimmersion g", "inst✝ : IsPreimmersion (f ≫ g)", "x : ↑↑X.toPresheafedSpace", "h : Function.Surjective ⇑(Scheme.Hom.stalkMap (f ≫ g) x)"], "goal": "Function.Surjective ⇑(Scheme.Hom.stalkMap f x)"}}
+{"state": {"context": ["f : Code → ℕ →. ℕ", "hf : Partrec₂ f", "cf : Code", "ef : cf.eval = fun n => (↑(decode n)).bind fun a => Part.map encode ((fun p => f p.1 p.2) a)", "c : Code", "e : (cf.curry (encode c)).eval = c.eval", "n : ℕ"], "goal": "c.eval n = f c n"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "β : Type u_2", "v : Mathlib.Vector α n", "f : α → β", "i : Fin n"], "goal": "(Mathlib.Vector.map f v).get i = f (v.get i)"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2"], "goal": "Filter.Tendsto (fun i => ‖Polynomial.eval (↑(newton_cau_seq hnorm) i) F‖) Filter.atTop (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁵ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝⁴ : NormedRing 𝕜", "inst✝³ : (i : α) → Module 𝕜 (E i)", "inst✝² : ∀ (i : α), BoundedSMul 𝕜 (E i)", "inst✝¹ : DecidableEq α", "inst✝ : Fact (1 ≤ p)", "hp : p ≠ ⊤", "f : ↥(lp E p)", "hp₀ : 0 < p", "hp' : 0 < p.toReal", "this : ∀ ε > 0, ∀ᶠ (x : Finset α) in Filter.atTop, dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε", "ε : ℝ", "hε : ε > 0", "s : Finset α", "hs : dist (∑ b ∈ s, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε ^ p.toReal"], "goal": "dist (∑ b ∈ s, lp.single p b (↑f b)) f ^ p.toReal < ε ^ p.toReal"}}
+{"state": {"context": ["G : Type u_1", "TopologicalSpace G", "InvolutiveInv G", "ContinuousInv G"], "goal": "⇑(Homeomorph.inv G) = Inv.inv"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "κ : ι → Sort u_7", "s : (i : ι) → κ i → Set α", "t : Set α", "i : ι", "j : κ i", "h : s i j ⊆ t"], "goal": "⋂ i, ⋂ j, s i j ⊆ t"}}
+{"state": {"context": ["m : Type u", "n : Type v", "α : Type w", "inst✝⁵ : DecidableEq n", "inst✝⁴ : Fintype n", "inst✝³ : DecidableEq m", "inst✝² : Fintype m", "inst✝¹ : CommRing α", "inst✝ : Nontrivial n", "i j j' : n", "hj' : j' ≠ j"], "goal": "∀ (i_1 : n), 0ᵀ.updateColumn j (Pi.single i 1) i_1 j' = 0"}}
+{"state": {"context": ["α✝ : Type u_1", "l₁ l₂ l : List α✝"], "goal": "(l₁ ++ l₂).Disjoint l ↔ l₁.Disjoint l ∧ l₂.Disjoint l"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_3", "RCLike 𝕜", "E : Type u_4", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "Fintype ι", "G : Type u_8", "NormedAddCommGroup G", "InnerProductSpace 𝕜 G", "b : OrthonormalBasis ι 𝕜 E", "L : E ≃ₗᵢ[𝕜] G", "i : ι"], "goal": "(b.map L) i = L (b i)"}}
+{"state": {"context": ["c : Char", "cs t : List Char", "i : Nat"], "goal": "go₂ (c :: cs ++ t) { byteIdx := i } { byteIdx := utf8Len (c :: cs) + i } = c :: cs"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "inst✝ : SFinite ν"], "goal": "∀ (f : α × β → ℝ≥0∞), Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CommGroup α", "TopologicalSpace α", "TopologicalGroup α", "f : β → α"], "goal": "(Multipliable fun b => (f b)⁻¹) ↔ Multipliable f"}}
+{"state": {"context": ["d : ℤ", "z : ℤ√d"], "goal": "(-z).re = -z.re"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "M : Type u_1", "inst✝¹ : Category.{u_2, u_1} M", "inst✝ : MonoidalCategory M", "F : MonoidalFunctor M (C ⥤ C)", "n : M", "X : C"], "goal": "F.ε.app ((F.obj n).obj X) = 𝟙 ((𝟙_ (C ⥤ C)).obj ((F.obj n).obj X)) ≫ (F.map (ρ_ n).inv).app X ≫ (F.μIso n (𝟙_ M)).inv.app X"}}
+{"state": {"context": ["Fq : Type u_1", "inst✝¹ : Fintype Fq", "inst✝ : Ring Fq", "d m : ℕ", "hm : Fintype.card Fq ^ d ≤ m", "b : Fq[X]", "A : Fin m.succ → Fq[X]", "hA : ∀ (i : Fin m.succ), (A i).degree < b.degree", "hb : b ≠ 0", "f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff (b.natDegree - ↑j.succ)", "this : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ)"], "goal": "∃ i₀ i₁, i₀ ≠ i₁ ∧ (A i₁ - A i₀).degree < ↑(b.natDegree - d)"}}
+{"state": {"context": ["ι✝ : Type u_1", "R✝ : Type u_2", "M✝ : Type u_3", "N : Type u_4", "inst✝⁹ : Finite ι✝", "inst✝⁸ : CommSemiring R✝", "inst✝⁷ : AddCommMonoid M✝", "inst✝⁶ : AddCommMonoid N", "inst✝⁵ : Module R✝ M✝", "inst✝⁴ : Module R✝ N", "ι : Type u_5", "R : Type u_6", "M : Type u_7", "inst✝³ : Fintype ι", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "v : ι → M", "w : ι → R"], "goal": "∑ x : ι, ((Pi.basisFun R ι).repr w) x • v x = ∑ i : ι, w i • v i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β", "c : α", "AddCommMonoid α", "DivisionSemiring γ", "Module γ α", "h : Function.Periodic f c", "a : γ"], "goal": "Function.Periodic (fun x => f (a • x)) (a⁻¹ • c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α"], "goal": "(∃ c, IsBigOWith c l f fun x => -g' x) ↔ ∃ c, IsBigOWith c l f g'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Denumerable α", "Denumerable β", "n : ℕ"], "goal": "Denumerable.ofNat (α × β) n = (Denumerable.ofNat α (Nat.unpair n).1, Denumerable.ofNat β (Nat.unpair n).2)"}}
+{"state": {"context": ["M : Type u_11", "N : Type u_12", "AddZeroClass N", "VAdd M N", "VAddCommClass M N N", "x : M", "y : N"], "goal": "y + (x +ᵥ 0) = x +ᵥ y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝⁴ : RCLike 𝕜", "inst✝³ : MeasurableSpace α", "μ : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : NormedAddCommGroup F", "f f' g : ↥(Lp E 2 μ)", "x : α", "hx : ↑↑(f + f') x = (↑↑f + ↑↑f') x"], "goal": "(fun a => ⟪↑↑(f + f') a, ↑↑g a⟫_𝕜) x = (fun a => ⟪↑↑f a + ↑↑f' a, ↑↑g a⟫_𝕜) x"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)", "i₁ : Fin (n + 3)", "p : P", "h : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p ∈ s.mongePlane i₁ i₂", "h' : ∀ (i₂ : Fin (n + 3)), i₁ ≠ i₂ → p -ᵥ s.mongePoint ∈ (Submodule.span ℝ {s.points i₁ -ᵥ s.points i₂})ᗮ ⊓ vectorSpan ℝ (Set.range s.points)", "hi : p -ᵥ s.mongePoint ∈ ⨅ i₂, (Submodule.span ℝ {s.points i₁ -ᵥ s.points ↑i₂})ᗮ"], "goal": "p -ᵥ s.mongePoint = 0"}}
+{"state": {"context": ["E : Type u_1", "V : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℂ E", "f : V → E", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : MeasurableSpace V", "inst✝³ : BorelSpace V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : FiniteDimensional ℝ V", "inst✝ : CompleteSpace E", "hf1 : Continuous f", "hf2 : HasCompactSupport f", "ε : ℝ", "hε : ε > 0", "R : ℝ", "hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0", "A : Set V := {v | ‖v‖ ≤ R + 1}", "mA : MeasurableSet A"], "goal": "∃ T, ∀ (w : V), T ≤ ‖w‖ → ‖∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v‖ < ε"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "φ : PowerSeries R"], "goal": "(PowerSeries.coeff R 0) (PowerSeries.X * φ) = 0"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "Monoid A", "Monoid B", "TopologicalSpace A", "TopologicalSpace B"], "goal": "PontryaginDual.map (ContinuousMonoidHom.one A B) = ContinuousMonoidHom.one (PontryaginDual B) (PontryaginDual A)"}}
+{"state": {"context": ["α : Type u_1", "p : PMF α", "s : Set α"], "goal": "∑' (a : α), s.indicator (⇑p) a ≠ ⊤"}}
+{"state": {"context": ["G : Type v", "Group G", "PseudoMetricSpace G", "IsometricSMul G G", "IsometricSMul Gᵐᵒᵖ G", "a b c : G"], "goal": "dist (a / b) (a / c) = dist b c"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a b c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "hab : IntervalIntegrable f μ a b", "hcd : IntervalIntegrable f μ c d", "hac : IntervalIntegrable f μ a c"], "goal": "∫ (x : ℝ) in a..b, f x ∂μ - ∫ (x : ℝ) in c..d, f x ∂μ = ∫ (x : ℝ) in d..b, f x ∂μ - ∫ (x : ℝ) in c..a, f x ∂μ"}}
+{"state": {"context": ["R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M₁ : Type u_10", "M₂ : Type u_11", "M₃ : Type u_12", "Semiring R₁", "Semiring R₂", "Semiring R₃", "AddCommMonoid M₁", "AddCommMonoid M₂", "AddCommMonoid M₃", "module_M₁ : Module R₁ M₁", "module_M₂ : Module R₂ M₂", "module_M₃ : Module R₃ M₃", "σ₁₂ : R₁ →+* R₂", "σ₂₃ : R₂ →+* R₃", "σ₁₃ : R₁ →+* R₃", "RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "f : M₂ →ₛₗ[σ₂₃] M₃", "hf : Function.Injective ⇑f"], "goal": "Function.Injective fun g => f.comp g"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : α → β → β", "op : α → α → α", "inst✝³ : Monoid β", "inst✝² : Monoid γ", "inst✝¹ : FunLike F β γ", "inst✝ : DecidableEq α", "s : Finset α", "a : α", "h : a ∈ s", "f : α → β", "comm : (↑s).Pairwise fun a b => Commute (f a) (f b)", "comm' : optParam (∀ x ∈ ↑(s.erase a), ∀ y ∈ ↑(s.erase a), x ≠ y → (fun a b => Commute (f a) (f b)) x y) ⋯"], "goal": "(s.erase a).noncommProd f comm' * f a = s.noncommProd f comm"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "n : ℕ", "a : α", "x : γ", "s : Set β"], "goal": "(κ a) (countablePartitionSet n x ×ˢ s) ≤ (ν a) (countablePartitionSet n x)"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : Semigroup M", "a : Stream' M", "i j : ℕ", "ij : i < j", "d : ℕ", "hd : j = i.succ + d"], "goal": "FP (Stream'.drop i a).tail (a.get j)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "A : Type u_4", "Ring A", "Algebra R A", "h2 : IsUnit 2", "f : M →ₗ[R] A"], "goal": "(∀ (x : M), f x * f x = (algebraMap R A) (Q x)) ↔\n  (LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f =\n    LinearMap.compr₂ (QuadraticMap.polarBilin Q) (Algebra.linearMap R A)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "BEq α", "LawfulBEq α", "f : α → β", "a : α", "as : List α", "h : a ∈ as"], "goal": "List.lookup a (List.map (fun x => (x, f x)) as) = some (f a)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_3, u_2} D", "L : CategoryTheory.Functor C D", "W : CategoryTheory.MorphismProperty C", "L.IsLocalization W", "W.HasRightCalculusOfFractions", "X Y : C", "f : L.obj X ⟶ L.obj Y"], "goal": "∃ φ, f = φ.map L ⋯"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "I : MulticospanIndex C", "K : Multifork I", "b : I.R"], "goal": "K.π.app (WalkingMulticospan.right b) = K.ι (I.sndTo b) ≫ I.snd b"}}
+{"state": {"context": ["f✝ f : StieltjesFunction", "x : ℝ"], "goal": "⨅ r, ↑f ↑↑r = ⨅ r, ↑f ↑r"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "𝕜 : Type u_4", "E : Type u_5", "inst✝³ : DecidableEq α", "inst✝² : Monoid α", "inst✝¹ : DecidableEq β", "inst✝ : Monoid β", "s t : Finset α", "n : ℕ", "a : α"], "goal": "ThreeGPFree ↑{a}"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "NormalizationMonoid α", "a : Associates α"], "goal": "normalize a.out = a.out"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι : Type u_12", "inst✝⁶ : SeminormedRing 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module 𝕜 E", "inst✝³ : SeminormedRing 𝕜₂", "inst✝² : AddCommGroup E₂", "inst✝¹ : Module 𝕜₂ E₂", "σ₁₂ : 𝕜 →+* 𝕜₂", "inst✝ : RingHomIsometric σ₁₂", "p✝ p : Seminorm 𝕜 E", "r₁ : ℝ", "hr₁ : r₁ ≠ 0", "r₂ : ℝ"], "goal": "Metric.closedBall 0 r₁ • p.ball 0 r₂ ⊆ p.ball 0 (r₁ * r₂)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "a : α", "b : β a", "s : AList β"], "goal": "(insert a b s).keys = a :: s.keys.erase a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "f g : α → Option β", "x : Option α", "h : ∀ (a : α), a ∈ x → f a = g a"], "goal": "x.bind f = x.bind g"}}
+{"state": {"context": ["p : Cardinal.{u_1} → Prop", "c : Cardinal.{u_1}", "h : ∀ (α : Type u_1), p (Cardinal.mk α)"], "goal": "p c"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "S : Submonoid M", "x : (↥S)ˣ"], "goal": "↑↑x⁻¹ * ↑↑x = 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "𝒜 : Finset (Finset α)", "s t : Finset α", "a : α", "k r : ℕ"], "goal": "t ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ s ∈ 𝒜, s ⊆ t ∧ t.card = s.card + k"}}
+{"state": {"context": ["X : Type u", "s : Set X", "TopologicalSpace X", "hs : IsOpen s"], "goal": "s ∩ frontier s = ∅"}}
+{"state": {"context": ["F : Type u_3", "NormedAddCommGroup F", "NormedSpace ℝ F", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "u : E → F", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p : ℝ≥0", "hp : (↑(FiniteDimensional.finrank ℝ E)).IsConjExponent p"], "goal": "MeasureTheory.eLpNorm u (↑p) μ ≤\n  ↑(MeasureTheory.eLpNormLESNormFDerivOneConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) 1 μ"}}
+{"state": {"context": ["α : Type u_1", "i : Nat", "l : List α"], "goal": "(List.drop i l).length = l.length - i"}}
+{"state": {"context": ["α : Type u_9", "s : Set (α → Prop)", "a : α"], "goal": "sSup s a ↔ ∃ r ∈ s, r a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "CategoryTheory.HasExt C", "X Y : C", "HasDerivedCategory C", "f : X ⟶ Y"], "goal": "(CategoryTheory.Abelian.Ext.mk₀ f).hom = CategoryTheory.ShiftedHom.mk₀ ↑0 ⋯ ((DerivedCategory.singleFunctor C 0).map f)"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "κ : ι → Sort u_3", "β : Type u_4", "SetLike α β", "l : LowerAdjoint SetLike.coe", "f : (i : ι) → κ i → α"], "goal": "l.toFun (⋃ i, ⋃ j, ↑(l.toFun ↑(f i j))) = l.toFun (⋃ i, ⋃ j, ↑(f i j))"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X"], "goal": "PerfectSpace X ↔ ∀ (x : X), (𝓝[≠] x).NeBot"}}
+{"state": {"context": ["α : Type u_5", "self : NonUnitalSeminormedRing α", "x y : α"], "goal": "dist x y = ‖x - y‖"}}
+{"state": {"context": ["α : Type u_1", "Inhabited α", "a : α"], "goal": "(some a).get! = a"}}
+{"state": {"context": ["R : Type u_4", "S : Type u_2", "M : Type u_3", "inst✝⁷ : CommRing R", "inst✝⁶ : Field S", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Algebra R S", "inst✝³ : Module R M", "inst✝² : Module S M", "inst✝¹ : IsScalarTower R S M", "B : BilinForm S M", "ι : Type u_1", "inst✝ : Finite ι", "hB : B.Nondegenerate", "b : Basis ι S M"], "goal": "B.dualSubmodule (B.flip.dualSubmodule (Submodule.span R (Set.range ⇑b))) = Submodule.span R (Set.range ⇑b)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "f : R →+* S", "x : S", "n : ℕ"], "goal": "Polynomial.eval₂ f x ↑n = ↑n"}}
+{"state": {"context": ["p : ℕ", "n : ℕ"], "goal": "0 ≤ padicValRat p ↑n"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "P : Type u_3", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid P", "inst✝¹ : Module R M", "inst✝ : Module R P", "N : Submodule R M", "h : ∀ s ≤ N, s.FG", "s : Submodule R ↥N", "f : ↥(Submodule.map N.subtype s) ≃ₗ[R] ↥s", "h₁ : (Submodule.map N.subtype s).FG", "h₂ : Submodule.map f ⊤ = ⊤"], "goal": "s.FG"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "inst✝ : Countable ι", "u : ι → UniformSpace α", "hcomp : ∀ (i : ι), CompleteSpace α", "hcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated", "ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i => UniformSpace.toTopologicalSpace) i ≤ t₀", "hut : ∀ (i : ι), UniformSpace.toTopologicalSpace = (fun i => UniformSpace.toTopologicalSpace) i"], "goal": "(𝓤 α).IsCountablyGenerated"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "DivisionRing k", "AddCommGroup V", "Module k V", "s t : Finset V", "hs : AffineIndependent k Subtype.val", "hst : ↑s ⊆ ↑(affineSpan k ↑t)"], "goal": "s.card ≤ t.card"}}
+{"state": {"context": ["M : Type u_1", "inst✝⁴ : Nonempty M", "inst✝³ : Semigroup M", "inst✝² : TopologicalSpace M", "inst✝¹ : CompactSpace M", "inst✝ : T2Space M", "continuous_mul_left : ∀ (r : M), Continuous fun x => x * r", "S : Set (Set M) := {N | IsClosed N ∧ N.Nonempty ∧ ∀ m ∈ N, ∀ m' ∈ N, m * m' ∈ N}", "c : Set (Set M)", "hcs : c ⊆ S", "hc : IsChain (fun x x_1 => x ⊆ x_1) c"], "goal": "(⋂₀ c).Nonempty"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "l : BoxIntegral.IntegrationParams", "I : BoxIntegral.Box ι", "c : ℝ≥0", "r : (ι → ℝ) → ↑(Set.Ioi 0)", "π : BoxIntegral.TaggedPrepartition I", "self : l.MemBaseSet I c r π"], "goal": "l.bHenstock = true → π.IsHenstock"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "F : MonoidalFunctor C D", "G : D ⥤ C", "h : F.toFunctor ⊣ G", "X Y : D"], "goal": "(monoidalAdjoint F h ⊗⋙ F.toLaxMonoidalFunctor).μ X Y ≫ h.counit.app (X ⊗ Y) = (h.counit.app X ⊗ h.counit.app Y) ≫ (LaxMonoidalFunctor.id D).μ X Y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "ι : Type u_3", "hm : MeasurableSpace α", "μ✝ : Measure α", "inst✝¹⁰ : TopologicalSpace α", "inst✝⁹ : BorelSpace α", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "g✝ : α → E", "l : Filter ι", "x₀ : α", "s t : Set α", "φ✝ : ι → α → ℝ", "a : E", "inst✝⁶ : CompleteSpace E", "F : Type u_4", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : FiniteDimensional ℝ F", "inst✝² : MeasurableSpace F", "inst✝¹ : BorelSpace F", "μ : Measure F", "inst✝ : μ.IsAddHaarMeasure", "φ : F → ℝ", "hφ : ∀ (x : F), 0 ≤ φ x", "h'φ : ∫ (x : F), φ x ∂μ = 1", "h : Tendsto (fun x => ‖x‖ ^ finrank ℝ F * φ x) (cobounded F) (𝓝 0)", "g : F → E", "hg : Integrable g μ", "h'g : ContinuousAt g 0", "I : Integrable φ μ"], "goal": "MeasurableSet (closedBall 0 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝³ : MeasurableSpace α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "p : ℝ≥0∞", "μ : Measure α", "inst✝ : Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "f : ↥(Lp E p μ)", "hfi' : Memℒp (↑↑f) p μ"], "goal": "∃ x, (∀ (n : ℕ), x n ∈ Set.range Subtype.val) ∧ Tendsto x atTop (𝓝 f)"}}
+{"state": {"context": ["R : Type u_1", "Zero R", "f : ArithmeticFunction R"], "goal": "f.toFun = ⇑f"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "𝒜 : Finset (Finset α)", "s t : Finset α", "a : α", "k r : ℕ", "h𝒜 : Set.Sized r ↑𝒜", "A : Finset α", "hA : A ∈ 𝒜", "i : α", "hi : i ∉ A", "h : insert i A ∈ ↑(∂⁺ 𝒜)"], "goal": "(insert i A).card = r + 1"}}
+{"state": {"context": ["α : Type ua", "u v : UniformSpace α"], "goal": "UniformSpace.toTopologicalSpace = UniformSpace.toTopologicalSpace ⊓ UniformSpace.toTopologicalSpace"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "I J : Submodule R P"], "goal": "IsFractional S I → IsFractional S J → IsFractional S (I ⊔ J)"}}
+{"state": {"context": ["α : Type u", "AddGroup α"], "goal": "Equiv.addLeft 0 = 1"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "OrderBot α", "LocallyFiniteOrder α", "b : α"], "goal": "Finset.Ico ⊥ ↑b = Finset.insertNone (Finset.Iio b)"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β₁ : Type u_3", "β₂ : Type u_4", "γ₁ : Type u_5", "γ₂ : Type u_6", "f₁ : α₁ → β₁ → Finset γ₁", "f₂ : α₂ → β₂ → Finset γ₂", "g₁ : α₁ → β₂ → Finset γ₁", "g₂ : α₁ → β₂ → Finset γ₂", "a : α₁ ⊕ α₂", "b : β₁ ⊕ β₂", "c : γ₁ ⊕ γ₂"], "goal": "c ∈ Finset.sumLexLift f₁ f₂ g₁ g₂ a b ↔\n  (∃ a₁ b₁ c₁, a = Sum.inl a₁ ∧ b = Sum.inl b₁ ∧ c = Sum.inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨\n    (∃ a₁ b₂ c₁, a = Sum.inl a₁ ∧ b = Sum.inr b₂ ∧ c = Sum.inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨\n      (∃ a₁ b₂ c₂, a = Sum.inl a₁ ∧ b = Sum.inr b₂ ∧ c = Sum.inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨\n        ∃ a₂ b₂ c₂, a = Sum.inr a₂ ∧ b = Sum.inr b₂ ∧ c = Sum.inr c₂ ∧ c₂ ∈ f₂ a₂ b₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : TopologicalSpace α", "inst✝³ : OpensMeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : TopologicalSpace β", "inst✝ : BorelSpace β", "f : α → β", "s : Set α", "μ : Measure α", "hf : ContinuousOn f s", "hs : MeasurableSet s", "a✝ : Nontrivial α", "inhabited_h : Inhabited α"], "goal": "AEMeasurable f (μ.restrict s)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : TopologicalSpace α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : μ.WeaklyRegular", "inst✝ : SigmaFinite μ", "f : α → ℝ≥0", "fint : Integrable (fun x => ↑(f x)) μ", "fmeas : AEMeasurable f μ", "ε : ℝ≥0", "εpos : 0 < ↑ε", "δ : ℝ≥0", "δpos : 0 < δ", "hδε : δ < ε", "int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ⊤", "g : α → ℝ≥0∞", "f_lt_g : ∀ (x : α), ↑(f x) < g x", "gcont : LowerSemicontinuous g", "gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ", "gint_ne : ∫⁻ (x : α), g x ∂μ ≠ ⊤", "g_lt_top : ∀ᵐ (x : α) ∂μ, g x < ⊤"], "goal": "∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∀ᵐ (x : α) ∂μ, g x < ⊤) ∧ Integrable (fun x => (g x).toReal) μ ∧ ∫ (x : α), (g x).toReal ∂μ < ∫ (x : α), ↑(f x) ∂μ + ↑ε"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrderedField α", "inst✝¹ : Ring β", "abv : β → α", "inst✝ : IsAbsoluteValue abv", "f g : ℕ → β", "a : ℕ → α", "n : ℕ", "hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m", "hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i", "ε : α", "ε0 : ε > 0", "i : ℕ", "hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2"], "goal": "∃ i, ∀ j ≥ i, abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) i) < ε"}}
+{"state": {"context": ["α : Type u_1", "m₁ m₂ : MeasurableSpace α", "s : Set α"], "goal": "MeasurableSet s ↔ MeasurableSet s ∧ MeasurableSet s"}}
+{"state": {"context": ["η : Type u_12", "Ms : η → Type u_13", "Ns : η → Type u_14", "(j : η) → Add (Ms j)", "(j : η) → Add (Ns j)", "es : (j : η) → Ms j ≃+ Ns j", "x : (j : η) → Ms j", "j : η"], "goal": "(AddEquiv.piCongrRight es) x j = (es j) (x j)"}}
+{"state": {"context": ["α : Type u_1", "d : DynkinSystem α", "s : Set (Set α)", "hs : IsPiSystem s"], "goal": "generate s ≤ ofMeasurableSpace (generateFrom s)"}}
+{"state": {"context": ["R : Type u", "inst✝ : EuclideanDomain R", "m k : R"], "goal": "m / k * k + m % k = m"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "S : Submonoid R", "M : Type u_6", "M' : Type u_7", "AddCommGroup M", "AddCommGroup M'", "Module R M", "Module R M'", "f : M →ₗ[R] M'", "IsLocalizedModule S f", "m₁ m₂ : M", "s : ↥S"], "goal": "IsLocalizedModule.mk' f (m₁ - m₂) s = IsLocalizedModule.mk' f m₁ s - IsLocalizedModule.mk' f m₂ s"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "F G H : CategoryTheory.MonoidalFunctor C D", "α : F ⟶ G", "β : G ⟶ H"], "goal": "(α ≫ β).toNatTrans = α.toNatTrans ≫ β.toNatTrans"}}
+{"state": {"context": ["i₁ i₂ : String.Pos"], "goal": "i₁ < i₂ ↔ i₁.byteIdx < i₂.byteIdx"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : CanonicallyLinearOrderedAddCommMonoid α", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a c d : α", "ha : AddLECancellable a", "h : a < a"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "N : Type u_2", "Zero N", "LinearOrder α", "Finite α", "LT N", "hwf : WellFoundedLT N"], "goal": "WellFoundedLT (Lex (α →₀ N))"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁶ : Field F", "inst✝⁵ : Field E", "inst✝⁴ : Algebra F E", "K : Type w", "inst✝³ : Field K", "inst✝² : Algebra F K", "inst✝¹ : Algebra E K", "inst✝ : IsScalarTower F E K", "h1 : IsPurelyInseparable F E", "h2 : IsPurelyInseparable E K", "q : ℕ", "h✝ : ExpChar F q"], "goal": "IsPurelyInseparable F K"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "l : List α", "x✝ : α", "hl : l.Nodup", "x : α", "hx : x ∈ l"], "goal": "l.formPerm x = x ↔ l.length ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α"], "goal": "∅ ⊂ s ↔ s.Nonempty"}}
+{"state": {"context": ["a✝ b✝ m n p a b : ℕ", "hab : a.Coprime b", "q : ℕ"], "goal": "(a * b).factorization q = (a.factorization + b.factorization) q"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_8", "Semiring R", "AddCommMonoid M", "Module R M", "e₁ e₂ : M ≃ₗ[R] M"], "goal": "↑(e₁ * e₂) = ↑e₁ * ↑e₂"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₁, u₂} D", "i : CategoryTheory.Functor D C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.CartesianClosed C", "h : ∀ (B : D) (A : C), (A ⟹ i.obj B) ∈ i.essImage"], "goal": "CategoryTheory.ExponentialIdeal i"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "K : ProperCone ℝ E"], "goal": "ProperCone.map (ContinuousLinearMap.id ℝ E) K = K"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t : Set X", "f : X → Y", "hs : IsLindelof s", "hf : ContinuousOn f s", "l : Filter Y", "lne : l.NeBot", "inst✝ : CountableInterFilter l", "ls : l ≤ 𝓟 (f '' s)", "this✝¹ : (comap f l ⊓ 𝓟 s).NeBot", "x : X", "hxs : x ∈ s", "hx : ClusterPt x (comap f l ⊓ 𝓟 s)", "this✝ : (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)).NeBot", "this : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f x) ⊓ l)"], "goal": "ClusterPt (f x) l"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "x : α", "inst : Batteries.OrientedCmp cmp", "t : Batteries.RBNode α"], "goal": "(Batteries.RBNode.insert cmp t x).reverse = Batteries.RBNode.insert (flip cmp) t.reverse x"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : GroupWithZero G₀", "a c d : G₀", "m n : ℕ", "hc : c ≠ 0", "h : 0⁻¹ * a = c"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "PredOrder α", "NoMinOrder α", "a : α"], "goal": "Set.Ioi (Order.pred a) = insert a (Set.Ioi a)"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommSemiring R", "S : Type v", "inst✝ : CommSemiring S", "p✝ q✝ : ℕ", "p q : R[X]", "hsep : p.Separable", "hq : 1 < multiplicity q p"], "goal": "q * q ∣ p"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIocDiv hp (a - p) b = toIocDiv hp a b + 1"}}
+{"state": {"context": ["K : Type u_1", "CommRing K"], "goal": "∀ (x x_1 : RatFunc K),\n  x.add x_1 =\n    match x, x_1 with\n    | { toFractionRing := p }, { toFractionRing := q } => { toFractionRing := p + q }"}}
+{"state": {"context": ["F : PFunctor.{u}", "X : Type u_1", "f✝ : X → ↑F X", "R : F.M → F.M → Prop", "inst✝ : Inhabited F.M", "bisim : IsBisimulation R", "ps : List F.Idx", "ps_ih : ∀ (a : F.A) (f f' : F.B a → F.M), R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩) → IsPath ps (M.mk ⟨a, f⟩) ∨ IsPath ps (M.mk ⟨a, f'⟩) → (isubtree ps (M.mk ⟨a, f⟩)).head = (isubtree ps (M.mk ⟨a, f'⟩)).head ∧ ∃ a_1 f_1 f'_1, isubtree ps (M.mk ⟨a, f⟩) = M.mk ⟨a_1, f_1⟩ ∧ isubtree ps (M.mk ⟨a, f'⟩) = M.mk ⟨a_1, f'_1⟩ ∧ ∀ (i : F.B a_1), R (f_1 i) (f'_1 i)", "a : F.A", "f f' : F.B a → F.M", "h₀ : R (M.mk ⟨a, f⟩) (M.mk ⟨a, f'⟩)", "i : F.B a", "hh : IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) ∨ IsPath (⟨a, i⟩ :: ps) (M.mk ⟨a, f'⟩)", "a₀ : F.A", "f₀ : F.B a₀ → F.M", "h : f i = M.mk ⟨a₀, f₀⟩", "f₁ : F.B a₀ → F.M", "h₁ : R (M.mk ⟨a₀, f₀⟩) (M.mk ⟨a₀, f₁⟩)", "h' : f' i = M.mk ⟨a₀, f₁⟩"], "goal": "IsPath ps (f i) ∨ IsPath ps (f' i)"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "∀ (i : 𝒰.J), CategoryTheory.Limits.HasPullback (𝒰.map i ≫ f) g", "i j : 𝒰.J"], "goal": "(AlgebraicGeometry.Scheme.Pullback.gluing 𝒰 f g).t i j = AlgebraicGeometry.Scheme.Pullback.t 𝒰 f g i j"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "S : Type u_2", "inst✝⁴ : CommRing S", "f : R →+* S", "I✝ J : Ideal S", "inst✝³ : Algebra R S", "inst✝² : Algebra.IsIntegral R S", "P : Ideal R", "inst✝¹ : P.IsPrime", "I : Ideal S", "inst✝ : I.IsPrime", "hIP : comap (algebraMap R S) I ≤ P", "Q' : Ideal (S ⧸ I)", "Q'_prime : Q'.IsPrime", "hQ' : comap (algebraMap (R ⧸ comap (algebraMap R S) I) (S ⧸ I)) Q' = map (Quotient.mk (comap (algebraMap R S) I)) P"], "goal": "comap (Quotient.mk (comap (algebraMap R S) I)) (map (Quotient.mk (comap (algebraMap R S) I)) P) = P"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "J : CategoryTheory.GrothendieckTopology C", "f : Y ⟶ X", "S R : CategoryTheory.Sieve X", "h : J.Covers S f"], "goal": "(∀ {Z : C} (g : Z ⟶ X), S.arrows g → J.Covers R g) → J.Covers R f"}}
+{"state": {"context": ["α : Type u", "Neg α", "x : Multiplicative α"], "goal": "Multiplicative.toAdd x⁻¹ = -Multiplicative.toAdd x"}}
+{"state": {"context": ["α : Type u_1", "NonUnitalSeminormedRing α", "a b : α"], "goal": "‖a * b‖₊ ≤ ‖a‖₊ * ‖b‖₊"}}
+{"state": {"context": ["S T : ℝ", "hT : 0 < T", "z τ : ℂ", "hz : |z.im| ≤ S", "hτ : T ≤ τ.im", "n : ℤ"], "goal": "↑|n| * -S ≤ -|↑n * z.im|"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "e : α ↪o β", "he : (Set.range ⇑e).OrdConnected", "x y : α"], "goal": "⇑e '' Set.Ico x y = Set.Ico (e x) (e y)"}}
+{"state": {"context": ["M : Type u_1", "inst✝¹ : Monoid M", "s : Set M", "A : Type u_2", "inst✝ : AddMonoid A", "t : Set A", "S : Submonoid M"], "goal": "IsSubmonoid ↑S"}}
+{"state": {"context": ["ι : Type w", "DecidableEq ι", "Fintype ι", "K : Type u", "L : Type v", "Field K", "Field L", "Algebra K L", "Module.Finite K L", "R : Type z", "CommRing R", "Algebra R K", "Algebra R L", "IsScalarTower R K L", "b : ι → L", "h : ∀ (i : ι), IsIntegral R (b i)"], "goal": "IsIntegral R (Algebra.discr K b)"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "s : Finset α"], "goal": "↑s = Set.univ ↔ s = Finset.univ"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Multiset α"], "goal": "s.Nodup ↔ ∀ (a : α), Multiset.count a s ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "p : α → Bool", "l : List α"], "goal": "List.countP p l = 0 ↔ ∀ (a : α), a ∈ l → ¬p a = true"}}
+{"state": {"context": ["a b c : Prop"], "goal": "a ∨ b ∧ c ↔ (a ∨ b) ∧ (a ∨ c)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommMonoidWithZero R", "n : ℕ", "χ : DirichletCharacter R n", "S : Type", "inst✝ : CommRing S", "m : ℕ", "ψ : DirichletCharacter S m", "hψ : ψ.Odd"], "goal": "ψ ↑(-1) = ↑(-1)"}}
+{"state": {"context": ["R : Type u_1", "η : Type u_2", "ιs : η → Type u_3", "Ms : η → Type u_4", "inst✝³ : Ring R", "inst✝² : (i : η) → AddCommGroup (Ms i)", "inst✝¹ : (i : η) → Module R (Ms i)", "inst✝ : DecidableEq η", "v : (j : η) → ιs j → Ms j", "hs : ∀ (i : η), LinearIndependent R (v i)", "hs' : ∀ (j : η), LinearIndependent R fun i => (stdBasis R Ms j) (v j i)", "j : η", "J : Set η", "a✝ : J.Finite", "hiJ : j ∉ J"], "goal": "Disjoint (span R (Set.range fun i => (stdBasis R Ms j) (v j i))) (⨆ i ∈ J, span R (Set.range fun i_1 => (stdBasis R Ms i) (v i i_1)))"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f✝ g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "f : X → ℝ", "hfi : Integrable f μ", "h_meas : NullMeasurableSet {x | 0 ≤ f x} μ"], "goal": "∫ (x : X) in {x | 0 ≤ f x}, ‖f x‖ ∂μ = ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "F : Type u_4", "MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "NormedAddCommGroup F", "mα : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure μ", "X : α → β", "Y : α → Ω", "mβ : MeasurableSpace β", "f : β × Ω → F", "NormedSpace ℝ F", "hY : AEMeasurable Y μ", "hf_int : MeasureTheory.Integrable f (MeasureTheory.Measure.map (fun a => (X a, Y a)) μ)"], "goal": "MeasureTheory.Integrable (fun x => ∫ (y : Ω), f (x, y) ∂(ProbabilityTheory.condDistrib Y X μ) x)\n  (MeasureTheory.Measure.map X μ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : Filter α", "f : α → β", "x : β", "h : Filter.Tendsto f l (pure x)"], "goal": "Filter.EventuallyConst f l"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E"], "goal": "DifferentiableWithinAt 𝕜 f Set.univ x ↔ DifferentiableAt 𝕜 f x"}}
+{"state": {"context": ["α : Type u", "SemilatticeInf α", "OrderTop α", "x y : α"], "goal": "(x ⨯ y) = x ⊓ y"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a b : R", "x y : M", "hv : LinearIndependent R v", "H : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0", "l : ι →₀ R", "hl : (Finsupp.total ι M R v) l = 0", "i : ι", "hn : ¬l i = 0"], "goal": "False"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "f' : F", "x : 𝕜"], "goal": "HasDerivAt f f' x ↔ Filter.Tendsto (fun t => t⁻¹ • (f (x + t) - f x)) (𝓝[≠] 0) (𝓝 f')"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "h : p.leadingCoeff * q.leadingCoeff ≠ 0", "hp : p ≠ 0", "hq : q ≠ 0"], "goal": "(p * q).coeff (p.natDegree + q.natDegree) ≠ 0"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "a : R"], "goal": "a ∈ Set.center R ↔\n  AddMonoidHom.mulLeft a = AddMonoidHom.mulRight a ∧\n    AddMonoidHom.mul.compr₂ (AddMonoidHom.mulLeft a) = AddMonoidHom.mul.comp (AddMonoidHom.mulLeft a) ∧\n      AddMonoidHom.mul.comp (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulLeft a) ∧\n        AddMonoidHom.mul.compr₂ (AddMonoidHom.mulRight a) = AddMonoidHom.mul.compl₂ (AddMonoidHom.mulRight a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "M : Type u_3", "AddCommMonoid M", "TopologicalSpace M", "T2Space M", "v : MeasureTheory.VectorMeasure α M", "Countable β", "f : β → Set α", "hf₁ : ∀ (i : β), MeasurableSet (f i)", "hf₂ : Pairwise (Disjoint on f)"], "goal": "↑v (⋃ i, f i) = ∑' (i : β), ↑v (f i)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_4, u_1} C", "inst✝² : Category.{u_3, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "inst✝ : W.HasLeftCalculusOfFractions", "X Y : C", "f f' : L.obj X ⟶ L.obj Y", "φ : W.LeftFraction X Y", "hφ : f = φ.map L ⋯", "φ' : W.LeftFraction X Y", "hφ' : f' = φ'.map L ⋯", "α : W.LeftFraction φ.Y' φ'.Y'", "hα : (RightFraction.mk φ.s ⋯ φ'.s).f ≫ α.s = (RightFraction.mk φ.s ⋯ φ'.s).s ≫ α.f", "ψ : W.LeftFraction₂ X Y := LeftFraction₂.mk (φ.f ≫ α.f) (φ'.f ≫ α.s) (φ'.s ≫ α.s) ⋯"], "goal": "∃ φ, f = φ.fst.map L ⋯ ∧ f' = φ.snd.map L ⋯"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "M : (i : ι) → Matroid (α i)", "I X : Set ((i : ι) × α i)", "hI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)", "x✝ : (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧ (∀ (x : ι) ⦃y : Set (α x)⦄, (M x).Indep y → y ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ y → Sigma.mk x ⁻¹' I = y) ∧ ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E", "hIX : ∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X", "h' : ∀ (x : ι) ⦃y : Set (α x)⦄, (M x).Indep y → y ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ y → Sigma.mk x ⁻¹' I = y", "h'' : ∀ (x : ι), Sigma.mk x ⁻¹' X ⊆ (M x).E"], "goal": "I ⊆ X"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "X✝ : Ω → ℝ", "μ✝ : Measure Ω", "c : ℝ", "X : Ω → ℝ", "μ : Measure Ω"], "goal": "variance (fun ω => c * X ω) μ = c ^ 2 * variance X μ"}}
+{"state": {"context": [], "goal": "∀ {X Y : SimplexCategory} (f : X ⟶ Y),\n  SimplexCategory.toTop.map f = { toFun := SimplexCategory.toTopMap f, continuous_toFun := ⋯ }"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → AddZeroClass (β i)", "x : Π₀ (i : ι), β i", "x✝ : x ∈ ⊤"], "goal": "0 ∈ AddSubmonoid.closure (⋃ i, Set.range (single i))"}}
+{"state": {"context": ["k : Type u_1", "inst✝² : Field k", "K : Type u_2", "inst✝¹ : Field K", "F : Type u_3", "inst✝ : Field F", "φ : K →+* ℂ", "hφ : ¬ComplexEmbedding.IsReal φ"], "goal": "¬ComplexEmbedding.IsReal (mk φ).embedding"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D", "G : CategoryTheory.Functor D C", "iso : e.inverse ≅ G", "X : C"], "goal": "(e.changeInverse iso).unitIso.hom.app X =\n  CategoryTheory.CategoryStruct.comp (e.unitIso.hom.app X) (iso.hom.app (e.functor.obj X))"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "x : A", "n : ℤ"], "goal": "(Polynomial.aeval x) (Polynomial.Chebyshev.U R n) = Polynomial.eval x (Polynomial.Chebyshev.U A n)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝¹ : SemilatticeInf α", "inst✝ : OrderTop α", "s s₁ : Finset β", "f✝ g✝ : β → α", "a : α", "f g : β → α", "hfg : ∀ a ∈ s₁, f a = g a"], "goal": "s₁.inf f = s₁.inf g"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "n m : ℕ", "inst✝¹ : Semiring R", "p✝ q r✝ : R[X]", "inst✝ : SMulZeroClass S R", "r : S", "p : R[X]", "i : ℕ", "hi : ¬r • p.coeff i = 0"], "goal": "¬p.coeff i = 0"}}
+{"state": {"context": ["G : Type u", "AddGroup G", "N : AddSubgroup G", "nN : N.Normal", "a b : G"], "goal": "↑(a + b) = ↑a + ↑b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : Ring R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "f : ℕ → R", "g : ℕ → M", "m n : ℕ", "hmn : m < n", "h₁ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range i, g i = ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i", "h₂ : ∑ i ∈ Ico (m + 1) n, f i • ∑ i ∈ range (i + 1), g i = ∑ i ∈ Ico m (n - 1), f i • ∑ i ∈ range (i + 1), g i + f (n - 1) • ∑ i ∈ range n, g i - f m • ∑ i ∈ range (m + 1), g i", "h₃ : ∑ i ∈ Ico (m + 1) n, f i • g i = ∑ i ∈ Ico (m + 1) n, f i • ((range (i + 1)).sum g - (range i).sum g)", "h₄ : ∑ i ∈ Ico m (n - 1), f i • ∑ i ∈ range (i + 1), g i + f (n - 1) • ∑ i ∈ range n, g i - f m • ∑ i ∈ range (m + 1), g i - ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i = f (n - 1) • (range n).sum g - f m • (range (m + 1)).sum g + ∑ i ∈ Ico m (n - 1), (f i • (range (i + 1)).sum g - f (i + 1) • (range (i + 1)).sum g)"], "goal": "f m • g m + (∑ i ∈ Ico m (n - 1), f i • ∑ i ∈ range (i + 1), g i + f (n - 1) • ∑ i ∈ range n, g i - f m • ∑ i ∈ range (m + 1), g i - ∑ i ∈ Ico m (n - 1), f (i + 1) • ∑ i ∈ range (i + 1), g i) = f (n - 1) • ∑ i ∈ range n, g i - f m • ∑ i ∈ range m, g i - ∑ i ∈ Ico m (n - 1), (f (i + 1) - f i) • ∑ i ∈ range (i + 1), g i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : Group G", "a b c d : G", "n : ℤ", "h : b * a = c"], "goal": "a = b⁻¹ * c"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : α → ℝ", "hf : Measurable f"], "goal": "Measurable fun x => ENNReal.ofReal (f x)"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "hC : Pretriangulated C", "T : Triangle C", "hT : T ∈ distinguishedTriangles", "X : C", "f : T.obj₂ ⟶ X", "hf : T.mor₁ ≫ f = 0"], "goal": "∃ g, f = T.mor₂ ≫ g"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁹ : LinearOrderedField R", "inst✝⁸ : LinearOrderedField R'", "inst✝⁷ : AddCommGroup E", "inst✝⁶ : AddCommGroup F", "inst✝⁵ : LinearOrderedAddCommGroup α", "inst✝⁴ : Module R E", "inst✝³ : Module R F", "inst✝² : Module R α", "inst✝¹ : OrderedSMul R α", "s : Set E", "i j : ι", "c : R", "t : Finset ι", "w : ι → R", "z : ι → E", "inst✝ : Fintype ι", "f : ι → R"], "goal": "(convexHull R) (Set.range fun i j => if i = j then 1 else 0) = stdSimplex R ι"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "w x y z : α", "inst✝ : GeneralizedBooleanAlgebra α", "s : x ⊓ y ⊔ z = x", "i : x ⊓ y ⊓ z = ⊥"], "goal": "x \\ y = z"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "p : Type u_5", "q : Type u_6", "m' : o → Type u_7", "n' : o → Type u_8", "p' : o → Type u_9", "R : Type u_10", "S : Type u_11", "α✝ : Type u_12", "β : Type u_13", "inst✝⁴ : DecidableEq o", "inst✝³ : Zero α✝", "inst✝² : Zero β", "α : Type u_14", "inst✝¹ : AddMonoid α", "inst✝ : StarAddMonoid α", "M : (i : o) → Matrix (m' i) (n' i) α"], "goal": "(blockDiagonal' M)ᴴ = blockDiagonal' fun k => (M k)ᴴ"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "γ : Type u_5", "δ : Type u_7", "f : Filter α", "g : Filter β", "m : α' → β → γ", "n : α → α'", "m' : β → α → δ", "n' : δ → γ", "h_left_anticomm : ∀ (a : α) (b : β), m (n a) b = n' (m' b a)"], "goal": "Filter.map₂ m (Filter.map n f) g = Filter.map n' (Filter.map₂ m' g f)"}}
+{"state": {"context": ["α : Type u", "x : FreeCommRing α"], "goal": "∃ s, x.IsSupported ↑s"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E"], "goal": "‖0‖₊ = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : Preorder α", "inst✝⁴ : Preorder β", "inst✝³ : OrderTop α", "inst✝² : OrderBot β", "inst✝¹ : LocallyFiniteOrder α", "inst✝ : LocallyFiniteOrder β", "a a₁ a₂ : α", "b b₁ b₂ : β"], "goal": "Icc (inlₗ a₁) (inlₗ a₂) = map (Embedding.inl.trans toLex.toEmbedding) (Icc a₁ a₂)"}}
+{"state": {"context": ["X : (CategoryTheory.Limits.WidePushoutShape CategoryTheory.Limits.WalkingPair)ᵒᵖ"], "goal": "CategoryTheory.Limits.walkingSpanOpEquiv.functor.obj X = Opposite.unop X"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : Zero α", "s : Finset ι", "f : (i : ι) → i ∈ s → α", "i✝ i : ι", "hi : i ∈ ↑(indicator s f).support"], "goal": "i ∈ ↑s"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "Group α", "a : α"], "goal": "a⁻ᵐ ≤ 1 ↔ a⁻¹ ≤ 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "Preorder β", "s : Set α", "f g : α → β", "a : α", "heq : f =ᶠ[𝓝[s] a] g", "hmem : a ∈ s"], "goal": "IsLocalExtrOn f s a ↔ IsLocalExtrOn g s a"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "K : Type u_3", "L : Type u_4", "inst✝¹⁶ : EuclideanDomain R", "inst✝¹⁵ : CommRing S", "inst✝¹⁴ : IsDomain S", "inst✝¹³ : Field K", "inst✝¹² : Field L", "inst✝¹¹ : Algebra R K", "inst✝¹⁰ : IsFractionRing R K", "inst✝⁹ : Algebra K L", "inst✝⁸ : FiniteDimensional K L", "inst✝⁷ : Algebra.IsSeparable K L", "algRL : Algebra R L", "inst✝⁶ : IsScalarTower R K L", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra S L", "ist : IsScalarTower R S L", "iic : IsIntegralClosure S R L", "abv : AbsoluteValue R ℤ", "ι : Type u_5", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "bS : Basis ι R S", "adm : abv.IsAdmissible", "inst✝¹ : Infinite R", "inst✝ : DecidableEq R"], "goal": "(algebraMap R S) (∏ m ∈ finsetApprox bS adm, m) ≠ 0"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "DecidableEq ι", "β : ι → Type u_4", "i : ι", "f : (j : ι) → α j", "a : α i", "t : (j : ι) → α j → Set (β j)"], "goal": "(Set.univ.pi fun j => t j (Function.update f i a j)) = {x | x i ∈ t i a} ∩ {i}ᶜ.pi fun j => t j (f j)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s✝ : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "s : Finset P", "w : P → k"], "goal": "(affineCombination k s.attach Subtype.val) (w ∘ Subtype.val) = (affineCombination k s id) w"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "m : Type u_4", "n : Type u_5", "Fintype m", "M : Matrix m n R"], "goal": "Mᵀ.mulVecLin = M.vecMulLinear"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "f' : F", "x : 𝕜", "s t : Set 𝕜", "h : HasDerivWithinAt f f' t x", "hst : t ∈ 𝓝[s] x"], "goal": "HasDerivWithinAt f f' s x"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "R : Type v", "inst✝ : CommRing R", "d : n → R"], "goal": "∑ σ : Perm n, ↑↑(sign σ) * ∏ i : n, diagonal d (σ i) i = ∏ i : n, d i"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "p : Submodule R M"], "goal": "(∃ N, ↑N = p) ↔ ∀ (x : L), ∀ m ∈ p, ⁅x, m⁆ ∈ p"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α"], "goal": "a :: l ≠ l"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "σ : Type u_4", "inst✝⁹ : Semiring A", "inst✝⁸ : DecidableEq ι", "inst✝⁷ : CanonicallyOrderedAddCommMonoid ι", "inst✝⁶ : SetLike σ A", "inst✝⁵ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝⁴ : GradedRing 𝒜", "a : A", "n✝ i : ι", "inst✝³ : Sub ι", "inst✝² : OrderedSub ι", "inst✝¹ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "n : ι", "inst✝ : Decidable (i ≤ n)", "b : ↥(𝒜 i)"], "goal": "↑(((decompose 𝒜) (a * ↑b)) n) = if i ≤ n then ↑(((decompose 𝒜) a) (n - i)) * ↑b else 0"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a b : α", "DenselyOrdered α"], "goal": "(Finset.Ioo a b).Nonempty ↔ a < b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "f g : α → E", "s t : Set α", "μ ν : Measure α", "hf : IntegrableOn f s μ", "ht : NullMeasurableSet t μ", "h't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0", "u : Set α := {x | x ∈ s ∧ f x ≠ 0}", "hu : IntegrableOn f u μ", "v : Set α := toMeasurable μ u", "A : IntegrableOn f v μ", "B : IntegrableOn f (t \\ v) μ"], "goal": "t ⊆ v ∪ t \\ v"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "inst✝² : DecidableRel r", "inst✝¹ : DecidableEq α", "inst✝ : IsRefl α r", "x : α", "xs : List α", "h : (decide ¬r x x) = true"], "goal": "False"}}
+{"state": {"context": ["x y z : ℝ", "n : ℕ", "hx : 1 < x", "hyz : y < z"], "goal": "log x * y < log x * z"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : CommMonoid α", "x y z : α", "H : IsRelPrime x (y * z)"], "goal": "IsRelPrime x y"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "s : Finset α", "H : s.Nonempty", "x : α", "H2 : ∀ y ∈ s, y ≤ x"], "goal": "s.max' H ≤ x"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "p : α", "hp : Prime p"], "goal": "Irreducible p"}}
+{"state": {"context": ["r r₁ r₂ : ℝ≥0", "x y : ℝ"], "goal": "↑(sInf ∅) = sInf (toReal '' ∅)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : T2Space X", "inst✝ : CompactSpace X", "x : X", "hs : IsClosed (⋂ s, ↑s)", "a b : Set X", "ha : IsClosed a", "hb : IsClosed b", "hab : ⋂ s, ↑s ⊆ a ∪ b", "ab_disj : Disjoint a b", "u v : Set X", "hu : IsOpen u", "hv : IsOpen v", "hau : a ⊆ u", "hbv : b ⊆ v", "huv : Disjoint u v"], "goal": "⋂ s, ↑s ⊆ a ∨ ⋂ s, ↑s ⊆ b"}}
+{"state": {"context": ["o : Ordinal.{u_1}"], "goal": "let_fun this := ⋯; HEq (mk (Quotient.out o).α (Quotient.out o).α (fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)) fun o₂ => nim (typein (fun x x_1 => x < x_1) o₂)).moveRight fun i => nim (typein (fun x x_1 => x < x_1) i)"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "inst✝²² : NontriviallyNormedField 𝕜₁", "𝕜₂ : Type u_2", "inst✝²¹ : NontriviallyNormedField 𝕜₂", "σ : 𝕜₁ →+* 𝕜₂", "iσ : RingHomIsometric σ", "B : Type u_3", "F₁ : Type u_4", "inst✝²⁰ : NormedAddCommGroup F₁", "inst✝¹⁹ : NormedSpace 𝕜₁ F₁", "E₁ : B → Type u_5", "inst✝¹⁸ : (x : B) → AddCommGroup (E₁ x)", "inst✝¹⁷ : (x : B) → Module 𝕜₁ (E₁ x)", "inst✝¹⁶ : TopologicalSpace (TotalSpace F₁ E₁)", "F₂ : Type u_6", "inst✝¹⁵ : NormedAddCommGroup F₂", "inst✝¹⁴ : NormedSpace 𝕜₂ F₂", "E₂ : B → Type u_7", "inst✝¹³ : (x : B) → AddCommGroup (E₂ x)", "inst✝¹² : (x : B) → Module 𝕜₂ (E₂ x)", "inst✝¹¹ : TopologicalSpace (TotalSpace F₂ E₂)", "inst✝¹⁰ : TopologicalSpace B", "e₁ e₁' : Trivialization F₁ TotalSpace.proj", "e₂ e₂' : Trivialization F₂ TotalSpace.proj", "inst✝⁹ : (x : B) → TopologicalSpace (E₁ x)", "inst✝⁸ : FiberBundle F₁ E₁", "inst✝⁷ : (x : B) → TopologicalSpace (E₂ x)", "ita : ∀ (x : B), TopologicalAddGroup (E₂ x)", "inst✝⁶ : FiberBundle F₂ E₂", "inst✝⁵ : VectorBundle 𝕜₁ F₁ E₁", "inst✝⁴ : VectorBundle 𝕜₂ F₂ E₂", "inst✝³ : MemTrivializationAtlas e₁", "inst✝² : MemTrivializationAtlas e₁'", "inst✝¹ : MemTrivializationAtlas e₂", "inst✝ : MemTrivializationAtlas e₂'", "h₁ : Continuous ⇑(compSL F₁ F₂ F₂ σ (RingHom.id 𝕜₂))", "h₂ : Continuous ⇑(compSL F₁ F₁ F₂ (RingHom.id 𝕜₁) σ).flip"], "goal": "ContinuousOn (continuousLinearMapCoordChange σ e₁ e₁' e₂ e₂') (e₁.baseSet ∩ e₂.baseSet ∩ (e₁'.baseSet ∩ e₂'.baseSet))"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f : ℂ → E", "z : ℂ", "ε : ℝ", "hε : ε > 0"], "goal": "DiffContOnCl ℂ (fun z => invInterpStrip f z ε) (verticalStrip 0 1)"}}
+{"state": {"context": ["θ : Real.Angle"], "goal": "2 • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ ↑π"}}
+{"state": {"context": ["Γ : Type u_1", "inst✝² : Inhabited Γ", "Λ : Type u_2", "inst✝¹ : Inhabited Λ", "σ : Type u_3", "inst✝ : Inhabited σ", "n : ℕ", "enc : Γ → Vector Bool n", "dec : Vector Bool n → Γ", "enc0 : enc default = Vector.replicate n false", "M : Λ → Stmt₁", "R : ListBlank Γ", "a : Γ", "L : ListBlank Γ"], "goal": "∀ {L' R' : ListBlank Bool} {l₁ l₂ : List Bool}, (enc a).toList = l₁.reverseAux l₂ → (Tape.move Dir.left)^[l₁.length] (Tape.mk' (ListBlank.append l₁ L') (ListBlank.append l₂ R')) = Tape.mk' L' (ListBlank.append (enc a).toList R')"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Ring S", "inst✝⁸ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : IsDomain B", "inst✝⁴ : Algebra A B", "K : Type u_6", "inst✝³ : Field K", "inst✝² : Algebra A S", "S' : Type u_7", "inst✝¹ : Ring S'", "inst✝ : Algebra A S'", "pb : PowerBasis A S", "x : B"], "goal": "map (algebraMap A B) (minpoly A pb.gen) ≠ 0"}}
+{"state": {"context": ["X : Type u_1", "inst✝³ : TopologicalSpace X", "𝕜 : Type u_2", "inst✝² : RCLike 𝕜", "s : Set 𝕜", "inst✝¹ : Zero ↑s", "h0 : ↑0 = 0", "inst✝ : CompactSpace ↑s", "h0' : 0 ∈ s"], "goal": "⇑toContinuousMapHom ⁻¹' ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0'⟩)) = Set.univ"}}
+{"state": {"context": ["n k m : ℕ"], "goal": "(n + k).dist (m + k) = n.dist m"}}
+{"state": {"context": ["ι : Type u_1", "M₀ : Type u_4", "MulZeroClass M₀", "s : Set ι", "f g : ι → M₀"], "goal": "(s.indicator fun i => f i * g i) = fun i => s.indicator f i * s.indicator g i"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹ : Category.{u_5, u_1} C", "inst✝ : Category.{?u.41420, u_2} D", "F G H : C ⥤ Cat", "α : F ⟶ G", "β : G ⟶ H"], "goal": "∀ (X : Grothendieck F), (map (α ≫ β)).obj X = (map α ⋙ map β).obj X"}}
+{"state": {"context": ["ι : Type u_1", "F : Filter ι", "f : ι → ℝ", "h : Bornology.IsBounded (Set.range f)"], "goal": "Filter.IsBounded (fun x x_1 => x ≤ x_1) (Filter.map f F)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "DivisionRing k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : Set P", "h : FiniteDimensional.finrank k V = 2"], "goal": "Coplanar k s"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : Zero α", "s : Finset ι", "f : ι →₀ α", "t : ι →₀ Finset α", "ht : t.support ⊆ s", "i : ι", "h : f i ∈ t i", "hi : i ∈ f.support", "H : t i = 0"], "goal": "f i = 0"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "PartialOrder Γ", "AddMonoid R", "x : HahnSeries Γ Rᵃᵒᵖ"], "goal": "(AddOpposite.unop (HahnSeries.addOppositeEquiv x)).orderTop = x.orderTop"}}
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+{"state": {"context": [], "goal": "RespectsIso @FinitePresentation"}}
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+{"state": {"context": ["n m : ℕ", "a b c : Fin n"], "goal": "a ≤ b → ↑a = ↑b ∨ a < b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "B : Type u_4", "Monoid R", "Monoid S", "Star A", "Star B", "NonUnitalNonAssocSemiring A", "NonUnitalNonAssocSemiring B", "MulAction R S", "DistribMulAction S A", "DistribMulAction S B", "DistribMulAction R A", "DistribMulAction R B", "IsScalarTower R S A", "IsScalarTower R S B", "f : A →⋆ₙₐ[S] B", "x : A"], "goal": "(NonUnitalStarAlgHom.restrictScalars R f) x = f x"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R"], "goal": "MvPolynomial.coeff 0 1 = 1"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G", "a : G", "m n : ℕ", "h : n ≤ m"], "goal": "(m - n) • a = m • a + -(n • a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : Option α", "f : (a : α) → a ∈ x → Option β", "y : β"], "goal": "x.pbind f = some y ↔ ∃ z H, f z H = some y"}}
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+{"state": {"context": ["M : Type u_1", "Monoid M", "x : M"], "goal": "x ∈ powers x"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "c : ℝ", "X : Ω → ℝ", "μ : MeasureTheory.Measure Ω"], "goal": "ProbabilityTheory.evariance (fun ω => c * X ω) μ = ENNReal.ofReal (c ^ 2) * ProbabilityTheory.evariance X μ"}}
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+{"state": {"context": ["x : ℂ"], "goal": "Complex.tan (x - ↑Real.pi) = Complex.tan x"}}
+{"state": {"context": ["F : Sort u_2", "α : Sort u_3", "β : Sort u_4", "γ : Sort u_5", "iF : EquivLike F β γ", "f : α → β", "e : F"], "goal": "Function.Injective (⇑e ∘ f) ↔ Function.Injective f"}}
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+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.SFinite μ"], "goal": "MeasurableSet μ.sigmaFiniteSet"}}
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+{"state": {"context": ["μ : ℝ", "v : ℝ≥0"], "goal": "MeasureTheory.Integrable (ProbabilityTheory.gaussianPDFReal μ v) MeasureTheory.volume"}}
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+{"state": {"context": ["ι α β : Type u", "c : Cardinal.{u}", "l✝ l : Filter α"], "goal": "∀ (S : Set (Set α)), #↑S < ℵ₀ → (∀ s ∈ S, s ∈ l) → ⋂₀ S ∈ l"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ : C", "inst✝ : HasPullbacks C", "X Y : C", "f : X ⟶ Y", "x : Over X", "y : Over Y", "v : x ⟶ (pullback f).obj y"], "goal": "(v.left ≫ pullback.fst y.hom f) ≫ y.hom = ((map f).obj x).hom"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁵ : MeasurableSpace α", "μ ν : Measure α", "inst✝⁴ : TopologicalSpace β", "inst✝³ : TopologicalSpace γ", "inst✝² : TopologicalSpace δ", "inst✝¹ : SemilatticeSup β", "inst✝ : ContinuousSup β", "f g : α →ₘ[μ] β", "a✝ : α", "ha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝"], "goal": "↑f a✝ ≤ ↑(f ⊔ g) a✝"}}
+{"state": {"context": ["G : Type u_6", "AddGroup G", "H : Type u_9", "AddGroup H", "f : G → H", "map_div : ∀ (a b : G), f (a + -b) = f a + -f b"], "goal": "⇑(AddMonoidHom.ofMapAddNeg f map_div) = f"}}
+{"state": {"context": ["α : Type u_1", "BEq α", "L : List (List α)", "a : α"], "goal": "List.count a L.join = Nat.sum (List.map (List.count a) L)"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁶ : DivisionRing K", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module K V", "A : Type u_1", "inst✝³ : Semiring A", "inst✝² : Module A V", "inst✝¹ : SMul K A", "inst✝ : IsScalarTower K A V", "h : finrank K V = 1", "this : Nontrivial V", "S : Submodule A V", "hn : ¬restrictScalars K S = restrictScalars K ⊥"], "goal": "restrictScalars K S = restrictScalars K ⊤"}}
+{"state": {"context": ["α : Type u_2", "Zero α", "a : α"], "goal": "Finset.singletonZeroHom a = {a}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s t : Finset α"], "goal": "(Iic s).card = 2 ^ s.card"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ"], "goal": "(deriv fun x => -cos x) = sin"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{u₂, u₁} C", "I : Type v₁", "CategoryTheory.Category.{v₂, v₁} I", "F : CategoryTheory.Functor I C", "CategoryTheory.Limits.HasColimit F", "CategoryTheory.Limits.HasLimitsOfShape Iᵒᵖ (Type u₂)", "A : C", "i : I"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.colimitHomIsoLimitYoneda F A).inv\n    ((CategoryTheory.coyoneda.map (CategoryTheory.Limits.colimit.ι F i).op).app A) =\n  CategoryTheory.Limits.limit.π (F.op.comp (CategoryTheory.yoneda.obj A)) (Opposite.op i)"}}
+{"state": {"context": ["α : Type u_2", "GroupWithZero α"], "goal": "0⁻¹ = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "X : C"], "goal": "(α_ (𝟙_ C) (𝟙_ C) X).hom ≫ 𝟙_ C ◁ (β_ X (𝟙_ C)).inv ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ (λ_ X).hom ▷ 𝟙_ C =\n  (λ_ (𝟙_ C)).hom ▷ X ≫ (β_ X (𝟙_ C)).inv"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "T : D", "Y' : C", "S : CategoryTheory.Functor C D", "f : CategoryTheory.CostructuredArrow S T", "g : Y' ⟶ f.left"], "goal": "(f.homMk' g).left = g"}}
+{"state": {"context": ["a : ℝ"], "goal": "#↑(Set.Iio a) = 𝔠"}}
+{"state": {"context": ["α : Type u", "DistribLattice α", "a b c : α"], "goal": "(a ⊔ b) ⊓ c = a ⊓ c ⊔ b ⊓ c"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "P₂ : Type u_3", "V₁ : Type u_6", "V₂ : Type u_7", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AffineSpace V₂ P₂", "e : P₁ ≃ᵃ[k] P₂", "a b : P₁", "c : k"], "goal": "e ((AffineMap.lineMap a b) c) = (AffineMap.lineMap (e a) (e b)) c"}}
+{"state": {"context": ["k : Type u", "Field k", "n : ℕ"], "goal": "AlgebraicClosure.Step k (n + 1) = AlgebraicClosure.AdjoinMonic (AlgebraicClosure.Step k n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Primcodable α", "Primcodable β", "Primcodable γ", "p : α → Prop", "q : β → Prop", "r : γ → Prop", "h : OneOneEquiv p q"], "goal": "p ≤₁ r ↔ q ≤₁ r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Sub α", "inst✝ : Bot α", "a b : WithTop α"], "goal": "a - b = ⊤ ↔ a = ⊤ ∧ b ≠ ⊤"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_3", "M : Type u_5", "M₂ : Type u_7", "Semiring R", "Semiring R₂", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R₂ M₂", "σ₁₂ : R →+* R₂", "F : Type u_9", "FunLike F M M₂", "SemilinearMapClass F σ₁₂ M M₂", "RingHomSurjective σ₁₂", "f : F"], "goal": "GaloisConnection (Submodule.map f) (Submodule.comap f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "a : α", "b : β", "p : α × β → Prop"], "goal": "(∃ x ∈ s ×ˢ t, p x) ↔ ∃ x ∈ s, ∃ y ∈ t, p (x, y)"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "IsNormalSubgroup (IsSubgroup.trivial G)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "x✝ : R"], "goal": "inverse (↑x + x✝) - ↑x⁻¹ + ↑x⁻¹ * x✝ * ↑x⁻¹ = inverse (↑x + x✝) - (∑ i ∈ range 2, (-↑x⁻¹ * x✝) ^ i) * ↑x⁻¹"}}
+{"state": {"context": ["M : Type u", "Monoid M", "C : Type v", "CategoryTheory.Category.{w, v} C", "X : C", "f : M →* CategoryTheory.End X"], "goal": "∀ (x : CategoryTheory.SingleObj M), (CategoryTheory.SingleObj.functor f).obj x = X"}}
+{"state": {"context": ["r r₁ r₂ : ℝ≥0", "x y : ℝ", "s : Set ℝ≥0", "hs : s.Nonempty"], "goal": "↑(sSup s) = sSup (toReal '' s)"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α✝ : Type u", "β : Type v", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "α : Type u_1", "l : Filter α", "x : G", "u : α → G", "A : Tendsto (fun x_1 => x) l (𝓝 x)", "h : Tendsto u l (𝓝 x)"], "goal": "Tendsto (fun x_1 => u x_1 / x) l (𝓝 1)"}}
+{"state": {"context": ["k : Type u_1", "E : Type u_2", "PE : Type u_3", "inst✝³ : OrderedRing k", "inst✝² : OrderedAddCommGroup E", "inst✝¹ : Module k E", "inst✝ : OrderedSMul k E", "a a' b b' : E", "r r' : k", "h : r < 1"], "goal": "(lineMap a b) r < b ↔ (lineMap a b) r < (lineMap a b) 1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C"], "goal": "CategoryTheory.Square.arrowArrowEquivalence'.counitIso =\n  CategoryTheory.Iso.refl (CategoryTheory.Square.fromArrowArrowFunctor'.comp CategoryTheory.Square.toArrowArrowFunctor')"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "CategoryTheory.MonoidalCategory E", "B : Type u₀", "CategoryTheory.Category.{v₀, u₀} B", "CategoryTheory.MonoidalCategory B", "F : CategoryTheory.LaxMonoidalFunctor B C", "G : CategoryTheory.LaxMonoidalFunctor D E", "X Y : B × D"], "goal": "(F.prod G).μ X Y = (F.μ X.1 Y.1, G.μ X.2 Y.2)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t✝ : Set α", "inst✝ : Preorder α", "s t : Set α", "H : ∀ a ∈ s, ∀ b ∈ t, a < b", "a✝ : ℕ", "h : t.chainHeight = ↑a✝"], "goal": "(s ∪ t).chainHeight ≤ s.chainHeight + ↑a✝"}}
+{"state": {"context": ["r : ℚ"], "goal": "↑r.num / ↑r.den = r"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "S.HasHomology", "CategoryTheory.Limits.HasKernel S.g", "CategoryTheory.Limits.HasCokernel S.f"], "goal": "S.Exact ↔\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι S.g) (CategoryTheory.Limits.cokernel.π S.f) = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "s : Set X", "f : X → Y", "hf : Embedding f"], "goal": "IsLocallyClosed s ↔ ∃ s', IsLocallyClosed s' ∧ s' ∩ Set.range f = f '' s"}}
+{"state": {"context": ["ι : Type u_1", "I : Box ι", "inst✝ : Fintype ι"], "goal": "↑I =ᶠ[ae volume] Box.Icc I"}}
+{"state": {"context": ["V : Type u", "Quiver V", "a b c : V", "p : Quiver.Path a b", "q₁ q₂ : Quiver.Path b c"], "goal": "p.comp q₁ = p.comp q₂ ↔ q₁ = q₂"}}
+{"state": {"context": ["X✝ Y Z : Scheme", "𝒰✝ : X✝.OpenCover", "f✝ : X✝ ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g", "X W : Scheme", "𝒰 : X.OpenCover", "f : W ⟶ X", "x : ↑↑W.toPresheafedSpace"], "goal": "x ∈ Set.range ⇑((PreservesPullback.iso forgetToTop f (𝒰.map (𝒰.f (f.val.base x)))).hom ≫ pullback.fst (forgetToTop.map f) (forgetToTop.map (𝒰.map (𝒰.f (f.val.base x)))))"}}
+{"state": {"context": ["G : Type u_1", "Group G", "tG : Monoid.IsTorsion G", "bounded : (Set.range fun g => orderOf g).Finite"], "goal": "Monoid.ExponentExists G"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "ℱ : MeasureTheory.Filtration ℕ m0", "s : ℕ → Set Ω", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "hs : ∀ (n : ℕ), MeasurableSet (s n)", "n : ℕ"], "goal": "MeasureTheory.Integrable (MeasureTheory.BorelCantelli.process s n) μ"}}
+{"state": {"context": ["E : Type u", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "F : Type v", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℂ F", "f : ℂ → F", "hf : Differentiable ℂ f", "hb : Bornology.IsBounded (range f)", "c : ℂ", "C : ℝ", "C₀ : C > 0", "hC : ∀ (z : ℂ), ‖f z‖ ≤ C", "ε : ℝ", "ε₀ : 0 < ε"], "goal": "‖deriv f c‖ ≤ ε"}}
+{"state": {"context": ["T : ℝ", "g : ℝ → ℝ", "hg : Function.Periodic g T", "h_int : ∀ (t₁ t₂ : ℝ), IntervalIntegrable g MeasureTheory.volume t₁ t₂", "h₀ : 0 < ∫ (x : ℝ) in 0 ..T, g x", "hT : 0 < T"], "goal": "Filter.Tendsto (fun t => ∫ (x : ℝ) in 0 ..t, g x) Filter.atTop Filter.atTop"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "r : ℝ", "n : ℤ"], "goal": "↑(r ^ n) = ↑r ^ n"}}
+{"state": {"context": ["α : Type u", "uniformSpace : UniformSpace α", "CompleteSpace α", "s : Set α", "h : IsClosed s"], "goal": "IsComplete s"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b c : α", "n : ℤ", "a : α"], "goal": "a < a + p ∧ ∃ z, a + p = a + z • p"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {r : α → α → Prop} {β : Type u_2} {s : β → β → Prop} {b₁ b₂ : β},\n  Sum.Lex r s (Sum.inr b₁) (Sum.inr b₂) ↔ s b₁ b₂"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "self : CompactSpace X"], "goal": "IsCompact Set.univ"}}
+{"state": {"context": ["a : ENNReal"], "goal": "⊤ / a = if a = ⊤ then 0 else ⊤"}}
+{"state": {"context": ["α : Type u", "inst✝¹ : Mul α", "β : Type v", "inst✝ : Semigroup β", "f✝ f : α →ₙ* β", "x : AssocQuotient α"], "goal": "∀ (a b : α), AssocRel α a b → f a = f b"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "t : MeasureTheory.SignedMeasure α", "f : α → ℝ", "hf : Measurable f", "htμ : t ⟂ᵥ μ.toENNRealVectorMeasure"], "goal": "(t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x)) ⟂ₘ\n  t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "Ring R", "AddCommGroup A", "Module R A", "StarAddMonoid R", "StarAddMonoid A", "StarModule R A", "r : R", "hr : r ∈ skewAdjoint R", "a : A", "ha : a ∈ skewAdjoint A"], "goal": "IsSelfAdjoint (r • a)"}}
+{"state": {"context": ["α : Type u_1", "s✝¹ t✝ : Finset α", "n : ?m.30699", "s✝ t : Finset α", "inst✝ : DecidableEq (Finset α)", "s : Finset α"], "goal": "s.powerset = (range (s.card + 1)).biUnion fun i => powersetCard i s"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "n p : ℕ", "f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ", "f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ", "hle : 1 ≤ n + 1", "hne : ↑p + 1 + 1 ≠ 0", "h1 : ∀ (r : ℚ), r * (↑p + 1 + 1) * ↑n ^ p.succ / (↑p + 1 + 1) = r * ↑n ^ p.succ"], "goal": "∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i"}}
+{"state": {"context": ["D : GlueData", "α : Type u", "inst✝ : TopologicalSpace α", "J : Type u", "U : J → Opens α", "i : (ofOpenSubsets U).J", "x : ↑(of α)", "hx : x ∈ U i", "j : (ofOpenSubsets U).J", "hy : x ∈ U j"], "goal": "(ofOpenSubsets U).Rel ⟨i, ⟨x, hx⟩⟩ ⟨j, ⟨x, hy⟩⟩"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "s : Set α", "γ : Type u_3", "LinearOrder γ", "TopologicalSpace γ", "OrderTopology γ", "δ : Type u_4", "LinearOrder δ", "TopologicalSpace δ", "OrderTopology δ", "g : γ → δ", "f : α → γ", "hg : Continuous g", "hf : LowerSemicontinuousOn f s", "gmon : Antitone g"], "goal": "UpperSemicontinuousOn (g ∘ f) s"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "l : Filter α", "p : ι → Prop", "s : ι → Set α", "h : l.HasBasis p s", "V : Set α", "hV : V ∈ l"], "goal": "l.HasBasis (fun i => p i ∧ s i ⊆ V) s"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "W : α → Sort w", "Z : β → Sort z", "h₁ : α ≃ β", "h₂ : (b : β) → W (h₁.symm b) ≃ Z b"], "goal": "⇑(h₁.piCongr' h₂) = fun f b => (h₂ b) (f (h₁.symm b))"}}
+{"state": {"context": ["a b : Ordinal.{u_1}"], "goal": "¬a ≤ b → ¬a.toPGame ≤ b.toPGame"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹¹ : CommRing R", "inst✝¹⁰ : Ring S", "inst✝⁹ : Algebra R S", "K : Type u_4", "L : Type u_5", "F : Type u_6", "inst✝⁸ : Field K", "inst✝⁷ : Field L", "inst✝⁶ : Field F", "inst✝⁵ : Algebra K L", "inst✝⁴ : Algebra K F", "ι : Type w", "E : Type u_7", "inst✝³ : Field E", "inst✝² : Algebra K E", "inst✝¹ : FiniteDimensional K L", "inst✝ : IsGalois K L", "x : L"], "goal": "(L →ₐ[K] AlgebraicClosure L) → AlgebraicClosure L"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β → γ", "a : α", "b : Option β"], "goal": "Option.map₂ f (some a) b = Option.map (fun b => f a b) b"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "G : C", "∀ (A : C), CategoryTheory.Limits.HasCoproduct fun x => G"], "goal": "CategoryTheory.IsSeparator G ↔ ∀ (A : C), CategoryTheory.Epi (CategoryTheory.Limits.Sigma.desc fun f => f)"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "P : Type u_5", "Ring 𝕜", "AddCommGroup V", "Module 𝕜 V", "TopologicalSpace P", "AddTorsor V P", "s t : Set P", "h : s ⊆ t"], "goal": "intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t"}}
+{"state": {"context": [], "goal": "@List.dropLast = @List.dropLastTR"}}
+{"state": {"context": ["α : Type u", "Infinite α", "s : Set α", "hf : s.Finite"], "goal": "sᶜ ∈ Filter.hyperfilter α"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "ι : Type u_3", "hm : MeasurableSpace α", "μ : Measure α", "inst✝⁶ : TopologicalSpace α", "inst✝⁵ : BorelSpace α", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "g : α → E", "l : Filter ι", "x₀ : α", "s t : Set α", "φ✝ : ι → α → ℝ", "a : E", "inst✝² : CompleteSpace E", "inst✝¹ : MetrizableSpace α", "inst✝ : IsLocallyFiniteMeasure μ", "hs : IsCompact s", "hμ : ∀ (u : Set α), IsOpen u → x₀ ∈ u → 0 < μ (u ∩ s)", "c : α → ℝ", "hc : ContinuousOn c s", "h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀", "hnc : ∀ x ∈ s, 0 ≤ c x", "hnc₀ : 0 < c x₀", "h₀ : x₀ ∈ s", "hmg : IntegrableOn g s μ", "hcg : ContinuousWithinAt g s x₀", "φ : ℕ → α → ℝ := fun n x => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ * c x ^ n"], "goal": "Tendsto (fun n => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) atTop (𝓝 (g x₀))"}}
+{"state": {"context": ["E : Type u_1", "𝕜 : Type u_2", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : Module 𝕜 E", "inst✝¹ : TopologicalSpace E", "inst✝ : ContinuousSMul 𝕜 E", "s : Set E", "hs : s ∈ 𝓝 0", "c : 𝕜", "hc : 1 < ‖c‖", "cn_ne : ∀ (n : ℕ), c ^ n ≠ 0", "A : ∀ (x : E), ∀ᶠ (n : ℕ) in atTop, x ∈ c ^ n • s"], "goal": "#↑s = #E"}}
+{"state": {"context": ["n : ℕ+", "h : n < 2"], "goal": "1 ≤ n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "TopologicalSpace α", "LinearOrder α", "OrderClosedTopology α", "f g : β → α", "TopologicalSpace β", "TopologicalSpace γ", "(x : β) → Decidable (f x ≤ g x)", "f' g' : β → γ", "hf : Continuous f", "hg : Continuous g", "hf' : ContinuousOn f' {x | f x ≤ g x}", "hg' : ContinuousOn g' {x | g x ≤ f x}", "hfg : ∀ (x : β), f x = g x → f' x = g' x"], "goal": "Continuous fun x => if f x ≤ g x then f' x else g' x"}}
+{"state": {"context": ["x : ℝ", "n : ℕ", "hx : 0 ≤ x", "hn : n ≠ 0"], "goal": "(x ^ (↑n)⁻¹) ^ n = x"}}
+{"state": {"context": ["a : UnitAddCircle"], "goal": "Tendsto (fun s => hurwitzZeta a s - 1 / (s - 1) / s.Gammaℝ) (𝓝 1) (𝓝 (hurwitzZeta a 1))"}}
+{"state": {"context": ["p : ℕ", "hp✝ : Fact (Nat.Prime p)", "k : Type u_1", "inst✝¹ : Field k", "inst✝ : CharP k p", "hp : ¬IsUnit ↑p", "a : 𝕎 k", "ha : a.coeff 0 ≠ 0", "b : 𝕎 k", "hb : b.coeff 0 ≠ 0", "n✝ : ℕ", "ha0 : (⇑verschiebung)^[n✝ + 1] a ≠ 0", "n : ℕ", "hb0 : (⇑verschiebung)^[n + 1] b ≠ 0", "hab : 1 = 0"], "goal": "IsUnit ((⇑verschiebung)^[n✝ + 1] a) ∨ IsUnit ((⇑verschiebung)^[n + 1] b)"}}
+{"state": {"context": ["n : ℕ+", "K : Type u", "L : Type v", "Field K", "CommRing L", "IsDomain L", "Algebra K L", "IsCyclotomicExtension {n} K L", "ζ : L", "hζ : IsPrimitiveRoot ζ ↑n"], "goal": "(IsPrimitiveRoot.powerBasis K hζ).gen ∈ Algebra.adjoin K {ζ - 1}"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝³ : TopologicalSpace G", "inst✝² : Group G", "inst✝¹ : TopologicalGroup G", "inst✝ : TopologicalSpace α", "f : α → G", "s✝ : Set α", "x✝ : α", "x : G", "s : Set G"], "goal": "x ∈ closure s ↔ ∀ U ∈ 𝓝 1, ∃ y ∈ s, y / x ∈ U"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "s : NonUnitalSubsemiring R"], "goal": "NonUnitalSubsemiring.closure ↑s = s"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "𝕜' : Type u_4", "NormedField 𝕜'", "NormedAlgebra 𝕜 𝕜'", "CompleteSpace 𝕜'", "n : ℕ∞"], "goal": "ContDiffOn 𝕜 n Inv.inv {0}ᶜ"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "p : α → Prop", "f : (a : α) → p a → β", "l : List α", "H : ∀ (a : α), a ∈ l → p a"], "goal": "pmap f l H = map (fun x => f ↑x ⋯) l.attach"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝¹¹ : CommSemiring ι", "inst✝¹⁰ : Module ι (Additive ℤˣ)", "inst✝⁹ : DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)", "inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)", "inst✝⁵ : (i : ι) → Module R (𝒜 i)", "inst✝⁴ : (i : ι) → Module R (ℬ i)", "inst✝³ : DirectSum.GRing 𝒜", "inst✝² : DirectSum.GRing ℬ", "inst✝¹ : DirectSum.GAlgebra R 𝒜", "inst✝ : DirectSum.GAlgebra R ℬ", "x : (⨁ (i : ι), 𝒜 i) ⊗[R] ⨁ (i : ι), ℬ i", "r : R", "i✝¹ : ι", "x✝¹ : 𝒜 i✝¹", "i✝ : ι", "x✝ : ℬ i✝"], "goal": "((lof R ι 𝒜 i✝¹) x✝¹ * (lof R ι 𝒜 0) (DirectSum.GAlgebra.toFun r)) ⊗ₜ[R] ((lof R ι ℬ i✝) x✝ * 1) = (r • (lof R ι 𝒜 i✝¹) x✝¹) ⊗ₜ[R] (lof R ι ℬ i✝) x✝"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : ℝ → E", "T : Set ℝ", "hT : T ⊆ Set.Ioi 0", "hT' : MeasurableSet T", "hfc : MeasureTheory.AEStronglyMeasurable f (MeasureTheory.volume.restrict (Set.Ioi 0))", "s : ℂ"], "goal": "MeasureTheory.IntegrableOn (fun t => ↑t ^ (s - 1) • f t) T MeasureTheory.volume ↔\n  MeasureTheory.IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) T MeasureTheory.volume"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ : ℝ", "x₀ : M", "hx : I.IsInteriorPoint x₀", "left✝ : ContinuousAt (fun x => { proj := x, snd := v x }) x₀", "hv : ContDiffWithinAt ℝ 1 (↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘ (fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm) (range ↑I) (↑(extChartAt I x₀) x₀)"], "goal": "∃ γ, γ t₀ = x₀ ∧ IsIntegralCurveAt γ v t₀"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "Field K", "Field L", "Algebra K L", "S : IntermediateField K L", "x : ↥S"], "goal": "↑(-x) = -↑x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "One β", "f : α → β", "s : Set α", "hs : ∀ a ∈ frontier s, f a = 1", "hf : ContinuousOn f (closure s)"], "goal": "Continuous (s.mulIndicator f)"}}
+{"state": {"context": ["R : Type u", "EuclideanDomain R", "DecidableEq R", "x : R"], "goal": "EuclideanDomain.lcm 0 x = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Group α", "inst✝² : LinearOrder α", "a✝¹ b✝¹ : α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a✝ b✝ c : α", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b : α", "ba : b ≤ a"], "goal": "max a b / min a b = mabs (a / b)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_3", "M : Type u_5", "M₂ : Type u_7", "Semiring R", "Semiring R₂", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R₂ M₂", "σ₁₂ : R →+* R₂", "F : Type u_9", "FunLike F M M₂", "SemilinearMapClass F σ₁₂ M M₂", "RingHomSurjective σ₁₂", "f : F", "hf : Function.Surjective ⇑f", "p q : Submodule R₂ M₂"], "goal": "Submodule.map f (Submodule.comap f p ⊔ Submodule.comap f q) = p ⊔ q"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "W : Submodule R (Module.Dual R M)"], "goal": "Function.Injective ⇑W.quotDualCoannihilatorToDual.flip"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CommGroup α", "inst✝ : DecidableEq α", "A B C : Finset α", "hAB : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card", "hA : A.Nonempty", "n : ℕ", "ih : ↑(A * B ^ n).card ≤ (↑(A * B).card / ↑A.card) ^ n * ↑A.card"], "goal": "↑((A * B).card * (A * B ^ n).card) ≤ ↑(A * B).card * ((↑(A * B).card / ↑A.card) ^ n * ↑A.card)"}}
+{"state": {"context": ["R : Type u_1", "CommMonoid R", "R' : Type u_2", "CommMonoidWithZero R'", "χ : MulChar R R'"], "goal": "χ = 1 ↔ ∀ (a : Rˣ), χ ↑a = 1"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "I✝ : MultispanIndex C", "K : Multicofork I✝", "I : MultispanIndex C", "P : C", "π : (b : I.R) → I.right b ⟶ P", "w : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)"], "goal": "∀ ⦃X Y : WalkingMultispan I.fstFrom I.sndFrom⦄ (f : X ⟶ Y), I.multispan.map f ≫ (fun x => match x with | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a) | WalkingMultispan.right b => π b) Y = (fun x => match x with | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a) | WalkingMultispan.right b => π b) X ≫ ((Functor.const (WalkingMultispan I.fstFrom I.sndFrom)).obj P).map f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "s : Set X", "hs : IsSigmaCompact s", "hf : ContinuousOn f s"], "goal": "IsSigmaCompact (f '' s)"}}
+{"state": {"context": ["R : Type u_2", "V : Type u_3", "W : Type u_4", "NormedAddCommGroup V", "NormedAddCommGroup W", "NormedField R", "NormedSpace R V", "NormedSpace R W", "f : V →ᴬ[R] W"], "goal": "⇑f = ⇑f.contLinear + Function.const V (f 0)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "fa : AnalyticOn 𝕜 f Set.univ"], "goal": "Continuous f"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "ι : Type u_5", "CommSemiring R", "AddCommMonoid M", "Module R M", "Fintype ι", "DecidableEq ι", "b : Basis ι R M", "ij : ι × ι"], "goal": "b.end ij = (Matrix.toLin b b) ((Matrix.stdBasis R ι ι) ij)"}}
+{"state": {"context": ["C : CategoryTheory.Cat"], "goal": "CategoryTheory.Quiv.forget.obj C = CategoryTheory.Quiv.of ↑C"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "h : MeasureTheory.NullMeasurableSet s μ", "hs : μ s ≠ ⊤"], "goal": "μ sᶜ = μ Set.univ - μ s"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝² : Field R", "p✝ q : R[X]", "inst✝¹ : CommSemiring k", "inst✝ : DecidableEq R", "ϕ : R →+* k", "f g : R[X]", "α : k", "hα : eval₂ ϕ α (EuclideanDomain.gcd f g) = 0", "p : R[X]", "hp : f = EuclideanDomain.gcd f g * p"], "goal": "eval₂ ϕ α f = 0"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f✝ : ℕ → Ω → ℝ", "N✝ n m : ℕ", "ω✝ : Ω", "f : ℕ → Ω → ℝ", "N : ℕ", "ω : Ω", "hab : a < b", "h : ¬∃ n, upperCrossingTime a b f N n ω = N"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "π : α → Type u_11", "i : Set α", "s t : (a : α) → Set (π a)", "x : (i : α) → π i", "a : α", "ha : a ∈ i", "hx : (∀ i_1 ∈ i, x i_1 ∈ s i_1) ∧ ∀ x_1 ∈ i, eval x_1 x ∈ s x_1 → eval x_1 x ∈ t x_1"], "goal": "x a ∈ t a"}}
+{"state": {"context": ["R : Type u", "Semiring R", "StrongRankCondition R", "α : Type u_1", "β : Type u_2", "Fintype α", "Fintype β", "f : (α →₀ R) →ₗ[R] β →₀ R", "i : Function.Injective ⇑f"], "goal": "Fintype.card α ≤ Fintype.card β"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommSemiring R", "S : Type v", "inst✝¹ : CommSemiring S", "p q : ℕ", "inst✝ : ExpChar R p", "f : R[X]", "n✝ : ℕ", "n_ih : map (frobenius R p ^ n✝) ((expand R (p ^ n✝)) f) = f ^ p ^ n✝"], "goal": "f ^ p ^ (n✝ + 1) = map (frobenius R p ^ (n✝ + 1)) ((expand R (p ^ (n✝ + 1))) f)"}}
+{"state": {"context": ["α : Type u_1", "Add α", "PartialOrder α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a b c d : α", "hac : a ≤ c", "hbd : b ≤ d"], "goal": "a + b = c + d ↔ a = c ∧ b = d"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝ : CancelMonoidWithZero M₀", "a b c : M₀"], "goal": "a * b = b ↔ a * b = 1 * b"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "Module R M", "Module R N", "Module R P", "r : R", "f : N →ₗ[R] P"], "goal": "LinearMap.lTensor M (r • f) = r • LinearMap.lTensor M f"}}
+{"state": {"context": ["α : Type u_1", "NormedAddCommGroup α", "Lattice α", "self : HasSolidNorm α", "x y : α"], "goal": "|x| ≤ |y| → ‖x‖ ≤ ‖y‖"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b : B", "f g : a ⟶ b", "CategoryTheory.BicategoricalCoherence f g"], "goal": " ⊗𝟙 = ⊗𝟙 ≫ (ρ_ g).inv"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "S : Type u_2", "inst✝ : CommRing S", "f : R →+* S", "I J : Ideal S", "P : Ideal R[X]", "Pb : P ≠ ⊥", "hP : ∀ (x : R), C x ∈ P → x = 0", "m : ↥P", "hm : m ≠ 0", "pp0 : Polynomial.map (Quotient.mk (comap C P)) ↑m = 0"], "goal": "comap C P = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "hab : a ≤ b", "hcd : c ≤ d", "h₁ : a ≤ d", "h₂ : c ≤ b"], "goal": "Ico a b ∪ Ico c d = Ico (min a c) (max b d)"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x y : α", "s t : Set α", "Φ : α → β", "r : ℝ≥0∞", "h : r < infEdist x s"], "goal": "∀ ⦃a : α⦄, a ∈ closedBall x r → a ∉ s"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "G : D ⥤ C", "adj : F ⊣ G", "X' X : C", "Y Y' : D", "f : F.obj X ⟶ Y", "g : Y ⟶ Y'"], "goal": "(adj.homEquiv X Y') (f ≫ g) = (adj.homEquiv X Y) f ≫ G.map g"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "a : G₀"], "goal": "IsUnit a ↔ a ≠ 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : Mul β", "f g : FreeMagma α →ₙ* β", "h : ⇑f ∘ of = ⇑g ∘ of", "x x✝ y✝ : FreeMagma α", "a✝¹ : f x✝ = g x✝", "a✝ : f y✝ = g y✝"], "goal": "f (x✝ * y✝) = g (x✝ * y✝)"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "x y : α"], "goal": "ConnectedComponents.mk x = ConnectedComponents.mk y ↔ x ∈ connectedComponent y"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "ι : Sort u_1", "hι : Nonempty ι", "S : ι → Subfield K", "hS : Directed (fun x x_1 => x ≤ x_1) S"], "goal": "↑(⨆ i, S i) = ⋃ i, ↑(S i)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "S : C", "E : CategoryTheory.PreOneHypercover S", "E.HasPullbacks", "mem₀ : E.sieve₀ ∈ J.sieves S", "mem₁' : ∀ (i₁ i₂ : E.I₀), E.sieve₁' i₁ i₂ ∈ J.sieves (CategoryTheory.Limits.pullback (E.f i₁) (E.f i₂))"], "goal": "(CategoryTheory.GrothendieckTopology.OneHypercover.mk' E mem₀ mem₁').toPreOneHypercover = E"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.Category.{w, v} J", "F : J ⥤ Type u", "c : CategoryTheory.Limits.Cone F"], "goal": "Nonempty (CategoryTheory.Limits.IsLimit c) ↔ ∀ s ∈ F.sections, ∃! x, ∀ (j : J), c.π.app j x = s j"}}
+{"state": {"context": ["R : Type u_1", "Semiring R"], "goal": "∀ (x x_1 : R[X]),\n  Polynomial.mul x x_1 =\n    match x, x_1 with\n    | { toFinsupp := a }, { toFinsupp := b } => { toFinsupp := a * b }"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "s : SignedMeasure α", "μ : Measure α", "inst✝ : SigmaFinite μ", "h : s.totalVariation ≪ μ.toENNRealVectorMeasure.ennrealToMeasure"], "goal": "μ.toENNRealVectorMeasure.ennrealToMeasure = μ"}}
+{"state": {"context": ["f g : CircleDeg1Lift", "n : ℕ", "this : 0 < ↑n + 1"], "goal": "dist ((f ^ (n + 1)) 0) (↑(n + 1) * τ f) ≤ 1"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X : C"], "goal": "(CategoryTheory.Monad.id C).obj X = X"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v, u₁} C", "D : Type u₂", "inst✝² : Category.{v, u₂} D", "S : D", "T : C ⥤ D", "inst✝¹ : HasLimits C", "inst✝ : PreservesLimits T", "A : StructuredArrow S T"], "goal": "∀ {a b : Subobject A}, { toFun := fun P => ⟨projectSubobject P, ⋯⟩, invFun := fun P => liftSubobject ↑P ⋯, left_inv := ⋯, right_inv := ⋯ } a ≤ { toFun := fun P => ⟨projectSubobject P, ⋯⟩, invFun := fun P => liftSubobject ↑P ⋯, left_inv := ⋯, right_inv := ⋯ } b ↔ a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α", "h : a ∈ l"], "goal": "∃ n, l.get? n = some a"}}
+{"state": {"context": ["α : Type u_3", "TopologicalSpace α", "self : LocallyConnectedSpace α", "x : α"], "goal": "(𝓝 x).HasBasis (fun s => IsOpen s ∧ x ∈ s ∧ IsConnected s) id"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasCoproducts Cᵒᵖ"], "goal": "CategoryTheory.Limits.HasProducts C"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = 0"], "goal": "x = y ↔ ‖x‖ = ‖y‖"}}
+{"state": {"context": ["α : Type u_2", "ConditionallyCompleteLinearOrderBot α", "s : Set α", "x : α", "hs : IsLUB s x", "hx : ¬Order.IsSuccLimit x"], "goal": "x ∈ s"}}
+{"state": {"context": ["R : Type u_1", "n : ℕ", "inst✝² : CommRing R", "a✝ b x y : R", "p : ℕ", "inst✝¹ : IsDomain R", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "hp : Prime ↑p", "hp1 : Odd p", "hxy : ↑p ∣ x - y", "hx : ¬↑p ∣ x", "a : ℕ", "h_ind : multiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (↑p) (x - y) + ↑a"], "goal": "¬↑p ∣ x ^ p ^ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : Type u_3", "n : Type u_4", "o : Type u_5", "m' : α → Type u_6", "n' : α → Type u_7", "R : Type v", "inst✝⁷ : CommRing R", "M N : Matrix m m R", "b : m → α", "inst✝⁶ : DecidableEq m", "inst✝⁵ : Fintype m", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq α", "inst���¹ : Fintype α", "inst✝ : LinearOrder α", "h : M.BlockTriangular b", "a : α", "x✝ : a ∈ univ", "ha : a ∉ image b univ"], "goal": "(M.toSquareBlock b a).det = 1"}}
+{"state": {"context": ["F : Type u → Type u", "q : QPF F", "α : Type u", "g : α → F α", "x : α"], "goal": "(QPF.corecF g x).dest = (QPF.P F).map (QPF.corecF g) (QPF.repr (g x))"}}
+{"state": {"context": [], "goal": "Measurable Real.arcsin"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "s t : TopologicalSpace.PositiveCompacts α", "h : ↑s = ↑t"], "goal": "s = t"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "PartialOrder α", "CommMonoid M", "f : α → M", "LocallyFiniteOrderTop α", "a : α"], "goal": "(∏ x ∈ Finset.Ioi a, f x) * f a = ∏ x ∈ Finset.Ici a, f x"}}
+{"state": {"context": ["α : Type u_1", "l : List α"], "goal": "l.length = 1 ↔ ∃ a, l = [a]"}}
+{"state": {"context": ["m : Type u_1", "n✝ : Type u_2", "inst✝⁴ : DecidableEq n✝", "inst✝³ : Fintype n✝", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "R : Type v", "inst✝ : CommRing R", "n : ℕ", "k✝ : Fin (n + 1)", "k : Fin n", "ih : ∀ (c : Fin n → R), (∀ (i : Fin n), k.castSucc < i.succ → c i = 0) → ∀ {M N : Matrix (Fin n.succ) (Fin n.succ) R}, (∀ (j : Fin n.succ), M 0 j = N 0 j) → (∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c i * M i.castSucc j) → M.det = N.det", "c : Fin n → R", "hc : ∀ (i : Fin n), k.succ < i.succ → c i = 0", "M N : Matrix (Fin n.succ) (Fin n.succ) R", "h0 : ∀ (j : Fin n.succ), M 0 j = N 0 j", "hsucc : ∀ (i : Fin n) (j : Fin n.succ), M i.succ j = N i.succ j + c i * M i.castSucc j", "M' : Matrix (Fin n.succ) (Fin n.succ) R := M.updateRow k.succ (N k.succ)", "hM' : M' = M.updateRow k.succ (N k.succ)", "hM : M = M'.updateRow k.succ (M' k.succ + c k • M k.castSucc)", "k_ne_succ : k.castSucc ≠ k.succ", "M_k : M k.castSucc = M' k.castSucc", "i : Fin n", "j : Fin n.succ"], "goal": "M' i.succ j = N i.succ j + update c k 0 i * M' i.castSucc j"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f : X ⟶ Y", "F : CategoryTheory.Limits.MonoFactorisation f", "Y' : C", "g : Y ⟶ Y'", "CategoryTheory.Mono g"], "goal": "(F.compMono g).e = F.e"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.Limits.HasCoequalizers C", "(X : C) →\n    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (CategoryTheory.MonoidalCategory.tensorLeft X)", "(X : C) →\n    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}\n      (CategoryTheory.MonoidalCategory.tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N : Bimod Y Z"], "goal": "M.whiskerLeft (𝟙 N) = 𝟙 (M.tensorBimod N)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "M₂ : Type u_3", "inst✝⁷ : TopologicalSpace M", "inst✝⁶ : TopologicalSpace M₂", "inst✝⁵ : Ring R", "inst✝⁴ : AddCommGroup M", "inst✝³ : TopologicalAddGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup M₂", "inst✝ : Module R M₂", "x✝¹ : M →L[R] M", "x✝ : M"], "goal": "(Ring.inverse x✝¹) x✝ = x✝¹.inverse x✝"}}
+{"state": {"context": ["I : Type u", "inst✝²⁵ : AddMonoid I", "C : Type u_1", "inst✝²⁴ : Category.{u_2, u_1} C", "inst✝²³ : MonoidalCategory C", "X₁ X₂ X₃ X₄ : GradedObject I C", "inst✝²² : X₁.HasTensor X₂", "inst✝²¹ : X₂.HasTensor X₃", "inst✝²⁰ : X₃.HasTensor X₄", "inst✝¹⁹ : (tensorObj X₁ X₂).HasTensor X₃", "inst✝¹⁸ : X₁.HasTensor (tensorObj X₂ X₃)", "inst✝¹⁷ : (tensorObj X₂ X₃).HasTensor X₄", "inst✝¹⁶ : X₂.HasTensor (tensorObj X₃ X₄)", "inst✝¹⁵ : (tensorObj (tensorObj X₁ X₂) X₃).HasTensor X₄", "inst✝¹⁴ : (tensorObj X₁ (tensorObj X₂ X₃)).HasTensor X₄", "inst✝¹³ : X₁.HasTensor (tensorObj (tensorObj X₂ X₃) X₄)", "inst✝¹² : X₁.HasTensor (tensorObj X₂ (tensorObj X₃ X₄))", "inst✝¹¹ : (tensorObj X₁ X₂).HasTensor (tensorObj X₃ X₄)", "inst✝¹⁰ : X₁.HasGoodTensor₁₂Tensor X₂ X₃", "inst✝⁹ : X₁.HasGoodTensorTensor₂₃ X₂ X₃", "inst✝⁸ : X₁.HasGoodTensor₁₂Tensor (tensorObj X₂ X₃) X₄", "inst✝⁷ : X₁.HasGoodTensorTensor₂₃ (tensorObj X₂ X₃) X₄", "inst✝⁶ : X₂.HasGoodTensor₁₂Tensor X₃ X₄", "inst✝⁵ : X₂.HasGoodTensorTensor₂₃ X₃ X₄", "inst✝⁴ : (tensorObj X₁ X₂).HasGoodTensor₁₂Tensor X₃ X₄", "inst✝³ : (tensorObj X₁ X₂).HasGoodTensorTensor₂₃ X₃ X₄", "inst✝² : X₁.HasGoodTensor₁₂Tensor X₂ (tensorObj X₃ X₄)", "inst✝¹ : X₁.HasGoodTensorTensor₂₃ X₂ (tensorObj X₃ X₄)", "inst✝ : X₁.HasTensor₄ObjExt X₂ X₃ X₄", "j i₁ i₂ i₃ i₄ : I", "h : i₁ + i₂ + i₃ + i₄ = j"], "goal": "(α_ (X₁ i₁) (X₂ i₂) (X₃ i₃ ⊗ X₄ i₄)).inv ≫ (α_ (X₁ i₁ ⊗ X₂ i₂) (X₃ i₃) (X₄ i₄)).inv ≫ ιTensorObj₃' X₁ X₂ X₃ i₁ i₂ i₃ (i₁ + i₂ + i₃) ⋯ ▷ X₄ i₄ ≫ ιTensorObj (tensorObj (tensorObj X₁ X₂) X₃) X₄ (i₁ + i₂ + i₃) i₄ j ⋯ = ιTensorObj₄ X₁ X₂ X₃ X₄ i₁ i₂ i₃ i₄ j h ≫ (associator X₁ X₂ (tensorObj X₃ X₄)).inv j ≫ (associator (tensorObj X₁ X₂) X₃ X₄).inv j"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "x : E", "r : ℝ", "hr✝ : r ≠ 0", "hr : 0 < r"], "goal": "interior (closedBall x r) ⊆ ball x r"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "p : G.Walk u v"], "goal": "p.reverse.toSubgraph = p.toSubgraph"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "E' : Type u_3", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "F : Type u_4", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "H : Type u_5", "TopologicalSpace H", "H' : Type u_6", "TopologicalSpace H'", "I : ModelWithCorners 𝕜 E H", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_9", "TopologicalSpace M", "ChartedSpace H M", "M' : Type u_10", "TopologicalSpace M'", "ChartedSpace H' M'", "e : E ≃ₘ⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F", "f : M' → M"], "goal": "Smooth I' (I.transDiffeomorph e) f ↔ Smooth I' I f"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "inst✝³ : Semiring R", "φ : ι → Type u_3", "inst✝² : (i : ι) → AddCommMonoid (φ i)", "inst✝¹ : (i : ι) → Module R (φ i)", "inst✝ : DecidableEq ι", "I J : Set ι", "h : Disjoint I J"], "goal": "∀ ⦃x : (i : ι) → φ i⦄, ((∀ i ∈ Iᶜ, x i = 0) ∧ ∀ i ∈ Jᶜ, x i = 0) → ∀ (a : ι), x a = 0 a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν✝ ν μ : Measure α", "inst✝¹ : IsFiniteMeasure ν", "inst✝ : ν.HaveLebesgueDecomposition μ", "r : ℝ≥0∞", "hr : r ≠ ⊤", "h : (r.toNNReal • ν).rnDeriv μ =ᶠ[ae μ] r.toNNReal • ν.rnDeriv μ"], "goal": "(r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ"}}
+{"state": {"context": ["α : Type u_1", "s t : Finset α", "h : t.card ≤ s.card"], "goal": "s ⊆ t ↔ s = t"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "InvolutiveNeg α", "s : Finset α"], "goal": "s.preimage (fun x => -x) ⋯ = -s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrderedField α", "Ring β", "abv : β → α", "f g : CauSeq β abv", "h : ∀ (i : ℕ), ↑f i = ↑g i"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u_1"], "goal": "(EuclideanSpace.measurableEquiv ι).toEquiv = WithLp.equiv 2 ((i : ι) → (fun x => ℝ) i)"}}
+{"state": {"context": ["p : ℕ", "hp : Prime p", "this : p = p - 2 + 2"], "goal": "(p - 2 + 2).minFac :: ((p - 2 + 2) / (p - 2 + 2).minFac).primeFactorsList = [p - 2 + 2]"}}
+{"state": {"context": ["𝕜 : Type u", "hnorm : NontriviallyNormedField 𝕜", "E : Type v", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "TopologicalAddGroup E", "ContinuousSMul 𝕜 E", "F' : Type x", "AddCommGroup F'", "Module 𝕜 F'", "TopologicalSpace F'", "TopologicalAddGroup F'", "ContinuousSMul 𝕜 F'", "CompleteSpace 𝕜", "T2Space E", "FiniteDimensional 𝕜 E", "f : E →ₗ[𝕜] F'"], "goal": "LinearMap.ker (LinearMap.toContinuousLinearMap f) = LinearMap.ker f"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "m : Ordnode α", "y : α", "r : Ordnode α"], "goal": "(l.node4R x m y r).dual = r.dual.node4L y m.dual x l.dual"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "π : ι → Set (Set α)", "i : ι", "s : Set α"], "goal": "s ∈ piiUnionInter π {i} ↔ s ∈ π i ∪ {univ}"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝² : Nonempty Ω", "m0 : MeasurableSpace Ω", "μ : FiniteMeasure Ω", "inst✝¹ : TopologicalSpace Ω", "inst✝ : OpensMeasurableSpace Ω", "γ : Type u_2", "F : Filter γ", "μs : γ → FiniteMeasure Ω", "μs_lim : Tendsto (ProbabilityMeasure.toFiniteMeasure ∘ fun i => (μs i).normalize) F (𝓝 μ.normalize.toFiniteMeasure)", "mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 μ.mass)", "f : Ω →ᵇ ℝ≥0", "h_mass : ¬μ.mass = 0"], "goal": "Tendsto (fun i => (μs i).mass * (μs i).normalize.toFiniteMeasure.testAgainstNN f) F (𝓝 (μ.mass * μ.normalize.toFiniteMeasure.testAgainstNN f))"}}
+{"state": {"context": ["L : Language", "M : Type u_1", "inst✝¹ : L.Structure M", "N : Type u_2", "inst✝ : L.Structure N", "f : M ↪[L] N", "s : L.Substructure M", "t : Finset N", "h : (closure L).toFun ↑t = Substructure.map f.toHom s", "hf : Function.Injective ⇑f.toHom", "x : N", "hx : x ∈ ↑t"], "goal": "x ∈ range ⇑f"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "NonUnitalSemiring α", "Fintype n", "StarRing α", "A : Matrix m n α", "x : n → α"], "goal": "x ᵥ* Aᴴ = star (A *ᵥ star x)"}}
+{"state": {"context": ["n m : ℕ", "P : MvPFunctor.{u} n", "Q : Fin2 n → MvPFunctor.{u} m", "α✝ β : TypeVec.{u} m", "α : TypeVec.{u} n", "a : P.A", "f : P.B a ⟹ α", "i : Fin2 n", "x : α i"], "goal": "x ∈ supp ⟨a, f⟩ i ↔ x ∈ f i '' univ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "inst✝² : TopologicalSpace α", "inst✝¹ : SecondCountableTopology α", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : IsLocallyFiniteMeasure μ", "h✝ : Nonempty α", "inhabited_h : Inhabited α", "S : Set (Set α) := {s | IsOpen s ∧ μ s < ⊤}", "T : Set (Set α)", "T_count : T.Countable", "TS : T ⊆ S", "hT : ⋃₀ T = univ", "T_ne : T.Nonempty", "f : ℕ → Set α", "hf : T = range f", "fS : ∀ (n : ℕ), f n ∈ S"], "goal": "⋃ i, f i = univ"}}
+{"state": {"context": ["R : Type u", "inst✝³ : NonUnitalSemiring R", "inst✝² : PartialOrder R", "inst✝¹ : StarRing R", "inst✝ : StarOrderedRing R", "x y : R"], "goal": "∀ (x y : R), x ≤ y → star x ≤ star y"}}
+{"state": {"context": ["α : Type u", "l : List α", "n m : ℕ", "hml : m < l.length", "hm : m < (drop (n % l.length) l).length"], "goal": "(drop (n % l.length) l ++ take (n % l.length) l)[m]? = l[(m + n) % l.length]?"}}
+{"state": {"context": ["ι : Type v", "β : ι → Type w", "(i : ι) → AddCommMonoid (β i)", "DecidableEq ι", "(i : ι) → (x : β i) → Decidable (x ≠ 0)", "x : DirectSum ι fun i => β i"], "goal": "∑ i ∈ DFinsupp.support x, (DirectSum.of β i) (x i) = x"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α"], "goal": "toIcoMod hp (-a) b = p - toIocMod hp a (-b)"}}
+{"state": {"context": ["D : GlueData", "i j : D.J", "U : Set ↑(D.U i)"], "goal": "Function.Bijective ((forget TopCat).map (D.t i j))"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "S : Type u_4", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing A", "inst✝⁸ : Ring B", "inst✝⁷ : CommRing S", "inst✝⁶ : Algebra R A", "inst✝⁵ : Algebra R B", "f✝ : R →+* S", "M : Type u_5", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Module A M", "inst✝¹ : IsScalarTower R A M", "inst✝ : NoZeroSMulDivisors A M", "N : Submodule R M", "hN : N ≠ ⊥", "hN' : N.FG", "x : A", "hx : ∀ n ∈ N, x • n ∈ N", "A' : Subalgebra R A := { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯, algebraMap_mem' := ⋯ }", "f : ↥A' →ₐ[R] Module.End R ↥N := AlgHom.ofLinearMap { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯", "a : M", "ha₁ : a ∈ N", "ha₂ : a ≠ 0", "this : Function.Injective ⇑f"], "goal": "IsIntegral R x"}}
+{"state": {"context": ["Ω : Type u_1", "Ω' : Type u_2", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSpace Ω'", "ν : FiniteMeasure Ω", "f : Ω → Ω'", "f_aemble : AEMeasurable f ↑ν", "A : Set Ω'", "A_mble : MeasurableSet A", "this : ↑(ν.map f) A = ↑ν (f ⁻¹' A)"], "goal": "(ν.map f) A = ν (f ⁻¹' A)"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "a b : α", "h : a ⩿ b"], "goal": "Set.Ioc a b ⊆ {b}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁴ : AddMonoidWithOne α", "inst✝³ : PartialOrder α", "inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : ZeroLEOneClass α", "inst✝ : CharZero α", "m n : ℕ"], "goal": "1 ≤ ↑n ↔ 1 ≤ n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "R : Type u_4", "Zero X", "Zero Y", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace R", "CommSemiring R", "StarRing R", "TopologicalSemiring R", "ContinuousStar R", "f : ContinuousMapZero X Y", "g : ContinuousMapZero Y R"], "goal": "(ContinuousMapZero.nonUnitalStarAlgHom_precomp R f) g = g.comp f"}}
+{"state": {"context": ["X : Type u_1", "LinearOrder X", "TopologicalSpace X", "OrderTopology X", "a : X", "s t : Set X", "hd : Disjoint s (closure t)", "ha : a ∈ s"], "goal": "(s.ordSeparatingSet t).ordConnectedSectionᶜ ∈ 𝓝[≤] a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "l : Filter α", "f : α → β", "lc : Filter γ", "g : γ → α", "hg : Filter.Tendsto g lc l"], "goal": "(↑f).compTendsto g hg = ↑(f ∘ g)"}}
+{"state": {"context": ["M : Type u_5", "CommMonoid M", "s : List M"], "goal": "(List.map (fun i => {i}) s).prod = {s.prod}"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : AddCancelMonoid A", "inst✝ : HasAntidiagonal A", "p q : A × A", "n : A", "hp : p ∈ antidiagonal n", "hq : q ∈ antidiagonal n"], "goal": "p = q ↔ p.1 = q.1"}}
+{"state": {"context": ["xl xr : Type u", "x y : PGame"], "goal": "-x ‖ y ↔ x ‖ -y"}}
+{"state": {"context": ["J : Type w", "C : Type uC", "inst✝³ : Category.{uC', uC} C", "inst✝² : HasZeroMorphisms C", "D : Type uD", "inst✝¹ : Category.{uD', uD} D", "inst✝ : HasZeroMorphisms D", "F f : J → C", "t : Bicone f", "ht : IsLimit t.toCone", "j : J", "j' : Discrete J"], "goal": "t.ι j ≫ t.toCone.π.app j' = (Fan.mk (f j) fun j' => if h : j = j' then eqToHom ⋯ else 0).π.app j'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Ring α", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "a : ℕ", "b : ℤ"], "goal": "multiplicity a b.natAbs = multiplicity (↑a) b"}}
+{"state": {"context": ["R : Type u", "Semiring R", "a : R", "m : ℕ"], "goal": "((Polynomial.monomial m) a).natDegree ≤ m"}}
+{"state": {"context": ["n d k : ℕ"], "goal": "ThreeAPFree ↑(Behrend.sphere n d k)"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : MonoidalCategory C", "P : C → Prop", "inst✝² : MonoidalPredicate P", "P' : C → Prop", "inst✝¹ : MonoidalPredicate P'", "inst✝ : BraidedCategory C", "X Y : FullSubcategory P"], "goal": "(fullMonoidalSubcategoryInclusion P).map (β_ X Y).hom = inv ((fullMonoidalSubcategoryInclusion P).μ X Y) ≫ (β_ ((fullMonoidalSubcategoryInclusion P).obj X) ((fullMonoidalSubcategoryInclusion P).obj Y)).hom ≫ (fullMonoidalSubcategoryInclusion P).μ Y X"}}
+{"state": {"context": ["a b m n k : ℕ", "p : ℕ → Prop", "H' : ¬p 0", "H : ∃ n, p n"], "goal": "∃ n, ¬p n ∧ p (n + 1)"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} n → Type u_1", "q : MvQPF F", "α : TypeVec.{u} n", "i : Fin2 n", "x : F α"], "goal": "MvFunctor.supp x i = {u | ∀ (a : (MvQPF.P F).A) (f : (MvQPF.P F).B a ⟹ α), MvQPF.abs ⟨a, f⟩ = x → u ∈ f i '' Set.univ}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : NormedAddCommGroup β", "f g✝ : ι → α → β", "p : ℝ≥0∞", "g : α → β", "hp_ne_top : p ≠ ⊤", "hg : Memℒp g p μ", "ε : ℝ≥0", "hε : 0 < ε", "hε_top : ¬↑ε = ⊤", "s : Set α", "left✝ : MeasurableSet s", "hμs : μ s < ⊤", "hgε : eLpNorm (sᶜ.indicator g) p μ < ↑ε"], "goal": "∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator ((fun x => g) i)) p μ ≤ ↑ε"}}
+{"state": {"context": ["V : Type u", "inst✝¹ : Category.{v, u} V", "inst✝ : HasZeroMorphisms V", "C : CochainComplex V ℕ", "i : ℕ"], "goal": "C.d 0 (i + 2) = 0"}}
+{"state": {"context": ["I : Type u", "LinearOrder I", "IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "l : Profinite.NobelingProof.Products I", "o : Ordinal.{u}", "hl : ∀ i ∈ ↑l, Profinite.NobelingProof.ord I i < o"], "goal": "Profinite.NobelingProof.Products.eval C '' {m | m < l} =\n  ⇑(Profinite.NobelingProof.πs C o) ''\n    (Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.π C fun x => Profinite.NobelingProof.ord I x < o) ''\n      {m | m < l})"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "s : Set α", "h : s.IsWF", "a : α"], "goal": "(insert a s).IsWF"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "E : ι → Type u_3", "F : Type u_4", "NormedField 𝕜", "(i : ι) → TopologicalSpace (E i)", "(i : ι) → AddCommGroup (E i)", "(i : ι) → Module 𝕜 (E i)", "AddCommGroup F", "Module 𝕜 F", "TopologicalSpace F", "TopologicalAddGroup F"], "goal": "(𝓝 0).HasBasis (fun SV => Bornology.IsVonNBounded 𝕜 SV.1 ∧ SV.2 ∈ 𝓝 0) fun SV => {f | Set.MapsTo (⇑f) SV.1 SV.2}"}}
+{"state": {"context": ["M : Type u", "inst✝² : AddCommGroup M", "inst✝¹ : Module ℤ M", "inst✝ : Finite ℤ M", "hM : IsTorsion ℤ M", "ι : Type", "w✝ : Fintype ι", "p : ι → ℤ", "h : ∀ (i : ι), Irreducible (p i)", "e : ι → ℕ", "l : M ≃ₗ[ℤ] ⨁ (i : ι), ℤ ⧸ Submodule.span ℤ {p i ^ e i}"], "goal": "_root_.Finite M"}}
+{"state": {"context": ["R : Type u", "inst✝³ : CommRing R", "R' : Type v", "inst✝² : CommMonoid R'", "C : Type v", "inst✝¹ : CommMonoid C", "n : ℕ", "inst✝ : NeZero n", "ψ : AddChar (ZMod n) C", "hψ : ψ.IsPrimitive", "a : ZMod n", "h : ψ a = 1"], "goal": "¬ψ.mulShift a ≠ 1"}}
+{"state": {"context": ["n p : ℕ"], "goal": "p ∈ n.primeFactorsList ↔ Prime p ∧ p ∣ n ∧ n ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommGroupWithZero R", "f g : ArithmeticFunction R", "hf : f.IsMultiplicative", "hg : g.IsMultiplicative", "m n : ℕ", "cop : m.Coprime n"], "goal": "(f.pdiv g) (m * n) = (f.pdiv g) m * (f.pdiv g) n"}}
+{"state": {"context": ["E : Type u_2", "AddCommGroup E", "Module ℝ E", "s : Set E", "x : E", "TopologicalSpace E", "TopologicalAddGroup E", "ContinuousSMul ℝ E", "hc : Convex ℝ s", "hs₀ : s ∈ 𝓝 0"], "goal": "gauge s x ≤ 1 ↔ x ∈ closure s"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "s : ι → Set α", "a : α", "h : ∀ (i : ι), a ∈ s i"], "goal": "a ∈ ⋂ i, s i"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "CategoryTheory.Limits.HasPullbacks C", "f g : CategoryTheory.Under X", "h : f ⟶ g", "CategoryTheory.Mono h"], "goal": "CategoryTheory.Mono h.right"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝¹ : (i : α) → NormedAddCommGroup (E i)", "ι : Type u_3", "inst✝ : PseudoMetricSpace α", "g : α → �� → ℝ", "K : ℝ≥0", "hg : ∀ (i : ι), LipschitzWith K fun x => g x i", "a₀ a : α", "M : ℝ", "hM : M ∈ upperBounds (Set.range fun i => ‖g a₀ i‖)"], "goal": "Memℓp (g a) ⊤"}}
+{"state": {"context": ["p q : ℚ", "hq : 0 ≤ q", "hp : 0 ≤ p"], "goal": "(q + p).toNNRat = q.toNNRat + p.toNNRat"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y p₂ : P", "h : p₂ ∈ s", "a✝ : x ∉ s", "hy : y ∉ s"], "goal": "(y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)) ↔ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)"}}
+{"state": {"context": ["S : LightProfinite", "n m : ℕ", "h : n ≤ m"], "goal": "⇑(S.transitionMapLE h) ∘ ⇑(S.proj m) = ⇑(S.proj m ≫ S.diagram.map (Opposite.op (homOfLE h)))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β", "μ✝ ν ν₁ ν₂ : Measure α", "s t : Set α", "X : Type u_5", "inst✝¹ : MeasurableSpace X", "μ : Measure X", "inst✝ : IsFiniteMeasure μ", "Es : ℕ → Set X", "Es_mble : ∀ (i : ℕ), MeasurableSet (Es i)", "Es_disj : Pairwise fun n m => Disjoint (Es n) (Es m)", "decr : Antitone fun n => ⋃ i, ⋃ (_ : i ≥ n), Es i"], "goal": "Tendsto (⇑μ ∘ fun n => ⋃ i, ⋃ (_ : i ≥ n), Es i) atTop (𝓝 0)"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "D : Type u'", "inst✝¹ : Category.{v', u'} D", "inst✝ : Preadditive D", "n₁ n₂ : ℤ", "K : CochainComplex C ℤ", "x✝² x✝¹ : ℤ", "x✝ : (ComplexShape.up ℤ).Rel x✝² x✝¹"], "goal": "(XIsoOfEq K ⋯).hom ≫ (n₂.negOnePow • n₁.negOnePow • K.d (x✝² + n₂ + n₁) (x✝¹ + n₂ + n₁)) = ((n₁ + n₂).negOnePow • K.d (x✝² + (n₁ + n₂)) (x✝¹ + (n₁ + n₂))) ≫ (XIsoOfEq K ⋯).hom"}}
+{"state": {"context": ["R : Type u", "CommRing R", "E : EllipticCurve R"], "goal": "E.variableChange WeierstrassCurve.VariableChange.id = E"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "n✝ : ℕ", "P : ℕ → Prop", "this : IsPoly p fun {R} [CommRing R] x => select P x + select (fun i => ¬P i) x", "R : Type u_1", "R._inst : CommRing R", "x : 𝕎 R", "n m : ℕ", "x✝ : m ∈ Finset.range (n + 1)"], "goal": "(if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)"}}
+{"state": {"context": ["L : FirstOrder.Language", "α : Type u'", "k m n : ℕ", "km : k ≤ m", "mn : m ≤ n"], "goal": "FirstOrder.Language.BoundedFormula.castLE mn ∘ FirstOrder.Language.BoundedFormula.castLE km =\n  FirstOrder.Language.BoundedFormula.castLE ⋯"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝ : MonoidWithZero M₀", "x y : M₀", "h : IsUnit x"], "goal": "inverse x * (x * y) = y"}}
+{"state": {"context": ["v : ℤ", "h : 1 * v = -1"], "goal": "1 = 1 ∧ v = -1 ∨ 1 = -1 ∧ v = 1"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "s : Set A"], "goal": "(NonUnitalAlgebra.adjoin R s).toSubmodule = Submodule.span R ↑(Subsemigroup.closure s)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "B : Type u_3", "Semiring B", "TopologicalSpace B", "Algebra R A", "Algebra R B"], "goal": "↑(ContinuousAlgHom.snd R A B) = AlgHom.snd R A B"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimit (CategoryTheory.Limits.Types.constPUnitFunctor C)"], "goal": "CategoryTheory.IsConnected C ↔\n  Nonempty (CategoryTheory.Limits.colimit (CategoryTheory.Limits.Types.constPUnitFunctor C) ≅ PUnit.{w + 1})"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "P : Algebra.Generators R S", "R' : Type u_1", "S' : Type u_2", "CommRing R'", "CommRing S'", "Algebra R' S'", "P' : Algebra.Generators R' S'", "Algebra R R'", "Algebra S S'", "Algebra R S'", "IsScalarTower R R' S'", "IsScalarTower R S S'", "f : P.Hom P'"], "goal": "f.comp (Algebra.Generators.Hom.id P) = f"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "f : ↥(MeasureTheory.Lp.simpleFunc E p μ)"], "goal": "↑(MeasureTheory.Lp.simpleFunc.toSimpleFunc f) =ᵐ[μ] ↑↑↑f"}}
+{"state": {"context": ["p : Nat.Primes", "s : ℂ", "hs : 1 < s.re"], "goal": "‖↑↑p ^ (-s)‖ ≤ 1 / 2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Ring α", "F : Type u_1", "Ring β", "FunLike F α β", "RingHomClass F α β", "f : F", "u : αˣ"], "goal": "(Units.map ↑f) (-u) = -(Units.map ↑f) u"}}
+{"state": {"context": ["p : ℕ → Prop", "DecidablePred p"], "goal": "Nat.count p (Nat.nth p 0) = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "μ ν ν₁ ν₂ : Measure α", "s✝ t : Set α", "γ : Type u_5", "f g : α → γ", "s : Set α", "inst✝ : DecidablePred fun x => x ∈ s", "hs_zero : μ s = 0"], "goal": "(fun x => if x ∈ s then f x else g x) =ᶠ[ae μ] g"}}
+{"state": {"context": ["α : Type u_1", "Finite α", "h : Nontrivial α"], "goal": "1 < Nat.card α"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "v : E → E", "t₀ : ℝ", "x₀ : E", "inst✝ : CompleteSpace E", "hv : ContDiffAt ℝ 1 v x₀", "ε : ℝ", "hε : ε > 0", "w✝² : ℝ≥0", "w✝¹ w✝ : ℝ", "hpl : IsPicardLindelof (fun x => v) (t₀ - ε) t₀ (t₀ + ε) x₀ w✝² w✝¹ w✝"], "goal": "∃ f, f t₀ = x₀ ∧ ∃ ε > 0, ∀ t ∈ Ioo (t₀ - ε) (t₀ + ε), HasDerivAt f (v (f t)) t"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a : α", "h₁ : 0 < a", "h₂ : a < 1"], "goal": "1 < a⁻¹"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "TopologicalSpace R", "TopologicalSpace M", "CommSemiring R", "AddCommMonoid M", "Module R M", "x : tsze R M"], "goal": "(TrivSqZeroExt.sndCLM R M) x = x.snd"}}
+{"state": {"context": ["G : Type u_1", "Mul G", "self : UniqueProds G", "A B : Finset G"], "goal": "A.Nonempty → B.Nonempty → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0"}}
+{"state": {"context": [], "goal": "(Sheaf.finestTopologySingle (yoneda.obj (ULift.{u, 0} Bool))).sieves = typesGrothendieckTopology.sieves"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "FiniteDimensional ℝ E"], "goal": "Besicovitch.multiplicity E ≤ 5 ^ FiniteDimensional.finrank ℝ E"}}
+{"state": {"context": ["a b : ℝ", "f : ℝ → ℝ", "c : ContinuousOn f (Set.Icc a b)", "ε : ℝ", "pos : 0 < ε", "f' : C(↑(Set.Icc a b), ℝ) := { toFun := fun x => f ↑x, continuous_toFun := ⋯ }"], "goal": "∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : TopologicalSpace α", "x : EReal", "hx : x < ⊤", "y : ℚ", "hxy : x < ↑↑y"], "goal": "∃ i', True ∧ Ioi ↑i' ⊆ Ioi x"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "a : α", "ha : 0 ≤ a"], "goal": "↑⌈a⌉₊ < a + 1"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "p a b : α", "c : α", "h : a ≡ b [PMOD p]"], "goal": "c - a ≡ c - b [PMOD p]"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "CompactSpace Y"], "goal": "IsClosedMap Prod.fst"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β", "g : γ → α", "s : Set γ", "t : Set β"], "goal": "Set.MapsTo f (g '' s) t ↔ Set.MapsTo (f ∘ g) s t"}}
+{"state": {"context": ["α : Type u_1", "P : Set α → Prop", "m : (s : Set α) → P s → ℝ≥0∞", "P0 : P ∅", "m0 : m ∅ P0 = 0", "PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)", "mU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯", "msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯", "m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂", "s : Set α"], "goal": "⨅ t, ⨅ (ht : P t), ⨅ (_ : s ⊆ t), m t ht ≤ (inducedOuterMeasure m P0 m0) s"}}
+{"state": {"context": ["K : Type u_1", "Field K", "t : ℕ", "Invertible ↑t"], "goal": "¬ringChar K ∣ t"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "I : CategoryTheory.Limits.MulticospanIndex C", "CategoryTheory.Limits.HasProduct I.left", "CategoryTheory.Limits.HasProduct I.right", "K₁ K₂ : CategoryTheory.Limits.Multifork I", "f : K₁ ⟶ K₂"], "goal": "(I.toPiForkFunctor.map f).hom = f.hom"}}
+{"state": {"context": ["n : ℕ", "self : Fin n"], "goal": "(Fin.valOrderEmb n) self = ↑self"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : CompleteSpace E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "μ ν : Measure α", "s t : Set α", "f g : α → E", "h : s =ᶠ[ae μ] t"], "goal": "⨍ (x : α) in s, f x ∂μ = ⨍ (x : α) in t, f x ∂μ"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "NontriviallyNormedField 𝕜₁", "NormedField 𝕜₂", "σ₁₂ : 𝕜₁ →+* 𝕜₂", "M₁ : Type u_3", "M₂ : Type u_4", "SeminormedAddCommGroup M₁", "AddCommGroup M₂", "NormedSpace 𝕜₁ M₁", "Module 𝕜₂ M₂", "UniformSpace M₂", "UniformAddGroup M₂", "ContinuousConstSMul 𝕜₂ M₂", "T2Space M₂", "CompleteSpace M₂"], "goal": "(compactOperator σ₁₂ M₁ M₂).topologicalClosure = compactOperator σ₁₂ M₁ M₂"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "f : CategoryTheory.End X"], "goal": "IsUnit f ↔ CategoryTheory.IsIso f"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "f : α → E", "g : α → F", "l : Filter α", "hf : Tendsto f l (𝓝 1)", "op : E → F → G", "A : ℝ", "h_op : ∀ (x : E) (y : F), ‖op x y‖ ≤ A * ‖x‖ * ‖y‖", "C : ℝ", "hC : ∀ᶠ (x : ℝ) in map (Norm.norm ∘ g) l, (fun x x_1 => x ≤ x_1) x C"], "goal": "Tendsto (fun x => op (f x) (g x)) l (𝓝 1)"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "inst✝² : Group G", "inst✝¹ : MulAction G α", "inst✝ : DecidableEq α", "S : Set (Equiv.Perm α)", "hS : ∀ f ∈ S, f.IsSwap", "supp : Set α", "fin : supp.Finite"], "goal": "∀ {f : Equiv.Perm α}, (∀ (x : α), f x ∈ orbit (↥(closure S)) x) → (fixedBy α f)ᶜ ⊆ supp → f ∈ closure S"}}
+{"state": {"context": ["x : PGame", "h : x ⧏ 0"], "goal": "x⁻¹ = -(-x).inv'"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve.Affine R", "Nontrivial R"], "goal": "W.polynomial.natDegree = 2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α ≃. β"], "goal": "f.symm.trans f = PEquiv.ofSet {b | (f.symm b).isSome = true}"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "ι : Type w", "b : Finset ι", "h : Basis { x // x ∈ b } R M"], "goal": "FiniteDimensional.finrank R M = b.card"}}
+{"state": {"context": ["α : Type u_2", "AddZeroClass α", "a : α"], "goal": "Set.singletonAddMonoidHom a = {a}"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "A : Type u_4", "inst✝² : DecidableEq ι", "inst✝¹ : AddMonoid ι", "inst✝ : Semiring M", "f g : M[ι]"], "goal": "(f * g).toDirectSum = f.toDirectSum * g.toDirectSum"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : Unit → α"], "goal": "Measurable f"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Group α", "a : α", "MulAction α β", "b : β", "k : ℤ"], "goal": "(MulAction.orbitZPowersEquiv a b).symm ↑k = ⟨a, ⋯⟩ ^ k • ⟨b, ⋯⟩"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F G : J ⥤ C", "s : CategoryTheory.Limits.Cocone G", "r t : CategoryTheory.Limits.Cocone F", "P : CategoryTheory.Limits.IsColimit t", "Q : CategoryTheory.Limits.IsColimit s", "w : F ≅ G"], "goal": "(P.coconePointsIsoOfNatIso Q w).inv ≫ P.desc r = Q.map r w.inv"}}
+{"state": {"context": ["α : Type u_1", "a b c : α", "MulOneClass α", "Zero α", "Preorder α", "MulPosMono α", "h : a < b * c", "hb : b ≤ 1", "hc : 0 ≤ c"], "goal": "a < c"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝² : CommMonoid M", "inst✝¹ : CommMonoid N", "inst✝ : DivisionCommMonoid G", "k l : ℕ", "ζ : G", "h : IsPrimitiveRoot ζ⁻¹ k"], "goal": "IsPrimitiveRoot ζ k"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "s t : SignedMeasure α", "f : α → ℝ", "hf : Measurable f", "hfi : Integrable f μ", "htμ : t ⟂ᵥ μ.toENNRealVectorMeasure", "hadd : s = t + μ.withDensityᵥ f", "this✝ : IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal (f x))", "this : IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal ((-f) x))"], "goal": "s = (JordanDecomposition.mk (t.toJordanDecomposition.posPart + μ.withDensity fun x => ENNReal.ofReal (f x)) (t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x)) ⋯).toSignedMeasure"}}
+{"state": {"context": ["α : Type u_4", "β : Type u_5", "r : α → β → Prop", "(a : α) → DecidablePred (r a)", "s₁ s₂ : Finset α", "t₁ t₂ : Finset β", "hs : s₂ ⊆ s₁", "ht : t₂ ⊆ t₁"], "goal": "Rel.interedges r s₂ t₂ ⊆ Rel.interedges r s₁ t₁"}}
+{"state": {"context": ["R : Type u_1", "M M₁ M₂ M₃ : Type u", "M' : Type v", "inst✝¹¹ : Ring R", "inst✝¹⁰ : AddCommGroup M", "inst✝⁹ : AddCommGroup M₁", "inst✝⁸ : AddCommGroup M₂", "inst✝⁷ : AddCommGroup M₃", "inst✝⁶ : AddCommGroup M'", "inst✝⁵ : Module R M", "inst✝⁴ : Module R M₁", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M'", "inst✝ : HasRankNullity.{u, u_1} R", "N : Submodule R M", "h : Module.rank R ↥N < Module.rank R M", "this : ∃ x, ∀ (a : R), ¬a = 0 → a • x ∉ N"], "goal": "∃ m, ∀ (r : R), r ≠ 0 → r • m ∉ N"}}
+{"state": {"context": ["R : Type u_1", "LinearOrderedField R", "M : Type u_2", "AddCommGroup M", "Module R M", "ι : Type u_3", "Fintype ι", "_i : FiniteDimensional R M", "x : Orientation R M ι", "f : M ≃ₗ[R] M", "h : Fintype.card ι = FiniteDimensional.finrank R M"], "goal": "(Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "σ : Type u_3", "inst✝² : Primcodable α", "inst✝¹ : Primcodable β", "inst✝ : Primcodable σ", "H : Nat.Primrec fun n => encode (decode n)", "f : α → List β", "g : α → σ", "h : α → σ × β → σ", "hf : Primrec f", "hg : Primrec g", "hh : Primrec₂ h", "this : Primcodable (List β) := prim H", "G : α → σ × List β → σ × List β := fun a IH => List.casesOn IH.2 IH fun b l => (h a (IH.1, b), l)", "hG : Primrec₂ G", "F : α → ℕ → σ × List β := fun a n => (G a)^[n] (g a, f a)"], "goal": "Primrec fun a => List.foldl (fun s b => h a (s, b)) (g a) (f a)"}}
+{"state": {"context": ["a b c : Nat"], "goal": "a % (b * c) % b = a % b"}}
+{"state": {"context": ["R : Type u_1", "P Q : Cubic R", "Semiring R", "h : P.toPoly = Q.toPoly"], "goal": "P.b = Q.b"}}
+{"state": {"context": [], "goal": "↑π ^ (1 / 2) = ↑√π"}}
+{"state": {"context": ["R : Type u", "Ring R", "s t : Set R", "h : s ⊆ t"], "goal": "Subring.closure s ≤ Subring.closure t"}}
+{"state": {"context": ["n m l : ℕ", "hln : l ≤ n"], "goal": "List.filter (fun x => decide (l ≤ x)) (List.Ico n m) = List.Ico n m"}}
+{"state": {"context": ["A : Type u_1", "CommMagma A", "self : IsCommJordan A", "a b : A"], "goal": "a * b * (a * a) = a * (b * (a * a))"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁸ : DivisionRing K", "inst✝⁷ : AddCommGroup V", "inst✝⁶ : Module K V", "W : Type u_1", "A : Type u_2", "inst✝⁵ : Semiring A", "inst✝⁴ : Module A V", "inst✝³ : AddCommGroup W", "inst✝² : Module K W", "inst✝¹ : Module A W", "inst✝ : LinearMap.CompatibleSMul V W K A", "h : finrank K W = 1", "f : V →ₗ[A] W", "w : f ≠ 0", "v : V", "n : f v ≠ 0 v", "c : K"], "goal": "∃ a, (↑K f) a = c • f v"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁸ : TopologicalSpace α", "inst✝⁷ : NormalSpace α", "inst✝⁶ : R1Space α", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : BorelSpace α", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "μ : Measure α", "p : ℝ≥0∞", "inst✝² : NormedSpace ℝ E", "inst✝¹ : WeaklyLocallyCompactSpace α", "inst✝ : μ.Regular", "hp : p ≠ ⊤", "f✝ : α → E", "hf : Memℒp f✝ p μ", "ε✝ : ℝ≥0∞", "hε✝ : ε✝ ≠ 0", "c : E", "t : Set α", "ht : MeasurableSet t", "htμ : μ t < ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ (f g : α → E), AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → eLpNorm f p μ ≤ δ → eLpNorm g p μ ≤ δ → eLpNorm (f + g) p μ < ε", "η : ℝ≥0", "ηpos : 0 < η", "hη : ∀ (s : Set α), μ s ≤ ↑η → eLpNorm (s.indicator fun _x => c) p μ ≤ δ", "hη_pos' : 0 < ↑η", "s : Set α", "st : s ⊆ t", "s_compact : IsCompact s", "s_closed : IsClosed s", "μs : μ (t \\ s) < ↑η", "hsμ : μ s < ⊤", "I1 : eLpNorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ", "k : Set α", "k_compact : IsCompact k", "sk : s ⊆ interior k", "f : α → E", "f_cont : Continuous f", "I2 : eLpNorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ δ", "_f_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖", "f_support : Function.support f ⊆ interior k", "f_mem : Memℒp f p μ", "I3 : eLpNorm (f - t.indicator fun _y => c) p μ ≤ ε", "x : α", "hx : x ∉ k"], "goal": "f x = 0"}}
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+{"state": {"context": ["z : ℤ", "hz : 0 ≤ z"], "goal": "Irrational √↑z ↔ ¬IsSquare z"}}
+{"state": {"context": ["K R : Type u", "V V₁ V₂ V₃ : Type v", "V' V'₁ : Type v'", "V'' : Type v''", "ι : Type w", "ι' : Type w'", "η : Type u₁'", "φ : η → Type u_1", "inst✝ : DivisionRing K", "aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)", "card_K : ℵ₀ < #K", "this✝ : Module.rank K (ℕ → K) < #K", "ιK : Type u", "bK : Basis ιK K (ℕ → K)", "L : Subfield K := Subfield.closure (range fun i => bK i.1 i.2)", "hLK : #↥L < #K", "this : Module (↥L)ᵐᵒᵖ K := Module.compHom K (RingHom.op L.subtype)", "ιL : Type u", "bL : Basis ιL (↥L)ᵐᵒᵖ K", "card_ιL : ℵ₀ ≤ #ιL", "e : ℕ ↪ ιL", "c : ιK →₀ K := bK.repr (⇑bL ∘ ⇑e)", "s : Finset ιK := c.support", "rep_e : ∑ a ∈ s, c a • bK a = ⇑bL ∘ ⇑e"], "goal": "False"}}
+{"state": {"context": ["M : Type u_1", "S : Set M", "AddCommSemigroup M"], "goal": "S.addCentralizer = Set.univ"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "RCLike 𝕜", "E : Type u_3", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "cplt : CompleteSpace E", "G : ι → Type u_4", "(i : ι) → NormedAddCommGroup (G i)", "(i : ι) → InnerProductSpace 𝕜 (G i)", "V : (i : ι) → G i →ₗᵢ[𝕜] E", "hV : OrthogonalFamily 𝕜 G V", "f : ↥(lp G 2)"], "goal": "hV.linearIsometry f = ∑' (i : ι), (V i) (↑f i)"}}
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+{"state": {"context": ["F : Type u", "inst✝¹ : Field F", "K : Type v", "inst✝ : Field K", "n : ℕ", "h : IsCoprime (X ^ n - 1) (C ↑n * X ^ (n - 1))", "hn' : ↑n = 0"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "Div α", "a b : Lex α"], "goal": "ofLex (a / b) = ofLex a / ofLex b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a : α", "f : α → β"], "goal": "Option.map f (some a) = some (f a)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "s : Set P", "p : P", "hp : p ∈ s"], "goal": "Submodule.span k (s -ᵥ s) = Submodule.span k ((fun x => p -ᵥ x) '' s)"}}
+{"state": {"context": ["B : Type u_1", "F : Type u_2", "E : B → Type u_3", "z : Bundle.TotalSpace F E"], "goal": "{ proj := z.proj, snd := z.snd } = z"}}
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+{"state": {"context": ["s : ℂ", "hs : 1 < s.re"], "goal": "Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (riemannZetaSummandHom ⋯) n))"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : CancelCommMonoidWithZero M", "q r : Associates M", "hr : r ∣ q", "n : ℕ", "hn : n + 1 ≠ 0", "c : Fin (n + 1 + 1) → Associates M", "h₁ : StrictMono c", "h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i", "i : Fin (n + 1 + 1)", "hp : Prime (c i)", "hp' : c i ∣ r", "hi : 1 < i"], "goal": "False"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s t : Multiset α", "h : s.Disjoint t"], "goal": "(s + t).dedup = s.dedup + t.dedup"}}
+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "α : Type u_4", "β : Type u_5", "l : Filter α", "m : Filter β", "s : Set α", "t : Set β", "x✝ : sᶜ ∉ l ∧ tᶜ ∉ m", "hs : sᶜ ∉ l", "ht : tᶜ ∉ m"], "goal": "∃ᶠ (x : α) in l, ∃ᶠ (y : β) in m, (x, y) ∈ s ×ˢ t"}}
+{"state": {"context": ["𝕜 : Type u_1", "NormedField 𝕜", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "c : 𝕜", "hc : 1 < ‖c‖", "x : E"], "goal": "egauge 𝕜 (Metric.ball 0 1) x ≤ ↑‖c‖₊ * ↑‖x‖₊"}}
+{"state": {"context": ["M : Type u_2", "AddCommGroup M", "A : Type u_7", "CommRing A", "Module A M", "f : M →ₗ[A] M", "hf : IsUnit f"], "goal": "IsUnit (LinearMap.det f)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_4", "ι : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "TopologicalSpace M", "AddCommMonoid N", "Module R N", "TopologicalSpace N", "f : M [⋀^ι]→L[R] N", "DecidableEq ι", "m : ι → M", "i : ι", "x : M"], "goal": "(f.toContinuousLinearMap m i) x = f (Function.update m i x)"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y f' : ℝ", "hfc : ConcaveOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hf' : HasDerivAt f f' y"], "goal": "f' ≤ slope f x y"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme"], "goal": "QuasiSeparatedSpace ↑↑X.toPresheafedSpace ↔ AlgebraicGeometry.QuasiSeparated (CategoryTheory.Limits.terminal.from X)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "α β : X ≅ Y"], "goal": "α.symm = β.symm ↔ α = β"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M", "S : Submonoid M", "x y : M"], "goal": "x ∈ S → y ∈ S → x * y ∈ S"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "s : Set ↑(X.presheaf.obj (Opposite.op ⊤))", "hs : Ideal.span s = ⊤", "hs₂ : ∀ i ∈ s, AlgebraicGeometry.IsAffineOpen (X.basicOpen i)"], "goal": "AlgebraicGeometry.IsAffine X"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "R₁ R₂ : ComposableArrows C 3", "φ : R₁ ⟶ R₂", "hR₁ : (mk₂ (R₁.map' 1 2 ⋯ ⋯) (R₁.map' 2 3 ⋯ ⋯)).Exact", "hR₂ : (mk₂ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).Exact", "hR₂' : R₂.map' 1 3 ⋯ ⋯ = 0", "h₀ : Epi (app' φ 0 ⋯)", "h₂✝ : Epi (app' φ 2 ⋯)", "h₃✝ : Mono (app' φ 3 ⋯)", "A : C", "g₁ : A ⟶ R₂.obj' 1 ⋯", "A₁ : C", "π₁ : A₁ ⟶ A", "w✝¹ : Epi π₁", "f₂ : A₁ ⟶ R₁.obj' 2 ⋯", "h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 ⋯", "h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0", "A₂ : C", "π₂ : A₂ ⟶ A₁", "w✝ : Epi π₂", "f₁ : A₂ ⟶ ((mk₂ (R₁.map' 1 2 ⋯ ⋯) (R₁.map' 2 3 ⋯ ⋯)).sc ⋯ 0 ⋯).X₁", "h₃ : π₂ ≫ f₂ = f₁ ≫ ((mk₂ (R₁.map' 1 2 ⋯ ⋯) (R₁.map' 2 3 ⋯ ⋯)).sc ⋯ 0 ⋯).f"], "goal": "∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ g₁ = x ≫ app' φ 1 ⋯"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "n : ℤ"], "goal": "(↑n + 1) * T R (n + 1) = X * U R n - (1 - X ^ 2) * derivative (U R n)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "σ : Type u_3", "n : ℕ", "x : Fin n → R", "inst✝ : CommSemiring R", "f : MvPolynomial (Fin (n + 1)) R", "q : MvPolynomial (Fin n) R"], "goal": "(eval x) (Polynomial.eval q ((finSuccEquiv R n) f)) = (eval fun i => Fin.cases ((eval x) q) x i) f"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "ι : Type u_3", "Semiring R", "AddCommMonoid M", "Module R M", "φ : ι → Type u_4", "b : (i : ι) → Basis (φ i) R M"], "goal": "⇑(Finsupp.basis b) = fun ix => Finsupp.single ix.fst ((b ix.fst) ix.snd)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x✝ : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "x : M", "r : Rˣ"], "goal": "span R {↑r • x} = span R {x}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : TopologicalSpace H", "inst✝⁶ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁵ : NormedAddCommGroup E'", "inst✝⁴ : NormedSpace 𝕜 E'", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝¹ : ChartedSpace H M", "inst✝ : ChartedSpace H' M'", "x y : M", "hy : y ∈ (extChartAt I x).source"], "goal": "y ∈ (chartAt H x).source"}}
+{"state": {"context": ["M : Type u_3", "Monoid M", "Preorder M", "CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "CovariantClass M M (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "n : ℕ", "hn : n ≠ 0"], "goal": "StrictMono fun x => x ^ n"}}
+{"state": {"context": ["q : ℚ", "q_pos : 0 < q", "q_num_pos : 0 < q.num", "q_num_abs_eq_q_num : ↑q.num.natAbs = q.num", "q_inv : ℚ := ↑q.den / ↑q.num", "q_inv_def : q_inv = ↑q.den / ↑q.num"], "goal": "(fract q⁻¹).num < q.num"}}
+{"state": {"context": ["α : Type u", "β : Type v", "OrderedAddCommMonoid β", "f g : α → β", "a : α", "l : Filter α", "hf : IsMaxFilter f l a", "hg : IsMaxFilter g l a"], "goal": "IsMaxFilter (fun x => f x + g x) l a"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "R X Y Z : C", "f : X ⟶ Y", "a b : R ⟶ X", "inst✝¹ : IsIso a", "inst✝ : Mono f"], "goal": "a = a ≫ 𝟙 X"}}
+{"state": {"context": ["R : Type u", "inst✝⁷ : CommSemiring R", "S : Submonoid R", "M : Type v", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "T : Type u_1", "inst✝⁴ : CommSemiring T", "inst✝³ : Algebra R T", "inst✝² : IsLocalization S T", "A : Type u_2", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "x : T", "a₁ s₁ : A", "a₂ s₂ : ↥S"], "goal": "x • mk a₁ a₂ * mk s₁ s₂ = x • (mk a₁ a₂ * mk s₁ s₂)"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝ : HasCoequalizers C", "R S : Mon_ C", "P : Bimod R S"], "goal": "(ρ_ (P.X ⊗ S.X)).inv ≫ (((α_ P.X S.X (𝟙_ C)).hom ≫ P.X ◁ S.X ◁ S.one) ≫ P.X ◁ S.mul) ≫ coequalizer.π (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) = coequalizer.π (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ≫ 𝟙 (coequalizer (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul))"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Preorder α", "n✝ : ℕ", "f : Fin (n✝ + 1) → α", "a : α", "n : ℕ", "x : { x // x ≠ last n }"], "goal": "some ((finSuccAboveOrderIso (last n)).symm x) = some ((↑x).castLT ⋯)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "K : TopologicalSpace.PositiveCompacts α"], "goal": "TopologicalSpace.PositiveCompacts.map id ⋯ ⋯ K = K"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "Preorder α"], "goal": "s.chainHeight = ⨆ i ∈ s, (s ∩ Set.Ici i).chainHeight"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "DecidableEq σ", "s s' : σ", "n n' : ℕ"], "goal": "MvPolynomial.coeff (Finsupp.single s' n') (MvPolynomial.X s ^ n) = if s = s' ∧ n = n' ∨ n = 0 ∧ n' = 0 then 1 else 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "h₀ : S₁.X₁ ⟶ S₂.X₁", "h₀_f : CategoryTheory.CategoryStruct.comp h₀ S₂.f = 0", "h₁ : S₁.X₂ ⟶ S₂.X₁", "h₂ : S₁.X₃ ⟶ S₂.X₂", "h₃ : S₁.X₃ ⟶ S₂.X₃", "g_h₃ : CategoryTheory.CategoryStruct.comp S₁.g h₃ = 0"], "goal": "(S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₃ = CategoryTheory.CategoryStruct.comp h₂ S₂.g + h₃"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "AddCommGroup F", "Module 𝕜 F", "B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜"], "goal": "WithSeminorms B.toSeminormFamily"}}
+{"state": {"context": ["G : Type u_2", "MeasurableSpace G", "Neg G", "MeasurableNeg G", "μ : MeasureTheory.Measure G", "μ.IsNegInvariant"], "goal": "MeasureTheory.MeasurePreserving Neg.neg μ μ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasBinaryCoproducts C", "A : C", "f : CategoryTheory.Over A"], "goal": "CategoryTheory.Over.coprod.obj f = f.coprodObj"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f g : Perm α", "l : List (Perm α)", "x : α", "hx : ∀ f ∈ l, x ∉ f.support"], "goal": "x ∉ l.prod.support"}}
+{"state": {"context": ["x y✝ z : ℝ", "n : ℕ", "hx : 0 ≤ x", "y : ℝ"], "goal": "x⁻¹ ^ y = (x ^ y)⁻¹"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M", "p : Submodule R M", "R₂ : Type u_3", "M₂ : Type u_4", "Ring R₂", "AddCommGroup M₂", "Module R₂ M₂", "τ₁₂ : R →+* R₂", "f g : M ⧸ p →ₛₗ[τ₁₂] M₂", "h : f.comp p.mkQ = g.comp p.mkQ"], "goal": "f = g"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "M : Type u_3", "inst✝⁶ : Field K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : AddCommGroup M", "inst✝² : Module K M", "inst✝¹ : LieRingModule L M", "inst✝ : Module.Finite K L", "Φ : LinearMap.BilinForm K L", "hΦ_nondeg : Φ.Nondegenerate", "hΦ_inv : LinearMap.BilinForm.lieInvariant L Φ", "hΦ_refl : Φ.IsRefl", "hL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥↑I", "I : LieIdeal K L", "hI : I ∈ {I | IsAtom I}"], "goal": "sSup ({I | IsAtom I} \\ {I}) ≤ orthogonal Φ hΦ_inv I"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : RCLike 𝕜", "μ : Measure ℝ", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : CompleteSpace E", "H : Type u_3", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "a b ε : ℝ", "bound : ℝ → ℝ", "F : H → ℝ → E", "F' : H → ℝ → H →L[𝕜] E", "x₀ : H", "ε_pos : 0 < ε", "hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))", "hF'_meas : AEStronglyMeasurable (F' x₀) (μ.restrict (Ι a b))", "h_bound : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, ‖F' x_1 x‖ ≤ bound x", "h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), ∀ x_1 ∈ ball x₀ ε, HasFDerivAt (fun x_2 => F x_2 x) (F' x_1 x) x_1", "hF_int : IntegrableOn (F x₀) (Ι a b) μ", "bound_integrable : IntegrableOn bound (Ι a b) μ"], "goal": "HasFDerivAt (fun x => ∫ (t : ℝ) in a..b, F x t ∂μ) (∫ (t : ℝ) in a..b, F' x₀ t ∂μ) x₀"}}
+{"state": {"context": ["G : Type u_2", "AddGroup G", "s : Set G", "p : (g : G) → g ∈ AddSubgroup.closure s → Prop", "mem : ∀ (x : G) (hx : x ∈ s), p x ⋯", "inv_mem : ∀ (x : G) (hx : x ∈ s), p (-x) ⋯", "one : p 0 ⋯", "mul : ∀ (x y : G) (hx : x ∈ AddSubgroup.closure s) (hy : y ∈ AddSubgroup.closure s), p x hx → p y hy → p (x + y) ⋯", "x : G", "h : x ∈ AddSubgroup.closure s"], "goal": "p x h"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Ring R", "n : ℕ", "h : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n"], "goal": "eval 1 (cyclotomic n R) = 1"}}
+{"state": {"context": ["L : Language", "ι : Type v", "inst✝⁴ : Preorder ι", "G : ι → Type w", "inst✝³ : (i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "inst✝² : IsDirected ι fun x x_1 => x ≤ x_1", "inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)", "inst✝ : Nonempty ι", "S : L.Substructure (DirectLimit G f)", "S_fg : S.FG", "A : Finset (DirectLimit G f)", "A_closure : (Substructure.closure L).toFun ↑A = S"], "goal": "∃ i T, Substructure.map (of L ι G f i).toHom T = S"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_4", "m : Type u_9", "n : Type u_10", "CommSemiring R", "AddCommMonoid α", "AddCommMonoid β", "Module R α", "Module R β", "Fintype m", "Fintype n", "A : Matrix m m α", "B : Matrix n n β"], "goal": "(Matrix.kroneckerMap (TensorProduct.tmul R) A B).trace = A.trace ⊗ₜ[R] B.trace"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W✝ : Affine R", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "hx : x₁ = x₂", "hy : y₁ ≠ W.negY x₂ y₂"], "goal": "(3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.negY x₁ y₁) = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / evalEval x₁ y₁ W.polynomialY"}}
+{"state": {"context": ["α : Type u_2", "f g : Equiv.Perm α", "x : α", "DecidableEq α", "Fintype α", "hf : f.IsCycle", "hg : g.IsCycle", "h : ∀ x ∈ f.support ∩ g.support, f x = g x", "hx : f x = g x", "hx' : x ∈ f.support"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedRing α", "FloorRing α", "TopologicalSpace α", "n : ℤ"], "goal": "ContinuousOn (fun x => ↑⌊x⌋) (Set.Ico (↑n) (↑n + 1))"}}
+{"state": {"context": ["M : Type u_1", "AddCommMonoid M", "S : AddSubmonoid M", "N : Type u_2", "AddCommMonoid N", "P : Type u_3", "AddCommMonoid P", "f : S.LocalizationMap N", "g : M →+ P", "hg : ∀ (y : ↥S), IsAddUnit (g ↑y)", "x : M", "y : ↥S"], "goal": "(f.lift hg) (f.mk' x y) = g x + ↑(-(IsAddUnit.liftRight (g.restrict S) hg) y)"}}
+{"state": {"context": ["η : Type u_12", "Ms : η → Type u_13", "(j : η) → Add (Ms j)"], "goal": "(AddEquiv.piCongrRight fun j => AddEquiv.refl (Ms j)) = AddEquiv.refl ((j : η) → Ms j)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : CancelCommMonoidWithZero α", "eif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a", "uif : ∀ (f g : Multiset α), (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g", "p : α", "this : DecidableEq α := Classical.decEq α", "hpi : Irreducible p", "a b : α", "x✝ : p ∣ a * b", "x : α", "hx : a * b = p * x", "hab0 : ¬a * b = 0", "hx0 : x ≠ 0", "ha0 : a ≠ 0", "hb0 : b ≠ 0", "fx : Multiset α", "hfx : (∀ b ∈ fx, Irreducible b) ∧ fx.prod ~ᵤ x", "fa : Multiset α", "hfa : (∀ b ∈ fa, Irreducible b) ∧ fa.prod ~ᵤ a", "fb : Multiset α", "hfb : (∀ b ∈ fb, Irreducible b) ∧ fb.prod ~ᵤ b", "h : Multiset.Rel Associated (p ::ₘ fx) (fa + fb)"], "goal": "p ∣ a ∨ p ∣ b"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "X Y : C", "f : X ⟶ Y", "W.ContainsIdentities"], "goal": "(CategoryTheory.MorphismProperty.RightFraction.ofHom W f).f = f"}}
+{"state": {"context": ["M : Type u", "Monoid M", "x : Mˣ"], "goal": "((Units.toAut M) x).inv = (CategoryTheory.SingleObj.toEnd M) ↑x⁻¹"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁴ : DivisionRing K", "inst✝³ : AddCommGroup V", "inst✝² : Module K V", "s t : Submodule K V", "inst✝¹ : FiniteDimensional K ↥s", "inst✝ : FiniteDimensional K ↥t"], "goal": "finrank K ↥(s ⊔ t) ≤ finrank K ↥s + finrank K ↥t"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁴ : Group G", "inst✝³ : Group G'", "inst✝² : Group G''", "A : Type u_4", "inst✝¹ : AddGroup A", "N : Type u_5", "inst✝ : Group N", "f✝ f : G →* N", "H : Subgroup G", "h : f.ker ≤ H"], "goal": "comap f (map f H) = H"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrderedField α", "inst✝¹ : Ring β", "abv : β → α", "inst✝ : IsAbsoluteValue abv", "f g : ℕ → β", "a : ℕ → α", "ha : IsCauSeq abs fun m => ∑ n ∈ range m, abv (f n)", "hb : IsCauSeq abv fun m => ∑ n ∈ range m, g n", "ε : α", "ε0 : 0 < ε"], "goal": "∃ i, ∀ j ≥ i, abv ((∑ k ∈ range j, f k) * ∑ k ∈ range j, g k - ∑ n ∈ range j, ∑ m ∈ range (n + 1), f m * g (n - m)) < ε"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "n p : ℕ", "f : ℕ → ℚ := fun i => bernoulli i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ", "f' : ℕ → ℚ := fun i => bernoulli' i * ↑(p.succ.succ.choose i) * ↑n ^ (p.succ.succ - i) / ↑p.succ.succ", "hle : 1 ≤ n + 1"], "goal": "∑ k ∈ Ico 1 n.succ, ↑k ^ p.succ = ∑ i ∈ range p.succ.succ, f' i"}}
+{"state": {"context": ["M : Type u_2", "β : Type u_3", "α : Type u_4", "MeasurableSpace M", "MeasurableSpace β", "AddMonoid M", "AddAction M β", "MeasurableVAdd M β", "MeasurableSpace α", "f : α → β", "μ : MeasureTheory.Measure α", "c : M", "hc : IsAddUnit c"], "goal": "AEMeasurable (fun x => c +ᵥ f x) μ ↔ AEMeasurable f μ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "CategoryTheory.BraidedCategory D"], "goal": "(CategoryTheory.Monoidal.commMonFunctorCategoryEquivalence C D).functor =\n  CategoryTheory.Monoidal.CommMonFunctorCategoryEquivalence.functor"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "MulOneClass M", "MulOneClass N", "Monoid P", "f : M →* P", "g : N →* P", "x : N"], "goal": "(Monoid.Coprod.lift f g) (Monoid.Coprod.inr x) = g x"}}
+{"state": {"context": ["ι : Type u_1", "Finite ι", "μ : MeasureTheory.Measure (ι → ℝ)", "MeasureTheory.IsLocallyFiniteMeasure μ", "J : BoxIntegral.Box ι"], "goal": "μ.toBoxAdditive J = (μ ↑J).toReal"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y : C", "S✝ R : Sieve X", "J : GrothendieckTopology C", "i : X ⟶ Y", "inst✝ : IsIso i", "S : Sieve Y"], "goal": "Sieve.pullback i S ∈ J.sieves X ↔ S ∈ J.sieves Y"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "W X Y Z : C", "f : X ⟶ Y", "g : X ⟶ Z", "t : PushoutCocone f g"], "goal": "t.ι.app WalkingSpan.zero = f ≫ t.inl"}}
+{"state": {"context": ["n : ℕ"], "goal": "(2 ^ n).testBit n = true"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "Semiring k", "Add G", "A : Type u₃", "NonUnitalNonAssocSemiring A", "Module k A", "IsScalarTower k A A", "SMulCommClass k A A", "f : Multiplicative G →ₙ* A", "a : AddMonoidAlgebra k G"], "goal": "((AddMonoidAlgebra.liftMagma k) f) a = Finsupp.sum a fun m t => t • f (Multiplicative.ofAdd m)"}}
+{"state": {"context": ["t : Type u → Type u → Type u", "inst✝⁵ : Bitraversable t", "β : Type u", "F G : Type u → Type u", "inst✝⁴ : Applicative F", "inst✝³ : Applicative G", "inst✝² : LawfulBitraversable t", "inst✝¹ : LawfulApplicative F", "inst✝ : LawfulApplicative G", "α₀ α₁ β₀ β₁ : Type u", "f : α₀ → F α₁", "f' : β₀ → G β₁", "x : t α₀ β₀"], "goal": "bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ map f' ∘ pure) x = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x"}}
+{"state": {"context": ["α : Type u", "Group α", "LT α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b : α"], "goal": "a⁻¹ < b⁻¹ ↔ b < a"}}
+{"state": {"context": ["α : Type u_5", "f : α → α", "n : ℕ", "hf : Function.Involutive f", "hn : Odd n"], "goal": "f^[n] = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m₁ m₂ : MeasurableSpace α", "f : α → β"], "goal": "MeasurableSpace.map f (m₁ ⊓ m₂) = MeasurableSpace.map f m₁ ⊓ MeasurableSpace.map f m₂"}}
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+{"state": {"context": ["ι : Sort u_1", "f : ι → ℕ", "IsEmpty ι"], "goal": "⨅ i, ↑(f i) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "M : Type u_3", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "S : Type u_4", "inst✝⁴ : Monoid S", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : DistribMulAction S M", "sR : Set R", "s✝ : Set S", "N✝ : Submodule R M", "inst✝ : SMulCommClass S R M", "s : Set S", "N : Submodule R M"], "goal": "s • N = ⨆ a ∈ s, a • N"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁶ : Category.{?u.266427, u_1} C", "inst✝⁵ : Category.{?u.266431, u_2} D", "inst✝⁴ : Preadditive C", "inst✝³ : Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "inst✝² : HasHomotopyCofiber φ", "H : C ⥤ D", "inst✝¹ : H.Additive", "inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)", "n m : ℤ", "hnm : n + 1 = m"], "goal": "((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map ((inl φ).v m n ⋯) + (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map ((inr φ).f n)) ≫ (H.map ((↑(fst φ)).v n m ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ + H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) = 𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "MulOneClass M", "MulOneClass N", "S : Submonoid M", "F : Type u_4", "FunLike F M N", "mc : MonoidHomClass F M N", "T : Submonoid N", "f : F"], "goal": "S ≤ Submonoid.comap f T → Submonoid.map f S ≤ T"}}
+{"state": {"context": ["X : Type u", "x : X", "TopologicalSpace X", "f : Filter X", "H : f ≤ 𝓝 x", "_hf : f.NeBot"], "goal": "ClusterPt x f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν : Measure α", "s✝ t✝ s t : SignedMeasure α", "μ : Measure α", "inst✝¹ : s.HaveLebesgueDecomposition μ", "inst✝ : t.HaveLebesgueDecomposition μ", "hst : (s + -t).HaveLebesgueDecomposition μ"], "goal": "(s - t).rnDeriv μ =ᶠ[ae μ] s.rnDeriv μ - t.rnDeriv μ"}}
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+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "self : Comon_ C"], "goal": "self.comul ≫ self.X ◁ self.counit = (ρ_ self.X).inv"}}
+{"state": {"context": ["α : Type u_1", "MulOneClass α", "LE α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b : α", "h : 1 ≤ b"], "goal": "a ≤ b * a"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ", "r : ℂ", "h : -1 < r.re ∨ r ≠ -1 ∧ 0 ∉ [[a, b]]"], "goal": "∫ (x : ℝ) in a..b, ↑x ^ r = (↑b ^ (r + 1) - ↑a ^ (r + 1)) / (r + 1)"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "x y : V", "h : ⟪x, y⟫_ℝ = 0", "h0 : x = 0 ∨ y ≠ 0"], "goal": "‖y‖ / Real.tan (InnerProductGeometry.angle x (x + y)) = ‖x‖"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "C : Type u_2", "D : Type u_3", "CategoryTheory.Category.{u_4, u_2} C", "CategoryTheory.Category.{u_5, u_3} D", "CategoryTheory.Preadditive C", "CategoryTheory.Preadditive D", "CategoryTheory.Linear R C", "CategoryTheory.Linear R D", "F : CategoryTheory.Functor C D", "F.Additive", "CategoryTheory.Functor.Linear R F", "X Y : C"], "goal": "∀ (a : X ⟶ Y), (CategoryTheory.Functor.mapLinearMap R F) a = (↑F.mapAddHom).toFun a"}}
+{"state": {"context": ["m : ℤ"], "goal": "‖↑m‖ = ‖m‖"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.HasShift C ℤ", "X₁ X₂ : C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasBinaryBiproduct X₁ X₂"], "goal": "(CategoryTheory.Pretriangulated.binaryProductTriangleIsoBinaryBiproductTriangle X₁ X₂).hom.hom₁ = 𝟙 X₁"}}
+{"state": {"context": ["α : Type u_2", "ConditionallyCompleteLattice α", "a b : α", "h : a ≤ b", "S : Set ↑(Set.Icc a b)", "hS : S.Nonempty"], "goal": "↑(sInf S) = sInf (Subtype.val '' S)"}}
+{"state": {"context": ["ι : Type u", "f : ι → AlgebraicGeometry.Scheme"], "goal": "∀ (a : ι), (AlgebraicGeometry.disjointGlueData' f).U a = f a"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "Ring S", "Algebra R S", "B : PowerBasis R S"], "goal": "Algebra.adjoin R {B.gen} = ⊤"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "Semiring R", "AddCommMonoid M", "Module R M", "S T : Submodule R M", "x : M"], "goal": "x ∈ S → x ∈ S ⊔ T"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : α → Set β", "Preorder α", "x : α"], "goal": "⋃ y, ⋃ (_ : y ≤ x), Set.Accumulate s y = ⋃ y, ⋃ (_ : y ≤ x), s y"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : (i : ι) → TopologicalSpace (π i)", "x✝ y✝ z : X", "s✝ : Set X", "f✝ g : X → Y", "t : Set (SeparationQuotient X)", "f : X → Y → α", "hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d", "x : X", "y : Y", "s : Set (SeparationQuotient X × SeparationQuotient Y)", "l : Filter α"], "goal": "Tendsto (uncurry (lift₂ f hf)) (𝓝[s] (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝[Prod.map mk mk ⁻¹' s] (x, y)) l"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "c x : P", "R : ℝ"], "goal": "dist (EuclideanGeometry.inversion c R x) c = R ^ 2 / dist x c"}}
+{"state": {"context": ["α β : Type u", "n : ℕ"], "goal": "↑n.succ = succ ↑n"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M"], "goal": "∀ (a : M), MulOpposite.opHomeomorph a = MulOpposite.op a"}}
+{"state": {"context": ["a m n p : ℕ", "ha : a ≠ 0"], "goal": "a.factorizationLCMRight 0 ∣ 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : E → ℂ", "x : E", "s : Set E", "hf : DifferentiableWithinAt ℂ f s x", "hxs : UniqueDiffWithinAt ℂ s x"], "goal": "fderivWithin ℂ (fun x => Complex.cosh (f x)) s x = Complex.sinh (f x) • fderivWithin ℂ f s x"}}
+{"state": {"context": ["α : Type u_1", "OrderedAddCommGroup α", "s t : UpperSet α"], "goal": "↑(s - t) = ↑s - ↑t"}}
+{"state": {"context": ["α : Type u", "s t u : Set α"], "goal": "s ∩ symmDiff t u = symmDiff (s ∩ t) (s ∩ u)"}}
+{"state": {"context": ["E : Type u_1", "AddCommGroup E", "TopologicalSpace E", "TopologicalAddGroup E", "T2Space E", "Module ℝ E", "ContinuousSMul ℝ E", "FiniteDimensional ℝ E", "x : E"], "goal": "(𝓝 x).HasBasis (fun r => 0 < r) (Euclidean.ball x)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p q r : R[X]", "n : ℕ", "hn : n ∈ (p * q).support"], "goal": "p.trailingDegree + q.trailingDegree ≤ ↑n"}}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "A : Type w", "B : Type u₁", "C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝⁶ : CommSemiring C", "inst✝⁵ : CommSemiring D", "inst✝⁴ : CommSemiring E", "inst✝³ : Algebra C D", "inst✝² : Algebra C E", "inst✝¹ : Algebra D E", "inst✝ : IsScalarTower C D E", "S : Set E"], "goal": "Subalgebra.restrictScalars C (adjoin D S) = Subalgebra.restrictScalars C (adjoin (↥(Subalgebra.map (IsScalarTower.toAlgHom C D E) ⊤)) S)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PseudoMetricSpace α", "MeasurableSpace α", "OpensMeasurableSpace α", "MeasurableSpace β", "f : β → α", "hf : Measurable f", "s : Set α"], "goal": "Measurable fun x => Metric.infNndist (f x) s"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "b : Basis ι R M"], "goal": "Submodule.span R (Set.range ⇑b) = ⊤"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "ι₁ : Sort u_6", "f : ι₁ → Submodule R M", "ι₂ : Sort u_7", "g : ι₂ → Submodule R M"], "goal": "(⨅ i, f i).colon (⨆ j, g j) = ⨅ i, ⨅ j, (f i).colon (g j)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedSemiring α", "inst✝ : FloorSemiring α", "a✝ : α", "n✝ n a : ℕ"], "goal": "⌈↑n⌉₊ ≤ a ↔ n ≤ a"}}
+{"state": {"context": ["x y : ℝ"], "goal": "sin x ^ 2 ≤ 1"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "S S' S'' : D", "Y Y' Y'' : C", "T T' : C ⥤ D", "f f' : StructuredArrow S T", "g : f.right ≅ f'.right", "w : autoParam (f.hom ≫ T.map g.hom = f'.hom) _auto✝"], "goal": "(Functor.fromPUnit S).map (eqToIso ⋯).hom ≫ f'.hom = f.hom ≫ T.map g.hom"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝ : PseudoEMetricSpace α", "δ✝ : ℝ", "s : Set α", "x✝ : α", "E : Set α", "x : α", "h : x ∉ closure E", "ε : ℝ", "ε_pos : 0 < ε", "ε_lt : ENNReal.ofReal ε < infEdist x E", "δ : ℝ", "hδ : δ < ε"], "goal": "x ∉ thickening δ E"}}
+{"state": {"context": ["α : Type u_1", "m : Type u_3", "R : Type v", "CommRing R", "M : Matrix m m R", "b : m → α", "DecidableEq m", "Fintype m", "LinearOrder α", "Invertible M", "hM : M.BlockTriangular b", "k : α"], "goal": "(M.toBlock (fun i => b i < k) fun i => b i < k)⁻¹ = M⁻¹.toBlock (fun i => b i < k) fun i => b i < k"}}
+{"state": {"context": ["cs : List Char"], "goal": "{ data := cs }.utf8ByteSize = String.utf8Len cs"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "G : Type u_6", "k : Type u_7", "R : Type u_8", "inst✝¹ : CommMonoid M", "inst✝ : OrderedCommMonoid N", "f g : ι → N", "s t : Finset ι", "i j : ι", "hf : ∀ i ∈ s, 1 ≤ f i", "hi : i ∈ s", "hj : j ∈ s", "hne : i ≠ j"], "goal": "∏ k ∈ cons i {j} ⋯, f k ≤ ∏ k ∈ s, f k"}}
+{"state": {"context": ["α : Type u"], "goal": "Function.Injective FreeGroup.invRev"}}
+{"state": {"context": ["n : Type u_1", "𝕜 : Type u_2", "RCLike 𝕜", "Fintype n", "DecidableEq n", "A : Matrix n n 𝕜", "hA : A.IsHermitian", "f : ℝ → ℝ"], "goal": "cfc f A = hA.cfc f"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "p q : ℝ", "hpq : p.IsConjExponent q", "f g : ι → ℝ≥0", "A B : ℝ≥0", "hf_sum : HasSum (fun a => f a ^ p) (A ^ p)", "hg_sum : HasSum (fun a => g a ^ q) (B ^ q)", "C : ℝ≥0", "hC : C ≤ A * B", "H : HasSum (fun i => f i * g i) C"], "goal": "∃ C, 0 ≤ C ∧ C ≤ ↑A * ↑B ∧ HasSum (fun i => ↑(f i) * ↑(g i)) C"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "s : σ"], "goal": "(MvPowerSeries.constantCoeff σ R) (MvPowerSeries.X s) = 0"}}
+{"state": {"context": ["R : Type w₁", "A : Type w₂", "B : Type w₃", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Algebra R A", "inst✝² : CommRing B", "inst✝¹ : Algebra R B", "I : Ideal A", "h : I.FG", "inst✝ : FinitePresentation R A", "n : ℕ", "f : MvPolynomial (Fin n) R →ₐ[R] A", "hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG"], "goal": "∃ n f, Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "inst✝¹ : Inhabited Λ", "σ : Type u_4", "inst✝ : Inhabited σ", "L L' : ListBlank ((k : K) → Option (Γ k))", "l : List ((k : K) → Option (Γ k))"], "goal": "ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } (ListBlank.mk l)) = ListBlank.mk l"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "x : E", "hf : HasStrictFDerivAt f f' x", "c : F"], "goal": "HasStrictFDerivAt (fun y => c + f y) f' x"}}
+{"state": {"context": ["L : FirstOrder.Language", "α : Type u'", "n : ℕ", "φ : L.BoundedFormula α n", "h : φ.IsQF"], "goal": "φ.not.IsQF"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a b c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "hab : IntervalIntegrable f μ a b", "hbc : IntervalIntegrable f μ b c", "hac : IntervalIntegrable f μ a c"], "goal": "Disjoint (Ioc b a) (Ioc c b)"}}
+{"state": {"context": ["a b c : Cardinal.{u_1}", "h : a + b = c + b", "hb : b < ℵ₀"], "goal": "a = c"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "a a₁ a₂ b c : G", "inst✝ : Group G", "s : Set G", "hs : IsSubgroup s"], "goal": "(∀ (x : G), x ∈ s ↔ x = 1) ↔ ∀ x ∈ s, x = 1"}}
+{"state": {"context": [], "goal": "UpperHalfPlane.I.re = 0"}}
+{"state": {"context": ["V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "X₀ X₁ X₂ : V", "d₀ : X₀ ⟶ X₁", "d₁ : X₁ ⟶ X₂", "s : d₀ ≫ d₁ = 0", "succ : (S : CategoryTheory.ShortComplex V) → (X₄ : V) ×' (d₂ : S.X₃ ⟶ X₄) ×' S.g ≫ d₂ = 0"], "goal": "(CochainComplex.mk X₀ X₁ X₂ d₀ d₁ s succ).X 1 = X₁"}}
+{"state": {"context": ["ι : Sort u_1", "M : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝ : Mul M", "S : Set (Subsemigroup M)", "hS : DirectedOn (fun x x_1 => x ≤ x_1) S", "x : M"], "goal": "x ∈ ↑(sSup S) ↔ x ∈ ⋃ s ∈ S, ↑s"}}
+{"state": {"context": [], "goal": "Ordinal.nfp (HMul.hMul 0) = id"}}
+{"state": {"context": ["s : ℂ", "hs : 0 < s.re", "n : ℕ", "hn : n ≠ 0", "this✝ : ∀ (x : ℝ), x = x / ↑n * ↑n", "this : ∫ (x : ℝ) in 0 ..↑n, ↑((1 - x / ↑n) ^ n) * ↑(x / ↑n * ↑n) ^ (s - 1) = ↑n • ∫ (x : ℝ) in 0 / ↑n..↑n / ↑n, ↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1)", "x : ℝ", "hx : x ∈ Ioc 0 1"], "goal": "↑n ^ s * (↑x ^ (s - 1) * (1 - ↑x) ^ ↑n) = ↑↑n * (↑((1 - x) ^ n) * ↑(x * ↑n) ^ (s - 1))"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "(Set.Ioi a).OrdConnected"}}
+{"state": {"context": ["ι : Type u_2", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Preadditive V", "c : ComplexShape ι", "C : HomologicalComplex V c"], "goal": "CategoryTheory.Limits.IsZero ((HomotopyCategory.quotient V c).obj C) ↔ Nonempty (Homotopy (𝟙 C) 0)"}}
+{"state": {"context": ["α : Type u", "n : ℕ", "s : Stream' α"], "goal": "Stream'.take n s ++ [s.get n] = Stream'.take (n + 1) s"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "s : Set M", "p : M → Prop", "x : M", "h : x ∈ AddSubmonoid.closure s", "mem : ∀ x ∈ s, p x", "one : p 0", "mul : ∀ (x y : M), p x → p y → p (x + y)"], "goal": "p x"}}
+{"state": {"context": ["X : Type u_2", "Y : Sort u_3", "Z : Type u_1", "f : X → Y", "g : Z → Y", "p₁ p₂ : Pullback f g", "he : (p₁, p₂).1.snd = (p₁, p₂).2.snd"], "goal": "⟨(p₁, p₂), he⟩ ∈ map_fst ⁻¹' pullbackDiagonal f ↔ ⟨(p₁, p₂), he⟩ ∈ pullbackDiagonal snd"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "o : Part α", "h : o.Dom", "f : α → Part β", "b : β"], "goal": "(∃ a ∈ o, b ∈ f a) ↔ b ∈ f (o.get h)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : Preorder α", "inst✝⁴ : Preorder β", "inst✝³ : OrderTop α", "inst✝² : OrderBot β", "inst✝¹ : LocallyFiniteOrder α", "inst✝ : LocallyFiniteOrder β", "a a₁ a₂ : α", "b b₁ b₂ : β"], "goal": "Ioc (inrₗ b₁) (inrₗ b₂) = map (Embedding.inr.trans toLex.toEmbedding) (Ioc b₁ b₂)"}}
+{"state": {"context": ["f : ℂ →ₗᵢ[ℝ] ℂ", "h₂ : ∀ (z : ℂ), (f z).re = z.re", "z : ℂ"], "goal": "(f z).im = z.im ∨ (f z).im = -z.im"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝³ : Category.{u_5, u_1} C", "inst✝² : Category.{?u.34532, u_2} D", "inst✝¹ : Category.{?u.34536, u_3} E", "P : Cᵒᵖ ⥤ Type u_4", "X B : C", "π : X ⟶ B", "inst✝ : HasPullback π π"], "goal": "MapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯ ≫ (Types.equalizerIso (P.map (pullback.fst π π).op) (P.map (pullback.snd π π).op)).inv = equalizer.lift (P.map π.op) ⋯"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Primcodable α", "Primcodable β", "p : β → Prop", "DecidablePred p", "hp : PrimrecPred p", "f : α → β", "h : ∀ (a : α), p (f a)", "hf : Primrec f"], "goal": "Primrec fun a => ⟨f a, ⋯⟩"}}
+{"state": {"context": ["R : Type u_1", "StrictOrderedCommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "x y : M"], "goal": "SameRay R (-x) y ↔ SameRay R x (-y)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : TopologicalSpace α", "a : ℝ", "h : 0 < a"], "goal": "ContinuousAt (fun p => p.1 * p.2) (⊤, ↑a)"}}
+{"state": {"context": ["k : Type u", "CommRing k", "G : Type u", "Group G", "A : Rep k G", "g : G", "x : CoeSort.coe A"], "goal": "(A.ρ g) ((A.ρ g⁻¹) x) = x"}}
+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "n : ℕ", "inst✝ : Monoid G", "h : QuasiIso (((forget₂ (Rep k G) (ModuleCat k)).mapHomologicalComplex (ComplexShape.down ℕ)).map (εToSingle₀ k G) ≫ (HomologicalComplex.singleMapHomologicalComplex (forget₂ (Rep k G) (ModuleCat k)) (ComplexShape.down ℕ) 0).hom.app (Rep.trivial k G k))"], "goal": "QuasiIso (((forget₂ (Rep k G) (ModuleCat k)).mapHomologicalComplex (ComplexShape.down ℕ)).map (εToSingle₀ k G))"}}
+{"state": {"context": ["α β : Type u", "c : Cardinal.{u}", "f : α → β", "hf : ∀ (b : β), Cardinal.mk ↑(f ⁻¹' {b}) ≤ c"], "goal": "Cardinal.mk α ≤ Cardinal.mk β * c"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "inst✝ : DecidableEq V", "u v : V", "p : G.Walk u v", "h : p.IsEulerian"], "goal": "p.IsTrail"}}
+{"state": {"context": ["α : Type u_3", "AddMonoid α", "s : Multiset α", "comm : {x | x ∈ s}.Pairwise AddCommute", "m : α", "h : ∀ x ∈ s, x = m"], "goal": "s.noncommSum comm = Multiset.card s • m"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β", "g : α → γ", "l : List α"], "goal": "(List.map f l).zip (List.map g l) = List.map (fun a => (f a, g a)) l"}}
+{"state": {"context": ["R S Rₘ Sₘ : Type u", "inst✝¹⁵ : CommRing R", "inst✝¹⁴ : CommRing S", "inst✝¹³ : CommRing Rₘ", "inst✝¹² : CommRing Sₘ", "M : Submonoid R", "inst✝¹¹ : Algebra R S", "inst✝¹⁰ : Algebra R Sₘ", "inst✝⁹ : Algebra S Sₘ", "inst✝⁸ : Algebra R Rₘ", "inst✝⁷ : Algebra Rₘ Sₘ", "inst✝⁶ : IsScalarTower R Rₘ Sₘ", "inst✝⁵ : IsScalarTower R S Sₘ", "inst✝⁴ : IsLocalization M Rₘ", "inst✝³ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ", "inst✝² : FormallySmooth R Sₘ", "Q : Type u", "inst✝¹ : CommRing Q", "inst✝ : Algebra Rₘ Q", "I : Ideal Q", "e : I ^ 2 = ⊥", "f : Sₘ →ₐ[Rₘ] Q ⧸ I", "this✝ : Algebra R Q := ((algebraMap Rₘ Q).comp (algebraMap R Rₘ)).toAlgebra", "this : IsScalarTower R Rₘ Q := IsScalarTower.of_algebraMap_eq' rfl"], "goal": "∃ a, (Ideal.Quotient.mkₐ Rₘ I).comp a = f"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.InitialMonoClass C", "I X : C", "hI : CategoryTheory.Limits.IsInitial I", "f : I ⟶ X"], "goal": "CategoryTheory.Mono f"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s t : Finset ι", "i j : ι", "v r r' : ι → F", "hvs : Set.InjOn v ↑s", "hrr' : (interpolate s v) r = (interpolate s v) r'", "x✝ : ι", "hi : x✝ ∈ s"], "goal": "r x✝ = r' x✝"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝² : CommMonoid M", "inst✝¹ : CommMonoid N", "inst✝ : DivisionCommMonoid G", "k l : ℕ", "ζ : M", "f : F", "h✝ h : IsPrimitiveRoot ζ k", "h0 : 0 < k"], "goal": "ζ * ζ ^ (k - 1) = 1"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w : ι → k", "p : ι → P", "h : ∑ i ∈ s, w i = 1", "b₁ b₂ : P"], "goal": "b₁ -ᵥ b₂ + ∑ x ∈ s, (w x • (p x -ᵥ b₁) - w x • (p x -ᵥ b₂)) = 0"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "Zero M", "Zero N", "f : M → N", "hf : f 0 = 0"], "goal": "Finsupp.mapRange f hf 0 = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁✝ l₂✝ : List α", "p : α → Bool", "l : List α", "a : α", "al : a ∈ l", "pa : p a = true", "w✝ : α", "l₁ l₂ : List α", "left✝¹ : ∀ (b : α), b ∈ l₁ → ¬p b = true", "left✝ : p w✝ = true", "h₁ : l = l₁ ++ w✝ :: l₂", "h₂ : eraseP p l = l₁ ++ l₂"], "goal": "l₁.length + l₂.length + 1 = l₁.length + (w✝ :: l₂).length"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X : SimplicialObject C", "q : ℕ"], "goal": "𝟙 K[X] - (P q + P q ≫ Hσ q) = 𝟙 K[X] - P q - P q ≫ Hσ q"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type u_1", "L.Structure M", "L.Structure N"], "goal": "L.ElementarilyEquivalent M N ↔ ∀ (φ : L.Sentence), M ⊨ φ ↔ N ⊨ φ"}}
+{"state": {"context": ["x n : ℂ"], "goal": "(if x = 0 then if n = 0 then 1 else 0 else if x.arg = π then (cexp ((starRingEnd ℂ) (log x) * n))⁻¹ else (cexp (log x * n))⁻¹) = if x.arg = π then if x = 0 then if n = 0 then 1 else 0 else (cexp ((starRingEnd ℂ) (log x) * n))⁻¹ else if x = 0 then if n = 0 then 1 else 0 else (cexp (log x * n))⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "𝕜 : Type u_3", "inst✝³ : RCLike 𝕜", "inst✝² : NormedSpace 𝕜 E", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "x✝ : E", "s : Set E", "hs : IsOpen s", "h's : IsPreconnected s", "hf' : UniformCauchySeqOn f' l s", "hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y", "x₀ x : E", "hx₀ : x₀ ∈ s", "hx : x ∈ s", "hfg : Cauchy (map (fun n => f n x₀) l)", "this : l.NeBot", "t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n => f n y) l)}", "A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t"], "goal": "s ⊆ t"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α", "h : Bornology.IsBounded s", "a : ℝ", "c : α"], "goal": "∃ r, a < r ∧ s ⊆ Metric.ball c r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r r₁ r₂ : α → α → Prop", "s t : Set α", "a✝ b✝ x : α", "inst✝ : PartialOrder α", "hs : IsAntichain (fun x x_1 => x ≤ x_1) s", "a : α", "ha : a ∈ s", "b : α", "x✝ : b ∈ ↑(upperClosure s)", "hba : (fun x x_1 => x ≤ x_1) b a", "c : α", "hc : c ∈ s", "hcb : c ≤ b"], "goal": "(fun x x_1 => x ≤ x_1) a b"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsJacobson R", "P : Ideal R[X]", "hP : P.IsMaximal", "inst✝ : Nontrivial R", "hR : IsJacobson R", "hP' : ∀ (x : R), C x ∈ P → x = 0", "P' : Ideal R := comap C P", "pX : R[X]", "hpX : pX ∈ P", "hp0 : Polynomial.map (Quotient.mk (comap C P)) pX ≠ 0", "a : R ⧸ P' := (Polynomial.map (Quotient.mk P') pX).leadingCoeff", "M : Submonoid (R ⧸ P') := Submonoid.powers a", "φ : R ⧸ P' →+* R[X] ⧸ P := quotientMap P C ⋯", "hP'_prime : P'.IsPrime", "hM : 0 ∉ M", "M' : Submonoid (R[X] ⧸ P) := Submonoid.map φ M"], "goal": "(quotientMap P C ⋯).IsIntegral"}}
+{"state": {"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "hcont✝ : ContinuousWithinAt g (Ici a) a", "hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x", "g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x", "hg : Tendsto g atTop (𝓝 l)", "hcont : ContinuousOn g (Ici a)"], "goal": "Tendsto (fun i => ∫ (x : ℝ) in a..id i, ‖g' x‖) atTop (𝓝 (l - g a))"}}
+{"state": {"context": ["α : Type u_4", "s : Multiset α"], "goal": "s.Nodup ↔ Multiset.Pairwise (fun x x_1 => x ≠ x_1) s"}}
+{"state": {"context": ["R : Type u_1", "Semiring R"], "goal": "ε * ε = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : α → β → Prop", "l₁ : List α", "l₂ : List β"], "goal": "List.Forall₂ R l₁ l₂ ↔\n  l₁.length = l₂.length ∧ ∀ (i : ℕ) (h₁ : i < l₁.length) (h₂ : i < l₂.length), R (l₁.get ⟨i, h₁⟩) (l₂.get ⟨i, h₂⟩)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "OrderedAddCommMonoid α", "TopologicalSpace α", "OrderClosedTopology α", "g : ι → α", "a : α", "h : ∀ (i : ι), g i ≤ 0", "ha : HasSum g a"], "goal": "a ≤ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrderedField α", "Ring β", "abv : β → α", "f : CauSeq β abv"], "goal": "IsCauSeq abv ↑f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : OrderedCommGroup α", "s t : Set α", "a : α"], "goal": "s * ↑(upperClosure t) = ↑(upperClosure (s * t))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β"], "goal": "Set.SurjOn f s t ↔ ∃ t' g, t ⊆ t' ∧ Function.Surjective g ∧ ∀ (x : ↑s), f ↑x = ↑(g x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "f✝ g✝ : Filter α", "s t : Set α", "f g : Filter α"], "goal": "f ⊓ g = ⊥ ↔ ∃ U ∈ f, ∃ V ∈ g, U ∩ V = ∅"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "z : ℤ", "a : α"], "goal": "z + 1 ≤ ⌈a⌉ ↔ ↑z < a"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "Algebra R A", "M : Type u_3", "AddCommMonoid M", "Module A M", "Module R M", "IsScalarTower R A M", "r : R", "a : A", "m : M"], "goal": "a • r • m = r • a • m"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "G : Type u_4", "inst✝¹ : PartialOrder α", "inst✝ : Group G", "f₁ f₂ : G →* α ≃o α", "h : α → α", "H : ∀ (x : α), IsLUB (range fun g' => (f₁ g')⁻¹ ((f₂ g') x)) (h x)", "g : G", "y : α", "this : IsLUB (range ((⇑(f₁ g) ∘ fun g' => (f₁ g')⁻¹ ((f₂ g') y)) ∘ ⇑(Equiv.mulRight g))) ((f₁ g) (h y))"], "goal": "IsLUB (range fun g' => (f₁ g')⁻¹ ((f₂ g') ((f₂ g) y))) ((f₁ g) (h y))"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort x", "CompleteLattice α", "CompleteLattice β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "f : ι → β"], "goal": "u (iInf f) = ⨅ i, u (f i)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G", "nH : H.Normal", "a b : G", "h : a * b ∈ H"], "goal": "b * a ∈ H"}}
+{"state": {"context": ["e : ℝ", "d : ℕ", "n : ℤ", "inv : Invertible ↑d", "eq : e = ↑n * ⅟↑d", "h₁ : decide (0 < ↑n / ↑d) = true", "h₂ : decide (↑n / ↑d < 1) = true", "h₁' : 0 < ↑n / ↑d"], "goal": "Real.log (↑n / ↑d) ≠ 0"}}
+{"state": {"context": ["J : Type u₁", "inst✝² : Category.{v₁, u₁} J", "K : Type u₂", "inst✝¹ : Category.{v₂, u₂} K", "inst✝ : IsPreconnected J", "p : Set J", "j₀ : J", "h0 : j₀ ∈ p", "h1 : ∀ {j₁ j₂ : J}, (j₁ ⟶ j₂) → (j₁ ∈ p ↔ j₂ ∈ p)", "j : J", "aux : ∀ (j₁ j₂ : J), (j₁ ⟶ j₂) → { down := j₁ ∈ p } = { down := j₂ ∈ p } := fun j₁ j₂ f => congrArg ULift.up (Iff.eq (h1 f))"], "goal": "j ∈ p"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m m0 : MeasurableSpace α✝", "α : Type u_5", "inst✝¹ : MeasurableSpace α", "A : Set α", "ι : Type u_6", "L : Filter ι", "inst✝ : L.IsCountablyGenerated", "As : ι → Set α", "μ : Measure α", "A_mble : MeasurableSet A", "As_mble : ∀ (i : ι), MeasurableSet (As i)", "B : Set α", "B_mble : MeasurableSet B", "B_finmeas : μ B ≠ ⊤", "As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B", "h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A"], "goal": "∀ᵐ (a : α) ∂μ, Tendsto (fun n => (As n).indicator 1 a) L (𝓝 (A.indicator 1 a))"}}
+{"state": {"context": ["n : ℕ"], "goal": "3 ∣ n ↔ 3 ∣ (Nat.digits 10 n).sum"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : SemilatticeSup α", "a✝ b✝ c d a b : α"], "goal": "a ⊔ b = b ⊔ a"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂", "x : M₂"], "goal": "x ∈ map (LinearMap.snd R M M₂) (snd R M M₂)"}}
+{"state": {"context": ["b x : Ordinal.{u_1}", "hb : 1 < b", "hx : x ≠ 0", "t : Ordinal.{u_1} := sInf {o | x < b ^ o}", "this : x < b ^ t", "h : t = 0"], "goal": "succ (log b x) = sInf {o | x < b ^ o}"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_4", "Fintype ι", "OrderedAddCommMonoid M"], "goal": "Monotone fun f => ∑ i : ι, f i"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "f : Perm (Fin n)", "x✝ : (_ : Fin n) × Fin n", "a b : Fin n", "hab : ⟨a, b⟩ ∈ finPairsLT n", "h : ¬f⁻¹ b < f⁻¹ a"], "goal": "(if f⁻¹ ⟨a, b⟩.fst ≤ f⁻¹ ⟨a, b⟩.snd then -1 else 1) = if f (f⁻¹.signBijAux ⟨a, b⟩).fst ≤ f (f⁻¹.signBijAux ⟨a, b⟩).snd then -1 else 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_4", "s : Set α", "t : Set β", "g : δ → β"], "goal": "s ×ˢ (g ⁻¹' t) = (fun p => (p.1, g p.2)) ⁻¹' s ×ˢ t"}}
+{"state": {"context": ["n : ℤ"], "goal": "‖n‖ = |↑n|"}}
+{"state": {"context": ["n : ℕ"], "goal": "(2 ^ n).testBit n = true"}}
+{"state": {"context": ["α : Type u_1", "Inv α", "a : α"], "goal": "toLex a⁻¹ = (toLex a)⁻¹"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f g : E → ℝ", "hf : Differentiable ℝ f", "hg : Differentiable ℝ g", "h : ∀ (x : E), f x ≠ 0"], "goal": "Differentiable ℝ fun x => f x ^ g x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "LinearOrder β", "f g : α → β", "hf : Monotone f", "hg : Monotone g"], "goal": "Monotone fun x => min (f x) (g x)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s t : Set α"], "goal": "maximals r (s ∪ t) ⊆ maximals r s ∪ maximals r t"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type u_1", "N : Type u_2", "L.Structure M", "L.Structure N", "f : M ≃[L] N"], "goal": "f.toElementaryEmbedding.toEmbedding = f.toEmbedding"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p q : ℕ", "f : R[X]"], "goal": "(Polynomial.expand R (p * q)) f = (Polynomial.expand R p) ((Polynomial.expand R q) f)"}}
+{"state": {"context": ["α : Type u_3", "UniformSpace α", "AddGroup α", "UniformAddGroup α"], "goal": "∀ (a : α), UniformSpace.Completion.toCompl a = ↑α a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Fintype α", "Fintype β", "A : Matrix α β ℤ", "hn : Fintype.card α < Fintype.card β", "hm : 0 < Fintype.card α", "hA : A ≠ 0"], "goal": "∃ t,\n  t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑(Fintype.card β) * ‖A‖) ^ (↑(Fintype.card α) / (↑(Fintype.card β) - ↑(Fintype.card α)))"}}
+{"state": {"context": ["α : Type u_1", "s : Multiset α", "p q : α → Prop", "DecidablePred p", "DecidablePred q", "h : ∀ (b : α), p b → q b"], "goal": "Multiset.filter p s ≤ Multiset.filter q s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "k : ℕ"], "goal": "U R (-↑(k + 1) + 2) = 2 * X * U R (-↑(k + 1) + 1) - U R (-↑(k + 1))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝¹ : Countable ι", "f g : α → β", "m m' : MeasurableSpace α", "inst✝ : TopologicalSpace β", "hf : StronglyMeasurable f", "h_mono : m' ≤ m", "f_approx : ℕ → α →ₛ β := fun n => { toFun := ↑(hf.approx n), measurableSet_fiber' := ⋯, finite_range' := ⋯ }"], "goal": "StronglyMeasurable f"}}
+{"state": {"context": ["n : ℕ", "p : Fin n → Prop", "DecidablePred p", "i : Fin n"], "goal": "i ∈ Fin.find p → ∀ {j : Fin n}, j < i → ¬p j"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "f g : α → ℝ", "hf : AEStronglyMeasurable' m f μ", "hfg : Integrable (f * g) μ", "hg : Integrable g μ", "this✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]", "this : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]"], "goal": "μ[AEStronglyMeasurable'.mk f hf * g|m] =ᶠ[ae μ] f * μ[g|m]"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "s t : Set α", "hs_sup : SupClosed s", "hs_inf : InfClosed s", "ht_sup : SupClosed t", "ht_inf : InfClosed t"], "goal": "{ carrier := s, supClosed' := hs_sup, infClosed' := hs_inf } ≤\n    { carrier := t, supClosed' := ht_sup, infClosed' := ht_inf } ↔\n  s ⊆ t"}}
+{"state": {"context": ["E : ℕ → Type u_1", "inst✝¹ : (n : ℕ) → TopologicalSpace (E n)", "inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)", "x y : (n : ℕ) → E n", "s : Set ((n : ℕ) → E n)", "hs : IsClosed s", "hne : s.Nonempty", "H : longestPrefix x s < firstDiff x y", "xs : x ∉ s", "ys : y ∉ s", "l_eq : longestPrefix y s = longestPrefix x s"], "goal": "x ∈ cylinder y (longestPrefix x s)"}}
+{"state": {"context": ["α : Type u_1", "Monoid α", "Preorder α", "a b : αˣ"], "goal": "↑a < ↑b ↔ a < b"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "a b : E", "r : ℝ", "h : b ∈ Metric.ball a r"], "goal": "‖b‖ < ‖a‖ + r"}}
+{"state": {"context": ["x : ℝ"], "goal": "(↑(Real.arctan x)).re ≤ π / 2"}}
+{"state": {"context": ["x : ℝ", "x_large : 512 ≤ x", "f : ℝ → ℝ := fun x => log x + √(2 * x) * log (2 * x) - log 4 / 3 * x", "hf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)"], "goal": "x * (2 * x) ^ √(2 * x) * 4 ^ (2 * x / 3) ≤ 4 ^ x"}}
+{"state": {"context": ["α : Type u_2", "Semiring α", "a b : α", "ha : Even a", "hab : a ∣ b"], "goal": "Even b"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : Set (Topology.WithLowerSet α)"], "goal": "IsLowerSet (⇑Topology.WithLowerSet.toLowerSet ⁻¹' s) ↔ IsOpen s"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "β : Type u_2", "γ : Type u_3", "hd₁ : α", "tl₁ : Mathlib.Vector α n", "hd₂ : β", "tl₂ : Mathlib.Vector β n", "f : α → β → γ"], "goal": "Mathlib.Vector.map₂ f (hd₁ ::ᵥ tl₁) (hd₂ ::ᵥ tl₂) = f hd₁ hd₂ ::ᵥ Mathlib.Vector.map₂ f tl₁ tl₂"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "s t : Finmap β"], "goal": "s.entries = t.entries → s = t"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "α : Type u_3", "β : Type u_4", "E : Type u_5", "inst✝⁶ : Group G", "inst✝⁵ : Group H", "inst✝⁴ : MulAction G α", "inst✝³ : MeasurableSpace α", "inst✝² : MulAction H β", "inst✝¹ : MeasurableSpace β", "inst✝ : NormedAddCommGroup E", "s t : Set α", "μ : Measure α", "ν : Measure β", "h : IsFundamentalDomain G s μ", "f : α ≃ β", "hf : QuasiMeasurePreserving (⇑f.symm) ν μ", "e : H ≃ G", "hef : ∀ (g : H), Semiconj (⇑f) (fun x => e g • x) fun x => g • x", "g : G", "x : β"], "goal": "f.symm ((fun x => e.symm g • x) x) = (fun x => g • x) (f.symm x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l : α → β", "u : β → α", "SemilatticeSup α", "SemilatticeSup β", "gi : GaloisInsertion l u", "a b : β"], "goal": "l (u a ⊔ u b) = a ⊔ b"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "f : ι → α → ℝ≥0∞", "s : Finset α", "Nonempty ι", "h : ∀ (t : Finset α) (i j : ι), ∃ k, ∀ a ∈ t, f k a ≤ f i a ∧ f k a ≤ f j a"], "goal": "⨅ i, ∑ a ∈ s, f i a = ∑ a ∈ s, ⨅ i, f i a"}}
+{"state": {"context": ["n c d : ℕ", "hn : 2 * (c ^ 2 + 1) + d ≥ 2 * (c ^ 2 + 1)", "c0 : c > 0"], "goal": "c ^ (2 * (c ^ 2 + 1) + d) < (2 * (c ^ 2 + 1) + d - 1)!"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Preorder α", "a b : α"], "goal": "some ⁻¹' Iio ⊤ = univ"}}
+{"state": {"context": ["α : Sort u_2", "β : α → Sort u_1", "a : (i : α) ×' β i"], "goal": "(Equiv.psigmaEquivSigmaPLift β) a = ⟨{ down := a.fst }, { down := a.snd }⟩"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "P₂ : Type u_3", "V₁ : Type u_6", "V₂ : Type u_7", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AffineSpace V₂ P₂", "e : P₁ ≃ᵃ[k] P₂"], "goal": "(↑e).linear = ↑e.linear"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "𝕜 : Type u_4", "NontriviallyNormedField 𝕜", "NormedRing A", "NormedAlgebra 𝕜 A", "CompleteSpace A", "StarRing A", "NormedRing B", "NormedAlgebra 𝕜 B", "CompleteSpace B", "StarRing B", "ψ : A →⋆ₐ[𝕜] B", "φ : ↑(WeakDual.characterSpace 𝕜 B)"], "goal": "(WeakDual.CharacterSpace.compContinuousMap ψ) φ =\n  WeakDual.CharacterSpace.equivAlgHom.symm ((WeakDual.CharacterSpace.equivAlgHom φ).comp ψ.toAlgHom)"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "pf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a"], "goal": "WfDvdMonoid α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Fintype α", "DecidableEq β", "f : α ↪ β", "a : α"], "goal": "f.invOfMemRange ⟨f a, ⋯⟩ = a"}}
+{"state": {"context": ["f : Type u₀ → Type u₁", "EquivFunctor f", "α : Type u₀"], "goal": "EquivFunctor.mapEquiv f (Equiv.refl α) = Equiv.refl (f α)"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "t₁ : UniformSpace α", "t₂ : UniformSpace β", "a : Set (α × α)", "b : Set (β × β)", "ha : a ∈ 𝓤 α", "hb : b ∈ 𝓤 β"], "goal": "{p | (p.1.1, p.2.1) ∈ a ∧ (p.1.2, p.2.2) ∈ b} ∈ comap (fun p => (p.1.1, p.2.1)) (𝓤 α) ⊓ comap (fun p => (p.1.2, p.2.2)) (𝓤 β)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "G : Type u_5", "inst✝² : Group G", "inst✝¹ : Fintype G", "S : Set G", "inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k"], "goal": "∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "TopologicalSpace α", "TopologicalSpace β", "TopologicalSpace γ", "TopologicalSpace δ", "f : C(γ, δ)", "g : C(β, γ)", "h : C(α, β)"], "goal": "(f.comp g).comp h = f.comp (g.comp h)"}}
+{"state": {"context": ["R : Type u_2", "NonAssocSemiring R", "S : Subsemiring R"], "goal": "↑S.op = MulOpposite.unop ⁻¹' ↑S"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝³ : AddCommGroup E", "inst✝² : Module ℝ E", "s t : Set E", "x : E", "a : ℝ", "inst✝¹ : TopologicalSpace E", "inst✝ : ContinuousSMul ℝ E", "ε : ℝ", "hs : ε • s ∈ 𝓝 0", "hε : 0 < ε"], "goal": "∀ᶠ (x : E) in 𝓝 0, gauge s x ∈ Icc 0 ε"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "G : SimpleGraph α", "a b : α", "p : G.Walk a b", "hp : p.IsHamiltonian"], "goal": "p.IsPath"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x y✝ z : E", "δ ε : ℝ", "hδ : 0 < δ", "hε : 0 ≤ ε", "h : dist x z < ε + δ", "y : E", "yz : dist z y ≤ ε", "xy : dist y x < δ"], "goal": "∃ y, dist x y < δ ∧ dist y z ≤ ε"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R✝ : Sheaf J RingCat", "R : Cᵒᵖ ⥤ RingCat", "M₁ M₂ : PresheafOfModules R", "f : M₁ ⟶ M₂", "N : PresheafOfModules R", "hN : Presheaf.IsSheaf J N.presheaf", "inst✝² : J.WEqualsLocallyBijective AddCommGrp", "inst✝¹ : Presheaf.IsLocallySurjective J f.hom", "inst✝ : Presheaf.IsLocallyInjective J f.hom", "ψ : M₁ ⟶ N", "φ : M₂.presheaf ⟶ N.presheaf", "hφ : f.hom ≫ φ = ψ.hom", "X : Cᵒᵖ", "r : ↑(R.obj X)", "y : ↑(M₂.presheaf.obj X)", "Y : C", "p : Y ⟶ Opposite.unop X", "x : (forget AddCommGrp).obj (M₁.presheaf.obj (Opposite.op Y))", "hx : (f.hom.app (Opposite.op Y)) x = (M₂.presheaf.map p.op) y", "eq : (ψ.hom.app (Opposite.op Y)) ((R.map p.op) r • x) = (R.map p.op) r • (ψ.hom.app (Opposite.op Y)) x"], "goal": "(N.presheaf.map p.op) ((φ.app X) (r • y)) = (N.presheaf.map p.op) (r • (φ.app X) y)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a : α", "s : Set α", "x : α", "m : ℤ", "hms : x - ↑m ∈ s", "hm0 : 0 ≤ x - ↑m", "hm1 : x - ↑m < 1"], "goal": "fract x ∈ s"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : Measure α", "f✝ f g : α → ℝ"], "goal": "∫ (x : α), rexp (g x) ∂μ.tilted f = (∫ (x : α), rexp ((f + g) x) ∂μ) / ∫ (x : α), rexp (f x) ∂μ"}}
+{"state": {"context": ["G : Type u_1", "Group G", "A B : Subgroup G", "d : HNNExtension.NormalWord.TransversalPair G A B", "g : G"], "goal": "(HNNExtension.NormalWord.ofGroup g).toList = []"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "S : D", "T : CategoryTheory.Functor C D", "X Y : CategoryTheory.StructuredArrow S T", "h : X = Y"], "goal": "(CategoryTheory.eqToHom h).right = CategoryTheory.eqToHom ⋯"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "P : Algebra.Generators R S", "R' : Type u'", "S' : Type v'", "CommRing R'", "CommRing S'", "Algebra R' S'", "P' : Algebra.Generators R' S'", "Algebra R R'", "Algebra S S'", "Algebra R S'", "IsScalarTower R R' S'", "IsScalarTower R S S'", "f g : P.Hom P'", "c : P.Ring"], "goal": "↑((f.subToKer g) c) = f.toAlgHom c - g.toAlgHom c"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "p : ι → P", "j : ι", "hp : LinearIndependent k fun i => p ↑i -ᵥ p j", "w : ι → kˣ"], "goal": "LinearIndependent k fun i => ↑(w ↑i) • (p ↑i -ᵥ p j)"}}
+{"state": {"context": ["R : Type u", "M : Type v", "M₂ : Type w", "Semiring R", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂"], "goal": "LinearMap.fst R M M₂ ∘ₗ LinearMap.inl R M M₂ = LinearMap.id"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f g : E → F", "f' g' : E →L[𝕜] F", "x : E", "L : Filter E", "hf : HasFDerivAtFilter f f' x L", "hg : HasFDerivAtFilter g g' x L"], "goal": "HasFDerivAtFilter (fun x => f x - g x) (f' - g') x L"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "s t : Set α", "f g : α → β", "inst✝ : DecidablePred fun x => x ∈ s", "hs : MeasurableSet s"], "goal": "s.piecewise f g =ᶠ[ae (μ.restrict sᶜ)] g"}}
+{"state": {"context": ["n : ℤ"], "goal": "(2 * n).negOnePow = 1"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "r : R", "f : R[X]", "n✝¹ n✝ : ℕ"], "goal": "((taylor 0 ∘ₗ monomial n✝¹) 1).coeff n✝ = ((LinearMap.id ∘ₗ monomial n✝¹) 1).coeff n✝"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝³ : Category.{u_4, u_1} C", "inst✝² : Category.{u_3, u_2} D", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "inst✝ : W.HasLeftCalculusOfFractions", "X Y : C", "E : MorphismProperty.LeftFraction.Localization W ≌ D := uniq (MorphismProperty.LeftFraction.Localization.Q W) L W", "e : MorphismProperty.LeftFraction.Localization.Q W ⋙ E.functor ≅ L := compUniqFunctor (MorphismProperty.LeftFraction.Localization.Q W) L W", "g : W.LeftFraction X Y"], "goal": "e.inv.app X ≫ E.functor.map (MorphismProperty.LeftFraction.Localization.Hom.mk g) ≫ e.hom.app Y = g.map L ⋯"}}
+{"state": {"context": ["M : Type u_1", "Mul M", "s : Set M"], "goal": "Subsemigroup.closure (Subtype.val ⁻¹' s) = ⊤"}}
+{"state": {"context": ["b : Prop", "p : b → Prop"], "goal": "Exists p → b"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G"], "goal": "⊥.index = Nat.card G"}}
+{"state": {"context": ["X : RingCat"], "goal": "⇑(𝟙 X) = id"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s : Simplex ℝ P 0", "i : Fin 1"], "goal": "s.circumcenter = s.points i"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "One α", "One β"], "goal": "1 ×ˢ 1 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "f : α → β", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.Integrable (fun a => ‖f a‖) μ ↔ MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : CompleteSpace E", "inst✝⁴ : NormedSpace ℝ E", "a✝ b✝ : ℝ", "μ : Measure ℝ", "f : ℝ → E", "inst✝³ : RCLike 𝕜", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "a b : ℝ", "φ : ℝ → F →L[𝕜] E", "hφ : IntervalIntegrable φ μ a b", "v : F"], "goal": "(∫ (x : ℝ) in a..b, φ x ∂μ) v = ∫ (x : ℝ) in a..b, (φ x) v ∂μ"}}
+{"state": {"context": ["z : ℂ", "n : ℕ", "hn : n ≠ 0", "aux : ∀ (a b c d : ℂ), a * b * (c * d) = a * c * (b * d)", "this✝ : ↑n ^ z * ↑n ^ (1 - z) = ↑n", "this : ∀ (j : ℕ), (z + ↑(j + 1)) * (1 - z + ↑j) = ↑((j + 1) ^ 2) * (1 - z ^ 2 / (↑j + 1) ^ 2)"], "goal": "↑n * ↑n ! ^ 2 / ((∏ x ∈ Finset.range n, (z + ↑(x + 1)) * (1 - z + ↑x)) * (z * (↑n + 1 - z))) = ↑n / (↑n + 1 - z) * (1 / (z * ∏ j ∈ Finset.range n, (1 - z ^ 2 / (↑j + 1) ^ 2)))"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.WellPowered C", "CategoryTheory.Limits.HasWidePullbacks C", "A : C", "s : Set (CategoryTheory.Subobject A)", "f : CategoryTheory.Subobject A", "k : ∀ g ∈ s, f ≤ g"], "goal": "f ≤ CategoryTheory.Subobject.sInf s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "i : Fin n"], "goal": "Iio ↑i.castSucc = map (castSuccEmb.trans valEmbedding) (Iio i)"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : Jᵒᵖ ⥤ Cᵒᵖ", "c : CategoryTheory.Limits.Cone F.unop"], "goal": "(CategoryTheory.Limits.coconeOfConeUnop c).ι = CategoryTheory.NatTrans.removeUnop c.π"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a✝ b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "a : R"], "goal": "(p + C a).natDegree = p.natDegree"}}
+{"state": {"context": ["L : FirstOrder.Language", "α : Type u'", "n : ℕ", "φ : L.BoundedFormula α n", "h : φ.IsAtomic"], "goal": "φ.IsPrenex"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹⁶ : CommSemiring R", "R' : Type u_2", "inst✝¹⁵ : Monoid R'", "R'' : Type u_3", "inst✝¹⁴ : Semiring R''", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "S : Type u_8", "T : Type u_9", "inst✝¹³ : AddCommMonoid M", "inst✝¹² : AddCommMonoid N", "inst✝¹¹ : AddCommMonoid P", "inst✝¹⁰ : AddCommMonoid Q", "inst✝⁹ : AddCommMonoid S", "inst✝⁸ : AddCommMonoid T", "inst✝⁷ : Module R M", "inst✝⁶ : Module R N", "inst✝⁵ : Module R P", "inst✝⁴ : Module R Q", "inst✝³ : Module R S", "inst✝² : Module R T", "inst✝¹ : DistribMulAction R' M", "inst✝ : Module R'' M", "f : M ≃ₗ[R] M", "g : N ≃ₗ[R] N", "n : ℕ"], "goal": "congr f g ^ n = congr (f ^ n) (g ^ n)"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Bool", "l : List α", "h : ∀ (x : α), x ∈ l → p x = true → q x = true"], "goal": "countP p l ≤ countP q l"}}
+{"state": {"context": ["α : Type u_1", "KleeneAlgebra α", "a : α"], "goal": "1 ≤ a∗"}}
+{"state": {"context": ["I : Type u_1", "A : Type u_2", "X : I → Type u_3", "inst✝¹ : (i : I) → TopologicalSpace (X i)", "inst✝ : TopologicalSpace A", "f g : (i : I) → C(A, X i)", "S : Set A", "homotopies : (i : I) → (f i).HomotopyRel (g i) S", "t : ↑unitInterval", "x : A", "hx : x ∈ S"], "goal": "{ toFun := fun x => __src✝.toFun (t, x), continuous_toFun := ⋯ } x = (ContinuousMap.pi f) x"}}
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+{"state": {"context": ["α : Type u_1", "s : Set (Set α)"], "goal": "⋃₀ s = ⋃ i, ↑i"}}
+{"state": {"context": ["g' g : ℝ → ℝ", "a b : ℝ", "hcont : ContinuousOn g (Set.Icc a b)", "hderiv : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x", "g'pos : ∀ x ∈ Set.Ioo a b, 0 ≤ g' x"], "goal": "MeasureTheory.IntegrableOn g' (Set.Ioc a b) MeasureTheory.volume"}}
+{"state": {"context": ["α : Type u", "ι : Sort x", "f : ι → Filter α"], "goal": "(iSup f).sets = ⋂ i, (f i).sets"}}
+{"state": {"context": ["R : Type u", "S : Type v", "M : Type w", "inst✝⁷ : CommSemiring R", "inst✝⁶ : Semiring S", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : Module S M", "inst✝² : SMulCommClass S R M", "inst✝¹ : SMul R S", "inst✝ : IsScalarTower R S M", "x : R", "h : IsUnit ((algebraMap R (End S M)) x)", "m m' : M", "H : m = x • m'"], "goal": "x • m' = ↑h.unit m'"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : α → Set β"], "goal": "⋃ i, s i = Set.range fun a => ↑a.snd"}}
+{"state": {"context": ["ι : Type u_1", "c : ComplexShape ι", "C : Type u_3", "CategoryTheory.Category.{u_4, u_3} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "K : HomologicalComplex C c", "i : Option ι", "hi : i = none"], "goal": "CategoryTheory.Limits.IsZero (HomologicalComplex.extend.X K i)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "a b c : α"], "goal": "min (a - b) (a - c) = a - max b c"}}
+{"state": {"context": ["m n : ℕ", "h : m ≤ n"], "goal": "n.gcd (n - m) = n.gcd m"}}
+{"state": {"context": ["V : Type u_1", "V₁ : Type u_2", "V₂ : Type u_3", "V₃ : Type u_4", "inst✝³ : SeminormedAddCommGroup V", "inst✝² : SeminormedAddCommGroup V₁", "inst✝¹ : SeminormedAddCommGroup V₂", "inst✝ : SeminormedAddCommGroup V₃", "f✝ g✝ f : NormedAddGroupHom V₁ V₂", "K : AddSubgroup V₂", "C C' : ℝ", "h : f.SurjectiveOnWith K C", "H : C ≤ C'", "g : V₁", "k_in : f g ∈ K", "hg : ‖g‖ ≤ C * ‖f g‖"], "goal": "∃ g_1, f g_1 = f g ∧ ‖g_1‖ ≤ C' * ‖f g‖"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "K : Type u_4", "A : Type u_5", "A' : Type u_6", "A'' : Type u_7", "V : Type u", "V' : Type u_8", "x : ι → A", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing A'", "inst✝³ : CommRing A''", "inst✝² : Algebra R A", "inst✝¹ : Algebra R A'", "inst✝ : Algebra R A''", "a b : R", "n : ℕ", "H : ∀ (s : Finset A), (AlgebraicIndependent R fun i => ↑i) → s.card ≤ n", "s : Set A", "li : AlgebraicIndependent R Subtype.val", "t : Finset ↑s"], "goal": "AlgebraicIndependent R fun i => ↑i"}}
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+{"state": {"context": ["R : Type u", "Semiring R", "n m : ℕ", "r s : R"], "goal": "(Polynomial.monomial n) r * (Polynomial.monomial m) s = (Polynomial.monomial (n + m)) (r * s)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "x : M", "f : Module.Dual R M", "y : M"], "goal": "(Module.preReflection x f) y = y - f y • x"}}
+{"state": {"context": ["A : Type u₁", "B : Type u₂", "C : Type u₃", "D : Type u₄", "E : Type u₅", "F : Type u₆", "inst✝⁵ : Category.{v₁, u₁} A", "inst✝⁴ : Category.{v₂, u₂} B", "inst✝³ : Category.{v₃, u₃} C", "inst✝² : Category.{v₄, u₄} D", "inst✝¹ : Category.{v₅, u₅} E", "inst✝ : Category.{v₆, u₆} F", "G₁ : A ⥤ C", "G₂ : C ⥤ E", "H₁ : B ⥤ D", "H₂ : D ⥤ F", "L₁ : A ⥤ B", "R₁ : B ⥤ A", "L₂ : C ⥤ D", "R₂ : D ⥤ C", "L₃ : E ⥤ F", "R₃ : F ⥤ E", "adj₁ : L₁ ⊣ R₁", "adj₂ : L₂ ⊣ R₂", "adj₃ : L₃ ⊣ R₃", "α : G₁ ⋙ L₂ ⟶ L₁ ⋙ H₁", "β : G₂ ⋙ L₃ ⟶ L₂ ⋙ H₂", "b : B"], "goal": "adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫ R₃.map (β.app (G₁.obj (R₁.obj b))) ≫ R₃.map (H₂.map (α.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b))) = adj₃.unit.app (G₂.obj (G₁.obj (R₁.obj b))) ≫ R₃.map (β.app (G₁.obj (R₁.obj b))) ≫ R₃.map (H₂.map (L₂.map (adj₂.unit.app (G₁.obj (R₁.obj b))))) ≫ R₃.map (H₂.map (adj₂.counit.app (L₂.obj (G₁.obj (R₁.obj b))))) ≫ R₃.map (H₂.map (α.app (R₁.obj b))) ≫ R₃.map (H₂.map (H₁.map (adj₁.counit.app b)))"}}
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+{"state": {"context": ["α : Type u_2", "One α", "f : Filter α"], "goal": "f ≤ 1 ↔ 1 ∈ f"}}
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+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝³ : CancelCommMonoidWithZero R", "inst✝² : UniqueFactorizationMonoid R", "inst✝¹ : NormalizationMonoid R", "inst✝ : DecidableRel Dvd.dvd", "a b : R", "n : ℕ", "ha : Irreducible a", "hb : b ≠ 0"], "goal": "a ^ n ∣ b ↔ replicate n (normalize a) ≤ normalizedFactors b"}}
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+{"state": {"context": ["Ω : Type u_1", "MeasureTheory.MeasureSpace Ω", "MeasureTheory.IsProbabilityMeasure ℙ", "X : Ω → ℝ", "hX : MeasureTheory.AEStronglyMeasurable X ℙ"], "goal": "ProbabilityTheory.evariance X ℙ = (∫⁻ (ω : Ω), ↑(‖X ω‖₊ ^ 2)) - ENNReal.ofReal ((∫ (a : Ω), X a) ^ 2)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "B : LinearMap.BilinForm R M", "l r : M →ₗ[R] M"], "goal": "(B.compRight r).compLeft l = B.comp l r"}}
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+{"state": {"context": ["P : Type u_2", "MetricSpace P", "p c : P", "ps : Set P", "r : ℝ", "hp : p ∈ ps", "hps : ps ⊆ Metric.sphere { center := c, radius := r }.center { center := c, radius := r }.radius"], "goal": "dist p c = r"}}
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+{"state": {"context": ["N : ℕ", "x✝ : SL(2, ℤ)"], "goal": "x✝ ∈ Gamma 1 ↔ x✝ ∈ ⊤"}}
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+{"state": {"context": [], "goal": "SetTheory.PGame.birthday 0 = 0"}}
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+{"state": {"context": ["z w : ℂ"], "goal": "↑(z + w) = ↑z + ↑w"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "S : CategoryTheory.Subgroupoid C", "Sw : S.IsWide", "c : C"], "goal": "CategoryTheory.CategoryStruct.id c ∈ S.arrows c c"}}
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+{"state": {"context": ["M : Type u_4", "Add M", "c : AddCon M", "x : M"], "goal": "Quot.mk (⇑c) x = ↑x"}}
+{"state": {"context": ["x : ℝ", "hx : x ∈ Icc (-(π / 2)) (π / 2)"], "goal": "0 ≤ x + π / 2"}}
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+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝¹ : CommSemiring R", "inst✝ : Nontrivial R", "i j : σ →₀ ℕ"], "goal": "(1 = 0 ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "F : Type u_2", "SeminormedAddCommGroup F", "NormedSpace 𝕜 F", "fr : RestrictScalars ℝ 𝕜 F →ₗ[ℝ] ℝ", "x : F"], "goal": "fr.extendTo𝕜 x = ↑(fr x) - RCLike.I * ↑(fr (RCLike.I • x))"}}
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+{"state": {"context": ["n : ℕ", "NeZero n", "k : Fin n"], "goal": "0 * k = 0"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_2", "AddCommMonoid M", "Module R M"], "goal": "Function.LeftInverse ⇑TensorAlgebra.ιInv ⇑(TensorAlgebra.ι R)"}}
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+{"state": {"context": ["ι : Sort u_1", "M : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝ : Mul M", "S : ι → Subsemigroup M", "i : ι"], "goal": "∀ {x : M}, x ∈ S i → x ∈ iSup S"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "x : α"], "goal": "IsUnit x ↔ ∀ (y : α), x ∣ y"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "s✝ : Set E", "f✝ : E → E", "f'✝ : E → E →L[ℝ] E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "s : Set ℝ", "f f' : ℝ → ℝ", "hs : MeasurableSet s", "hf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x", "hf : InjOn f s", "g : ℝ → F"], "goal": "IntegrableOn g (f '' s) volume ↔ IntegrableOn (fun x => |f' x| • g (f x)) s volume"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace ℝ E", "hE : CompleteSpace E", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : SMulCommClass ℝ 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : CompleteSpace F", "G : Type u_5", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "m0 : MeasurableSpace α", "μ : Measure α", "f : α → ℝ", "r : ℝ", "hfint : Integrable f μ", "hfint' : 0 ≤ ∫ (x : α), f x ∂μ", "hf : ∀ᵐ (ω : α) ∂μ, f ω ≤ r"], "goal": "eLpNorm f 1 μ ≤ 2 * μ univ * ENNReal.ofReal r"}}
+{"state": {"context": ["n d : ℕ", "hnd : d ∣ n"], "goal": "(n / d).totient = (Finset.filter (fun k => n.gcd k = d) (Finset.range n)).card"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "c : ComplexShape ι", "c' : ComplexShape ι'", "e : c.Embedding c'", "e.IsTruncGE", "j : ι", "k' : ι'", "h : c'.Rel (e.f j) k'"], "goal": "∃ k, e.f k = k'"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "X : CategoryTheory.SimplicialObject C", "Y Z : C", "q n : ℕ", "φ : Y ⟶ X _[n + 1]", "v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ", "f : Z ⟶ Y"], "goal": "AlgebraicTopology.DoldKan.HigherFacesVanish q (f ≫ φ)"}}
+{"state": {"context": ["x : ℝ"], "goal": "↑x ≠ 0 ↔ x ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "PseudoEMetricSpace α", "PseudoEMetricSpace β", "K : ℝ≥0", "f : α → β", "hf : AntilipschitzWith K f", "s : Set α"], "goal": "EMetric.diam s ≤ ↑K * EMetric.diam (f '' s)"}}
+{"state": {"context": ["h : IsSquare (-1)"], "goal": "∃ x y, 1 = x ^ 2 + y ^ 2"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "GroupWithZero α", "s : Finset α", "hs : s.Nonempty"], "goal": "s / 0 = 0"}}
+{"state": {"context": ["P : Type u_1", "LE P", "IsDirected P fun x x_1 => x ≤ x_1", "Nonempty P", "I : Order.Ideal P", "I.IsMaximal"], "goal": "IsCoatom I"}}
+{"state": {"context": ["n : ℕ", "n.AtLeastTwo"], "goal": "↑(OfNat.ofNat n) = OfNat.ofNat n"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X Y : C", "f f' : X ⟶ Y", "inst✝² : HasImage f", "inst✝¹ : HasImage f'", "inst✝ : HasEqualizers C", "h : f = f'"], "goal": "factorThruImage f ≫ ι f = factorThruImage f ≫ lift (MonoFactorisation.mk (image f') (ι f') (factorThruImage f') ⋯) ≫ ι f'"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "P : CategoryTheory.MorphismProperty C", "hP : P.StableUnderBaseChange", "X Y S : C", "f : X ⟶ S", "g : Y ⟶ S", "CategoryTheory.Limits.HasPullback f g", "H : P f"], "goal": "P (CategoryTheory.Limits.pullback.snd f g)"}}
+{"state": {"context": ["V : Type u_1", "W : Type u_2", "f : V ↪ W"], "goal": "Monotone (SimpleGraph.map f)"}}
+{"state": {"context": ["α : Type u", "s : Computation α", "a : α", "n : ℕ", "h : s.think.Results a n"], "goal": "∃ m, s.Results a m ∧ n = m + 1"}}
+{"state": {"context": ["S : Type u_1", "R : Type u_2", "A : Type u_3", "CommSemiring S", "CommSemiring R", "NonUnitalSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "Algebra S R", "DistribMulAction S A", "IsScalarTower S R A"], "goal": "algebraMap S (Unitization R A) = (Unitization.inlRingHom R A).comp (algebraMap S R)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β"], "goal": "s ∩ f ⁻¹' t ⊆ f ⁻¹' (f '' s ∩ t)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x✝ : M", "p✝ p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "p : (x : M) → x ∈ span R s → Prop", "zero : p 0 ⋯", "add : ∀ (x : M) (hx : x ∈ span R s) (y : M) (hy : y ∈ span R s), p x hx → p y hy → p (x + y) ⋯", "smul_mem : ∀ (r : R) (x : M) (h : x ∈ s), p (r • x) ⋯", "x : M", "hx : x ∈ span R s"], "goal": "p x hx"}}
+{"state": {"context": ["α : Sort v", "β : α → Sort u", "a₁ a₂ : α", "b : (x : α) → β x", "h : a₁ = a₂"], "goal": "HEq\n  (let x := a₁;\n  b x)\n  (let x := a₂;\n  b x)"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : Monoid G", "a b x✝ y : G", "n✝ m : ℕ", "x : G", "n : ℕ"], "goal": "x ^ (n % orderOf x) = x ^ (n % orderOf x + orderOf x * (n / orderOf x))"}}
+{"state": {"context": ["f : CircleDeg1Lift", "x : ℝ"], "goal": "f (x + 1) = f x + 1"}}
+{"state": {"context": ["E : Type u_6", "SeminormedGroup E", "a : E"], "goal": "dist a 1 = ‖a‖"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "MulZeroClass α", "a : WithBot α"], "goal": "a * ⊥ = if a = 0 then 0 else ⊥"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝⁴ : MeasurableSpace β✝", "μ ν ν₁ ν₂ : Measure α", "s t : Set α", "β : Type u_5", "inst✝³ : TopologicalSpace β", "inst✝² : T1Space β", "inst✝¹ : SecondCountableTopology β", "inst✝ : Nonempty β", "f : α → β", "m : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure", "h : ∀ (b : β), m {b}ᶜ ≠ 0", "inhabited_h : Inhabited β", "this : m univ ≠ 0", "b : β", "hb : ∀ t ∈ 𝓝 b, 0 < m t", "a : β", "hab : a ≠ b", "ha : ∀ t ∈ 𝓝 a, 0 < m t"], "goal": "∃ a b, a ≠ b ∧ (∀ s ∈ 𝓝 a, 0 < μ (f ⁻¹' s)) ∧ ∀ t ∈ 𝓝 b, 0 < μ (f ⁻¹' t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "h : 2 ≤ n", "this : 1 ≤ n / 2", "m : ℕ", "e : n / 2 = m.succ"], "goal": "(match (n.bodd, n / 2) with | (false, m) => Option.map Sum.inl (decode m) | (fst, m) => Option.map Sum.inr (decode m)) = none"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "p : α → Prop", "hs : MeasurableSet s"], "goal": "(∀ᵐ (x : α) ∂μ.restrict s, p x) ↔ ∀ᵐ (x : α) ∂μ, x ∈ s → p x"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : Balanced C", "𝒢 : Set C", "h𝒢 : IsSeparating 𝒢", "X Y : C", "f : X ⟶ Y", "hf : ∀ G ∈ 𝒢, ∀ (h : G ⟶ Y), ∃! h', h' ≫ f = h"], "goal": "IsIso f"}}
+{"state": {"context": ["n : ℕ", "f : ℕ → ℕ", "hf' : ∀ k < n, Squarefree (f k)", "hn : n ≠ 0", "hf : StrictMonoOn f (Set.Iio n)", "this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n", "𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)", "hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))"], "goal": "Set.InjOn (fun s => ∏ p ∈ s, p) ↑(𝒜 \\\\ 𝒜)"}}
+{"state": {"context": [], "goal": "Filter.Tendsto (fun n => ↑n ! / ↑n ^ n) Filter.atTop (𝓝 0)"}}
+{"state": {"context": ["G : Type u", "AddGroup G", "N : AddSubgroup G", "nN : N.Normal", "M : Type x", "AddMonoid M", "f g : G ⧸ N →+ M", "h : f.comp (QuotientAddGroup.mk' N) = g.comp (QuotientAddGroup.mk' N)"], "goal": "f = g"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M", "p : Submodule R M", "R₂ : Type u_3", "M₂ : Type u_4", "Ring R₂", "AddCommGroup M₂", "Module R₂ M₂", "τ₁₂ : R →+* R₂", "q : Submodule R₂ M₂", "f : M →ₛₗ[τ₁₂] M₂", "h : p ≤ Submodule.comap f q", "x : M"], "goal": "(p.mapQ q f h) (Submodule.Quotient.mk x) = Submodule.Quotient.mk (f x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "s t : SignedMeasure α", "f : α → ℝ", "hf : Measurable f", "hfi : Integrable f μ", "htμ : t ⟂ᵥ μ.toENNRealVectorMeasure", "hadd : s = t + μ.withDensityᵥ f", "this : IsFiniteMeasure (μ.withDensity fun x => ENNReal.ofReal ((-f) x))"], "goal": "IsFiniteMeasure (t.toJordanDecomposition.negPart + μ.withDensity fun x => ENNReal.ofReal (-f x))"}}
+{"state": {"context": ["K : Type u₁", "Field K", "v : Valuation K ℝ≥0", "O : Type u₂", "CommRing O", "Algebra O K", "hv : v.Integers O", "p : ℕ", "x y : ModP K v O hv p"], "goal": "ModP.preVal K v O hv p (x + y) ≤ max (ModP.preVal K v O hv p x) (ModP.preVal K v O hv p y)"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "a : G₀"], "goal": "a⁻¹ = 0 ↔ a = 0"}}
+{"state": {"context": ["x y : USize"], "goal": "x.toNat = y.toNat → x = y"}}
+{"state": {"context": ["x : ℝ", "hx : Liouville x", "n : ℕ", "H : ¬∃ᶠ (b : ℕ) in atTop, ∃ a, x ≠ ↑a / ↑b ∧ |x - ↑a / ↑b| < 1 / ↑b ^ n"], "goal": "False"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X"], "goal": "Continuous ULift.up"}}
+{"state": {"context": [], "goal": "1 ≤ Real.pi / 2"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "σ : Type u_3", "AddCommMonoid M", "SemilatticeSup M", "OrderBot M", "w : σ → M", "φ : MvPolynomial σ R", "d : σ →₀ ℕ", "hd : d ∈ φ.support"], "goal": "(Finsupp.weight w) d ≤ MvPolynomial.weightedTotalDegree w φ"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "Group G", "Group G'", "f : G →* G'", "hf : Function.Surjective ⇑f", "n : ℕ"], "goal": "Subgroup.map f (derivedSeries G n) = derivedSeries G' n"}}
+{"state": {"context": ["α : Sort u_1", "β : α → Sort u_2", "γ : (a : α) → β a → Sort u_3", "δ : (a : α) → (b : β a) → γ a b → Sort u_4", "ε : (a : α) → (b : β a) → (c : γ a b) → δ a b c → Sort u_5", "p q : (a : α) → (b : β a) → (c : γ a b) → (d : δ a b c) → ε a b c d → Prop", "h : ∀ (a : α) (b : β a) (c : γ a b) (d : δ a b c) (e : ε a b c d), p a b c d e ↔ q a b c d e"], "goal": "(∃ a b c d e, p a b c d e) ↔ ∃ a b c d e, q a b c d e"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "a : α", "m n : ℕ"], "goal": "iterate f a (m + n) = iterate f a m ++ iterate f (f^[m] a) n"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s s₁ s₂ : Finset α", "a✝ : α", "f g : α → β", "a : Multiset α", "i : Multiset (Multiset α)", "l : List (Multiset α)"], "goal": "(sum ⟦l⟧).Disjoint a ↔ ∀ b ∈ ⟦l⟧, b.Disjoint a"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "hS : S.Exact", "hS' : S.ShortExact", "v : ι → ↑S.X₁", "β : Type u_4", "u : ι ⊕ β → ↑S.X₂", "huv : u ∘ Sum.inl = ⇑S.f ∘ v", "hv : ⊤ ≤ span R (range v)", "hw : ⊤ ≤ span R (range (⇑S.g ∘ u ∘ Sum.inr))", "m : ↑S.X₂", "a✝ : m ∈ ⊤", "cm : β →₀ R", "hm : (cm.sum fun i a => a • (⇑S.g ∘ u ∘ Sum.inr) i) = S.g m", "m' : ↑S.X₂ := cm.sum fun j a => a • u (Sum.inr j)", "n : ↑S.X₁", "cn : ι →₀ R", "hnm : (cn.sum fun a b => S.f (b • v a)) = m - m'", "hn : (cn.sum fun i a => a • v i) = n", "hn' : (cn.sum fun a b => b • S.f (v a)) = cn.sum fun a b => b • u (Sum.inl a)"], "goal": "m' ∈ span R (range u)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "f : α → ℝ", "s : Set α", "a : α", "hf : ContinuousWithinAt f s a", "h₀ : f a ≠ 0"], "goal": "ContinuousWithinAt (fun x => Real.log (f x)) s a"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p₁ p₂ p₃ : P"], "goal": "EuclideanGeometry.angle p₁ p₂ p₃ = 0 ↔ p₁ ≠ p₂ ∧ Wbtw ℝ p₂ p₁ p₃ ∨ p₃ ≠ p₂ ∧ Wbtw ℝ p₂ p₃ p₁"}}
+{"state": {"context": ["M : Type u_1", "AddMonoid M", "x : M"], "goal": "0 ∈ multiples x"}}
+{"state": {"context": ["G : Type u_1", "TopologicalSpace G", "InvolutiveNeg G", "ContinuousNeg G"], "goal": "⇑(Homeomorph.neg G) = Neg.neg"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "S : Submonoid R", "H : ⊤ ≤ LinearMap.ker (mkLinearMap S M)", "x : M", "s : ↥S"], "goal": "Function.uncurry (IsLocalizedModule.mk' (mkLinearMap S M)) (x, s) = 0"}}
+{"state": {"context": ["l : List ℤ", "n : ℕ"], "goal": "jacobiSym l.prod n = (List.map (fun a => jacobiSym a n) l).prod"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "t : Batteries.RBNode α"], "goal": "Batteries.RBNode.Ordered cmp t.setRed ↔ Batteries.RBNode.Ordered cmp t"}}
+{"state": {"context": ["α : Type u", "x y : α", "L1 L2 : List α"], "goal": "{ head := x, tail := L1 } * { head := y, tail := L2 } = { head := x, tail := L1 ++ y :: L2 }"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.MonoidalPreadditive C", "P : C", "Q R : C", "J : Type u_2", "s : Finset J", "g : J → (Q ⟶ R)"], "goal": "CategoryTheory.MonoidalCategory.whiskerLeft P (∑ j ∈ s, g j) =\n  ∑ j ∈ s, CategoryTheory.MonoidalCategory.whiskerLeft P (g j)"}}
+{"state": {"context": ["n : ℕ"], "goal": "n.descFactorial 0 = 1"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "G₀ : Type u_3", "M₀ : Type u_4", "R : Type u_5", "inst✝ : MulZeroClass M₀", "s✝ t : Set ι", "f✝ g✝ : ι → M₀", "i : ι", "s : Set ι", "f g : ι → M₀", "x✝ : ι", "h✝ : x✝ ∉ s"], "goal": "0 = 0 * 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "s : AffineSubspace ℝ P", "ps : Set P", "h : ps ⊆ ↑s", "Nonempty ↥s", "n : ℕ", "FiniteDimensional ℝ ↥s.direction", "hd : FiniteDimensional.finrank ℝ ↥s.direction = n", "hc : EuclideanGeometry.Cospherical ps", "sx₁ sx₂ : Affine.Simplex ℝ P n", "hsx₁ : Set.range sx₁.points ⊆ ps", "hsx₂ : Set.range sx₂.points ⊆ ps"], "goal": "sx₁.circumradius = sx₂.circumradius"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "t = s ∩ t ↔ t ⊆ s"}}
+{"state": {"context": ["m : Type u", "n : Type v", "α : Type w", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "inst✝² : DecidableEq m", "inst✝¹ : Fintype m", "inst✝ : CommRing α", "A : Matrix n n α", "b : n → α", "i j : n"], "goal": "1 = det 1"}}
+{"state": {"context": ["F : Type u → Type w", "G : Type v → Type u", "α : Type v", "x y : Functor.Comp F G α"], "goal": "x.run = y.run → x = y"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "X : C", "E : CategoryTheory.PreOneHypercover X", "F : CategoryTheory.Functor C D", "i₁ i₂ : E.I₀"], "goal": "(E.map F).I₁ i₁ i₂ = E.I₁ i₁ i₂"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "A : ι → Type u_3", "AddMonoid ι", "GradedMonoid.GMonoid A", "l : List α", "f : α → GradedMonoid A"], "goal": "(List.map f l).prod =\n  GradedMonoid.mk (l.dProdIndex fun i => (f i).fst) (l.dProd (fun i => (f i).fst) fun i => (f i).snd)"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "a b : α", "h : a ≤ b"], "goal": "Set.Icc a b \\ Set.Ioo a b = {a, b}"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝¹ : HasAffineProperty P Q", "X✝ Y✝ : Scheme", "f✝ : X✝ ⟶ Y✝", "hP' : Q.StableUnderBaseChange", "X Y S : Scheme", "f : X ⟶ S", "g : Y ⟶ S", "inst✝ : IsAffine S", "H : Q g"], "goal": "P (pullback.fst f g)"}}
+{"state": {"context": ["b : ℕ", "b_ge_two : 2 ≤ b", "p : ℕ", "p_prime : Prime p", "p_gt_two : 2 < p", "not_dvd : ¬p ∣ b * (b ^ 2 - 1)", "A : ℕ := (b ^ p - 1) / (b - 1)", "B : ℕ := (b ^ p + 1) / (b + 1)", "hi_A : 1 < A", "hi_B : 1 < B"], "goal": "(A * B).FermatPsp b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "a : R", "x : σ"], "goal": "degreeOf x (C a) = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f f₁ f₂ : Filter α", "g g₁ g₂ : Filter β", "m : α → β", "m' : β → γ", "s : Set α", "t : Set β"], "goal": "∅ ∈ map m f → ∅ ∈ f"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "A : Type u_3", "inst✝¹⁶ : Category.{u_4, u_1} C", "inst✝¹⁵ : HasZeroObject C", "inst✝¹⁴ : HasShift C ℤ", "inst✝¹³ : Preadditive C", "inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝¹¹ : Pretriangulated C", "inst✝¹⁰ : Category.{?u.108053, u_2} D", "inst✝⁹ : HasZeroObject D", "inst✝⁸ : HasShift D ℤ", "inst✝⁷ : Preadditive D", "inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive", "inst✝⁵ : Pretriangulated D", "inst✝⁴ : Category.{u_5, u_3} A", "inst✝³ : Abelian A", "F : C ⥤ A", "inst✝² inst✝¹ : F.IsHomological", "inst✝ : F.ShiftSequence ℤ", "T T' : Triangle C", "hT✝ : T ∈ distinguishedTriangles", "hT' : T' ∈ distinguishedTriangles", "φ : T ⟶ T'", "n₀ n₁ : ℤ", "h✝ : n₀ + 1 = n₁", "X Y : C", "f : X ⟶ Y", "Z : C", "g : Y ⟶ Z", "h : Z ⟶ (shiftFunctor C 1).obj X", "hT : Triangle.mk f g h ∈ distinguishedTriangles"], "goal": "F.homologicalKernel.W f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "Z : D", "F : CategoryTheory.Functor C D", "M : (x : C) → F.obj x ⟶ Z", "hM : ∀ (x y : C) (f : x ⟶ y), CategoryTheory.CategoryStruct.comp (F.map f) (M y) = M x"], "goal": "(CategoryTheory.WithTerminal.liftStar F M hM).inv = CategoryTheory.CategoryStruct.id Z"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "α : Type u_1", "p₂ : α → Prop"], "goal": "∀ {α_1 : Type u_1} {q : α_1 → Prop} {f : m (α → α_1)} [inst : Applicative m] [inst_1 : LawfulApplicative m] {x : m α},\n  SatisfiesM (fun f => ∀ {a : α}, p₂ a → q (f a)) f → SatisfiesM p₂ x → SatisfiesM q (Seq.seq f fun x_1 => x)"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalSemiring R", "s : Set R"], "goal": "NonUnitalSubsemiring.center R ≤ NonUnitalSubsemiring.centralizer s"}}
+{"state": {"context": ["n : ℕ"], "goal": "0 < n → n.zeckendorf = n.greatestFib :: (n - Nat.fib n.greatestFib).zeckendorf"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "NormedField 𝕜", "AddCommGroup V", "Module 𝕜 V", "e : ENorm 𝕜 V"], "goal": "↑e 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "σ : Type u_4", "inst✝³ : Primcodable α", "inst✝² : Primcodable β", "inst✝¹ : Primcodable γ", "inst✝ : Primcodable σ", "p : α → ℕ →. Bool", "hp : Partrec₂ p", "n✝ : ℕ", "a : α", "e : decode n✝ = Option.some a", "n : ℕ"], "goal": "Part.map (fun x => decide ((bif x then 0 else 1) = 0)) (p a n) = p a n"}}
+{"state": {"context": ["a : Prop"], "goal": "¬(¬a ∧ a)"}}
+{"state": {"context": ["n : ℕ", "f : ℕ → ℕ", "hf' : ∀ k < n, Squarefree (f k)", "hn : n ≠ 0", "hf : StrictMonoOn f (Set.Iio n)", "this : ∀ i < n, ∀ j < n, f i < (f i).gcd (f j) * n", "𝒜 : Finset (Finset ℕ) := image (fun n => (f n).primeFactors) (Iio n)", "hf'' : ∀ i < n, ∀ (j : ℕ), Squarefree (f i / (f i).gcd (f j))"], "goal": "∀ a ∈ 𝒜 \\\\ 𝒜, (fun s => ∏ p ∈ s, p) a ∈ Ioo 0 n"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "hm : 0 < m"], "goal": "CategoryTheory.IsIso (AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec 𝒜 f)"}}
+{"state": {"context": ["G : Type u_1", "AddMonoid G", "y : AddUnits G"], "goal": "addOrderOf ↑y = addOrderOf y"}}
+{"state": {"context": ["C : Type u", "inst✝⁸ : Category.{v, u} C", "X Y : C", "D : Type u₂", "inst✝⁷ : Category.{w, u₂} D", "F : C ⥤ D", "A✝ A' B B' : C", "inst✝⁶ : HasBinaryCoproduct A✝ B", "inst✝⁵ : HasBinaryCoproduct A' B'", "inst✝⁴ : HasBinaryCoproduct (F.obj A✝) (F.obj B)", "inst✝³ : HasBinaryCoproduct (F.obj A') (F.obj B')", "inst✝² : HasBinaryCoproducts C", "inst✝¹ : HasBinaryCoproducts D", "A : C", "inst✝ : ∀ (B : C), IsIso (coprodComparison F A B)"], "goal": "IsIso { app := fun B => coprodComparison F A B, naturality := ⋯ }"}}
+{"state": {"context": ["α : Type u_1", "M₁ M₂ : Matroid α"], "goal": "M₁ = M₂✶ ↔ M₁✶ = M₂"}}
+{"state": {"context": ["P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop", "h₁ : RespectsIso P", "h₂ : ∀ ⦃R S T : Type u⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) → P Algebra.TensorProduct.includeLeftRingHom", "R S R' S' : Type u", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing S", "inst✝⁸ : CommRing R'", "inst✝⁷ : CommRing S'", "inst✝⁶ : Algebra R S", "inst✝⁵ : Algebra R R'", "inst✝⁴ : Algebra R S'", "inst✝³ : Algebra S S'", "inst✝² : Algebra R' S'", "inst✝¹ : IsScalarTower R S S'", "inst✝ : IsScalarTower R R' S'", "h : Algebra.IsPushout R S R' S'", "H : P (algebraMap R S)", "e : TensorProduct R R' S ≃ₛₗ[id R'] S' := ⋯.equiv", "f' : TensorProduct R R' S →ₐ[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')", "this : ∀ (x : TensorProduct R R' S), e x = f' x"], "goal": "P (algebraMap R' S')"}}
+{"state": {"context": ["α : Type u_1", "ord : Ord α", "inst : Batteries.OrientedOrd α"], "goal": "Batteries.OrientedOrd α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "x : α", "y : β", "r : ℝ", "z : α × β"], "goal": "z ∈ ball x r ×ˢ ball y r ↔ z ∈ ball (x, y) r"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "p : P"], "goal": "↑(affineSpan k {p}) = {p}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x : α", "inst✝ : AddSemigroup α", "h1 : Periodic f c₁", "h2 : Periodic f c₂"], "goal": "Periodic f (c₁ + c₂)"}}
+{"state": {"context": ["G : Type w", "TopologicalSpace G", "Group G", "TopologicalGroup G", "s t : Set G", "ht : IsClosed t", "hs : IsCompact s"], "goal": "IsClosed (s * t)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J₁ : CategoryTheory.GrothendieckTopology C", "X : C", "S : CategoryTheory.Sieve X"], "goal": "S ≤ J₁.close S"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "s : Set A"], "goal": "∃ s_1, zeroLocus 𝒜 s = zeroLocus 𝒜 s_1"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "s : MeasureTheory.SignedMeasure α", "μ : MeasureTheory.Measure α", "r : ℝ"], "goal": "(r • s).singularPart μ = r • s.singularPart μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "s : Set E"], "goal": "balancedCore 𝕜 s ⊆ balancedCoreAux 𝕜 s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f✝ f₁ f₂ : Filter α", "g g₁ g₂ : Filter β", "m : α → β", "m' : β → γ", "s : Set α", "t : Set β", "f : α → β", "F : Filter β", "x : α"], "goal": "(∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A) ↔ ∀ B ∈ F, f x ∈ B"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommSemiring R", "M : Submonoid R", "S : Type u_2", "inst✝⁵ : CommSemiring S", "inst✝⁴ : Algebra R S", "P : Type u_3", "inst✝³ : CommSemiring P", "A : Type u_4", "inst✝² : CommRing A", "inst✝¹ : IsDomain A", "I : Ideal R", "hI : I.IsPrime", "inst✝ : IsLocalization.AtPrime S I", "x : R", "h : optParam (LocalRing S) ⋯"], "goal": "(algebraMap R S) x ∉ LocalRing.maximalIdeal S ↔ x ∉ I"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "n : ℕ", "φ ψ : R⟦X⟧"], "goal": "∑ p ∈ antidiagonal (single () n), (MvPowerSeries.coeff R p.1) φ * (MvPowerSeries.coeff R p.2) ψ = ∑ p ∈ antidiagonal n, (coeff R p.1) φ * (coeff R p.2) ψ"}}
+{"state": {"context": ["α : Type u_4", "β : Type u_5", "a : Part α", "Decidable a.Dom", "b : β", "f : α → β"], "goal": "a.toOption.elim b f = if h : a.Dom then f (a.get h) else b"}}
+{"state": {"context": ["α : Type u", "inst✝² : UniformSpace α", "β : Type v", "γ : Type w", "inst✝¹ : UniformSpace β", "inst✝ : UniformSpace γ", "s : Set (CauchyFilter α × CauchyFilter α)", "t : Set (α × α)", "_h : t ∈ 𝓤 α"], "goal": "gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : Countable ι", "inst✝ : T2Space α", "s : ι → Set α", "hs : ∀ (n : ι), AnalyticSet (s n)", "i₀ : ι", "β : ι → Type", "hβ : (n : ι) → TopologicalSpace (β n)", "h'β : ∀ (n : ι), PolishSpace (β n)", "f : (n : ι) → β n → α", "f_cont : ∀ (n : ι), Continuous (f n)", "f_range : ∀ (n : ι), range (f n) = s n", "γ : Type u_2 := (n : ι) → β n", "t : Set γ := ⋂ n, {x | f n (x n) = f i₀ (x i₀)}"], "goal": "AnalyticSet (⋂ n, s n)"}}
+{"state": {"context": ["a b : ℕ", "ha : Odd a", "hb : Odd b"], "goal": "jacobiSym (↑a) b = (-1) ^ (a / 2 * (b / 2)) * jacobiSym (↑b) a"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Lattice α", "Lattice β", "f : LatticeHom α β"], "goal": "f.withTop.toSupHom = f.withTop"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝¹ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "c c₁ c₂ : ℝ≥0", "ε ε₁ ε��� : ℝ", "π₁✝ π₂✝ : TaggedPrepartition I", "inst✝ : CompleteSpace F", "h : Integrable I l f vol", "π₁ π₂ : Prepartition I", "hU : π₁.iUnion = π₂.iUnion"], "goal": "Tendsto (integralSum f vol) (l.toFilteriUnion I π₂) (𝓝 (∑ J ∈ π₂.boxes, integral J l f vol))"}}
+{"state": {"context": ["E : Type u", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℂ E", "F : Type v", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℂ F", "inst✝ : StrictConvexSpace ℝ F", "f : E → F", "U : Set E", "c : E", "hc : IsPreconnected U", "ho : IsOpen U", "hd : DifferentiableOn ℂ f U", "hcU : c ∈ U", "hm : IsMaxOn (norm ∘ f) U c", "x : E", "hx : x ∈ U", "H₁ : ‖f x‖ = ‖f c‖", "H₂ : ‖f x + f c‖ = ‖f c + f c‖"], "goal": "‖f x + Function.const E (f c) x‖ = ‖f x‖ + ‖Function.const E (f c) x‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "CompleteLattice β", "f : α ≃o β", "s : Set α"], "goal": "f (sSup s) = ⨆ a ∈ s, f a"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{?u.84186, u_1} C", "inst✝³ : Category.{?u.84190, u_2} D", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "S : ShortComplex C", "inst✝ : HasZeroObject C", "s : S.Splitting", "hi : IsLimit (KernelFork.ofι S.f ⋯)", "f' : (KernelFork.ofι S.f ⋯).pt ⟶ (KernelFork.ofι S.f ⋯).pt := hi.lift (KernelFork.ofι S.f ⋯)", "hf' : f' = 𝟙 (KernelFork.ofι S.f ⋯).pt", "wπ : f' ≫ 0 = 0", "hπ : IsColimit (CokernelCofork.ofπ 0 wπ)"], "goal": "S.LeftHomologyData"}}
+{"state": {"context": ["k : Type u", "inst✝ : Field k", "n : ℕ", "x : Step k n"], "goal": "((ofStep k (n + 1)).comp (toStepSucc k n)) x = (ofStep k n) x"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝⁶ : AddCommMonoid R✝", "inst✝⁵ : Pow R✝ ℕ", "inst✝⁴ : BinomialRing R✝", "R : Type u_2", "inst✝³ : NonAssocSemiring R", "inst✝² : Pow R ℕ", "inst✝¹ : NatPowAssoc R", "inst✝ : BinomialRing R", "r : R", "k : ℕ"], "goal": "((r + 1) ^ 1 + (↑k).smeval (r + 1)) * (ascPochhammer ℕ k).smeval (r + 1) = (k + 1).factorial • multichoose (r + 1) k + r ^ 1 * (ascPochhammer ℕ k).smeval (r ^ 1 + smeval 1 r)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v', u'} D", "f : CategoryTheory.Functor C D", "hf : f.FullyFaithful", "X : C", "i : X ≅ X"], "goal": "((hf.autMulEquivOfFullyFaithful X) i).inv = f.map i.inv"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "a : α", "h : 0 < ⌊a⌋₊"], "goal": "0 < a"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "DecidableRel r", "IsTotal α r", "IsTrans α r", "l l' : List α"], "goal": "List.Sorted r l → List.Sorted r l' → List.Sorted r (List.merge (fun x x_1 => decide (r x x_1)) l l')"}}
+{"state": {"context": ["G : Type u", "self : AddLeftCancelSemigroup G", "a b c : G"], "goal": "a + b = a + c → b = c"}}
+{"state": {"context": [], "goal": "Filter.Tendsto Stirling.stirlingSeq Filter.atTop (𝓝 √π)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "l : List α", "n : Nat"], "goal": "(List.map f l)[n]? = Option.map f l[n]?"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹⁵ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹⁴ : TopologicalSpace M", "inst✝¹³ : ChartedSpace H M", "inst✝¹² : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹¹ : NormedAddCommGroup E'", "inst✝¹⁰ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁹ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁸ : TopologicalSpace M'", "inst✝⁷ : ChartedSpace H' M'", "inst✝⁶ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝⁵ : NormedAddCommGroup E''", "inst✝⁴ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝³ : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝² : TopologicalSpace M''", "inst✝¹ : ChartedSpace H'' M''", "inst✝ : SmoothManifoldWithCorners I'' M''", "s : Set M", "x✝ : M", "c✝ c : M'", "x : M"], "goal": "HasMFDerivAt I I' (fun x => c) x 0"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁷ : NormedRing A", "inst✝⁶ : StarRing A", "inst✝⁵ : CstarRing A", "inst✝⁴ : CompleteSpace A", "inst✝³ : NormedAlgebra ℂ A", "inst✝² : StarModule ℂ A", "inst✝¹ : PartialOrder A", "inst✝ : StarOrderedRing A", "a : A", "this : star a * a ≤ (algebraMap ℝ A) ‖star a * a‖"], "goal": "star a * a ≤ (algebraMap ℝ A) (‖a‖ ^ 2)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f✝ : α → β", "E I✝ s : Set α", "M : Matroid α", "N : Matroid β", "f : α → β", "h_range : N.E ⊆ range f", "hf : InjOn f (f ⁻¹' N.E)", "I : Set α", "hI : I �� f ⁻¹' N.E", "x✝ : ∃ I₀, (N.Indep (f '' I₀) ∧ InjOn f I₀) ∧ f '' I = f '' I₀", "I₀ : Set α", "hI₀ : N.Indep (f '' I₀)", "right✝ : InjOn f I₀", "hII₀ : f '' I = f '' I₀"], "goal": "N.Indep (f '' I)"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁵ : NormedAddCommGroup E", "inst✝¹⁴ : NormedAddCommGroup E'", "inst✝¹³ : NormedAddCommGroup E''", "inst✝¹² : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NormedSpace 𝕜 E", "inst✝⁹ : NormedSpace 𝕜 E'", "inst✝⁸ : NormedSpace 𝕜 E''", "inst✝⁷ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝⁶ : MeasurableSpace G", "μ ν : Measure G", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : AddGroup G", "inst✝³ : MeasurableAdd₂ G", "inst✝² : MeasurableNeg G", "inst✝¹ : SFinite μ", "inst✝ : μ.IsAddRightInvariant", "h1 : f =ᶠ[ae μ] f'", "h2 : g =ᶠ[ae μ] g'", "x : G"], "goal": "(f ⋆[L, μ] g) x = (f' ⋆[L, μ] g') x"}}
+{"state": {"context": ["α : Type u_1", "p : α → Bool", "y : α", "xs : List α", "w : xs.get? (List.findIdx p xs) = some y"], "goal": "p y = true"}}
+{"state": {"context": ["K : Type u", "Field K", "n : ℕ", "hζ : (primitiveRoots n K).Nonempty", "a : K", "H : Irreducible (Polynomial.X ^ n - Polynomial.C a)"], "goal": "Polynomial.IsSplittingField K K[n√a] (Polynomial.X ^ n - Polynomial.C a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : α ⊕ β → Sort u_3", "p : ((ab : α ⊕ β) → γ ab) → Prop"], "goal": "(∀ (fab : (ab : α ⊕ β) → γ ab), p fab) ↔\n  ∀ (fa : (val : α) → γ (Sum.inl val)) (fb : (val : β) → γ (Sum.inr val)), p fun t => Sum.rec fa fb t"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "CompleteLattice α", "CompleteLattice β", "CompleteLattice γ", "f : FrameHom β γ", "g : FrameHom α β"], "goal": "⇑(f.comp g) = ⇑f ∘ ⇑g"}}
+{"state": {"context": [], "goal": "-ψ = φ⁻¹"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞"], "goal": "⨍⁻ (x : α), f x ∂μ = ∫⁻ (x : α), f x ∂(μ Set.univ)⁻¹ • μ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : Filter α", "g : Filter β", "m : α → β → γ", "s : Set γ"], "goal": "s ∈ (Filter.map m f).seq g ↔ ∃ t u, t ∈ g ∧ u ∈ f ∧ ∀ x ∈ u, ∀ y ∈ t, m x y ∈ s"}}
+{"state": {"context": ["𝒮 : Type u₁", "CategoryTheory.Category.{v₁, u₁} 𝒮", "𝒳 : CategoryTheory.BasedCategory 𝒮", "𝒴 : CategoryTheory.BasedCategory 𝒮", "𝒵 : CategoryTheory.BasedCategory 𝒮", "F : CategoryTheory.BasedFunctor 𝒳 𝒴", "G H : CategoryTheory.BasedFunctor 𝒴 𝒵", "α : G ⟶ H"], "goal": "(CategoryTheory.BasedCategory.whiskerLeft F α).toNatTrans = CategoryTheory.whiskerLeft F.toFunctor α.toNatTrans"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "g : β → γ", "s : Stream'.WSeq α"], "goal": "Stream'.WSeq.map (g ∘ f) s = Stream'.WSeq.map g (Stream'.WSeq.map f s)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι✝ : Sort y", "s t u : Set α", "inst✝¹⁰ : TopologicalSpace α", "inst✝⁹ : MeasurableSpace α", "inst✝⁸ : BorelSpace α", "inst✝⁷ : TopologicalSpace β", "inst✝⁶ : MeasurableSpace β", "inst✝⁵ : BorelSpace β", "inst✝⁴ : MeasurableSpace δ", "inst✝³ : LinearOrder α", "inst✝² : OrderTopology α", "inst✝¹ : SecondCountableTopology α", "ι : Sort u_5", "inst✝ : Countable ι", "f : ι → δ → α", "hf : ∀ (i : ι), Measurable (f i)", "hα : Nonempty α"], "goal": "MeasurableSet {b | BddAbove (range fun i => f i b)}"}}
+{"state": {"context": ["α : Type u_1", "F' : Type u_7", "E'' : Type u_9", "SeminormedAddCommGroup F'", "NormedAddCommGroup E''", "g' : α → F'", "f'' : α → E''", "l : Filter α", "h : f'' =O[l] g'", "hf : ∃ᶠ (x : α) in l, f'' x ≠ 0"], "goal": "¬g' =o[l] f''"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "self : CategoryTheory.Subgroupoid C", "c d : C", "p : c ⟶ d"], "goal": "p ∈ self.arrows c d → CategoryTheory.Groupoid.inv p ∈ self.arrows d c"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedAddCommGroup α", "LinearOrderedAddCommGroup β", "s : Set ι", "f : ι → α", "g₁ g₂ : ι → β", "h₁ : MonovaryOn f g₁ s", "h₂ : MonovaryOn f g₂ s"], "goal": "MonovaryOn f (g₁ + g₂) s"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π : BoxIntegral.TaggedPrepartition I", "r : (ι → ℝ) → ↑(Set.Ioi 0)", "Fintype ι", "h : π.IsSubordinate r", "π' : BoxIntegral.Prepartition I"], "goal": "(π.infPrepartition π').IsSubordinate r"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a p : α", "ha0 : a ≠ 0", "hp : Irreducible p", "hd : p ∣ a"], "goal": "⟨Associates.mk p, ⋯⟩ ∈ factors' a"}}
+{"state": {"context": ["s : Substring"], "goal": "s.Valid → (s.isEmpty = true ↔ s.toString = \"\")"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R"], "goal": "C (C 4 * X ^ 3 + C (W.a₁ ^ 2 + 4 * W.a₂) * X ^ 2 + C (2 * (2 * W.a₄ + W.a₁ * W.a₃)) * X + C (W.a₃ ^ 2 + 4 * W.a₆)) = (C (C 2) * Y + C (C W.toAffine.a₁ * X + C W.toAffine.a₃)) ^ 2 - 4 * (Y ^ 2 + C (C W.toAffine.a₁ * X + C W.toAffine.a₃) * Y - C (X ^ 3 + C W.toAffine.a₂ * X ^ 2 + C W.toAffine.a₄ * X + C W.toAffine.a₆))"}}
+{"state": {"context": ["α : Type u", "Monoid α", "a b : α", "Invertible b"], "goal": "a * b * ⅟b = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "H : Type u_2", "TopologicalSpace H", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "TopologicalSpace G", "ChartedSpace H G", "AddGroup G", "LieAddGroup I G"], "goal": "Smooth I I fun x => -x"}}
+{"state": {"context": ["x : ℝ", "hx : x ≤ 0", "y : ℂ", "hlt : x < 0"], "goal": "↑x ^ y = (-↑x) ^ y * cexp (↑π * I * y)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "op : β → β → β", "hc : Std.Commutative op", "ha : Std.Associative op", "f : α → β", "b : β", "s : Finset α", "a : α", "inst✝ : LinearOrder β", "c x y z : β"], "goal": "min y z < x ↔ x > y ∨ x > z"}}
+{"state": {"context": ["α : Type u_1", "AddMonoidWithOne α", "Preorder α", "ZeroLEOneClass α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1"], "goal": "0 ≤ 3"}}
+{"state": {"context": ["α : Type u_5", "self : ConditionallyCompleteLinearOrderBot α", "x : α"], "goal": "⊥ ≤ x"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_2", "Group M", "Ring R", "MulSemiringAction M R", "a : M", "S : Subring R", "x : R"], "goal": "a • x ∈ a • S ↔ x ∈ S"}}
+{"state": {"context": ["n : ℕ", "q : ℝ", "hn' : 3 ≤ n", "hq' : 1 < q", "hn : 0 < n"], "goal": "eval q (cyclotomic n ℝ) < (q + 1) ^ φ n"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "s : Multiset α"], "goal": "Multiset.filter p s ≤ s"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "f g : PadicSeq p", "hf : ¬f ≈ 0", "hg : ¬g ≈ 0", "hfg : ¬f * g ≈ 0"], "goal": "ℕ"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝⁴ : DivisionRing k", "inst✝³ : AddCommGroup V", "inst✝² : Module k V", "inst✝¹ : AffineSpace V P", "s : Set P", "inst✝ : FiniteDimensional k ↥(vectorSpan k s)"], "goal": "Collinear k s ↔ finrank k ↥(vectorSpan k s) ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : DistribLattice α", "inst✝² : BoundedOrder α", "inst✝¹ : DecidableEq α", "inst✝ : DecidableRel fun x x_1 => x ≤ x_1", "s t : Finset α", "a : α", "hs : a ∈ upperClosure ↑s", "ht : a ∈ upperClosure ↑t"], "goal": "((image (Function.uncurry fun x x_1 => x ⊔ x_1) (filter (fun x => x ≤ a) s ×ˢ filter (fun x => x ≤ a) t)).inf' ⋯ fun x => id x) = (filter (fun x => x ≤ a) s ×ˢ filter (fun x => x ≤ a) t).inf' ⋯ fun i => id i.1 ⊔ id i.2"}}
+{"state": {"context": ["C : Type u_2", "CategoryTheory.Category.{u_1, u_2} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "S₁.HasHomology", "S₂.HasHomology", "φ : S₁ ⟶ S₂", "CategoryTheory.ShortComplex.QuasiIso φ"], "goal": "CategoryTheory.ShortComplex.QuasiIso (CategoryTheory.ShortComplex.opMap φ)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "K : Subgroup G", "S : Set G", "hSK : Subgroup.IsComplement S ↑K", "g k : G", "h : k ∈ K"], "goal": "hSK.equiv (g * k) = ((hSK.equiv g).1, (hSK.equiv g).2 * ⟨k, h⟩)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "b : Basis ι R M", "m : M"], "goal": "m ∈ Submodule.span R (⇑b '' ↑(b.repr m).support)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "ℰ : Type u₂", "CategoryTheory.Category.{v₁, u₂} ℰ", "A : CategoryTheory.Functor C ℰ", "X : ℰ"], "goal": "∀ {X_1 Y : Cᵒᵖ} (f : X_1 ⟶ Y) (a : (CategoryTheory.yoneda.obj X).obj (A.op.obj X_1)),\n  ((CategoryTheory.Presheaf.restrictedYoneda A).obj X).map f a = CategoryTheory.CategoryStruct.comp (A.map f.unop) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "G : Type u_5", "inst✝¹⁶ : TopologicalSpace α", "inst✝¹⁵ : AddCommGroup α", "inst✝¹⁴ : TopologicalAddGroup α", "inst✝¹³ : TopologicalSpace β", "inst✝¹² : AddCommGroup β", "inst✝¹¹ : TopologicalAddGroup β", "inst✝¹⁰ : TopologicalSpace γ", "inst✝⁹ : AddCommGroup γ", "inst✝⁸ : TopologicalAddGroup γ", "inst✝⁷ : TopologicalSpace δ", "inst✝⁶ : AddCommGroup δ", "inst✝⁵ : TopologicalAddGroup δ", "inst✝⁴ : UniformSpace G", "inst✝³ : AddCommGroup G", "inst✝² : UniformAddGroup G", "inst✝¹ : T0Space G", "inst✝ : CompleteSpace G", "e : β →+ α", "de : DenseInducing ⇑e", "f : δ →+ γ", "df : DenseInducing ⇑f", "φ : β →+ δ →+ G", "hφ : Continuous fun p => (φ p.1) p.2", "W' : Set G", "W'_nhd : W' ∈ 𝓝 0", "x₀ : α", "y₁ : δ", "Nx : Filter α := 𝓝 x₀", "ee : β × β → α × α := fun u => (e u.1, e u.2)", "lim1 : Tendsto (fun a => (a.2 - a.1, y₁)) (comap (⇑e) Nx ×ˢ comap (⇑e) Nx) (𝓝 (0, y₁))"], "goal": "∃ U₂ ∈ comap (⇑e) (𝓝 x₀), ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p => (φ p.1) p.2) (x' - x, y₁) ∈ W'"}}
+{"state": {"context": ["E : Type u_1", "AddCommGroup E", "Module ℝ E", "TopologicalSpace E", "ContinuousAdd E", "ContinuousSMul ℝ E", "h : 1 < Module.rank ℝ E", "x : E"], "goal": "IsConnected {x}ᶜ"}}
+{"state": {"context": ["R : Type u", "CommRing R", "x : R"], "goal": "RingHom.ker (Polynomial.evalRingHom x) = Ideal.span {Polynomial.X - Polynomial.C x}"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : ↥(MeasureTheory.Lp.simpleFunc ℝ 1 μ)"], "goal": "↑(MeasureTheory.L1.SimpleFunc.posPart f) = MeasureTheory.Lp.posPart ↑f"}}
+{"state": {"context": [], "goal": "Finset.univ = {()}"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "Fact (FiniteDimensional.finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "Complex.abs ((o.kahler x) y) = ‖x‖ * ‖y‖"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f g : E → F", "p pf pg : FormalMultilinearSeries 𝕜 E F", "x : E", "r r' : ℝ≥0∞", "hf : HasFPowerSeriesOnBall f pf x r"], "goal": "r ≤ pf.radius"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "p : α → Prop", "x y : Subtype p"], "goal": "edist x y = edist ↑x ↑y"}}
+{"state": {"context": ["X : Type u_1", "R : Type u_2", "inst✝³ : TopologicalSpace X", "inst✝² : Semiring R", "inst✝¹ : TopologicalSpace R", "inst✝ : TopologicalSemiring R", "s : Set X", "f : C(X, R)"], "goal": "f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0"}}
+{"state": {"context": ["α : Type u_1", "Zero α", "a : α"], "goal": "AddOpposite.op a = 0 ↔ a = 0"}}
+{"state": {"context": ["M : Type u_3", "A : Type u_4", "CommMonoid M", "s : Set M", "hs : IsSubmonoid s", "f : A → M", "t : Finset A"], "goal": "(∀ b ∈ t, f b ∈ s) → ∏ b ∈ t, f b ∈ s"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : TopologicalSpace E", "inst✝³ : AddCommGroup E", "inst✝² : TopologicalAddGroup E", "inst✝¹ : Module ℝ E", "inst✝ : ContinuousSMul ℝ E", "s : Set E", "hs₀ : 0 ∈ s", "hs₁ : Convex ℝ s", "hs₂ : IsOpen s", "x₀ : E", "hx₀ : x₀ ∉ s", "f : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯", "y : ℝ", "hx : y • x₀ ∈ f.domain", "h : y ≤ 0"], "goal": "y ≤ gauge s (y • x₀)"}}
+{"state": {"context": ["α : Type u_3", "Preorder α", "LocallyFiniteOrderBot α", "a : α", "Fintype ↑(Set.Iic a)"], "goal": "(Set.Iic a).toFinset = Finset.Iic a"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "CommGroup α", "s : Finset α", "t : Finset α", "ht : t.Nonempty"], "goal": "∃ u, u.card * t.card ≤ (s * t).card ∧ s ⊆ u * t / t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ≥0∞", "h : ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0", "φ : α →ₛ ℝ≥0", "hle : ∀ (x : α), ↑(↑φ x) ≤ f x", "b : ℝ≥0∞", "hbφ : b < (SimpleFunc.map ofNNReal φ).lintegral μ + ε", "hb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → (SimpleFunc.map ofNNReal i).lintegral μ ≤ b", "ψ : α →ₛ ℝ≥0", "hψ : ∀ (x : α), ↑(↑ψ x) ≤ f x", "this : (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ⊤", "x : α"], "goal": "↑(φ + (ψ - φ)) x ≤ max (↑φ x) (↑ψ x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : InnerProductSpace 𝕜 E", "inst✝⁴ : InnerProductSpace 𝕜 F", "inst✝³ : InnerProductSpace 𝕜 G", "inst✝² : FiniteDimensional 𝕜 E", "inst✝¹ : FiniteDimensional 𝕜 F", "inst✝ : FiniteDimensional 𝕜 G", "T : E →ₗ[𝕜] E", "x : E"], "goal": "im ⟪x, (adjoint T) (T x)⟫_𝕜 = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝² : Ring α", "inst✝¹ : Ring β", "f : α → β", "hf✝ : IsRingHom f", "x✝ y✝ : α", "hf : IsRingHom f", "γ : Type u_1", "inst✝ : Ring γ", "g : β → γ", "hg : IsRingHom g", "x y : α"], "goal": "(g ∘ f) (x + y) = (g ∘ f) x + (g ∘ f) y"}}
+{"state": {"context": ["𝕜 : Type u", "A : Type v", "Field 𝕜", "Ring A", "Algebra 𝕜 A", "IsAlgClosed 𝕜", "a : A", "p : Polynomial 𝕜", "hdeg : 0 < p.degree"], "goal": "spectrum 𝕜 ((Polynomial.aeval a) p) = (fun x => Polynomial.eval x p) '' spectrum 𝕜 a"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "t : Set α", "s : Set α", "ht : MeasurableSet t"], "goal": "μ.restrict (s ∩ t) + μ.restrict (s \\ t) = μ.restrict s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "inst✝² : OrderedRing E", "inst✝¹ : TopologicalSpace E", "inst✝ : OrderClosedTopology E", "l : E", "f : ℕ → E", "hfl : Tendsto (fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)", "hfm : Monotone f", "k : ℕ"], "goal": "∑ i ∈ Finset.range (2 * k + 1), (-1) ^ i * f i ≤ l"}}
+{"state": {"context": ["X Y : Type u", "f : X ⟶ Y"], "goal": "typeToBoolAlgOp.map f =\n  Quiver.Hom.op\n    (let __src := { toFun := ⇑(CompleteLatticeHom.setPreimage f), map_sup' := ⋯, map_inf' := ⋯ };\n    { toFun := ⇑(CompleteLatticeHom.setPreimage f), map_sup' := ⋯, map_inf' := ⋯, map_top' := ⋯, map_bot' := ⋯ })"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "MeasureTheory.SFinite ν", "f : α × β → ℝ≥0∞"], "goal": "Measurable f → ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (x : α), ∫⁻ (y : β), f (x, y) ∂ν ∂μ"}}
+{"state": {"context": ["x y : ℝ", "hx : 0 ≤ x", "hy : 0 ≤ y", "h : √x = y"], "goal": "y * y = x"}}
+{"state": {"context": ["K : Type u", "inst✝¹ : Field K", "L : Type u_1", "inst✝ : CommGroupWithZero L", "φ : K[X] →*₀ L", "hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰", "f : RatFunc K"], "goal": "(liftMonoidWithZeroHom φ hφ) f = φ f.num / φ f.denom"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "a : α", "DecidableEq α", "ha : a ∉ s"], "goal": "s ⋖ insert a s"}}
+{"state": {"context": ["G : Type u_1", "inst✝ : Mul G", "A B : Finset G", "h : 1 < A.card * B.card → ∃ p1 ∈ A ×ˢ B, ∃ p2 ∈ A ×ˢ B, p1 ≠ p2 ∧ UniqueMul A B p1.1 p1.2 ∧ UniqueMul A B p2.1 p2.2", "hA : A.Nonempty", "hB : B.Nonempty", "hc : ¬(A.card ≤ 1 ∧ B.card ≤ 1)"], "goal": "∃ a ∈ A, ∃ b ∈ B, UniqueMul A B a b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a : α", "b₁ b₂ : β", "GroupWithZero α", "Preorder α", "Preorder β", "MulAction α β", "PosSMulStrictMono α β", "PosSMulReflectLT α β", "ha : 0 < a"], "goal": "a⁻¹ • b₁ < b₂ ↔ b₁ < a • b₂"}}
+{"state": {"context": ["G : Type w", "Group G", "TopologicalSpace G", "TopologicalGroup G", "x : G"], "goal": "Filter.comap (fun x_1 => x_1 / x) (𝓝 1) = 𝓝 x"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "FiniteDimensional 𝕜 E", "n : ℕ", "hn : FiniteDimensional.finrank 𝕜 E = n", "i : Fin n"], "goal": "Module.End.HasEigenvalue T ↑(hT.eigenvalues hn i)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "s t : Associates α", "d : Associates α", "eq : t = s * d"], "goal": "s.factors ≤ s.factors + d.factors"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "TopologicalSpace.NoetherianSpace α"], "goal": "(irreducibleComponents α).Finite"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "P : Finset α → Prop", "s : Finset α", "hP : ∀ ⦃s t : Finset α⦄, P s → s ⊆ t → P t"], "goal": "s ∈ minimals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬P (s.erase x)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Finset α"], "goal": "Finset.Ioo s t = Finset.filter (fun x => s ⊂ x) t.ssubsets"}}
+{"state": {"context": ["ι : Type u_4", "f : ι → Ordinal.{max u_5 u_4}", "a : Ordinal.{max u_4 u_5}", "ha : a < Ordinal.mex f"], "goal": "∃ i, f i = a"}}
+{"state": {"context": ["m n : ℕ", "α✝ : Fin (n + 1) → Type u", "x : α✝ (last n)", "q : (i : Fin (n + 1)) → α✝ i", "p : (i : Fin n) → α✝ i.castSucc", "i : Fin n", "y : α✝ i.castSucc", "z : α✝ (last n)", "α : Type u_1", "a : α", "as : Fin n → α", "bs : Fin m → α"], "goal": "append (cons a as) bs = cons a (append as bs) ∘ cast ⋯"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "X : Type u", "x : X"], "goal": "((TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)).compr₂ (λ_ ((free R).obj X)).hom) 1 ∘ₗ Finsupp.lsingle x = ((TensorProduct.mk R ↑(𝟙_ (ModuleCat R)) ↑((free R).obj X)).compr₂ ((ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom)) 1 ∘ₗ Finsupp.lsingle x"}}
+{"state": {"context": ["α : Type u", "l : List α"], "goal": "l.getLast?.isSome = true ↔ l ≠ []"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "X Y Z : C", "D : Type u₂", "inst✝³ : Category.{v₂, u₂} D", "inst✝² : WellPowered C", "inst✝¹ : HasCoproducts C", "inst✝ : HasImages C", "A : C", "s : Set (Subobject A)", "f : Subobject A", "k : ∀ g ∈ s, g ≤ f"], "goal": "underlying.obj (sSup s) ⟶ underlying.obj f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : 𝕜 → E", "a b : 𝕜", "h : b ≠ a"], "goal": "ContinuousAt (dslope f a) b ↔ ContinuousAt f b"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "u v w : Fin 3 → R"], "goal": "(crossProduct u) ((crossProduct v) w) = (crossProduct ((crossProduct u) v)) w + (crossProduct v) ((crossProduct u) w)"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "ConditionallyCompleteLinearOrderBot ι", "a b : ℝ", "f : ι → Ω → ℝ", "N : ι", "n m : ℕ", "ω : Ω", "hnm : n ≤ m"], "goal": "MeasureTheory.upperCrossingTime a b f N n ω ≤ MeasureTheory.upperCrossingTime a b f N m ω"}}
+{"state": {"context": ["ι : Type u_2", "α : ι → Type u_3", "Fintype ι", "(i : ι) → Dist (α i)", "f g : PiLp 0 α"], "goal": "dist f g = ↑⋯.toFinset.card"}}
+{"state": {"context": ["α✝ : Type u", "β✝ : Type v", "α : Type u_1", "β : Type u_2", "t : TopologicalSpace α", "b : Set (Set β)", "f : α → β", "h : ∀ a ∈ b, IsOpen (f ⁻¹' a)"], "goal": "∀ a ∈ b, IsOpen (f ⁻¹' a)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G", "x y : ↥H"], "goal": "↑(x + y) = ↑x + ↑y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁵ : TopologicalSpace α", "inst✝⁴ : LinearOrder α", "inst✝³ : DenselyOrdered α", "inst✝² : OrderTopology α", "inst✝¹ : TopologicalSpace β", "inst✝ : T2Space β", "f : α → β", "a b : α", "lb : β", "hab : a < b", "hb : Tendsto f (𝓝[<] b) (𝓝 lb)"], "goal": "b ∈ closure (Ioo a b)"}}
+{"state": {"context": ["α : Type u", "l : Filter α", "ι : Type u_2", "s : Finset ι", "f g : ι → Set α", "heq : ∀ i ∈ s, f i =ᶠ[l] g i"], "goal": "⋃ i ∈ s, f i =ᶠ[l] ⋃ i ∈ s, g i"}}
+{"state": {"context": ["A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "R : CategoryTheory.Functor B T", "L₁ L₂ : CategoryTheory.Functor A T", "l l' : L₁ ⟶ L₂", "h : l = l'", "X : CategoryTheory.Comma L₂ R"], "goal": "((CategoryTheory.Comma.mapLeftEq R l l' h).hom.app X).left = CategoryTheory.CategoryStruct.id X.left"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "J : Type u", "CategoryTheory.Category.{v, u} J", "T : CategoryTheory.Comonad C", "D : CategoryTheory.Functor J T.Coalgebra", "c : CategoryTheory.Limits.Cone (D.comp T.forget)"], "goal": "(CategoryTheory.Comonad.ForgetCreatesLimits'.newCone c).pt = c.pt"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G", "a b c d : G", "H : a - b = c - d"], "goal": "a = b ↔ c = d"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : FirstCountableTopology X", "inst✝ : R1Space X", "x y : X", "K : Set X", "hK : IsCompact K"], "goal": "y ∈ closure K ↔ ∃ x ∈ K, Inseparable x y"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasFiniteCoproducts C", "P : CategoryTheory.Idempotents.Karoubi (ChainComplex C ℕ)", "Δ : SimplexCategoryᵒᵖ"], "goal": "(AlgebraicTopology.DoldKan.Γ₂.obj P).p.app Δ =\n  (AlgebraicTopology.DoldKan.Γ₀.splitting P.X).desc Δ fun A =>\n    P.p.f (Opposite.unop A.fst).len ≫ ((AlgebraicTopology.DoldKan.Γ₀.splitting P.X).cofan Δ).inj A"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s✝ t s U : Set X"], "goal": "ContinuousOn U.boolIndicator s ↔ IsClopen (Subtype.val ⁻¹' U)"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α β : TypeVec.{u_1} n", "F' : TypeVec.{u_1} n → Type u", "inst✝¹ : MvFunctor F'", "inst✝ : LawfulMvFunctor F'", "R : β ⊗ β ⟹ repeat n Prop", "x : F' α", "f g : α ⟹ β", "h : α ⟹ Subtype_ R", "hh : subtypeVal R ⊚ h = (f ⊗' g) ⊚ prod.diag"], "goal": "(f ⊚ TypeVec.id) <$$> x = f <$$> x ∧ (prod.snd ⊚ subtypeVal R) <$$> h <$$> x = g <$$> x"}}
+{"state": {"context": ["a b : Prop", "h : b = False"], "goal": "(a → b) = ¬a"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "f : α → ℝ", "g : ℝ → ℝ", "s : Set α", "μ : Measure α", "f_nn : 0 ≤ᶠ[ae μ] f", "f_mble : AEMeasurable f μ", "cst : ℝ → ℝ := fun x => 1"], "goal": "∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddGroup E", "SMul 𝕜 E", "p q : Seminorm 𝕜 E", "x : E"], "goal": "(p ⊔ q) x = p x ⊔ q x"}}
+{"state": {"context": ["a : Ordinal.{u_1}", "n : ℕ", "a✝ : (HMul.hMul 0)^[n] a ≤ id a"], "goal": "0 ≤ a"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a b : α", "h : a ~ᵤ b", "hb : b ≠ 0", "ha : a ≠ 0"], "goal": "factors' a = factors' b"}}
+{"state": {"context": ["x y : ℝ"], "goal": "x ^ y = (↑x ^ ↑y).re"}}
+{"state": {"context": ["R : Type u", "a b : R", "m✝ n✝ : ℕ", "inst✝ : Semiring R", "p q : R[X]", "k l m n : ℕ", "u v : R", "hu : u ≠ 0", "hv : v ≠ 0"], "goal": "Finsupp.single k u + Finsupp.single l v = Finsupp.single m u + Finsupp.single n v ↔ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n"}}
+{"state": {"context": ["G : Type u_1", "Group G", "tG : Monoid.IsTorsion G", "H : Subgroup G"], "goal": "Monoid.IsTorsion ↥H"}}
+{"state": {"context": ["a b : Int"], "goal": "(a * b).fmod b = 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "inst✝ : FiniteDimensional ℝ V", "s : Set V", "h : affineSpan ℝ s = ⊤", "t : Finset V", "hts : ↑t ⊆ s", "b : AffineBasis ↑↑t ℝ V", "hb : ⇑b = Subtype.val"], "goal": "(interior ((convexHull ℝ) (range ⇑b))).Nonempty"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α", "x : α"], "goal": "𝓝 x = Filter.comap (fun y => (y, x)) (𝓤 α)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "sq : CategoryTheory.Square C"], "goal": "sq.flip.f₁₂ = sq.f₁₃"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "PseudoMetricSpace β", "δ : Type u_2", "TopologicalSpace δ", "DiscreteTopology δ", "f : α ↪ δ", "g₁ g₂ : BoundedContinuousFunction α β", "h₁ h₂ : BoundedContinuousFunction δ β"], "goal": "dist (BoundedContinuousFunction.extend f g₁ h₁) (BoundedContinuousFunction.extend f g₂ h₂) =\n  max (dist g₁ g₂) (dist (h₁.restrict (Set.range ⇑f)ᶜ) (h₂.restrict (Set.range ⇑f)ᶜ))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "TopologicalSpace α", "TopologicalSpace β", "TopologicalSpace γ", "g : β → γ", "f : α → β", "s : Set α", "t : Set β", "hg : ContinuousOn g t", "hf : ContinuousOn f s", "h : Set.MapsTo f s t"], "goal": "ContinuousOn (g ∘ f) s"}}
+{"state": {"context": ["K : Type u_1", "DivisionRing K", "p q : K[X]", "hp0 : p ≠ 0"], "goal": "q ∣ p * Polynomial.C p.leadingCoeff⁻¹ ↔ q ∣ p"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "R : Cᵒᵖ ⥤ RingCat", "J : Type u₂", "inst✝¹ : Category.{v₂, u₂} J", "F : J ⥤ PresheafOfModules R", "inst✝ : ∀ (X : Cᵒᵖ), Small.{v, max u₂ v} ↑((F ⋙ evaluation R X) ⋙ forget (ModuleCat ↑(R.obj X))).sections", "X Y Z : Cᵒᵖ", "f : X ⟶ Y", "g : Y ⟶ Z", "j : J"], "goal": "limMap (whiskerLeft F (restriction R (f ≫ g))) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j = (limMap (whiskerLeft F (restriction R f)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map f)) (F ⋙ evaluation R Y)).inv ≫ (ModuleCat.restrictScalars (R.map f)).map (limMap (whiskerLeft F (restriction R g)) ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map g)) (F ⋙ evaluation R Z)).inv) ≫ (ModuleCat.restrictScalarsComp'App (R.map f) (R.map g) (R.map (f ≫ g)) ⋯ (limit (F ⋙ evaluation R Z))).inv ≫ (preservesLimitIso (ModuleCat.restrictScalars (R.map (f ≫ g))) (F ⋙ evaluation R Z)).hom) ≫ limit.π (F ⋙ evaluation R Z ⋙ ModuleCat.restrictScalars (R.map (f ≫ g))) j"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "R : Type u_1", "CommRing R", "S : Type u_2", "Semiring S", "f : (k : ℕ) → S →+* TruncatedWittVector p k R", "f_compat : ∀ (k₁ k₂ : ℕ) (hk : k₁ ≤ k₂), (TruncatedWittVector.truncate hk).comp (f k₂) = f k₁", "g : S →+* 𝕎 R", "g_compat : ∀ (k : ℕ), (WittVector.truncate k).comp g = f k"], "goal": "WittVector.lift f f_compat = g"}}
+{"state": {"context": ["α : Type u_4", "TopologicalSpace α", "self : TopologicalSpace.Compacts α"], "goal": "IsCompact self.carrier"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I B X : Set α", "hB : M.Base B", "hBX : M.Basis (B ∩ X) X"], "goal": "M✶.Basis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)"}}
+{"state": {"context": ["α : Type u_1", "UniformSpace α", "p : UniformSpace.Completion α → Prop", "a : UniformSpace.Completion α", "hp : IsClosed {a | p a}", "ih : ∀ (a : α), p (↑α a)"], "goal": "p a"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.LocallyRingedSpace", "e : X ≅ Y", "y : ↑Y.toTopCat", "z : ↑(Y.presheaf.stalk (e.hom.val.base (e.inv.val.base y)))"], "goal": "(AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.inv y)\n    ((AlgebraicGeometry.LocallyRingedSpace.Hom.stalkMap e.hom (e.inv.val.base y)) z) =\n  (Y.presheaf.stalkSpecializes ⋯) z"}}
+{"state": {"context": ["R S : Type u_1", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "e : R ≃+* S", "x : S"], "goal": "e.toRingHom.IsIntegralElem (e (e.symm x))"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝⁴ : MeasurableSpace Ω", "inst✝³ : TopologicalSpace Ω", "inst✝² : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝¹ : IsProbabilityMeasure μ", "μs : ℕ → Measure Ω", "inst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)", "f : Ω →ᵇ ℝ", "f_nn : 0 ≤ f", "h_opens : ∀ (G : Set Ω), IsOpen G → μ G ≤ liminf (fun i => (μs i) G) atTop", "same : ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μ ≤ liminf (fun i => ∫⁻ (x : Ω), ENNReal.ofReal (f x) ∂μs i) atTop"], "goal": "∫ (x : Ω), f x ∂μ ≤ liminf (fun i => ∫ (x : Ω), f x ∂μs i) atTop"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁴ : Group G", "inst✝³ : Group G'", "inst✝² : Group G''", "A : Type u_4", "inst✝¹ : AddGroup A", "H K : Subgroup G", "N : Type u_5", "inst✝ : Group N", "f : N →* G", "x : N"], "goal": "x ∈ comap f H.normalizer → x ∈ (comap f H).normalizer"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "idx : Type u_2", "inst✝ : CommRing R", "hp : Fact (Nat.Prime p)", "Φ : MvPolynomial idx ℤ", "n : ℕ", "IH : ∀ m < n + 1, (map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((map (Int.castRingHom ℚ)) Φ) m", "key : (bind₁ fun b => (rename fun i => (b, i)) ((expand p) (W_ ℤ n))) Φ = (bind₁ fun i => (expand p) (wittStructureInt p Φ i)) (W_ ℤ n)"], "goal": "C ↑(p ^ (n + 1)) ∣ (bind₁ fun b => (rename fun i => (b, i)) (W_ ℤ (n + 1))) Φ - ∑ i ∈ Finset.range (n + 1), C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n + 1 - i)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "x : 𝕜", "s : Set 𝕜", "h : DifferentiableAt 𝕜 f x", "hxs : UniqueDiffWithinAt 𝕜 s x"], "goal": "derivWithin f s x = deriv f x"}}
+{"state": {"context": ["M : Type u_1", "MeasurableSpace M", "α : Type u_2", "m : MeasurableSpace α", "f : α → M", "Sup M", "MeasurableSup M", "hf : Measurable f", "c : M"], "goal": "Measurable fun x => f x ⊔ c"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "a : α"], "goal": "Filter.comap (fun x => dist x a) (𝓝 0) = 𝓝 a"}}
+{"state": {"context": ["Ω : Type u_1", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n : ℕ", "ω : Ω", "M : ℕ", "hNM : N ≤ M", "h : MeasureTheory.upperCrossingTime a b f N (n + 1) ω < N"], "goal": "MeasureTheory.upperCrossingTime a b f M (n + 1) ω = MeasureTheory.upperCrossingTime a b f N (n + 1) ω ∧\n  MeasureTheory.lowerCrossingTime a b f M n ω = MeasureTheory.lowerCrossingTime a b f N n ω"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "f₁ f₂ : α → β", "Preorder α", "Preorder β", "h₁ : StrictMonoOn f₁ s", "h : Set.EqOn f₁ f₂ s"], "goal": "StrictMonoOn f₂ s"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "s : Set αᵒᵈ"], "goal": "SupClosed (⇑OrderDual.toDual ⁻¹' s) ↔ InfClosed s"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M"], "goal": "⊤.cardQuot = 1"}}
+{"state": {"context": ["n✝ p n : ℕ", "h : 1 < p"], "goal": "(p - 1) * ∑ i ∈ range (log p n).succ, n / p ^ i.succ = n - (p.digits n).sum"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "DecidableEq β", "Group α", "MulAction α β", "t : Finset β", "s : Finset α"], "goal": "((fun x => x • t) '' ↑s).PairwiseDisjoint id → t.card ∣ (s • t).card"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y p₂ : P", "h : p₂ ∈ s"], "goal": "s.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)"}}
+{"state": {"context": ["α : Type u_1", "MonoidWithZero α", "a b : α", "h : a ~ᵤ b"], "goal": "a = 0 ↔ b = 0"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "p p' : AddSubgroup G"], "goal": "↑(p ⊓ p') = ↑p ∩ ↑p'"}}
+{"state": {"context": ["α : Type u", "s : Set α", "x : α"], "goal": "s ⊆ {x} ↔ s = ∅ ∨ s = {x}"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "E' : Type u_3", "inst✝¹ : NormedAddCommGroup E'", "inst✝ : NormedSpace 𝕜 E'", "f : E → E'", "s : Set E", "x : E"], "goal": "ContinuousWithinAt f s x ∧ DifferentiableWithinAt 𝕜 f s x ↔ DifferentiableWithinAt 𝕜 f s x"}}
+{"state": {"context": ["α : Type u_2", "ι : Sort u_5", "κ : ι → Sort u_6", "Div α", "s : (i : ι) → κ i → Set α", "t : Set α"], "goal": "(⋃ i, ⋃ j, s i j) / t = ⋃ i, ⋃ j, s i j / t"}}
+{"state": {"context": ["α : Type u", "Semiring α", "I : Ideal α"], "goal": "Ideal.span ↑I = I"}}
+{"state": {"context": ["ι : Type u_2", "Fintype ι", "b : ι → ℂ", "hb : ∀ (i : ι), 0 < (b i).re", "c : ι → ℂ"], "goal": "∫ (v : ι → ℝ), cexp (-∑ i : ι, b i * ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i)) =\n  ∏ i : ι, (↑π / b i) ^ (1 / 2) * cexp (c i ^ 2 / (4 * b i))"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝² : NontriviallyNormedField 𝕜", "inst✝¹ : NontriviallyNormedField 𝕜'", "inst✝ : NormedAlgebra 𝕜 𝕜'", "f : 𝕜 → 𝕜'", "x : 𝕜", "hdf : DifferentiableAt 𝕜 f x", "n : ℤ"], "goal": "logDeriv (fun x => f x ^ n) x = ↑n * logDeriv f x"}}
+{"state": {"context": ["R : Type u", "Semiring R", "Nontrivial R"], "goal": "Polynomial.X.degree = 1"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "t : (i : ι) → Set (α i)", "i : ι"], "goal": "Function.eval i '' Set.univ.pi t ⊆ t i"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "inst✝¹³ : NontriviallyNormedField 𝕜'", "σ : 𝕜 →+* 𝕜'", "σ' : 𝕜' →+* 𝕜", "inst✝¹² : RingHomInvPair σ σ'", "inst✝¹¹ : RingHomInvPair σ' σ", "inst✝¹⁰ : RingHomIsometric σ", "inst✝⁹ : RingHomIsometric σ'", "E : Type u_3", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F✝ : Type u_4", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace 𝕜' F✝", "f : E →SL[σ] F✝", "inst✝⁴ : CompleteSpace F✝", "inst✝³ : CompleteSpace E", "F : Type u_5", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "g : E →ₗ[𝕜] F", "hg : ∀ (u : ℕ → E) (x : E) (y : F), Tendsto u atTop (𝓝 x) → Tendsto (⇑g ∘ u) atTop (𝓝 y) → y = g x"], "goal": "Continuous ⇑g"}}
+{"state": {"context": ["𝕜 : Type u_1", "m : Type u_2", "n : Type u_3", "RCLike 𝕜", "Fintype m", "Fintype n", "DecidableEq n", "DecidableEq m", "A : Matrix m n 𝕜"], "goal": "‖Aᴴ‖ = ‖A‖"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "M : Type w", "M' : Type w₁", "inst✝¹² : CommRing R", "inst✝¹¹ : LieRing L", "inst✝¹⁰ : LieAlgebra R L", "inst✝⁹ : LieRing L'", "inst✝⁸ : LieAlgebra R L'", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M'", "inst✝² : Module R M'", "inst✝¹ : LieRingModule L M'", "inst✝ : LieModule R L M'", "f : L →ₗ⁅R⁆ L'", "I✝ : LieIdeal R L", "J : LieIdeal R L'", "I : LieIdeal R L", "h₁ : comap f ⊥ = ⊥", "h₂ : I ≤ ⊥"], "goal": "I ≤ ⊥"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "x : ℤ_[p]", "hx : x ≠ 0"], "goal": "x = ↑(unitCoeff hx) * ↑p ^ x.valuation.natAbs"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : ℕ → α → E", "hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ", "p : ℝ", "hp1 : 1 ≤ p", "B : ℕ → ℝ≥0∞", "hB : ∑' (i : ℕ), B i ≠ ⊤", "h : (∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ) ^ (1 / p) ≤ ∑' (i : ℕ), B i", "hp_pos : 0 < p", "h_integral : ∫⁻ (a : α), (∑' (i : ℕ), ↑‖f (i + 1) a - f i a‖₊) ^ p ∂μ < ⊤", "rpow_ae_lt_top : ∀ᵐ (x : α) ∂μ, (∑' (i : ℕ), ↑‖f (i + 1) x - f i x‖₊) ^ p < ⊤", "x : α", "hx : (∑' (i : ℕ), ↑‖f (i + 1) x - f i x‖₊) ^ p < ⊤"], "goal": "∑' (i : ℕ), ↑‖f (i + 1) x - f i x‖₊ < ⊤"}}
+{"state": {"context": ["M : Type u_2", "MulOneClass M", "S : Submonoid M"], "goal": "S.op.unop = S"}}
+{"state": {"context": ["M₀ : Type u_2", "MonoidWithZero M₀", "a : M₀"], "goal": "IsUnit (Ring.inverse a) ↔ IsUnit a"}}
+{"state": {"context": [], "goal": "Primrec₂ Nat.Partrec.Code.curry"}}
+{"state": {"context": ["α : Type u_1", "E' : Type u_6", "F' : Type u_7", "SeminormedAddCommGroup E'", "SeminormedAddCommGroup F'", "f' : α → E'", "g' : α → F'", "l : Filter α", "h : f' =Θ[l] g'"], "goal": "Filter.Tendsto (norm ∘ f') l Filter.atTop ↔ Filter.Tendsto (norm ∘ g') l Filter.atTop"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "σ : Type u_1", "CommSemiring R", "CommSemiring S₁", "f : R →+* S₁", "g : S₁ →+* R", "hf : Function.RightInverse ⇑f ⇑g"], "goal": "Function.RightInverse ⇑(MvPolynomial.map f) ⇑(MvPolynomial.map g)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "A B : Type u", "CommRing A", "Algebra R A", "CommRing B", "Algebra R B", "Algebra.Unramified R A", "e : A ≃ₐ[R] B"], "goal": "Algebra.Unramified R B"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℂ E"], "goal": "2 * Cardinal.lift.{0, u_1} (Module.rank ℂ E) = Cardinal.lift.{0, u_1} (2 * Module.rank ℂ E)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "M : R[T;T⁻¹] → Prop", "p : R[T;T⁻¹]", "h_C : ∀ (a : R), M (LaurentPolynomial.C a)", "h_add : ∀ {p q : R[T;T⁻¹]}, M p → M q → M (p + q)", "h_C_mul_T :\n  ∀ (n : ℕ) (a : R),\n    M (LaurentPolynomial.C a * LaurentPolynomial.T ↑n) → M (LaurentPolynomial.C a * LaurentPolynomial.T (↑n + 1))", "h_C_mul_T_Z :\n  ∀ (n : ℕ) (a : R),\n    M (LaurentPolynomial.C a * LaurentPolynomial.T (-↑n)) → M (LaurentPolynomial.C a * LaurentPolynomial.T (-↑n - 1))"], "goal": "M p"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n✝ : ℕ", "inst✝ : Ring R", "n : ℕ", "toFinsupp✝ : R[ℕ]"], "goal": "(-{ toFinsupp := toFinsupp✝ }).coeff n = -{ toFinsupp := toFinsupp✝ }.coeff n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "p✝ q✝ : R[X]", "K : Type u_1", "inst✝ : Field K", "p q : K[X]", "hpq : p ∣ q", "hq : q ≠ 0", "h₁ : q.natDegree ≤ p.natDegree"], "goal": "IsUnit q.leadingCoeff"}}
+{"state": {"context": ["α : Type u", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "DenselyOrdered α", "δ : Type u_1", "LinearOrder δ", "TopologicalSpace δ", "OrderClosedTopology δ", "a b : α", "hab : a ≤ b", "f : α → δ", "hf : ContinuousOn f (Set.Icc a b)"], "goal": "Set.Ico (f a) (f b) ⊆ f '' Set.Ico a b"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "BoundedOrder α", "x y : α", "h₁ : x ⊓ y ≤ ⊥", "h₂ : ⊤ ≤ x ⊔ y"], "goal": "IsCompl x y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : β", "f : α → β"], "goal": "none.casesOn' x f = x"}}
+{"state": {"context": ["α : Type u_1", "Nonempty α"], "goal": "∃ x, x ∈ Set.univ"}}
+{"state": {"context": ["a b : ℕ"], "goal": "Disjoint (Finset.range a) (Finset.map (addRightEmbedding a) (Finset.range b))"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "hf : AEMeasurable f μ", "g : α → ℝ≥0∞"], "goal": "∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ"}}
+{"state": {"context": ["m a b d : ℕ", "h : a ≡ b [MOD m]", "hdm : d ∣ m"], "goal": "d ∣ a ↔ d ∣ b"}}
+{"state": {"context": ["I : Type u", "β : Type u_2", "DecidableEq I", "One β", "i : I", "x : β", "i' : I"], "goal": "Pi.mulSingle i x i' = if i' = i then x else 1"}}
+{"state": {"context": ["p q : ℚ"], "goal": "(q + p).toNNRat ≤ q.toNNRat + p.toNNRat"}}
+{"state": {"context": ["R S : Type u_1", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "f : R →+* S", "s : Set R", "hs : Ideal.span s = ⊤", "H : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f) (Localization.awayMap f ↑r)", "this : Algebra R S := f.toAlgebra", "x : S"], "goal": "x ∈ (Algebra.ofId R S).range"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "β₂ : Type u_3", "γ : Type u_4", "ι : Sort u_5", "ι' : Sort u_6", "κ : ι → Sort u_7", "κ' : ι' → Sort u_8", "inst✝ : SupSet α", "f✝ g : ι → α", "s : Set β", "f : β → α"], "goal": "sSup (f '' s) = ⨆ a, f ↑a"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "hfc : ConvexOn ℝ S f", "hfd : ∀ x ∈ S, DifferentiableAt ℝ f x"], "goal": "MonotoneOn (deriv f) S"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝⁷ : AddCommMonoid M", "inst✝⁶ : Semiring R", "inst✝⁵ : Module R M", "inst✝⁴ : TopologicalSpace M", "inst✝³ : TopologicalSpace R", "inst✝² : SeparableSpace R", "inst✝¹ : ContinuousAdd M", "inst✝ : ContinuousSMul R M", "s : Set M", "hs : IsSeparable s"], "goal": "IsSeparable (⋃ n, (fun f => ∑ i : Fin n, (f i).1 • (f i).2) '' {f | ∀ (i : Fin n), (f i).2 ∈ s})"}}
+{"state": {"context": ["α : Type u_1", "A : α → PSet"], "goal": "(PSet.mk α A).Type = α"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "IsLocalization M S", "I : Ideal R", "hp : I.IsPrime", "hd : Disjoint ↑M ↑I"], "goal": "(Ideal.map (algebraMap R S) I).IsPrime"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "U : (Opens ���(ProjectiveSpectrum.top 𝒜))ᵒᵖ", "a b : (x : ↥(unop U)) → at ↑x", "ha : (isLocallyFraction 𝒜).pred a", "hb : (isLocallyFraction 𝒜).pred b", "x : ↥(unop U)", "Va : Opens ↑(ProjectiveSpectrum.top 𝒜)", "ma : ↑x ∈ Va", "ia : Va ⟶ unop U", "ja : ℕ", "ra : A", "ra_mem : ra ∈ 𝒜 ja", "sa : A", "sa_mem : sa ∈ 𝒜 ja", "hwa : ∀ (x : ↥Va), ↑⟨sa, sa_mem⟩ ∉ (↑x).asHomogeneousIdeal", "wa : ∀ (x : ↥Va), (fun x => a ((fun x => ⟨↑x, ⋯⟩) x)) x = HomogeneousLocalization.mk { deg := ja, num := ⟨ra, ra_mem⟩, den := ⟨sa, sa_mem⟩, den_mem := ⋯ }", "Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)", "mb : ↑x ∈ Vb", "ib : Vb ⟶ unop U", "jb : ℕ", "rb : A", "rb_mem : rb ∈ 𝒜 jb", "sb : A", "sb_mem : sb ∈ 𝒜 jb", "hwb : ∀ (x : ↥Vb), ↑⟨sb, sb_mem⟩ ∉ (↑x).asHomogeneousIdeal", "wb : ∀ (x : ↥Vb), (fun x => b ((fun x => ⟨↑x, ⋯⟩) x)) x = HomogeneousLocalization.mk { deg := jb, num := ⟨rb, rb_mem⟩, den := ⟨sb, sb_mem⟩, den_mem := ⋯ }", "y : ↥(Va ⊓ Vb)"], "goal": "(fun x => (a * b) ((fun x => ⟨↑x, ⋯⟩) x)) y = HomogeneousLocalization.mk { deg := ja + jb, num := ⟨ra * rb, ⋯⟩, den := ⟨sa * sb, ⋯⟩, den_mem := ⋯ }"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Algebra F E", "K : Type w", "inst✝¹ : Field K", "inst✝ : Algebra F K", "f : F[X]", "this : Multiset.card (f.aroots f.SplittingField) ≤ f.natDegree"], "goal": "f.natSepDegree ≤ f.natDegree"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "AddMonoid β", "DecidableEq α", "l : List α", "f : α → β", "comm : (↑l.toFinset).Pairwise fun a b => AddCommute (f a) (f b)", "hl : l.Nodup"], "goal": "l.toFinset.noncommSum f comm = (List.map f l).sum"}}
+{"state": {"context": ["R : Type u1", "inst✝⁴ : CommRing R", "M : Type u2", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "A : Type u_1", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "n : ℕ", "f : Fin n → ↑↑(LinearMap.range (ι R))", "hu : (List.ofFn fun i => ↑(f i)).prod ∈ ↑(LinearMap.range (ι R)) ^ n"], "goal": "(List.ofFn fun i => ↑(f i)).prod ∈ Set.range ⇑(ιMulti R n)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s' : Set α", "x : α", "p : Filter ι", "p' : Filter α", "g : ι → α", "l : Filter ι", "inst✝ : l.IsCountablyGenerated", "h : ∀ (u : ℕ → ι), Tendsto u atTop l → TendstoUniformlyOn (fun n => F (u n)) f atTop s"], "goal": "∀ (x : ℕ → ι × α), Tendsto x atTop (l ×ˢ 𝓟 s) → Tendsto ((fun q => (f q.2, F q.1 q.2)) ∘ x) atTop (𝓤 β)"}}
+{"state": {"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y"], "goal": "delayReflLeft 0 γ = (refl x).trans γ"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝³ : DecidableEq β", "inst✝² : GroupWithZero α", "inst✝¹ : MulAction α β", "s t : Finset β", "a : α", "b : β", "inst✝ : Fintype β", "ha : a ≠ 0"], "goal": "a • Set.univ = Set.univ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ≥0∞", "hf : AEMeasurable f μ", "g : α → ℝ≥0∞"], "goal": "∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ"}}
+{"state": {"context": ["M : Type u_1", "ι : Type u_2", "R : Type u_3", "inst✝³ : AddMonoid M", "inst✝² : DecidableEq ι", "inst✝¹ : AddMonoid ι", "inst✝ : CommSemiring R", "f : M →+ ι", "m : M", "r : R"], "goal": "Finsupp.single (Multiplicative.toAdd (Multiplicative.ofAdd m)) (r * 1) = ↑(GradedMonoid.mk (f m) ⟨Finsupp.single m r, ⋯⟩).snd"}}
+{"state": {"context": ["n : ℕ", "α : TypeVec.{u_1} (n + 1)", "p : α.Arrow (TypeVec.repeat (n + 1) Prop)"], "goal": "TypeVec.dropFun (TypeVec.toSubtype p) = TypeVec.toSubtype fun i => p i.fs"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VSub α β", "s t : Set β"], "goal": "(s -ᵥ t).Nonempty → t.Nonempty"}}
+{"state": {"context": ["G : Type u_1", "Group G", "X : Type u_2", "MulAction G X", "C : SubMulAction G X", "B : Set ↥C"], "goal": "MulAction.IsBlock G B ↔ MulAction.IsBlock G (Subtype.val '' B)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "inst✝ : Preorder α", "a : α", "ha : ∀ b ∈ s, b < a"], "goal": "(insert a s).chainHeight = s.chainHeight + 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "f : Ordinal.{u} → Ordinal.{max u v}", "x✝ : (∀ (a : Ordinal.{u}), f a < f (succ a)) ∧ ∀ (o : Ordinal.{u}), o.IsLimit → (o.bsup fun x x_1 => f x) = f o", "h₁ : ∀ (a : Ordinal.{u}), f a < f (succ a)", "h₂ : ∀ (o : Ordinal.{u}), o.IsLimit → (o.bsup fun x x_1 => f x) = f o", "o : Ordinal.{u}", "ho : o.IsLimit", "a : Ordinal.{max u v}"], "goal": "(o.bsup fun x x_1 => f x) ≤ a ↔ ∀ b < o, f b ≤ a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "S₁.HasHomology", "S₂.HasHomology"], "goal": "CategoryTheory.ShortComplex.homologyMap 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝² : DecidableEq α", "inst✝¹ : CommSemiring R", "f : α → R", "s : Finset α", "inst✝ : CommSemiring R", "x : α → R", "n : ℕ"], "goal": "s.sum x ^ n = ∑ k ∈ s.sym n, ↑(↑k).multinomial * (Multiset.map x ↑k).prod"}}
+{"state": {"context": ["p : ℕ", "a : ℚ", "hp : Fact (Nat.Prime p)", "H : 1 < padicNorm p a"], "goal": "¬a.isInt = true"}}
+{"state": {"context": ["α : Type u", "l : List α", "n k : ℕ", "hk : k < l.length"], "goal": "(l.rotate n).nthLe ((l.length - n % l.length + k) % l.length) ⋯ = l.nthLe k hk"}}
+{"state": {"context": ["𝕜 : Type u_1", "X : Type u_2", "E : Type u_3", "PseudoEMetricSpace X", "RCLike 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : X → X", "g l : X → E", "hf : LipschitzWith 1 f", "hg : UniformContinuous g", "hl : Continuous l"], "goal": "IsClosed {x | Filter.Tendsto (fun x_1 => birkhoffAverage 𝕜 f g x_1 x) Filter.atTop (𝓝 (l x))}"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "a : α", "ha : Irreducible a"], "goal": "¬IsSquare a"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_4", "TopologicalSpace α", "T3Space α", "NonUnitalNonAssocSemiring α", "TopologicalSemiring α", "f : ι → α", "g : κ → α", "s t : α", "hf : HasSum f s", "hg : HasSum g t", "hfg : Summable fun x => f x.1 * g x.2"], "goal": "HasSum (fun x => f x.1 * g x.2) (s * t)"}}
+{"state": {"context": ["R : Type uR", "M₂ : Type v₂", "M' : Type v'", "CommSemiring R", "AddCommMonoid M'", "AddCommMonoid M₂", "Module R M'", "Module R M₂", "k l n : ℕ", "s : Finset (Fin n)", "hk : s.card = k", "hl : sᶜ.card = l", "f : MultilinearMap R (fun x => M') (MultilinearMap R (fun x => M') M₂)", "x y : M'"], "goal": "((f fun x_1 => x) fun i => s.piecewise (fun x_1 => x) (fun x => y) ((sᶜ.orderEmbOfFin hl) i)) =\n  (f fun x_1 => x) fun x => y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_4", "CompleteLattice α", "f : β → γ → α", "s : Set β", "t : Set γ"], "goal": "sInf (Set.image2 f s t) = ⨅ a ∈ s, ⨅ b ∈ t, f a b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "CommRing S", "Algebra R S", "x y : S", "h : ↑(Algebra.adjoin R {x}) = ↑(Algebra.adjoin R {y})"], "goal": "conductor R x = conductor R y"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Multiset.attach 0 = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D"], "goal": "(CategoryTheory.Monoidal.transportedMonoidalUnitIso e).inv =\n  CategoryTheory.monoidalCounit (CategoryTheory.Monoidal.fromTransported e) e.symm.toAdjunction"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "y : E", "h : s =ᶠ[𝓝[{y}ᶜ] x] t", "this✝ : s =ᶠ[𝓝[≠] x] t", "this : 𝓝[s \\ {x}] x = 𝓝[t \\ {x}] x"], "goal": "fderivWithin 𝕜 f s x = fderivWithin 𝕜 f t x"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f✝ : α → β → β", "op : α → α → α", "inst✝² : Monoid β", "inst✝¹ : Monoid γ", "inst✝ : FunLike F β γ", "s : Finset α", "f : α → β", "comm : (↑s).Pairwise fun a b => Commute (f a) (f b)", "m : β", "h : ∀ x ∈ s, f x = m"], "goal": "∀ x ∈ Multiset.map f s.val, x = m"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "M : Type u_3", "CommRing K", "CommRing L", "CommRing M", "i : K →+* L", "j : K →+* M", "p : ℕ", "ExpChar K p", "ExpChar L p", "ExpChar M p", "PerfectRing L p", "IsPerfectClosure i p", "PerfectRing M p", "IsPerfectClosure j p", "x : K"], "goal": "(IsPerfectClosure.equiv i j p) (i x) = j x"}}
+{"state": {"context": ["R : Type u", "inst✝⁶ : CommSemiring R", "M : Type v", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "ι : Type w", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "κ : Type u_1", "inst✝¹ : DecidableEq κ", "inst✝ : Fintype κ", "b : Basis ι R M", "c : Basis κ R M", "g : M →ₗ[R] M", "f : (M →ₗ[R] M)ˣ"], "goal": "(trace R M) (↑f⁻¹ * (↑f * g)) = (trace R M) g"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a : α"], "goal": "¬a < a"}}
+{"state": {"context": ["B : Type u", "inst✝ : Bicategory B", "a b c : B", "f : a ⟶ b", "g : b ⟶ a", "η✝ : 𝟙 a ≅ f ≫ g", "ε✝ : g ≫ f ≅ 𝟙 b", "η : 𝟙 a ≅ f ≫ g", "ε : g ≫ f ≅ 𝟙 b"], "goal": "η.hom ▷ f ≫ ((α_ f g f).hom ≫ 𝟙 f ▷ (g ≫ f)) ≫ f ◁ (g ◁ ((ρ_ f).inv ≫ (f ◁ ε.inv ≫ (𝟙 f ▷ (g ≫ f) ≫ (α_ f g f).inv) ≫ η.inv ▷ f) ≫ (λ_ f).hom) ≫ ε.hom) = (λ_ f).hom ≫ (ρ_ f).inv"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_4", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "n : ℕ∞", "h : E ≃ₘ^n⟮𝓘(𝕜, E), 𝓘(𝕜, F)⟯ F", "hn : 1 ≤ n", "s : Set F"], "goal": "UniqueDiffOn 𝕜 (⇑h ⁻¹' s) ↔ UniqueDiffOn 𝕜 s"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "x : ℝ", "s : ℂ", "f : E"], "goal": "(↑(rexp (-x)) * cexp (-↑x) ^ (s - 1)) • f = cexp (-s * ↑x) • f"}}
+{"state": {"context": ["Γ✝ : Type u_1", "Γ' : Type u_2", "R : Type u_3", "V : Type u_4", "inst✝⁹ : PartialOrder Γ✝", "inst✝⁸ : PartialOrder Γ'", "inst✝⁷ : VAdd Γ✝ Γ'", "inst✝⁶ : IsOrderedCancelVAdd Γ✝ Γ'", "inst✝⁵ : AddCommMonoid V", "Γ : Type u_5", "inst✝⁴ : OrderedAddCommMonoid Γ", "inst✝³ : AddAction Γ Γ'", "inst✝² : IsOrderedCancelVAdd Γ Γ'", "inst✝¹ : MulZeroClass R", "inst✝ : SMulWithZero R V", "r : R", "x : HahnModule Γ' R V", "x✝ : Γ'"], "goal": "((of R).symm ((HahnSeries.single 0) r • x)).coeff x✝ = ((of R).symm (r • x)).coeff x✝"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : FiniteDimensional ℝ E", "s : Finset E", "hs : ∀ c ∈ s, ‖c‖ ≤ 2", "h : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "μ : Measure E := Measure.addHaar", "δ : ℝ := 1 / 2", "ρ : ℝ := 5 / 2", "ρpos : 0 < ρ", "A : Set E := ⋃ c ∈ s, ball c δ", "hA : A = ⋃ c ∈ s, ball c δ", "D : (↑s).Pairwise (Disjoint on fun c => ball c δ)", "A_subset : A ⊆ ball 0 ρ", "I : ↑s.card * ENNReal.ofReal (δ ^ finrank ℝ E) * μ (ball 0 1) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E) * μ (ball 0 1)", "J : ↑s.card * ENNReal.ofReal (δ ^ finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ finrank ℝ E)"], "goal": "s.card ≤ 5 ^ finrank ℝ E"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "DivisionRing K", "AddCommGroup V", "Module K V", "L : Type u_3", "W : Type u_4", "DivisionRing L", "AddCommGroup W", "Module L W", "σ : K →+* L", "τ : L →+* K", "RingHomInvPair σ τ", "f : V →ₛₗ[σ] W", "hf : Function.Injective ⇑f"], "goal": "Function.Injective (Projectivization.map f hf)"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "V₁ : Type u_6", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "b p' : P₁"], "goal": "(AffineEquiv.vaddConst k b).symm p' = p' -ᵥ b"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ : ℝ", "h : ∀ (t : ℝ), IsIntegralCurveAt γ v t", "t : ℝ"], "goal": "HasMFDerivAt 𝓘(ℝ, ℝ) I γ t (ContinuousLinearMap.smulRight 1 (v (γ t)))"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedRing α", "n : ℕ", "a b c : α"], "goal": "|a| ≤ |b| ↔ |a| * |a| ≤ |b| * |b|"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "k : Type u_1", "inst✝ : CommRing k", "n : ℕ", "mvpz : ↑p ^ (n + 1) = C (↑p ^ (n + 1))"], "goal": "wittPolyProd p (n + 1) = -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) + ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) + ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) + remainder p n"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a b c : α", "h : 0 < b"], "goal": "b⁻¹ * a ≤ c ↔ a ≤ c * b"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "w : ι → k", "p₁ p₂ : ι → P", "b : P"], "goal": "∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i) = (s.weightedVSubOfPoint p₁ b) w - (s.weightedVSubOfPoint p₂ b) w"}}
+{"state": {"context": ["a b✝ c d : ℝ≥0∞", "r p q b : ℝ≥0", "hb : 1 ≤ ↑r * ↑b"], "goal": "r⁻¹ ≤ b"}}
+{"state": {"context": ["b : ℤ", "k : ℕ", "L : List ℕ"], "goal": "Nat.ofDigits b L % ↑k = Nat.ofDigits (b % ↑k) L % ↑k"}}
+{"state": {"context": ["L : Language", "M : Type u_1", "inst✝ : L.Structure M"], "goal": "CG L M ↔ ∃ S, S.Countable ∧ (closure L).toFun S = ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p : FormalMultilinearSeries 𝕜 E F", "x y : E", "r✝ R r r' : ℝ≥0", "hr : ↑r + ↑r' < p.radius", "this : ∀ (n : ℕ), HasSum (fun s => ‖p (n - s.card + s.card)‖₊ * r ^ s.card * r' ^ (n - s.card)) (‖p n‖₊ * (r + r') ^ n)"], "goal": "Summable fun x => ‖p (x.fst - x.snd.card + x.snd.card)‖₊ * r ^ x.snd.card * r' ^ (x.fst - x.snd.card)"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "c : P", "R : ℝ", "x : P"], "goal": "EuclideanGeometry.inversion c R x = (AffineMap.lineMap c x) ((R / dist x c) ^ 2)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "CommMonoid M", "f : α → M", "s t : Set α", "hst : s ⊆ t", "ht : t.Finite"], "goal": "(∏ᶠ (i : α) (_ : i ∈ s), f i) * ∏ᶠ (i : α) (_ : i ∈ t \\ s), f i = ∏ᶠ (i : α) (_ : i ∈ t), f i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m inst✝³ : MeasurableSpace α", "inst✝² : MeasurableSpace β", "M : Type u_3", "inst✝¹ : AddCommMonoid M", "inst✝ : TopologicalSpace M", "v : VectorMeasure α M", "f : α → β", "hf : Measurable f", "s : Set β", "hs : MeasurableSet s"], "goal": "↑{ measureOf' := fun s => if MeasurableSet s then ↑v (f ⁻¹' s) else 0, empty' := ⋯, not_measurable' := ⋯, m_iUnion' := ⋯ } s = ↑v (f ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "q : ℝ≥0∞", "inst✝³ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝² : NormedRing 𝕜", "inst✝¹ : (i : α) → Module 𝕜 (E i)", "inst✝ : ∀ (i : α), BoundedSMul 𝕜 (E i)", "f : (i : α) → E i", "c : 𝕜", "hf : Memℓp f 0"], "goal": "Memℓp (c • f) 0"}}
+{"state": {"context": ["R : Type u", "Ring R", "p : R[X]"], "goal": "(-p).natTrailingDegree = p.natTrailingDegree"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "q : ℝ", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "hq0_ne : q ≠ 0", "hμ : μ ≠ 0", "f : α → F", "hf_zero : f =ᵐ[μ] 0"], "goal": "MeasureTheory.eLpNorm' f q μ = 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D", "G : CategoryTheory.Functor D C", "iso : e.inverse ≅ G", "X : D"], "goal": "(e.changeInverse iso).counitIso.hom.app X =\n  CategoryTheory.CategoryStruct.comp (e.functor.map (iso.inv.app X)) (e.counitIso.hom.app X)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f✝ p f : Perm α", "h : p.IsCycle ∧ ∀ a ∈ p.support, p a = f a", "x : α", "hx : x ∈ p.support"], "goal": "x ∈ f.support"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "FunLike F G H", "a : G", "b : H", "AddCommMonoid G", "AddMonoid H", "AddConstMapClass F G H a b", "f : F", "n : ℕ", "x : G"], "goal": "f (n • a + x) = f x + n • b"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_3", "LinearOrderedField R", "AddCommGroup E", "Module R E", "s t : Set E"], "goal": "(convexHull R) (s - t) = (convexHull R) s - (convexHull R) t"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "CategoryTheory.Preadditive C", "I₁ : Type u_2", "I₂ : Type u_3", "I₁₂ : Type u_4", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "K : HomologicalComplex₂ C c₁ c₂", "c₁₂ : ComplexShape I₁₂", "DecidableEq I₁₂", "TotalComplexShape c₁ c₂ c₁₂", "K.HasTotal c₁₂", "i₁₂ i₁₂' i₁₂'' : I₁₂"], "goal": "K.D₁ c₁₂ i₁₂ i₁₂' ≫ K.D₁ c₁₂ i₁₂' i₁₂'' = 0"}}
+{"state": {"context": ["L : FirstOrder.Language", "ι : Type v", "Preorder ι", "Nonempty ι", "IsDirected ι fun x x_1 => x ≤ x_1", "M : Type u_1", "L.Structure M", "S : ι →o L.Substructure M"], "goal": "(FirstOrder.Language.DirectLimit.liftInclusion S).toHom.range = ⨆ i, S i"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "CommRing S", "Algebra R S", "σ : Finset S"], "goal": "IsLocalization (Submonoid.comap (algebraMap (↥(Algebra.adjoin R ↑σ)) S) (IsUnit.submonoid S)) S ↔\n  ∀ (s : S), ∃ t ∈ Algebra.adjoin R ↑σ, IsUnit t ∧ s * t ∈ Algebra.adjoin R ↑σ"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝¹⁰ : TopologicalSpace α✝", "inst✝⁹ : TopologicalSpace β", "inst✝⁸ : LinearOrder α✝", "inst✝⁷ : LinearOrder β", "inst✝⁶ : OrderTopology α✝", "inst✝⁵ : OrderTopology β", "α : Type u_4", "inst✝⁴ : LinearOrder α", "inst✝³ : TopologicalSpace α", "inst✝² : DenselyOrdered α", "inst✝¹ : OrderTopology α", "inst✝ : FirstCountableTopology α", "x y : α", "hy : y < x", "hx : x ∉ Ioo y x"], "goal": "∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Ioo y x) ∧ Tendsto u atTop (𝓝 x)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Multiset α"], "goal": "(Finset.Ioo s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (Multiset.count i t + 1 - Multiset.count i s) - 2"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s : Set E", "f : E → F", "n : ℕ∞", "Hcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s", "Hdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s"], "goal": "ContDiffOn 𝕜 n f s"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : AddGroup R", "x : HahnSeries Γ R"], "goal": "(-x).orderTop = x.orderTop"}}
+{"state": {"context": ["𝕂 : Type u_1", "m : Type u_2", "n : Type u_3", "p : Type u_4", "n' : m → Type u_5", "𝔸 : Type u_6", "inst✝⁹ : RCLike 𝕂", "inst✝⁸ : Fintype m", "inst✝⁷ : DecidableEq m", "inst✝⁶ : Fintype n", "inst✝⁵ : DecidableEq n", "inst✝⁴ : (i : m) → Fintype (n' i)", "inst✝³ : (i : m) → DecidableEq (n' i)", "inst✝² : NormedCommRing 𝔸", "inst✝¹ : NormedAlgebra 𝕂 𝔸", "inst✝ : CompleteSpace 𝔸", "A : Matrix m m 𝔸"], "goal": "exp 𝕂 (-A) = Ring.inverse (exp 𝕂 A)"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "NormalSpace α", "R1Space α", "MeasurableSpace α", "BorelSpace α", "E : Type u_2", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "NormedSpace ℝ E", "WeaklyLocallyCompactSpace α", "μ.Regular", "f : α → E", "hf : MeasureTheory.Integrable f μ", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ MeasureTheory.Integrable g μ"}}
+{"state": {"context": ["F : Type v", "Field F", "W : WeierstrassCurve.Jacobian F", "P Q : Fin 3 → F", "n d : F", "hPz : P z ≠ 0", "hQz : Q z ≠ 0", "hd : d ≠ 0"], "goal": "W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2) (n / (P z * Q z * d)) =\n  (n ^ 2 + W.a₁ * n * P z * Q z * d - W.a₂ * P z ^ 2 * Q z ^ 2 * d ^ 2 - P x * Q z ^ 2 * d ^ 2 -\n      Q x * P z ^ 2 * d ^ 2) /\n    (P z * Q z * d) ^ 2"}}
+{"state": {"context": ["ι : Type u_1", "s : Finset ι", "(j : ι) → Decidable (j ∈ s)", "π : ι → Type u_3", "f g : (i : ι) → π i", "(i : ι) → Preorder (π i)", "h : g ≤ f"], "goal": "s.piecewise f g ∈ Set.Icc g f"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂ : Type w", "V₂ : Type w'", "M₃ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid M₂", "inst✝¹ : Module R M", "inst✝ : Module R M₂", "p : Submodule R M", "q : Submodule R M₂"], "goal": "map (snd R M M₂) (fst R M M₂) ≤ ⊥"}}
+{"state": {"context": ["η✝ : Type u_1", "α✝ : Type u_2", "ι : Type u_3", "κ✝ : Type u_4", "α : Type u_5", "κ : Type u_6", "η : Type u_7", "inst✝² : Finite α", "inst✝¹ : Finite κ", "inst✝ : Finite η", "val✝ : Fintype η"], "goal": "∃ ι x, ∀ (C : (ι → α) → κ), ∃ l, IsMono C l"}}
+{"state": {"context": ["α : Type u", "Group α", "a : α"], "goal": "IsUnit a"}}
+{"state": {"context": ["R : Type u", "M M₁ : Type v", "M' : Type v'", "ι : Type w", "inst✝⁸ : Ring R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup M'", "inst✝⁵ : AddCommGroup M₁", "inst✝⁴ : Module R M", "inst✝³ : Module R M'", "inst✝² : Module R M₁", "inst✝¹ : Nontrivial R", "inst✝ : NoZeroSMulDivisors R M", "h : Module.rank R M = 0"], "goal": "toNat (Module.rank R M) = 0"}}
+{"state": {"context": ["E : Type u_2", "SeminormedAddCommGroup E", "NormedSpace ℝ E", "δ : ℝ", "s : Set E", "x : E"], "goal": "EMetric.infEdist x (Metric.cthickening δ s) = EMetric.infEdist x s - ENNReal.ofReal δ"}}
+{"state": {"context": ["q : ℚ"], "goal": "Rat.sqrt (q * q) = |q|"}}
+{"state": {"context": ["K : Type u", "CommRing K", "IsDomain K", "L : Type u_1", "S : Type u_3", "Field L", "CommSemiring S", "Algebra S K[X]", "Algebra S L", "φ : K[X] →ₐ[S] L", "hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰", "n : K[X]", "d : ↥K[X]⁰"], "goal": "(RatFunc.liftAlgHom φ hφ) { toFractionRing := Localization.mk n d } = φ n / φ ↑d"}}
+{"state": {"context": ["J : Type u", "inst✝¹ : Category.{u_1, u} J", "F : J ⥤ Type v", "i j k : J", "s : Set (F.obj i)", "inst✝ : IsCofilteredOrEmpty J", "f : i ⟶ j"], "goal": "(∀ (i_1 : J) (j_1 : i_1 ⟶ j), range (F.map f) ⊆ range (F.map j_1)) ↔ ∀ ⦃k : J⦄ (g : k ⟶ i), range (F.map f) ⊆ range (F.map (g ≫ f))"}}
+{"state": {"context": ["ι : Type uι", "inst✝⁹ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝⁵ : SeminormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "E' : ι → Type u_1", "E'' : ι → Type u_2", "inst✝³ : (i : ι) → SeminormedAddCommGroup (E' i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E' i)", "inst✝¹ : (i : ι) → SeminormedAddCommGroup (E'' i)", "inst✝ : (i : ι) → NormedSpace 𝕜 (E'' i)", "g : (i : ι) → E' i →L[𝕜] E'' i", "f : (i : ι) → E i →L[𝕜] E' i", "x : ⨂[𝕜] (i : ι), E i", "m : (i : ι) → E i"], "goal": "∏ i : ι, ‖(f i) (m i)‖ ≤ (∏ i : ι, ‖f i‖) * ∏ i : ι, ‖m i‖"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "P : C → Prop", "CategoryTheory.MonoidalCategory.MonoidalPredicate P", "CategoryTheory.BraidedCategory C", "X Y : CategoryTheory.FullSubcategory P"], "goal": "(CategoryTheory.MonoidalCategory.fullBraidedSubcategoryInclusion P).μ X Y =\n  CategoryTheory.CategoryStruct.id (CategoryTheory.MonoidalCategory.tensorObj X.obj Y.obj)"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "_mΩ : MeasurableSpace Ω", "κ : ProbabilityTheory.Kernel α Ω", "μ : MeasureTheory.Measure α", "β : Type u_8", "m : MeasurableSpace β", "Div β", "MeasurableDiv₂ β", "f : ι → Ω → β", "ProbabilityTheory.IsMarkovKernel κ", "hf_indep : ProbabilityTheory.Kernel.iIndepFun (fun x => m) f κ μ", "hf_meas : ∀ (i : ι), Measurable (f i)", "i j k l : ι", "hik : i ≠ k", "hil : i ≠ l", "hjk : j ≠ k", "hjl : j ≠ l"], "goal": "ProbabilityTheory.Kernel.IndepFun (f i / f j) (f k / f l) κ μ"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "MetricSpace X", "MetricSpace Y", "f : GromovHausdorff.ProdSpaceFun X Y", "x y : X ⊕ Y", "fA : f ∈ GromovHausdorff.candidates X Y"], "goal": "0 ≤ f (x, y)"}}
+{"state": {"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "this : HasSum (fun n => -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑n ^ s / 2 + -(-I * ↑(↑n).sign) * cexp (-(2 * ↑π * I * ↑a * ↑n)) / ↑n ^ s / 2) (sinZeta (↑a) s)"], "goal": "(cexp (-(2 * ↑π * ↑a * ↑0) * I) - cexp (2 * ↑π * ↑a * ↑0 * I)) * I / 2 / ↑0 ^ s = -I * ↑(↑0).sign * cexp (2 * ↑π * I * ↑a * ↑0) / ↑0 ^ s / 2 + -(-I * ↑(↑0).sign) * cexp (-(2 * ↑π * I * ↑a * ↑0)) / ↑0 ^ s / 2"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedAddCommGroup F", "inst✝² : CompleteSpace E", "f : ℝ → E", "g : ℝ → F", "k : Set ℝ", "l : Filter ℝ", "inst✝¹ : l.NeBot", "inst✝ : TendstoIxxClass Icc l l", "hl : k ∈ l", "hd✝ : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x", "hf : Tendsto (fun x => ‖f x‖) l atTop", "hfg : deriv f =O[l] g", "C : ℝ", "hC₀ : 0 ≤ C", "s : Set ℝ", "hsl : s ∈ l", "hsub : ∀ x ∈ s, ∀ y ∈ s, [[x, y]] ⊆ k", "hfd : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], DifferentiableAt ℝ f z", "hg : ∀ x ∈ s, ∀ y ∈ s, ∀ z ∈ [[x, y]], ‖deriv f z‖ ≤ C * ‖g z‖", "hgi : IntegrableOn (fun x => C * ‖g x‖) k volume", "c : ℝ", "hc : c ∈ s", "d : ℝ", "hd : d ∈ s", "hlt : ‖f c‖ + ∫ (y : ℝ) in k, C * ‖g y‖ < ‖f d‖"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "Semiring α", "PartialOrder α", "a b c d : α", "ExistsAddOfLE α", "MulPosStrictMono α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "hba : b < a", "hdc : d < c"], "goal": "a * d + b * c < a * c + b * d"}}
+{"state": {"context": ["L : Language", "M : Type w", "inst✝¹ : Nonempty M", "inst✝ : L.Structure M"], "goal": "#((n : ℕ) × (L.sum L.skolem₁).Functions n) ≤ max ℵ₀ L.card"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "f : M → M'", "x : M", "s : Set M", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "x' : M", "y : M'", "hx : x' ∈ (chartAt H x).source", "hy : f x' ∈ (chartAt H' y).source"], "goal": "MDifferentiableWithinAt I I' f s x' ↔\n  ContinuousWithinAt f s x' ∧\n    DifferentiableWithinAt 𝕜 (↑(extChartAt I' y) ∘ f ∘ ↑(extChartAt I x).symm)\n      (↑(extChartAt I x).symm ⁻¹' s ∩ Set.range ↑I) (↑(extChartAt I x) x')"}}
+{"state": {"context": ["M : Type u_8", "AddCommMonoid M", "n : ℕ", "σ : Fin n →₀ M", "i : M"], "goal": "((Finsupp.cons i σ).sum fun x e => e) = i + σ.sum fun x e => e"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_10", "Neg R", "A : Matrix n l R", "B : Matrix n m R", "C : Matrix o l R", "D : Matrix o m R"], "goal": "-Matrix.fromBlocks A B C D = Matrix.fromBlocks (-A) (-B) (-C) (-D)"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "M : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : IsDomain R", "inst✝⁶ : Field K", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : Module.Finite R M", "inst✝² : Module.Free R M", "inst✝¹ : Module K M", "inst✝ : Module.Finite K M", "φ : Module.End K M", "this : [¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ¬⊥ < ker φ, ∀ (m : M), m ≠ 0 → φ m ≠ 0].TFAE"], "goal": "[¬φ.HasEigenvalue 0, ¬(minpoly K φ).IsRoot 0, constantCoeff (charpoly φ) ≠ 0, LinearMap.det φ ≠ 0, ker φ = ⊥, ∀ (m : M), φ m = 0 → m = 0].TFAE"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "k✝ : ℕ", "N✝ : LieSubmodule R L M", "M₂ : Type w₁", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "N : LieSubmodule R L M", "h₁ : N ≤ maxTrivSubmodule R L M", "k : ℕ", "hk : lowerCentralSeries R L (M ⧸ N) k = ⊥"], "goal": "lowerCentralSeries R L M k ≤ N"}}
+{"state": {"context": ["m : ℕ∞", "hm : m ≠ 0"], "goal": "ℵ₀ * ↑m = ℵ₀"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "f : ℕ → α", "m n : ℕ", "h : m ≤ n"], "goal": "edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1))"}}
+{"state": {"context": ["V : Type (u + 1)", "CategoryTheory.LargeCategory V", "G : MonCat", "CategoryTheory.Limits.HasZeroMorphisms V", "X Y : Action V G"], "goal": "Action.Hom.hom 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : LinearOrderedField α", "β : Type u_2", "inst✝² : Ring β", "abv : β → α", "inst✝¹ : IsAbsoluteValue abv", "inst✝ : IsComplete β abv", "f : CauSeq β abv", "h : f.lim = 0"], "goal": "f.LimZero"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : LinearOrderedField α", "l : Filter β", "f : β → α", "r : α", "hr : 0 < r"], "goal": "Tendsto (fun x => f x * r) l atBot ↔ Tendsto f l atBot"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : ℝ", "Ui Vj : Finset α"], "goal": "(Ui, Vj) ∈ P.parts.offDiag.attach.biUnion (SzemerediRegularity.distinctPairs hP G ε) → (Ui, Vj) ∈ (increment hP G ε).parts.offDiag"}}
+{"state": {"context": ["A : Type u_1", "inst✝⁵ : NormedRing A", "inst✝⁴ : StarRing A", "inst✝³ : CstarRing A", "inst✝² : CompleteSpace A", "inst✝¹ : NormedAlgebra ℂ A", "inst✝ : StarModule ℂ A", "a : A", "ha : IsSelfAdjoint a", "ha₁ : ∀ x ∈ σ ℝ a, 0 ≤ x", "ha₂ : ∀ x ∈ σ ℝ (-a), 0 ≤ x", "a✝ : Nontrivial A"], "goal": "σ ℝ a ⊆ {0}"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι✝ : Sort x", "l : Filter α", "ι : Type u_2", "s : Set ι", "hs : s.Finite", "f g : ι → Set α", "hle : ∀ i ∈ s, f i ≤ᶠ[l] g i", "this : Finite ↑s"], "goal": "⋂ i ∈ s, f i ≤ᶠ[l] ⋂ i ∈ s, g i"}}
+{"state": {"context": ["n : ℕ", "a : Fin (n + 1)"], "goal": "PredOrder.pred a = if a = 0 then 0 else a - 1"}}
+{"state": {"context": ["a b : Int", "n : Nat", "h : a + ↑n = b"], "goal": "a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α"], "goal": "Set.InjOn (fun a => insert a s) (↑s)ᶜ"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι✝ : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s✝ : Set E", "i j : ι✝", "c : R", "t : Finset ι✝", "w✝ : ι✝ → R", "z✝ : ι✝ → E", "ι : Sort u_8", "w : ι → R", "z : ι → E", "s : Set E", "hs : Convex R s", "h₀ : ∀ (i : ι), 0 ≤ w i", "h₁ : ∑ᶠ (i : ι), w i = 1", "hz : ∀ (i : ι), w i ≠ 0 → z i ∈ s", "hfin_w : (support (w �� PLift.down)).Finite"], "goal": "∑ᶠ (i : ι), w i • z i ∈ s"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "δ : ℝ", "s : Set α", "x : α"], "goal": "x ∈ Metric.thickening δ s ↔ EMetric.infEdist x s < ENNReal.ofReal δ"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "TopologicalSpace E", "TopologicalSpace X", "f : E → X", "s : Set X", "hf : IsCoveringMapOn f s", "x : E", "hx : f x ∈ s"], "goal": "ContinuousAt f x"}}
+{"state": {"context": ["n m : ℕ", "hzero : n ≠ 0", "h : m ∣ n"], "goal": "m.divisors ⊆ n.divisors"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c✝ : α", "n : ℤ", "a b c : α"], "goal": "toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝¹ : (i : α) → NormedAddCommGroup (E i)", "ι : Type u_3", "l : Filter ι", "inst✝ : l.NeBot", "_i : Fact (1 ≤ p)", "F : ℕ → ↥(lp E p)", "hF : CauchySeq F", "f : ↥(lp E p)", "hf : Tendsto (id fun i => ↑(F i)) atTop (𝓝 ↑f)"], "goal": "∀ (i : ℝ), 0 < i → ∀ᶠ (x : ℕ) in atTop, F x ∈ Metric.closedBall f i"}}
+{"state": {"context": ["m n a b c d✝ : ℤ", "h : a ≡ b [ZMOD n]", "d : ℕ", "hd : a ^ d ≡ b ^ d [ZMOD n]"], "goal": "a ^ d * a ≡ b ^ d * b [ZMOD n]"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α", "K : Set α", "ι' : Sort u_2", "ι : Sort u_3", "p : ι' → Prop", "V : ι' → Set (α × α)", "U : ι → Set α", "hbasis : (𝓤 α).HasBasis p V", "hK : IsCompact K", "hopen : ∀ (j : ι), IsOpen (U j)", "hcover : K ⊆ ⋃ j, U j"], "goal": "∃ i, p i ∧ ∀ x ∈ K, ∃ j, UniformSpace.ball x (V i) ⊆ U j"}}
+{"state": {"context": ["α : Type u", "s : Stream'.Seq α", "m n : ℕ"], "goal": "s.drop (m + n) = (s.drop m).drop n"}}
+{"state": {"context": ["D : Type u_1", "C : Type u_2", "inst✝³ : Groupoid D", "inst✝² : Category.{?u.4103, u_2} C", "inst✝¹ : MonoidalCategory C", "inst✝ : MonoidalClosed C", "F G : D ⥤ C", "X Y : D", "f : X ⟶ Y"], "goal": "(ihom.coev (F.obj Y)).app (G.obj X) ≫ (ihom (F.obj Y)).map (F.obj Y ◁ G.map f) = (ihom.coev (F.obj X)).app (G.obj X) ≫ (pre (CategoryTheory.inv (F.map f))).app (F.obj X ⊗ G.obj X) ≫ (ihom (F.obj Y)).map (F.map f ⊗ G.map f)"}}
+{"state": {"context": ["α : Type u", "x : α"], "goal": "Function.Injective (Stream'.Seq.cons x)"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "inst✝⁴ : DivisionRing K", "inst✝³ : AddCommGroup V", "inst✝² : AddCommGroup V'", "inst✝¹ : Module K V", "inst✝ : Module K V'", "v : ι → V", "s t : Set V", "x y z : V"], "goal": "LinearIndependent K (((fun o => o.casesOn' x v) ∘ ⇑(Equiv.optionEquivSumPUnit ι).symm) ∘ Sum.inl) ∧ ¬x = 0 ∧ (x ∈ span K (range (((fun o => o.casesOn' x v) ∘ ⇑(Equiv.optionEquivSumPUnit ι).symm) ∘ Sum.inl)) → x = 0) ↔ LinearIndependent K v ∧ x ∉ span K (range v)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X✝ Y✝ Z✝¹ X Y Z✝ : C", "f : X ⟶ Y", "g : Y ⟶ Z✝", "inst✝ : Mono (f ≫ g)", "Z : C", "a b : Z ⟶ X", "w : a ≫ f ≫ g = b ≫ f ≫ g"], "goal": "a = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹³ : Norm E", "inst✝¹² : Norm F", "inst✝¹¹ : Norm G", "inst✝¹⁰ : SeminormedAddCommGroup E'", "inst✝⁹ : SeminormedAddCommGroup F'", "inst✝⁸ : SeminormedAddCommGroup G'", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedAddCommGroup F''", "inst✝⁵ : NormedAddCommGroup G''", "inst✝⁴ : SeminormedRing R", "inst✝³ : SeminormedAddGroup E'''", "inst✝² : SeminormedRing R'", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g : α → F", "k : α → G", "f' : α → E'", "g' : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "h : ∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖"], "goal": "∀ᶠ (x : α) in l, ‖f x‖ ≤ ‖g x‖"}}
+{"state": {"context": ["α : Type u_1", "SeminormedAddCommGroup α", "f : ℕ → α", "h : CauchySeq fun n => ∑ k ∈ Finset.range n, f k"], "goal": "∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C"}}
+{"state": {"context": ["α : Type u_1", "R✝ : Type u_2", "M : Type u_3", "M₂ : Type u_4", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup M₂", "F : Type u_5", "inst✝⁵ : FunLike F M M₂", "inst✝⁴ : AddMonoidHomClass F M M₂", "f : F", "R : Type u_6", "S : Type u_7", "inst✝³ : DivisionRing R", "inst✝² : DivisionRing S", "inst✝¹ : Module R M", "inst✝ : Module S M₂", "x : M", "n : ℕ"], "goal": "f ((↑↑n)⁻¹ • x) = (↑↑n)⁻¹ • f x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "inst✝⁵ : MetricSpace X", "inst✝⁴ : CompactSpace X", "inst✝³ : Nonempty X", "inst✝² : MetricSpace Y", "inst✝¹ : CompactSpace Y", "inst✝ : Nonempty Y", "f✝ : GromovHausdorff.ProdSpaceFun X Y", "x y z t : X ⊕ Y", "f g : GromovHausdorff.Cb X Y", "cg : ℝ", "hcg : cg ∈ lowerBounds (range ⇑g)", "Hcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ g x", "cf : ℝ", "hcf : cf ∈ lowerBounds (range ⇑f)"], "goal": "⨆ x, ⨅ y, f (inl x, inr y) ≤ (⨆ x, ⨅ y, g (inl x, inr y)) + dist f g"}}
+{"state": {"context": ["m : Type u → Type u", "Monad m", "α β : Type u", "LawfulMonad m", "f : α → β → m β", "xs : FreeMonoid α"], "goal": "∀ (a : CategoryTheory.KleisliCat.mk m β),\n  (Monoid.foldrM.ofFreeMonoid f) xs a = flip (List.foldrM f) (FreeMonoid.toList xs) a"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "hS : S.Exact", "hS' : S.ShortExact", "v : ι → ↑S.X₁", "hv : LinearIndependent R v", "u : ι ⊕ ι' → ↑S.X₂", "hw : LinearIndependent R (⇑S.g ∘ u ∘ Sum.inr)", "hm : Mono S.f", "huv : u ∘ Sum.inl = ⇑S.f ∘ v"], "goal": "Disjoint (span R (range (u ∘ Sum.inr))) (LinearMap.ker S.g)"}}
+{"state": {"context": ["R : Type u_1", "x y : R", "n m p : ℕ", "Semiring R", "hp : n + m ≤ p + 1", "h_comm : Commute x y", "hx : x ^ n = 0"], "goal": "y ^ m ∣ (x + y) ^ p"}}
+{"state": {"context": ["F : Type u_1", "A : Type u_2", "B : Type u_3", "C : Type u_4", "D : Type u_5", "E : Type u_6", "inst✝¹⁹ : Monoid A", "inst✝¹⁸ : Monoid B", "inst✝¹⁷ : Monoid C", "inst✝¹⁶ : Monoid D", "inst✝¹⁵ : CommGroup E", "inst✝¹⁴ : TopologicalSpace A", "inst✝¹³ : TopologicalSpace B", "inst✝¹² : TopologicalSpace C", "inst✝¹¹ : TopologicalSpace D", "inst✝¹⁰ : TopologicalSpace E", "inst✝⁹ : TopologicalGroup E", "X : Type u_7", "Y : Type u_8", "inst✝⁸ : TopologicalSpace X", "inst✝⁷ : Group X", "inst✝⁶ : TopologicalGroup X", "inst✝⁵ : UniformSpace Y", "inst✝⁴ : CommGroup Y", "inst✝³ : UniformGroup Y", "inst✝² : T0Space Y", "inst✝¹ : CompactSpace Y", "inst✝ : LocallyCompactSpace X", "V : ℕ → Set Y", "hV : ∀ {n : ℕ} {x : Y}, x ∈ V n → x * x ∈ V n → x ∈ V (n + 1)", "hVo : (nhds 1).HasBasis (fun x => True) V", "U0 : Set X", "hU0c : IsCompact U0", "hU0o : U0 ∈ nhds 1", "U_aux : ℕ → ↑{S | S ∈ nhds 1} := fun t => Nat.rec ⟨U0, hU0o⟩ (fun x S => let h := ⋯; ⟨Classical.choose h, ⋯⟩) t", "U : ℕ → Set X := fun n => ↑(U_aux n)", "hU1 : ∀ (n : ℕ), U n ∈ nhds 1", "hU2 : ∀ (n : ℕ), U (n + 1) * U (n + 1) ⊆ U n", "hU3 : ∀ (n : ℕ), U (n + 1) ⊆ U n", "hU4 : ∀ (f : X →* Y), Set.MapsTo (⇑f) (U 0) (V 0) → ∀ (n : ℕ), Set.MapsTo (⇑f) (U n) (V n)"], "goal": "LocallyCompactSpace (ContinuousMonoidHom X Y)"}}
+{"state": {"context": ["ιa : Type u_1", "ιb : Type u_2", "inst✝¹⁰ : Fintype ιa", "inst✝⁹ : Fintype ιb", "R' : Type u_3", "Mᵢ : Type u_4", "N₁ : Type u_5", "N₂ : Type u_6", "inst✝⁸ : CommSemiring R'", "inst✝⁷ : AddCommGroup N₁", "inst✝⁶ : Module R' N₁", "inst✝⁵ : AddCommGroup N₂", "inst✝⁴ : Module R' N₂", "inst✝³ : AddCommMonoid Mᵢ", "inst✝² : Module R' Mᵢ", "inst✝¹ : DecidableEq ιa", "inst✝ : DecidableEq ιb", "a : MultilinearMap R' (fun x => Mᵢ) N₁", "b : MultilinearMap R' (fun x => Mᵢ) N₂", "σ : Quotient (QuotientGroup.leftRel (Perm.sumCongrHom ιa ιb).range)"], "goal": "∑ y ∈ Finset.filter (fun x => ⟦x⟧ = σ) Finset.univ, Perm.sign y • domDomCongr y (a.domCoprod b) = domCoprod.summand (alternatization a) (alternatization b) σ"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)"], "goal": "hζ.integralPowerBasis.dim = φ (↑p ^ k)"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "inst✝¹¹ : Category.{v, u} V", "inst✝¹⁰ : Preadditive V", "W : Type u_2", "inst✝⁹ : Category.{?u.75509, u_2} W", "inst✝⁸ : Preadditive W", "W₁ : Type u_3", "W₂ : Type u_4", "inst✝⁷ : Category.{u_6, u_3} W₁", "inst✝⁶ : Category.{u_7, u_4} W₂", "inst✝⁵ : HasZeroMorphisms W₁", "inst✝⁴ : HasZeroMorphisms W₂", "c : ComplexShape ι", "C D E : HomologicalComplex V c", "f g : C ⟶ D", "h k : D ⟶ E", "i : ι", "α : Type u_5", "inst✝³ : AddRightCancelSemigroup α", "inst✝² : One α", "inst✝¹ : DecidableEq α", "F : W₁ ⥤ W₂", "inst✝ : F.PreservesZeroMorphisms", "X : α → W₁", "d : (n : α) → X (n + 1) ⟶ X n", "sq : ∀ (n : α), d (n + 1) ≫ d n = 0"], "goal": "∀ (i j : α), (ComplexShape.down α).Rel i j → ((F.mapHomologicalComplex (ComplexShape.down α)).obj (of X d sq)).d i j ≫ eqToHom ⋯ = eqToHom ⋯ ≫ (of (fun n => F.obj (X n)) (fun n => F.map (d n)) ⋯).d i j"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "β : Type u", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : MeasurableSpace α", "inst✝² : OpensMeasurableSpace α", "inst✝¹ : HasBesicovitchCovering α", "μ : Measure α", "inst✝ : SFinite μ", "f : α → Set ℝ", "s : Set α", "hf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ).Nonempty", "R : α → ℝ", "hR : ∀ x ∈ s, 0 < R x", "g : α → Set ℝ := fun x => f x ∩ Ioo 0 (R x)", "hg : ∀ x ∈ s, ∀ δ > 0, (g x ∩ Ioo 0 δ).Nonempty", "r : α → ℝ", "t : Set α", "v_count : (graphOn r t).Countable", "vs : ∀ p ∈ graphOn r t, p.1 ∈ s", "vg : ∀ p ∈ graphOn r t, p.2 ∈ g p.1", "μv : μ (s \\ ⋃ p ∈ graphOn r t, closedBall p.1 p.2) = 0", "v_disj : (graphOn r t).PairwiseDisjoint fun p => closedBall p.1 p.2"], "goal": "∃ t r, t.Countable ∧ t ⊆ s ∧ (∀ x ∈ t, r x ∈ f x ∩ Ioo 0 (R x)) ∧ μ (s \\ ⋃ x ∈ t, closedBall x (r x)) = 0 ∧ t.PairwiseDisjoint fun x => closedBall x (r x)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "f : R[X]", "h : f.nextCoeff ≠ 0"], "goal": "f.eraseLead.natDegree + 1 = f.natDegree"}}
+{"state": {"context": ["a b : String"], "goal": "Levenshtein.stringLengthCost.substitute a b = max a.length b.length"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "I J : Ideal R", "M : Type u_2", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "M' : Type u_3", "inst✝² : AddCommGroup M'", "inst✝¹ : Module R M'", "f : M →ₗ[R] M'", "inst✝ : IsNoetherianRing R"], "goal": "{r | ∃ x, x ≠ 0 ∧ r ∈ (Submodule.span R {x}).annihilator} ⊆ ⋃ p ∈ associatedPrimes R M, ↑p"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "X Y : C", "inst✝¹ : HasTerminal C", "P : C", "inst✝ : HasBinaryProduct (⊤_ C) P"], "goal": "snd ≫ lift (terminal.from P) (𝟙 P) = 𝟙 ((⊤_ C) ⨯ P)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "n : R", "hn : n ≠ 0", "d : ↥R⁰", "hd_ne_zero : (algebraMap R K) ↑d ≠ 0", "h0 : spanSingleton R⁰ (mk' K n d) ≠ 0"], "goal": "∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) = spanSingleton R⁰ (mk' K n d)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "M : Type u_1", "CategoryTheory.Category.{u_2, u_1} M", "CategoryTheory.MonoidalCategory M", "F : CategoryTheory.MonoidalFunctor M (CategoryTheory.Functor C C)", "m n m' : M", "f : m ⟶ m'", "X : C"], "goal": "CategoryTheory.CategoryStruct.comp ((F.μIso m n).inv.app X) ((F.obj n).map ((F.map f).app X)) =\n  CategoryTheory.CategoryStruct.comp ((F.map (CategoryTheory.MonoidalCategory.whiskerRight f n)).app X)\n    ((F.μIso m' n).inv.app X)"}}
+{"state": {"context": ["p x t : ℝ", "aux : ∀ {a b c : ℝ}, a ∈ zmultiples b → c * a ∈ zmultiples (c * b)", "ht : t ≠ 0", "ht' : |t| ≠ 0"], "goal": "‖↑(t * x)‖ = |t| * ‖↑x‖"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : Measure α", "p q : Set α → Prop", "U : Set α", "ε : ℝ≥0∞", "H : μ.InnerRegularWRT p q", "hU : q U"], "goal": "μ U = ⨆ K, ⨆ (_ : K ⊆ U), ⨆ (_ : p K), μ K"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝³ : Countable ι", "m✝ : MeasurableSpace α", "μ✝ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "f g : α → β", "inst✝ : PseudoMetrizableSpace β", "m : MeasurableSpace α", "μ : ι → Measure α", "h : ∀ (i : ι), AEStronglyMeasurable f (μ i)", "this✝¹ : MeasurableSpace β := borel β", "this✝ : BorelSpace β", "t : ι → Set β", "t_sep : ∀ (i : ι), IsSeparable (t i)", "ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)"], "goal": "∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂sum μ, f x ∈ t"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "inst✝² : TopologicalSpace α", "inst✝¹ : UniformSpace β", "K : Set α", "V✝ : Set (β × β)", "f : C(α, β)", "ι : Type u₃", "p : Filter ι", "F : ι → C(α, β)", "inst✝ : WeaklyLocallyCompactSpace α", "h : ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a => (F i) a) (⇑f) p K", "V : Set (β × β)", "hV : V ∈ 𝓤 β", "x : α"], "goal": "∃ t ∈ 𝓝 x, ∀ᶠ (n : ι) in p, ∀ y ∈ t, (f y, (fun i a => (F i) a) n y) ∈ V"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "LinearOrderedField R", "AddCommGroup V", "Module R V", "AddTorsor V P", "x y z : P"], "goal": "Wbtw R x y z ↔ z = y ∨ x ∈ ⇑(AffineMap.lineMap z y) '' Set.Ici 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hε₁ : ε ≤ 1", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "hUVne : U ≠ V", "hUV : ¬G.IsUniform ε U V", "t : (star hP G ε hU V).product (star hP G ε hV U) ⊆ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts", "hε : 0 ≤ ε", "sp : ∀ (a b : Finset (Finset α)), a.product b = a ×ˢ b", "this : ↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / 25 + ↑((star hP G ε hU V).product (star hP G ε hV U)).card / ↑((chunk hP G ε hU).parts.product (chunk hP G ε hV).parts).card * (3 / 4 * ε) ^ 2 ≤ (∑ i ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, (fun x => ↑(G.edgeDensity x.1 x.2)) i ^ 2) / ↑((chunk hP G ε hU).parts.product (chunk hP G ε hV).parts).card"], "goal": "↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / 25 + ↑(star hP G ε hU V).card * ↑(star hP G ε hV U).card / 16 ^ P.parts.card * (9 / 16) * ε ^ 2 ≤ (∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2) ^ 2) / 16 ^ P.parts.card"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "inst✝¹ : SeminormedAddCommGroup M", "inst✝ : SeminormedAddCommGroup N", "S : AddSubgroup M", "h : ↑S.topologicalClosure = univ", "x : M", "hker : x ∈ S.topologicalClosure"], "goal": "‖S.normedMk x‖ ≤ 0 * ‖x‖"}}
+{"state": {"context": ["C✝ : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C✝", "ℰ : Type u₂", "inst✝³ : Category.{v₁, u₂} ℰ", "A✝ A : C✝ ⥤ ℰ", "inst✝² : yoneda.HasPointwiseLeftKanExtension A", "C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "P : Cᵒᵖ ⥤ Type v₁", "I : Type v₁", "inst✝ : SmallCategory I", "F : I ⥤ C", "c : Cocone (F ⋙ yoneda)", "hc : IsColimit c", "isc : IsColimit ((Over.forget c.pt).mapCocone (colimit.cocone ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))) := PreservesColimit.preserves (colimit.isColimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt))"], "goal": "Over.mk (𝟙 c.pt) ≅ colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙ CostructuredArrow.toOver yoneda c.pt)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "P : Ideal R", "hP : P.IsPrime", "p q : R[X]", "hfl : (p * q).leadingCoeff ∉ P", "hfP : ∀ (n : ℕ), ↑n < (p * q).degree → (p * q).coeff n ∈ P", "hfd0✝ : 0 < (p * q).degree", "h0 : (p * q).coeff 0 ∉ P ^ 2", "hu : (p * q).IsPrimitive", "hf0 : p * q ≠ 0", "hf : map (mk P) p * map (mk P) q = C ((mk P) (p * q).leadingCoeff) * X ^ (p * q).natDegree", "hfd0 : 0 < (p * q).natDegree", "m : ℕ", "b c : (R ⧸ P)[X]", "hbc : C ((mk P) (p * q).leadingCoeff) = b * c", "hp : map (mk P) p = b * X ^ m", "hpql0 : (mk P) (p * q).leadingCoeff ≠ 0", "hp0 : p ≠ 0", "hq0 : q ≠ 0", "hbc0 : b.degree = 0 ∧ c.degree = 0", "hmp : m ≤ p.natDegree", "hmnd : m + 0 = (p * q).natDegree", "hq : map (mk P) q = c * X ^ 0", "hmn : 0 < m → 0 < 0 → False", "hnq : 0 ≤ q.natDegree", "hpmqn : p.natDegree = m ∧ q.natDegree = 0"], "goal": "IsUnit p ∨ IsUnit q"}}
+{"state": {"context": ["α : Type u_1", "m : Set α → ℝ≥0∞", "m_empty : m ∅ = 0", "s t : Set α", "h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ⊤", "f : ℕ → Set α", "hf : s ∪ t ⊆ ⋃ i, f i", "μ : OuterMeasure α := OuterMeasure.ofFunction m m_empty", "he : ¬∃ i, (s ∩ f i).Nonempty ∧ (t ∩ f i).Nonempty", "I : Set α → Set ℕ := fun s => {i | (s ∩ f i).Nonempty}", "hd : Disjoint (I s) (I t)", "hI : ∀ u ⊆ s ∪ t, μ u ≤ ∑' (i : ↑(I u)), μ (f ↑i)"], "goal": "μ s + μ t ≤ ∑' (i : ℕ), m (f i)"}}
+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "n✝ : ℕ", "inst✝ : Monoid G", "n : ℕ", "x : ((classifyingSpaceUniversalCover G).obj (Opposite.op (SimplexCategory.mk (n + 1)))).V", "x✝¹ : Fin (n + 2)", "x✝ : x✝¹ ∈ Finset.univ"], "goal": "((-1) ^ ↑x✝¹ • Finsupp.lmapDomain k k ((classifyingSpaceUniversalCover G).map (SimplexCategory.δ x✝¹).op).hom) (Finsupp.single x 1) = Finsupp.single (x ∘ x✝¹.succAbove) ((-1) ^ ↑x✝¹)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "S : ConvexCone 𝕜 E"], "goal": "S.Pointed ↔ ¬S.Blunt"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s✝ t✝ u : Set α", "m : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "s : Set α", "t : Set β", "x : α", "y : β", "hx : (x, y).1 ∈ s", "hy : (x, y).2 ∈ t", "hst : MeasurableSet (s ×ˢ t)"], "goal": "MeasurableSet s ∧ MeasurableSet t"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "f : ι → Filter α"], "goal": "𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i)"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "CommRing R", "LieRing L", "LieAlgebra R L", "D1 D2 : LieDerivation R L L"], "goal": "↑⁅D1, D2⁆ = ⁅↑D1, ↑D2⁆"}}
+{"state": {"context": ["μ : ℝ", "v : ℝ≥0", "h : v ≠ 0", "hfm : AEStronglyMeasurable (gaussianPDFReal μ v) ℙ", "hf : 0 ≤ᶠ[ae ℙ] gaussianPDFReal μ v"], "goal": "(√(2 * π * ↑v))⁻¹ * ∫ (x : ℝ), rexp (-(↑v)⁻¹ * 2⁻¹ * x ^ 2) = 1"}}
+{"state": {"context": ["Y : Type v", "Z : Type u_1", "TopologicalSpace Y", "TopologicalSpace Z", "ι : Type u_5", "κ : Type u_6", "f : ι → Y", "g : κ → Z", "hf : DenseRange f", "hg : DenseRange g"], "goal": "DenseRange (Prod.map f g)"}}
+{"state": {"context": ["x : ℝ≥0", "z : ℝ", "hx : 1 ≤ x", "hz : z ≤ 0"], "goal": "x ^ z ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "f g : α → ℝ≥0∞", "s : Set α", "hf : ContinuousOn f s", "hg : ContinuousOn g s", "h₁ : ∀ x ∈ s, f x ≠ 0 ∨ g x ≠ ⊤", "h₂ : ∀ x ∈ s, g x ≠ 0 ∨ f x ≠ ⊤"], "goal": "ContinuousOn (fun x => f x * g x) s"}}
+{"state": {"context": ["α : Sort u_1", "γ : Sort u_4", "r : α → α → Prop", "f : α → γ", "h : ∀ (a₁ a₂ : α), r a₁ a₂ → f a₁ = f a₂", "a : α"], "goal": "Quot.lift f h (Quot.mk r a) = f a"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝⁵ : CommRing P", "inst✝⁴ : Algebra R P", "loc : IsLocalization S P", "R₁ : Type u_3", "inst✝³ : CommRing R₁", "K : Type u_4", "inst✝² : Field K", "inst✝¹ : Algebra R₁ K", "inst✝ : IsFractionRing R₁ K", "x✝ : P"], "goal": "x✝ ∈ spanSingleton S 0 ↔ x✝ ∈ 0"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝³ : Field K", "inst✝² : LieRing L", "inst✝¹ : LieAlgebra K L", "inst✝ : Module.Finite K L", "hLK : ↑(finrank K L) ≤ #K", "U : LieSubalgebra K L", "x : L", "hxU : x ∈ U", "y : L", "hyU : y ∈ U", "Ex : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩", "Ey : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y, ⋯⟩", "hUle : U ≤ ↑Ex", "hmin : ∀ E ≤ Ex, Ex ≤ E", "E : LieSubmodule K (↥U) L := let __src := engel K x; { toSubmodule := __src.toSubmodule, lie_mem := ⋯ }", "hx₀ : x ≠ 0"], "goal": "Ex ≤ Ey"}}
+{"state": {"context": ["n : ℕ", "n0 : n ≠ 0", "g a' b' : ℕ", "g0' : 0 < g", "co : a'.Coprime b'", "h : (a' * g) ^ n ∣ (b' * g) ^ n", "g0 : (a' * g).gcd (b' * g) > 0"], "goal": "a' * g ∣ b' * g"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Abelian C", "S : ShortComplex C", "γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f", "f' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯", "hf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ", "wπ : f' ≫ cokernel.π (kernel.ι γ) = 0", "e : Abelian.image S.f ≅ kernel γ := S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))", "he : e.hom ≫ kernel.ι γ = S.abelianImageToKernel"], "goal": "S.LeftHomologyData"}}
+{"state": {"context": ["G : Type u_3", "InvolutiveNeg G", "a b : G"], "goal": "-a = b ↔ a = -b"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "Y Z : C", "f : Y ⟶ Z", "P : CategoryTheory.ProjectiveResolution Y", "Q : CategoryTheory.ProjectiveResolution Z"], "goal": "CategoryTheory.CategoryStruct.comp ((CategoryTheory.ProjectiveResolution.lift f P Q).f 0) (Q.π.f 0) =\n  CategoryTheory.CategoryStruct.comp (P.π.f 0) f"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R"], "goal": "content 1 = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝⁷ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "E' : Type u_5", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝² : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹ : TopologicalSpace M'", "inst✝ : ChartedSpace H' M'", "s : Set H", "x : H", "f : H → H'", "e : PartialHomeomorph H H", "he : e ∈ contDiffGroupoid ⊤ I", "hx : x ∈ e.source", "h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) ((↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x)))", "this✝¹ : ↑I x = (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I (↑e x))", "this✝ : ↑I (↑e x) ∈ ↑I.symm ⁻¹' e.target ∩ range ↑I", "this : ContDiffWithinAt 𝕜 ⊤ (↑I ∘ ↑e.symm ∘ ↑I.symm) (↑I.symm ⁻¹' e.target ∩ range ↑I) (↑I (↑e x))"], "goal": "↑I.symm ⁻¹' e.target ∩ range ↑I ∩ ↑I ∘ ↑e.symm ∘ ↑I.symm ⁻¹' (↑I.symm ⁻¹' s ∩ range ↑I) ∈ 𝓝[↑I.symm ⁻¹' (↑e.symm ⁻¹' s) ∩ range ↑I] ↑I (↑e x)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "S' : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : CommSemiring S", "inst✝⁴ : Algebra S R", "inst✝³ : CommSemiring S'", "inst✝² : Algebra S' R", "inst✝¹ : Algebra S S'", "inst✝ : IsScalarTower S S' R", "I : Ideal R", "x : R", "hx : x ∈ I"], "goal": "I.cotangentEquivIdeal.symm ⟨(Submodule.mkQ (I ^ 2)) x, ⋯⟩ = I.toCotangent ⟨x, hx⟩"}}
+{"state": {"context": ["α : Type u_2", "l : Type u_8", "m : Type u_9", "n : Type u_10", "p : Type u_11", "Distrib α", "A : Matrix l m α", "B₁ B₂ : Matrix n p α"], "goal": "Matrix.kroneckerMap (fun x x_1 => x * x_1) A (B₁ + B₂) =\n  Matrix.kroneckerMap (fun x x_1 => x * x_1) A B₁ + Matrix.kroneckerMap (fun x x_1 => x * x_1) A B₂"}}
+{"state": {"context": ["S X₁ X₂ : Type u", "f : S ⟶ X₁", "g : S ⟶ X₂", "c : PushoutCocone f g", "hc : IsColimit c", "h₁ : Function.Injective f", "x₁ : X₁", "x₂ : X₂", "this : Mono f"], "goal": "(Pushout.cocone f g).inl x₁ = (Pushout.cocone f g).inr x₂ ↔ ∃ s, f s = x₁ ∧ g s = x₂"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : Fin 3 → F", "hPz : P z = 0", "hQz : Q z = 0", "hPx : IsUnit (P x)", "hPy : IsUnit (P y)", "hQx : IsUnit (Q x)", "hQy : IsUnit (Q y)", "hP : P y ^ 2 = P x ^ 3 ∧ (3 * P x ^ 2 ≠ 0 ∨ 2 * P y ≠ 0 ∨ W.a₁ * P x * P y ≠ 0)", "hQ : Q y ^ 2 = Q x ^ 3 ∧ (3 * Q x ^ 2 ≠ 0 ∨ 2 * Q y ≠ 0 ∨ W.a₁ * Q x * Q y ≠ 0)"], "goal": "(fun m => m • Q) (hPy.unit / hPx.unit * (hQx.unit / hQy.unit)) = P"}}
+{"state": {"context": ["J : Type w", "K : Type u_1", "C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "f : J → C", "inst✝¹ : HasBiproduct f", "p : J → Prop", "inst✝ : HasBiproduct (Subtype.restrict p f)", "j j✝ : Subtype p"], "goal": "(if h : ↑j = ↑j✝ then eqToHom ⋯ else 0) = if h : j = j✝ then eqToHom ⋯ else 0"}}
+{"state": {"context": ["α : Type u", "inst✝ : Monoid α", "a b c x : α", "u₁ u₂ : αˣ"], "goal": "x /ₚ u₁ /ₚ u₂ = x /ₚ (u₂ * u₁)"}}
+{"state": {"context": ["L : Type u_1", "CompleteLattice L", "u : L"], "goal": "(Locale.openOfElementHom L) u = {x | x u}"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C"], "goal": "CategoryTheory.Square.arrowArrowEquivalence.unitIso =\n  CategoryTheory.Iso.refl (CategoryTheory.Functor.id (CategoryTheory.Square C))"}}
+{"state": {"context": ["α : Type u", "n : ℕ", "f : Fin n → α", "i : ℕ", "h : i < (List.ofFn f).length"], "goal": "(List.ofFn f).nthLe i h = f ⟨i, ⋯⟩"}}
+{"state": {"context": ["α : Sort u_4", "β : α → Sort u_3"], "goal": "Nonempty (PSigma β) ↔ ∃ a, Nonempty (β a)"}}
+{"state": {"context": ["R : Type u_1", "MonoidWithZero R", "f : ArithmeticFunction R"], "goal": "f.IsMultiplicative ↔ f 1 = 1 ∧ ∀ {m n : ℕ}, m ≠ 0 → n ≠ 0 → m.Coprime n → f (m * n) = f m * f n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "i : α → β", "di✝ : DenseInducing i", "inst✝ : TopologicalSpace δ", "f : γ → α", "g : γ → δ", "h : δ → β", "d : δ", "a : α", "di : DenseInducing i", "H : Tendsto h (𝓝 d) (𝓝 (i a))", "comm : h ∘ g = i ∘ f", "lim1 : map f (comap g (𝓝 d)) ≤ comap i (map h (𝓝 d))", "lim2 : comap i (map h (𝓝 d)) ≤ comap i (𝓝 (i a))"], "goal": "Tendsto f (comap g (𝓝 d)) (𝓝 a)"}}
+{"state": {"context": ["x y : ↥GL(2, ℝ)⁺", "z : ℍ"], "goal": "denom (x * y) z = (↑(↑↑x 1 0) * (num y z / denom y z) + ↑(↑↑x 1 1)) * denom y z"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "t : A", "n : ℕ", "hnp1 : IsUnit ↑(n + 1)!"], "goal": "∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (1 / ↑n !)"}}
+{"state": {"context": ["X : Type u_1", "EMetricSpace X", "ι : Type u_2", "I : Set ι", "hI : I.Finite", "s : Set X", "t : ι → Set X"], "goal": "IsMetricSeparated s (⋃ i ∈ I, t i) ↔ ∀ i ∈ I, IsMetricSeparated s (t i)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "p : ι → P", "j : ι", "w₁ w₂ : ι → k", "hw : ∀ (i : ι), i ≠ j → w₁ i = w₂ i"], "goal": "∑ i ∈ s, w₁ i • (p i -ᵥ p j) = ∑ i ∈ s, w₂ i • (p i -ᵥ p j)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : α → β → Prop", "ca : Computation α", "b : β"], "goal": "Computation.LiftRel R ca (Computation.pure b) ↔ ∃ a, a ∈ ca ∧ R a b"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "h : Symmetric r"], "goal": "Symmetric rᶜ"}}
+{"state": {"context": ["ι : Type u_1", "I J : BoxIntegral.Box ι", "π : BoxIntegral.Prepartition I", "πi : (J : BoxIntegral.Box ι) → BoxIntegral.Prepartition J", "hJ : J ∈ π.biUnion πi"], "goal": "J ≤ π.biUnionIndex πi J"}}
+{"state": {"context": ["E : Type u_2", "PseudoEMetricSpace E", "f : ℝ → E", "C : ℝ≥0", "s : Set ℝ", "hf : LipschitzOnWith C f s"], "goal": "LocallyBoundedVariationOn f s"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "x : 𝕜"], "goal": "logDeriv id x = 1 / x"}}
+{"state": {"context": ["V✝ : Type u_1", "V₁ : Type u_2", "V₂ : Type u_3", "V₃ : Type u_4", "inst✝⁵ : SeminormedAddCommGroup V✝", "inst✝⁴ : SeminormedAddCommGroup V₁", "inst✝³ : SeminormedAddCommGroup V₂", "inst✝² : SeminormedAddCommGroup V₃", "f g : NormedAddGroupHom V₁ V₂", "V : Type u_5", "inst✝¹ : NormedAddCommGroup V", "inst✝ : Nontrivial V", "x : V", "hx : x ≠ 0"], "goal": "∃ x, ‖x‖ ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : Measure α", "f g : α → F'", "s : Set α", "hm : m ≤ m0", "hμm_not : ¬SigmaFinite (μ.trim hm)"], "goal": "¬(SigmaFinite (μ.trim hm) ∧ Integrable f μ)"}}
+{"state": {"context": ["α : Type u_1", "r : Rel α α", "β : Type u_2", "s✝ : Rel β β", "s : RelSeries r", "x : α", "irrefl : Irreflexive r"], "goal": "s.length ≠ 0 ↔ {x | x ∈ s}.Nontrivial"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : TopologicalSpace Y", "inst✝³ : TopologicalSpace Z", "inst✝² : TopologicalSpace W", "inst✝¹ : TopologicalSpace ε", "inst✝ : TopologicalSpace ζ", "f : X → Y", "g : Z → W", "hf : _root_.Embedding f", "hg : _root_.Embedding g", "x✝¹ x✝ : X × Z", "x₁ : X", "z₁ : Z", "x₂ : X", "z₂ : Z"], "goal": "Prod.map f g (x₁, z₁) = Prod.map f g (x₂, z₂) → (x₁, z₁) = (x₂, z₂)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "s✝ : Set α", "x✝ y z : α", "s : ℕ → Set α", "h0 : IsComplete (s 0)", "hs : ∀ (n : ℕ), IsClosed (s n)", "h's : ∀ (n : ℕ), IsBounded (s n)", "h : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty", "h' : Tendsto (fun n => diam (s n)) atTop (𝓝 0)", "u : ℕ → α := fun N => ⋯.some", "I : ∀ (n N : ℕ), n ≤ N → u N ∈ s n", "this : CauchySeq u", "x : α", "xlim : Tendsto (fun n => u n) atTop (𝓝 x)", "n p : ℕ", "hp : p ∈ Ici n"], "goal": "u p ∈ s n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "TopologicalSpace β", "𝕜 : Type u_5", "TopologicalSpace 𝕜", "VAdd 𝕜 β", "ContinuousVAdd 𝕜 β", "f : α → 𝕜", "hf : MeasureTheory.StronglyMeasurable f", "c : β"], "goal": "MeasureTheory.StronglyMeasurable fun x => f x +ᵥ c"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "MeasurableSpace X", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : X → E", "μ : MeasureTheory.Measure X", "PartialOrder X", "x : X", "MeasureTheory.NoAtoms μ"], "goal": "∫ (t : X) in Set.Iic x, f t ∂μ = ∫ (t : X) in Set.Iio x, f t ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : CompleteLattice α", "inst✝ : CompleteLattice β", "f : CompleteLatticeHom α β", "L : CompleteSublattice α", "a b : ↥L"], "goal": "Codisjoint a b ↔ Codisjoint ↑a ↑b"}}
+{"state": {"context": ["R : Type u", "Ring R", "self : RingFilterBasis R", "U : Set R"], "goal": "U ∈ AddGroupFilterBasis.toFilterBasis.sets → ∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V * V ⊆ U"}}
+{"state": {"context": ["J : Type w", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "Unique J", "f : J → C"], "goal": "(CategoryTheory.Limits.limitBiconeOfUnique f).isBilimit.isLimit = (CategoryTheory.Limits.limitConeOfUnique f).isLimit"}}
+{"state": {"context": ["A : Type u_2", "AddMonoid A", "TopologicalSpace A", "x : A"], "goal": "(ContinuousAddMonoidHom.id A) x = x"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_5", "ι' : Sort u_6", "κ : ι → Sort u_7", "κ' : ι' → Sort u_8", "CompleteLattice α", "f : (i : ι) → κ i → α", "g : (i' : ι') → κ' i' → α", "h : ∀ (i : ι') (j : κ' i), ∃ i' j', f i' j' ≤ g i j"], "goal": "⨅ i, ⨅ j, f i j ≤ ⨅ i, ⨅ j, g i j"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝³ : Group G", "inst✝² : Group G'", "inst✝¹ : Group G''", "A : Type u_4", "inst✝ : AddGroup A", "H K : Subgroup G", "k : Set G", "x y : G", "hy : y ∈ closure {x}", "n : ℤ"], "goal": "∃ n_1, x ^ n_1 = (x ^ n)⁻¹"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "v w : F", "h : ‖v‖ = ‖w‖", "R : F ≃ₗᵢ[ℝ] F := reflection (Submodule.span ℝ {v - w})ᗮ"], "goal": "R v + R v = w + w"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "PseudoMetricSpace β", "One β", "f : BoundedContinuousFunction α β"], "goal": "(∀ (x : α), f x = 1) ↔ f = 1"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommRing R", "n✝ : ℕ", "inst✝² : IsDomain R", "n : ℕ", "K : Type u_2", "inst✝¹ : Field K", "μ : K", "inst✝ : NeZero ↑n", "hnpos : 0 < n", "hμn : orderOf μ ∣ n", "hnμ : orderOf μ ≠ n", "ho : 0 < orderOf μ", "i : ℕ", "hio : i ∣ orderOf μ", "key : i < n", "key' : i ∣ n", "hni : {i, n} ⊆ n.divisors", "k : K[X]", "hk : cyclotomic i K = (X - C μ) * k", "j : K[X]", "hj : cyclotomic n K = (X - C μ) * j", "this : (∏ x ∈ n.divisors \\ {i, n}, cyclotomic x K) * ((X - C μ) * k * ((X - C μ) * j)) = X ^ n - 1", "hn : Squarefree (X ^ n - 1)"], "goal": "False"}}
+{"state": {"context": ["G : Type u_3", "AddCommGroup G", "a b c : G"], "goal": "a + c + (b - c) = a + b"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "F : Type uF", "NormedAddCommGroup E", "NormedAddCommGroup E'", "NormedAddCommGroup F", "f : G → E", "g : G → E'", "NontriviallyNormedField 𝕜", "NormedSpace 𝕜 E", "NormedSpace 𝕜 E'", "NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "MeasurableSpace G", "μ : MeasureTheory.Measure G", "NormedSpace ℝ F", "AddGroup G", "TopologicalSpace G", "TopologicalAddGroup G", "T2Space G", "hcf : HasCompactSupport f", "hcg : HasCompactSupport g"], "goal": "HasCompactSupport (MeasureTheory.convolution f g L μ)"}}
+{"state": {"context": ["J : Type v'", "inst✝⁵ : Category.{u', v'} J", "C : Type u", "inst✝⁴ : Category.{v, u} C", "K : Type u_1", "inst✝³ : Category.{?u.306779, u_1} K", "D : Type u_2", "inst✝² : Category.{?u.306786, u_2} D", "n : ℕ", "f : Fin (n + 1) → C", "c₁ : Cofan fun i => f i.succ", "c₂ : BinaryCofan (f 0) c₁.pt", "t₁ : IsVanKampenColimit c₁", "t₂ : IsVanKampenColimit c₂", "inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i", "inst✝ : HasFiniteCoproducts C", "F : Discrete (Fin (n + 1)) ⥤ C", "c : Cocone F", "α : F ⟶ Discrete.functor f", "i : c.pt ⟶ (extendCofan c₁ c₂).pt", "e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i", "hα : NatTrans.Equifibered α", "Hc : IsColimit c", "j : Fin (n + 1)"], "goal": "IsColimit (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ }))"}}
+{"state": {"context": ["p k n : ℕ", "hp : Fact (Nat.Prime p)", "h : k ≤ n"], "goal": "(p - 1) * padicValNat p (n.choose k) = (p.digits k).sum + (p.digits (n - k)).sum - (p.digits n).sum"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ"], "goal": "ContinuousOn sin [[a, b]]"}}
+{"state": {"context": ["α : Type u_1", "s : Set α", "Preorder α", "NoMaxOrder α"], "goal": "Set.Unbounded (fun x x_1 => x < x_1) s ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s"}}
+{"state": {"context": ["ε : ℝ", "hε : 0 < ε"], "goal": "‖log ↑0‖ ≤ ↑0 ^ ε / ε"}}
+{"state": {"context": ["α : Type u_1", "Encodable α", "a : α"], "goal": "Encodable.decode₂ α (Encodable.encode a) = some a"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "M' : Type u_3", "F : Type u_4", "G : Type u_5", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "N✝ N₁ N₂ P P₁ P₂ N : Submodule R M"], "goal": "Module.annihilator R (M ⧸ N) = N.colon ⊤"}}
+{"state": {"context": ["α : Type u_1", "MonoidWithZero α", "a : α"], "goal": "Associates.mk a ≠ 0 ↔ a ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : Filter α", "LE β"], "goal": "(fun x x_1 => x ≤ x_1) = Filter.Germ.LiftRel fun x x_1 => x ≤ x_1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Preorder α", "Zero β", "Preorder β", "Zero γ", "Preorder γ", "SMul α β", "SMul α γ", "f : β → γ", "SMulPosReflectLT α γ", "hf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂", "smul : ∀ (a : α) (b : β), f (a • b) = a • f b", "zero : f 0 = 0"], "goal": "SMulPosReflectLT α β"}}
+{"state": {"context": ["α : Type u_1", "a : α", "z : Sym2 α", "h : a ∈ z"], "goal": "Sym2.Mem.other h ∈ z"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "x : α"], "goal": "connectedComponent x ⊆ ⋂ Z, ↑Z"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2", "hnsol : Polynomial.eval a F ≠ 0", "k : ℕ", "_h : 0 < k + 2"], "goal": "‖newton_seq (k + 2) - newton_seq (k + 1)‖ < ‖newton_seq (k + 1) - a‖"}}
+{"state": {"context": ["α : Type u_1", "xs : List α", "a b : α", "h : s(a, b) ∈ xs.sym2"], "goal": "a ∈ xs"}}
+{"state": {"context": ["s : Finset ℕ", "N : ℕ", "hN : ¬Nat.Prime N"], "goal": "Nat.factoredNumbers (insert N s) = Nat.factoredNumbers s"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "α : Type u_2", "inst✝ : Mul α", "J : Subgroup G", "g : G", "H K : Subgroup G", "a h : G", "hh : h ∈ ↑H", "k : G", "hk : k ∈ ↑K", "hb : h * a * k ∈ doset a ↑H ↑K"], "goal": "doset (h * a * k) ↑H ↑K = doset a ↑H ↑K"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedCommGroup α", "a✝ b✝ a b : α", "ab : b ≤ a"], "goal": "mabs (a * b) = mabs a * mabs b ↔ 1 ≤ a ∧ 1 ≤ b ∨ a ≤ 1 ∧ b ≤ 1"}}
+{"state": {"context": ["a b : Nat"], "goal": "a.succ * b.succ = a * b + a + b + 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "h : Function.Injective f"], "goal": "Function.RightInverse (Set.rangeFactorization f) (Set.rangeSplitting f)"}}
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+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I✝ : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "f : InfinitePlace K → ℝ≥0", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ", "h : minkowskiBound K 1 < volume (convexBodyLT K f)", "a : 𝓞 K", "h_mem : (algebraMap (𝓞 K) K) a ∈ ↑1", "h_nz : (algebraMap (𝓞 K) K) a ≠ 0", "h_bd : ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) a) < ↑(f w)"], "goal": "∃ a, a ≠ 0 ∧ ∀ (w : InfinitePlace K), w ↑a < ↑(f w)"}}
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+{"state": {"context": ["α : Type u_1", "S T : Set (Set α)"], "goal": "⋂₀ (S ∪ T) = ⋂₀ S ∩ ⋂₀ T"}}
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+{"state": {"context": ["α : Type u", "s t u : Set α", "h : s ⊆ u", "d : Disjoint u t"], "goal": "Disjoint s t"}}
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+{"state": {"context": ["x : ℝ", "hx : 0 < x"], "goal": "0 < Real.log x ↔ 1 < x"}}
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+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "x : E"], "goal": "√(1 + ‖x‖ ^ 2) ≤ 1 + ‖x‖"}}
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+{"state": {"context": ["α : Sort u_1", "q : Prop", "p : α → Prop", "Decidable q"], "goal": "(∀ (x : α), p x ∨ q) ↔ (∀ (x : α), p x) ∨ q"}}
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+{"state": {"context": ["R : Type u", "M : Type v", "M₂ : Type w", "Semiring R", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂"], "goal": "LinearMap.snd R M M₂ ∘ₗ LinearMap.inl R M M₂ = 0"}}
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+{"state": {"context": ["E : Type u_1", "SeminormedAddGroup E", "s t : Set E", "hs : Bornology.IsBounded s", "ht : Bornology.IsBounded t"], "goal": "Bornology.IsBounded (s - t)"}}
+{"state": {"context": ["α : Type u", "P : α → Prop", "a : α", "s : Set α"], "goal": "(∀ x ∈ insert a s, P x) ↔ P a ∧ ∀ x ∈ s, P x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "K : Type u_3", "L : Type u_4", "inst✝¹⁶ : EuclideanDomain R", "inst✝¹⁵ : CommRing S", "inst✝¹⁴ : IsDomain S", "inst✝¹³ : Field K", "inst✝¹² : Field L", "inst✝¹¹ : Algebra R K", "inst✝¹⁰ : IsFractionRing R K", "inst✝⁹ : Algebra K L", "inst✝⁸ : FiniteDimensional K L", "inst✝⁷ : Algebra.IsSeparable K L", "algRL : Algebra R L", "inst✝⁶ : IsScalarTower R K L", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra S L", "ist : IsScalarTower R S L", "iic : IsIntegralClosure S R L", "abv : AbsoluteValue R ℤ", "ι : Type u_5", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "bS : Basis ι R S", "adm : abv.IsAdmissible", "inst✝¹ : Infinite R", "inst✝ : DecidableEq R", "a : S", "b : R", "hb : b ≠ 0", "dim_pos : 0 < Fintype.card ι"], "goal": "∃ q, ∃ r ∈ finsetApprox bS adm, abv ((Algebra.norm R) (r • a - b • q)) < abv ((Algebra.norm R) ((algebraMap R S) b))"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "Ring R", "Invertible 2", "AddCommGroup V", "Module R V", "x y : V"], "goal": "midpoint R (x - y) (x + y) = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : CommSemigroup G", "a b c : G"], "goal": "a * (b * c) = b * (c * a)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "Semiring R", "AddCommMonoid M", "Module R M", "s : Set (Submodule R M)"], "goal": "(sSup s).toAddSubmonoid = sSup (Submodule.toAddSubmonoid '' s)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "δ : Type u_2", "π : ι → Type u_3", "r : α → α → Prop", "inst✝³ : Preorder α", "inst✝² : Nonempty α", "inst✝¹ : NoMinOrder α", "inst✝ : NoMaxOrder α", "inhabited_h : Inhabited α"], "goal": "∃ f, StrictMono f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ : Measure α", "f : α → ℝ", "f_intble : Integrable f μ", "f_nn : 0 ≤ᶠ[ae μ] f"], "goal": "(fun a => (μ {a_1 | a < f a_1}).toReal) =ᶠ[ae (volume.restrict (Ioi 0))] fun a => (μ {a_1 | a ≤ f a_1}).toReal"}}
+{"state": {"context": ["m : ℕ", "hx : ↑m ≠ ⊤", "n : ℕ", "h : ↑m + 1 ≤ ↑n"], "goal": "↑m < ↑n"}}
+{"state": {"context": ["ξ : ℚ"], "goal": "{q | |ξ - q| < 1 / ↑q.den ^ 2}.Finite"}}
+{"state": {"context": ["p : ℕ", "a : ZMod p", "ha : a ^ (p - 1) = 1", "hd : ∀ (q : ℕ), Nat.Prime q → q ∣ p - 1 → a ^ ((p - 1) / q) ≠ 1"], "goal": "Nat.Prime p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "γ₂ : Type u_4", "δ : Type u_5", "ι : Sort y", "s t u : Set α", "inst✝² : TopologicalSpace α", "inst✝¹ : T1Space α", "inst✝ : Countable α"], "goal": "borel α = ⊤"}}
+{"state": {"context": ["p b : ℝ", "hb : 0 < b", "hp : 1 < p"], "goal": "Tendsto (fun x => -x - -b * x ^ p) atTop atTop"}}
+{"state": {"context": [], "goal": "∀ {X Y : SimplexCategory} (f : X ⟶ Y), SimplexCategory.skeletalFunctor.map f = SimplexCategory.Hom.toOrderHom f"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : (-o).oangle y x = ↑(π / 2)"], "goal": "‖x‖ / ((-o).oangle y (y + x)).tan = ‖y‖"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "PseudoMetricSpace β", "AddMonoid β", "BoundedAdd β", "ContinuousAdd β", "CompactSpace α", "f g : C(α, β)"], "goal": "BoundedContinuousFunction.mkOfCompact (f + g) =\n  BoundedContinuousFunction.mkOfCompact f + BoundedContinuousFunction.mkOfCompact g"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_7", "M : Type u_8", "M₂ : Type u_10", "Semiring R", "Semiring S", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂", "Module S M", "Module S M₂", "LinearMap.CompatibleSMul M M₂ R S"], "goal": "Function.Injective (LinearEquiv.restrictScalars R)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r r₁ r₂ : α → α → Prop", "s t : Set α", "a b x : α", "h : s.Subsingleton"], "goal": "maximals r s = s"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "Module R N", "Module R P", "Module R Q", "g : N →ₗ[R] P"], "goal": "↑(TensorProduct.comm R P Q) ∘ₗ LinearMap.rTensor Q g ∘ₗ ↑(TensorProduct.comm R Q N) = LinearMap.lTensor Q g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : ℕ → α → ℝ≥0∞", "hf : ∀ (n : ℕ), Measurable (f n)", "h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a", "s : Set α", "hs : {a | ¬∀ (i : ℕ), f i a ≤ f i.succ a} ⊆ s ∧ MeasurableSet s ∧ μ s = 0", "g : ℕ → α → ℝ≥0∞ := fun n a => if a ∈ s then 0 else f n a", "g_eq_f : ∀ᵐ (a : α) ∂μ, ∀ (n : ℕ), g n a = f n a", "n : ℕ", "a : α"], "goal": "g n a ≤ g (n + 1) a"}}
+{"state": {"context": ["x y : ℝ", "hy : 0 < y"], "goal": "x ≤ Real.log y ↔ Real.exp x ≤ y"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "p : ι → P", "ha : AffineIndependent k p"], "goal": "AffineIndependent k fun x => ↑x"}}
+{"state": {"context": ["S : Type u_1", "inst✝² : CommRing S", "inst✝¹ : IsDomain S", "P : Ideal S", "P_prime : P.IsPrime", "hP : P ≠ ⊥", "inst✝ : IsDedekindDomain S", "i : ℕ", "ih : cardQuot (P ^ i) = cardQuot P ^ i", "this : P ^ (i + 1) < P ^ i", "a : S", "a_mem : a ∈ P ^ i", "a_not_mem : a ∉ P ^ (i + 1)", "f g : (c : S) → c ∈ P ^ i → S", "hg : ∀ (c : S) (hc : c ∈ P ^ i), g c hc ∈ P ^ (i + 1)", "hf : ∀ (c : S) (hc : c ∈ P ^ i), a * f c hc + g c hc = c"], "goal": "↥(map (mkQ (P ^ i.succ)) (P ^ i)) ≃ S ⧸ P"}}
+{"state": {"context": ["x y z : ℝ", "n : ℕ", "hx1 : 0 < x", "hx2 : x ≤ 1", "hz : z ≤ 0"], "goal": "1 ≤ x ^ z"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "Zero M", "f : α ≃ β", "a : α", "b : M"], "goal": "Finsupp.equivMapDomain f (Finsupp.single a b) = Finsupp.single (f a) b"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y z : V", "hx : x ≠ 0", "hy : y ≠ 0", "hz : z ≠ 0"], "goal": "(o.kahler y) z ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "OrderedSemiring α", "a b : α", "ha : 0 < a", "hb : 0 ≤ b"], "goal": "0 ≤ a * b"}}
+{"state": {"context": ["R : Type u", "EuclideanDomain R", "p q : R", "hpq : q ∣ p"], "goal": "p / q ∣ p"}}
+{"state": {"context": ["x : ℝ", "hx : 0 < x", "a : ℝ", "h1 : I * ↑(1 / x) = -1 / (I * ↑x)", "hx' : I * ↑x ≠ 0"], "goal": "cexp (-↑π * ↑a ^ 2 * ↑x) * (1 / (-I * (I * ↑x)) ^ (1 / 2) * cexp (-↑π * I * (↑a * I * ↑x) ^ 2 / (I * ↑x)) * jacobiTheta₂ (↑a * I * ↑x / (I * ↑x)) (-1 / (I * ↑x))) = ↑(1 / x ^ (1 / 2)) * jacobiTheta₂ (↑a) (I * ↑(1 / x))"}}
+{"state": {"context": ["X : Type u_1", "MeasurableSpace X", "s : Set X", "μ : MeasureTheory.Measure X", "f : X → ℝ", "hf : 0 ≤ᵐ[μ.restrict s] f", "hfi : MeasureTheory.IntegrableOn f s μ"], "goal": "∫ (x : X) in s, f x ∂μ = 0 ↔ f =ᵐ[μ.restrict s] 0"}}
+{"state": {"context": ["M : Type u_4", "Monoid M", "l₁ l₂ : List M"], "goal": "(l₁ ++ l₂).prod = l₁.prod * l₂.prod"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "g : α → ℝ≥0∞", "hg : Measurable g"], "goal": "μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g"}}
+{"state": {"context": ["o : Ordinal.{u_2}", "h : ω ≤ o"], "goal": "(Cardinal.aleph' (Order.succ o)).IsRegular"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "R : Type u_7", "α : Type v", "Semiring R", "AddCommMonoid α", "Module R α"], "goal": "∀ (a : Matrix m n α), (Matrix.transposeLinearEquiv m n R α) a = (Matrix.transposeAddEquiv m n α).toFun a"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝² : EMetricSpace X", "inst✝¹ : EMetricSpace Y", "inst✝ : SecondCountableTopology X", "s : Set X"], "goal": "⨆ x, limsup dimH (𝓝[s] x).smallSets = dimH s"}}
+{"state": {"context": ["G₀ : Type u_1", "α : Type u_2", "inst✝² : GroupWithZero G₀", "inst✝¹ : Bornology G₀", "inst✝ : MulAction G₀ α", "s t u : Set α", "S : Set (Set α)", "a : G₀", "ha : a ≠ 0"], "goal": "s ⊆ univ"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "E : ι → Type wE", "E₁ : ι → Type wE₁", "G : Type wG", "Fintype ι", "NontriviallyNormedField 𝕜", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "(i : ι) → SeminormedAddCommGroup (E₁ i)", "(i : ι) → NormedSpace 𝕜 (E₁ i)", "SeminormedAddCommGroup G", "NormedSpace 𝕜 G", "g : ContinuousMultilinearMap 𝕜 E₁ G", "f : (i : ι) → E i →L[𝕜] E₁ i"], "goal": "(ContinuousMultilinearMap.compContinuousLinearMapL f) g = g.compContinuousLinearMap f"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Mul M", "Mul N", "ι : Sort u_5", "f : M →ₙ* N", "s : ι → Subsemigroup M"], "goal": "Subsemigroup.map f (iSup s) = ⨆ i, Subsemigroup.map f (s i)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r"], "goal": "asympBound g a b =O[atTop] T"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "DecidableEq α", "s : Multiset α", "f : α → β", "x : { x // x ∈ s }"], "goal": "Multiset.map (fun j => f ↑j) (s.attach.erase x) = Multiset.map f (s.erase ↑x)"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "Field K", "Field L", "Algebra K L", "S : IntermediateField K L", "x : L", "hx : x ∈ S"], "goal": "S.val ⟨x, hx⟩ = x"}}
+{"state": {"context": ["α : Type u_1", "m : Set α → ℝ≥0∞", "s : Set α", "h₀ : m ∅ = 0", "hs : ∀ (t : Set α), m (t ∩ s) + m (t \\ s) ≤ m t", "t : Set α", "f : ℕ → Set α", "hf : t ⊆ ⋃ i, f i"], "goal": "∑' (i : ℕ), m ((fun i => f i ∩ s) i) + ∑' (i : ℕ), m ((fun i => f i \\ s) i) ≤ ∑' (i : ℕ), m (f i)"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G"], "goal": "H.index = 2 ↔ ∃ a, ∀ (b : G), Xor' (b * a ∈ H) (b ∈ H)"}}
+{"state": {"context": ["N : ℕ", "j : ℝ", "hj : 0 < j", "c : ℝ", "hc : 1 < c"], "goal": "∑ i ∈ Finset.filter (fun x => j < c ^ x) (Finset.range N), 1 / (c ^ i) ^ 2 ≤ c ^ 3 * (c - 1)⁻¹ / j ^ 2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝ : TopologicalSpace α", "s✝ t u✝ v✝ s : Set α", "hs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y", "u v : Set α", "u_op : IsOpen u", "v_op : IsOpen v", "hsuv : s ⊆ u ∪ v", "x : α", "x_in_s : x ∈ s", "x_in_u : x ∈ u", "H : s ∩ (u ∩ v) = ∅", "y : α", "y_in_s : y ∈ s", "y_in_v : y ∈ v", "hy : y ∉ u", "this : ContinuousOn u.boolIndicator s"], "goal": "False"}}
+{"state": {"context": ["X Y : AddCommMonCat", "f : X ⟶ Y"], "goal": "(CategoryTheory.forget AddCommMonCat).map f = ⇑f"}}
+{"state": {"context": ["α : Sort u_1", "DecidableEq α", "a b x : α", "h : (Equiv.swap a b) x ≠ x"], "goal": "x = a ∨ x = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "G : Type u_5", "inst✝³ : TopologicalSpace G", "inst✝² : CommGroup G", "inst✝¹ : TopologicalGroup G", "f : α → G", "inst✝ : T2Space G", "a : G", "hβ : Infinite β"], "goal": "∏' (x : β), a = a ^ Nat.card β"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "TopologicalSpace R", "TopologicalSpace M", "CommSemiring R", "AddCommMonoid M", "Module R M", "x : tsze R M"], "goal": "(TrivSqZeroExt.fstCLM R M) x = x.fst"}}
+{"state": {"context": ["α : Type u_1", "V : Type u_2", "P : Type u_3", "SeminormedAddCommGroup V", "PseudoMetricSpace P", "NormedAddTorsor V P", "PseudoEMetricSpace α", "f : α → V", "g : α → P", "Kf Kg : ℝ≥0", "hf : LipschitzWith Kf f", "hg : LipschitzWith Kg g"], "goal": "LipschitzWith (Kf + Kg) (f +ᵥ g)"}}
+{"state": {"context": ["g : ↥GL(2, ℝ)⁺", "z : ℍ"], "goal": "↑↑g 0 0 * z.im * (↑↑g 1 0 * z.re + ↑↑g 1 1) / Complex.normSq (↑(↑↑g 1 0) * ↑z + ↑(↑↑g 1 1)) - (↑↑g 0 0 * z.re + ↑↑g 0 1) * (↑↑g 1 0 * z.im) / Complex.normSq (↑(↑↑g 1 0) * ↑z + ↑(↑↑g 1 1)) = (↑↑g).det * z.im / Complex.normSq (↑(↑↑g 1 0) * ↑z + ↑(↑↑g 1 1))"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q✝ : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : α → F", "p q : ℝ", "hq_pos : 0 < q", "x : α"], "goal": "ENNReal.ofReal ‖‖f x‖ ^ q‖ ^ p = ENNReal.ofReal ‖f x‖ ^ (p * q)"}}
+{"state": {"context": ["L : Language", "κ : Cardinal.{w}", "T : L.Theory", "h : κ.Categorical T", "h1 : ℵ₀ ≤ κ", "h2 : lift.{w, max u v} L.card ≤ lift.{max u v, w} κ", "hS : T.IsSatisfiable", "hT : ∀ (M : T.ModelType), Infinite ↑M", "φ : L.Sentence", "w✝ : T.ModelType", "h✝ : #↑w✝ = κ", "MF : T.ModelType", "hMF : ¬↑MF ⊨ φ", "MT : T.ModelType", "hMT : ↑MT ⊨ φ", "this✝ : Infinite ↑MT", "this : Infinite ↑MF", "NT : Bundled L.Structure", "MNT : ↑MT ≅[L] ↑NT", "hNT : #↑NT = κ", "NF : Bundled L.Structure", "MNF : ↑MF ≅[L] ↑NF", "hNF : #↑NF = κ", "TF : ↑(ElementarilyEquivalent.toModel T MNT) ≃[L] ↑(ElementarilyEquivalent.toModel T MNF)"], "goal": "↑MF ⊨ φ"}}
+{"state": {"context": ["α : Sort u_3", "p : α → Sort u_4", "∀ (x : α), Subsingleton (p x)", "q : (x : α) → p x → Prop", "b : Prop", "h₂ : ∃! x, ∃! h, q x h", "h₁ : ∀ (x : α) (h : p x), q x h → (∀ (y : α) (hy : p y), q y hy → y = x) → b"], "goal": "b"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "s t : Set α", "hs_sup : SupClosed s", "hs_inf : InfClosed s", "ht_sup : SupClosed t", "ht_inf : InfClosed t"], "goal": "{ carrier := s, supClosed' := hs_sup, infClosed' := hs_inf } <\n    { carrier := t, supClosed' := ht_sup, infClosed' := ht_inf } ↔\n  s ⊂ t"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "CommRing R", "LieRing L", "LieAlgebra R L", "LieAlgebra.IsSimple R L", "I : LieIdeal R L"], "goal": "IsAtom I ↔ I = ⊤"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : (nilradical R).IsPrime", "inst✝¹ : CommRing S", "inst✝ : IsDomain S", "φ : R →+* S", "f : R[X]", "hm : f.Monic", "R' : Type u_1 := R ⧸ nilradical R", "ψ : R' →+* S := Ideal.Quotient.lift (nilradical R) φ ⋯", "ι : R →+* R' := algebraMap R R'", "hi : Irreducible (Polynomial.map ψ (Polynomial.map ι f))"], "goal": "Irreducible f"}}
+{"state": {"context": ["p : ℝ≥0∞", "𝕜 : Type u_1", "ι : Type u_2", "Finite ι", "Ring 𝕜"], "goal": "(PiLp.basisFun p 𝕜 ι).map (WithLp.linearEquiv p 𝕜 (ι → 𝕜)) = Pi.basisFun 𝕜 ι"}}
+{"state": {"context": ["R : Type w₁", "A : Type w₂", "B : Type w₃", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Algebra R A", "inst✝² : CommRing B", "inst✝¹ : Algebra R B", "inst✝ : FinitePresentation R A", "e : A ≃ₐ[R] B", "n : ℕ", "f : MvPolynomial (Fin n) R →ₐ[R] A", "hf : Surjective ⇑f ∧ (RingHom.ker f.toRingHom).FG"], "goal": "(RingHom.ker ((↑e).comp f).toRingHom).FG"}}
+{"state": {"context": ["α : Type u", "β : Type v", "a b : α", "inst✝ : CommSemiring α", "I : Ideal α", "s : Set α", "n : ℕ", "f : ↑s →₀ α", "hf✝ : (Finsupp.total (↑s) α α Subtype.val) f = 1", "hf : ∑ a ∈ f.support, f a * ↑a = 1"], "goal": "1 ∈ span ((fun x => x ^ (n + 1)) '' s)"}}
+{"state": {"context": ["a : Prop"], "goal": "¬¬a ↔ a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝¹ : OmegaCompletePartialOrder α", "inst✝ : CompleteLattice β"], "goal": "Continuous (sSup ∅)"}}
+{"state": {"context": [], "goal": "IsGδ {x | Liouville x}"}}
+{"state": {"context": ["V : Type u_1", "G G' : SimpleGraph V", "M M' : G.Subgraph", "v w : V", "inst✝ : (v : V) → Fintype ↑(M.neighborSet v)"], "goal": "M.IsMatching ↔ ∀ v ∈ M.verts, M.degree v = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "R : Type u_4", "inst✝ : NormedRing R", "x : R", "h : ‖x‖ < 1"], "goal": "Tendsto (fun n => x ^ n) atTop (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "𝒜 ℬ : Set (Set α)", "h : 𝒜 ⊆ ℬ", "hℬ : MeasureTheory.IsSetAlgebra ℬ"], "goal": "MeasureTheory.generateSetAlgebra 𝒜 ⊆ ℬ"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "P : StrongFEPair E", "r : ℝ", "this : (fun x => P.f x - (P.ε * ↑(x ^ (-P.k))) • 0) =O[𝓝[>] 0] fun x => x ^ r"], "goal": "P.f =O[𝓝[>] 0] fun x => x ^ r"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝¹ : LinearOrderedField 𝕜", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : 𝕜", "s t : Finset α", "a b : α", "h : ¬G.IsUniform ε s t"], "goal": "(G.nonuniformWitnesses ε s t).2 ⊆ t"}}
+{"state": {"context": ["x : ℝ*", "hi : x.Infinite"], "goal": "x.st = 0"}}
+{"state": {"context": ["n : ℕ", "E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f g : (Fin n → ℂ) → E", "c : Fin n → ℂ", "R : Fin n → ℝ", "hf : TorusIntegrable f c R", "hg : TorusIntegrable g c R"], "goal": "(∯ (x : Fin n → ℂ) in T(c, R), f x + g x) = (∯ (x : Fin n → ℂ) in T(c, R), f x) + ∯ (x : Fin n → ℂ) in T(c, R), g x"}}
+{"state": {"context": ["α : Type u_1", "EMetricSpace α", "s : Set α", "Finite ↑s"], "goal": "0 < s.einfsep"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "n : Type w", "inst✝¹ : DecidableEq n", "inst✝ : Fintype n", "i j : n", "k : ℕ", "x : R"], "goal": "(polyEquivTensor R (Matrix n n R)).symm ((Algebra.TensorProduct.comm R R[X] (Matrix n n R)) ((monomial k) x ⊗ₜ[R] stdBasisMatrix i j 1)) = (monomial k) (stdBasisMatrix i j x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "m : Multiset α", "x : α", "n : ℕ"], "goal": "(∃ a, a ::ₘ m ≤ replicate (n + 1) x) ↔ m ≤ replicate n x"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "MeasurableSpace X", "TopologicalSpace X", "NormedAddCommGroup E", "f : X → E", "μ : MeasureTheory.Measure X", "hf : MeasureTheory.LocallyIntegrable f μ", "s : Set X"], "goal": "MeasureTheory.LocallyIntegrableOn f s μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "hf : Function.Injective f", "u s : Set α"], "goal": "(fun x => ↑((Equiv.Set.image f s hf).symm x)) ⁻¹' u = Subtype.val ⁻¹' (f '' u)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α"], "goal": "∀ (x : α), Levenshtein.defaultCost.delete x = 1"}}
+{"state": {"context": ["ι : Sort u_1", "R : Type u_2", "inst✝ : Ring R", "s : Set R"], "goal": "sInf {a | s ⊆ MulOpposite.op ⁻¹' ↑a} = sInf {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : Filter α", "g : Set α → Filter β", "hg : Monotone g", "s : Set β"], "goal": "s ∈ f.lift g ↔ ∃ t ∈ f, s ∈ g t"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "R₁ R₂ : ComposableArrows C 3", "φ : R₁ ⟶ R₂", "hR₁ : (mk₂ (R₁.map' 1 2 ⋯ ⋯) (R₁.map' 2 3 ⋯ ⋯)).Exact", "hR₂ : (mk₂ (R₂.map' 0 1 ⋯ ⋯) (R₂.map' 1 2 ⋯ ⋯)).Exact", "hR₂' : R₂.map' 1 3 ⋯ ⋯ = 0", "h₀ : Epi (app' φ 0 ⋯)", "h₂✝ : Epi (app' φ 2 ⋯)", "h₃✝ : Mono (app' φ 3 ⋯)", "A : C", "g₁ : A ⟶ R₂.obj' 1 ⋯", "A₁ : C", "π₁ : A₁ ⟶ A", "w✝¹ : Epi π₁", "f₂ : A₁ ⟶ R₁.obj' 2 ⋯", "h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 ⋯", "h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0", "A₂ : C", "π₂ : A₂ ⟶ A₁", "w✝ : Epi π₂", "f₁ : A₂ ⟶ R₁.obj 1", "h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)"], "goal": "∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ g₁ = x ≫ app' φ 1 ⋯"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_4", "Fintype ι", "V : ι → Submodule 𝕜 E", "∀ (i : ι), CompleteSpace ↥(V i)", "hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ", "x : E", "hx : x ∈ iSup V"], "goal": "∑ i : ι, ↑((orthogonalProjection (V i)) x) = x"}}
+{"state": {"context": ["n : ℤ"], "goal": "|↑n.negOnePow| = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "NontriviallyNormedField 𝕜", "NonUnitalNormedRing A", "NormedSpace 𝕜 A", "SMulCommClass 𝕜 A A", "IsScalarTower 𝕜 A A", "StarRing A", "CstarRing A", "a : 𝓜(𝕜, A)"], "goal": "‖a.toProd.1‖ = ‖a.toProd.2‖"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Mul M", "Mul N", "S : Subsemigroup M", "T : Subsemigroup N", "f : M →ₙ* N"], "goal": "S ≤ Subsemigroup.comap f T → Subsemigroup.map f S ≤ T"}}
+{"state": {"context": ["A : Type u_1", "Mul A", "self : IsJordan A", "a b : A"], "goal": "a * a * (a * b) = a * (a * a * b)"}}
+{"state": {"context": ["α : Type u_2", "AddMonoid α", "n : ℕ", "a : α", "s : Fin n → Set α"], "goal": "a ∈ (List.ofFn s).sum ↔ ∃ f, (List.ofFn fun i => ↑(f i)).sum = a"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "n : ℕ", "X : CategoryTheory.SimplicialObject C"], "goal": "(AlgebraicTopology.DoldKan.natTransPInfty_f C n).app X = AlgebraicTopology.DoldKan.PInfty.f n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : Inducing f", "h : IsClosed (Set.range f)"], "goal": "IsClosedMap f"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "CategoryTheory.Category.{v₁, u₁} 𝒮", "CategoryTheory.Category.{v₂, u₂} 𝒳", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ⟶ b", "p.IsCartesian f φ"], "goal": "CategoryTheory.Functor.IsCartesian.map p f φ φ = 𝟙 a"}}
+{"state": {"context": ["o : Ordinal.{u}"], "goal": "Ordinal.lsub (Ordinal.typein fun x x_1 => x < x_1) = o"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "A✝ : Type u_3", "S : Type u_4", "inst✝¹ : Group G", "inst✝ : AddGroup A✝", "s : Set G", "A B C : Subgroup G", "h : A ≤ C", "x✝ : G"], "goal": "(∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝) ↔ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C"}}
+{"state": {"context": ["R : Type u_1", "n : ℕ", "x y : ℤ", "hxy : 2 ∣ x - y", "hx : ¬2 ∣ x", "hy : Odd y", "d : ℕ"], "goal": "multiplicity 2 (x ^ (d + d) - y ^ (d + d)) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑(d + d)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : StrictOrderedCommRing R", "M : Type u_2", "N : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : AddCommGroup N", "inst✝¹ : Module R M", "inst✝ : Module R N", "x y : M", "u : Rˣ", "hu : ↑u < 0", "v : Ray R M"], "goal": "u • v = -v"}}
+{"state": {"context": ["X : RingedSpace", "U : Opens ↑↑X.toPresheafedSpace", "f : ↑(X.presheaf.obj (op U))", "x : ↥U", "h : IsUnit ((X.presheaf.germ x) f)", "g' : ↑(X.presheaf.stalk ↑x)", "heq : (X.presheaf.germ x) f * g' = 1"], "goal": "∃ V i, ∃ (_ : ↑x ∈ V), IsUnit ((X.presheaf.map i.op) f)"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s t : Set α", "hs : MeasureTheory.NullMeasurableSet s μ", "hd : MeasureTheory.AEDisjoint μ s t"], "goal": "μ (s ∪ t) = μ s + μ t"}}
+{"state": {"context": [], "goal": "Tendsto (fun x => log (1 + 1 / x)) atTop (𝓝 0)"}}
+{"state": {"context": ["M : Type u_2", "MonoidWithZero M", "Nontrivial M"], "goal": "0 ∉ M⁰"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C"], "goal": "CategoryTheory.AsSmall.equiv.functor = CategoryTheory.AsSmall.up"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "s : Set X"], "goal": "Bornology.IsBounded s ↔ ∃ t, IsCompact t ∧ s ⊆ t"}}
+{"state": {"context": ["M : Type u_4", "AddMonoid M", "l : List M", "a : M"], "goal": "(a :: l).sum = a + l.sum"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "x : S", "I : Ideal R", "inst✝³ : IsDomain R", "inst✝² : IsIntegrallyClosed R", "inst✝¹ : IsDedekindDomain S", "inst✝ : NoZeroSMulDivisors R S", "hI : I.IsMaximal", "hI' : I ≠ ⊥", "hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤", "hx' : _root_.IsIntegral R x", "hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))", "mem_norm_factors : normalize (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x)) ∈ normalizedFactors (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))"], "goal": "Irreducible (Ideal.map (algebraMap R S) I)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "P : C", "π : Y ⟶ P", "w : CategoryTheory.CategoryStruct.comp f π = CategoryTheory.CategoryStruct.comp g π"], "goal": "∀ (X_1 : CategoryTheory.Limits.WalkingParallelPair),\n  (CategoryTheory.Limits.Cofork.ofπ π w).ι.app X_1 =\n    CategoryTheory.Limits.WalkingParallelPair.casesOn X_1 (CategoryTheory.CategoryStruct.comp f π) π"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "s : Set α", "s_mble : MeasurableSet s"], "goal": "MeasureTheory.Measure.count s = ⊤ ↔ s.Infinite"}}
+{"state": {"context": ["f g : CircleDeg1Lift", "hf : Continuous ⇑f", "z : ℝ", "hz : ∀ (x : ℝ), f x < x + z", "x : ℝ", "hx : ∀ y ∈ Icc 0 1, f y - y ≤ f x - x"], "goal": "∀ (x_1 : ℝ), f x_1 ≤ x_1 + (f x - x)"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "D₂ : Type u_3", "H : Type u_4", "inst✝⁵ : Category.{u_5, u_1} C₁", "inst✝⁴ : Category.{u_6, u_2} C₂", "inst✝³ : Category.{u_7, u_3} D₂", "inst✝² : Category.{?u.53917, u_4} H", "W₁ : MorphismProperty C₁", "W₂ : MorphismProperty C₂", "Φ : LocalizerMorphism W₁ W₂", "L₂ : C₂ ⥤ D₂", "inst✝¹ : L₂.IsLocalization W₂", "inst✝ : Φ.HasLeftResolutions", "X₂ : D₂", "this : L₂.EssSurj"], "goal": "X₂ ∈ (Φ.functor ⋙ L₂).essImage"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : AddGroup R", "inst✝² : Fintype R", "R' : Type u_2", "inst✝¹ : CommRing R'", "inst✝ : IsDomain R'", "ψ : AddChar R R'", "hψ : ψ ≠ 1", "b : R", "hb : ψ b ≠ 1", "h₁ : ∑ a : R, ψ (b + a) = ∑ a : R, ψ a"], "goal": "∑ a : R, ψ a = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Finset α", "t : α → Finset β", "DecidableEq β"], "goal": "(s.biUnion t).Nonempty ↔ ∃ x ∈ s, (t x).Nonempty"}}
+{"state": {"context": ["E : Type u_1", "ι : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "U K : Set ℂ", "z : ℂ", "M r δ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "hr : 0 < r", "hf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M", "hM : 0 ≤ M", "h1 : ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2"], "goal": "‖cderiv r f z‖ ≤ M / r"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "n m : ℕ", "inst✝ : Semiring R", "p q r : R[X]", "h : p * q ≠ 0", "hp : p ≠ 0", "hq : q ≠ 0"], "goal": "p.natTrailingDegree + q.natTrailingDegree ≤ (p * q).natTrailingDegree"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s : LowerSet α"], "goal": "↑s = Set.univ ↔ s = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq β", "f g : α → β", "s : Finset α", "t : Finset β", "a : α", "b c : β", "hf : Injective f"], "goal": "f a ∈ image f s ↔ a ∈ s"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁸ : TopologicalSpace X", "inst✝⁷ : Zero X", "A : Type u_2", "inst✝⁶ : TopologicalSpace A", "inst✝⁵ : NonUnitalRing A", "inst✝⁴ : StarRing A", "inst✝³ : Module ℝ A", "inst✝² : IsScalarTower ℝ A A", "inst✝¹ : SMulCommClass ℝ A A", "inst✝ : TopologicalRing A", "φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A", "f : C(X, ℝ)₀"], "goal": "{ toFun := fun f => φ f.toNNReal - φ (-f).toNNReal, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯ }.toFun (star f) = star ({ toFun := fun f => φ f.toNNReal - φ (-f).toNNReal, map_smul' := ⋯, map_zero' := ⋯, map_add' := ⋯, map_mul' := ⋯ }.toFun f)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝¹ : PseudoEMetricSpace α", "x✝² y z : α", "ε✝ ε₁ ε₂ : ℝ≥0∞", "s t✝ : Set α", "inst✝ : PseudoEMetricSpace β", "f : α → β", "t : Set β", "a : α", "b : β", "ε : ℝ≥0∞", "x✝¹ : 0 < ε", "δ : ℝ≥0∞", "x✝ : 0 < δ", "x : α"], "goal": "edist x a < δ → x ∈ s → edist (f x) b < ε ∧ f x ∈ t ↔ x ∈ s → edist x a < δ → f x ∈ t ∧ edist (f x) b < ε"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "E : Type u_3", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedRing 𝕜", "NormedAddCommGroup E", "MulActionWithZero 𝕜 E", "BoundedSMul 𝕜 E", "p q r : ℝ≥0∞", "f : α → E", "φ : α → 𝕜", "hf : MeasureTheory.Memℒp f r μ", "hφ : MeasureTheory.Memℒp φ q μ", "hpqr : 1 / p = 1 / q + 1 / r"], "goal": "MeasureTheory.Memℒp (φ • f) p μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f g : α → ℝ≥0∞", "hμ : ¬μ univ = 0", "hg : AEMeasurable g μ", "hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "h : ∀ᵐ (x : α) ∂μ, f x < g x"], "goal": "∫⁻ (x : α), f x ∂μ < ∫⁻ (x : α), g x ∂μ"}}
+{"state": {"context": ["R₁ : Type u_1", "A : Type u_3", "B : Type u_4", "CommSemiring R₁", "CommRing A", "Algebra R₁ A", "Semiring B", "Algebra R₁ B", "I : Ideal A", "f : A →ₐ[R₁] B", "hI : ∀ a ∈ I, f a = 0"], "goal": "(Ideal.Quotient.liftₐ I f hI).comp (Ideal.Quotient.mkₐ R₁ I) = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace α", "inst✝¹ : NormedAddCommGroup E", "f✝ g : α → E", "s✝ t : Set α", "μ✝ ν : Measure α", "inst✝ : MeasurableSpace β", "e : α → β", "he : MeasurableEmbedding e", "f : β → E", "μ : Measure β", "s : Set β", "hs : s ⊆ range e"], "goal": "IntegrableOn f s μ ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) (Measure.comap e μ)"}}
+{"state": {"context": ["a✝ b✝ c d m n k : ℕ", "p q : ℕ → Prop", "a b : ℕ"], "goal": "a % b + a / b * b = a"}}
+{"state": {"context": ["ι : Type u_1", "S : Type u_3", "R : Type u_4", "DecidableEq ι", "AddMonoid ι", "CommSemiring S", "Semiring R", "Algebra S R", "A : ι → Submodule S R", "SetLike.GradedMonoid A", "s : S"], "goal": "↑(DirectSum.GAlgebra.toFun s) = (algebraMap S R) s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "G : Type u_4", "M : Type u_5", "N : Type u_6", "inst✝² : CommMonoid M", "inst✝¹ : CommMonoid N", "f✝ g✝ : α → M", "a b✝ : α", "s✝ t : Set α", "inst✝ : DecidableEq β", "s : Finset α", "f : α → M", "g : α → β", "b : β", "h : b ∈ mulSupport fun b => (filter (fun a => g a = b) s).prod f"], "goal": "(filter (fun a => g a = b) s).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "p : ℝ≥0∞", "q : ℝ", "μ : Measure α", "f✝ g✝ f g : α → E", "hf : AEStronglyMeasurable f μ", "hg : AEStronglyMeasurable g μ", "hp : 1 ≤ p"], "goal": "eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ p₄ : P", "h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄"], "goal": "EuclideanGeometry.Concyclic {p₁, p₂, p₃, p₄} ∨ Collinear ℝ {p₁, p₂, p₃, p₄}"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "HasDerivedCategory C", "S : CategoryTheory.ShortComplex C", "hS : S.ShortExact"], "goal": "hS.singleTriangleIso.inv.hom₃ = ((DerivedCategory.singleFunctorsPostcompQIso C).inv.hom 0).app S.X₃"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "TopologicalAddGroup G", "K₀ : TopologicalSpace.PositiveCompacts G"], "goal": "0 < (MeasureTheory.Measure.haar.addHaarContent K₀).outerMeasure (closure ↑K₀)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "N : Type u_5", "AddGroup N", "H : AddSubgroup G", "K : AddSubgroup N"], "goal": "↑(H.prod K) = ↑H ×ˢ ↑K"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "G : Type u_4", "M' : Type u_5", "inst✝⁸ : NormedField R", "inst✝⁷ : SeminormedAddCommGroup M", "inst✝⁶ : SeminormedAddCommGroup N", "inst✝⁵ : SeminormedAddCommGroup G", "inst✝⁴ : NormedSpace R M", "inst✝³ : NormedSpace R N", "inst✝² : NormedSpace R G", "inst✝¹ : NormedAddCommGroup M'", "inst✝ : NormedSpace R M'", "c cf : R", "hcf : cf ≠ 0", "lif : M →ₗᵢ[R] N", "cg : R", "hcg : cg ≠ 0", "lig : N →ₗᵢ[R] G"], "goal": "(cg • lig.toContinuousLinearMap).comp (cf • lif.toContinuousLinearMap) = (cg * cf) • (lig.comp lif).toContinuousLinearMap"}}
+{"state": {"context": ["α : Type u_1", "M N : Matroid α", "h : Disjoint M.E N.E", "B : Set α"], "goal": "(M.disjointSum N h).Base B ↔ M.Base (B ∩ M.E) ∧ N.Base (B ∩ N.E) ∧ B ⊆ M.E ∪ N.E"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "l : List (Sigma β)"], "goal": "List.lookupAll a l = [] ↔ ∀ (b : β a), ⟨a, b⟩ ∉ l"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C"], "goal": "(CategoryTheory.ShortComplex.Hom.id S).τ₁ = CategoryTheory.CategoryStruct.id S.X₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "s : Set α", "p : α → Prop", "f g : α → β", "(a : α) → Decidable (p a)", "hpf : ∀ a ∈ s ∩ frontier {a | p a}, Filter.Tendsto f (𝓝[s ∩ {a | p a}] a) (𝓝 (if p a then f a else g a))", "hpg : ∀ a ∈ s ∩ frontier {a | p a}, Filter.Tendsto g (𝓝[s ∩ {a | ¬p a}] a) (𝓝 (if p a then f a else g a))", "hf : ContinuousOn f (s ∩ {a | p a})", "hg : ContinuousOn g (s ∩ {a | ¬p a})"], "goal": "ContinuousOn (fun a => if p a then f a else g a) s"}}
+{"state": {"context": ["ι : Type u_3", "α : ι → Type u_1", "β : ι → Type u_2"], "goal": "∀ (a : (i : ι) × α i ⊕ (i : ι) × β i),\n  (Equiv.sigmaSumDistrib α β).symm a = Sum.elim (Sigma.map id fun x => Sum.inl) (Sigma.map id fun x => Sum.inr) a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Seq α", "n : ℕ"], "goal": "((↑s).drop n).head = Computation.pure (s.get? n)"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_3", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : TopologicalSpace B", "b : B", "v : F"], "goal": "(Trivialization.coordChangeL 𝕜 (trivialization B F) (trivialization B F) b) v = (ContinuousLinearEquiv.refl 𝕜 F) v"}}
+{"state": {"context": [], "goal": "↑ArithmeticFunction.zeta * ArithmeticFunction.vonMangoldt = ArithmeticFunction.log"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "Z : D", "F : CategoryTheory.Functor C D", "hZ : CategoryTheory.Limits.IsInitial Z", "X Y : CategoryTheory.WithInitial C", "f : X ⟶ Y"], "goal": "(CategoryTheory.WithInitial.liftToInitial F hZ).map f =\n  match X, Y, f with\n  | CategoryTheory.WithInitial.of x, CategoryTheory.WithInitial.of y, f => F.map (CategoryTheory.WithInitial.down f)\n  | CategoryTheory.WithInitial.star, CategoryTheory.WithInitial.of x, x_1 => hZ.to (F.obj x)\n  | CategoryTheory.WithInitial.star, CategoryTheory.WithInitial.star, x => CategoryTheory.CategoryStruct.id Z"}}
+{"state": {"context": ["α : Type u", "n' : Type uₙ", "NonUnitalNonAssocSemiring α", "v : Fin 0 → α", "w : n' → α"], "goal": "Matrix.vecMulVec v w = ![]"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "IsMax a → Order.IsPredLimit a"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "x : α"], "goal": "EMetric.ball x ⊤ = Set.univ"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f f' : 𝕜 → F", "h : ∀ (x : 𝕜), HasDerivAt f (f' x) x"], "goal": "deriv f = f'"}}
+{"state": {"context": ["s : ℂ", "hs : ∀ (m : ℕ), s ≠ -↑m", "n : ℕ := ⌊1 - s.re⌋₊ + 1", "hn : 1 - s.re < ↑n"], "goal": "DifferentiableAt ℂ Gamma s"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "S : Set E", "α : E", "p : F[X]", "K : IntermediateField F E", "hp : Splits (algebraMap F ↥K) p"], "goal": "Algebra.adjoin F (p.rootSet E) = K.val.range ↔ K.toSubalgebra = Algebra.adjoin F (p.rootSet E)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "AddCommMonoid β", "s : Finset α", "f : α → β", "a : α", "hp : ∑ x ∈ s, f x = 0", "h1 : ∀ x ∈ s, x ≠ a → f x = 0", "x : α"], "goal": "x ∈ s → f x = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι✝ → Sort u_7", "κ₁ : ι✝ → Sort u_8", "κ₂ : ι✝ → Sort u_9", "κ' : ι' → Sort u_10", "s t u : Set α", "f : α → β", "ι : Type u_11", "Es : ι → Set α", "Es_union : ⋃ i, Es i = univ", "Es_disj : Pairwise fun i j => Disjoint (Es i) (Es j)", "I : Set ι", "x : α", "i : ι", "hix : x ∈ Es i", "obs : ∀ (J : Set ι), x ∈ ⋃ j ∈ J, Es j ↔ i ∈ J", "obs' : ∀ (J : Set ι), x ∈ (⋃ j ∈ J, Es j)ᶜ ↔ i ∉ J"], "goal": "x ∈ (⋃ i ∈ I, Es i)ᶜ ↔ x ∈ ⋃ i ∈ Iᶜ, Es i"}}
+{"state": {"context": ["α : Type u_1", "l₁ l₂ l₃ : List α"], "goal": "l₂ <:+: l₁ ++ (l₂ ++ l₃)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v w : V", "p : G.Walk u v", "h : G.Adj v w"], "goal": "p.concat h = p.append (SimpleGraph.Walk.cons h SimpleGraph.Walk.nil)"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "k : Type u_1", "inst✝² : Field k", "inst✝¹ : CharP k p", "inst✝ : IsAlgClosed k", "a₁ a₂ : 𝕎 k", "ha₁ : a₁.coeff 0 ≠ 0", "ha₂ : a₂.coeff 0 ≠ 0", "h : frobeniusRotation p ha₁ ha₂ = 0"], "goal": "False"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁸ : Category.{v₁, u₁} C", "inst✝⁷ : MonoidalCategory C", "inst✝⁶ : BraidedCategory C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : MonoidalCategory D", "inst✝³ : BraidedCategory D", "E : Type u₃", "inst✝² : Category.{v₃, u₃} E", "inst✝¹ : MonoidalCategory E", "inst✝ : BraidedCategory E", "X₁ X₂ : C", "this : (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv ≫ (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv = (α_ X₁ X₂ (𝟙_ C)).hom ≫ X₁ ◁ (ρ_ X₂).inv ▷ 𝟙_ C"], "goal": "(ρ_ (X₁ ⊗ X₂)).hom = (X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv ≫ ((α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv ≫ X₁ ◁ (β_ X₂ (𝟙_ C)).hom ▷ 𝟙_ C ≫ X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).hom ≫ (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).inv) ≫ ((ρ_ X₁).hom ⊗ (ρ_ X₂).hom)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "P : G.Partition", "v : V"], "goal": "v ∈ P.partOfVertex v"}}
+{"state": {"context": ["α : Type u", "s : Set (Set α)", "i : (Quotient.out (aleph 1).ord).α"], "goal": "∅ ∈ let i := i; let S := ⋃ j, generateMeasurableRec s ↑j; s ∪ {∅} ∪ compl '' S ∪ range fun f => ⋃ n, ↑(f n)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "x : M"], "goal": "UniqueMDiffWithinAt I Set.univ x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "E'' : Type u_8", "NormedAddCommGroup E''", "NormedSpace 𝕜 E''", "H'' : Type u_9", "TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "TopologicalSpace M''", "ChartedSpace H'' M''", "f : M → M'", "s : Set M", "n : ℕ∞", "t : Set M'", "g : M' → M''", "x : M", "hg : ContMDiffWithinAt I' I'' n g t (f x)", "hf : ContMDiffWithinAt I I' n f s x", "st : Set.MapsTo f s t"], "goal": "ContMDiffWithinAt I I'' n (g ∘ f) s x"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M M' : Bimod W X", "f : M ⟶ M'", "N : Bimod X Y", "P : Bimod Y Z"], "goal": "M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ f.hom ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ colimit.ι (parallelPair (M'.actRight ▷ coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)) ((α_ M'.X X.X (coequalizer (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft))).hom ≫ M'.X ◁ TensorBimod.actLeft N P)) WalkingParallelPair.one = M.X ◁ coequalizer.π (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft) ≫ (((PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ▷ P.X) ((α_ N.X Y.X P.X).hom ≫ N.X ◁ P.actLeft)).inv ≫ coequalizer.desc ((α_ M.X N.X P.X).inv ≫ coequalizer.π (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ▷ P.X ≫ coequalizer.π (TensorBimod.actRight M N ▷ P.X) ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ TensorBimod.X M N ◁ P.actLeft)) ⋯) ≫ colimMap (parallelPairHom (TensorBimod.actRight M N ▷ P.X) ((α_ (coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft)) Y.X P.X).hom ≫ coequalizer (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) ◁ P.actLeft) (TensorBimod.actRight M' N ▷ P.X) ((α_ (coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft)) Y.X P.X).hom ≫ coequalizer (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) ◁ P.actLeft) (colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷ Y.X ▷ P.X) (colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X X.X N.X).hom ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X) ((α_ M'.X X.X N.X).hom ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷ P.X) ⋯ ⋯)) ≫ coequalizer.desc (AssociatorBimod.homAux M' N P) ⋯"}}
+{"state": {"context": ["A : Type u_1", "M : Type u_3", "AddMonoid A", "Monoid M", "ψ : AddChar A M"], "goal": "ψ ≠ 1 ↔ ∃ x, ψ x ≠ 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "E'' : Type u_8", "NormedAddCommGroup E''", "NormedSpace 𝕜 E''", "H'' : Type u_9", "TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "TopologicalSpace M''", "ChartedSpace H'' M''", "f : M → M'", "s : Set M", "t : Set M'", "g : M' → M''", "x : M", "hg : SmoothWithinAt I' I'' g t (f x)", "hf : SmoothWithinAt I I' f s x"], "goal": "SmoothWithinAt I I'' (g ∘ f) (s ∩ f ⁻¹' t) x"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "AddCommMonoid ι", "DecidableEq ι", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ι → Submodule R A", "GradedAlgebra 𝒜", "x : Submonoid A"], "goal": "HomogeneousLocalization.mk 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommSemiring R", "p : ℕ", "x : R", "hp : ¬1 < p"], "goal": "x ∈ pNilradical R p ↔ ∃ n, x ^ p ^ n = 0"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝⁴ : FunLike F α β", "inst✝³ : Bot α", "inst✝² : Bot β", "inst✝¹ : Bot γ", "inst✝ : Bot δ", "g : BotHom β γ", "f₁ f₂ : BotHom α β", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "a : α"], "goal": "g (f₁ a) = g (f₂ a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "UniformSpace α", "AddGroup α", "UniformAddGroup α", "ι : Type u_3", "l : Filter ι", "f f' : ι → β → α", "g g' : β → α", "hf : TendstoUniformly f g l", "hf' : TendstoUniformly f' g' l"], "goal": "TendstoUniformly (f + f') (g + g') l"}}
+{"state": {"context": ["J : Type v'", "inst✝⁷ : Category.{u', v'} J", "C : Type u", "inst✝⁶ : Category.{v, u} C", "K : Type u_1", "inst✝⁵ : Category.{?u.162150, u_1} K", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_2} D", "Gl : C ⥤ D", "Gr : D ⥤ C", "adj : Gl ⊣ Gr", "inst✝³ : Gr.Full", "inst✝² : Gr.Faithful", "F : J ⥤ D", "c : Cocone (F ⋙ Gr)", "H : IsUniversalColimit c", "inst✝¹ : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)", "inst✝ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl", "this✝¹ : PreservesLimitsOfSize.{?u.164955, ?u.164954, u_3, v, u_2, u} Gr", "this✝ : PreservesColimitsOfSize.{u', v', v, u_3, u, u_2} Gl", "F' : J ⥤ D", "c' : Cocone F'", "α : F' ⟶ (F ⋙ Gr) ⋙ Gl", "f : c'.pt ⟶ (Gl.mapCocone c).pt", "h : α ≫ (Gl.mapCocone c).ι = c'.ι ≫ (Functor.const J).map f", "hα : NatTrans.Equifibered α", "hc' : ∀ (j : J), IsPullback (c'.ι.app j) (α.app j) f ((Gl.mapCocone c).ι.app j)", "this : HasPullback (Gl.map (Gr.map f)) (Gl.map (adj.unit.app c.pt))", "α' : F' ⟶ F := α ≫ (F.associator Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom", "hα' : NatTrans.Equifibered α'", "hadj : ∀ (X : C), Gl.map (adj.unit.app X) = inv (adj.counit.app (Gl.toPrefunctor.1 ((𝟭 C).obj X)))"], "goal": "Nonempty (IsColimit c')"}}
+{"state": {"context": ["b : Ordinal.{u_4}"], "goal": "0 % b = 0"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "E : Type u_3", "inst✝⁷ : TopologicalSpace X", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : Module ℝ E", "inst✝⁴ : NormalSpace X", "inst✝³ : ParacompactSpace X", "inst✝² : TopologicalSpace E", "inst✝¹ : ContinuousAdd E", "inst✝ : ContinuousSMul ℝ E", "t : X → Set E", "ht : ∀ (x : X), Convex ℝ (t x)", "U : X → Set X", "hU : ∀ (x : X), U x ∈ 𝓝 x", "g : X → X → E", "hgc : ∀ (x : X), ContinuousOn (g x) (U x)", "hgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y", "f : PartitionOfUnity X X", "hf : f.IsSubordinate fun x => interior (U x)"], "goal": "∃ g, ∀ (x : X), g x ∈ t x"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing S", "I : Ideal R", "inst✝¹ : Algebra R S", "inst✝ : Algebra.IsIntegral R S", "hR : IsJacobson R"], "goal": "IsJacobson S"}}
+{"state": {"context": ["α : Type u", "Preorder α", "s : Set α", "a : α", "h : IsLeast s a"], "goal": "IsGLB s a"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b c : B", "f : a ⟶ b", "g : b ⟶ c"], "goal": "(CategoryTheory.Bicategory.rightUnitor (CategoryTheory.CategoryStruct.comp f g)).inv =\n  CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.Bicategory.whiskerLeft f (CategoryTheory.Bicategory.rightUnitor g).inv)\n    (CategoryTheory.Bicategory.associator f g (CategoryTheory.CategoryStruct.id c)).inv"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : Filter α", "h : μ.FiniteAtFilter f"], "goal": "∀ᶠ (s : Set α) in f.smallSets, μ s < ⊤"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u_3", "B : Type u_4", "inst✝¹³ : CommRing A", "inst✝¹² : CommRing B", "inst✝¹¹ : Algebra A B", "inst✝¹⁰ : Field K", "inst✝⁹ : Field L", "inst✝⁸ : Algebra A K", "inst✝⁷ : IsFractionRing A K", "inst✝⁶ : Algebra B L", "inst✝⁵ : Algebra K L", "inst✝⁴ : Algebra A L", "inst✝³ : IsScalarTower A B L", "inst✝² : IsScalarTower A K L", "inst✝¹ : IsIntegralClosure B A L", "inst✝ : FiniteDimensional K L", "σ : B →ₐ[A] B", "x : B"], "goal": "((galRestrictHom A K L B).symm σ) ((algebraMap B L) x) = (algebraMap B L) (σ x)"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeSup α", "s t : Finset α", "a : α", "DecidableEq α"], "goal": "a ∈ lowerClosure ↑(s ∪ t) ↔ a ∈ lowerClosure ↑s ∨ a ∈ lowerClosure ↑t"}}
+{"state": {"context": ["n : ℕ", "h0 : n ≠ 0"], "goal": "IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁴ : TopologicalSpace X", "inst✝³ : MeasurableSpace X", "inst✝² : OpensMeasurableSpace X", "ι : Type u_2", "L : Filter ι", "μ : Measure X", "inst✝¹ : IsProbabilityMeasure μ", "μs : ι → Measure X", "inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)", "h : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i => ∫ (x : X), f x ∂μs i) L", "f : X →ᵇ ℝ", "hL : L.NeBot", "obs : Bornology.IsBounded (Set.range fun i => ∫ (x : X), f x ∂μs i)", "bdd_above : IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i", "bdd_below : IsBoundedUnder (fun x x_1 => x ≥ x_1) L fun i => ∫ (x : X), f x ∂μs i"], "goal": "autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) L fun i => ∫ (x : X), f x ∂μs i) _auto✝"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "h5 : 5 ≤ card α", "f : Perm α", "hf : f.IsThreeCycle", "hi : Function.Injective ⇑(alternatingGroup α).subtype"], "goal": "alternatingGroup α ≤ closure (⇑(alternatingGroup α).subtype '' Group.conjugatesOfSet {⟨f, ⋯⟩})"}}
+{"state": {"context": ["G : Type u_1", "SubNegMonoid G", "a b : G"], "goal": "a - b = a + -b"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P", "h : ∡ p₁ p₂ p₃ = ↑(π / 2)"], "goal": "∡ p₂ p₃ p₁ = ↑(Real.arcsin (dist p₁ p₂ / dist p₁ p₃))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G₀ : Type u_3", "inst✝² : TopologicalSpace G₀", "inst✝¹ : GroupWithZero G₀", "inst✝ : ContinuousMul G₀", "a : G₀", "ha : a ≠ 0"], "goal": "map (fun x => x * a) (𝓝 1) = 𝓝 a"}}
+{"state": {"context": ["K : Type u_1", "Field K", "Fintype K"], "goal": "↑q = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s✝ : Multiset α", "a : α", "s t : Multiset α"], "goal": "s.Disjoint (ndinsert a t) ↔ a ∉ s ∧ s.Disjoint t"}}
+{"state": {"context": ["x : ℝ*"], "goal": "x.Infinite → ∀ (r : ℝ), x ≠ ↑r"}}
+{"state": {"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "TotalComplexShape c₁ c₂ c₁₂", "i₁ i₁' : I₁", "i₂ i₂' : I₂", "h₁ : c₁.Rel i₁ i₁'", "h₂ : c₂.Rel i₂ i₂'"], "goal": "c₁.ε₂ c₂ c₁₂ (i₁, i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂') = -c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂)"}}
+{"state": {"context": ["k : Type u_1", "inst✝² : Field k", "σ : Type u_2", "inst✝¹ : IsAlgClosed k", "inst✝ : Finite σ", "I : Ideal (MvPolynomial σ k)", "p : MvPolynomial σ k", "hp : p ∈ I.jacobson", "x : σ → k", "hx : x ∈ zeroLocus I"], "goal": "(eval x) p = 0"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "MulOneClass M", "MulOneClass N", "f : M →* N"], "goal": "f 1 = 1"}}
+{"state": {"context": ["α✝ : Type u_1", "r✝ : α✝ → α✝ → Prop", "α : Type u_2", "h : ∀ x < #α, 2 ^ x < #α", "ha : #α ≠ 0", "h' : (#α).IsStrongLimit", "r : α → α → Prop", "wo : IsWellOrder α r", "hr : (#α).ord = type r"], "goal": "#{ s // #↑s < (#α).ord.cof } = #α"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "OrderedSemiring 𝕜", "AddCommMonoid E", "AddCommMonoid F", "Module 𝕜 E", "Module 𝕜 F", "x : E", "s : Set E", "hs : StarConvex 𝕜 x s", "f : E →ₗ[𝕜] F"], "goal": "StarConvex 𝕜 (f x) (⇑f '' s)"}}
+{"state": {"context": ["α : Type u", "a : α"], "goal": "Stream'.corec id id a = Stream'.const a"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "X Y : C", "f : X ⟶ Y", "inst✝¹ : HasKernel f", "X' Y' : C", "f' : X' ⟶ Y'", "inst✝ : HasKernel f'", "p : X ≅ X'", "q : Y ≅ Y'", "w : f ≫ q.hom = p.hom ≫ f'"], "goal": "f' ≫ q.inv = p.inv ≫ f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "W : Type u_4", "ι : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace W", "f : X → Y", "g : Y → Z", "hf : IsProperMap f", "hg : IsProperMap g", "ℱ : Filter X", "z : Z", "h : MapClusterPt z ℱ (g ∘ f)"], "goal": "∃ x, (g ∘ f) x = z ∧ ClusterPt x ℱ"}}
+{"state": {"context": ["α β : Sort u", "a : α", "b : β", "h : HEq a b"], "goal": "HEq b a"}}
+{"state": {"context": ["α : Type u_1", "ms : MeasurableSpace α", "μ : MeasureTheory.Measure α"], "goal": "ms ≤ μ.caratheodory"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "StarRing R", "TrivialStar R", "AddCommMonoid A", "StarAddMonoid A", "Module R A", "StarModule R A", "TopologicalSpace A", "ContinuousStar A"], "goal": "∀ (a : A), (starL' R) a = star a"}}
+{"state": {"context": ["N k n : ℕ"], "goal": "n ∈ N.smoothNumbersUpTo k ↔ n ≤ N ∧ n ∈ k.smoothNumbers"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "A : Type u_3", "inst✝¹⁶ : Category.{u_4, u_1} C", "inst✝¹⁵ : HasZeroObject C", "inst✝¹⁴ : HasShift C ℤ", "inst✝¹³ : Preadditive C", "inst✝¹² : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝¹¹ : Pretriangulated C", "inst✝¹⁰ : Category.{?u.108053, u_2} D", "inst✝⁹ : HasZeroObject D", "inst✝⁸ : HasShift D ℤ", "inst✝⁷ : Preadditive D", "inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive", "inst✝⁵ : Pretriangulated D", "inst✝⁴ : Category.{u_5, u_3} A", "inst✝³ : Abelian A", "F : C ⥤ A", "inst✝² inst✝¹ : F.IsHomological", "inst✝ : F.ShiftSequence ℤ", "T T' : Triangle C", "hT : T ∈ distinguishedTriangles", "hT' : T' ∈ distinguishedTriangles", "φ : T ⟶ T'", "n₀ n₁ : ℤ", "h : n₀ + 1 = n₁", "X Y : C", "f : X ⟶ Y"], "goal": "F.homologicalKernel.W f ↔ ∀ (n : ℤ), IsIso ((F.shift n).map f)"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "α : ↥(solvableByRad F E)", "n : ℕ", "hn : n ≠ 0", "hα : solvableByRad.P (α ^ n)"], "goal": "solvableByRad.P α"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasWideEqualizers C", "T : C", "hT : ∀ (X : C), Nonempty (T ⟶ X)", "endos : Type v := T ⟶ T", "i : wideEqualizer id ⟶ T := wideEqualizer.ι id", "this : Nonempty endos"], "goal": "HasInitial C"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝⁷ : Countable ι", "f✝ g✝ : α✝ → β", "mα : MeasurableSpace α✝", "inst✝⁶ : TopologicalSpace β", "α : Type u_5", "E : Type u_6", "x✝ : MeasurableSpace α", "inst✝⁵ : AddGroup E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : ContinuousAdd E", "inst✝ : PseudoMetrizableSpace E", "g f : α → E", "hg : Measurable g", "φ : ℕ → α →ₛ E", "hφ : ∀ (x : α), Tendsto (fun n => ↑(φ n) x) atTop (𝓝 (f x))", "this : Tendsto (fun n x => ↑(φ n) x + g x) atTop (𝓝 (f + g))"], "goal": "Measurable (f + g)"}}
+{"state": {"context": ["m n : Nat"], "goal": "Int.negSucc m + ↑n = Int.subNatNat n m.succ"}}
+{"state": {"context": ["α₁ : Type u_6", "α₂ : Type u_7", "β₁ : Type u_8", "β₂ : Type u_9", "t₁ : Set β₁", "t₂ : Set β₂", "f₁ : α₁ → β₁", "f₂ : α₂ → β₂", "g₁ : β₁ → α₁", "g₂ : β₂ → α₂", "h₁ : Set.RightInvOn g₁ f₁ t₁", "h₂ : Set.RightInvOn g₂ f₂ t₂"], "goal": "Set.RightInvOn (fun x => (g₁ x.1, g₂ x.2)) (fun x => (f₁ x.1, f₂ x.2)) (t₁ ×ˢ t₂)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C"], "goal": "CategoryTheory.ShortComplex.Hom.τ₃ 0 = 0"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "X Y : C", "inst✝ : HasRightDual X", "f : Y ⟶ Xᘁ"], "goal": "(ρ_ Y).inv ≫ Y ◁ η_ X Xᘁ ≫ (α_ Y X Xᘁ).inv ≫ (f ▷ X ≫ ε_ X Xᘁ) ▷ Xᘁ = 𝟙 Y ⊗≫ (Y ◁ η_ X Xᘁ ≫ f ▷ (X ⊗ Xᘁ)) ⊗≫ ε_ X Xᘁ ▷ Xᘁ"}}
+{"state": {"context": ["R : Type u_3", "CommRing R"], "goal": "Quaternion.re 1 = 1"}}
+{"state": {"context": ["k : Type u_1", "E : Type u_2", "LinearOrderedField k", "OrderedAddCommGroup E", "Module k E", "OrderedSMul k E", "a b : E"], "goal": "a ≤ midpoint k a b ↔ a ≤ b"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : C"], "goal": "(CategoryTheory.MonoidalFunctor.diag C).μ X Y =\n  𝟙\n    (CategoryTheory.MonoidalCategory.tensorObj ((CategoryTheory.Functor.diag C).obj X)\n      ((CategoryTheory.Functor.diag C).obj Y))"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : LinearOrderedRing α", "inst✝² : FloorRing α", "z : ℤ", "a : α", "k : Type u_4", "inst✝¹ : LinearOrderedField k", "inst✝ : FloorRing k", "b : k", "m n : ℕ", "hn : n > 0", "hn' : 0 < ↑n"], "goal": "0 ≤ ↑(m % n) / ↑n"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "α : Type u_12", "DecidableEq l", "DecidableEq m", "Zero α", "v : l ⊕ m → α"], "goal": "(Matrix.diagonal v).toBlocks₁₂ = 0"}}
+{"state": {"context": ["G : Type u_1", "Group G", "N : Type u_5", "Group N", "f : G →* N", "g : N →* G", "hl : Function.LeftInverse ⇑g ⇑f", "hr : Function.RightInverse ⇑g ⇑f", "H : Subgroup G"], "goal": "Subgroup.map f H = Subgroup.comap g H"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β → Option γ", "inst✝¹ : BEq α", "inst✝ : Hashable α", "m : Imp α β", "H : m.WF", "g₁ : AssocList α β → List (α × γ) := fun l => List.filterMap (fun x => Option.map (fun x_1 => (x.fst, x_1)) (f x.fst x.snd)) l.toList", "H1 : ∀ (l : AssocList α β) (n : ULift Nat) (acc : AssocList α γ), filterMap.go f acc l n = (((g₁ l).reverse ++ acc.toList).toAssocList, { down := n.down + (g₁ l).length })", "g : AssocList α β → AssocList α γ := fun l => (g₁ l).reverse.toAssocList", "M : Type (max u_3 u_1) → Type (max u_3 u_1) := StateT (ULift Nat) Id", "H2 : ∀ (l : List (AssocList α β)) (n : ULift Nat), List.mapM (filterMap.go f AssocList.nil) l n = (List.map g l, { down := n.down + Nat.sum (List.map (fun x => x.toList.length) (List.map g l)) })", "H3 : ∀ (l : List (α × β)), (List.map (fun a => a.fst) (List.filterMap (fun x => match x with | (a, b) => Option.map (fun x => (a, x)) (f a b)) l)).Sublist (List.map (fun x => x.fst) l)"], "goal": "∀ (bk : List (AssocList α γ)) (sz : Nat) (h : 0 < bk.length), pure { data := List.map (fun l => (g₁ l).reverse.toAssocList) m.buckets.val.data } { down := Nat.sum (List.map ((fun x => x.toList.length) ∘ fun l => (g₁ l).reverse.toAssocList) m.buckets.val.data) } = ({ data := bk }, { down := sz }) → { size := sz, buckets := ⟨{ data := bk }, h⟩ }.WF"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N P Q : Bimod Y Z", "f : N ⟶ P", "g : P ⟶ Q"], "goal": "(M.whiskerLeft (f ≫ g)).hom = (M.whiskerLeft f ≫ M.whiskerLeft g).hom"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "CategoryTheory.Abelian C", "CategoryTheory.HasInjectiveResolutions C", "CategoryTheory.Abelian D", "F : CategoryTheory.Functor C D", "F.Additive", "CategoryTheory.Limits.PreservesFiniteLimits F"], "goal": "F.rightDerivedZeroIsoSelf.inv = F.toRightDerivedZero"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : Module ℝ E", "s✝ t : Set E", "x✝ : E", "a✝ : ℝ", "α : Type u_4", "inst✝⁴ : LinearOrderedField α", "inst✝³ : MulActionWithZero α ℝ", "inst✝² : OrderedSMul α ℝ", "inst✝¹ : MulActionWithZero α E", "inst✝ : IsScalarTower α ℝ (Set E)", "s : Set E", "a : α", "ha : 0 ≤ a", "x : E", "ha' : 0 < a"], "goal": "gauge s (a • x) = a • gauge s x"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "ℰ : Type u₂", "inst✝³ : Category.{v₁, u₂} ℰ", "A✝ A : C ⥤ ℰ", "inst✝² : yoneda.HasPointwiseLeftKanExtension A", "D : Type u₂", "inst✝¹ : Category.{v₁, u₂} D", "F : C ⥤ D", "inst✝ : ∀ (P : Cᵒᵖ ⥤ Type v₁), F.op.HasLeftKanExtension P", "X✝ : C", "G : (Cᵒᵖ ⥤ Type v₁) ⥤ Dᵒᵖ ⥤ Type v₁", "φ : F ⋙ yoneda ⟶ yoneda ⋙ G", "P : Cᵒᵖ ⥤ Type v₁", "X : C", "x : P.obj (Opposite.op X)"], "goal": "yonedaEquiv (yonedaEquiv.symm x ≫ presheafHom φ P) = (G.map (yonedaEquiv.symm x)).app (Opposite.op (F.obj X)) ((φ.app X).app (Opposite.op (F.obj X)) (𝟙 (Opposite.unop (Opposite.op (F.obj X)))))"}}
+{"state": {"context": ["n : ℕ", "h : ↑(n - 1)! = -1", "h0 : n ≠ 0", "h1 : 1 < n", "h2 : ¬Prime n", "m : ℕ", "hm1 : m ∣ n", "hm2 : 1 < m", "hm3 : m < n", "hm : m ∣ (n - 1)!"], "goal": "n ∣ (n - 1)! + 1"}}
+{"state": {"context": ["α : Type u_1", "AddZeroClass α", "Preorder α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a b : α", "ha : 0 ≤ a", "hb : 0 < b"], "goal": "0 < a + b"}}
+{"state": {"context": ["x : ZFSet", "h : ∅ ∉ x", "y : ZFSet", "yx : y ∈ x"], "goal": "↑x.choice ′ ↑y ∈ ↑y"}}
+{"state": {"context": ["M : Type u_1", "CommMonoidWithZero M", "S : Submonoid M", "x : ↥S"], "goal": "Localization.mk 0 x = 0"}}
+{"state": {"context": ["w n : Nat", "x : BitVec w"], "goal": "x.shiftLeftZeroExtend n = BitVec.zeroExtend (w + n) x <<< n"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "n : ℕ"], "goal": "↑n < p.degree ↔ n < p.natDegree"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : LocallyFiniteOrder α", "a b c : α", "inst✝ : DecidableEq α", "h : a < b"], "goal": "insert a (Ioo a b) = Ico a b"}}
+{"state": {"context": ["n : ℕ"], "goal": "harmonic (n + 1) = harmonic n + (↑(n + 1))⁻¹"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Mul α", "Mul β", "e : α ≃* β", "a : WithOne α"], "goal": "e.withOneCongr a = (WithOne.map e.toMulHom) a"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "NonAssocSemiring R", "NonAssocSemiring A", "f : R →+* A", "h : Function.Injective ⇑f", "CharZero R"], "goal": "CharZero A"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n : ℕ", "M : Fin n.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝¹¹ : Semiring R", "inst✝¹⁰ : (i : Fin n.succ) → AddCommMonoid (M i)", "inst✝⁹ : (i : ι) → AddCommMonoid (M₁ i)", "inst✝⁸ : AddCommMonoid M₂", "inst✝⁷ : AddCommMonoid M₃", "inst✝⁶ : AddCommMonoid M'", "inst✝⁵ : (i : Fin n.succ) → Module R (M i)", "inst✝⁴ : (i : ι) → Module R (M₁ i)", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M'", "f f' : MultilinearMap R M₁ M₂", "inst✝ : DecidableEq ι", "m : (i : ι) → M₁ i", "t✝ : Finset ι", "i : ι", "t : Finset ι", "hit : i ∉ t", "Hrec : ∀ (m' : (i : ι) → M₁ i), f (t.piecewise (m + m') m') = ∑ s ∈ t.powerset, f (s.piecewise m m')", "m' : (i : ι) → M₁ i"], "goal": "f ((insert i t).piecewise (m + m') m') = ∑ s ∈ (insert i t).powerset, f (s.piecewise m m')"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "ι : Sort u_5", "κ : ι → Sort u_6", "VAdd α β", "s : (i : ι) → κ i → Set α", "t : Set β"], "goal": "(⋂ i, ⋂ j, s i j) +ᵥ t ⊆ ⋂ i, ⋂ j, s i j +ᵥ t"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "CategoryTheory.Category.{v₁, u₁} 𝒮", "CategoryTheory.Category.{v₂, u₂} 𝒳", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ⟶ b", "p.IsStronglyCartesian f φ", "R' : 𝒮", "a' : 𝒳", "g : R' ⟶ R", "f' : R' ⟶ S", "hf' : f' = g ≫ f", "φ' : a' ⟶ b", "p.IsHomLift f' φ'"], "goal": "CategoryTheory.Functor.IsStronglyCartesian.map p f φ hf' φ' ≫ φ = φ'"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x✝¹ : M", "p✝ p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "ι : Sort u_9", "p : ι → Submodule R M", "x✝ : M", "hx✝ : x✝ ∈ span R (⋃ i, ↑(p i))", "x : M", "hx : x ∈ ⋃ i, ↑(p i)"], "goal": "x ∈ AddSubmonoid.closure (⋃ i, ↑(p i))"}}
+{"state": {"context": ["α : Type u_1", "k : Heap α → Heap α", "n : Nat", "le : α → α → Bool", "res : FindMin α", "s✝ : Heap α", "m : Nat", "hk : ∀ (s : Heap α), (k s).realSize = m + s.realSize", "hres : Batteries.BinomialHeap.Imp.FindMin.HasSize res n", "r : Nat", "a : α", "c : HeapNode α", "s : Heap α", "eq : n = m + (cons r a c s).realSize"], "goal": "Batteries.BinomialHeap.Imp.FindMin.HasSize (findMin le (k ∘ cons r a c) s (if le res.val a = true then res else { before := k, val := a, node := c, next := s })) n"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "s : Finset α", "f : α → ℚ", "hf : ∀ a ∈ s, 0 ≤ f a"], "goal": "(∏ a ∈ s, f a).toNNRat = ∏ a ∈ s, (f a).toNNRat"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝¹ : TopologicalSpace α", "s✝ t u v : Set α", "inst✝ : PreconnectedSpace α", "s : ι → Set α", "h_nonempty : ∀ (i : ι), (s i).Nonempty", "h_disj : Pairwise (Disjoint on s)", "h_open : ∀ (i : ι), IsOpen (s i)", "h_Union : ⋃ i, s i = univ", "i : ι"], "goal": "IsOpen (⋃ i_1, s i_1 \\ s i)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "NormedAddCommGroup F", "𝕜 : Type u_5", "NontriviallyNormedField 𝕜", "NormedSpace 𝕜 E", "NormedSpace 𝕜 F", "Fact (1 ≤ p)", "L : E →L[𝕜] F", "f : ↥(MeasureTheory.Lp E p μ)"], "goal": "↑↑((ContinuousLinearMap.compLpL p μ L) f) =ᵐ[μ] fun a => L (↑↑f a)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁸ : Semiring R", "inst✝⁷ : Semiring R₂", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup M₂", "inst✝³ : Module R₂ M₂", "τ₁₂ : R →+* R₂", "inst✝² : RingHomSurjective τ₁₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F τ₁₂ M M₂", "ι : Type u_9", "s : Set ι", "p : ι → Submodule R M", "x✝¹ : ↥(⨆ i ∈ s, p i)", "x : M", "hx : x ∈ ⨆ i ∈ s, p i", "x✝ : ⟨x, hx⟩ ∈ ⊤", "i : ι", "hi : i ∈ s"], "goal": "map (⨆ i ∈ s, p i).subtype (comap (⨆ i ∈ s, p i).subtype (p i)) = p i"}}
+{"state": {"context": ["α : Type u", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "DenselyOrdered α", "s : Set α"], "goal": "IsPreconnected s ↔ s.OrdConnected"}}
+{"state": {"context": ["q₁ q₂ : ℚ"], "goal": "(q₁ * q₂).num = q₁.num * q��.num / ↑((q₁.num * q₂.num).natAbs.gcd (q₁.den * q₂.den))"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "F : J ⥤ C", "e : C ≌ D"], "goal": "(CategoryTheory.Limits.Cocones.functorialityEquivalence F e).counitIso =\n  CategoryTheory.NatIso.ofComponents (fun c => CategoryTheory.Limits.Cocones.ext (e.counitIso.app c.pt) ⋯) ⋯"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "AddMonoid M", "AddMonoid N", "f : M →+ N", "h : ∀ (x : M), IsAddUnit (f x)", "x : M"], "goal": "↑(-(IsAddUnit.liftRight f h) x) + f x = 0"}}
+{"state": {"context": ["α : Type u", "AddMonoid α", "u : AddUnits α", "a : α"], "goal": "a + ↑(-u) = 0 ↔ a = ↑u"}}
+{"state": {"context": ["R : Type u", "K : Type u'", "M : Type v", "V : Type v'", "M₂✝ : Type w", "V₂ : Type w'", "M₃✝ : Type y", "V₃ : Type y'", "M₄ : Type z", "ι : Type x", "M₅ : Type u_1", "M₆ : Type u_2", "inst✝¹² : Semiring R", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : AddCommMonoid M₂✝", "inst✝⁹ : AddCommMonoid M₃✝", "inst✝⁸ : AddCommMonoid M₄", "inst✝⁷ : Module R M", "inst✝⁶ : Module R M₂✝", "inst✝⁵ : Module R M₃✝", "inst✝⁴ : Module R M₄", "M₂ : Type u_3", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "M₃ : Type u_4", "inst✝¹ : AddCommGroup M₃", "inst✝ : Module R M₃", "f : M →ₗ[R] M₃", "g : M₂ →ₗ[R] M₃", "hd : Disjoint (range f) (range g)", "y : M", "z : M₂", "h : f y + g z = 0"], "goal": "f y = 0 ∧ g z = 0"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝¹⁰ : MeasurableSpace α", "inst✝⁹ : MeasurableSpace α'", "inst✝⁸ : MeasurableSpace β", "inst✝⁷ : MeasurableSpace β'", "inst✝⁶ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : SFinite ν", "inst✝³ : NormedSpace ℝ E", "inst✝² : SFinite μ", "E' : Type u_7", "inst✝¹ : NormedAddCommGroup E'", "inst✝ : NormedSpace ℝ E'", "hE : CompleteSpace E"], "goal": "∀ (f : α × β → E), Integrable f (μ.prod ν) → ∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ"], "goal": "HurwitzZeta.completedHurwitzZetaEven₀ a (1 - s) = HurwitzZeta.completedCosZeta₀ a s"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "J : Type u₂", "inst✝ : Category.{v₂, u₂} J", "X : Type v₂", "α : Type u_1", "Z : α → C", "c c' : Cofan Z", "f : Fan fun x => op (Z x)", "hc : IsColimit c", "hc' : IsColimit c'", "hf : IsLimit f"], "goal": "∀ (j : Discrete α), c'.ι.app j ≫ ((opCoproductIsoProduct' hc hf).hom ≫ (opCoproductIsoProduct' hc' hf).inv).unop = c'.ι.app j ≫ (hc.coconePointUniqueUpToIso hc').op.inv.unop"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "f : α → ℝ", "μ : Measure α", "f_nn : 0 ≤ᶠ[ae μ] f", "f_mble : AEMeasurable f μ", "p : ℝ", "p_pos : 0 < p"], "goal": "ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) = ENNReal.ofReal p * ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))"}}
+{"state": {"context": ["a✝ b✝ c d m n k : ℕ", "p q : ℕ → Prop", "a : ℕ", "ha : 2 ≤ a", "b : ℕ", "x✝ : 2 ≤ b + 3"], "goal": "a + (b + 3) ≤ a * (b + 3)"}}
+{"state": {"context": ["ι : Type u", "r : ι → ι → Prop", "IsWellOrder ι r", "f : ι → Ordinal.{max u v}"], "goal": "(Ordinal.type r).blsub (Ordinal.bfamilyOfFamily' r f) = Ordinal.lsub f"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{?u.2513, u_1} C", "D : Type u_2", "inst✝⁴ : Category.{?u.2520, u_2} D", "G : C ⥤ D", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "A : Type v", "inst✝³ : Category.{u, v} A", "inst✝² : G.LocallyCoverDense K", "inst✝¹ : G.IsLocallyFull K", "inst✝ : G.IsLocallyFaithful K", "X : C", "S : Sieve X", "hS : S ∈ (fun X S => K.sieves (G.obj X) (Sieve.functorPushforward G S)) X", "S' : Sieve X", "H' : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f S' ∈ (fun X S => K.sieves (G.obj X) (Sieve.functorPushforward G S)) Y"], "goal": "∀ ⦃Y : D⦄ ⦃f : Y ⟶ G.obj X⦄, (Sieve.functorPushforward G S).arrows f → Sieve.pullback f (Sieve.functorPushforward G S') ∈ K.sieves Y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "e : α ≃o β", "a : α"], "goal": "⇑e '' Set.Iic a = Set.Iic (e a)"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "TopologicalSpace R", "Star R", "ContinuousStar R", "TopologicalSpace α", "f : α → R", "hf : Continuous f"], "goal": "Continuous fun x => star (f x)"}}
+{"state": {"context": ["ι : Type u_1", "R✝ : Type u_2", "M : Type u_3", "inst��¹² : CommRing R✝", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R✝ M", "N : ι → Submodule R✝ M", "inst✝⁹ : DecidableEq ι", "h : IsInternal N", "R : Type u_4", "M₁ : Type u_5", "M₂ : Type u_6", "inst✝⁸ : CommSemiring R", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : Module R M₁", "N₁ : ι → Submodule R M₁", "h₁ : IsInternal N₁", "inst✝⁵ : AddCommMonoid M₂", "inst✝⁴ : Module R M₂", "N₂ : ι → Submodule R M₂", "h₂ : IsInternal N₂", "κ₁ : ι → Type u_7", "κ₂ : ι → Type u_8", "inst✝³ : (i : ι) → Fintype (κ₁ i)", "inst✝² : ∀ (i : ι), Finite (κ₂ i)", "inst✝¹ : (i : ι) → DecidableEq (κ₁ i)", "inst✝ : Fintype ι", "b₁ : (i : ι) → Basis (κ₁ i) R ↥(N₁ i)", "b₂ : (i : ι) → Basis (κ₂ i) R ↥(N₂ i)", "f : M₁ →ₗ[R] M₂", "hf : ∀ (i : ι), MapsTo ⇑f ↑(N₁ i) ↑(N₂ i)", "i : ι", "snd✝¹ : κ₂ i", "j : ι", "snd✝ : κ₁ j"], "goal": "(toMatrix (h₁.collectedBasis b₁) (h₂.collectedBasis b₂)) f ⟨i, snd✝¹⟩ ⟨j, snd✝⟩ = Matrix.blockDiagonal' (fun i => (toMatrix (b₁ i) (b₂ i)) (f.restrict ⋯)) ⟨i, snd✝¹⟩ ⟨j, snd✝⟩"}}
+{"state": {"context": ["M : Type u_3", "G : Type u_6", "H : Type u_7", "AddZeroClass M", "AddCommGroup G", "AddCommGroup H", "φ : G →+ H", "ψ : M →+ G"], "goal": "φ.comp (-ψ) = -φ.comp ψ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "u : ι → UniformSpace γ"], "goal": "UniformFun.filter α γ (⨅ i, 𝓤 γ) = ⨅ i, 𝓤 (α →ᵤ γ)"}}
+{"state": {"context": [], "goal": "Primrec fun n => 2 * n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "M : Type u_1", "AddCommGroup M", "Module R M", "Module S M", "IsScalarTower R S M", "m : Ω[S⁄R] →ₗ[S] M"], "goal": "∀ (a : S), ((KaehlerDifferential.linearMapEquivDerivation R S) m) a = m ((KaehlerDifferential.D R S) a)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "hp : Fact (1 ≤ p)", "H : ∀ (f : ℕ → α → E), (∀ (n : ℕ), Memℒp (f n) p μ) → ∀ (B : ℕ → ℝ≥0∞), ∑' (i : ℕ), B i < ⊤ → (∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N) → ∃ f_lim, Memℒp f_lim p μ ∧ Tendsto (fun n => eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)", "B : ℕ → ℝ := fun n => (1 / 2) ^ n", "hB_pos : ∀ (n : ℕ), 0 < B n", "f : ℕ → ↥(Lp E p μ)", "hf : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) < B N", "M : ℝ", "hB : HasSum B M", "B1 : ℕ → ℝ≥0∞ := fun n => ENNReal.ofReal (B n)", "hB1_has : HasSum B1 (ENNReal.ofReal M)"], "goal": "∃ f_lim, Memℒp f_lim p μ ∧ Tendsto (fun n => eLpNorm (↑↑(f n) - f_lim) p μ) atTop (𝓝 0)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ₁ φ₂ : S₁ ⟶ S₂", "self : CategoryTheory.ShortComplex.Homotopy φ₁ φ₂"], "goal": "φ₁.τ₃ = self.h₃ + CategoryTheory.CategoryStruct.comp self.h₂ S₂.g + φ₂.τ₃"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : OrderedAddCommMonoid M", "f : ℕ → M", "u : ℕ → ℕ", "hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m", "h_pos : ∀ (n : ℕ), 0 < u n", "hu : Monotone u", "n : ℕ", "ihn : ∑ k ∈ Ico (u 0) (u n), f k ≤ ∑ k ∈ range n, (u (k + 1) - u k) • f (u k)"], "goal": "∑ k ∈ Ico (u n) (u (n + 1)), f k ≤ (u (n + 1) - u n) • f (u n)"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝¹ : Field F", "W : Jacobian F", "inst✝ : Nontrivial R", "X Y : R", "h : W'.toAffine.Nonsingular X Y", "h0 : fromAffine (Affine.Point.some h) = 0"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a₁ a₂ : α", "b : β", "SMul α β", "Preorder α", "Preorder β", "Zero β", "SMulPosMono α β", "SMulPosReflectLE α β", "hb : 0 < b"], "goal": "a₁ • b ≤ a₂ • b ↔ a₁ ≤ a₂"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "inst✝⁵ : Ring R", "inst✝⁴ : Ring S", "M : Type u_4", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "m : Submodule R M", "N : Type u_5", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "x✝ : Nontrivial R", "h : ∀ (x : R), x ≠ 0 → Function.Surjective ⇑(LinearMap.toSpanSingleton R R x)", "x : R", "hx : x ≠ 0", "y : R", "hyx : y * x = 1", "hy : y ≠ 0"], "goal": "IsUnit x"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "l : List α", "a : α"], "goal": "↑a ≤ l.maximum ↔ ∃ b, b ∈ l ∧ a ≤ b"}}
+{"state": {"context": ["R : Type u_2", "NonAssocSemiring R", "s : Set Rᵐᵒᵖ"], "goal": "(Subsemiring.closure s).unop = Subsemiring.closure (MulOpposite.op ⁻¹' s)"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝ : Semiring R", "φ : MvPowerSeries σ R", "s : σ"], "goal": "(coeff R 0) (X s * φ) = 0"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "n✝ : ℕ", "α : Type u_1", "β : Type u_2", "C : G.Coloring α", "inst✝ : Nonempty V", "n : ℕ", "hc : G.Colorable n", "m : ℕ", "hm : m ∈ {n | G.Colorable n}", "h' : m < Nat.succ 0"], "goal": "False"}}
+{"state": {"context": ["β : Type u_2", "mβ : MeasurableSpace β", "Ω : Type u_5", "F : Type u_6", "mΩ : MeasurableSpace Ω", "X : Ω → β", "μ : MeasureTheory.Measure Ω", "TopologicalSpace F", "f : Ω → F", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.AEStronglyMeasurable (fun x => f x.2) (MeasureTheory.Measure.map (fun ω => (X ω, ω)) μ)"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasZeroMorphisms C", "X Y : C", "inst✝ : Simple Y", "i : X ≅ Y", "Y✝ : C", "f : Y✝ ⟶ X", "m : Mono f", "this : Mono (f ≫ i.hom)"], "goal": "IsIso f ↔ f ≠ 0"}}
+{"state": {"context": ["J : Type u", "CategoryTheory.Category.{v, u} J", "CategoryTheory.IsFilteredOrEmpty J"], "goal": "CategoryTheory.FinallySmall J ↔ ∃ s, ∃ (_ : Small.{v, u} ↑s), ∀ (i : J), ∃ j ∈ s, Nonempty (i ⟶ j)"}}
+{"state": {"context": ["num : Int", "den g : Nat", "den_nz : den ≠ 0", "e : g = num.natAbs.gcd den"], "goal": "(num / ↑g).natAbs.Coprime (den / g)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "inst✝¹ : DivisionMonoid α", "inst✝ : HasDistribNeg α", "a : α", "n : ℤ", "h : Even n"], "goal": "(-1) ^ n = 1"}}
+{"state": {"context": ["a b : ℝ", "hab : a < b", "l : Filter ℝ", "f f' g g' : ℝ → ℝ", "hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x", "hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x", "hg' : ∀ x ∈ Ioo a b, g' x ≠ 0", "hfa : Tendsto f (𝓝[>] a) (𝓝 0)", "hga : Tendsto g (𝓝[>] a) (𝓝 0)", "hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l", "sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b", "hg : ∀ x ∈ Ioo a b, g x ≠ 0", "c : ℝ → ℝ", "hc : ∀ x ∈ Ioo a b, c x ∈ Ioo a x ∧ f x * g' (c x) = g x * f' (c x)", "this : ∀ x ∈ Ioo a b, ((fun x' => f' x' / g' x') ∘ c) x = f x / g x"], "goal": "Tendsto (fun x => f x / g x) (𝓝[>] a) l"}}
+{"state": {"context": ["α : Type u_1", "l : Lists' α true"], "goal": "Lists'.ofList l.toList = l"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "H : Type u_2", "inst✝⁴ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝³ : TopologicalSpace M", "inst✝² : ChartedSpace H M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "this✝ : LocallyCompactSpace H", "this : LocallyCompactSpace M"], "goal": "MetrizableSpace M"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "f g : α → β", "hf : Integrable f μ", "hg : Integrable g μ"], "goal": "Integrable (f - g) μ"}}
+{"state": {"context": ["E : Type u", "NormedAddCommGroup E", "NormedSpace ℂ E", "CompleteSpace E", "f : ℂ → E", "z w : ℂ", "Hd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])", "Hi :\n  MeasureTheory.IntegrableOn (fun z => Complex.I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) Complex.I)\n    ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) MeasureTheory.volume"], "goal": "(((∫ (x : ℝ) in z.re..w.re, f (↑x + ↑z.im * Complex.I)) - ∫ (x : ℝ) in z.re..w.re, f (↑x + ↑w.im * Complex.I)) +\n      Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * Complex.I)) -\n    Complex.I • ∫ (y : ℝ) in z.im..w.im, f (↑z.re + ↑y * Complex.I) =\n  ∫ (x : ℝ) in z.re..w.re,\n    ∫ (y : ℝ) in z.im..w.im,\n      Complex.I • (fderiv ℝ f (↑x + ↑y * Complex.I)) 1 - (fderiv ℝ f (↑x + ↑y * Complex.I)) Complex.I"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "a : α", "l r : List α", "h1 : (a :: l).length - 1 - 0 < ((a :: l).reverseAux r).length", "x✝ : 0 < (a :: l).length", "aux : ∀ (h1 : 0 + l.length < (l.reverseAux (a :: r)).length) (h2 : 0 < (a :: r).length), (l.reverseAux (a :: r)).get ⟨0 + l.length, h1⟩ = (a :: r).get ⟨0, h2⟩"], "goal": "((a :: l).reverseAux r)[(a :: l).length - 1 - 0] = (a :: l)[0]"}}
+{"state": {"context": ["T : Type u", "CategoryTheory.Category.{v, u} T", "X Y : T", "i : X ≅ Y", "p : CategoryTheory.Arrow T", "sq : CategoryTheory.Arrow.mk i.hom ⟶ p"], "goal": "CategoryTheory.CategoryStruct.comp i.inv (CategoryTheory.CategoryStruct.comp sq.left p.hom) = sq.right"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "LocallyCompactSpace E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "α : Type u_4", "TopologicalSpace α", "MeasurableSpace α", "MeasurableSpace E", "OpensMeasurableSpace α", "OpensMeasurableSpace E", "f : α → E → F", "CompleteSpace F", "hf : Continuous (Function.uncurry f)"], "goal": "Measurable fun p => fderiv 𝕜 (f p.1) p.2"}}
+{"state": {"context": ["R : Type u", "Ring R", "P : Type (max u v)", "AddCommGroup P", "Module R P", "huniv :\n  ∀ {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M]\n    [inst_3 : Module R N] (f : M →ₗ[R] N) (g : P →ₗ[R] N), Function.Surjective ⇑f → ∃ h, f ∘ₗ h = g"], "goal": "Module.Projective R P"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y : Mon_ C", "M N : Bimod X Y", "f : M ⟶ N"], "goal": "f.hom ▷ Y.X ≫ colimit.ι (parallelPair (N.actRight ▷ Y.X) ((α_ N.X Y.X Y.X).hom ≫ N.X ◁ Y.mul)) WalkingParallelPair.one = f.hom ▷ Y.X ≫ (((ρ_ (N.X ⊗ Y.X)).inv ≫ N.actRight ▷ 𝟙_ C) ≫ N.X ◁ Y.one) ≫ coequalizer.π (N.actRight ▷ Y.X) ((α_ N.X Y.X Y.X).hom ≫ N.X ◁ Y.mul)"}}
+{"state": {"context": ["R : Type u", "inst✝⁴ : CommSemiring R", "A : Type v", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "n : Type w", "inst✝¹ : DecidableEq n", "inst✝ : Fintype n", "M : Matrix n n R"], "goal": "∑ p : n × n, (algebraMap R A) (M p.1 p.2) ⊗ₜ[R] stdBasisMatrix p.1 p.2 1 = 1 ⊗ₜ[R] M"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "inst✝ : Nonempty α", "this : DecidableEq ι", "i : ι", "s : Finset ι", "hi : i ∉ s", "ih : ∀ (f : ι → Finset (Finset α)), (∀ i ∈ s, (↑(f i)).Intersecting) → s.card ≤ Fintype.card α → (s.biUnion f).card ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - s.card)", "f : ι → Finset (Finset α)", "hf : ∀ i_1 ∈ cons i s hi, (↑(f i_1)).Intersecting", "hs : (cons i s hi).card ≤ Fintype.card α", "f' : ι → Finset (Finset α) := fun j => if hj : j ∈ cons i s hi then ⋯.choose else ∅", "hf₁ : ∀ j ∈ cons i s hi, f j ⊆ f' j ∧ 2 * (f' j).card = 2 ^ Fintype.card α ∧ (↑(f' j)).Intersecting", "hf₂ : ∀ j ∈ cons i s hi, IsUpperSet ↑(f' j)"], "goal": "2 ^ Fintype.card α + (2 ^ Fintype.card α - 2 ^ (Fintype.card α - s.card)) ≤ 2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - (cons i s hi).card))"}}
+{"state": {"context": ["α : Type u_1", "a : α"], "goal": "Batteries.DList.singleton a = Batteries.DList.lazy_ofList { fn := fun x => [a] }"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : CompleteSpace E", "F : Type u_3", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "f : E → F", "f' : E →L[𝕜] F", "a : E", "hf : HasStrictFDerivAt f f' a", "hf' : range f' = ⊤", "hker : (LinearMap.ker f').ClosedComplemented"], "goal": "(implicitFunctionDataOfComplemented f f' hf hf' hker).rightDeriv.comp (LinearMap.ker f').subtypeL = ContinuousLinearMap.id 𝕜 ↥(LinearMap.ker f')"}}
+{"state": {"context": ["m : ℕ", "h : 2 ≤ 2 + m"], "goal": "(finRotate (2 + m)).IsCycle"}}
+{"state": {"context": ["x y : ℂ"], "goal": "Complex.cosh (x + y) = Complex.cosh x * Complex.cosh y + Complex.sinh x * Complex.sinh y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : NormedSpace ℝ E", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ℝ F", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace ℝ F'", "inst✝⁶ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : NormedSpace 𝕜 E", "G'' : Type u_7", "G' : Type u_8", "inst✝³ : NormedLatticeAddCommGroup G'", "inst✝² : NormedSpace ℝ G'", "inst✝¹ : NormedLatticeAddCommGroup G''", "inst✝ : NormedSpace ℝ G''", "T : Set α → G'' →L[ℝ] G'", "h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0", "h_add : FinMeasAdditive μ T", "hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G''), 0 ≤ x → 0 ≤ (T s) x", "f g : ↥(simpleFunc G'' 1 μ)", "hfg : 0 ≤ g - f"], "goal": "0 ≤ setToL1S T (g - f)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ✝ : Measure α", "inst✝ : NormedAddCommGroup β", "f✝ g : ι → α → β", "p✝ : ℝ≥0∞", "x✝ : MeasurableSpace α", "f : ι → α → β", "p : ℝ≥0∞", "μ : Measure α"], "goal": "UnifTight f p μ ↔ ∀ ⦃ε : ℝ≥0∞⦄, 0 < ε → ∃ s, μ s ≠ ⊤ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "i : α", "β : α → Type u_2", "f : β i × ((j : { j // j ≠ i }) → β ↑j)", "x : { j // j ≠ i }"], "goal": "((fun f => (f i, fun j => f ↑j)) ((fun f j => if h : j = i then ⋯ ▸ f.1 else f.2 ⟨j, h⟩) f)).2 x = f.2 x"}}
+{"state": {"context": ["S : CategoryTheory.ShortComplex Ab"], "goal": "S.Exact ↔ Function.Exact ⇑S.f ⇑S.g"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type uG", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "f : E → F →L[𝕜] 𝕜", "g : E → G", "n : ℕ∞", "hf : ContDiff 𝕜 n f", "hg : ContDiff 𝕜 n g"], "goal": "ContDiff 𝕜 n fun x => (f x).smulRight (g x)"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W"], "goal": "cs.length 1 = 0"}}
+{"state": {"context": ["α : Type u", "AddMonoidWithOne α", "n : ℕ", "n.AtLeastTwo", "m : α"], "goal": "OfNat.ofNat n = ↑m ↔ OfNat.ofNat n = m"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "V : ℕ → Set (α × α)", "hV : ∀ (n : ℕ), V n ∈ 𝓤 α", "u : ℕ → α", "hu : CauchySeq u"], "goal": "∃ φ, StrictMono φ ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V n"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "Zero M", "Zero N", "f : M → N", "hf : f 0 = 0", "g : α →₀ M"], "goal": "(Finsupp.mapRange f hf g).support ⊆ g.support"}}
+{"state": {"context": ["H : Type u", "H' : Type u_1", "M : Type u_2", "M' : Type u_3", "TopologicalSpace H", "TopologicalSpace M", "ChartedSpace H M", "TopologicalSpace H'", "TopologicalSpace M'", "ChartedSpace H' M'", "x : M × M'"], "goal": "chartAt (ModelProd H H') x = (chartAt H x.1).prod (chartAt H' x.2)"}}
+{"state": {"context": ["z✝ w✝ : ℍ", "r✝ R : ℝ", "z : ℍ", "r : ℝ", "w : ℂ", "h : dist w ↑(z.center r) ≤ z.im * Real.sinh r"], "goal": "dist (↑(z.center r)) w ≤ z.im * Real.sinh r"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "j : ℕ"], "goal": "cs.wordProd ω * ((cs.wordProd (drop (j + 1) ω))⁻¹ * (Option.map cs.simple (ω.get? j)).getD 1 * cs.wordProd (drop (j + 1) ω)) = cs.wordProd (take j ω ++ drop (j + 1) ω)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a✝ o✝ : Ordinal.{u}", "f✝ : (b : Ordinal.{u}) → b < o✝ → Ordinal.{u}", "a o o' : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{u}", "hf : a.IsFundamentalSequence o f", "g : (b : Ordinal.{u}) → b < o' → Ordinal.{u}", "hg : o.IsFundamentalSequence o' g", "i : Ordinal.{u}", "hi : i < o'"], "goal": "g i hi < o'.blsub g"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "𝒜 : Finset (Finset α)", "r : ℕ", "h𝒜 : Set.Sized r ↑𝒜", "i : DecidableRel fun x x_1 => x ⊆ x_1 := fun x x_1 => Classical.dec ((fun x x_2 => x ⊆ x_2) x x_1)", "s : Finset α", "hs : s ∈ ∂ 𝒜", "t : Finset α", "ht : t ∈ 𝒜 ∧ s ⊆ t", "this : ∅ ∉ 𝒜"], "goal": "t ∈ image (fun a => insert a s) sᶜ"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "p : PProd α β"], "goal": "⟨p.fst, p.snd⟩ = p"}}
+{"state": {"context": ["ι : Type u_6", "α : Type u_8", "DecidableEq ι", "Fintype ι", "CommSemiring α", "f g : ι → α"], "goal": "∏ a : ι, (f a + g a) = ∑ t : Finset ι, (∏ a ∈ t, f a) * ∏ a ∈ tᶜ, g a"}}
+{"state": {"context": ["n : ℕ", "n2 : 2 ≤ n", "np : ¬Nat.Prime n"], "goal": "∃ m, m ∣ n ∧ 2 ≤ m ∧ m < n"}}
+{"state": {"context": ["α : Type u_1", "p : Set α → Prop", "t : Set α", "S : Set (Set α)", "Sct : S.Countable", "Sp : ∀ s ∈ S, p s", "hS : ∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y"], "goal": "HasCountableSeparatingOn (↑t) (fun u => ∃ v, p v ∧ Subtype.val ⁻¹' v = u) univ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "g : β → γ", "x : FreeAbelianGroup α"], "goal": "(map (g ∘ f)) x = (map g) ((map f) x)"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁴ : Category.{u_5, u_1} C", "inst✝³ : Preadditive C", "I₁ : Type u_2", "I₂ : Type u_3", "I₁₂ : Type u_4", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "K L M : HomologicalComplex₂ C c₁ c₂", "φ : K ⟶ L", "e : K ≅ L", "ψ : L ⟶ M", "c₁₂ : ComplexShape I₁₂", "inst✝² : DecidableEq I₁₂", "inst✝¹ : TotalComplexShape c₁ c₂ c₁₂", "inst✝ : K.HasTotal c₁₂", "i₁₂ i���₂' : I₁₂", "h₁₂ : ¬c₁₂.Rel i₁₂ i₁₂'", "i₁ : I₁", "i₂ : I₂", "h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂"], "goal": "K.d₁ c₁₂ i₁ i₂ i₁₂' = 0"}}
+{"state": {"context": ["Γ : Subgroup SL(2, ℤ)", "k : ℤ", "α : Type u_2", "SMul α ℂ", "IsScalarTower α ℂ ℂ", "f : SlashInvariantForm Γ k", "n : α", "z : ℍ"], "goal": "(n • f) z = n • f z"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y : C", "a : X ⟶ Y"], "goal": "a - a - (0 - a) = a"}}
+{"state": {"context": ["M : Type u_2", "AddMonoid M", "a : M", "n : ℕ"], "goal": "n • a + a = a + n • a"}}
+{"state": {"context": ["α : Type u_1", "s t : Set α", "x✝ : α × α", "a : α", "x : α × α"], "goal": "x ∈ (s ∩ t).offDiag ↔ x ∈ s.offDiag ∩ t.offDiag"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "α : Type v", "Fintype n", "Fintype o", "Mul α", "AddCommMonoid α", "p : Type u_10", "q : Type u_11", "M : Matrix m n α", "N : Matrix n p α", "e₁ : l → m", "e₂ : o → n", "e₃ : q → p", "he₂ : Function.Bijective e₂"], "goal": "(M * N).submatrix e₁ e₃ = M.submatrix e₁ e₂ * N.submatrix e₂ e₃"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "x : Y", "hx : x ∈ e.target"], "goal": "∀ᶠ (y : X) in 𝓝 (↑e.symm x), ↑e.symm (↑e y) = y"}}
+{"state": {"context": ["x✝¹ y✝¹ z✝¹ w x✝ y✝ z✝ x : ℝ", "hx : 0 ≤ x", "y z : ℝ", "H : 0 = x"], "goal": "x ^ y * x ^ z ≤ x ^ (y + z)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "A : Type w", "B : Type u₁", "M : Type v₁", "inst✝⁶ : CommSemiring R", "inst✝⁵ : Semiring A", "inst✝⁴ : Algebra R A", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : Module A M", "inst✝ : IsScalarTower R A M", "hsur : Function.Surjective ⇑(algebraMap R A)", "X : Set M"], "goal": "restrictScalars R (span A X) = span R X"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "T T' : Triangle C", "φ : T ⟶ T'", "hT : T ∈ distinguishedTriangles", "hT' : T' ∈ distinguishedTriangles", "h₁ : IsIso φ.hom₁", "h₃ : IsIso φ.hom₃", "this : Mono φ.hom₂", "A : C"], "goal": "Function.Surjective fun x => x ≫ φ.hom₂"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "c : ℝ", "hc : c < 1", "h₁ : (∀ a' < 1, ∀ᶠ (b : ℝ) in atTop, a' < 1 - ε b) ∧ ∀ a' > 1, ∀ᶠ (b : ℝ) in atTop, 1 - ε b < a'"], "goal": "∀ᶠ (x : ℝ) in atTop, c < 1 - ε x"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "inst✝¹ : IsLocalAtSource P", "X Y U V : Scheme", "f : X ⟶ Y", "g : U ⟶ Y", "inst✝ : IsOpenImmersion g", "𝒰 : X.OpenCover", "H : ∀ (i : 𝒰.J), P (𝒰.map i ≫ f)", "i : 𝒰.J"], "goal": "P (𝒰.map i ≫ f)"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "E : Type u₃", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Category.{v₃, u₃} E", "F : C × D ⥤ E", "X X' : C", "f : X ⟶ X'", "Y Y' : D", "g : Y ⟶ Y'"], "goal": "F.map (f, 𝟙 Y) ≫ F.map (𝟙 X', g) = F.map (f, g)"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "TopologicalSpace E", "TopologicalSpace X", "f : E → X", "hf : IsCoveringMap f", "hf' : Function.Surjective f"], "goal": "QuotientMap f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "NormedSpace ℝ E", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace ℝ G", "f g : G → E", "s : Set G", "x : G", "hf : DifferentiableWithinAt ℝ f s x", "hg : DifferentiableWithinAt ℝ g s x"], "goal": "DifferentiableWithinAt ℝ (fun x => ⟪f x, g x⟫_𝕜) s x"}}
+{"state": {"context": ["n p : ℕ", "hn : n ≠ 0", "pp : Nat.Prime p", "h : p ∣ n"], "goal": "p ∣ p ^ n.factorization p"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "SemilatticeSup α", "s t : Finset α", "p : α → Prop"], "goal": "(∀ c ∈ s ⊻ t, p c) ↔ ∀ a ∈ s, ∀ b ∈ t, p (a ⊔ b)"}}
+{"state": {"context": ["𝕜 : Type u_1", "S : Type u_2", "R : Type u_3", "M : Type u_4", "inst✝¹⁸ : Field 𝕜", "inst✝¹⁷ : CharZero 𝕜", "inst✝¹⁶ : Ring R", "inst✝¹⁵ : AddCommGroup M", "inst✝¹⁴ : Algebra 𝕜 R", "inst✝¹³ : Module 𝕜 M", "inst✝¹² : Module R M", "inst✝¹¹ : Module Rᵐᵒᵖ M", "inst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M", "inst✝⁹ : IsScalarTower 𝕜 R M", "inst✝⁸ : IsScalarTower 𝕜 Rᵐᵒᵖ M", "inst✝⁷ : TopologicalSpace R", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : TopologicalRing R", "inst✝⁴ : TopologicalAddGroup M", "inst✝³ : ContinuousSMul R M", "inst✝² : ContinuousSMul Rᵐᵒᵖ M", "inst✝¹ : T2Space R", "inst✝ : T2Space M", "m : M"], "goal": "MulOpposite.op (inr m).fst • (inr m).snd = (inr m).fst • (inr m).snd"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s t : Finset α", "x : α × α"], "goal": "s.diag ∪ s.offDiag = filter (fun a => a.1 = a.2) (s ×ˢ s) ∪ filter (fun a => ¬a.1 = a.2) (s ×ˢ s)"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "inst✝¹ : Fintype F", "inst✝ : DecidableEq F", "hF : ringChar F ≠ 2"], "goal": "(quadraticChar F) (-1) = χ₄ ↑(Fintype.card F)"}}
+{"state": {"context": ["a : Prop"], "goal": "¬(a ↔ ¬a)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C"], "goal": "∀ {X Y : CategoryTheory.Arrow (CategoryTheory.Arrow C)} (φ : X ⟶ Y),\n  (CategoryTheory.Square.fromArrowArrowFunctor'.map φ).τ₂ = φ.left.right"}}
+{"state": {"context": ["α : Type u", "DivisionRing α", "x : α", "hx1 : x ≠ 1", "hx0 : x ≠ 0", "n : ℕ"], "goal": "∑ i ∈ Finset.range n, x⁻¹ ^ i = (x - 1)⁻¹ * (x - x⁻¹ ^ n * x)"}}
+{"state": {"context": ["n : ℕ", "b : Fin n"], "goal": "Finset.Iio b = Finset.fin n (Finset.Iio ↑b)"}}
+{"state": {"context": ["q : ℚ", "d : ℤ", "hd : d ≠ 0", "qdf : q = 0 /. d"], "goal": "∃ c, 0 = c * q.num ∧ d = c * ↑q.den"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CompleteLattice α", "inst✝ : IsCompactlyGenerated α", "S : Set α", "hS : BooleanGenerators S", "T₁ T₂ : Set α", "hT₁ : T₁ ⊆ S", "hT₂ : T₂ ⊆ S", "X : Set α", "hX : X ⊆ S", "hX' : sSup T₁ ⊓ sSup T₂ = sSup X", "I : α", "hI : I ∈ X"], "goal": "I ≤ sSup (T₁ ∩ T₂)"}}
+{"state": {"context": ["w x✝ y✝ z✝ : ℝ", "x y : ℝ≥0", "z : ℝ", "hz : 0 < z"], "goal": "x ^ z⁻¹ < y ↔ x < y ^ z"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "a : α"], "goal": "IsUnit (Associates.mk a) ↔ IsUnit a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u₂", "CategoryTheory.Category.{w, u₂} D", "F : C ⥤ D", "CategoryTheory.Limits.HasBinaryProducts C", "CategoryTheory.Limits.HasBinaryProducts D", "A : C", "∀ (B : C), CategoryTheory.IsIso (CategoryTheory.Limits.prodComparison F A B)"], "goal": "(CategoryTheory.Limits.prodComparisonNatIso F A).hom = CategoryTheory.Limits.prodComparisonNatTrans F A"}}
+{"state": {"context": ["V : Type (u + 1)", "CategoryTheory.LargeCategory V", "CategoryTheory.MonoidalCategory V", "H : Grp", "X : Action V ((CategoryTheory.forget₂ Grp MonCat).obj H)", "CategoryTheory.RightRigidCategory V", "h : ↑H"], "goal": "Xᘁ.ρ h = X.ρ h⁻¹ᘁ"}}
+{"state": {"context": ["α : Type u", "a : α", "G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "g : G", "hg : ∀ (x : G), x ∈ zpowers g", "g' : G'", "hg' : orderOf g' ∣ orderOf g"], "goal": "g ^ Classical.choose ⋯ = g"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : ℤ → R", "f : R →+* S", "b c d : R"], "goal": "normEDS b c d 3 = c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f fa : α → α", "fb : β → β", "x y : α", "m n✝ : ℕ", "hx : x ∈ periodicPts f", "n : ℕ"], "goal": "IsPeriodicPt f (minimalPeriod f x) (f^[n] x)"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "κ : Type u_5", "α : ι → Type u_6", "inst✝³ : DecidableEq ι", "inst✝² : DecidableEq κ", "inst✝¹ : Fintype ι", "inst✝ : (i : ι) → DecidableEq (α i)", "s : (i : ι) → Finset (α i)", "i : ι", "a : α i", "ha : a ∈ s i"], "goal": "(filter (fun f => f i = a) (piFinset s)).card = ∏ j ∈ univ.erase i, (s j).card"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq β", "inst✝² : DistribLattice α", "inst✝¹ : OrderBot α", "inst✝ : DecidableRel Disjoint", "s t u v : Finset α"], "goal": "∀ (s t u : Finset α), s ○ t ○ u = s ○ (t ○ u)"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "γ : Type u_5", "δ : Type u_7", "f : α → β → γ", "s : Set α", "t : Set β", "g : γ → δ", "f' : β → α' → δ", "g' : α → α'", "h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a)"], "goal": "g '' Set.image2 f s t = Set.image2 f' t (g' '' s)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : PartialOrder α", "s : Set α", "x✝ y : α", "hs : IsAntichain (fun x x_1 => x ≤ x_1) s", "x : α", "hx : x ∈ s", "z : α", "hy : x ∈ s", "hz : z = x"], "goal": "z ∈ s"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "e : G ≃L[𝕜] E"], "goal": "ContDiffOn 𝕜 n (f ∘ ⇑e) (?m.449353 ⁻¹' univ) ↔ ContDiffOn 𝕜 n f univ"}}
+{"state": {"context": ["Γ : Type u_1", "Λ : Type u_2", "Inhabited Λ", "σ : Type u_3", "Inhabited σ", "M : Λ → Turing.TM1.Stmt Γ Λ σ", "Fintype σ", "S : Finset Λ", "ss : Turing.TM1.Supports M S"], "goal": "Turing.TM0.Supports (Turing.TM1to0.tr M) ↑(Turing.TM1to0.trStmts M S)"}}
+{"state": {"context": ["α : Type u", "s : Set (Set α)", "i : (Quotient.out (aleph 1).ord).α"], "goal": "#↑(generateMeasurableRec s i) ≤ max (#↑s) 2 ^ ℵ₀"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End R M", "μ : R", "h : f.HasEigenvalue μ", "n : ℕ"], "goal": "(f ^ n).HasEigenvalue (μ ^ n)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f✝ g : Perm α", "x y : α", "inst✝³ : DecidableRel f✝.SameCycle", "inst✝² : DecidableRel g.SameCycle", "f : Perm α", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "h : f.SameCycle x y", "hx : f x ≠ x"], "goal": "∃ i < (f.cycleOf x).support.card, (f ^ i) x = y"}}
+{"state": {"context": ["n : Nat", "j : Fin n"], "goal": "j.succ.castSucc = j.castSucc.succ"}}
+{"state": {"context": ["α : Type u_1", "OrderedSemiring α", "h : 1 ∈ Set.Icc 0 1"], "goal": "⟨1, h⟩ = 1"}}
+{"state": {"context": ["p : ℝ≥0∞", "𝕜 : Type u_1", "ι : Type u_2", "α : ι → Type u_3", "β✝ : ι → Type u_4", "inst✝⁷ : Fact (1 ≤ p)", "inst✝⁶ : Fintype ι", "inst✝⁵ : Semiring 𝕜", "inst✝⁴ : (i : ι) → SeminormedAddCommGroup (α i)", "inst✝³ : (i : ι) → SeminormedAddCommGroup (β✝ i)", "inst✝² : (i : ι) → Module 𝕜 (α i)", "inst✝¹ : (i : ι) → Module 𝕜 (β✝ i)", "c : 𝕜", "β : Type u_5", "inst✝ : SeminormedAddCommGroup β", "hp : p ≠ ⊤", "b : β", "h : p = ⊤"], "goal": "‖(WithLp.equiv p (ι → β)).symm (Function.const ι b)‖₊ = ↑(Fintype.card ι) ^ (1 / p).toReal * ‖b‖₊"}}
+{"state": {"context": ["α : Type u_4"], "goal": "IsEmpty (ULift.{u_5, u_4} α) ↔ IsEmpty α"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : DecidableEq α", "inst✝¹ : Fintype α", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "s✝ : Finset α", "P : Finpartition univ", "ε : ℝ", "hε : 0 < ε", "hP₁ : P.IsEquipartition", "hP₃ : P.parts.card ≤ bound (ε / 8) ⌈4 / ε⌉₊", "x y z : α", "s : Finset α", "hX : s ∈ P.parts", "Y : Finset α", "hY : Y ∈ P.parts", "xX : x ∈ s", "yY : y ∈ Y", "nXY : s ≠ Y", "uXY : G.IsUniform (ε / 8) s Y", "dXY : ε / 4 ≤ ↑(G.edgeDensity s Y)", "Z : Finset α", "hZ : Z ∈ P.parts", "zZ : z ∈ Z", "Z' : Finset α", "hZ' : Z' ∈ P.parts", "zZ' : z ∈ Z'", "hX' : s ∈ P.parts", "xX' : x ∈ s", "nXZ : s ≠ Z", "uXZ : G.IsUniform (ε / 8) s Z", "dXZ : ε / 4 ≤ ↑(G.edgeDensity s Z)", "hY' : Y ∈ P.parts", "yY' : y ∈ Y", "nYZ : Y ≠ Z'", "uYZ : G.IsUniform (ε / 8) Y Z'", "dYZ : ε / 4 ≤ ↑(G.edgeDensity Y Z')"], "goal": "triangleRemovalBound ε * ↑(Fintype.card α) ^ 3 ≤ ↑(G.cliqueFinset 3).card"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : C ⥤ D", "X Y Z : C", "f : X ⟶ Y", "g : X ⟶ Z"], "goal": "(CategoryTheory.Limits.spanCompIso F f g).hom.app CategoryTheory.Limits.WalkingSpan.zero =\n  𝟙 ((CategoryTheory.Limits.span f g ⋙ F).obj CategoryTheory.Limits.WalkingSpan.zero)"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "H K : Subgroup G", "S T : Set G", "α : Type u_2", "inst✝ : MulAction G α", "a : α", "h1 : ∀ (h : ↥H), h • a = a → h = 1", "h2 : ∀ (g : G), ∃ h, h • g • a = a", "h : ↥H", "h' : ↑↑H", "g : G", "hg : g • a = a", "hh : h • (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2) • a = a", "hh' : (↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2)) • a = a"], "goal": "(h', ⟨g, hg⟩) = (h⁻¹, ⟨↑h * (↑(h', ⟨g, hg⟩).1 * ↑(h', ⟨g, hg⟩).2), hh'⟩)"}}
+{"state": {"context": ["α : Type u", "HasSubset α", "a b : α", "IsAntisymm α fun x x_1 => x ⊆ x_1"], "goal": "a ⊆ b → b ⊆ a → a = b"}}
+{"state": {"context": ["b n : ℕ"], "goal": "(b + 2).digits (n + 1) = (n + 1) % (b + 2) :: (b + 2).digits ((n + 1) / (b + 2))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝³ : OrderedSemiring 𝕜", "inst✝² : AddCommGroup E", "inst✝¹ : Module 𝕜 E", "s✝ t : Set E", "inst✝ : Subsingleton ι", "p : ι → E", "s : Set ι", "x : ι", "hx : p x ∈ (convexHull 𝕜) (p '' s)", "this : ((convexHull 𝕜) (p '' s)).Nonempty"], "goal": "x ∈ s"}}
+{"state": {"context": ["o : Ordinal.{u_4}", "h : o.IsLimit"], "goal": "0 < o"}}
+{"state": {"context": ["a✝ b✝ c d : ℝ≥0∞", "r p q : ℝ≥0", "b : ℝ", "hb : 0 ≤ b", "a : ℝ≥0"], "goal": "↑a ≤ ENNReal.ofReal b ↔ (↑a).toReal ≤ b"}}
+{"state": {"context": ["n : ℕ", "p : ℕ → Prop", "hn : n ≤ sInf {m | p m}", "m : ℕ", "hm : m ∈ {m | p (m + n)}"], "goal": "sInf {m | p (m + n)} + n = sInf {m | p m}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "hs : MeasurableSet s"], "goal": "Measurable fun a => κ.compProdFun η a s"}}
+{"state": {"context": ["E : Type u", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "s : Set E", "a y : E", "h : IsLocalMinOn f s a", "hf : HasFDerivWithinAt f f' s a", "hy : y ∈ posTangentConeAt s a", "hy' : -y ∈ posTangentConeAt s a"], "goal": "f' y = 0"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "W X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "f' : W ⟶ X", "inst✝² : HasPullback f g", "inst✝¹ : HasPullback f' (pullback.fst f g)", "inst✝ : HasPullback (f' ≫ f) g"], "goal": "(pullbackRightPullbackFstIso f g f').inv ≫ pullback.fst f' (pullback.fst f g) ≫ f' = pullback.fst (f' ≫ f) g ≫ f'"}}
+{"state": {"context": ["f : ℕ → ℝ≥0", "hf : Summable f"], "goal": "Tendsto f atTop (𝓝 0)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "s : Set E", "h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ DifferentiableOn 𝕜 f (s ∩ u)"], "goal": "DifferentiableOn 𝕜 f s"}}
+{"state": {"context": ["ιa : Type u_1", "ιb : Type u_2", "Fintype ιa", "Fintype ιb", "R' : Type u_3", "Mᵢ : Type u_4", "N₁ : Type u_5", "N₂ : Type u_6", "CommSemiring R'", "AddCommGroup N₁", "Module R' N₁", "AddCommGroup N₂", "Module R' N₂", "AddCommMonoid Mᵢ", "Module R' Mᵢ", "DecidableEq ιa", "DecidableEq ιb", "a : Mᵢ [⋀^ιa]→ₗ[R'] N₁", "b : Mᵢ [⋀^ιb]→ₗ[R'] N₂", "σ : Equiv.Perm.ModSumCongr ιa ιb", "v : ιa ⊕ ιb → Mᵢ", "i j : ιa ⊕ ιb", "hv : v i = v j", "hij : i ≠ j"], "goal": "(AlternatingMap.domCoprod.summand a b σ) v + (AlternatingMap.domCoprod.summand a b (Equiv.swap i j • σ)) v = 0"}}
+{"state": {"context": ["α : Type u_1", "CommGroup α", "DecidableEq α", "A B C : Finset α"], "goal": "(A * C).card * B.card ≤ (A * B).card * (B * C).card"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Preadditive C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "CategoryTheory.Pretriangulated C", "T₁ : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ", "hT₁ : T₁ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C", "T₂ : CategoryTheory.Pretriangulated.Triangle Cᵒᵖ", "e : T₂ ≅ T₁"], "goal": "T₂ ∈ CategoryTheory.Pretriangulated.Opposite.distinguishedTriangles C"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "m : List I", "hm : List.Chain' (fun x x_1 => x > x_1) m", "h : ⟨m, hm⟩ ∈ {m | m < nil}"], "goal": "⟨m, hm⟩ ∈ ∅"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "a : α", "l₁ l₂ : List α"], "goal": "(a :: l₁).diff l₂ <+~ a :: l₁.diff l₂"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing S", "I : Ideal R", "y : R", "inst✝¹ : Algebra R S", "inst✝ : Away y S", "H : IsJacobson R", "P' : Ideal S", "hP'✝ : P'.IsPrime", "hP' : (comap (algebraMap R S) P').IsPrime", "hPM : Disjoint ↑(powers y) ↑(comap (algebraMap R S) P')"], "goal": "P'.jacobson ≤ P'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "DecidableEq α", "DecidableEq β", "f : α → α → β", "s t : Finset α", "hf : ∀ (a b : α), f a b = f b a"], "goal": "Finset.image₂ f (s ∪ t) (s ∩ t) ⊆ Finset.image₂ f s t"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "xs✝¹ ys : List α", "n✝¹ : ℕ", "β : Type u_3", "f : α → β", "n✝ : ℕ", "xs✝ : List α", "n : ℕ", "x : α", "xs : List α"], "goal": "map (fun p => f x ::ₛ p) (map (Sym.map f) (List.sym n (x :: xs))) ++ map (Sym.map f) (List.sym (n + 1) xs) = map (Sym.map f) (List.sym (n + 1) (x :: xs))"}}
+{"state": {"context": ["q : ℚ", "q_pos : 0 < q", "q_num_pos : 0 < q.num", "q_num_abs_eq_q_num : ↑q.num.natAbs = q.num", "q_inv : ℚ := ↑q.den / ���q.num", "q_inv_def : q_inv = ↑q.den / ↑q.num", "q_inv_eq : q⁻¹ = q_inv"], "goal": "(fract q⁻¹).num < q.num"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s s₁ s₂ : Finset α", "a : α", "f g : α → β", "inst✝ : CommMonoid β", "h : Disjoint s₁ s₂"], "goal": "∏ x ∈ s₁.disjUnion s₂ h, f x = fold (fun x x_1 => x * x_1) 1 (fun x => f x) (s₁.disjUnion s₂ h)"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁷ : Ring R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : AddCommGroup M''", "inst✝³ : Module R M", "inst✝² : Module R M'", "inst✝¹ : Module R M''", "a b : R", "x✝ y : M", "inst✝ : Nontrivial R", "hv : LinearIndependent R v", "x : ι", "f : ι →₀ R", "h : x ∉ ↑f.support", "p : v x ∉ span R (v '' ↑f.support)"], "goal": "(Finsupp.total ι M R v) f ≠ v x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "K L : CochainComplex C ℕ", "f : K ⟶ L", "HomologicalComplex.HasHomology K 0", "HomologicalComplex.HasHomology L 0", "(HomologicalComplex.sc' K 0 0 1).HasHomology", "(HomologicalComplex.sc' L 0 0 1).HasHomology"], "goal": "QuasiIsoAt f 0 ↔\n  CategoryTheory.ShortComplex.QuasiIso ((HomologicalComplex.shortComplexFunctor' C (ComplexShape.up ℕ) 0 0 1).map f)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Field α", "n k e : α", "h2 : e ≠ 0", "h3 : n * e = k"], "goal": "k * e⁻¹ = n"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "OrderedRing R", "AddCommGroup V", "Module R V", "AddTorsor V P", "x y : P"], "goal": "Wbtw R x y y"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "f✝ g : Filter α", "s✝ t✝ : Set α", "s : Finset α", "f : α → Filter β", "t : Set β"], "goal": "t ∈ ⨅ a ∈ s, f a ↔ ∃ p, (∀ a ∈ s, p a ∈ f a) ∧ t = ⋂ a ∈ s, p a"}}
+{"state": {"context": ["a : Primrec fun a => ofNat (ℕ × Code) a.length", "k : Primrec fun a => (ofNat (ℕ × Code) a.1.length).1"], "goal": "Primrec fun p => (fun a n => Nat.casesOn (ofNat (ℕ × Code) a.length).1 Option.none fun k' => Code.recOn (ofNat (ℕ × Code) a.length).2 (some 0) (some n.succ) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cf) n let y ← Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x ← Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cg) n Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cf) x) (fun cf cg x x => let z := (unpair n).1; Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cf) z) fun y => do let i ← Nat.Partrec.Code.lup a (k', (ofNat (ℕ × Code) a.length).2) (Nat.pair z y) Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair n).1; let m := (unpair n).2; do let x ← Nat.Partrec.Code.lup a ((ofNat (ℕ × Code) a.length).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a (k', (ofNat (ℕ × Code) a.length).2) (Nat.pair z (m + 1))) p.1 p.2"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "β : Type u_3", "ι : Type u_4", "inst✝ : TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s s₁ s₂ : Set α", "u : Set τ", "hu : u ∈ f"], "goal": "ω f ϕ s ⊆ closure (image2 ϕ u s)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "inst✝ : NoetherianSpace α", "H : ∀ y < ⊥, ∃ S, S.Finite ∧ (∀ t ∈ S, IsIrreducible ↑t) ∧ y = sSup S"], "goal": "∃ S, S.Finite ∧ (∀ t ∈ S, IsIrreducible ↑t) ∧ ⊥ = sSup S"}}
+{"state": {"context": ["n : Nat", "h : n.isPowerOfTwo"], "goal": "n > 0"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "s t u : Set α", "x y : α", "Φ : α → β"], "goal": "infDist x {y} = dist x y"}}
+{"state": {"context": ["R : Type u_3", "S : Type u_4", "A : Type u_5", "Semifield R", "Field S", "NonUnitalRing A", "Module R A", "Module S A", "Algebra R S", "a : A", "f : S → R", "hf : Function.LeftInverse f ⇑(algebraMap R S)", "h : quasispectrum S a ⊆ Set.range ⇑(algebraMap R S)"], "goal": "QuasispectrumRestricts a f"}}
+{"state": {"context": ["G : Type u_1", "X : Type u_2", "inst✝⁹ : TopologicalSpace G", "inst✝⁸ : TopologicalSpace X", "inst✝⁷ : Group G", "inst✝⁶ : TopologicalGroup G", "inst✝⁵ : MulAction G X", "inst✝⁴ : SigmaCompactSpace G", "inst✝³ : BaireSpace X", "inst✝² : T2Space X", "inst✝¹ : ContinuousSMul G X", "inst✝ : IsPretransitive G X", "U : Set G", "hU : U ∈ 𝓝 1", "x : X", "V : Set G", "V_mem : V ∈ 𝓝 1", "V_closed : IsClosed V", "V_symm : V⁻¹ = V", "VU : V * V ⊆ U", "s : Set G", "s_count : s.Countable", "hs : ⋃ g ∈ s, g • V = univ", "K : ℕ → Set G := compactCovering G", "F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}", "n : ℕ", "g : G", "hg : g ∈ s", "hi : (interior (F (n, ⟨g, hg⟩))).Nonempty", "I : (interior ((g • V) • {x})).Nonempty"], "goal": "U • {x} ∈ 𝓝 x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : Option α → β → γ", "b : β", "bs : List β"], "goal": "List.map₂Right' f [] (b :: bs) = (f none b :: List.map (f none) bs, [])"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : Fintype α", "s t : Finset α", "a b : α", "inst✝ : DecidableEq α", "h : s.card = t.card"], "goal": "(fun x => x + t.dens) (s \\ t).dens = (fun x => x + t.dens) (t \\ s).dens"}}
+{"state": {"context": ["x : ℝ", "hx : x ≤ 0"], "goal": "x⁻¹.toNNReal = x.toNNReal⁻¹"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : CancelCommMonoidWithZero α", "inst✝³ : UniqueFactorizationMonoid α", "inst✝² : DecidableEq (Associates α)", "inst✝¹ : (p : Associates α) → Decidable (Irreducible p)", "inst✝ : DecidableEq (Associates α)", "a b p : Associates α", "hp : Irreducible p", "hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d", "ha : ¬a = 0", "hb : ¬b = 0"], "goal": "p.count a.factors + p.count b.factors = p.count a.factors ∨ p.count a.factors + p.count b.factors = p.count b.factors"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_2, u_1} C", "inst✝⁴ : HasZeroObject C", "inst✝³ : HasShift C ℤ", "inst✝² : Preadditive C", "inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝ : Pretriangulated C", "S : Subcategory C", "Y Z : C", "g : Y ⟶ Z"], "goal": "(∃ Z_1 g_1 h, ∃ (_ : Triangle.mk g g_1 h ∈ distinguishedTriangles), S.P Z_1) ↔ ∃ X f h, ∃ (_ : Triangle.mk f g h ∈ distinguishedTriangles), S.P X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : TopologicalSpace α", "t : Set α", "a : α", "s : Set α"], "goal": "t ∈ 𝓝[s] a ↔ ∃ u, IsOpen u ∧ a ∈ u ∧ u ∩ s ⊆ t"}}
+{"state": {"context": ["G : Type u_10", "AddGroup G", "a : G"], "goal": "⇑(Equiv.addRight a) = fun x => x + a"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : CompactSpace X", "L : Set C(X, ℝ)", "nA : L.Nonempty", "inf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L", "sup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L", "sep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y", "f : C(X, ℝ)", "ε : ℝ", "pos : 0 < ε", "nX : Nonempty X", "g : X → X → C(X, ℝ)", "hg : ∀ (x y : X), g x y ∈ L", "w₁ : ∀ (x y : X), (g x y) x = f x", "w₂ : ∀ (x y : X), (g x y) y = f y", "U : X → X → Set X := fun x y => {z | f z - ε < (g x y) z}", "U_nhd_y : ∀ (x y : X), U x y ∈ 𝓝 y", "ys : X → Finset X := fun x => ⋯.choose", "ys_w : ∀ (x : X), ⋃ y ∈ ys x, U x y = ⊤ := fun x => Exists.choose_spec (CompactSpace.elim_nhds_subcover (U x) (U_nhd_y x))", "ys_nonempty : ∀ (x : X), (ys x).Nonempty"], "goal": "∃ x, dist x f < ε ∧ x ∈ L"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "φ✝ φ ψ : R⟦X⟧"], "goal": "∀ (i : ℕ), ↑i < min φ.order ψ.order → (coeff R i) (φ + ψ) = 0"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "x : α", "n : ℕ", "hn : 0 < n", "hx : Function.IsPeriodicPt f n x"], "goal": "0 < Function.minimalPeriod f x"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝⁴ : Monoid R", "inst✝³ : MulAction R M", "inst✝² : Monoid M", "inst✝¹ : IsScalarTower R M M", "inst✝ : SMulCommClass R M M", "p : SubMulAction R M"], "goal": "↑p ^ 0 ⊆ ↑(p ^ 0)"}}
+{"state": {"context": ["n : ℕ", "d : ℕ", "a : Fin n → ℕ"], "goal": "(Behrend.map d) a = ∑ i : Fin n, a i * d ^ ↑i"}}
+{"state": {"context": ["s : ℝ", "hs : -1 ≤ s", "hs' : s ≠ 0", "p : ℝ", "hp : 1 < p"], "goal": "1 + p * s < (1 + s) ^ p"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "α : Type u_10", "MulZeroClass α", "f g : X → α"], "goal": "(tsupport fun x => f x * g x) ⊆ tsupport g"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a b : R", "x✝ y : M", "hv : LinearIndependent R v", "i : ι", "x : ↥(span R (range v))", "hx : ↑x = v i"], "goal": "(Finsupp.total ι M R v) (Finsupp.single i 1) = ↑x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n : ℕ", "K : Type u_1", "inst✝ : DivisionRing K", "p q : K[X]", "h : q ≠ 0", "h₁ : q.leadingCoeff⁻¹ ≠ 0"], "goal": "(p * C q.leadingCoeff⁻¹).degree = p.degree"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "F : CategoryTheory.ComposableArrows C 0"], "goal": "∃ X, F = CategoryTheory.ComposableArrows.mk₀ X"}}
+{"state": {"context": ["n : ℕ", "x : ZMod n"], "goal": "(-x).cast = -x.cast % ↑n"}}
+{"state": {"context": ["A : Type u_1", "TopologicalSpace A", "Ring A", "StarRing A", "Algebra ℂ A", "ContinuousFunctionalCalculus ℂ IsStarNormal", "ContinuousFunctionalCalculus ℝ IsSelfAdjoint", "UniqueContinuousFunctionalCalculus ℝ A", "UniqueContinuousFunctionalCalculus ℂ A", "a : A", "ha : IsSelfAdjoint a"], "goal": "cfcHom ha = SpectrumRestricts.starAlgHom (cfcHom ⋯) ⋯"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : InnerProductSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace 𝕜 F", "inst✝¹ : CompleteSpace E", "inst✝ : CompleteSpace F", "A : E →L[𝕜] F", "p : Submodule 𝕜 E", "hp : Dense ↑p", "x : ↥((↑A).toPMap p).adjoint.domain", "y : ↥((↑(adjoint A)).toPMap ⊤).domain", "hxy : ↑x = ↑y", "v : ↥((↑A).toPMap p).domain"], "goal": "⟪↑((↑(adjoint A)).toPMap ⊤) y, ↑v⟫_𝕜 = ⟪↑x, ↑((↑A).toPMap p) v⟫_𝕜"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "Nontrivial R", "OreLocalization.OreSet (nonZeroDivisors R)", "NoZeroDivisors R", "r : R", "s : ↥(nonZeroDivisors R)"], "goal": "(r /ₒ s)⁻¹ = if hr : r = 0 then 0 else ↑s /ₒ ⟨r, ⋯⟩"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VAdd α β", "t : Set β", "a : α", "x : β"], "goal": "x ∈ a +ᵥ t ↔ ∃ y ∈ t, a +ᵥ y = x"}}
+{"state": {"context": ["α : Type u_1", "p : α → Bool", "xs : List α", "w : List.findIdx p xs < xs.length"], "goal": "p (xs.get ⟨List.findIdx p xs, w⟩) = true"}}
+{"state": {"context": ["L : Language", "L' : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'", "γ : Type u_3", "n l : ℕ", "φ ψ : L.BoundedFormula α l", "θ : L.BoundedFormula α l.succ", "v : α → M", "xs : Fin l → M", "R : L.Relations 2", "t₁ t₂ : L.Term (α ⊕ Fin l)"], "goal": "(R.boundedFormula₂ t₁ t₂).Realize v xs ↔ RelMap R ![realize (Sum.elim v xs) t₁, realize (Sum.elim v xs) t₂]"}}
+{"state": {"context": ["G : Type u", "inst✝¹ : Group G", "inst✝ : Finite ↑(commutatorSet G)"], "goal": "Group.rank ↥(commutator G) ≤ Nat.card ↑(commutatorSet G)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝ : CompleteLattice α", "f g : Filter β", "p q : β → Prop", "u v : β → α"], "goal": "∀ (x : β), p x ∧ q x → p x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f : Perm α", "s : Finset α", "hf : f.IsCycleOn ↑s", "h✝ : (↑s).Nontrivial"], "goal": "(↑(range s.card)).PairwiseDisjoint fun k => map { toFun := fun i => (i, (f ^ k) i), inj' := ⋯ } s"}}
+{"state": {"context": ["𝕜 : Type u", "hnorm : NontriviallyNormedField 𝕜", "E : Type v", "inst✝¹⁷ : AddCommGroup E", "inst✝¹⁶ : Module 𝕜 E", "inst✝¹⁵ : TopologicalSpace E", "inst✝¹⁴ : TopologicalAddGroup E", "inst✝¹³ : ContinuousSMul 𝕜 E", "F : Type w", "inst✝¹² : AddCommGroup F", "inst✝¹¹ : Module 𝕜 F", "inst✝¹⁰ : TopologicalSpace F", "inst✝⁹ : TopologicalAddGroup F", "inst✝⁸ : ContinuousSMul 𝕜 F", "F' : Type x", "inst✝⁷ : AddCommGroup F'", "inst✝⁶ : Module 𝕜 F'", "inst✝⁵ : TopologicalSpace F'", "inst✝⁴ : TopologicalAddGroup F'", "inst✝³ : ContinuousSMul 𝕜 F'", "inst✝² : CompleteSpace 𝕜", "inst✝¹ : T2Space E", "ι : Type v", "inst✝ : Fintype ι", "ξ : Basis ι 𝕜 E", "this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E", "this : UniformAddGroup E := comm_topologicalAddGroup_is_uniform"], "goal": "Continuous ⇑ξ.equivFun"}}
+{"state": {"context": ["α : Type u", "CanonicallyOrderedCommSemiring α", "a : α", "ha : 0 < a", "n : ℕ"], "goal": "0 < a ^ n"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "F : C ⥤ D", "W X Y Z : C", "f : W ⟶ X", "g : W ⟶ Y", "h : X ⟶ Z", "i : Y ⟶ Z", "inst✝ : ReflectsColimit (span f g) F", "e : f ≫ h = g ≫ i", "H : IsPushout (F.map f) (F.map g) (F.map h) (F.map i)"], "goal": "((Cocones.precompose (spanCompIso F f g).inv).obj (F.mapCocone (PushoutCocone.mk h i ⋯))).ι.app WalkingCospan.left ≫ ?refine_1.hom = H.cocone.ι.app WalkingCospan.left"}}
+{"state": {"context": ["α : Type u_1", "s : List α", "x : α", "hn : 0 < (List.permutations'Aux x s).length := ⋯"], "goal": "(List.permutations'Aux x s).get ⟨0, hn⟩ = x :: s"}}
+{"state": {"context": ["α : Type u_1", "a : α", "f : α →₀ ℕ"], "goal": "f.multinomial = (f.sum fun x => id).choose (f a) * (f.update a 0).multinomial"}}
+{"state": {"context": ["G : Type u", "inst✝¹ : AddCommGroup G", "inst✝ : Finite G", "val✝ : Fintype G"], "goal": "∃ ι x p, ∃ (_ : ∀ (i : ι), Nat.Prime (p i)), ∃ e, Nonempty (G ≃+ ⨁ (i : ι), ZMod (p i ^ e i))"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "IsDedekindDomain R", "I : (FractionalIdeal R⁰ (FractionRing R))ˣ"], "goal": "ClassGroup.mk0 ⟨ClassGroup.integralRep ↑I, ⋯⟩ = ClassGroup.mk I"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "AddGroup G", "AddAction G α", "MeasurableSpace α", "s t : Set α", "μ : MeasureTheory.Measure α", "MeasurableSpace G", "MeasurableVAdd G α", "MeasureTheory.VAddInvariantMeasure G α μ", "Countable G", "hs : MeasureTheory.IsAddFundamentalDomain G s μ", "ht : MeasureTheory.NullMeasurableSet t μ", "hd : Pairwise (MeasureTheory.AEDisjoint μ on fun g => (g +ᵥ t) ∩ s)"], "goal": "μ t ≤ μ s"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "hxs : UniqueDiffWithinAt 𝕜 s x", "h : ¬DifferentiableWithinAt 𝕜 f s x"], "goal": "¬DifferentiableWithinAt 𝕜 (fun y => -f y) s x"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "ι : Sort u_1", "t : ι → Set (PrimeSpectrum R)"], "goal": "PrimeSpectrum.vanishingIdeal (⋃ i, t i) = ⨅ i, PrimeSpectrum.vanishingIdeal (t i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "𝕜 : Type u_4", "inst✝¹ : NormedDivisionRing 𝕜", "inst✝ : CompleteSpace 𝕜", "r : 𝕜", "hr : ‖r‖ < 1", "A : Summable fun n => ↑n * r ^ n"], "goal": "HasSum (fun n => ↑n * r ^ n) (r / (1 - r) ^ 2)"}}
+{"state": {"context": ["F : Type u_1", "R : Type u", "A : Type v", "B : Type w", "inst✝¹⁰ : CommSemiring R", "inst✝⁹ : NonUnitalNonAssocSemiring A", "inst✝⁸ : Module R A", "inst✝⁷ : IsScalarTower R A A", "inst✝⁶ : SMulCommClass R A A", "inst✝⁵ : NonUnitalNonAssocSemiring B", "inst✝⁴ : Module R B", "inst✝³ : IsScalarTower R B B", "inst✝² : SMulCommClass R B B", "inst✝¹ : FunLike F A B", "inst✝ : NonUnitalAlgHomClass F R A B", "f : A →ₙₐ[R] B"], "goal": "↑(NonUnitalSubalgebra.map f ⊥) = ↑⊥"}}
+{"state": {"context": ["n m a : ℕ", "h : n ≤ m", "ha : m - n ∣ a", "p : ℕ", "pp : Prime p", "hn : p ∣ n * a + 1", "hm : p ∣ m * a + 1", "this✝ : p ∣ (m - n) * a", "this : p ∣ a"], "goal": "p = 1"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "F : Type u_4", "MeasurableSpace X", "μ : MeasureTheory.Measure X", "NormedAddCommGroup E", "NormedAddCommGroup F", "NormedSpace ℝ E", "NormedSpace ℝ F", "CompleteSpace F", "f : X → E × F", "hf : MeasureTheory.Integrable f μ"], "goal": "(∫ (x : X), f x ∂μ).1 = ∫ (x : X), (f x).1 ∂μ"}}
+{"state": {"context": ["M : Type u_2", "AddZeroClass M", "x : AddSubmonoid M"], "goal": "↑x.op = AddOpposite.unop ⁻¹' ↑x"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "S T : Set (Set α)", "h_gen : m0 = generateFrom S", "h_inter : IsPiSystem S", "h_sub : T ⊆ S", "hc : T.Countable", "hU : ⋃₀ T = univ", "htop : ∀ s ∈ T, μ s ≠ ⊤", "h_eq : ∀ s ∈ S, μ s = ν s"], "goal": "∀ t ∈ T, ∀ s ∈ S, μ (s ∩ t) = ν (s ∩ t)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "TopologicalSpace α", "Topology.IsLawson α"], "goal": "TopologicalSpace.IsTopologicalBasis (Topology.IsLawson.lawsonBasis α)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "E' : Type u_3", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H : Type u_4", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "H' : Type u_5", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "N : Type u_6", "TopologicalSpace N", "ChartedSpace H N", "G : Type u_10", "Zero G", "TopologicalSpace G", "ChartedSpace H' G"], "goal": "⇑0 = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : LocallyCompactSpace X", "inst✝ : RegularSpace X", "K U : Set X", "hK : IsCompact K", "hU : IsOpen U", "hKU : K ⊆ U", "L : Set X", "L_compact : IsCompact L", "L_closed : IsClosed L", "KL : K ⊆ interior L", "LU : L ⊆ U"], "goal": "∃ V, IsOpen V ∧ K ⊆ V ∧ closure V ⊆ U ∧ IsCompact (closure V)"}}
+{"state": {"context": ["G : Type u_1", "X : Type u_2", "inst✝¹⁴ : TopologicalSpace G", "inst✝¹³ : TopologicalSpace X", "inst✝¹² : Group G", "inst✝¹¹ : TopologicalGroup G", "inst✝¹⁰ : MulAction G X", "inst✝⁹ : SigmaCompactSpace G", "inst✝⁸ : BaireSpace X", "inst✝⁷ : T2Space X", "inst✝⁶ : ContinuousSMul G X", "inst✝⁵ : IsPretransitive G X", "H : Type u_3", "inst✝⁴ : Group H", "inst✝³ : TopologicalSpace H", "inst✝² : BaireSpace H", "inst✝¹ : T2Space H", "inst✝ : ContinuousMul H", "f : G →* H", "hf : Surjective ⇑f", "h'f : Continuous ⇑f", "A : MulAction G H := MulAction.compHom H f", "this : ContinuousSMul G H"], "goal": "IsOpenMap ⇑f"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "DivisionRing K", "AddCommGroup V", "Module K V", "v : Projectivization K V"], "goal": "FiniteDimensional.finrank K ↥v.submodule = 1"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y : ℝ", "hfc : StrictConcaveOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hfd : DifferentiableAt ℝ f y"], "goal": "deriv f y < slope f x y"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "u v : V"], "goal": "G.dist u v = sInf (Set.range SimpleGraph.Walk.length)"}}
+{"state": {"context": ["R : Type u", "ι : Type v", "M₁ : ι → Type w₁", "M₂ : Type w₂", "Semiring R", "(i : ι) → AddCommMonoid (M₁ i)", "AddCommMonoid M₂", "(i : ι) → Module R (M₁ i)", "Module R M₂", "(i : ι) → TopologicalSpace (M₁ i)", "TopologicalSpace M₂", "f : ContinuousMultilinearMap R M₁ M₂"], "goal": "⇑f.toMultilinearMap = ⇑f"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝² : CommRing R", "inst✝¹ : Ring S", "f : R →+* S", "I : Type u_3", "inst✝ : Finite I", "e : I → R", "he : OrthogonalIdempotents e", "i j : I", "hij : i ≠ j"], "goal": "IsCoprime (1 - e i) (1 - e j)"}}
+{"state": {"context": ["n : ℕ"], "goal": "↑(WeierstrassCurve.expCoeff n) = if Even n then ↑n / 2 else ↑n"}}
+{"state": {"context": ["R : Type u_2", "CommRing R", "S : Type u_3", "CommRing S", "Algebra R S", "r : R", "IsLocalization.Away r S"], "goal": "(IsLocalization.Away.auxHom S r).comp (IsLocalization.Away.auxInv S r) = RingHom.id S"}}
+{"state": {"context": ["R : Type u", "S : Type v", "K : Type w", "inst✝¹ : CommRing R", "g : R[X]", "inst✝ : Field K", "f : K[X]", "hf : f ≠ 0"], "goal": "minpoly K (root f) = f * C f.leadingCoeff⁻¹"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X : SimplicialObject C", "Y : C", "n q : ℕ", "φ : Y ⟶ X _[n + 1]", "v : HigherFacesVanish q φ", "j : Fin (n + 1)", "hj₁ : n + 1 ≤ ↑j + (q + 1)"], "goal": "(φ ≫ (𝟙 (X _[n + 1]) + (Hσ q).f (n + 1))) ≫ X.δ j.succ = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "s : Set α", "t : Set β", "hst : ↑s ≃ ↑t"], "goal": "s.Finite ↔ t.Finite"}}
+{"state": {"context": ["K : Type u", "V : Type v", "Field K", "AddCommGroup V", "Module K V", "FiniteDimensional K V", "Φ : Subspace K (Module.Dual K V)"], "goal": "FiniteDimensional.finrank K ↥(Submodule.dualCoannihilator Φ) =\n  FiniteDimensional.finrank K ↥(Submodule.dualAnnihilator Φ)"}}
+{"state": {"context": ["x : ℂ"], "goal": "Complex.cos ((starRingEnd ℂ) x) = (starRingEnd ℂ) (Complex.cos x)"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "s : Finset α", "CommMonoid β", "f : α → β", "p : α → Prop", "DecidablePred p", "h : ∀ x ∈ s, p x"], "goal": "∏ x ∈ Finset.subtype p s, f ↑x = ∏ x ∈ s, f x"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "s ⊆ tᶜ ↔ Disjoint s t"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℂ E", "V : Type u_2", "W : Type u_3", "inst✝⁸ : NormedAddCommGroup V", "inst✝⁷ : NormedSpace ℝ V", "inst✝⁶ : NormedAddCommGroup W", "inst✝⁵ : NormedSpace ℝ W", "L : V →L[ℝ] W →L[ℝ] ℝ", "f : V → E", "inst✝⁴ : SecondCountableTopology V", "inst✝³ : MeasurableSpace V", "inst✝² : BorelSpace V", "μ✝ : Measure V", "inst✝¹ : FiniteDimensional ℝ V", "μ : Measure V", "inst✝ : μ.IsAddHaarMeasure", "K N : ℕ∞", "hf : ContDiff ℝ N f", "h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ", "k n : ℕ", "hk : ↑k ≤ K", "hn : ↑n ≤ N", "w : W", "I : ∀ p ∈ Finset.range (k + 1) ×ˢ Finset.range (n + 1), Integrable (fun v => ‖v‖ ^ p.1 * ‖iteratedFDeriv ℝ p.2 f v‖) μ"], "goal": "0 ≤ᶠ[ae μ] fun a => ‖iteratedFDeriv ℝ n (fun v => fourierPowSMulRight L f v k) a‖"}}
+{"state": {"context": ["F : Type u_1", "E : Type u_2", "Field F", "Ring E", "Algebra F E", "hr : FiniteDimensional.finrank F E = 2"], "goal": "IsSimpleOrder (Subalgebra F E)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E", "𝔸 : Type u_6", "NormedRing 𝔸", "NormedAlgebra 𝕜 𝔸", "a : E → 𝔸", "a' : E →L[𝕜] 𝔸", "ha : HasStrictFDerivAt a a' x", "b : 𝔸"], "goal": "HasStrictFDerivAt (fun y => a y * b) (a'.smulRight b) x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K : ℝ≥0", "f : α → β", "hf : AntilipschitzWith K f", "x y : α"], "goal": "edist x y / ↑K ≤ edist (f x) (f y)"}}
+{"state": {"context": ["α : Type u", "p : α → Prop", "f : Filter α", "hp : ∃ᶠ (x : α) in f, p x"], "goal": "∃ x, p x"}}
+{"state": {"context": ["F : Type u", "Field F", "r : F"], "goal": "↑(RatFunc.C r) = HahnSeries.C r"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "M₂ : Type w₁", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "N N' : LieSubmodule R L M", "I J : LieIdeal R L", "N₂ : LieSubmodule R L M₂"], "goal": "⁅I, ⊥⁆ = ⊥"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop"], "goal": "m * m < n * n ↔ m < n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ✝ : Kernel α β", "f : α → β → ℝ≥0∞", "κ : Kernel α β", "inst✝ : IsSFiniteKernel κ", "a✝¹ : α", "s✝ : Set β", "a✝ : MeasurableSet s✝"], "goal": "((κ.withDensity 1) a✝¹) s✝ = (κ a✝¹) s✝"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "f : α → γ", "g : β → δ", "p : α × β"], "goal": "Prod.map f g p = (f p.1, g p.2)"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E", "inst✝⁵ : Algebra F E", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "inst✝² : Algebra E K", "inst✝¹ : IsScalarTower F E K", "x : K", "hsep : IsSeparable F x", "inst✝ : IsPurelyInseparable F E", "hi : IsIntegral F x", "hi' : IsIntegral E x", "hsep' : IsSeparable E x", "this✝² : Algebra.IsSeparable F ↥F⟮x⟯", "this✝¹ : Algebra.IsSeparable E ↥E⟮x⟯", "this✝ : Algebra.IsAlgebraic F ↥F⟮x⟯", "this : Algebra.IsAlgebraic E ↥E⟮x⟯"], "goal": "(Polynomial.map (algebraMap F E) (minpoly F x)).natDegree = (minpoly E x).natDegree"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_3", "Semiring R", "S : Type u_4", "Monoid S", "AddCommMonoid M", "Module R M", "DistribMulAction S M", "s : Set S", "N : Submodule R M", "r : S", "m : M", "mem1 : r ∈ s", "mem2 : m ∈ N"], "goal": "r • m ∈ s • N"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "β : Type u_3", "AddGroup G", "AddAction G α", "AddAction G β"], "goal": "AddAction.orbitRel G (α × β) ≤ Setoid.comap Prod.snd (AddAction.orbitRel G β)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "M : Type u_4", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : AddCommMonoid M", "inst✝⁹ : Module R M", "inst✝⁸ : Semiring A", "inst✝⁷ : Semiring B", "inst✝⁶ : Module A M", "inst✝⁵ : Module B M", "inst✝⁴ : Algebra R A", "inst✝³ : Algebra R B", "inst✝² : IsScalarTower R A M", "inst✝¹ : IsScalarTower R B M", "inst✝ : SMulCommClass A B M", "p : AddSubmonoid M", "hA : ∀ (a : A) {m : M}, m ∈ p → a • m ∈ p", "hB : ∀ (b : B) {m : M}, m ∈ p → b • m ∈ p", "ab : A ⊗[R] B", "m : M", "x✝ : m ∈ { carrier := ↑p, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"], "goal": "0 • m ∈ { carrier := ↑p, add_mem' := ⋯, zero_mem' := ⋯ }.carrier"}}
+{"state": {"context": ["R S : Type u", "M : Type v", "M' : Type v'", "M₁ : Type v", "ι : Type w", "ι' : Type w'", "η : Type u₁'", "φ : η → Type u_1", "inst✝¹² : Ring R", "inst✝¹¹ : CommRing S", "inst✝¹⁰ : AddCommGroup M", "inst✝⁹ : AddCommGroup M'", "inst✝⁸ : AddCommGroup M₁", "inst✝⁷ : Module R M", "inst✝⁶ : Module R M'", "inst✝⁵ : Module R M₁", "inst✝⁴ : StrongRankCondition R", "inst✝³ : Module.Free R M", "inst✝² : Module.Free R M'", "m : Type v", "n : Type w", "inst✝¹ : Finite m", "inst✝ : Finite n", "val✝¹ : Fintype m", "val✝ : Fintype n"], "goal": "Module.rank R (Matrix m n R) = lift.{max v w u, v} #m * lift.{max v w u, w} #n"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : IsLocallyFiniteMeasure μ", "inst✝ : IsUnifLocDoublingMeasure μ", "p : ℕ → Prop", "s : ℕ → Set α", "r : ℕ → ℝ", "hr : Tendsto r atTop (𝓝 0)", "hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < r i"], "goal": "blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Finite α", "inst✝ : DecidableEq α", "f : Perm α", "hf : f.IsCycle", "val✝ : Fintype α", "x : α", "hx : f x ≠ x", "hy : ∀ ⦃y : α⦄, f y ≠ y → f.SameCycle x y"], "goal": "∀ (y : Cycle α), (fun s => ∃ (h : s.Nodup), s.formPerm h = f) y → y = ↑(f.toList x)"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_3", "f : ι → Filter α"], "goal": "(iInf f).smallSets = ⨅ i, (f i).smallSets"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrderedField α", "Ring β", "abv : β → α", "IsAbsoluteValue abv", "x : β", "n : ℕ"], "goal": "CauSeq.const abv (x ^ n) = CauSeq.const abv x ^ n"}}
+{"state": {"context": ["ι : Sort u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁴ : OrderedSemiring 𝕜", "inst✝³ : AddCommMonoid E", "inst✝² : Module 𝕜 E", "inst✝¹ : AddCommMonoid F", "inst✝ : Module 𝕜 F", "ts : Set (TopologicalSpace E)", "h : ∀ t ∈ ts, LocallyConvexSpace 𝕜 E", "this : TopologicalSpace E := sInf ts", "x : E"], "goal": "(𝓝 x).HasBasis ((fun x If => If.1.Finite ∧ ∀ i ∈ If.1, If.2 i ∈ 𝓝 x ∧ Convex 𝕜 (If.2 i)) x) ((fun x If => ⋂ i ∈ If.1, If.2 i) x)"}}
+{"state": {"context": ["R : Type uR", "M₁ : Type uM₁", "M₂ : Type uM₂", "M₃ : Type uM₃", "M₄ : Type uM₄", "inst✝⁹ : CommRing R", "inst✝⁸ : AddCommGroup M₁", "inst✝⁷ : AddCommGroup M₂", "inst✝⁶ : AddCommGroup M₃", "inst✝⁵ : AddCommGroup M₄", "inst✝⁴ : Module R M₁", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M₄", "inst✝ : Invertible 2", "Q₂ : QuadraticForm R M₂"], "goal": "comp Q₂ ↑(TensorProduct.lid R M₂) = QuadraticForm.tmul QuadraticMap.sq Q₂"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "f : R[X]", "h : f.eraseLead = 0"], "goal": "f.nextCoeff = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝ : CompleteDistribLattice α", "f : Filter β", "p q : β → Prop", "u : β → α", "g : Filter β"], "goal": "limsup u (f ⊔ g) = limsup u f ⊔ limsup u g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "p : α → Prop", "f : (a : α) → p a → β", "x : Option α", "a : α", "h : ∀ (a : α), a ∈ x → p a", "ha✝ : a ∈ x", "ha : x = some a"], "goal": "pmap f x h = some (f a ⋯)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s₁ s₂ : Multiset α", "t₁ t₂ : Multiset β", "hs : s₁ ≤ s₂", "ht : t₁ < t₂"], "goal": "s₁.disjSum t₁ < s₂.disjSum t₂"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹ : CommSemiring R", "inst✝ : AddMonoid M", "S : Set R[M]", "hS : adjoin R S = ⊤"], "goal": "S ⊆ ↑(adjoin R (⋃ f ∈ S, of' R M '' ↑f.support))"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "p : Submodule R E", "q : { q // IsCompl p q }"], "goal": "↑(p.isComplEquivProj q) = p.linearProjOfIsCompl ↑q ⋯"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "inst✝³ : CommRing A", "inst✝² : Algebra R S", "inst✝¹ : Algebra R A", "inst✝ : Algebra.FinitePresentation R A", "f : A →ₐ[R] S", "hf : Function.Surjective ⇑f", "t : Finset A", "ht : Ideal.span ↑t = ⊤", "H : ∀ (g : { x // x ∈ t }), Algebra.FinitePresentation R (Localization.Away (f ↑g))", "g : ↑↑t", "f' : Localization.Away ↑g →ₐ[R] Localization.Away (f ↑g) := Localization.awayMapₐ f ↑g"], "goal": "(ker (Localization.awayMap f.toRingHom ↑g)).FG"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "f : ι → α", "p : α → Prop"], "goal": "(∀ a ∈ Set.range f, p a) ↔ ∀ (i : ι), p (f i)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "hs : HasFDerivWithinAt f f' s x", "ht : HasFDerivWithinAt f f' t x"], "goal": "HasFDerivAtFilter f f' x (𝓝[s] x ⊔ 𝓝[t] x)"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_3", "ι : Type u_5", "CommRing R", "Nontrivial R", "LieRing L", "LieAlgebra R L", "Module.Finite R L", "Module.Free R L", "Fintype ι", "DecidableEq ι", "b : Basis ι R L"], "goal": "((↑(LieAlgebra.ad R L)).polyCharpoly b).coeff (LieAlgebra.rank R L) ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "c : ℝ≥0∞", "f : α → ℝ≥0∞"], "goal": "∫⁻ (a : α), f a ∂c • μ = c * ∫⁻ (a : α), f a ∂μ"}}
+{"state": {"context": ["α : Type u_2", "HeytingAlgebra α", "a : α"], "goal": "a ⇨ aᶜ = aᶜ"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "Lattice α", "Lattice β", "Lattice γ", "g : LatticeHom β γ", "f₁ f₂ : LatticeHom α β", "hg : Function.Injective ⇑g"], "goal": "g.comp f₁ = g.comp f₂ ↔ f₁ = f₂"}}
+{"state": {"context": ["C : ℕ", "u : ℕ → ℕ", "h_pos : ∀ (n : ℕ), 0 < u n", "hu_strict : StrictMono u", "hC_nonzero : C ≠ 0", "h_succ_diff : SuccDiffBounded C u", "f : ℕ → ℝ≥0", "hf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m", "this : ∀ (k : ℕ), ↑(u (k + 1)) - ↑(u k) = ↑(↑(u (k + 1)) - ↑(u k))"], "goal": "(Summable fun k => ↑(↑(u (k + 1)) - ↑(u k)) * ↑(f (u k))) ↔ Summable fun i => ↑(f i)"}}
+{"state": {"context": ["n : ℕ", "c : CompositionAsSet n"], "goal": "c.blocks.sum = n"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : Type u_5", "A' : Type u_6", "x : ι → A", "CommRing R", "CommRing A", "CommRing A'", "Algebra R A", "Algebra R A'", "f : A →ₐ[R] A'", "hfv : AlgebraicIndependent R (⇑f ∘ x)"], "goal": "AlgebraicIndependent R x"}}
+{"state": {"context": ["A : Type u_4", "CommRing A", "IsDomain A", "K : Type u_5", "CommRing K", "Algebra A K", "IsFractionRing A K"], "goal": "IsDomain K"}}
+{"state": {"context": ["k : ℕ"], "goal": "(k + 1 + 1)! = (k + 1 + 1)‼ * (k + 1)‼"}}
+{"state": {"context": ["N k : ℕ"], "goal": "(N.roughNumbersUpTo k).card ≤ ∑ p ∈ N.succ.primesBelow \\ k.primesBelow, N / p"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝² : Semiring R", "r : R[X]", "inst✝¹ : Semiring S", "f : R →+* S", "x : S", "inst✝ : Semiring T", "p q : R[ℕ]", "hf : ∀ (k : ℕ), Commute (f (q k)) x"], "goal": "(liftNC ↑f ⇑((powersHom S) x)) (p * q) = (liftNC ↑f ⇑((powersHom S) x)) p * (liftNC ↑f ⇑((powersHom S) x)) q"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "q : ↑↑(Spec A⁰_ f).toPresheafedSpace", "a b : A", "ha : a ∈ carrier f_deg q", "hb : b ∈ carrier f_deg q", "i : ℕ", "g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }"], "goal": "∀ j ∈ range (m + m + 1), Localization.mk ((m + m).choose j • ((proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m + m - j))) ⟨f ^ (i + i), ⋯⟩ = (algebraMap (A⁰_ f) (Localization.Away f)) (g j)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "PartialOrder α", "f : β → β → α", "comm : ∀ (a b : β), f a b ≤ f b a", "a b : β"], "goal": "f a b = f b a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "iso : E ≃L[𝕜] F", "f : F → G", "s : Set F"], "goal": "DifferentiableOn 𝕜 (f ∘ ⇑iso) (⇑iso ⁻¹' s) ↔ DifferentiableOn 𝕜 f s"}}
+{"state": {"context": ["C : Type u_5", "D : Type u_3", "H : Type u_4", "CategoryTheory.Category.{u_6, u_5} C", "CategoryTheory.Category.{u_1, u_3} D", "CategoryTheory.Category.{u_2, u_4} H", "RF : CategoryTheory.Functor D H", "F : CategoryTheory.Functor C H", "L : CategoryTheory.Functor C D", "α : F ⟶ L.comp RF", "W : CategoryTheory.MorphismProperty C", "L.IsLocalization W", "RF.IsRightDerivedFunctor α W", "G : CategoryTheory.Functor D H", "β : F ⟶ L.comp G"], "goal": "G.IsRightDerivedFunctor β W ↔ CategoryTheory.IsIso (RF.rightDerivedDesc α W G β)"}}
+{"state": {"context": [], "goal": "∀ᶠ (x : ℝ) in residual ℝ, Liouville x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i : ι", "K.HasHomology i"], "goal": "K.ExactAt i ↔ CategoryTheory.Limits.IsZero (K.homology i)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ε : Type u_5", "ζ : Type u_6", "r : α → α → Prop", "a b : α", "h✝ : Equivalence r", "h : EqvGen r a b"], "goal": "r a b"}}
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+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "A : Set M", "α : Type u₁", "x : α → M"], "goal": "x ∉ ⊥"}}
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+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "UniformSpace α", "UniformSpace β", "UniformSpace γ", "f : α × β → γ"], "goal": "UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f"}}
+{"state": {"context": ["σ : Type u_1", "s : Set σ", "R : Type u_3", "CommRing R", "F : MvPolynomial σ ℤ", "hF : ↑F.vars ⊆ s"], "goal": "∃ f, ∀ (x : σ → R), f (x ∘ Subtype.val) = (MvPolynomial.aeval x) F"}}
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+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "hm : 0 < m", "q : ↑↑(Spec A⁰_ f).toPresheafedSpace", "i : ℕ", "e_1✝ : HomogeneousLocalization 𝒜 (Submonoid.powers f) = ↑(CommRingCat.of (A⁰_ f))"], "goal": "Localization.mk 0 ⟨f ^ i, ⋯⟩ = 0"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : Rel α β"], "goal": "r.inv.inv = r"}}
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+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "S : Type u_2", "inst✝ : CommRing S", "f : R →+* S", "I J : Ideal S", "P : Ideal R[X]"], "goal": "Function.Injective ⇑(quotientMap (map (mapRingHom (Quotient.mk (comap C P))) P) (mapRingHom (Quotient.mk (comap C P))) ⋯)"}}
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+{"state": {"context": ["α : Type u_1", "Max α", "a b : α"], "goal": "max (some a) (some b) = some (max a b)"}}
+{"state": {"context": ["R : Type u_4", "Ring R", "r : R", "k : ℤ"], "goal": "↑k * r ∈ AddSubgroup.zmultiples r"}}
+{"state": {"context": ["A : Type u_2", "K : Type u_3", "CommRing A", "Field K", "Algebra A K", "IsFractionRing A K", "IsDedekindDomain A", "I : FractionalIdeal A⁰ K", "hI : I ≠ 0"], "goal": "StrictMono fun x => I * x"}}
+{"state": {"context": ["α β : Type u", "f : α → FreeGroup β", "x : FreeGroup α"], "goal": "x⁻¹ >>= f = (x >>= f)⁻¹"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β", "h : Set.MapsTo f s t"], "goal": "Filter.Tendsto f (𝓟 s) (𝓟 t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "f g : α → β", "hf : Monotone f", "hg : Antitone g"], "goal": "Antitone fun x => Set.Ioo (f x) (g x)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "SupSet α", "s : Set β", "f : β → α"], "goal": "sSup (f '' s) = ⨆ a, f ↑a"}}
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+{"state": {"context": ["α : Type u", "p : α", "s : Set α"], "goal": "(FreeCommRing.of p).IsSupported s ↔ p ∈ s"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "MeasurableSpace α", "f : MeasureTheory.SimpleFunc α (β → γ)", "g : MeasureTheory.SimpleFunc α β", "a : α"], "goal": "↑(f.seq g) a = ↑f a (↑g a)"}}
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+{"state": {"context": ["α : Type u", "Zero α", "β : Type u_1", "f : α → β"], "goal": "WithBot.map f 0 = ↑(f 0)"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "p : α → Prop", "inst✝ : DecidablePred p", "s : Multiset α", "_l : List α"], "goal": "countP p (Quot.mk Setoid.r _l) = 0 ↔ ∀ a ∈ Quot.mk Setoid.r _l, ¬p a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "f : α → α", "x : α", "hf : StrictMono f", "hx : x < f x"], "goal": "StrictMono fun n => f^[n] x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Computation α", "f : α → Computation β", "g : β → Computation γ"], "goal": "(s.bind f).bind g = s.bind fun x => (f x).bind g"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "CommRing R", "Field K", "Algebra R K", "IsFractionRing R K", "x : Kˣ"], "goal": "↑((toPrincipalIdeal R K) x) = FractionalIdeal.spanSingleton R⁰ ↑x"}}
+{"state": {"context": ["R : Type u_3", "Γ₀ : Type u_4", "LinearOrderedAddCommGroupWithTop Γ₀", "Ring R", "v : AddValuation R Γ₀", "x y : R", "h : v x < v y"], "goal": "v (x + y) = v x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : (i : ι) → TopologicalSpace (π i)", "x y z : X", "s : Set X", "f✝ g✝ : X → Y", "f g : (i : ι) → π i"], "goal": "(f ~ᵢ g) ↔ ∀ (i : ι), f i ~ᵢ g i"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "IsNormalSubgroup (IsSubgroup.center G)"}}
+{"state": {"context": ["ι : Type u_1", "I J : WithBot (BoxIntegral.Box ι)"], "goal": "↑I ⊆ ↑J ↔ I ≤ J"}}
+{"state": {"context": ["G : Type u_1", "Group G", "s : Set G", "hs : IsNormalSubgroup s"], "goal": "IsNormalAddSubgroup s"}}
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+{"state": {"context": ["α : Type u", "x y : ULift.{v, u} α", "Add α"], "goal": "(x + y).down = x.down + y.down"}}
+{"state": {"context": ["R : Type u_1", "r : R", "inst✝ : CommRing R", "P✝ P : R[X]", "N : ℕ", "hN : P.natDegree ≤ N", "h : ∀ (i : ℕ), IsNilpotent ((reflect N P).coeff i)", "i : ℕ", "hi : i ≤ N"], "goal": "IsNilpotent (P.coeff i)"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕝 : Type u_2", "E : Type u_3", "F : Type u_4", "β : Type u_5", "inst✝⁵ : OrderedSemiring 𝕜", "inst✝⁴ : TopologicalSpace E", "inst✝³ : TopologicalSpace F", "inst✝² : AddCancelCommMonoid E", "inst✝¹ : ContinuousAdd E", "inst✝ : Module 𝕜 E", "s : Set E", "hs : StrictConvex 𝕜 s", "z x : E", "hx : x ∈ (fun x => z + x) ⁻¹' s", "y : E", "hy : y ∈ (fun x => z + x) ⁻¹' s", "hxy : x ≠ y", "a b : 𝕜", "ha : 0 < a", "hb : 0 < b", "hab : a + b = 1"], "goal": "a • x + b • y ∈ (fun b => z + b) ⁻¹' interior s"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Ring R", "n : ℤ"], "goal": "(↑n).natDegree = 0"}}
+{"state": {"context": ["α : Type u", "Order.Coframe α", "s : Set α", "b : α"], "goal": "sInf s ⊔ b = ⨅ a ∈ s, a ⊔ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "R : Type u_4", "inst✝ : NormedRing R", "k : ℕ", "r₁ : R", "r₂ : ℝ", "h : ‖r₁‖ < r₂", "h0 : 0 < ‖r₁‖"], "goal": "(fun n => ↑n ^ k * r₁ ^ n) =o[atTop] fun n => r₂ ^ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "x : α", "s : Set α", "f : α → β", "ctsf : ContinuousWithinAt f s x"], "goal": "𝓝[s] x ≤ Filter.comap f (𝓝[f '' s] f x)"}}
+{"state": {"context": [], "goal": "∀ {R S : Type u_1} [inst : CommRing R] [inst_1 : CommRing S] (e : R ≃+* S), Function.Surjective ⇑e.toRingHom"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "X' : Type u_4", "Y : Type u_5", "Z : Type u_6", "α : Type u_7", "α' : Type u_8", "β : Type u_9", "β' : Type u_10", "γ : Type u_11", "𝓕 : Type u_12", "tX : TopologicalSpace X", "tY : TopologicalSpace Y", "tZ : TopologicalSpace Z", "uα : UniformSpace α", "uβ : UniformSpace β", "uγ : UniformSpace γ", "F : ι → β → α", "S : Set β", "h : UniformEquicontinuousOn F S"], "goal": "UniformEquicontinuousOn (F ∘ rangeSplitting F) S"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "X₁ X₂ : C"], "goal": "(λ_ X₁).hom ⊗ (λ_ X₂).hom =\n  CategoryTheory.tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ (λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂) ≫ (λ_ (X₁ ⊗ X₂)).hom"}}
+{"state": {"context": ["x y : Unit"], "goal": "x = y"}}
+{"state": {"context": ["ι : Type w", "Fintype ι", "A : Type u", "B : Type v", "CommRing A", "CommRing B", "Algebra A B", "b : Basis ι A B", "z : B"], "goal": "Algebra.traceMatrix A ⇑b *ᵥ b.equivFun z = fun i => (Algebra.trace A B) (z * b i)"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "CategoryTheory.Category.{v, u} C", "AddGroup A", "CategoryTheory.HasShift C A", "n : A", "X : C"], "goal": "(CategoryTheory.shiftFunctor C (-n)).map ((CategoryTheory.shiftFunctorCompIsoId C (-n) n ⋯).inv.app X) =\n  (CategoryTheory.shiftFunctorCompIsoId C n (-n) ⋯).inv.app ((CategoryTheory.shiftFunctor C (-n)).obj X)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "e : ↑s ↪ ↑t"], "goal": "s.encard ≤ t.encard"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s s' : Cycle α", "a✝ : List α", "h : Nodup (Quot.mk Setoid.r a✝)"], "goal": "(reverse (Quot.mk Setoid.r a✝)).formPerm ⋯ = (formPerm (Quot.mk Setoid.r a✝) h)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "δ : Type u_3", "r✝ : α → β → Prop", "p : γ → δ → Prop", "r : α → α → Prop", "inst✝² : IsTrans α r", "inst✝¹ : IsSymm α r", "x : α", "inst✝ : DecidablePred (r x)", "t : Multiset α", "h : Rel r 0 t"], "goal": "countP (r x) 0 = countP (r x) t"}}
+{"state": {"context": ["E : ℕ → Type u_1", "x y : (n : ℕ) → E n", "n : ℕ", "hn : n ≤ firstDiff x y"], "goal": "x ∈ cylinder y n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "h₁ : Embedding f", "h₂ : IsClosedMap f"], "goal": "ClosedEmbedding f"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "M N : Submodule R A", "h : M.FG", "n : ℕ"], "goal": "span R ↑{1} = M ^ Nat.zero"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "p : ℝ", "hp : 1 < p"], "goal": "∀ (x : E), ContinuousAt (fderiv ℝ fun x => ‖x‖ ^ p) x"}}
+{"state": {"context": ["α : Type u", "m n : ℕ", "a : Fin m → α", "b : Fin n → α"], "goal": "List.ofFn (Fin.append a b) = List.ofFn a ++ List.ofFn b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "f : α → β", "g : γ → δ", "x : α ⊕ γ"], "goal": "(Sum.map f g x).isLeft = x.isLeft"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "c : ClosureOperator α", "x : α"], "goal": "c.IsClosed x → c x = x"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁹ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝² : NormedAddCommGroup G'", "inst✝¹ : NormedSpace 𝕜 G'", "ε : ℝ", "inst✝ : CompleteSpace E", "f : E → F", "f' : E ≃L[𝕜] F", "a : E", "hf : HasStrictFDerivAt f (↑f') a"], "goal": "∃ s, a ∈ s ∧ IsOpen s ∧ ApproximatesLinearOn f (↑f') s (‖↑f'.symm‖₊⁻¹ / 2)"}}
+{"state": {"context": ["α : Sort u_3", "β : Sort u_4", "γ : Sort u_5", "f : α → β → γ", "P : Prop", "Decidable P", "a b : α", "c d : β"], "goal": "f (if P then a else b) (if P then c else d) = if P then f a c else f b d"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_8", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "(LinearEquiv.refl R M).symm = LinearEquiv.refl R M"}}
+{"state": {"context": ["x✝ y✝ x : ℝ", "hex : rexp 1 ≤ x", "y : ℝ", "hey : rexp 1 ≤ y", "hxy : x ≤ y"], "goal": "log y / y ≤ log x / x"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "τ : Type u_3", "S : Type u_4", "inst✝³ : CommSemiring R", "inst✝² : CommSemiring S", "inst✝¹ : Fintype σ", "inst✝ : Fintype τ", "n : ℕ"], "goal": "esymm σ R n = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, (monomial (Finsupp.single i 1)) 1"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "m n : ℕ", "h : m ≤ n"], "goal": "I ^ n ≤ I ^ m"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "X Y Z : C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "A : C", "B : D", "e : MonoOver A ≌ MonoOver B"], "goal": "𝟭 (Subobject A) = ThinSkeleton.map (𝟭 (MonoOver A))"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "LE α", "LE β", "e : α ≃o β"], "goal": "e.symm = e.symm.toEquiv"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "LocallyFiniteOrder α", "a b c d : α", "h₁ : min a b ≤ max c d", "h₂ : min c d ≤ max a b"], "goal": "Finset.Ico a b ∪ Finset.Ico c d = Finset.Ico (min a c) (max b d)"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)"], "goal": "Padic.addValuationDef 0 = ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²⁰ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁹ : NormedAddCommGroup E", "inst✝¹⁸ : NormedSpace 𝕜 E", "E' : Type u_3", "inst✝¹⁷ : NormedAddCommGroup E'", "inst✝¹⁶ : NormedSpace 𝕜 E'", "F : Type u_4", "inst✝¹⁵ : NormedAddCommGroup F", "inst✝¹⁴ : NormedSpace 𝕜 F", "H : Type u_5", "inst✝¹³ : TopologicalSpace H", "H' : Type u_6", "inst✝¹² : TopologicalSpace H'", "G : Type u_7", "inst✝¹¹ : TopologicalSpace G", "G' : Type u_8", "inst✝¹⁰ : TopologicalSpace G'", "I : ModelWithCorners 𝕜 E H", "I' : ModelWithCorners 𝕜 E' H'", "J : ModelWithCorners 𝕜 F G", "J' : ModelWithCorners 𝕜 F G'", "M : Type u_9", "inst✝⁹ : TopologicalSpace M", "inst✝⁸ : ChartedSpace H M", "M' : Type u_10", "inst✝⁷ : TopologicalSpace M'", "inst✝⁶ : ChartedSpace H' M'", "N : Type u_11", "inst✝⁵ : TopologicalSpace N", "inst✝⁴ : ChartedSpace G N", "N' : Type u_12", "inst✝³ : TopologicalSpace N'", "inst✝² : ChartedSpace G' N'", "n : ℕ∞", "inst✝¹ : SmoothManifoldWithCorners I M", "inst✝ : SmoothManifoldWithCorners J N", "h : M ≃ₘ^n⟮I, J⟯ N", "hn : 1 ≤ n", "s : Set M", "hs : UniqueMDiffOn I s"], "goal": "UniqueMDiffOn J (⇑h '' s)"}}
+{"state": {"context": ["G : Type u_6", "AddGroup G", "x : G"], "goal": "addOrderOf x ∣ Nat.card G"}}
+{"state": {"context": ["f g : MulRingNorm ℚ"], "goal": "(∀ (n : ℕ), f ↑n = g ↑n) ↔ ∀ (n : ℤ), f ↑n = g ↑n"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasLimits C", "ι : Type v", "F : Discrete ι ⥤ SheafedSpace C", "inst✝ : HasColimit F", "i : Discrete ι", "this : colimit.ι (F ⋙ forget C) i ≫ (HasColimit.isoOfNatIso Discrete.natIsoFunctor).hom = Discrete.natIsoFunctor.hom.app i ≫ colimit.ι (Discrete.functor ((F ⋙ forget C).obj ∘ Discrete.mk)) i"], "goal": "OpenEmbedding ⇑(colimit.ι (F ⋙ forget C) i ≫ (preservesColimitIso (forget C) F).inv)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Fintype α", "Fintype β", "f : α → β", "h : Function.Surjective f", "h' : ¬Function.Injective f"], "goal": "Fintype.card β < Fintype.card α"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : AddCommMonoid M", "inst✝⁹ : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝⁸ : CommRing R₁", "inst✝⁷ : AddCommGroup M₁", "inst✝⁶ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝⁵ : Field K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M₂' : Type u_7", "inst✝² : AddCommMonoid M₂'", "inst✝¹ : Module R M₂'", "inst✝ : FiniteDimensional K V", "B : BilinForm K V", "W : Submodule K V", "b₁ : B.IsRefl", "b₂ : (B.restrict W).Nondegenerate", "this : W ⊓ B.orthogonal W = ⊥"], "goal": "finrank K V ≤ finrank K V + finrank K ↥(W ⊓ B.orthogonal ⊤)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ : List α", "a t : α", "ts is : List α", "IH2✝ : (is.permutationsAux []).length + [].length ! = (is.length + [].length)!", "IH2 : (is.permutationsAux []).length + 1 = is.length !", "IH1 : (ts.permutationsAux (t :: is)).length + is.length ! * is.length.succ = (ts.length + is.length)! * (ts.length + is.length).succ"], "goal": "((t :: ts).permutationsAux is).length + is.length ! = ((t :: ts).length + is.length)!"}}
+{"state": {"context": ["α : Type u_1", "l : List (List (List α))"], "goal": "l.join.join = (List.map List.join l).join"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "hm : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace α", "BorelSpace α", "NormedAddCommGroup E", "NormedSpace ℝ E", "g : α → E", "x₀ : α", "s : Set α", "CompleteSpace E", "TopologicalSpace.MetrizableSpace α", "MeasureTheory.IsLocallyFiniteMeasure μ", "μ.IsOpenPosMeasure", "hs : IsCompact s", "c : α → ℝ", "hc : ContinuousOn c s", "h'c : ∀ y ∈ s, y ≠ x₀ → c y < c x₀", "hnc : ∀ x ∈ s, 0 ≤ c x", "hnc₀ : 0 < c x₀", "h₀ : x₀ ∈ closure (interior s)", "hmg : MeasureTheory.IntegrableOn g s μ", "hcg : ContinuousWithinAt g s x₀"], "goal": "Filter.Tendsto (fun n => (∫ (x : α) in s, c x ^ n ∂μ)⁻¹ • ∫ (x : α) in s, c x ^ n • g x ∂μ) Filter.atTop (𝓝 (g x₀))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "ρ : Measure (α × ℝ)", "r : ℝ"], "goal": "ρ.IicSnd r ≤ ρ.fst"}}
+{"state": {"context": ["α : Type u_1"], "goal": "∀ {x : MeasurableSpace α} {f : α → ℝ},\n  MeasureTheory.StronglyMeasurable f → MeasureTheory.StronglyMeasurable fun x => (f x).toNNReal"}}
+{"state": {"context": ["L : Language", "L' : Language", "M : Type w", "inst✝ : L.Structure M", "α : Type w'", "ie : IsEmpty α"], "goal": "((LHom.id L).sumElim (LHom.ofIsEmpty (constantsOn α) L)).comp (L.lhomWithConstants α) = LHom.id L"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹¹ : CommRing R", "inst✝¹⁰ : CommRing S", "inst✝⁹ : CommRing T", "inst✝⁸ : Algebra R S", "inst✝⁷ : Algebra R T", "K : Type u_4", "L : Type u_5", "inst✝⁶ : Field K", "inst✝⁵ : Field L", "inst✝⁴ : Algebra K L", "ι κ : Type w", "inst✝³ : Fintype ι", "F : Type u_6", "inst✝² : Field F", "inst✝¹ : Algebra K S", "inst✝ : Algebra K F", "x : L", "hx : ∃ s, Nonempty (Basis { x_1 // x_1 ∈ s } K ↥K⟮x⟯)"], "goal": "IsIntegral K x"}}
+{"state": {"context": ["α : Type u_5", "NonUnitalNonAssocRing α", "x : CentroidHom α"], "goal": "(-x).toEnd = -x.toEnd"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "W✝ : Affine R", "f : R →+* S", "F : Type u", "inst✝ : Field F", "W : Affine F", "x y : F", "h : W.Nonsingular x y"], "goal": "ClassGroup.mk (XYIdeal' ⋯) * ClassGroup.mk (XYIdeal' h) = 1"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "S : Set (Sublattice α)"], "goal": "↑(sInf S) = ⋂ L ∈ S, ↑L"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "NontriviallyNormedField 𝕜", "NonUnitalNormedRing A", "NormedSpace 𝕜 A", "SMulCommClass 𝕜 A A", "IsScalarTower 𝕜 A A", "StarRing 𝕜", "StarRing A", "StarModule 𝕜 A", "NormedStarGroup A", "a : A"], "goal": "DoubleCentralizer.coeHom a = ↑𝕜 a"}}
+{"state": {"context": ["n : ℕ", "F✝ : TypeVec.{u} (n + 1) → Type u", "q✝ : MvQPF F✝", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "x : Cofix F α", "R : Cofix F α → Cofix F α → Prop := fun x y => abs y.repr = x"], "goal": "∀ (x y : Cofix F α), R x y → LiftR' (α.RelLast' R) x.dest y.dest"}}
+{"state": {"context": ["this : HasDerivAt Gamma ↑(-√π * (γ + 2 * Real.log 2)) ↑(1 / 2)"], "goal": "HasDerivAt Gamma (-↑√π * (↑γ + 2 * log 2)) (1 / 2)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "MeasurableSingletonClass α", "s : Finset α"], "goal": "MeasurableSet ↑s"}}
+{"state": {"context": ["E : Type u_5", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "v : E", "s : Set ℝ"], "goal": "μH[1] ((fun r => r • v) '' s) = ‖v‖₊ • μH[1] s"}}
+{"state": {"context": ["G : Type v", "AddGroup G", "PseudoEMetricSpace G", "IsometricVAdd G G", "IsometricVAdd Gᵃᵒᵖ G"], "goal": "∀ (a : G), (IsometryEquiv.neg G) a = -a"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "P Q : CategoryTheory.Idempotents.Karoubi C", "f : P.Hom Q"], "goal": "CategoryTheory.CategoryStruct.comp P.p f.f = f.f"}}
+{"state": {"context": ["x y : SetTheory.PGame", "h : x ≤ y", "i : x.LeftMoves"], "goal": "(x.moveLeft i).LF y"}}
+{"state": {"context": ["G : Type u_1", "Mul G", "A B : Finset G", "a0 b0 : G"], "goal": "UniqueMul (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } B) (Finset.map { toFun := MulOpposite.op, inj' := ⋯ } A)\n    (MulOpposite.op b0) (MulOpposite.op a0) ↔\n  UniqueMul A B a0 b0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C", "S : CategoryTheory.ShortComplex.SnakeInput C"], "goal": "CategoryTheory.ShortComplex.SnakeInput.functorL₂'.obj S = S.L₂'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "NonAssocSemiring α", "NonAssocSemiring β", "f : α →+* β", "I : Type u_3", "h : I → α"], "goal": "∀ (a : I), (f.compLeft I) h a = (⇑f ∘ h) a"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β₁ : Type u_3", "β₂ : Type u_4", "γ₁ : Type u_5", "γ₂ : Type u_6", "f f₁ g₁ : α₁ → β₁ → Finset γ₁", "g f₂ g₂ : α₂ → β₂ → Finset γ₂", "a : α₁ ⊕ α₂", "b : β₁ ⊕ β₂", "c : γ₁ ⊕ γ₂", "c₂✝ : γ₂", "w✝¹ : α₁", "w✝ : β₁", "c₂ : γ₁", "left✝¹ : a = inl w✝¹", "left✝ : b = inl w✝", "h : inr c₂✝ = inl c₂", "right✝ : c₂ ∈ f w✝¹ w✝"], "goal": "False"}}
+{"state": {"context": ["α : Type v", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "s : Set α", "hs : IsConnected s", "a b : α", "ha : a ∈ s", "hb : b ∈ s"], "goal": "Set.Icc a b ⊆ s"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "inst✝⁴ : MulOneClass M", "inst✝³ : MulOneClass N", "inst✝² : MulOneClass P", "S✝ : Submonoid M", "A : Type u_4", "inst✝¹ : SetLike A M", "hA : SubmonoidClass A M", "S' : A", "F : Type u_5", "inst✝ : FunLike F M N", "mc : MonoidHomClass F M N", "f : F", "S : Submonoid N", "h : S ≤ MonoidHom.mrange f"], "goal": "map f (comap f S) = S"}}
+{"state": {"context": ["𝕂 : Type u_1", "m : Type u_2", "𝔸 : Type u_6", "RCLike 𝕂", "Fintype m", "DecidableEq m", "NormedRing 𝔸", "NormedAlgebra 𝕂 𝔸", "CompleteSpace 𝔸", "U : (Matrix m m 𝔸)ˣ", "A : Matrix m m 𝔸"], "goal": "NormedSpace.exp 𝕂 (↑U * A * ↑U⁻¹) = ↑U * NormedSpace.exp 𝕂 A * ↑U⁻¹"}}
+{"state": {"context": ["J : Type v", "inst✝ : SmallCategory J", "X✝ Y✝ Z : TopCat", "X Y S : TopCat", "f : X ⟶ S", "g : Y ⟶ S", "H₁ : Embedding ⇑f", "H₂ : Embedding ⇑g", "e_2✝ : ↑((cospan f g).obj WalkingCospan.one) = ↑S"], "goal": "⇑(limit.π (cospan f g) WalkingCospan.one) = ⇑g ∘ ⇑(pullback.snd f g)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "NonUnitalSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A"], "goal": "NonUnitalSubalgebra.centralizer R Set.univ = NonUnitalSubalgebra.center R A"}}
+{"state": {"context": ["ctx : Nat.Linear.Context", "fuel : Nat", "p₁ p₂ : Nat.Linear.Poly"], "goal": "Nat.Linear.Poly.denote ctx (Nat.Linear.Poly.combineAux fuel p₁ p₂) =\n  Nat.Linear.Poly.denote ctx p₁ + Nat.Linear.Poly.denote ctx p₂"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "S : Type u_4", "M : Type u_5", "M₁ : Type u_6", "M₂ : Type u_7", "M₃ : Type u_8", "N : Type u_9", "N₁ : Type u_10", "N₂ : Type u_11", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : AddCommGroup N₁", "inst✝⁴ : Module R M", "inst✝³ : Module R N₁", "inst✝² : Monoid S", "inst✝¹ : DistribMulAction S M", "inst✝ : SMulCommClass R S M", "f' : M →ₗ[R] M", "n : ℕ"], "goal": "f' ^ (n + 1) = (f' ^ n) ∘ₗ f'"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra R L", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : LieRingModule L M", "inst✝ : LieModule R L M", "k✝ : ℕ", "N N₁ N₂ : LieSubmodule R L M", "k : ℕ"], "goal": "k = 0 + k"}}
+{"state": {"context": ["M : Type u_2", "Mul M", "e₁ e₂ : MulAut M"], "goal": "⇑(e₁ * e₂) = ⇑e₁ ∘ ⇑e₂"}}
+{"state": {"context": ["p n : ℕ"], "goal": "padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "x p : ℝ", "hp : 1 < p"], "goal": "p * ‖x‖ ^ (p - 2) * x = ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) 1"}}
+{"state": {"context": ["a b c d : ℕ"], "goal": "a / b * (c / d) ≤ a * c / (b * d)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LE α", "LE β", "b : β"], "goal": "(OrderIso.sumLexDualAntidistrib α β).symm (Sum.inl (OrderDual.toDual b)) = OrderDual.toDual (Sum.inr b)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "s : Set α", "a b c d : α"], "goal": "(a < b ∨ c < b) ∧ a < c ↔ a < b ∧ a < c"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "SmoothManifoldWithCorners I' M'", "f : M → M'", "s : Set M", "x : M", "n : ℕ"], "goal": "ContMDiffWithinAt I I' (↑n) f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ContMDiffOn I I' (↑n) f u"}}
+{"state": {"context": ["α : Type u", "DecidableEq α", "l : List α", "H : l.Nodup", "i : Fin l.length"], "goal": "List.indexOf (l.get i) l = ↑i"}}
+{"state": {"context": ["m n : ℕ", "IH : shiftLeft' true m n ≠ 0 → (shiftLeft' true m n).size = m.size + n", "h : ¬bit true 0 = 0", "s0 : shiftLeft' true m n = 0", "this : shiftLeft' true m n + 1 = 1"], "goal": "(size 0).succ = (m.size + n).succ"}}
+{"state": {"context": ["X Y : Stonean", "f : X ⟶ Y", "h : Epi f", "y : (forget Stonean).obj Y", "hy : ∀ (a : (forget Stonean).obj X), f a ≠ y", "C : Set ((forget Stonean).obj Y) := Set.range ⇑f", "hC : IsClosed C", "U : Set ((forget Stonean).obj Y) := Cᶜ"], "goal": "False"}}
+{"state": {"context": ["M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "R : Type u_9", "S : Type u_10", "Semiring S", "SMul R M₁", "Module S M₁", "SMul R M₂", "Module S M₂", "LinearMap.CompatibleSMul M₁ M₂ R S", "f : M₁ →L[S] M₂", "c : R", "x : M₁"], "goal": "f (c • x) = c • f x"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_5, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Preadditive D", "F G : CategoryTheory.Functor C D", "ι : Type u_3", "s : Finset ι", "X : C", "α : ι → (F ⟶ G)"], "goal": "(∑ i ∈ s, α i).app X = ∑ i ∈ s, (α i).app X"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "PredOrder α", "a b : α"], "goal": "Order.pred (max a b) = max (Order.pred a) (Order.pred b)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I X J : Set α", "hI : M.Basis' I X", "hJ : M.Basis' J X"], "goal": "I.encard = J.encard"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {cut : α → Ordering} {t : Batteries.RBNode α} {path : Batteries.RBNode.Path α}\n  {t' : Batteries.RBNode α} {path' : Batteries.RBNode.Path α},\n  Batteries.RBNode.zoom cut t path = (t', path') →\n    Batteries.RBNode.Path.Zoomed cut path → Batteries.RBNode.Path.Zoomed cut path'"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Monoid β", "DecidableEq α", "s : Finset α", "a : α", "f : α → β", "comm : (↑(insert a s)).Pairwise fun a b => Commute (f a) (f b)", "ha : a ∉ s"], "goal": "(insert a s).noncommProd f comm = s.noncommProd f ⋯ * f a"}}
+{"state": {"context": ["y : ℝ", "hy : y ≥ 1", "τ : ℂ", "hτ : τ.im = y"], "goal": "0 < 1 - rexp (-π)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F : CategoryTheory.Functor C C", "X : C"], "goal": "(CategoryTheory.MonoidalCategory.rightUnitor F).hom.app X =\n  CategoryTheory.CategoryStruct.id\n    ((CategoryTheory.MonoidalCategory.tensorObj F (𝟙_ (CategoryTheory.Functor C C))).obj X)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "a : α", "r✝ r : List (List α)", "l : List α", "ih : sublistsAux a r = r.bind fun l => [l, a :: l]"], "goal": "foldl (fun r l => r ++ [l, a :: l]) (foldl (fun r l => r ++ [l, a :: l]) [] r) [l] = sublistsAux a r ++ [l, a :: l]"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace ℝ F", "inst✝⁶ : NormedAddCommGroup F'", "inst✝⁵ : NormedSpace ℝ F'", "inst✝⁴ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "G' : Type u_7", "G'' : Type u_8", "inst✝³ : NormedLatticeAddCommGroup G''", "inst✝² : NormedSpace ℝ G''", "inst✝¹ : NormedLatticeAddCommGroup G'", "inst✝ : NormedSpace ℝ G'", "T T' : Set α → E →L[ℝ] G''", "hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : E), (T s) x ≤ (T' s) x", "f : α →ₛ E", "hf : Integrable (↑f) μ"], "goal": "setToSimpleFunc T f ≤ setToSimpleFunc T' f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "PreconnectedSpace X", "f : X → Y", "hf : IsLocallyConstant f", "x y : X"], "goal": "f x = f y"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "σ : Type u_3", "AddCommMonoid M", "ι : Type u_4", "s : Finset ι", "φ : ι → MvPolynomial σ R", "n : M", "w : σ → M", "h : ∀ i ∈ s, MvPolynomial.IsWeightedHomogeneous w (φ i) n"], "goal": "MvPolynomial.IsWeightedHomogeneous w (∑ i ∈ s, φ i) n"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "σ : Type u_1", "CommSemiring R", "CommSemiring S₁", "f : R →+* S₁", "g : σ → S₁", "r : R"], "goal": "(MvPolynomial.eval₂Hom f g) (MvPolynomial.C r) = f r"}}
+{"state": {"context": ["I : Type u", "AddMonoid I", "C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.MonoidalCategory C", "X₁ X₂ X₃ : CategoryTheory.GradedObject I C", "X₂.HasTensor X₃", "X₁.HasTensor (CategoryTheory.GradedObject.Monoidal.tensorObj X₂ X₃)", "i₁ i₂ i₃ j : I", "h : i₁ + i₂ + i₃ = j", "i₂₃ : I", "h' : i₂ + i₃ = i₂₃"], "goal": "CategoryTheory.GradedObject.Monoidal.ιTensorObj₃ X₁ X₂ X₃ i₁ i₂ i₃ j h =\n  CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.MonoidalCategory.whiskerLeft (X₁ i₁)\n      (CategoryTheory.GradedObject.Monoidal.ιTensorObj X₂ X₃ i₂ i₃ i₂₃ h'))\n    (CategoryTheory.GradedObject.Monoidal.ιTensorObj X₁ (CategoryTheory.GradedObject.Monoidal.tensorObj X₂ X₃) i₁ i₂₃ j\n      ⋯)"}}
+{"state": {"context": ["ι : Sort u_1", "f : ι → ℝ≥0∞", "hf : ∀ (i : ι), f i ≠ ⊤"], "goal": "(iInf f).toReal = ⨅ i, (f i).toReal"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "S : CategoryTheory.ShortComplex C", "self : S.Splitting"], "goal": "CategoryTheory.CategoryStruct.comp S.f self.r = CategoryTheory.CategoryStruct.id S.X₁"}}
+{"state": {"context": ["R : Type u_1", "n : ℕ", "inst✝² : CommRing R", "a b x y : R", "p : ℕ", "inst✝¹ : IsDomain R", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "hp : Prime ↑p", "hp1 : Odd p", "hx : ¬↑p ∣ y", "k : R", "hk : x - y = ↑p * k"], "goal": "¬↑p ∣ y ^ (p - 1)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "CategoryTheory.MonoidalCategory E", "F : CategoryTheory.LaxMonoidalFunctor C D", "G : CategoryTheory.LaxMonoidalFunctor C E", "X Y : C"], "goal": "(F.prod' G).μ X Y = (F.μ X Y, G.μ X Y)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroMorphisms D", "S : CategoryTheory.ShortComplex C", "h : S.LeftHomologyData", "F : CategoryTheory.Functor C D", "F.PreservesZeroMorphisms", "h.IsPreservedBy F"], "goal": "(h.map F).π = F.map h.π"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f g : CategoryTheory.Arrow C", "CategoryTheory.Limits.HasImage f.hom", "CategoryTheory.Limits.HasImage g.hom", "sq : f ⟶ g", "map : CategoryTheory.Limits.image f.hom ⟶ CategoryTheory.Limits.image g.hom", "map_ι : map ≫ CategoryTheory.Limits.image.ι g.hom = CategoryTheory.Limits.image.ι f.hom ≫ sq.right := by aesop_cat", "map' : CategoryTheory.Limits.image f.hom ⟶ CategoryTheory.Limits.image g.hom", "map_ι' : map' ≫ CategoryTheory.Limits.image.ι g.hom = CategoryTheory.Limits.image.ι f.hom ≫ sq.right"], "goal": "(map = map') = True"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R", "f : R →+* S", "n : ℤ"], "goal": "(W.map f).Φ n = Polynomial.map f (W.Φ n)"}}
+{"state": {"context": ["Ω✝ : Type u_1", "inst✝⁷ : PseudoEMetricSpace Ω✝", "inst✝⁶ : MeasurableSpace Ω✝", "inst✝⁵ : OpensMeasurableSpace Ω✝", "Ω : Type u_2", "ι : Type u_3", "L : Filter ι", "inst✝⁴ : L.NeBot", "inst✝³ : MeasurableSpace Ω", "inst✝² : PseudoEMetricSpace Ω", "inst✝¹ : OpensMeasurableSpace Ω", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "μs : ι → Measure Ω", "h : ∀ {E : Set Ω}, MeasurableSet E → μ (frontier E) = 0 → Tendsto (fun i => (μs i) E) L (𝓝 (μ E))", "F : Set Ω", "F_closed : IsClosed F", "ex : ∃ rs, Tendsto rs atTop (𝓝 0) ∧ ∀ (n : ℕ), 0 < rs n ∧ μ (frontier (Metric.thickening (rs n) F)) = 0", "rs : ℕ → ℝ := Classical.choose ex", "rs_lim : Tendsto rs atTop (𝓝 0)"], "goal": "limsup (fun i => (μs i) F) L ≤ μ F"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : Semiring R", "inst✝ : Semiring S", "f : R →+* S", "p : R[X][Y]", "q : R[X]", "r : R"], "goal": "eval (f r) (eval (map f q) (map (mapRingHom f) p)) = f (eval r (eval q p))"}}
+{"state": {"context": ["α : Type u_1", "Group α", "LinearOrder α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α"], "goal": "(mabs a)⁻¹ ≤ a"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Group α", "MulAction α β", "a : α", "s : Set β"], "goal": "(a • s).Infinite ↔ s.Infinite"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : NormedAddCommGroup E'", "inst✝¹⁰ : NormedAddCommGroup E''", "inst✝⁹ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝⁸ : NontriviallyNormedField 𝕜", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : NormedSpace 𝕜 E'", "inst✝⁵ : NormedSpace 𝕜 E''", "inst✝⁴ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝³ : MeasurableSpace G", "μ ν : Measure G", "inst✝² : AddGroup G", "inst✝¹ : MeasurableAdd G", "inst✝ : MeasurableNeg G", "x₀ : G", "h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ", "hmf : AEStronglyMeasurable f μ", "hmg : AEStronglyMeasurable g (Measure.map (fun t => x₀ - t) μ)", "x : G"], "goal": "‖(L (f x)) (g (x₀ - x))‖ ≤ ‖L‖ * (fun x => ‖f x‖) x * (fun x => ‖g x‖) (x₀ - x)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝⁴ : CommSemiring R", "inst✝³ : NonUnitalSemiring A", "inst✝² : Module R A", "inst✝¹ : IsScalarTower R A A", "inst✝ : SMulCommClass R A A", "x : ↥(unitsFstOne R A)"], "goal": "↑↑((fun x => ⟨{ val := 1 + ↑(equiv.symm ↑x), inv := 1 + ↑(equiv.symm ↑x⁻¹), val_inv := ⋯, inv_val := ⋯ }, ⋯⟩) ((fun x => { val := equiv (↑↑x).snd, inv := equiv (↑↑x⁻¹).snd, val_inv := ⋯, inv_val := ⋯ }) x)) = ↑↑x"}}
+{"state": {"context": ["p : ℕ", "hp : Prime p", "this : Fact (Prime p)", "natAbs_iff : ∀ {a b c d : ℤ} {k : ℕ}, a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k", "hm : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p", "a b c d : ℕ", "hmin : ∀ m < 1, ¬(m < p ∧ 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p)", "hmp : 1 < p", "hm₀ : 0 < 1", "habcd : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 1 * p"], "goal": "∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = p"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝⁷ : Field K", "inst✝⁶ : CharZero K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : IsKilling K L", "inst✝² : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝¹ : H.IsCartanSubalgebra", "inst✝ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "n : ℕ", "hn : ↑(chainBotCoeff (⇑α) β) < ↑n"], "goal": "rootSpace H (-↑n • ⇑α + ⇑β) = ⊥"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f g : E → F", "p pf pg : FormalMultilinearSeries 𝕜 E F", "x : E", "r r'✝ : ℝ≥0∞", "r' : ℝ≥0", "hf : HasFPowerSeriesOnBall f p x r", "h : ↑r' < r", "a : ℝ", "ha : a ∈ Ioo 0 1", "C : ℝ", "hC : C > 0", "hp : ∀ (n : ℕ), ‖p n‖ * ↑r' ^ n ≤ C * a ^ n", "y : E", "hy : y ∈ Metric.ball 0 ↑r'", "n : ℕ", "yr' : ‖y‖ < ↑r'"], "goal": "‖f (x + y) - p.partialSum n y‖ ≤ C / (1 - a) * (a * (‖y‖ / ↑r')) ^ n"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "inst✝⁵ : AddCommMonoid ι", "inst✝⁴ : DecidableEq ι", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ι → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "x : Submonoid A", "f : A", "m : ι", "hf : f ∈ 𝒜 m", "z : Away 𝒜 f", "k : ℕ", "hk : f ^ k = den z", "k' : ℕ", "hk' : k ≤ k'"], "goal": "∃ x ∈ 𝒜 (k' • m), (algebraMap A (at Submonoid.powers f)) x = f ^ k' • val z"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "FunLike F (Set α) ℝ≥0∞", "MeasureTheory.OuterMeasureClass F α", "μ : F", "ι : Sort u_4", "Countable ι", "p : α → ι → Prop"], "goal": "(∀ᵐ (a : α) ∂μ, ∀ (i : ι), p a i) ↔ ∀ (i : ι), ∀ᵐ (a : α) ∂μ, p a i"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "X✝ Y✝ Z : C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "B A : C", "X Y : Subobject B", "f : A ⟶ B", "inst✝ : Mono f", "h₁ : X ≤ Y", "h₂ : Y ≤ mk f"], "goal": "underlying.map (h₁.hom ≫ h₂.hom) ≫ (underlyingIso f).hom = underlying.map ⋯.hom ≫ (underlyingIso f).hom"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C", "X Y : C", "f : X ⟶ Y"], "goal": "(CategoryTheory.Limits.cokernel.π f.op).unop =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.cokernelOpUnop f).hom\n    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι f) (CategoryTheory.eqToHom ⋯))"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "S : Type u_3", "CommSemiring S", "Algebra R S", "B : Type u_4", "Ring B", "TopologicalSpace B", "Algebra R B", "Algebra S B", "IsScalarTower R S B", "C : Type u_5", "Ring C", "TopologicalSpace C", "Algebra R C", "Algebra S C", "IsScalarTower R S C", "f : B →A[S] C"], "goal": "↑(ContinuousAlgHom.restrictScalars R f) = AlgHom.restrictScalars R ↑f"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M₁ : Type u_2", "M₂ : Type u_3", "M : Type u_4", "AddCommMonoid M₁", "AddCommMonoid M₂", "AddCommMonoid M", "Module R M₁", "Module R M₂", "Module R M", "f : M₁ →ₗ[R] M₂ →ₗ[R] M", "h : IsTensorProduct f", "C : M → Prop", "m : M", "h0 : C 0", "htmul : ∀ (x : M₁) (y : M₂), C ((f x) y)", "hadd : ∀ (x y : M), C x → C y → C (x + y)"], "goal": "C m"}}
+{"state": {"context": ["p : ℝ", "hp₀ : 0 < p", "hp₁ : p < 1", "hp₀' : 0 < 1 / p"], "goal": "StrictConcaveOn ℝ≥0 univ fun x => x ^ p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "f✝ : α → β", "s t a b : Set α", "f : α → β", "h : a ⊆ b"], "goal": "f '' a ⊆ f '' b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Add β", "f g : AddHom (FreeAddSemigroup α) β", "h : ⇑f ∘ FreeAddSemigroup.of = ⇑g ∘ FreeAddSemigroup.of"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u_1", "E : ι → Type u_2", "inst✝ : (i : ι) → MetricSpace (E i)", "i : ι", "x : E i", "j : ι", "y : E j", "z : E i"], "goal": "dist ⟨i, x⟩ ⟨i, z⟩ ≤ dist ⟨i, x⟩ ⟨j, y⟩ + dist ⟨j, y⟩ ⟨i, z⟩"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : NormedDivisionRing 𝕜", "α : Type u_2", "E : Type u_3", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "c : 𝕜", "s✝ t : Set E", "x✝ y : E", "r : ℝ≥0∞", "hc : c ≠ 0", "s : Set E", "x : E"], "goal": "egauge 𝕜 (c • s) x = egauge 𝕜 s x / ↑‖c‖₊"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedSemifield α", "a : α", "m n : ℤ", "hx : 1 < a"], "goal": "a ^ m < a ^ n ↔ m < n"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R", "s : NonUnitalSubring R"], "goal": "↑s.toAddSubgroup = ↑s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "A B : C", "X : C", "CategoryTheory.Closed A", "CategoryTheory.Closed B", "f : B ⟶ A"], "goal": "CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev A).app X)\n    ((CategoryTheory.MonoidalClosed.pre f).app (CategoryTheory.MonoidalCategory.tensorObj A X)) =\n  CategoryTheory.CategoryStruct.comp ((CategoryTheory.ihom.coev B).app X)\n    ((CategoryTheory.ihom B).map (CategoryTheory.MonoidalCategory.whiskerRight f X))"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "K : Type u_5", "CommRing K", "Algebra R K", "IsFractionRing R K", "x : R"], "goal": "(algebraMap R K) x = 0 ↔ x = 0"}}
+{"state": {"context": ["ι : Type uι", "Fintype ι", "𝕜 : Type u𝕜", "NontriviallyNormedField 𝕜", "E : ι → Type uE", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)"], "goal": "BddAbove\n  {p |\n    ∃ G x x_1,\n      p =\n        (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp\n          (PiTensorProduct.toDualContinuousMultilinearMap G)}"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_4", "CommRing A", "CommRing B", "Algebra A B", "IsDomain A", "IsIntegrallyClosed A", "IsDomain B", "IsIntegrallyClosed B", "Module.Finite A B", "NoZeroSMulDivisors A B", "Algebra.IsSeparable (FractionRing A) (FractionRing B)", "x : B"], "goal": "(Algebra.intNorm A B) x = 0 ↔ x = 0"}}
+{"state": {"context": ["α : Type u_1", "a b c : α", "MulOneClass α", "Zero α", "Preorder α", "PosMulStrictMono α", "bc : b ≤ c", "ha : a < 1", "b0 : 0 < b"], "goal": "b * a < c"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "C D : ℝ≥0", "f g : E → ℝ", "s✝ : Set E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "s : Set E", "hs : s.Countable", "B : Basis ↑(Basis.ofVectorSpaceIndex ℝ E) ℝ E"], "goal": "∀ᵐ (x : E) ∂μ, ∃ L, ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v"}}
+{"state": {"context": ["α : Type u", "m n : ℕ", "f : Fin (m * n) → α"], "goal": "ofFn f = (ofFn fun i => ofFn fun j => f ⟨m * ↑i + ↑j, ⋯⟩).join"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type w'", "L.Structure M", "L.Structure N", "P : Type u_1", "L.Structure P", "Q : Type u_2", "L.Structure Q", "f : L.Equiv M N", "g : L.Equiv N P", "h : L.Equiv P Q"], "goal": "(h.comp g).comp f = h.comp (g.comp f)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "inst✝ : T1Space α", "s : Closeds α", "x : α", "ht : IsAtom {x}"], "goal": "↑(Closeds.closure {x}) = {x}"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "T1Space X", "x y : X", "h : x ⤳ y"], "goal": "x = y"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "C c : ℝ", "hc_mem : c ∈ Set.Ioo 0 1", "hc : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c * ↑n ≤ ↑(r i n)", "n : ℕ", "hn₁ : ⌈C / c⌉₊ ≤ n", "hn₂ : ∀ (i : α), c * ↑n ≤ ↑(r i n)"], "goal": "∀ (i : α), C ≤ ↑(r i n)"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p} ℚ K", "this : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K"], "goal": "NumberField.discr K = (-1) ^ ((↑p - 1) / 2) * ↑↑p ^ (↑p - 2)"}}
+{"state": {"context": ["J : Type u'", "CategoryTheory.Category.{v', u'} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimitsOfShape J TopCat", "∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)", "F : J ⥤ AlgebraicGeometry.PresheafedSpace C", "j : J"], "goal": "((AlgebraicGeometry.PresheafedSpace.colimitCocone F).ι.app j).base =\n  CategoryTheory.Limits.colimit.ι (F ⋙ AlgebraicGeometry.PresheafedSpace.forget C) j"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝² : Lattice α", "inst✝¹ : OrderBot α", "s✝ t✝ : Finset ι", "f : ι → α", "i✝ : ι", "inst✝ : DecidableEq ι", "s : Finset ι'", "g : ι' → ι", "hs : s.SupIndep (f ∘ g)", "t : Finset ι", "ht : t ⊆ Finset.image g s", "i : ι'", "hi : i ∈ s", "hit : g i ∉ t", "this : DecidableEq ι'", "j : ι", "hjt : j ∈ t"], "goal": "j ∈ Finset.image g (s.erase i)"}}
+{"state": {"context": ["M : Type u_2", "CommMonoid M", "MeasurableSpace M", "MeasurableMul₂ M", "μ ν : MeasureTheory.Measure M", "MeasureTheory.SFinite μ", "MeasureTheory.SFinite ν"], "goal": "μ.mconv ν = ν.mconv μ"}}
+{"state": {"context": ["E : Type u_1", "f✝ : ℝ → E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f f' : ℝ → ℝ", "g : ℝ → E", "a : ℝ", "hf : ContinuousOn f (Ici a)", "hft : Tendsto f atTop atTop", "hff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x", "hg_cont : ContinuousOn g (f '' Ioi a)", "hg1 : IntegrableOn g (f '' Ici a) volume", "hg2 : IntegrableOn (fun x => f' x • (g ∘ f) x) (Ioi a) volume", "eq : ∀ (b : ℝ), a < b → ∫ (x : ℝ) in a..b, f' x • (g ∘ f) x = ∫ (u : ℝ) in f a..f b, g u", "t2 : Tendsto (fun i => ∫ (x : ℝ) in a..id i, f' x • (g ∘ f) x) atTop (𝓝 (∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x ∂volume))", "this : Ioi (f a) ⊆ f '' Ici a", "t1 : Tendsto ((fun i => ∫ (x : ℝ) in f a..id i, g x) ∘ f) atTop (𝓝 (∫ (x : ℝ) in Ioi (f a), g x ∂volume))"], "goal": "∫ (x : ℝ) in Ioi a, f' x • (g ∘ f) x = ∫ (u : ℝ) in Ioi (f a), g u"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "p : ℕ", "ExpChar R p", "h : Function.Injective ⇑(frobenius R p)"], "goal": "pNilradical R p = ⊥"}}
+{"state": {"context": ["m n : ℕ", "α : Type u_1", "a : Fin n → α"], "goal": "Fin.repeat m a ∘ Fin.rev = Fin.repeat m (a ∘ Fin.rev)"}}
+{"state": {"context": ["α : Type u_1", "M✝ : Matroid α", "B I✝ J✝ X Y : Set α", "e : α", "M : Matroid α", "hXY : X ⊆ Y", "hY : autoParam (Y ⊆ M.E) _auto✝", "I : Set α", "hI : M.Basis I X", "J : Set α", "hJ : M.Basis J Y", "hIJ : I ⊆ J"], "goal": "∃ I J, M.Basis I X ∧ M.Basis J Y ∧ I ⊆ J"}}
+{"state": {"context": ["α : Type u_1", "Zero α", "One α", "Neg α", "β : Type u_2", "One β", "Neg β", "Zero β", "f : α → β", "h₁ : f 1 = 1", "h₂ : f 0 = 0", "h₃ : f (-1) = -1", "s : SignType"], "goal": "f ↑s = ↑s"}}
+{"state": {"context": ["X : Type u_2", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "f : C(X, Y)", "A B : DiscreteQuotient Y", "h : A ≤ B"], "goal": "DiscreteQuotient.comap f A ≤ DiscreteQuotient.comap f B"}}
+{"state": {"context": ["α : Type u", "One α", "a : α", "LT α"], "goal": "↑a < 1 ↔ a < 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "OrderedAddCommGroup E", "Module 𝕜 E", "OrderedSMul 𝕜 E"], "goal": "↑(PointedCone.positive 𝕜 E) = ConvexCone.positive 𝕜 E"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "K₁ K₂ : Submodule 𝕜 E", "h : K₁ ≤ K₂"], "goal": "K₁ᗮᗮ ≤ K₂ᗮᗮ"}}
+{"state": {"context": ["p : ℝ≥0∞", "ι : Type u_2", "β : ι → Type u_4", "Fact (1 ≤ p)", "Fintype ι", "(i : ι) → SeminormedAddCommGroup (β i)", "DecidableEq ι", "i : ι", "b₁ b₂ : β i"], "goal": "nndist ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₁))\n    ((WithLp.equiv p ((i : ι) → β i)).symm (Pi.single i b₂)) =\n  nndist b₁ b₂"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type u_1", "f✝ : α → β", "ι : Sort u_2", "tα : TopologicalSpace α", "tβ : TopologicalSpace β", "f : β → α"], "goal": "tβ = induced f tα ↔ ∀ (b : β), 𝓝 b = comap f (𝓝 (f b))"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "σ : Type u_4", "inst✝⁶ : CommSemiring R", "inst✝⁵ : CommSemiring A", "inst✝⁴ : CommSemiring B", "inst✝³ : Algebra R A", "inst✝² : Algebra A B", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R A B", "x : σ → A", "p : MvPolynomial σ R"], "goal": "(eval₂Hom (algebraMap R B) (⇑(algebraMap A B) ∘ x)) p = (eval₂Hom (algebraMap R B) fun i => (algebraMap A B) (x i)) p"}}
+{"state": {"context": ["ιA : Type u_1", "ιB : Type u_2", "A : ιA → Type u_3", "M : ιB → Type u_4", "AddMonoid ιA", "VAdd ιA ιB", "GradedMonoid.GMonoid A", "(i : ιB) → AddMonoid (M i)", "self : DirectSum.GdistribMulAction A M", "i : ιA", "j : ιB", "a : A i"], "goal": "GradedMonoid.GSMul.smul a 0 = 0"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "s : Set R"], "goal": "↑(NonUnitalSubsemiring.closure s) = ↑(AddSubmonoid.closure ↑(Subsemigroup.closure s))"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "Countable ι", "f : ι → Set α", "hn : Pairwise (Disjoint on f)", "h : ∀ (i : ι), MeasurableSet (f i)"], "goal": "μ (⋃ i, f i) = ∑' (i : ι), μ (f i)"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "c : ComplexShape ι", "C₁ C₂ : HomologicalComplex V c", "h : C₁ = C₂", "n : ι"], "goal": "(CategoryTheory.eqToHom h).f n = CategoryTheory.eqToHom ⋯"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "δ : Type u_2", "π : ι → Type u_3", "r : α → α → Prop", "inst✝ : Preorder α", "f : ℕ → α", "hf : Monotone f", "n a : ℕ", "h1 : f n < f a", "h2 : f a < f (n + 1)"], "goal": "False"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n : ℕ", "inst✝ : Ring R", "toFinsupp✝ : R[ℕ]"], "goal": "(match { toFinsupp := -toFinsupp✝ } with | { toFinsupp := p } => p.support) = match { toFinsupp := toFinsupp✝ } with | { toFinsupp := p } => p.support"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s : Set P", "hs : Cospherical s", "p : Fin 3 → P", "hps : Set.range p ⊆ s", "hpi : Function.Injective p", "v : V", "hv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0"], "goal": "False"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "n : ℕ", "h₁ : (1, n) ∈ antidiagonal n.succ", "h₂ : ↑n + 1 ≠ 0", "h₃ : (1 + n).choose n = n + 1", "H : ∑ k ∈ antidiagonal n.succ, ↑((k.1 + k.2).choose k.2) / (↑k.2 + 1) * bernoulli' k.1 = 1"], "goal": "∑ k ∈ antidiagonal (n + 1), ↑((k.1 + k.2).choose k.2) / (↑k.2 + 1) * bernoulli k.1 = 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P n", "i₁ i₂ : Fin (n + 1)", "h : i₁ ≠ i₂", "hc : {i₁, i₂}.card = 2", "W : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})", "h_faces : ↑((EuclideanGeometry.orthogonalProjection W) s.circumcenter) = ↑((s.face hc).orthogonalProjectionSpan s.circumcenter)"], "goal": "Finset ?m.329269"}}
+{"state": {"context": ["𝕜 : Type u_1", "G : Type u_2", "H : Type u_3", "inst✝⁶ : MeasurableSpace G", "inst✝⁵ : MeasurableSpace H", "inst✝⁴ : DivisionMonoid G", "inst✝³ : MeasurableMul G", "inst✝² : MeasurableInv G", "μ✝ μ : Measure G", "inst✝¹ : μ.IsInvInvariant", "inst✝ : μ.IsMulLeftInvariant", "g : G"], "goal": "MeasurePreserving (fun t => g * t⁻¹) μ μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : LocallyCompactSpace E", "F : Type u_3", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "α : Type u_4", "inst✝⁴ : TopologicalSpace α", "inst✝³ : MeasurableSpace α", "inst✝² : MeasurableSpace E", "inst✝¹ : OpensMeasurableSpace α", "inst✝ : OpensMeasurableSpace E", "f : α → E → F", "K✝ : Set (E →L[𝕜] F)", "hf : Continuous (Function.uncurry f)", "K : Set (E →L[𝕜] F)", "hK : IsComplete K", "this : {p | DifferentiableAt 𝕜 (f p.1) p.2 ∧ fderiv 𝕜 (f p.1) p.2 ∈ K} = {p | p.2 ∈ D (f p.1) K}", "x✝¹ x✝ : ℕ"], "goal": "MeasurableSet (⋂ i, ⋂ (_ : i ≥ x✝), ⋂ i_1, ⋂ (_ : i_1 ≥ x✝), {x | x.2 ∈ B (f x.1) K ((1 / 2) ^ i) ((1 / 2) ^ i_1) ((1 / 2) ^ x✝¹)})"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_4", "κ : ι → Sort u_7", "f : α → β", "s : (i : ι) → κ i → Set β"], "goal": "f ⁻¹' ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, f ⁻¹' s i j"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : ℂ → E", "hd : DiffContOnCl ℂ f {z | 0 < z.re}", "hexp : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)", "hre : Asymptotics.SuperpolynomialDecay Filter.atTop expR fun x => ‖f ↑x‖", "him : ∃ C, ∀ (x : ℝ), ‖f (↑x * Complex.I)‖ ≤ C"], "goal": "Set.EqOn f 0 {z | 0 ≤ z.re}"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_3", "m : MeasurableSpace Ω", "Preorder ι", "𝒢 : MeasureTheory.Filtration ι m", "τ η : Ω → ι", "i : ι", "s : Set Ω", "DecidablePred fun x => x ∈ s", "hτ_st : MeasureTheory.IsStoppingTime 𝒢 τ", "hη_st : MeasureTheory.IsStoppingTime 𝒢 η", "hτ : ∀ (ω : Ω), i ≤ τ ω", "hη : ∀ (ω : Ω), i ≤ η ω", "hs : MeasurableSet s"], "goal": "MeasureTheory.IsStoppingTime 𝒢 (s.piecewise τ η)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t u : Finset α", "P : Finpartition s", "a : α", "ha : a ∈ s"], "goal": "∃ t ∈ P.parts, a ∈ t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m inst✝³ : MeasurableSpace α", "inst✝² : MeasurableSpace β", "M : Type u_3", "inst✝¹ : AddCommMonoid M", "inst✝ : TopologicalSpace M", "v : VectorMeasure α M", "f : α → β", "hf : Measurable f", "s : Set β", "hs : MeasurableSet s"], "goal": "↑(v.map f) s = ↑v (f ⁻¹' s)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "P : CategoryTheory.Subobject Y", "f : X ⟶ Y", "h : P.Factors f"], "goal": "P.factorThru f h = 0 ↔ f = 0"}}
+{"state": {"context": ["x : ℝ", "hx : x ≠ 0"], "goal": "Real.exp (Real.log x) = |x|"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "l l₁ l₂ : List (Sigma β)", "inst✝ : DecidableEq α", "nd₁ : l₁.NodupKeys", "nd₂ : l₂.NodupKeys"], "goal": "(l₁.kunion l₂).NodupKeys"}}
+{"state": {"context": ["R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M : Type u_9", "M₂ : Type u_11", "M₃ : Type u_12", "Semiring R₁", "Semiring R₂", "Semiring R₃", "AddCommMonoid M", "AddCommMonoid M₂", "AddCommMonoid M₃", "Module R₁ M", "Module R₂ M₂", "Module R₃ M₃", "σ₁₂ : R₁ →+* R₂", "σ₂₃ : R₂ →+* R₃", "σ₁₃ : R₁ →+* R₃", "RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "f : M →ₛₗ[σ₁₂] M₂"], "goal": "LinearMap.comp 0 f = 0"}}
+{"state": {"context": ["n : ℕ", "c : Composition n"], "goal": "∑ i : Fin c.length, c.blocksFun i = n"}}
+{"state": {"context": ["a : ℝ"], "goal": "Isometry fun y => UpperHalfPlane.mk { re := a, im := rexp y } ⋯"}}
+{"state": {"context": ["G : Type u_2", "AddGroup G", "A B C : AddSubgroup G", "h : A ≤ C"], "goal": "↑A + ↑(B ⊓ C) = (↑A + ↑B) ∩ ↑C"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B B₁ B₂ : Set α", "hB : autoParam (B ⊆ M.E) _auto✝", "x✝ : (∃ B_1, M.Base B_1 ∧ Disjoint B B_1) ∧ ∀ ⦃y : Set α⦄, y ⊆ M.E → ∀ (x : Set α), M.Base x ∧ Disjoint y x → B ⊆ y → B = y", "B' : Set α", "hB' : M.Base B'", "hBB' : Disjoint B B'", "h : ∀ ⦃y : Set α⦄, y ⊆ M.E → ∀ (x : Set α), M.Base x ∧ Disjoint y x → B ⊆ y → B = y"], "goal": "B ⊆ M.E \\ B'"}}
+{"state": {"context": ["R : Type u", "CommRing R", "p : PrimeSpectrum R", "r : R"], "goal": "(algebraMap R ↑((AlgebraicGeometry.Spec.structureSheaf R).presheaf.stalk p)) r =\n  (AlgebraicGeometry.StructureSheaf.toStalk R p) r"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "x y : α", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "hf : f.IsCycle", "hg : g.IsCycle", "h : f.support ⊆ g.support", "h' : ∀ x ∈ f.support, f x = g x"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁵ : Fintype ι", "inst✝⁴ : Fintype ι'", "inst✝³ : AddCommGroup E", "inst✝² : Module ℝ E", "inst✝¹ : AddCommGroup F", "inst✝ : Module ℝ F", "b : Basis ι ℝ E"], "goal": "parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Icc 0 1}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : α ⊕ β"], "goal": "x.isLeft = false ↔ x.isRight = true"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "U V : TopologicalSpace.Opens α", "i : U ≤ V"], "goal": "OpenEmbedding (Set.inclusion ⋯)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedAddCommGroup α", "OrderedAddCommGroup β", "s : Set ι", "f₁ f₂ : ι → α", "g : ι → β", "h₁ : MonovaryOn f₁ g s", "h₂ : AntivaryOn f₂ g s"], "goal": "MonovaryOn (f₁ - f₂) g s"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "k m n : ℕ", "u v w : R", "hkm : k < m", "hmn : m < n", "hu : u ≠ 0"], "goal": "(Polynomial.trinomial k m n u v w).trailingCoeff = u"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y✝ : C", "S R : Sieve X", "J : GrothendieckTopology C", "Hss : S ≤ R", "sjx : S ∈ J.sieves X", "Y : C", "f : Y ⟶ X", "hf : S.arrows f"], "goal": "Sieve.pullback f S ≤ Sieve.pullback f R"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "LinearOrder α", "OrderClosedTopology α", "f g : β → α", "TopologicalSpace β", "x₀ : β", "hf : ContinuousAt f x₀", "hg : ContinuousAt g x₀", "hfg : f x₀ < g x₀"], "goal": "∀ᶠ (x : β) in 𝓝 x₀, f x < g x"}}
+{"state": {"context": ["x y z : ℤ", "h : PythagoreanTriple x y z", "hc : x.gcd y = 1", "hx : x % 2 = 1", "hy : y % 2 = 0"], "goal": "x % 2 = 0 ∧ y % 2 = 1 ∨ x % 2 = 1 ∧ y % 2 = 0"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "IsLocalization M S", "x y : R", "hy : y ∈ M"], "goal": "(algebraMap R S) y * IsLocalization.mk' S x ⟨y, hy⟩ = (algebraMap R S) x"}}
+{"state": {"context": [], "goal": "List.finRange 0 = []"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "s : ι → MeasurableSpace Ω", "m m0 : MeasurableSpace Ω", "κ : Kernel α Ω", "μα : Measure α", "μ : Measure Ω", "t : Set Ω", "h_indep : ∀ᵐ (a : α) ∂μα, (κ a) (t ∩ t) = (κ a) t * (κ a) t", "a : α", "ha : (κ a) (t ∩ t) = (κ a) t * (κ a) t"], "goal": "(κ a) t = 0 ∨ (κ a) t = 1 ∨ (κ a) t = ⊤"}}
+{"state": {"context": ["n : Nat", "m : Nat", "i : Fin n"], "goal": "Fin.cast ⋯ (i.addNat m) = Fin.natAdd m i"}}
+{"state": {"context": ["α : Type u_1", "BEq α", "l : List α"], "goal": "[] ∪ l = l"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.Category.{w, v} J", "F : J ⥤ Type u", "CategoryTheory.Limits.HasLimit F", "x y : CategoryTheory.Limits.limit F"], "goal": "x = y ↔ ∀ (j : J), CategoryTheory.Limits.limit.π F j x = CategoryTheory.Limits.limit.π F j y"}}
+{"state": {"context": ["R : Type v", "CommRing R", "A : Matrix (Fin 1) (Fin 1) R"], "goal": "A.det = A 0 0"}}
+{"state": {"context": ["G : Type u_3", "Group G", "x : G", "m : ℤ", "n : ℤ", "h : x ^ n = 1"], "goal": "x ^ m = x ^ (m % n)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "RCLike 𝕜", "NormedAddCommGroup E", "NormedAddCommGroup F", "NormedAddCommGroup G", "InnerProductSpace 𝕜 E", "InnerProductSpace 𝕜 F", "InnerProductSpace 𝕜 G", "CompleteSpace E", "CompleteSpace G", "CompleteSpace F", "A : F →L[𝕜] G", "B : E →L[𝕜] F"], "goal": "ContinuousLinearMap.adjoint (A.comp B) = (ContinuousLinearMap.adjoint B).comp (ContinuousLinearMap.adjoint A)"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "m1 : MeasurableSpace β", "f : α → β", "hf : MeasurableEmbedding f", "μ : Measure β", "s : Set β", "t : Set α", "ht : MeasurableSet t"], "goal": "(Measure.comap f (μ.restrict s)) t = ((Measure.comap f μ).restrict (f ⁻¹' s)) t"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.IsIdempotentComplete C", "CategoryTheory.Limits.HasFiniteCoproducts C"], "goal": "AlgebraicTopology.DoldKan.Compatibility.τ₀ =\n  AlgebraicTopology.DoldKan.Compatibility.τ₁ CategoryTheory.Idempotents.DoldKan.isoN₁\n    CategoryTheory.Idempotents.DoldKan.isoΓ₀ AlgebraicTopology.DoldKan.N₁Γ₀"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "i j : ZMod 2"], "goal": "evenOdd Q i * evenOdd Q j ≤ evenOdd Q (i + j)"}}
+{"state": {"context": ["F : Type u", "E : Type v", "Field F", "Field E", "Algebra F E", "K : Type v", "Field K", "Algebra F K", "i : E ≃ₐ[F] K"], "goal": "Field.insepDegree F E = Field.insepDegree F K"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "U : Opens ↑(PrimeSpectrum.Top R)", "s : ↑((structureSheaf R).val.obj (op U))", "x : ↥U", "V : Opens ↑(PrimeSpectrum.Top R)", "hxV : ↑x ∈ V.carrier", "iVU : V ⟶ U", "f g : R", "hVDg : V ≤ PrimeSpectrum.basicOpen g", "s_eq : const R f g V hVDg = ((structureSheaf R).val.map iVU.op) s", "h : R", "hxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h", "hDhV : PrimeSpectrum.basicOpen h ≤ V"], "goal": "∃ f g i, ↑x ∈ PrimeSpectrum.basicOpen g ∧ const R f g (PrimeSpectrum.basicOpen g) ⋯ = ((structureSheaf R).val.map i.op) s"}}
+{"state": {"context": ["β : Type u_2", "AddMonoid β", "f : ℕ →+ β", "n : ℕ"], "goal": "f n = n • f 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "l : List α", "a m : α", "h : l ≠ []", "head✝ : α", "tail✝ : List α", "x✝ : head✝ :: tail✝ ≠ []"], "goal": "(head✝ :: tail✝).maximum ≠ ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "UniformSpace α", "UniformSpace β", "r : Set (β × β)", "s : Set α", "hs : IsCompact s", "f : α → β", "hf : ∀ a ∈ s, ContinuousAt f a", "hr : r ∈ 𝓤 β"], "goal": "{x | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type u_1", "N : Type u_2", "L.Structure M", "e : M ≃ N"], "goal": "e.inducedStructureEquiv.toEquiv = e"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "f : R", "s : ↑((structureSheaf R).val.obj (op (PrimeSpectrum.basicOpen f)))", "ι : Type u := ↥(PrimeSpectrum.basicOpen f)", "t : Finset ι", "a h : ι → R", "iDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f", "ah_ha : ∀ i ∈ t, ∀ j ∈ t, a i * h j = h i * a j", "s_eq : ∀ i ∈ t, ((structureSheaf R).val.map (iDh i).op) s = const R (a i) (h i) (PrimeSpectrum.basicOpen (h i)) ⋯", "ht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ i ∈ t, ↑(PrimeSpectrum.basicOpen (h i))", "n : ℕ", "b : ι →₀ R", "b_supp : b ∈ Finsupp.supported R R ↑t", "hb : ∑ i ∈ t, b i * h i = f ^ (n + 1)", "tt : Type u := ↑↑t", "i : ι", "hi : i ∈ ↑t", "j : ι", "hj : j ∈ t"], "goal": "b j * a j * h i = a i * (b j * h j)"}}
+{"state": {"context": ["G : Type u_2", "Group G", "S : Set (Subgroup Gᵐᵒᵖ)"], "goal": "(sSup S).unop = sSup (Subgroup.op ⁻¹' S)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "CommMonoid α", "Fintype ι", "f : { i // i ∈ Finset.univ } → α"], "goal": "∏ i ∈ Finset.univ.attach, f i = ∏ i : ι, f ⟨i, ⋯⟩"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α"], "goal": "UniformContinuous ⇑Multiplicative.ofAdd"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "f : G →g G'", "f' : G' →g G''", "u v u' v' : V", "p : G.Walk u v"], "goal": "(Walk.map f p).reverse = Walk.map f p.reverse"}}
+{"state": {"context": ["M : Type u_2", "α : Sort u_4", "AddCommMonoid M", "f : α → M", "a : α", "ha : ∀ (x : α), x ≠ a → f x = 0"], "goal": "∑ᶠ (x : α), f x = f a"}}
+{"state": {"context": ["J : Type u", "inst✝ : Category.{?u.10050, u} J", "F : J ⥤ Type v", "i j✝ k : J", "s : Set (F.obj i)", "j : J"], "goal": "(fun x => ⟨𝟙 (F.obj j) ↑x, ⋯⟩) = 𝟙 ↑(⋂ f, F.map f ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t u : Finset α", "P : Finpartition s", "a : α", "ha : a ∈ s"], "goal": "a ∈ P.part a"}}
+{"state": {"context": ["α : Type u_2", "AddMonoid α", "s t : Set α", "hst : s ⊆ t", "n : ℕ"], "goal": "n • s ⊆ n • t"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "i j k : ι"], "goal": "∀ {X Y : HomologicalComplex C c} (f : X ⟶ Y), ((HomologicalComplex.shortComplexFunctor' C c i j k).map f).τ₁ = f.f i"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : ℝ → E", "C : ℝ", "hf : DifferentiableOn ℝ f (Set.Icc 0 1)", "bound : ∀ x ∈ Set.Ico 0 1, ‖derivWithin f (Set.Icc 0 1) x‖ ≤ C"], "goal": "‖f 1 - f 0‖ ≤ C"}}
+{"state": {"context": ["x₁ x₂ x₃ y₁ y₂ y₃ : SetTheory.PGame", "h₁ : Surreal.Multiplication.P3 x₃ x₂ y₂ y₃", "h₂ : Surreal.Multiplication.P3 x₁ x₃ y₂ y₁"], "goal": "Surreal.Multiplication.P1 x₁ x₂ x₃ y₁ y₂ y₃"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "𝕜 : Type u_4", "E : Type u_5", "s : Finset ℕ", "k n a b : ℕ", "h : b ≤ a"], "goal": "addRothNumber (Ico a b) = rothNumberNat (b - a)"}}
+{"state": {"context": ["A : Type u_1", "R : Type u_2", "B : Type u_3", "CommRing R", "CommSemiring A", "CommRing B", "Algebra R B", "Algebra A B", "IsIntegralClosure A R B"], "goal": "Function.Injective ⇑(algebraMap A B)"}}
+{"state": {"context": ["G : Type w", "TopologicalSpace G", "AddGroup G", "TopologicalAddGroup G", "K : Set G", "hK : IsCompact K"], "goal": "K + closure {0} = closure K"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "Sup α", "Sup β", "Sup γ", "f : SupHom β γ", "g : SupHom α β"], "goal": "⇑(f.comp g) = ⇑f ∘ ⇑g"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "t : A", "n : ℕ", "hnp1 : IsUnit ↑(n + 1)!"], "goal": "(algebraMap ℚ A) ↑(n + 1)! * ∑ x ∈ range (n + 1), (PowerSeries.coeff A x) (PowerSeries.mk fun n => (aeval t) ((1 / ↑n !) • bernoulli n)) * (PowerSeries.coeff A (n + 1 - x)) (exp A - 1) = t ^ n * (algebraMap ℚ A) (↑(n + 1)! * (1 / ↑n !))"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "f : E → F", "K : Set (E →L[𝕜] F)", "x : E", "hx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}", "e : ℕ", "this : 0 < (1 / 2) ^ e"], "goal": "x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : Semifield α", "inst✝³ : LinearOrder α", "inst✝² : PosMulReflectLT α", "inst✝¹ : ZeroLEOneClass α", "a b : α", "inst✝ : PosMulStrictMono α", "ha : 0 < a", "n : ℕ"], "goal": "0 < a ^ (-↑(n + 1))"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "M : Type u_7", "AddMonoid M", "s : Set G", "hs : AddSubgroup.closure s = ⊤", "f g : G →+ M", "h : Set.EqOn (⇑f) (⇑g) s"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u", "s : Stream'.Seq α"], "goal": "Stream'.Seq.nil.append s = s"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : ℝ × ℝ → E"], "goal": "∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p) = ∫ (p : ℝ × ℝ), f p"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "hg : S.g = 0", "c : CategoryTheory.Limits.CokernelCofork S.f", "hc : CategoryTheory.Limits.IsColimit c"], "goal": "(CategoryTheory.ShortComplex.RightHomologyData.ofIsColimitCokernelCofork S hg c hc).ι =\n  CategoryTheory.CategoryStruct.id c.pt"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕜₃ : Type u_3", "E : Type u_4", "Eₗ : Type u_5", "F : Type u_6", "Fₗ : Type u_7", "G : Type u_8", "Gₗ : Type u_9", "𝓕 : Type u_10", "inst✝¹⁷ : SeminormedAddCommGroup E", "inst✝¹⁶ : SeminormedAddCommGroup Eₗ", "inst✝¹⁵ : SeminormedAddCommGroup F", "inst✝¹⁴ : SeminormedAddCommGroup Fₗ", "inst✝¹³ : SeminormedAddCommGroup G", "inst✝¹² : SeminormedAddCommGroup Gₗ", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NontriviallyNormedField 𝕜₂", "inst✝⁹ : NontriviallyNormedField 𝕜₃", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : NormedSpace 𝕜 Eₗ", "inst✝⁶ : NormedSpace 𝕜₂ F", "inst✝⁵ : NormedSpace 𝕜 Fₗ", "inst✝⁴ : NormedSpace 𝕜₃ G", "inst✝³ : NormedSpace 𝕜 Gₗ", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "σ₁₃ : 𝕜 →+* 𝕜₃", "inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "inst✝¹ : FunLike 𝓕 E F", "inst✝ : RingHomIsometric σ₁₃", "f : E →SL[σ₁₂] F", "g : E →SL[σ₁₃] G", "h : ∀ (x : E), ‖f x‖ = ‖g x‖", "x : E"], "goal": "‖g x‖ ≤ ‖g‖ * ‖x‖"}}
+{"state": {"context": ["α : Type u_1", "BooleanRing α", "a : AsBoolAlg α"], "goal": "ofBoolAlg aᶜ = 1 + ofBoolAlg a"}}
+{"state": {"context": ["R : Type u_1", "N : Type u_2", "M : Type u_3", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid N", "inst✝² : Module R N", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "f i : N →ₗ[R] M", "K : Submodule R N", "m : ℕ", "heq : f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K", "hf : Surjective ⇑f", "hi : Injective ⇑i"], "goal": "ker f ≤ f.iterateMapComap i (0 + 1) K"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CancelCommMonoidWithZero R", "p : R", "n : ℕ", "hp : Prime p", "t u : Finset ℕ", "b c : R", "htus : t ∪ u = range n", "htu : Disjoint t u", "hx : (b * ∏ i ∈ t, p) * (c * ∏ i ∈ u, p) = b * c * p ^ n"], "goal": "∃ i j b_1 c_1, i + j = n ∧ b * c = b_1 * c_1 ∧ b * ∏ i ∈ t, p = b_1 * p ^ i ∧ c * ∏ i ∈ u, p = c_1 * p ^ j"}}
+{"state": {"context": ["M : Type u_1", "AddMonoid M", "h : IsAddUnit 0"], "goal": "h.addUnit = 0"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "l : Filter α", "f✝ : α → R", "r : R", "inst✝¹ : LinearOrderedAddCommGroup R", "inst✝ : Archimedean R", "f : α → ℤ", "hr : 0 < r", "hf : Tendsto f l atTop", "s : R"], "goal": "∀ᶠ (a : α) in l, s ≤ f a • r"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type w", "inst✝² : MetricSpace X", "inst✝¹ : MetricSpace Y", "Φ✝ : Z → X", "Ψ✝ : Z → Y", "ε✝ : ℝ", "inst✝ : Nonempty Z", "Φ : Z → X", "Ψ : Z → Y", "ε : ℝ", "ε0 : 0 < ε", "x : Y", "y : X", "h : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0"], "goal": "False"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C"], "goal": "∀ {X Y : CategoryTheory.CosimplicialObject.Augmented C} (η : X ⟶ Y),\n  (CategoryTheory.CosimplicialObject.Augmented.toArrow.map η).left = η.left"}}
+{"state": {"context": ["b : ℂ", "hb : b.re < 0", "c d : ℂ"], "goal": "∫ (x : ℝ), cexp (b * ↑x ^ 2 + c * ↑x + d) = (↑π / -b) ^ (1 / 2) * cexp (d - c ^ 2 / (4 * b))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.ConcreteCategory C", "CategoryTheory.Limits.HasLimits C", "CategoryTheory.ConcreteCategory.forget.ReflectsIsomorphisms", "CategoryTheory.Limits.PreservesLimits CategoryTheory.ConcreteCategory.forget", "X : TopCat", "F : TopCat.Sheaf C X", "U₁ U₂ V : TopologicalSpace.Opens ↑X", "i₁ : U₁ ⟶ V", "i₂ : U₂ ⟶ V", "hcover : V ≤ U₁ ⊔ U₂", "s t : (CategoryTheory.forget C).obj (F.val.obj (Opposite.op V))", "h₁ : (F.val.map i₁.op) s = (F.val.map i₁.op) t", "h₂ : (F.val.map i₂.op) s = (F.val.map i₂.op) t"], "goal": "s = t"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_3", "Semiring R", "AddCommGroup M", "Module R M", "S : Submodule R M"], "goal": "(-S).toAddSubmonoid = -S.toAddSubmonoid"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁵ : Group G", "inst✝¹⁴ : MeasurableSpace G", "inst✝¹³ : TopologicalSpace G", "inst✝¹² : TopologicalGroup G", "inst✝¹¹ : BorelSpace G", "inst✝¹⁰ : PolishSpace G", "Γ : Subgroup G", "inst✝⁹ : Countable ↥Γ", "inst✝⁸ : Γ.Normal", "inst✝⁷ : T2Space (G ⧸ Γ)", "inst✝⁶ : SecondCountableTopology (G ⧸ Γ)", "μ : Measure (G ⧸ Γ)", "ν : Measure G", "inst✝⁵ : ν.IsMulLeftInvariant", "inst✝⁴ : ν.IsMulRightInvariant", "inst✝³ : SigmaFinite ν", "inst✝² : μ.IsMulLeftInvariant", "inst✝¹ : SigmaFinite μ", "inst✝ : IsFiniteMeasure μ", "hasFun : HasFundamentalDomain (↥Γ.op) G ν", "h : covolume (↥Γ.op) G ν = μ univ", "s : Set G", "fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν"], "goal": "QuotientMeasureEqMeasurePreimage ν μ"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_5", "CommRing R", "AddCommGroup M", "Module R M", "NoZeroSMulDivisors R M", "a : R", "ha : a ≠ 0"], "goal": "LinearMap.ker ((LinearMap.lsmul R M) a) = ⊥"}}
+{"state": {"context": ["l : List ℕ"], "goal": "l.IsZeckendorfRep → (List.map Nat.fib l).sum.zeckendorf = l"}}
+{"state": {"context": ["g : SL(2, ℤ)", "z : ℍ", "hz : z ∈ 𝒟ᵒ", "hg : g • z ∈ 𝒟ᵒ", "c' : ℤ := ↑g 1 0", "c : ℝ := ↑c'", "hc : 0 < c ^ 4", "h₁ : 3 * 3 * c ^ 4 < 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4", "h₂ : (c * z.im) ^ 4 / normSq (denom (↑g) z) ^ 2 ≤ 1"], "goal": "9 * c ^ 4 < 16"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P : Fin 3 → R", "hPz : P z = 0"], "goal": "W'.negY P = -P y"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "I : TwoSidedIdeal R", "x : ↥I"], "goal": "I.subtypeMop x = MulOpposite.op ↑x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n : ℕ", "inst✝ : Semiring R", "p q r : R[X]", "ai : R", "au : ai * a = 1"], "goal": "(1 * p).natDegree = (C ai * (C a * p)).natDegree"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "W X Y Z : Mon_ C", "M : Bimod W X", "N : Bimod X Y", "P P' : Bimod Y Z", "f : P ⟶ P'"], "goal": "coequalizer.π ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷ P.X) ((α_ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X Y.X P.X).hom ≫ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁ P.actLeft) ≫ ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.whiskerLeft f).hom = coequalizer.π ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.actRight ▷ P.X) ((α_ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X Y.X P.X).hom ≫ { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯, actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ }.X ◁ P.actLeft) ≫ ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).hom ≫ M.whiskerLeft (N.whiskerLeft f) ≫ (isoOfIso { hom := AssociatorBimod.hom M N P', inv := AssociatorBimod.inv M N P', hom_inv_id := ⋯, inv_hom_id := ⋯ } ⋯ ⋯).inv).hom"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "AddGroup G", "AddGroup G'", "f : G →+ G'", "H : AddSubgroup G", "x : ↥H.toAddSubmonoid"], "goal": "↑((f.addSubgroupMap H) x) = f ↑x"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "R : Type u", "S : Type v", "r : R", "e : ℕ", "n m : σ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "s : Set σ"], "goal": "supported R s = (rename Subtype.val).range"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁷ : CommSemiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommMonoid N", "inst✝³ : Module R N", "Q : QuadraticMap R M N", "inst✝² : IsCancelAdd R", "inst✝¹ : NoZeroDivisors R", "inst✝ : CharZero R", "B : BilinMap R M R", "x y : M", "h : LinearMap.IsSymm B", "this : AddCancelMonoid R := let __src := inst✝²; let __src_1 := inferInstanceAs (AddCommMonoid R); AddCancelMonoid.mk ⋯"], "goal": "B.toQuadraticMap.IsOrtho x y ↔ LinearMap.IsOrtho B x y"}}
+{"state": {"context": ["R : Type u_1", "inst✝²⁷ : CommSemiring R", "S : Submonoid R", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝²⁶ : AddCommMonoid M", "inst✝²⁵ : AddCommMonoid M'", "inst✝²⁴ : AddCommMonoid M''", "A : Type u_5", "inst✝²³ : CommSemiring A", "inst✝²² : Algebra R A", "inst✝²¹ : Module A M'", "inst✝²⁰ : IsLocalization S A", "inst✝¹⁹ : Module R M", "inst✝¹⁸ : Module R M'", "inst✝¹⁷ : Module R M''", "inst✝¹⁶ : IsScalarTower R A M'", "f : M →ₗ[R] M'", "g✝¹ : M →ₗ[R] M''", "inst✝¹⁵ : IsLocalizedModule S f", "N : Type ?u.1107598", "N' : Type ?u.1107601", "inst✝¹⁴ : AddCommGroup N", "inst✝¹³ : AddCommGroup N'", "inst✝¹² : Module R N", "inst✝¹¹ : Module R N'", "g✝ : N →ₗ[R] N'", "inst✝¹⁰ : IsLocalizedModule S g✝", "M₀ : Type u_6", "M₀' : Type u_9", "inst✝⁹ : AddCommGroup M₀", "inst✝⁸ : AddCommGroup M₀'", "inst✝⁷ : Module R M₀", "inst✝⁶ : Module R M₀'", "f₀ : M₀ →ₗ[R] M₀'", "inst✝⁵ : IsLocalizedModule S f₀", "M₁ : Type u_7", "M₁' : Type u_8", "inst✝⁴ : AddCommGroup M₁", "inst✝³ : AddCommGroup M₁'", "inst✝² : Module R M₁", "inst✝¹ : Module R M₁'", "f₁ : M₁ →ₗ[R] M₁'", "inst✝ : IsLocalizedModule S f₁", "g : M₀ →ₗ[R] M₁", "x : LocalizedModule S M₀", "m : M₀", "s : ↥S"], "goal": "((map S f₀ f₁) g ∘ₗ ↑(iso S f₀)) (LocalizedModule.mk m 1) = (↑(iso S f₁) ∘ₗ (map S (mkLinearMap S M₀) (mkLinearMap S M₁)) g) (LocalizedModule.mk m 1)"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_3", "M : Type u_4", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M"], "goal": "LinearMap.flip (LieModule.traceForm R L M) = LieModule.traceForm R L M"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "self : SeqCompactSpace X"], "goal": "IsSeqCompact Set.univ"}}
+{"state": {"context": ["s : ℝ", "hs : 0 < s"], "goal": "Real.Gamma s * Real.Gamma (s + 1 / 2) = Real.Gamma (2 * s) * 2 ^ (1 - 2 * s) * √π"}}
+{"state": {"context": ["a : ℝ", "ha : a ∈ Ico 0 1", "p : ℝ", "hp : 0 < p", "hp' : (fun t => F_nat 0 a t - if a = 0 then 1 else 0) =O[atTop] fun t => rexp (-p * t)", "q : ℝ", "hq : 0 < q", "ha' : ↑a = 0 ↔ a = 0", "hq' : (fun t => F_nat 0 (1 - a) t) =O[atTop] fun t => rexp (-q * t)"], "goal": "(fun t => F_int 0 (↑a) t - if a = 0 then 1 else 0) =O[atTop] fun t => rexp (-min p q * t)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq β", "inst✝² : SemilatticeSup α", "inst✝¹ : OrderBot α", "inst✝ : DecidableRel Disjoint", "s s₁ s₂ t t₁ t₂ u : Finset α", "a b c a✝ : α"], "goal": "a✝ ∈ s ○ t ↔ a✝ ∈ t ○ s"}}
+{"state": {"context": ["n k : Nat", "h : n ≠ 0"], "goal": "k ≤ n.log2 ↔ 2 ^ k ≤ n"}}
+{"state": {"context": ["R : Type u", "CommRing R", "P : Fin 3 → R", "u : R"], "goal": "(u • P) x = u ^ 2 * P x ∧ (u • P) y = u ^ 3 * P y ∧ (u • P) z = u * P z"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : StrictOrderedCommRing R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y : P", "v : V", "hv : v ∈ s.direction"], "goal": "s.SOppSide (v +ᵥ x) y ↔ s.SOppSide x y"}}
+{"state": {"context": ["α : Type u_1", "Preorder α"], "goal": "Function.Injective Interval.pure"}}
+{"state": {"context": ["α : Type u_1", "SeminormedAddCommGroup α", "a : α"], "goal": "‖‖a‖‖₊ = ‖a‖₊"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "UniformSpace β", "Nonempty α", "Nonempty β"], "goal": "CompleteSpace (α × β) ↔ CompleteSpace α ∧ CompleteSpace β"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_6, u_1} C", "D : Type u_2", "inst✝⁴ : Category.{u_5, u_2} D", "E : Type u_3", "inst✝³ : Category.{?u.45377, u_3} E", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "L : GrothendieckTopology E", "A : Type u_4", "inst✝² : Category.{?u.45429, u_4} A", "G : C ⥤ D", "inst✝¹ : G.IsCoverDense K", "inst✝ : G.IsLocallyFull K", "ℱ : Dᵒᵖ ⥤ Type v", "ℱ' : SheafOfTypes K", "α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val", "X : D", "x : ℱ.obj (op X)", "Y : C", "f : G.obj Y ⟶ X"], "goal": "pushforwardFamily α x f ⋯ = α.app (op Y) (ℱ.map f.op x)"}}
+{"state": {"context": ["x y : ℝ"], "goal": "↑(cos x ^ 2) = ↑(1 / 2 + cos (2 * x) / 2)"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "f : ℝ → E", "a b : ℝ", "c ca cb : E", "l l' la la' lb lb' : Filter ℝ", "lt : Filter ι", "μ : Measure ℝ", "u v ua va ub vb : ι → ℝ", "inst✝¹ : l'.IsMeasurablyGenerated", "inst✝ : TendstoIxxClass Ioc l l'", "hfm : StronglyMeasurableAtFilter f l' μ", "hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)", "hl : μ.FiniteAtFilter l'", "hu : Tendsto u lt l", "hv : Tendsto v lt l", "hE : CompleteSpace E"], "goal": "(fun t => ∫ (x : ℝ) in u t..v t, f x ∂μ - ∫ (x : ℝ) in u t..v t, c ∂μ) =o[lt] fun t => ∫ (x : ℝ) in u t..v t, 1 ∂μ"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝³ : Field K", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : Algebra.IsIntegral K L", "σ τ : L ≃ₐ[K] L", "h_diff : σ ≠ τ", "hστ : σ⁻¹ * τ ≠ 1", "x : L", "hx : (σ⁻¹ * τ) x ≠ x", "E : IntermediateField K L := K⟮x⟯", "this : FiniteDimensional K ↥K⟮x⟯"], "goal": "∃ x, ∃ (_ : x ∈ E), ¬(σ⁻¹ * τ) x = x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "LinearOrder α", "Preorder β", "f : α → β", "s : Set α", "hf : StrictAntiOn f s", "a b : α", "ha : a ∈ s", "hb : b ∈ s"], "goal": "f a < f b ↔ b < a"}}
+{"state": {"context": ["θ ψ : Angle", "h : θ = ↑(π / 2) - ψ ∨ θ = ↑(π / 2) - ψ + ↑π"], "goal": "|θ.cos| = |ψ.sin|"}}
+{"state": {"context": ["n' : Type u_1", "inst✝² : DecidableEq n'", "inst✝¹ : Fintype n'", "R : Type u_2", "inst✝ : CommRing R", "A : M"], "goal": "IsUnit (A ^ 0).det ↔ IsUnit A.det ∨ 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_5", "CompleteLattice α", "κ : ι → Sort u_9", "f : ι → α"], "goal": "⨅ i, f i ≤ ⨅ i, ⨅ x, f i"}}
+{"state": {"context": ["R : Type u", "ι : Type v", "n : ℕ", "M✝ : Fin n.succ �� Type w", "M₁ : ι → Type w₁", "M₁' : ι → Type w₁'", "M₂ : Type w₂", "M₃ : Type w₃", "M₄ : Type w₄", "M : Type u_1", "inst✝⁷ : Fintype ι", "inst✝⁶ : CommRing R", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : TopologicalSpace R", "inst✝² : TopologicalSpace M", "inst✝¹ : ContinuousMul R", "inst✝ : ContinuousSMul R M", "z₁ z₂ : M"], "goal": "ContinuousMultilinearMap.mkPiRing R ι z₁ = ContinuousMultilinearMap.mkPiRing R ι z₂ ↔ z₁ = z₂"}}
+{"state": {"context": ["R : Type u_1", "AddZeroClass R"], "goal": "IsAddRegular 0"}}
+{"state": {"context": ["R : Type u_2", "NonUnitalRing R", "p : RingNorm R"], "goal": "p.toFun = ⇑p"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "a : ℤ", "h : legendreSym p a = -1", "x y : ZMod p", "hxy : x ^ 2 - ↑a * y ^ 2 = 0"], "goal": "x = 0 ∧ y = 0"}}
+{"state": {"context": ["α : Type u_3", "TopologicalSpace α", "s : Set α"], "goal": "MeasureTheory.AnalyticSet s = (s = ∅ ∨ ∃ f, Continuous f ∧ Set.range f = s)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m : MeasurableSpace α", "inst✝¹ : NormedAddCommGroup E", "μ : Measure α", "f✝ : α → E", "p q : ℝ≥0∞", "inst✝ : IsFiniteMeasure μ", "f : α → E", "hpq : p ≤ q", "hfq_m : AEStronglyMeasurable f μ", "hfq_lt_top : eLpNorm f q μ < ⊤", "hp0 : p ≠ 0", "hp_top : ¬p = ⊤", "hp_pos : 0 < p.toReal", "hq_top : ¬q = ⊤"], "goal": "eLpNorm f p μ < ⊤"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁷ : TopologicalSpace H", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "inst✝⁴ : TopologicalSpace H'", "inst✝³ : TopologicalSpace M'", "inst✝² : ChartedSpace H' M'", "inst✝¹ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e✝ e'✝ : PartialHomeomorph M H", "f✝ f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s✝ t : Set M", "x✝ : M", "Q : (H → H) → Set H → H → Prop", "inst✝ : ClosedUnderRestriction G", "s : Set H", "x : H", "f : H → H", "e' : PartialHomeomorph H H", "he'G : e' ∈ G", "he'x : x ∈ e'.source", "h : G.IsLocalStructomorphWithinAt f s x", "hx : ↑e' x ∈ ↑e'.symm ⁻¹' s", "hxs : x ∈ s", "e : PartialHomeomorph H H", "heG : e ∈ G", "hef : EqOn f (↑e.toPartialEquiv) (s ∩ e.source)", "hex : x ∈ e.source"], "goal": "↑e' x ∈ (e'.symm ≫ₕ e).source"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "CompleteSpace 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "CompleteSpace E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "FiniteDimensional 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "a : E", "hf : HasStrictFDerivAt f f' a", "hf' : LinearMap.range f' = ⊤", "α : Type u_4", "l : Filter α", "g₁ : α → F", "g₂ : α → ↥(LinearMap.ker f')", "h₁ : Filter.Tendsto g₁ l (𝓝 (f a))", "h₂ : Filter.Tendsto g₂ l (𝓝 0)"], "goal": "Filter.Tendsto (fun t => HasStrictFDerivAt.implicitFunction f f' hf hf' (g₁ t) (g₂ t)) l (𝓝 a)"}}
+{"state": {"context": ["n k : ℕ"], "goal": "(n + k).descFactorial k = (n + 1).ascFactorial k"}}
+{"state": {"context": ["x : ℝ"], "goal": "0 ≤ 0 ^ x"}}
+{"state": {"context": ["α : Type u_1", "I B X : Set α", "M : Matroid α", "hM : ∃ M_1, M.E = M_1.E ∧ ∀ (I : Set α), M_1.Indep I ↔ M.Indep I"], "goal": "∀ X ⊆ M.E, ExistsMaximalSubsetProperty M.Indep X"}}
+{"state": {"context": ["𝕂 : Type u_1", "m : Type u_2", "n : Type u_3", "𝔸 : Type u_6", "Fintype m", "DecidableEq m", "Fintype n", "DecidableEq n", "Field 𝕂", "Ring 𝔸", "TopologicalSpace 𝔸", "TopologicalRing 𝔸", "Algebra 𝕂 𝔸", "T2Space 𝔸", "v : m → Matrix n n 𝔸"], "goal": "NormedSpace.exp 𝕂 (Matrix.blockDiagonal v) = Matrix.blockDiagonal (NormedSpace.exp 𝕂 v)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "σ : Type u_4", "inst✝⁵ : Semiring A", "inst✝⁴ : DecidableEq ι", "inst✝³ : CanonicallyOrderedAddCommMonoid ι", "inst✝² : SetLike σ A", "inst✝¹ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝ : GradedRing 𝒜", "x✝¹ x✝ : A"], "goal": "↑(((decompose 𝒜) x✝¹ + (decompose 𝒜) x✝) 0) = ↑(((decompose 𝒜) x✝¹) 0) + ↑(((decompose 𝒜) x✝) 0)"}}
+{"state": {"context": ["C : Type u_1", "inst✝² : Category.{?u.71152, u_1} C", "inst✝¹ : Preadditive C", "K✝ K'✝ : ChainComplex C ℕ", "f✝ : K✝ ⟶ K'✝", "Δ✝ Δ'✝ Δ'' : SimplexCategory", "inst✝ : HasFiniteCoproducts C", "K K' : ChainComplex C ℕ", "f : K ⟶ K'", "Δ' Δ : SimplexCategoryᵒᵖ", "θ : Δ' ⟶ Δ"], "goal": "(obj K).map θ ≫ (fun Δ => (splitting K).desc Δ fun A => f.f (unop A.fst).len ≫ ((splitting K').cofan Δ).inj A) Δ = (fun Δ => (splitting K).desc Δ fun A => f.f (unop A.fst).len ≫ ((splitting K').cofan Δ).inj A) Δ' ≫ (obj K').map θ"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "Semiring A", "Algebra R A", "s : Set Aᵐᵒᵖ"], "goal": "(Algebra.adjoin R s).unop = Algebra.adjoin R (MulOpposite.op ⁻¹' s)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : αᵒᵈ"], "goal": "SimpleGraph.hasseDualIso a = OrderDual.ofDual a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "W✝ : Affine R", "f : R →+* S", "F : Type u", "inst✝ : Field F", "W : Affine F", "x₁ x₂ y₁ y₂ : F", "h₁ : W.Equation x₁ y₁", "h₂ : W.Equation x₂ y₂", "hxy : x₁ = x₂ → y₁ ≠ W.negY x₂ y₂", "sup_rw : ∀ (a b c d : Ideal W.CoordinateRing), a ⊔ (b ⊔ (c ⊔ d)) = a ⊔ d ⊔ b ⊔ c"], "goal": "XIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) * (XYIdeal W x₁ (C y₁) * XYIdeal W x₂ (C y₂)) = YIdeal W (linePolynomial x₁ y₁ (W.slope x₁ x₂ y₁ y₂)) * XYIdeal W (W.addX x₁ x₂ (W.slope x₁ x₂ y₁ y₂)) (C (W.addY x₁ x₂ y₁ (W.slope x₁ x₂ y₁ y₂)))"}}
+{"state": {"context": ["a✝ b✝ m n p a b : ℕ"], "goal": "(∀ (p : ℕ), a / p ^ a.factorization p ∣ b / p ^ b.factorization p) ↔ a ∣ b"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "s : Set E", "n : ℕ∞", "hf : ContDiffOn ℝ n f s"], "goal": "ContDiffOn ℝ n (fun x => Real.exp (f x)) s"}}
+{"state": {"context": ["C : Type u₁", "inst✝ : Category.{v₁, u₁} C", "P Q U : Cᵒᵖ ⥤ Type w", "X Y : C", "S : Sieve X", "R : Presieve X"], "goal": "(IsSeparatedFor P R ∧ ∀ (x : FamilyOfElements P R), x.Compatible → ∃ t, x.IsAmalgamation t) ↔ IsSheafFor P R"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m m0 : MeasurableSpace α", "E : Type u_5", "inst✝³ : NormedAddCommGroup E", "inst✝² : MeasurableSpace E", "inst✝¹ : OpensMeasurableSpace E", "μ : Measure α", "hm : m ≤ m0", "inst✝ : SigmaFinite (μ.trim hm)", "C : ℝ≥0∞", "f : Set α → ℝ≥0∞", "hf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ⊤ → f s ≤ C", "h_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)", "S : ℕ → Set α := spanningSets (μ.trim hm)", "hS_mono : Monotone S", "hS_meas : ∀ (n : ℕ), MeasurableSet (S n)"], "goal": "⨆ n, f (S n) ≤ C"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "DecidableRel fun x x_1 => x ∣ x_1", "a : α", "ha : ¬IsUnit a"], "goal": "multiplicity a 1 = 0"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "P : Type u_6", "AddZeroClass M", "AddZeroClass N", "AddZeroClass P", "g : N →+ P", "f₁ f₂ : M →+ N", "hg : Function.Injective ⇑g"], "goal": "g.comp f₁ = g.comp f₂ ↔ f₁ = f₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : PartialOrder β", "f : α → β", "h_mono : Monotone f", "h_surj : Surjective f", "a b : α", "hab : a < b"], "goal": "SurjOn f (Ico a b) (Ico (f a) (f b))"}}
+{"state": {"context": ["a x z✝ z : ℂ"], "goal": "z.arg ≤ π / 2 ↔ 0 ≤ z.re ∨ z.im < 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n✝ : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p q : R[X]", "ha : a ≠ 0", "n : ℕ"], "goal": "(C a * X ^ n).roots = n • {0}"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "f : S.LocalizationMap N", "g : M →* P", "hg : ∀ (y : ↥S), IsUnit (g ↑y)", "x : M", "v : P", "y : ↥S"], "goal": "g x * ↑((IsUnit.liftRight (g.restrict S) hg) y)⁻¹ = v ↔ g x = g ↑y * v"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : Zero R", "a b : Γ", "r : R", "x : HahnSeries Γ R"], "goal": "x ≠ 0 → x.leadingCoeff ≠ 0"}}
+{"state": {"context": ["G : SetTheory.PGame", "G.Impartial"], "goal": "G.Fuzzy 0 ↔ G.LF 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "x : E", "n : ℕ∞", "hf : ContDiffAt ℝ n f x"], "goal": "ContDiffAt ℝ n (fun x => Real.sinh (f x)) x"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "U : X.Opens", "V : (↑U).Opens", "x : ↥V"], "goal": "(↑U).presheaf.germ x ≫ (U.stalkIso ↑x).hom = X.presheaf.germ ⟨↑↑x, ⋯⟩"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "a : α"], "goal": "𝓝ˢ (Set.Ioi a) = 𝓟 (Set.Ioi a)"}}
+{"state": {"context": ["R : Type u1", "inst✝⁴ : CommRing R", "M : Type u2", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "A : Type u_1", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "g : ExteriorAlgebra R M →ₐ[R] A", "m : M"], "goal": "g ((ι R) m) * g ((ι R) m) = 0"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y : P", "hx : x ∈ s", "hy : y ∉ s", "h : x = y"], "goal": "y ∈ s"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "SemilatticeSup α", "OrderBot α", "DecidableRel Disjoint", "s₁ s₂ t₁ t₂ : Finset α", "hs : s₁ ⊆ s₂", "ht : t₁ ⊆ t₂"], "goal": "s₁.disjSups t₁ ⊆ s₂.disjSups t₂"}}
+{"state": {"context": ["α : Type u", "β : α → Type v"], "goal": "∅.entries = []"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : AddCommMonoid α", "inst✝ : LT α", "s : Finset ι", "f : ι → WithBot α"], "goal": "∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Add α", "s₁ s₂ t : Finset α"], "goal": "s₁ ∩ s₂ + t ⊆ (s₁ + t) ∩ (s₂ + t)"}}
+{"state": {"context": ["α : Type u_1", "Add α", "a b : αˢʸᵐ"], "goal": "SymAlg.unsym (a + b) = SymAlg.unsym a + SymAlg.unsym b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r r₁ r₂ : α → α → Prop", "s t : Set α", "a✝ b✝ x : α", "h : IsAntichain r s", "a : α", "ha : a ∈ s", "b : α", "hb : b ∈ s", "hab : r a b"], "goal": "r b a"}}
+{"state": {"context": ["f : ℕ → ℂ", "l : ℂ", "h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)", "M : ℝ", "hM : 1 < M", "s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i", "g : ℂ → ℂ := fun z => ∑' (n : ℕ), f n * z ^ n", "hm : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (∑ i ∈ range n, f i) l < ε"], "goal": "∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, x ∈ stolzSet M → dist x 1 < δ → dist (∑' (n : ℕ), f n * x ^ n) l < ε"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "m0 : MeasurableSpace α", "μ : Measure α", "v : VitaliFamily μ", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : SecondCountableTopology α", "inst✝² : BorelSpace α", "inst✝¹ : IsLocallyFiniteMeasure μ", "ρ✝ : Measure α", "inst✝ : IsLocallyFiniteMeasure ρ✝", "f : α → ℝ≥0∞", "hf : Measurable f", "h'f : ∫⁻ (y : α), f y ∂μ ≠ ⊤", "ρ : Measure α := μ.withDensity f", "this : IsFiniteMeasure ρ"], "goal": "∀ᵐ (x : α) ∂μ, Tendsto (fun a => (∫⁻ (y : α) in a, f y ∂μ) / μ a) (v.filterAt x) (𝓝 (f x))"}}
+{"state": {"context": ["E : Type u_1", "MeasurableSpace E", "μ : MeasureTheory.Measure E", "Ω : Type u_2"], "goal": "∀ {x : MeasurableSpace Ω} {ℙ : MeasureTheory.Measure Ω} {X : Ω → E} {s : Set E},\n  MeasureTheory.pdf.IsUniform X (MeasureTheory.toMeasurable μ s) ℙ μ ↔ MeasureTheory.pdf.IsUniform X s ℙ μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "hV : V ∈ P.parts", "hUV : U ≠ V", "h₂ : ¬G.IsUniform ε U V", "hX : G.nonuniformWitness ε U V ∈ P.nonuniformWitnesses G ε U", "q : G.nonuniformWitness ε U V \\ (star hP G ε hU V).biUnion id ⊆ (filter (fun B => B ⊆ G.nonuniformWitness ε U V ∧ B.Nonempty) (atomise U (P.nonuniformWitnesses G ε U)).parts).biUnion fun B => B \\ (filter (fun x => x ⊆ B) (chunk hP G ε hU).parts).biUnion id", "t : (filter (fun u => u ⊆ G.nonuniformWitness ε U V ∧ u.Nonempty) (atomise U (P.nonuniformWitnesses G ε U)).parts).card ≤ 2 ^ ((P.nonuniformWitnesses G ε U).card - 1)"], "goal": "(P.nonuniformWitnesses G ε U).card ≤ P.parts.card"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "s t : Set E", "hs : Convex 𝕜 s", "ht : Convex 𝕜 t", "hst : Disjoint s t", "S : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}", "C : Set E", "hC : C ∈ S", "hsC : s ⊆ C", "hCmax : ∀ a ∈ S, C ⊆ a → a = C", "x : E", "hx : x ∈ Cᶜ", "y : E", "hy : y ∈ Cᶜ", "z : E", "hz : z ∈ segment 𝕜 x y", "hzC : z ∈ C", "c : E", "hc : c ∈ Cᶜ"], "goal": "∃ a ∈ C, (segment 𝕜 c a ∩ t).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "ν μ : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure ν", "ν.HaveLebesgueDecomposition μ", "r : ℝ≥0"], "goal": "(r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "A : Type u_3", "p q : A → Prop", "inst✝²³ : Semifield R", "inst✝²² : StarRing R", "inst✝²¹ : MetricSpace R", "inst✝²⁰ : TopologicalSemiring R", "inst✝¹⁹ : ContinuousStar R", "inst✝¹⁸ : Semifield S", "inst✝¹⁷ : StarRing S", "inst✝¹⁶ : MetricSpace S", "inst✝¹⁵ : TopologicalSemiring S", "inst✝¹⁴ : ContinuousStar S", "inst✝¹³ : TopologicalSpace A", "inst✝¹² : Ring A", "inst��¹¹ : StarRing A", "inst✝¹⁰ : Algebra S A", "inst✝⁹ : ContinuousFunctionalCalculus S q", "inst✝⁸ : Algebra R S", "inst✝⁷ : Algebra R A", "inst✝⁶ : IsScalarTower R S A", "inst✝⁵ : StarModule R S", "inst✝⁴ : ContinuousSMul R S", "inst✝³ : CompleteSpace R", "inst✝² : ContinuousFunctionalCalculus R p", "inst✝¹ : UniqueContinuousFunctionalCalculus R A", "f : C(S, R)", "halg : UniformEmbedding ⇑(algebraMap R S)", "a : A", "hpa : p a", "hqa : q a", "h : SpectrumRestricts a ⇑f", "inst✝ : CompactSpace ↑(spectrum S a)"], "goal": "cfcHom hpa = starAlgHom (cfcHom hqa) h"}}
+{"state": {"context": ["𝕜 : Type u_1", "NormedDivisionRing 𝕜", "E : Type u_3", "AddCommGroup E", "Module 𝕜 E", "c : 𝕜", "s : Set E", "x : E", "h : c • x ∈ s", "hc : c ≠ 0"], "goal": "egauge 𝕜 s x ≤ ↑‖c‖₊⁻¹"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X₁ X₂ : Type u", "g₁ : Ring X₁", "g₂ : Ring X₂", "m₁ : Algebra R X₁", "m₂ : Algebra R X₂", "e : X₁ ≃ₐ[R] X₂"], "goal": "e.toAlgebraIso.hom = ↑e"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e e' : PartialEquiv α β", "s : Set α", "t : Set β", "(x : α) → Decidable (x ∈ s)", "(y : β) → Decidable (y ∈ t)", "H : e.IsImage s t", "H' : e'.IsImage s t"], "goal": "↑(e.piecewise e' s t H H').symm = t.piecewise ↑e.symm ↑e'.symm"}}
+{"state": {"context": ["α : Sort u", "self : WellFoundedRelation α"], "goal": "WellFounded WellFoundedRelation.rel"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "DecidableEq γ", "f : α → β → γ", "s : Finset α", "t t' : Finset β", "DecidableEq β", "hf : Function.Injective2 f"], "goal": "Finset.image₂ f s (t ∩ t') = Finset.image₂ f s t ∩ Finset.image₂ f s t'"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "A : Type u_3", "T : Type u_4", "B : Type u_5", "ι : Type u_6", "inst✝⁷ : Semiring R", "inst✝⁶ : Ring R'", "inst✝⁵ : AddZeroClass A", "inst✝⁴ : SemilatticeSup B", "inst✝³ : Add B", "inst✝² : OrderBot B", "D : A → B", "p q : R[A]", "hadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a2", "inst✝¹ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "inst✝ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "hD : Function.Injective D", "ap aq : A", "hp : supDegree D p ≤ D ap", "hq : supDegree D q ≤ D aq"], "goal": "aq ∉ q.support → (if ap + aq = ap + aq then p ap * q aq else 0) = 0"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "CategoryTheory.Limits.HasLimit F"], "goal": "CategoryTheory.Limits.limit.lift F (CategoryTheory.Limits.limit.cone F) = 𝟙 (CategoryTheory.Limits.limit F)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "r' : β → β → Prop", "s : Set α", "φ : r ↪r r'"], "goal": "IsAntichain r' (⇑φ '' s) ↔ IsAntichain r s"}}
+{"state": {"context": ["k G : Type u", "inst✝² : CommRing k", "inst✝¹ : Group G", "A : Rep k G", "inst✝ : A.IsTrivial", "x✝¹ : CoeSort.coe A", "x✝ : G"], "goal": "(dZero A) x✝¹ x✝ = 0 x✝¹ x✝"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NontriviallyNormedField 𝕜", "SeminormedAddCommGroup E", "NormedSpace 𝕜 E", "s : Set E"], "goal": "Bornology.IsBounded s ↔ ∃ a, s ⊆ a • Metric.closedBall 0 1"}}
+{"state": {"context": [], "goal": "4 * Real.arctan 5⁻¹ - Real.arctan 239⁻¹ = Real.pi / 4"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : Measure α", "f g : α → ℝ", "hf : AEStronglyMeasurable' m f μ", "hfg : Integrable (f * g) μ", "hg : Integrable g μ", "this✝ : μ[f * g|m] =ᶠ[ae μ] μ[AEStronglyMeasurable'.mk f hf * g|m]", "this : f * μ[g|m] =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * μ[g|m]"], "goal": "f * g =ᶠ[ae μ] AEStronglyMeasurable'.mk f hf * g"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕝 : Type u_2", "E : Type u_3", "F : Type u_4", "β : Type u_5", "inst✝¹⁰ : OrderedSemiring 𝕜", "inst✝⁹ : TopologicalSpace E", "inst✝⁸ : TopologicalSpace F", "inst✝⁷ : AddCommMonoid E", "inst✝⁶ : AddCommMonoid F", "inst✝⁵ : Module 𝕜 E", "inst✝⁴ : Module 𝕜 F", "s : Set E", "inst✝³ : Semiring 𝕝", "inst✝² : Module 𝕝 E", "inst✝¹ : Module 𝕝 F", "inst✝ : LinearMap.CompatibleSMul E F 𝕜 𝕝", "hs : StrictConvex 𝕜 s", "f : E →ₗ[𝕝] F", "hf : IsOpenMap ⇑f", "x : E", "hx : x ∈ s", "y : E", "hy : y ∈ s", "hxy : f x ≠ f y", "a b : 𝕜", "ha : 0 < a", "hb : 0 < b", "hab : a + b = 1"], "goal": "a • f x + b • f y ∈ interior (⇑f '' s)"}}
+{"state": {"context": ["R : Type u_3", "Γ : Type u_5", "LinearOrderedCancelAddCommMonoid Γ", "Semiring R", "NoZeroDivisors R", "x : HahnSeries Γ R", "n : ℕ"], "goal": "(x ^ n).order = n • x.order"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α �� β", "𝒜 𝒜₁ 𝒜₂ : Finset (Finset α)", "s t u : Finset α", "a b : α", "r : ℕ", "hcard : s.card ≤ t.card", "ha : a ∈ s", "ht : t.Nonempty", "m : α := t.min' ht", "h' : s ≠ t", "hwt : m ∈ t", "hws : m ∉ s", "hw : ∀ ⦃a : α⦄, m < a → (a ∈ s ↔ a ∈ t)", "haw : a < m"], "goal": "{ ofColex := s.erase a } ≤ { ofColex := t.erase m }"}}
+{"state": {"context": ["c : Turing.ToPartrec.Code", "v : List ℕ"], "goal": "∃ b,\n  Turing.PartrecToTM2.TrCfg (Turing.ToPartrec.stepNormal c Turing.ToPartrec.Cont.halt v) b ∧\n    Turing.Reaches₁ (Turing.TM2.step Turing.PartrecToTM2.tr) (Turing.PartrecToTM2.init c v) b"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "t₁ t₂ : TopologicalSpace α", "Topology.IsLowerSet α", "Topology.IsLower α"], "goal": "t₁ ≤ t₂"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : PartialOrder α", "a : ℕ →o α", "n m : ℕ", "h : n ≤ m"], "goal": "¬(a n ≤ a m ∧ a n ≠ a m) ↔ a n = a m"}}
+{"state": {"context": ["k : Type u_1", "Field k", "K : Type u_2", "Field K", "Algebra k K", "IsGalois k K", "h : Odd (FiniteDimensional.finrank k K)"], "goal": "IsUnramifiedAtInfinitePlaces k K"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁷ : _root_.RCLike 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : InnerProductSpace 𝕜 E", "inst✝³ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "inst✝² : CompleteSpace E", "ι : Type u_4", "inst✝¹ : SemilatticeSup ι", "U : ι → Submodule 𝕜 E", "inst✝ : ∀ (i : ι), CompleteSpace ↥(U i)", "hU : Monotone U", "x : E", "h✝ : Nonempty ι", "y : E := ↑((orthogonalProjection (⨆ i, U i).topologicalClosure) x)", "proj_x : ∀ (i : ι), (orthogonalProjection (U i)) x = (orthogonalProjection (U i)) y", "ε : ℝ", "hε : ε > 0"], "goal": "∃ I, ∀ i ≥ I, ‖↑((orthogonalProjection (U i)) y) - y‖ < ε"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹ : CommSemiring R", "inst✝ : Monoid M", "S : Set (MonoidAlgebra R M)", "hS : adjoin R S = ⊤"], "goal": "adjoin R (⇑(of R M) '' ⋃ f ∈ S, ↑f.support) = ⊤"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "f : X → Y → Z", "hf : ∀ (a : X) (b : Y) (c : X) (d : Y), (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d", "s : Set (SeparationQuotient X × SeparationQuotient Y)"], "goal": "ContinuousOn (Function.uncurry (SeparationQuotient.lift₂ f hf)) s ↔\n  ContinuousOn (Function.uncurry f) (Prod.map SeparationQuotient.mk SeparationQuotient.mk ⁻¹' s)"}}
+{"state": {"context": ["L R : List SetTheory.PGame"], "goal": "-SetTheory.PGame.ofLists L R = SetTheory.PGame.ofLists (List.map (fun x => -x) R) (List.map (fun x => -x) L)"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Sort u_3", "S : ι → Set α", "f : (i : ι) → ↑(S i) → β", "hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩", "dir : Directed (fun x x_1 => x ≤ x_1) S", "opi : (i : ι) → ↑(S i) → ↑(S i) → ↑(S i)", "opβ : β → β → β", "h : ∀ (i : ι) (x y : ↑(S i)), f i (opi i x y) = opβ (f i x) (f i y)", "op : ↑(iUnion S) → ↑(iUnion S) → ↑(iUnion S)", "hopi : ∀ (i : ι) (x y : ↑(S i)), inclusion ⋯ (opi i x y) = op (inclusion ⋯ x) (inclusion ⋯ y)", "x y : ↑(iUnion S)"], "goal": "iUnionLift S f hf (iUnion S) ⋯ (op x y) = opβ (iUnionLift S f hf (iUnion S) ⋯ x) (iUnionLift S f hf (iUnion S) ⋯ y)"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝² : LinearOrderedAddCommGroup 𝕜", "inst✝¹ : TopologicalSpace 𝕜", "inst✝ : OrderTopology 𝕜", "p : 𝕜", "hp : 0 < p", "x : 𝕜", "hx : 0 < x"], "goal": "(∃ n, n • p = x) ↔ ∃ n, n • p = x"}}
+{"state": {"context": ["ι : Type u_2", "ι' : Type u_3", "f : ι → ℝ", "g : ι' → ℝ", "hf : Summable f", "hg : Summable g", "hf' : 0 ≤ f", "hg' : 0 ≤ g"], "goal": "Summable fun x => f x.1 * g x.2"}}
+{"state": {"context": ["K : Type u_1", "RCLike K"], "goal": "↑RCLike.imCLM = RCLike.imLm"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "R : Type u_5", "inst✝¹¹ : MeasurableSpace X", "inst✝¹⁰ : TopologicalSpace X", "inst✝⁹ : MeasurableSpace Y", "inst✝⁸ : TopologicalSpace Y", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedAddCommGroup F", "f g : X → E", "μ : Measure X", "s : Set X", "inst✝⁵ : BorelSpace X", "inst✝⁴ : ConditionallyCompleteLinearOrder X", "inst✝³ : ConditionallyCompleteLinearOrder E", "inst✝² : OrderTopology X", "inst✝¹ : OrderTopology E", "inst✝ : SecondCountableTopology E", "hmono : MonotoneOn f s", "a b : X", "ha : IsLeast s a", "hb : IsGreatest s b", "hs : μ s ≠ ⊤", "h's : MeasurableSet s", "this✝¹ : MeasurableSpace E := borel E", "this✝ : BorelSpace E", "h✝ : s.Nonempty", "hbelow : BddBelow (f '' s)"], "goal": "IntegrableOn f s μ"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "CategoryTheory.Limits.HasColimit F", "c : CategoryTheory.Limits.Cocone F", "j : J"], "goal": "CategoryTheory.Limits.colimit.ι F j ≫ (CategoryTheory.Limits.colimit.coconeMorphism c).hom = c.ι.app j"}}
+{"state": {"context": ["α : Type u", "Mul α", "One α", "a : α"], "goal": "∀ {x : Invertible a}, a * ⅟a = 1"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "p : G.Walk u v"], "goal": "p.reverse.edges = p.edges.reverse"}}
+{"state": {"context": ["α : Type u_1", "SubtractionMonoid α", "a : α"], "goal": "-a = 0 ↔ a = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝⁵ : NontriviallyNormedField 𝕜", "inst✝⁴ : NonUnitalNormedRing A", "inst✝³ : NormedSpace 𝕜 A", "inst✝² : IsScalarTower 𝕜 A A", "inst✝¹ : SMulCommClass 𝕜 A A", "inst✝ : RegularNormedAlgebra 𝕜 A", "x : Unitization 𝕜 A"], "goal": "‖(algebraMap 𝕜 (A →L[𝕜] A)) x.fst + (mul 𝕜 A) x.snd‖ ≤ 2 * max ‖((addEquiv 𝕜 A) x).1‖ ‖((addEquiv 𝕜 A) x).2‖"}}
+{"state": {"context": ["α : Type u_5", "β : Type u_6", "LE α", "LE β"], "goal": "(OrderIso.sumComm α β).symm = OrderIso.sumComm β α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁷ : MeasurableSpace α", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : PseudoMetrizableSpace β", "inst✝⁴ : MeasurableSpace β", "inst✝³ : BorelSpace β", "ι : Type u_3", "inst✝² : Countable ι", "inst✝¹ : Nonempty ι", "μ : Measure α", "f : ι → α → β", "L : Filter ι", "inst✝ : L.IsCountablyGenerated", "hf : ∀ (n : ι), AEMeasurable (f n) μ", "h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)", "inhabited_h : Inhabited ι", "hL : L.NeBot", "p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)"], "goal": "∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝ : (i : α) → NormedAddCommGroup (E i)", "f : (i : α) → E i", "hfq : Memℓp f 0", "hpq : 0 ≤ 0"], "goal": "Memℓp f 0"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "NormalizedGCDMonoid α"], "goal": "Multiset.gcd 0 = 0"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ : R", "n : ℕ"], "goal": "(↑n).im = 0"}}
+{"state": {"context": ["α : Type u_1", "Ring α", "a : α", "n : ℕ"], "goal": "Mathlib.Meta.NormNum.IsNat a n → Mathlib.Meta.NormNum.IsRat a (Int.ofNat n) 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → β → Prop", "t : Finset β", "a : α", "DecidablePred (r a)"], "goal": "↑(Finset.bipartiteAbove r t a) = {b | b ∈ t ∧ r a b}"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "AddZeroClass M", "DecidableEq α", "g₁ g₂ : α →₀ M"], "goal": "(g₁ + g₂).support ⊆ g₁.support ∪ g₂.support"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : PartialOrder α", "inst✝ : TopologicalSpace β", "a : α", "f : α → β"], "goal": "ContinuousWithinAt f (Ioi a) a ↔ ContinuousWithinAt f (Ici a) a"}}
+{"state": {"context": ["α : Type u", "n : α → Filter α", "ι : α → Sort u_1", "p : (a : α) → ι a → Prop", "s : (a : α) → ι a → Set α", "hb : ∀ (a : α), (n a).HasBasis (p a) (s a)", "hpure : ∀ (a : α) (i : ι a), p a i → a ∈ s a i", "hopen : ∀ (a : α) (i : ι a), p a i → ∀ᶠ (x : α) in n a, s a i ∈ n x", "a : α", "t : TopologicalSpace α := TopologicalSpace.mkOfNhds n"], "goal": "𝓝 a ≤ n a"}}
+{"state": {"context": ["R : Type u", "ι : Type x", "Semiring R", "φ : ι → Type u_1", "ψ : ι → Type u_2", "χ : ι → Type u_3", "(i : ι) → AddCommMonoid (φ i)", "(i : ι) → Module R (φ i)", "(i : ι) → AddCommMonoid (ψ i)", "(i : ι) → Module R (ψ i)", "(i : ι) → AddCommMonoid (χ i)", "(i : ι) → Module R (χ i)", "e : (i : ι) → φ i ≃ₗ[R] ψ i", "f : (i : ι) → ψ i ≃ₗ[R] χ i"], "goal": "LinearEquiv.piCongrRight e ≪≫ₗ LinearEquiv.piCongrRight f = LinearEquiv.piCongrRight fun i => e i ≪≫ₗ f i"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "QuasiSeparatedSpace α", "s t : TopologicalSpace.CompactOpens α"], "goal": "↑(s ⊓ t) = ↑s ∩ ↑t"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "v w : V"], "goal": "G.IsBridge s(v, w) ↔ G.Adj v w ∧ ∀ ⦃u : V⦄ (p : G.Walk u u), p.IsCycle → s(v, w) ∉ p.edges"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s✝ s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p : Set γ", "f✝ f₁ f₂ f₃ : α → β", "g g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a✝ : α", "b : β", "f : α → β", "s : Set α", "a : α", "has : a ∉ s"], "goal": "(InjOn f s ∧ InjOn f {a} ∧ ∀ x ∈ s, ∀ y ∈ {a}, f x ≠ f y) ↔ InjOn f s ∧ f a ∉ f '' s"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x y z : E", "δ ε : ℝ", "hδ : 0 ≤ δ", "hε : 0 ≤ ε", "h : Disjoint (closedBall x δ) (closedBall y ε)", "hxy : ε + δ ≥ dist x y"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_2", "a b : ℝ", "f : ι → ℂ", "s : Finset ι", "ha₀ : 0 ≤ a", "ha : ↑a = ∑ i ∈ s, f i", "hb : ↑b = ∑ i ∈ s, ↑‖f i‖"], "goal": "a ≤ b"}}
+{"state": {"context": ["α : Type u_2", "Group α", "DecidableEq α", "s t : Finset α"], "goal": "((fun b => Finset.image (fun a => a * b) s) '' ↑t).PairwiseDisjoint id → s.card ∣ (s * t).card"}}
+{"state": {"context": ["ι : Type u_1", "I J : BoxIntegral.Box ι", "hJ : J ≤ I", "x : ι → ℝ", "h : x ∈ BoxIntegral.Box.Icc I"], "goal": "(BoxIntegral.TaggedPrepartition.single I J hJ x h).boxes.val = {J}"}}
+{"state": {"context": ["x : ENNReal"], "goal": "⊥ < x.log ↔ 0 < x"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedField F", "inst✝ : CompleteSpace F", "f : ℕ →*₀ F", "hsum : Summable fun x => ‖f x‖"], "goal": "HasProd (fun p => (1 - f ↑p)⁻¹) (∑' (n : ℕ), f n)"}}
+{"state": {"context": ["G : Type u", "Group G", "N : Subgroup G", "N.Normal", "hN : (Nat.card ↥N).Coprime N.index"], "goal": "∃ H, H.IsComplement' N"}}
+{"state": {"context": ["k : Type u", "CommRing k", "G : Grp", "X Y : Rep k ↑G", "f : ↥(X.ρ.linHom Y.ρ).invariants"], "goal": "((Representation.linHom.invariantsEquivRepHom X Y) f).hom = ↑f"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : PseudoMetricSpace X", "inst✝ : IsUltrametricDist X", "x y z : X", "r s : ℝ", "h : (ball x r ∩ ball y r).Nonempty", "h₁ : ball x r = ball h.some r"], "goal": "ball x r = ball y r"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "A : Type u_3", "S : Type u_4", "inst✝¹ : Group G", "inst✝ : AddGroup A", "s : Set G", "H N : Subgroup G", "hN : N.Normal", "x : G", "hx : x ∈ ↑(closure (↑H * ↑N))"], "goal": "x ∈ ↑H * ↑N"}}
+{"state": {"context": ["F : Type u_1", "inst✝³ : Field F", "E : Type u_2", "inst✝² : Field E", "inst✝¹ : Algebra F E", "α : E", "S : Set E", "K L : IntermediateField F E", "inst✝ : FiniteDimensional F E", "h : ∀ (x : E), finrank F ↥F⟮x⟯ ≤ 1"], "goal": "∀ (x : E), finrank F ↥F⟮x⟯ = 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoMetricSpace α", "f : β → α", "s : Set β"], "goal": "Bornology.IsBounded (f '' s) ↔ ∃ C, ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ C"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "w : ↑(G.commonNeighbors u v)"], "goal": "↑((G.walkLengthTwoEquivCommonNeighbors u v).symm w) = (SimpleGraph.Adj.toWalk ⋯).concat ⋯"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type y", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p✝ q r : R[X]", "inst✝ : CommSemiring S", "f : R →+* S", "x : S", "p : R[X]", "hq : eval₂ f x q = 0"], "goal": "eval₂ f x (p * q) = 0"}}
+{"state": {"context": ["k : ℕ"], "goal": "(2 ^ k).bitIndices = [k]"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "c : CategoryTheory.Limits.PullbackCone f g"], "goal": "c.π.app CategoryTheory.Limits.WalkingCospan.right = c.snd"}}
+{"state": {"context": ["m n k : ℤ"], "goal": "(m * n).lcm (k * n) = m.lcm k * n.natAbs"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "a b c d y✝ z : F", "r R : ℝ", "x : F", "hx : (fun x => c + x) x ≠ c", "this : HasFDerivAt (inversion c R) ((R ^ 2 * ((fun y => y⁻¹) ∘ fun x => ‖_root_.id x - c‖ ^ 2) (c + x)) • ContinuousLinearMap.id ℝ F + (R ^ 2 • -((‖_root_.id (c + x) - c‖ ^ 2) ^ 2)⁻¹ • 2 • ((innerSL ℝ) (_root_.id (c + x) - c)).comp (ContinuousLinearMap.id ℝ F)).smulRight (_root_.id (c + x) - c)) (c + x)", "y : F", "hy : y ∈ ↑(Submodule.span ℝ {x})ᗮ"], "goal": "((R ^ 2 * ((fun y => y⁻¹) ∘ fun x => ‖_root_.id x - c‖ ^ 2) (c + x)) • ContinuousLinearMap.id ℝ F + (R ^ 2 • -((‖_root_.id (c + x) - c‖ ^ 2) ^ 2)⁻¹ • 2 • ((innerSL ℝ) (_root_.id (c + x) - c)).comp (ContinuousLinearMap.id ℝ F)).smulRight (_root_.id (c + x) - c)) y = ((R / dist ((fun x => c + x) x) c) ^ 2 • ↑{ toLinearEquiv := (reflection (Submodule.span ℝ {(fun x => c + x) x - c})ᗮ).toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }) y"}}
+{"state": {"context": ["G : Type u_1", "Group G", "a x y : G"], "goal": "SemiconjBy a x⁻¹ y⁻¹ ↔ SemiconjBy a x y"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "f : α → Option α", "a : α", "l : List α", "h : f a = none"], "goal": "(match none with | some b => b :: l | none => (#[].push a).toListAppend (lookmap f l)) = a :: lookmap.go f l #[]"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν : Measure α", "s✝ t s : SignedMeasure α", "μ : Measure α", "r : ℝ≥0"], "goal": "((r • s).toJordanDecomposition.posPart.singularPart μ).toSignedMeasure - ((r • s).toJordanDecomposition.negPart.singularPart μ).toSignedMeasure = (r • s.toJordanDecomposition.posPart.singularPart μ).toSignedMeasure - (r • s.toJordanDecomposition.negPart.singularPart μ).toSignedMeasure"}}
+{"state": {"context": ["R : Type u", "K : Type v", "L : Type z", "p : R", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : Algebra K L", "inst✝⁶ : Algebra R L", "inst✝⁵ : Algebra R K", "inst✝⁴ : IsScalarTower R K L", "inst✝³ : Algebra.IsSeparable K L", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "B : PowerBasis K L", "hp : _root_.Prime p", "hBint : IsIntegral R B.gen", "z : L", "Q : R[X]", "hQ : (aeval B.gen) Q = p • z", "hzint : IsIntegral R z", "hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})", "this : Module.Finite K L := finite B", "P : R[X] := minpoly R B.gen", "n : ℕ", "hn : B.dim = n.succ", "finrank_K_L : FiniteDimensional.finrank K L = B.dim", "deg_K_P : (minpoly K B.gen).natDegree = B.dim", "deg_R_P : P.natDegree = B.dim", "f : ℕ → L", "hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ adjoin R {B.gen} ∧ (algebraMap R L) p * f i = B.gen ^ i", "aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n", "hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))", "r : R", "hr : (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))"], "goal": "(algebraMap R K) (Q.coeff 0 ^ n.succ * ((-1) ^ (n.succ * n) * (minpoly R B.gen).coeff 0 ^ n)) = (algebraMap R K) (p ^ n.succ * r)"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N : Bimod Y Z"], "goal": "(α_ M.X Y.X N.X).hom ≫ M.X ◁ coequalizer.π (Y.mul ▷ N.X) ((α_ Y.X Y.X N.X).hom ≫ Y.X ◁ N.actLeft) ≫ M.X ◁ coequalizer.desc N.actLeft ⋯ ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one = coequalizer.π (M.actRight ▷ Y.X) ((α_ M.X Y.X Y.X).hom ≫ M.X ◁ Y.mul) ▷ N.X ≫ coequalizer.desc M.actRight ⋯ ▷ N.X ≫ colimit.ι (parallelPair (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft)) WalkingParallelPair.one"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "f g : α → F", "hfg : f =ᵐ[μ] g"], "goal": "MeasureTheory.eLpNormEssSup f μ = MeasureTheory.eLpNormEssSup g μ"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : LinearOrderedField α", "l : Filter β", "f : β → α", "r : α", "inst✝ : l.NeBot", "h : Tendsto f l atBot"], "goal": "Tendsto (fun x => f x * r) l atTop ↔ r < 0"}}
+{"state": {"context": ["p q y : ℝ", "r : ℚ", "m✝ : ℤ", "n✝ : ℕ", "hp : 1 < p", "M : ℤ", "h : LiouvilleWith p ↑M", "n : ℕ", "hn : 0 < n", "m : ℤ", "hne : ↑M ≠ ↑m / ↑n", "hlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)", "hn' : 0 < ↑n"], "goal": "↑n ^ (-1) ≤ |↑M - ↑m / ↑n|"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "a : α", "c : α", "h : c < 0"], "goal": "(fun x => c * x) ⁻¹' Set.Iio a = Set.Ioi (a / c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "p : α × β"], "goal": "p.swap.1 = p.2"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "hm : 0 < m", "q :\n  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj\n          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace", "a : A", "n : ℕ", "hn : a ∈ 𝒜 n"], "goal": "a ∈ AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier f_deg q ↔\n  HomogeneousLocalization.mk { deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MetricSpace α", "β : Type u", "inst✝ : Nonempty β", "p : TauPackage β α", "N : ℕ", "hN : IsEmpty (SatelliteConfig α N p.τ)", "i : Ordinal.{u}", "IH : ∀ k < i, k < p.lastStep → p.color k < N", "hi : i < p.lastStep", "A : Set ℕ := ⋃ j, ⋃ (_ : (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty), {p.color ↑j}", "color_i : p.color i = sInf (univ \\ A)", "N_mem : N ∈ univ \\ A", "Inf_eq_N : sInf (univ \\ A) = N", "g : ℕ → Ordinal.{u}", "hg : ∀ k < N, g k < i ∧ (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧ k = p.color (g k)", "G : ℕ → Ordinal.{u} := fun n => if n = N then i else g n", "color_G : ∀ n ≤ N, p.color (G n) = n", "G_lt_last : ∀ n ≤ N, G n < p.lastStep", "fGn : ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))"], "goal": "False"}}
+{"state": {"context": ["ι : Type u", "X : Type v", "TopologicalSpace X", "s : Set X", "ρ : PartitionOfUnity ι X s", "x₀ : X"], "goal": "{i | x₀ ∈ tsupport ⇑(ρ i)}.Finite"}}
+{"state": {"context": ["α : Type u_6", "self : FilterBasis α", "x y : Set α"], "goal": "x ∈ self.sets → y ∈ self.sets → ∃ z ∈ self.sets, z ⊆ x ∩ y"}}
+{"state": {"context": ["p : ℝ≥0∞", "ι : Type u_1", "α : ι → Type u_2", "x y : PiLp p α", "h : ∀ (i : ι), x i = y i"], "goal": "x = y"}}
+{"state": {"context": ["α : Type u_1", "One α"], "goal": "MulOpposite.op 1 = 1"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "hμ : μ ≠ 0", "hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤"], "goal": "0 < μ {x | ⨍⁻ (a : α), f a ∂μ ≤ f x}"}}
+{"state": {"context": ["a b : ℤ", "hab : a = b"], "goal": "b ≤ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : EMetricSpace α", "x y z : α", "s t : Set α", "C : ℝ≥0∞", "sC : Set ℝ≥0∞", "inst✝ : Finite ↑s"], "goal": "0 < s.einfsep"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "ι : Type u_3", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t : Set X", "inst✝ : SigmaCompactSpace X", "f : ι → Set X", "hf : LocallyFinite f", "hne : ∀ (i : ι), (f i).Nonempty", "this : ∀ (n : ℕ), {i | (f i ∩ compactCovering X n).Nonempty}.Finite", "i : ι", "x✝ : i ∈ univ", "x : X", "hx : x ∈ f i", "n : ℕ", "hn : x ∈ compactCovering X n"], "goal": "i ∈ ⋃ i, {i_1 | (f i_1 ∩ compactCovering X i).Nonempty}"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁴ : MeasurableSpace X", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "f g : X → E", "s✝ t : Set X", "μ ν✝ : Measure X", "l l' : Filter X", "ι : Type u_5", "inst✝¹ : Countable ι", "inst✝ : SemilatticeSup ι", "s : ι → Set X", "hsm : ∀ (i : ι), MeasurableSet (s i)", "h_mono : Monotone s", "S : Set X := ⋃ i, s i", "hfi : IntegrableOn f S μ", "hSm : MeasurableSet S", "hsub : ∀ {i : ι}, s i ⊆ S", "ν : Measure X := μ.withDensity fun x => ↑‖f x‖₊", "hfi' : ν S < ⊤", "hν : ν = μ.withDensity fun x => ↑‖f x‖₊", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "this : ∀ᶠ (i : ι) in atTop, ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)", "i : ι", "hi : ν (s i) ∈ Icc (ν S - ↑ε) (ν S + ↑ε)"], "goal": "∫⁻ (a : X) in S \\ s i, ↑‖f a‖₊ ∂μ ≤ ↑ε"}}
+{"state": {"context": ["F : Type u", "inst✝¹ : Field F", "K : Type v", "inst✝ : Field K", "ι : Type u_1", "f : ι → F", "s : Finset ι", "H : ∀ x ∈ s, ∀ y ∈ s, f x = f y → x = y"], "goal": "(∏ i ∈ s, (X - C (f i))).Separable"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝³ : Countable ι", "m✝ : MeasurableSpace α", "μ✝ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "f g : α → β", "inst✝ : PseudoMetrizableSpace β", "m : MeasurableSpace α", "μ : ι → Measure α", "h : ∀ (i : ι), AEStronglyMeasurable f (μ i)", "this✝¹ : MeasurableSpace β := borel β", "this✝ : BorelSpace β", "t : ι → Set β", "t_sep : ∀ (i : ι), IsSeparable (t i)", "ht : ∀ (i : ι), f ⁻¹' t i ∈ ae (μ i)"], "goal": "∀ (b : ι), ∀ᵐ (x : α) ∂μ b, ∃ i, f x ∈ t i"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F G : CategoryTheory.Functor C (Type w)", "a : C", "z : (F ⨿ G).obj a"], "goal": "(CategoryTheory.FunctorToTypes.binaryCoproductEquiv F G a) z =\n  (CategoryTheory.FunctorToTypes.binaryCoproductIso F G).hom.app a z"}}
+{"state": {"context": ["α : Type u_1", "a : α"], "goal": "Quiver.SingleObj.toHom a = a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a : α"], "goal": "Set.Iio a ⊆ Set.Iic a"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "E : Type u₃", "inst✝ : Category.{v₃, u₃} E", "F G : C ⥤ D", "α : F ≅ G", "W X X' : D", "Y : C", "f : W ⟶ X", "g : X ⟶ G.obj Y", "f' : W ⟶ X'", "g' : X' ⟶ G.obj Y"], "goal": "f ≫ g ≫ α.inv.app Y = f' ≫ g' ≫ α.inv.app Y ↔ f ≫ g = f' ≫ g'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹¹ : RCLike 𝕜", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace 𝕜 F", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace 𝕜 F'", "inst✝⁶ : NormedSpace ℝ F'", "inst✝⁵ : CompleteSpace F'", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedAddCommGroup G'", "inst✝² : NormedSpace ℝ G'", "inst✝¹ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "hm : m ≤ m0", "inst✝ : SigmaFinite (μ.trim hm)", "f✝ g : α → F'", "s : Set α", "f : ↥(Lp F' 1 μ)", "hs : MeasurableSet s", "S : ℕ → Set α := spanningSets (μ.trim hm)", "hS_meas : ∀ (i : ℕ), MeasurableSet (S i)", "hS_meas0 : ∀ (i : ℕ), MeasurableSet (S i)", "hs_eq : s = ⋃ i, S i ∩ s"], "goal": "∫ (x : α) in s, ↑↑((condexpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_2", "Monoid M", "Semiring R", "MulSemiringAction M R", "m : M", "S : Subsemiring R"], "goal": "↑(m • S) = m • ↑S"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : MeasureTheory.Measure α", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : α → ℝ", "g : α → E", "s : Set α", "hs : MeasurableSet s"], "goal": "∫ (x : α) in s, g x ∂μ.tilted f = ∫ (x : α) in s, (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕝 : Type u_2", "E : Type u_3", "F : Type u_4", "β : Type u_5", "inst✝⁶ : OrderedRing 𝕜", "inst✝⁵ : TopologicalSpace E", "inst✝⁴ : TopologicalSpace F", "inst✝³ : AddCommGroup E", "inst✝² : AddCommGroup F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "s t✝ : Set E", "x y : E", "h : StrictConvex 𝕜 s", "hx : x ∈ s", "hy : y ∈ s", "hxy : x ≠ y", "t : 𝕜", "ht₀ : 0 < t", "ht₁ : t < 1"], "goal": "x + t • (y - x) ∈ (fun θ => x + θ • (y - x)) '' Ioo 0 1"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J"], "goal": "Symmetric CategoryTheory.Zigzag"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α"], "goal": "Iio b \\ Iic a = Ioo a b"}}
+{"state": {"context": ["R : Type u", "Γ₀ : Type v", "Ring R", "LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "r : R"], "goal": "r ∈ v.integer ↔ v r ≤ 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "f : M → M'", "s : Set M", "x : M", "n : ℕ∞", "h : ContMDiffWithinAt I I' n f s x", "ht : s ∈ 𝓝 x"], "goal": "ContMDiffAt I I' n f x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "s : Set 𝕜", "x : 𝕜", "n : ℕ∞", "m : ℕ", "h : ContDiffWithinAt 𝕜 n f s x", "hmn : ↑m < n", "hs : UniqueDiffOn 𝕜 (insert x s)"], "goal": "DifferentiableWithinAt 𝕜 (iteratedDerivWithin m f s) s x"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "z : K"], "goal": "0 ≤ z ↔ ∃ x ≥ 0, ↑x = z"}}
+{"state": {"context": ["n : Nat"], "goal": "list (n + 1) = List.map castSucc (list n) ++ [last n]"}}
+{"state": {"context": ["q a✝ b✝ c✝ : ℚ", "a b c : ℤ", "h : ¬c = 0"], "goal": "(a + b) /. c = a /. c + b /. c"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "B : Type u_3", "Semiring B", "TopologicalSpace B", "Algebra R A", "Algebra R B", "f : A →A[R] B", "f' : A → B", "h : f' = ⇑f"], "goal": "⇑(f.copy f' h) = f'"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "e : S₁ ≅ S₂", "h₁ : S₁.LeftHomologyData", "h₂ : S₂.LeftHomologyData"], "goal": "(CategoryTheory.ShortComplex.cyclesMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.cyclesMap' e.hom h₁ h₂"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "inst✝² : Ring R", "inst✝¹ : Ring S", "inst✝ : Ring T", "s : Set R", "x✝¹ : R", "h : x✝¹ ∈ closure s", "x : R", "x✝ : (fun x => ∃ L, (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1) ∧ (List.map List.prod L).sum = x) x", "L : List (List R)", "hL : (∀ t ∈ L, ∀ y ∈ t, y ∈ s ∨ y = -1) ∧ (List.map List.prod L).sum = x"], "goal": "∀ (head : List R) (tail : List (List R)), (List.map List.prod (List.map (List.cons (-1)) tail)).sum = -(List.map List.prod tail).sum → (List.map List.prod (List.map (List.cons (-1)) (head :: tail))).sum = -(List.map List.prod (head :: tail)).sum"}}
+{"state": {"context": ["n : ℕ", "E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : (Fin n → ℂ) → E", "c : Fin n → ℂ", "R : Fin n → ℝ"], "goal": "(∯ (x : Fin n → ℂ) in T(c, R), -f x) = -∯ (x : Fin n → ℂ) in T(c, R), f x"}}
+{"state": {"context": ["a b✝ : ℝ", "f : ℝ → ℝ", "c : ContinuousOn f (Set.Icc a b✝)", "ε : ℝ", "pos : 0 < ε", "f' : C(↑(Set.Icc a b✝), ℝ) := { toFun := fun x => f ↑x, continuous_toFun := ⋯ }", "p : ℝ[X]", "b : ‖p.toContinuousMapOn (Set.Icc a b✝) - f'‖ < ε"], "goal": "∃ p, ∀ x ∈ Set.Icc a b✝, |Polynomial.eval x p - f x| < ε"}}
+{"state": {"context": ["R✝ : Type u_1", "A : Type u_2", "B : Type u_3", "M✝ : Type u_4", "N : Type u_5", "inst✝⁸ : Semiring R✝", "inst✝⁷ : AddCommMonoid M✝", "inst✝⁶ : Module R✝ M✝", "inst✝⁵ : AddCommMonoid N", "inst✝⁴ : Module R✝ N", "R : Type u_6", "M : Type u_7", "inst✝³ : CommSemiring R", "inst✝² : AddCommMonoid M", "inst✝¹ : Module R M", "inst✝ : Finite R M", "f : End R M", "h : ∀ (m : M), ∃ n, (f ^ n) m = 0"], "goal": "IsNilpotent f"}}
+{"state": {"context": ["J : Type w", "CategoryTheory.Category.{w', w} J", "A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "L : CategoryTheory.Functor A T", "R : CategoryTheory.Functor B T", "F : CategoryTheory.Functor J (CategoryTheory.Comma L R)", "CategoryTheory.Limits.PreservesColimit (F.comp (CategoryTheory.Comma.fst L R)) L", "c₁ : CategoryTheory.Limits.Cocone (F.comp (CategoryTheory.Comma.fst L R))", "t₁ : CategoryTheory.Limits.IsColimit c₁", "c₂ : CategoryTheory.Limits.Cocone (F.comp (CategoryTheory.Comma.snd L R))", "j : J"], "goal": "((CategoryTheory.Comma.coconeOfPreserves F t₁ c₂).ι.app j).left = c₁.ι.app j"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s : Finset ι", "v : ι → F", "i : ι", "hvs : Set.InjOn v ↑s", "hi : i ∈ s"], "goal": "∀ a ∈ s.erase i, (v i - v a)⁻¹ ≠ 0"}}
+{"state": {"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "h : I = 0", "z : K"], "goal": "‖{ toFun := __spread✝⁻⁰.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := __spread✝⁻⁰.invFun, left_inv := ⋯, right_inv := ⋯ } z‖ = ‖z‖"}}
+{"state": {"context": ["a b m n p : ℕ", "f : ℕ →₀ ℕ", "hf : ∀ p ∈ f.support, Prime p", "h : ∀ x ∈ f.support, x ^ f x ≠ 0"], "goal": "(f.prod fun x x_1 => x ^ x_1).factorization = f"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "IsIrrefl α r"], "goal": "{c | ∃ S, Set.Unbounded r S ∧ #↑S = c}.Nonempty"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "f : E →ₗ[ℝ] E", "hf : LinearMap.det f ≠ 0", "s : Set E"], "goal": "μ (⇑f ⁻¹' s) = ENNReal.ofReal |(LinearMap.det f)⁻¹| * μ s"}}
+{"state": {"context": [], "goal": "Tendsto (fun x => rexp (-π * x)) atTop (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "x y z : ℝ", "l : Filter α", "n : ℕ", "C : ℝ", "hC₁ : 1 ≤ C", "hC₀ : 0 < C", "this : 0 < (rexp 1 * C)⁻¹", "N : ℕ", "hN : ∀ k ≥ N, ↑k ^ n < rexp ↑k / (rexp 1 * C)"], "goal": "∃ ia, True ∧ ∀ x ∈ Set.Ioi ia, rexp x / x ^ n ∈ Set.Ici C"}}
+{"state": {"context": ["A : Type u", "B : Type v", "C : Type z", "ι : Type w", "inst✝⁸ : DecidableEq ι", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : CommRing C", "inst✝³ : Algebra A C", "ι' : Type u_1", "inst✝² : Fintype ι'", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι'", "b : ι → B", "P : Matrix ι ι A"], "goal": "discr A (P.map ⇑(algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b"}}
+{"state": {"context": ["B : Type u_2", "F : Type u_3", "E : B → Type u_4", "TopologicalSpace B", "TopologicalSpace F", "(x : B) → Zero (E x)", "e : Pretrivialization F Bundle.TotalSpace.proj", "z : Bundle.TotalSpace F E", "hz : z.proj ∈ e.baseSet"], "goal": "e.symm z.proj (↑e z).2 = z.snd"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "a b : ℝ", "f : ℝ → E", "hab : b < a", "hf : f =O[atTop] fun x => x ^ (-a)", "t : ℝ", "ht : 0 < t"], "goal": "(fun x => x ^ (a - b) • x ^ (-a)) t = (fun x => x ^ (-b)) t"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : ℝ → E", "p : ℝ", "hp : p ≠ 0", "S : Set ℝ := Ioi 0", "a1 : ∀ x ∈ S, HasDerivWithinAt (fun t => t ^ p) (p * x ^ (p - 1)) S x", "a2 : InjOn (fun x => x ^ p) S", "a3 : (fun t => t ^ p) '' S = S"], "goal": "IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) volume ↔ IntegrableOn f (Ioi 0) volume"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "inst✝¹ : Monoid M", "inst✝ : Monoid N"], "goal": "(MonoidHom.fst M N).comp toProd = fst"}}
+{"state": {"context": ["R : Type u", "inst✝⁵ : Ring R", "ι : Type v", "inst✝⁴ : Preorder ι", "G : ι → Type w", "inst✝³ : (i : ι) → CommRing (G i)", "f : (i j : ι) → i ≤ j → G i → G j", "f' : (i j : ι) → i ≤ j → G i →+* G j", "P : Type u₁", "inst✝² : CommRing P", "g✝ : (i : ι) → G i →+* P", "Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g✝ j) (f i j hij x) = (g✝ i) x", "inst✝¹ : Nonempty ι", "inst✝ : IsDirected ι fun x x_1 => x ≤ x_1", "injective : ∀ (i : ι) (a : G i), (g✝ i) a = 0 → a = 0", "i✝ : ι", "g : G i✝", "hz : (g✝ i✝) g = 0"], "goal": "(of G f i✝) g = 0"}}
+{"state": {"context": ["ι : Sort u_1", "s : ι → ℝ≥0"], "goal": "↑(⨅ i, s i) = ⨅ i, ↑(s i)"}}
+{"state": {"context": ["f : ℕ → ℂ"], "goal": "AnalyticOn ℂ (LSeries f) {s | LSeries.abscissaOfAbsConv f < ↑s.re}"}}
+{"state": {"context": ["m n : Nat"], "goal": "n * m = 0 ↔ n = 0 ∨ m = 0"}}
+{"state": {"context": ["x : ℝ"], "goal": "0 ≤ expNegInvGlue x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "f✝ : ι → α", "s t : Set α", "f : α → β"], "goal": "f '' s ∪ f '' sᶜ = range f"}}
+{"state": {"context": ["R : Type u", "L₁ : Type v", "CommRing R", "LieRing L₁", "LieAlgebra R L₁", "L L' : LieSubalgebra R L₁", "h : ↑L = ↑L'", "x : ↥L"], "goal": "↑((LieEquiv.ofEq L L' h) x) = ↑x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "Φ : CategoryTheory.Functor C C", "ε : CategoryTheory.Functor.id C ⟶ Φ", "J : Type u", "LinearOrder J", "IsWellOrder J fun x x_1 => x < x_1", "OrderBot J", "j : J", "iter₁ iter₂ : CategoryTheory.Functor.Iteration ε j", "f g : iter₁.Hom iter₂", "h : f.natTrans = g.natTrans"], "goal": "f = g"}}
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+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "DecidableEq α", "β : Type u_2", "s : β → β → Prop", "p q : α × β", "w : ¬Prod.Lex r s p q"], "goal": "¬r p.fst q.fst ∧ (p.fst ≠ q.fst ∨ ¬s p.snd q.snd)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "c c₁ c₂ c₃ s t : Set α", "a✝ b✝ x y : α", "h : IsChain r univ", "a b : α"], "goal": "r a b ∨ a = b ∨ r b a"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : m → n → Type v", "M N : DMatrix m n α"], "goal": "(∀ (i : m) (j : n), M i j = N i j) → M = N"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "a : α"], "goal": "Sym2.diag a ∈ s.sym2 ↔ a ∈ s"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "s : Subfield K", "x : K", "hx : x ∈ s", "n : ℤ"], "goal": "n • x ∈ s"}}
+{"state": {"context": ["a b c x : ℕ"], "goal": "x ∈ image (⇑(addLeftEmbedding a)) (Ico 0 b) ↔ x ∈ Ico a (a + b)"}}
+{"state": {"context": ["N : Type u_1", "G : Type u_2", "H : Type u_3", "Group N", "Group G", "Group H", "φ : G →* MulAut N", "F : N ⋊[φ] G →* H"], "goal": "F = SemidirectProduct.lift (F.comp SemidirectProduct.inl) (F.comp SemidirectProduct.inr) ⋯"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : TopologicalSpace X", "s : Set (OnePoint X)", "t : Set X", "Y : Type u_2", "inst✝ : TopologicalSpace Y", "f : OnePoint X → Y", "x : X"], "goal": "ContinuousAt f ↑x ↔ ContinuousAt (f ∘ some) x"}}
+{"state": {"context": ["x y : ℝ≥0∞", "z : ℝ", "hz : 0 < z"], "goal": "x ^ (1 / z) ≤ y ↔ x ≤ y ^ z"}}
+{"state": {"context": ["a✝ b✝ c d : ℝ≥0∞", "r p q : ℝ≥0", "a b : ℝ≥0∞"], "goal": "(a / b).toReal = a.toReal / b.toReal"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{?u.247016, u_1} C", "n m : ℕ", "F G : ComposableArrows C n", "f g : ComposableArrows C 3", "app₀ : f.obj' 0 ⋯ ⟶ g.obj' 0 ⋯", "app₁ : f.obj' 1 ⋯ ⟶ g.obj' 1 ⋯", "app₂ : f.obj' 2 ⋯ ⟶ g.obj' 2 ⋯", "app₃ : f.obj' 3 ⋯ ⟶ g.obj' 3 ⋯", "w₀ : f.map' 0 1 ⋯ ⋯ ≫ app₁ = app₀ ≫ g.map' 0 1 ⋯ ⋯", "w₁ : f.map' 1 2 ⋯ ⋯ ≫ app₂ = app₁ ≫ g.map' 1 2 ⋯ ⋯", "w₂ : f.map' 2 3 ⋯ ⋯ ≫ app₃ = app₂ ≫ g.map' 2 3 ⋯ ⋯"], "goal": "2 < 3 + 1"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "f : MeasureTheory.SimpleFunc β γ", "g : α → β", "hgm : Measurable g"], "goal": "↑(f.comp g hgm) = ↑f ∘ g"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_3", "M : Type u_5", "M₂ : Type u_7", "Semiring R", "Semiring R₂", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R₂ M₂", "τ₁₂ : R →+* R₂", "τ₂₁ : R₂ →+* R", "RingHomInvPair τ₁₂ τ₂₁", "RingHomInvPair τ₂₁ τ₁₂", "e : M ≃ₛₗ[τ₁₂] M₂", "p : Submodule R M"], "goal": "(Submodule.orderIsoMapComap e) p = Submodule.comap e.symm p"}}
+{"state": {"context": [], "goal": "Real.cos (Real.pi / 16) = √(2 + √(2 + √2)) / 2"}}
+{"state": {"context": [], "goal": "Set.SurjOn Real.log (Set.Iio 0) Set.univ"}}
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+{"state": {"context": ["L : Language", "M : Type u_1", "N : Type u_2", "P : Type u_3", "Q : Type u_4", "inst✝³ : L.Structure M", "inst✝² : L.Structure N", "inst✝¹ : L.Structure P", "inst✝ : L.Structure Q", "φ : M ↪ₑ[L] N", "n : ℕ", "f : L.Functions n", "x : Fin n → M"], "goal": "φ (funMap f x) = funMap f (⇑φ ∘ x)"}}
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+{"state": {"context": ["x : ℝ*", "h0 : x ≠ 0", "hi : x⁻¹.Infinitesimal", "hn : x < 0"], "goal": "x.Infinite"}}
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+{"state": {"context": [], "goal": "RingHom.OfLocalizationSpan fun {X Y} [CommRing X] [CommRing Y] f => Function.Surjective ⇑f"}}
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+{"state": {"context": ["R : Type u_6", "S : Type u_7", "K : Type u_8", "L : Type u_9", "CommRing R", "CommRing S", "Field K", "CommRing L", "Algebra R S", "Algebra R K", "Algebra R L", "Algebra S L", "Algebra K L", "IsScalarTower R S L", "IsScalarTower R K L", "IsFractionRing S L", "Algebra.IsIntegral R S"], "goal": "Algebra.IsAlgebraic K L"}}
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+{"state": {"context": ["G : Type w", "TopologicalSpace G", "Inv G", "ContinuousInv G", "x y : G", "h : x ⤳ y"], "goal": "x⁻¹ ⤳ y⁻¹"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "f g : CategoryTheory.ComposableArrows C 2", "φ φ' : f ⟶ g", "h₀ : CategoryTheory.ComposableArrows.app' φ 0 ⋯ = CategoryTheory.ComposableArrows.app' φ' 0 ⋯", "h₁ : CategoryTheory.ComposableArrows.app' φ 1 ⋯ = CategoryTheory.ComposableArrows.app' φ' 1 ⋯", "h₂ : CategoryTheory.ComposableArrows.app' φ 2 ⋯ = CategoryTheory.ComposableArrows.app' φ' 2 ⋯"], "goal": "φ = φ'"}}
+{"state": {"context": ["A : Type u", "inst✝¹ : AddCommGroup A", "inst✝ : DivisibleBy A ℤ", "m : ℤ", "g : ↥(Submodule.span ℤ {m}) →ₗ[ℤ] A", "h0 : m ≠ 0", "gₘ : A := g ⟨m, ⋯⟩"], "goal": "∃ g', ∀ (x : ℤ) (mem : x ∈ Submodule.span ℤ {m}), g' x = g ⟨x, mem⟩"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "K : Type u_2", "inst✝⁴ : Field K", "inst✝³ : Algebra R K", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "f : R[X]", "hf : f.Monic", "g : K[X]", "hg : g ∣ map (algebraMap R K) f", "g_ne_0 : g ≠ 0"], "goal": "∃ g', map (algebraMap R K) g' = g * C g.leadingCoeff⁻¹"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq β", "inst✝² : SemilatticeSup α", "inst✝¹ : OrderBot α", "inst✝ : DecidableRel Disjoint", "s s₁ s₂ t t₁ t₂ u : Finset α", "a b c : α"], "goal": "∀ ⦃x : α⦄, (∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x) → ∃ a ∈ s, ∃ b ∈ t, a ⊔ b = x"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "inst✝¹ : Group G", "hp : Fact (Nat.Prime p)", "inst✝ : Finite (Sylow p G)", "P : Sylow p G", "hP : (↑P).relindex (↑P).normalizer ≠ 0"], "goal": "¬p ∣ (↑P).relindex (↑P).normalizer * Nat.card (Sylow p G)"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℂ", "c : ℂ"], "goal": "HasSum f c ↔ HasSum (fun x => (f x).re) c.re ∧ HasSum (fun x => (f x).im) c.im"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "Group G", "MulAction G α", "s t : Set α", "g : G", "t_ss_s : t ⊆ s", "s_in_fixedBy : s ∈ MulAction.fixedBy (Set α) g"], "goal": "g • t ⊆ s"}}
+{"state": {"context": ["R : Type u", "inst✝⁵ : CommRing R", "M : Type u", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "N : Type u", "inst✝² : AddCommGroup N", "inst✝¹ : Module R N", "ι : Type u", "inst✝ : Fintype ι", "m : ι → M", "n : ι → N", "hm : span R (Set.range m) = ⊤", "hmn : ∑ i : ι, m i ⊗ₜ[R] n i = 0", "G : (ι →₀ R) →ₗ[R] M := Finsupp.total ι M R m", "hG : G = Finsupp.total ι M R m", "G_basis_eq : ∀ (i : ι), G (Finsupp.single i 1) = m i", "G_surjective : Surjective ⇑G", "en : (ι →₀ R) ⊗[R] N := ∑ i : ι, Finsupp.single i 1 ⊗ₜ[R] n i", "hen : en = ∑ i : ι, Finsupp.single i 1 ⊗ₜ[R] n i", "en_mem_ker : en ∈ ker (rTensor N G)"], "goal": "VanishesTrivially R m n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "S₁.HasHomology", "S₂.HasHomology", "φ : S₁ ⟶ S₂", "h₁ : S₁.LeftHomologyData", "h₂ : S₂.LeftHomologyData"], "goal": "CategoryTheory.CategoryStruct.comp h₁.homologyIso.inv (CategoryTheory.ShortComplex.homologyMap φ) =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.leftHomologyMap' φ h₁ h₂) h₂.homologyIso.inv"}}
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+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ", "K : CochainComplex C ℤ", "n m : ℤ", "α : CochainComplex.HomComplex.Cocycle K F m", "β : CochainComplex.HomComplex.Cochain K G n", "h : n + 1 = m", "eq : CochainComplex.HomComplex.δ n m β + (↑α).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ = 0"], "goal": "↑(CochainComplex.mappingCone.liftCocycle φ α β h eq) = CochainComplex.mappingCone.liftCochain φ (↑α) β h"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "l✝ : List α", "x✝ : α", "l : List α", "x : α", "h : l.formPerm x ≠ x"], "goal": "x ∈ {y | l.formPerm y ≠ y}"}}
+{"state": {"context": ["X : Type u_3", "Y : ?m.22013", "A✝ : Type ?u.22022", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace A✝", "E : Type u_1", "A : Type u_2", "inst✝¹ : TopologicalSpace E", "inst✝ : TopologicalSpace A", "p : E → X", "sep : IsSeparatedMap p", "inj : IsLocallyInjective p", "s : Set A", "hs : PreconnectedSpace ↑s", "g g₁ g₂ : A → E", "h₁ : Continuous (s.restrict g₁)", "h₂ : Continuous (s.restrict g₂)", "he : s.restrict (p ∘ g₁) = s.restrict (p ∘ g₂)", "a : A", "has : a ∈ s", "ha : g₁ a = g₂ a"], "goal": "s.restrict g₁ = s.restrict g₂"}}
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+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E ≃L[𝕜] F", "g : F → E", "a : F", "hg : ContinuousAt g a", "hf : HasFDerivAt f (↑f') (g a)", "hfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y"], "goal": "HasFDerivAt g (↑f'.symm) a"}}
+{"state": {"context": ["K : Type u_1", "DivisionSemiring K", "q : ℚ≥0", "z : ℤ", "hq : ↑q.num ≠ 0"], "goal": "↑(q ^ z) = ↑q ^ z"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "B : LinearMap.BilinForm R M", "a : R", "x y : M"], "goal": "(B x) (a • y) = a * (B x) y"}}
+{"state": {"context": ["G : Type u", "inst✝ : Monoid G", "p : ℕ", "hp : Nat.Prime p", "this : Fact (Nat.Prime p)", "h : p ∈ (exponent G).factorization.support", "he : 0 < exponent G", "g : G", "hg : g ^ (exponent G / p) ≠ 1"], "goal": "∃ g, orderOf g = p ^ (exponent G).factorization p"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hC : IsClosed C", "hsC : contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1", "l : ↑(MaxProducts C ho)", "q : Products I", "hq : q < (↑l).Tail", "this : Inhabited I"], "goal": "List.Chain' (fun x x_1 => x > x_1) (term I ho :: ↑q)"}}
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+{"state": {"context": ["α : Type u_1", "DecidableEq α"], "goal": "Sup.sup = Union.union"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : UniformSpace α", "inst✝¹ : Group α", "inst✝ : UniformGroup α"], "goal": "𝓤 α = comap (fun x => x.1 / x.2) (𝓝 1)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "S : Submonoid R", "M : Type u_2", "M' : Type u_3", "AddCommMonoid M", "AddCommMonoid M'", "Module R M", "Module R M'", "f : M →ₗ[R] M'", "IsLocalizedModule S f"], "goal": "∀ (a : M'),\n  (IsLocalizedModule.iso S f).symm a = (Equiv.ofBijective ⇑(IsLocalizedModule.fromLocalizedModule S f) ⋯).symm a"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M"], "goal": "AddSubmonoid.map (AddMonoidHom.id M) S = S"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y : F"], "goal": "‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Preadditive C", "X Y : C", "f g : X ⟶ Y", "c : CokernelCofork (f - g)"], "goal": "f ≫ Cofork.π c = g ≫ Cofork.π c"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "MeasurableSpace G", "MeasurableSpace α", "Group G", "MulAction G α", "MeasurableSMul G α", "c : G"], "goal": "(MeasurableEquiv.smul c).toEquiv = MulAction.toPerm c"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "J : Type u", "CategoryTheory.Category.{v, u} J", "T : CategoryTheory.Comonad C", "D : CategoryTheory.Functor J T.Coalgebra", "c : CategoryTheory.Limits.Cocone (D.comp T.forget)", "t : CategoryTheory.Limits.IsColimit c"], "goal": "(CategoryTheory.Comonad.ForgetCreatesColimits'.coconePoint D c t).a =\n  t.desc (CategoryTheory.Comonad.ForgetCreatesColimits'.newCocone D c)"}}
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+{"state": {"context": ["G : Type u_1", "AddGroup G", "a b : G", "h : G"], "goal": "AddCommute (h + a + -h) (h + b + -h) ↔ AddCommute a b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "E : Type u_6", "MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "NormedAddCommGroup E", "MeasureTheory.SFinite ν", "NormedSpace ℝ E", "MeasureTheory.SFinite μ", "f : α × β → E", "hf : MeasureTheory.Integrable f (μ.prod ν)"], "goal": "∫ (z : α × β), f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), f (x, y) ∂ν ∂μ"}}
+{"state": {"context": ["H : Type u_5", "NormedAddCommGroup H", "InnerProductSpace ℝ H", "y : H", "s : Set H"], "goal": "y ∈ s.innerDualCone ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫_ℝ"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_3", "n : Type u_4", "Monoid R", "AddMonoid α", "DistribMulAction R α", "k : R", "A : Matrix n n α", "ha : A.IsDiag"], "goal": "(k • A).IsDiag"}}
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+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "hm : m ≤ m0", "f : α → E", "hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → MeasureTheory.IntegrableOn f s μ", "hf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = 0", "hf : MeasureTheory.FinStronglyMeasurable f (μ.trim hm)"], "goal": "f =ᵐ[μ] 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4"], "goal": "LipschitzWith 2 (Subtype.val ∘ fun p => p.1 - p.2)"}}
+{"state": {"context": ["m n : ℕ"], "goal": "↑(m * (n + 1)) = ↑m * ↑(n + 1)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "M : ModuleCat R", "Module.Flat R ↑M", "C : CategoryTheory.ShortComplex (ModuleCat R)", "hC : C.Exact"], "goal": "(C.map (CategoryTheory.MonoidalCategory.tensorRight M)).Exact"}}
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+{"state": {"context": [], "goal": "∀ {α : Type u_1} {s₁ t₁ s₂ t₂ : List α}, s₁ ++ t₁ = s₂ ++ t₂ → t₁.length = t₂.length → s₁ = s₂ ∧ t₁ = t₂"}}
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+{"state": {"context": ["G : Type u_1", "P : Type u_2", "AddGroup G", "AddTorsor G P", "x y : P"], "goal": "(Equiv.pointReflection x) y = x -ᵥ y +ᵥ x"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "S : Submonoid R", "A : Type u_2", "Semiring A", "Algebra R A", "a₁ a₂ : A", "s₁ s₂ : ↥S"], "goal": "LocalizedModule.mk a₁ s₁ * LocalizedModule.mk a₂ s₂ = LocalizedModule.mk (a₁ * a₂) (s₁ * s₂)"}}
+{"state": {"context": ["C : Type uC", "inst✝¹ : Category.{uC', uC} C", "W : MorphismProperty C", "D : Type uD", "inst✝ : Category.{uD', uD} D", "G : C ⥤ D", "hG : W.IsInvertedBy G", "F₁ F₂ : W.Localization ⥤ D", "τ : W.Q ⋙ F₁ ⟶ W.Q ⋙ F₂"], "goal": "∀ ⦃X Y : W.Localization⦄ (f : X ⟶ Y), F₁.map f ≫ NatTransExtension.app τ Y = NatTransExtension.app τ X ≫ F₂.map f"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n : ℕ", "M : Fin n.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝¹¹ : Semiring R", "inst✝¹⁰ : (i : Fin n.succ) → AddCommMonoid (M i)", "inst✝⁹ : (i : ι) → AddCommMonoid (M₁ i)", "inst✝⁸ : AddCommMonoid M₂", "inst✝⁷ : AddCommMonoid M₃", "inst✝⁶ : AddCommMonoid M'", "inst✝⁵ : (i : Fin n.succ) → Module R (M i)", "inst✝⁴ : (i : ι) → Module R (M₁ i)", "inst✝³ : Module R M₂", "inst✝² : Module R M₃", "inst✝¹ : Module R M'", "f f' : MultilinearMap R M₁ M₂", "inst✝ : Nonempty ι", "i : ι", "h✝ : i ∈ univ"], "goal": "f 0 = 0"}}
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+{"state": {"context": ["α : Type u", "β : α → Type v", "inst✝ : FinEnum α"], "goal": "(List.map (⇑equiv.symm) (List.finRange (card α))).Nodup"}}
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+{"state": {"context": ["M : Type u_1", "Mul M", "S : Subsemigroup M", "s : Set M", "hs : s = ↑S"], "goal": "S.copy s hs = S"}}
+{"state": {"context": ["z : ℍ"], "goal": "(↑z).im = z.im"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_3", "i : ι", "a : (i : ι) → α i"], "goal": "({i}.sigma fun i => {a i}) = {⟨i, a i⟩}"}}
+{"state": {"context": ["α : Type u_2", "OrderedCommGroup α", "s : NonemptyInterval α", "a : α", "ha : a ∈ s"], "goal": "a⁻¹ ∈ s⁻¹"}}
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+{"state": {"context": ["x y : ℝ"], "goal": "↑(cos (2 * x)) = ↑(cos x ^ 2 - sin x ^ 2)"}}
+{"state": {"context": ["α : Type u", "l : List α"], "goal": "l.rotate' l.length = l"}}
+{"state": {"context": ["X : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : CompactSpace X", "A : Subalgebra ℝ C(X, ℝ)", "f : ↥A", "g : ℝ[X]"], "goal": "(g.toContinuousMapOn (Set.Icc (-‖f‖) ‖f‖)).comp (↑f).attachBound ∈ A"}}
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+{"state": {"context": ["R : Type u", "CommSemiring R", "p : ℕ", "ExpChar R p", "x y : R"], "goal": "(frobenius R p) (x * y) = (frobenius R p) x * (frobenius R p) y"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "s : Set ι", "t : (i : ι) → Set (α i)", "∀ (i : ι), Nonempty (α i)"], "goal": "s.pi t = ∅ ↔ ∃ i ∈ s, t i = ∅"}}
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+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "a : ↥⊤"], "goal": "Subalgebra.topEquiv a = ↑a"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeInf α", "OrderTop α", "s : Finset β", "f : β → αᵒᵈ"], "goal": "OrderDual.ofDual (s.sup f) = s.inf (⇑OrderDual.ofDual ∘ f)"}}
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+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedField F", "inst✝ : CompleteSpace F", "f : ℕ →*₀ F", "p : ℕ", "hp : Nat.Prime p", "hsum : Summable fun x => ‖f x‖"], "goal": "(1 - f p)⁻¹ = ∑' (e : ℕ), f p ^ e"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "hf : Function.Injective f", "hf' : Continuous f", "T1Space Y"], "goal": "T1Space X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "f : α → β", "hf : AEMeasurable f μ"], "goal": "(map f μ).toOuterMeasure = ((OuterMeasure.map f) μ.toOuterMeasure).trim"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "DecidableEq α", "Semiring R", "x : α → R", "s : Finset α", "hc : (↑s).Pairwise fun i j => Commute (x i) (x j)", "n : ℕ"], "goal": "s.sum x ^ n = ∑ k : { x // x ∈ s.sym n }, ↑(↑↑k).multinomial * (Multiset.map x ↑↑k).noncommProd ⋯"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l : List α", "y : α", "hl : l.Nodup", "h✝ : (y :: l).getLast ⋯ ∈ l", "h : (y :: l).getLast ⋯ ∈ y :: l", "hy : (y :: l).getLast ⋯ ≠ y", "H : (y :: l).getLast ⋯ ∈ (y :: l).dropLast", "k : ℕ", "hk : k + 1 < (y :: l).dropLast.length", "hk' : (y :: l).dropLast.get ⟨k + 1, hk⟩ = (y :: l).getLast ⋯"], "goal": "False"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t u : Set Ω", "hsf : s.Finite", "h : ⋯.toFinset.card = hsf.toFinset.card"], "goal": "⋯.toFinset = hsf.toFinset"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : Filter β"], "goal": "(Filter.comap Prod.snd f).NeBot ↔ Nonempty α ∧ f.NeBot"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝² : MulOneClass α", "inst✝¹ : MulOneClass β", "f : α → β", "hf : IsMonoidHom f", "γ : Type u_1", "inst✝ : MulOneClass γ", "g : β → γ", "hg : IsMonoidHom g"], "goal": "g (f 1) = 1"}}
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+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "K : Type u_2", "inst✝⁵ : Field K", "inst✝⁴ : Algebra R K", "inst✝³ : IsFractionRing R K", "inst✝² : IsNoetherianRing R", "inst✝¹ : LocalRing R", "inst✝ : IsDomain R", "h : ¬IsField R", "this✝ : finrank (ResidueField R) (CotangentSpace R) = 1 ↔ finrank (ResidueField R) (CotangentSpace R) ≤ 1", "this : maximalIdeal R ≠ ⊥"], "goal": "DiscreteValuationRing R ↔ IsPrincipalIdealRing R"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "IsAsymm α r"], "goal": "IsIrrefl α r"}}
+{"state": {"context": ["α : Type u_1", "One α", "Pow α ℤ", "a : α", "n : ℤ"], "goal": "↑(a ^ n) = ↑a ^ n"}}
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+{"state": {"context": ["r : ℝ≥0"], "goal": "0 ≤ ↑r"}}
+{"state": {"context": ["n m : ℕ"], "goal": "n ≤ m + m.dist n"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddCommGroup E", "n : ℕ", "a : E"], "goal": "‖n • a‖ ≤ ↑n * ‖a‖"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "E I s : Set α", "M : Matroid α", "N : Matroid β", "hf : InjOn f M.E", "D : Set β"], "goal": "((∀ (x : Set α), M.Indep x → ¬D = f '' x) ∧ ∃ u ⊆ M.E, f '' u = D) ↔ ∃ D₀, (¬M.Indep D₀ ∧ D₀ ⊆ M.E) ∧ D = f '' D₀"}}
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+{"state": {"context": ["α : Type u_3", "E : Type u_6", "SeminormedAddGroup E", "TopologicalSpace α", "f : α → E"], "goal": "Continuous f → Continuous fun x => ‖f x‖"}}
+{"state": {"context": ["m : ℕ", "h : 2 ≤ 2 + m"], "goal": "(finRotate (m + 2)).IsCycle"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "s : Set P", "p : P", "hp : p ∈ s"], "goal": "vectorSpan k s = Submodule.span k ((fun x => p -ᵥ x) '' s)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "NonAssocSemiring α", "Preorder α", "NonAssocSemiring β", "Preorder β", "f : α ≃+*o β"], "goal": "f.toOrderRingHom = ↑f"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{u₂, u₁} C", "inst✝² : PreGaloisCategory C", "F : C ⥤ FintypeCat", "inst✝¹ : FiberFunctor F", "X Y : C", "f : X ⟶ Y", "inst✝ : Mono f", "h : Nat.card ↑(F.obj X) = Nat.card ↑(F.obj Y)"], "goal": "IsIso f"}}
+{"state": {"context": ["θ : ℝ"], "goal": "(↑θ).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : NonAssocSemiring R"], "goal": "∀ (x : ℕ), ↑x = 0 ↔ ringChar R ∣ x"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_4", "N' : Type u_3", "inst✝⁷ : CommRing R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup N", "inst✝³ : Module R N", "inst✝² : AddCommGroup N'", "inst✝¹ : Module R N'", "S : Submonoid R", "f : N →ₗ[R] N'", "inst✝ : IsLocalizedModule S f", "g : M →ₗ[R] N'", "σ : Finset M", "hσ : Submodule.span R ↑σ = ⊤", "τ : Finset ({ x // x ∈ σ } →₀ R)", "hτ : Submodule.span R ↑τ = LinearMap.ker (Finsupp.total { x // x ∈ σ } M R Subtype.val)", "π : ({ x // x ∈ σ } →₀ R) →ₗ[R] M := Finsupp.total { x // x ∈ σ } M R Subtype.val", "hπ : Function.Surjective ⇑π"], "goal": "∃ h s, f ∘ₗ h = s • g"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "inst✝¹ : (i : α) → NormedAddCommGroup (E i)", "inst✝ : Finite α", "f : (i : α) → E i"], "goal": "Memℓp f 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s✝ t u : Set α", "m : MeasurableSpace α", "mβ : MeasurableSpace β", "s : Set β"], "goal": "MeasurableSet (Sum.inr '' s) ↔ MeasurableSet s"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁵ : DivisionRing K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "V₂ : Type v'", "inst✝² : AddCommGroup V₂", "inst✝¹ : Module K V₂", "inst✝ : FiniteDimensional K V", "f : V →ₗ[K] V", "hinj : Injective ⇑f", "h : finrank K ↥(range f) = finrank K V"], "goal": "Surjective ⇑f"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "B : Type u_4", "Add A", "Add B", "Mul A", "Mul B", "SMul R A", "SMul R B", "Star A", "Star B", "e : A ≃⋆ₐ[R] B"], "goal": "EquivLike.inv e = ⇑e.symm"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v} → Ordinal.{max u v}", "a : Ordinal.{max u v}", "b : Ordinal.{max v u}"], "goal": "Ordinal.nfpFamily f a ≤ b ↔ ∀ (l : List ι), List.foldr f a l ≤ b"}}
+{"state": {"context": ["n : ℕ", "hn : 1 < n", "w z : ℂ"], "goal": "‖↑n ^ w‖ ≤ ‖↑n ^ z‖ ↔ w.re ≤ z.re"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "X : ℕ → C", "f : (n : ℕ) → X n ⟶ X (n + 1)", "n : ℕ"], "goal": "(CategoryTheory.Functor.ofSequence f).map (CategoryTheory.homOfLE ⋯) = f n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "x y z✝ : α", "δ ε ε₁ ε₂ : ℝ", "s : Set α", "h : dist x y ≤ ε₂ - ε₁", "z : α", "zx : z ∈ ball x ε₁"], "goal": "z ∈ ball y ε₂"}}
+{"state": {"context": ["l m r : List Char", "c : Char", "s : Substring"], "goal": "Substring.ValidFor l m r s → (s.contains c = true ↔ c ∈ m)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Abelian C", "X Y : C", "f : X ⟶ Y", "Z : C"], "goal": "[CategoryTheory.Epi f, CategoryTheory.Limits.cokernel.π f = 0, (CategoryTheory.ShortComplex.mk f 0 ⋯).Exact].TFAE"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁴ : Field F", "inst✝³ : Field E", "inst✝² : Algebra F E", "K : Type w", "inst✝¹ : Field K", "inst✝ : Algebra F K", "q : ℕ", "hF : ExpChar F q", "x : E", "this✝ : ExpChar E q", "this : ExpChar E[X] q"], "goal": "(minpoly F x).natSepDegree = 1 ↔ ∃ n, Polynomial.map (algebraMap F E) (minpoly F x) = (X - C x) ^ q ^ n"}}
+{"state": {"context": [], "goal": "∀ᶠ (n : ℕ) in Filter.atTop, 0 ≤ 1 + AkraBazziRecurrence.smoothingFn ↑n"}}
+{"state": {"context": ["J : Type w", "inst✝³ : SmallCategory J", "C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasLimitsOfShape J C", "inst✝ : MonoidalCategory C", "X : J ⥤ C", "j : J"], "goal": "(λ_ (limit X)).hom ≫ limit.π X j = (𝟙 (𝟙_ C) ⊗ 𝟙 (limit X)) ≫ (λ_ (limit X)).hom ≫ limit.π X j"}}
+{"state": {"context": ["X : TopCat"], "goal": "∀ (X_1 Y : Opens ↑((toTopCat X).obj ⊤)) (f : X_1 ⟶ Y), ⋯.functor.map f = eqToHom ⋯ ≫ (map (inclusionTopIso X).inv).map f ≫ eqToHom ⋯"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M"], "goal": "LinearMap.BilinMap.toQuadraticMap (LinearMap.dualProd R M) = 2 • QuadraticForm.dualProd R M"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "p : X → Prop"], "goal": "interior {x | p x} = {x | ∀ᶠ (y : X) in 𝓝 x, p y}"}}
+{"state": {"context": ["ξ : ℝ", "u v : ℤ", "hv : 2 ≤ v", "h : ContfracLegendre.Ass ξ u v", "hu₀ : 0 < u - ⌊ξ⌋ * v", "huv : u - ⌊ξ⌋ * v < v", "hξ₀ : 0 < fract ξ"], "goal": "|(fract ξ)⁻¹ - ↑v / (↑u - ↑⌊ξ⌋ * ↑v)| < ((↑u - ↑⌊ξ⌋ * ↑v) * (2 * (↑u - ↑⌊ξ⌋ * ↑v) - 1))⁻¹"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "t : Ordnode α", "x : α", "o₁ : WithBot α", "o₂ : WithTop α", "H : Ordnode.Valid' o₁ t o₂", "h₁ : Ordnode.nil.Bounded (↑x) o₂", "h₂ : Ordnode.All (fun x_1 => x_1 > x) t"], "goal": "Ordnode.Valid' (↑x) t o₂"}}
+{"state": {"context": ["x y : ℂ", "hre : |x.re| ≤ |y.re|", "him : |x.im| ≤ |y.im|"], "goal": "Complex.normSq x ≤ Complex.normSq y"}}
+{"state": {"context": ["ι : Type u_1", "ι₂ : Type u_2", "ι₃ : Type u_3", "R : Type u_4", "inst✝⁷ : CommSemiring R", "R₁ : Type u_5", "R₂ : Type u_6", "s : ι → Type u_7", "inst✝⁶ : (i : ι) → AddCommMonoid (s i)", "inst✝⁵ : (i : ι) → Module R (s i)", "M : Type u_8", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "E : Type u_9", "inst✝² : AddCommMonoid E", "inst✝¹ : Module R E", "F : Type u_10", "inst✝ : AddCommMonoid F", "motive : (⨂[R] (i : ι), s i) → Prop", "z : ⨂[R] (i : ι), s i", "smul_tprod : ∀ (r : R) (f : (i : ι) → s i), motive (r • (tprod R) f)", "add : ∀ (x y : ⨂[R] (i : ι), s i), motive x → motive y → motive (x + y)"], "goal": "motive z"}}
+{"state": {"context": ["R : Type u_2", "Semiring R", "f : HahnSeries ℕ R", "n : ℕ"], "goal": "(PowerSeries.coeff R n) (HahnSeries.toPowerSeries f) = f.coeff n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l₁ : Filter α", "l₂ : Filter β", "f : α →. β"], "goal": "Filter.PTendsto f l₁ l₂ ↔ Filter.RTendsto f.graph' l₁ l₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x : α", "inst✝ : Ring α", "h : Periodic f c", "n : ℤ"], "goal": "Periodic f (↑n * c)"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s : LowerSet α"], "goal": "s.carrier = ↑s"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop"], "goal": "(∀ x ∈ ∅, p x) ↔ True"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2"], "goal": "Sum.swap ∘ Sum.swap = id"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ ν : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure μ", "MeasureTheory.IsFiniteMeasure ν", "h : ¬μ ⟂ₘ ν"], "goal": "∃ ε,\n  0 < ε ∧\n    ∃ E,\n      MeasurableSet E ∧\n        0 < ν E ∧\n          MeasureTheory.VectorMeasure.restrict 0 E ≤\n            MeasureTheory.VectorMeasure.restrict (μ.toSignedMeasure - (ε • ν).toSignedMeasure) E"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "PreconnectedSpace X", "Nonempty Y", "f : LocallyConstant X Y"], "goal": "∃ y, f = LocallyConstant.const X y"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "Semiring R", "TopologicalSpace F", "TopologicalSpace B", "AddCommMonoid F", "Module R F", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module R (E x)", "e : Pretrivialization F Bundle.TotalSpace.proj", "Pretrivialization.IsLinear R e", "b : B"], "goal": "⇑(Pretrivialization.linearMapAt R e b) = fun y => if b ∈ e.baseSet then (↑e { proj := b, snd := y }).2 else 0"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "hf : ClosedEmbedding f"], "goal": "Filter.Tendsto f (Filter.cocompact X) (Filter.cocompact Y)"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : Fintype ι", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace ℝ F", "inst✝² : FiniteDimensional ℝ F", "inst✝¹ : MeasurableSpace F", "inst✝ : BorelSpace F", "b : OrthonormalBasis ι ℝ F", "this : Fact (finrank ℝ F = finrank ℝ F)", "o : Orientation ℝ F (Fin (finrank ℝ F)) := (stdOrthonormalBasis ℝ F).toBasis.orientation"], "goal": "volume (parallelepiped ⇑b) = 1"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "f : ℂ → E", "c : ℂ", "hd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z", "R : ℝ≥0", "hR0 : 0 < ↑R", "hRs : closedBall c ↑R ∩ {c}ᶜ ⊆ {x | (fun z => DifferentiableAt ℂ f z) x}", "hc : ContinuousOn f (closedBall c ↑R)"], "goal": "AnalyticAt ℂ f c"}}
+{"state": {"context": ["R : Type u1", "CommRing R", "M : Type u2", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "d : Module.Dual R M", "a : M"], "goal": "∀ (a_1 : CliffordAlgebra Q × CliffordAlgebra Q),\n  ((CliffordAlgebra.contractLeftAux Q d) a) a_1 = d a • a_1.1 - (CliffordAlgebra.ι Q) a * a_1.2"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_3", "t : (i : ι) → Set (α i)"], "goal": "∅.sigma t = ∅"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "IsEmpty V"], "goal": "G.chromaticNumber = 0"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "SeminormedRing 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : Seminorm 𝕜 E"], "goal": "p.comp LinearMap.id = p"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M"], "goal": "↑(Submonoid.center M) = Set.center M"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : 0 < y - x", "hyz : 0 < z - y", "hxz : 0 < z - x", "a : 𝕜 := (z - y) / (z - x)", "b : 𝕜 := (y - x) / (z - x)", "hy : a • x + b • z = y", "key : f (a • x + b • z) ≤ a • f x + b • f z"], "goal": "f y / (y - x) + f y / (z - y) ≤ f x / (y - x) + f z / (z - y)"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : DecidableEq R", "hp : Fact (Nat.Prime p)", "inst✝ : Invertible ↑p", "n : ℕ", "IH : ∀ m < n, constantCoeff (xInTermsOfW p R m) = 0"], "goal": "⅟↑p ^ n * ∑ x ∈ range n, constantCoeff (C (↑p ^ x) * xInTermsOfW p R x ^ p ^ (n - x)) = 0"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Add M", "Add N", "f : AddHom M N", "S : AddSubsemigroup M", "x : M", "hx : x ∈ S"], "goal": "f x ∈ AddSubsemigroup.map f S"}}
+{"state": {"context": ["α : Type u_4", "s : Finset α", "hs : s.Nonempty"], "goal": "∃ t a, ∃ (ha : a ∉ t), Finset.cons a t ha = s"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "htwo : ¬p ^ (k + 1) = 2", "this : NumberField K"], "goal": "hζ.toInteger - 1 ∣ ↑↑↑p"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "G : D ⥤ C", "h : G ⊣ F", "X' : Cᵒᵖ", "X Y : Dᵒᵖ", "f : F.op.obj X' ⟶ X", "g : X ⟶ Y"], "goal": "((h.homEquiv (Opposite.unop Y) (Opposite.unop X')).symm (g.unop ≫ f.unop)).op = ((h.homEquiv (Opposite.unop X) (Opposite.unop X')).symm f.unop).op ≫ (G.map g.unop).op"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a : α", "f : β → γ → ℝ≥0∞", "hf : Measurable (Function.uncurry f)", "F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)", "h : ∀ (a : β) (b : γ), ⨆ n, ↑(F n) (a, b) = f a b", "h_mono : Monotone F", "this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b)", "h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂η (a, b)", "n : ℕ"], "goal": "∫⁻ (a : β × γ), ↑(F n) a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑(F n) (a_1, c) ∂η (a, a_1) ∂κ a"}}
+{"state": {"context": ["α : Type u_1", "R : α → α → Prop", "DecidableRel R"], "goal": "List.destutter R [] = []"}}
+{"state": {"context": ["T : ℝ", "hT : Fact (0 < T)", "f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)"], "goal": "HasSum (fun i => fourierCoeff (↑↑f) i • fourierLp 2 i) f"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁶ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "inst✝⁴ : HasZeroMorphisms C", "inst✝³ : HasZeroMorphisms D", "F : C ⥤ D", "inst✝² : F.PreservesZeroMorphisms", "X Y : C", "inst✝¹ : HasBinaryBiproduct X Y", "inst✝ : PreservesBinaryBiproduct X Y F", "W : C", "f : W ⟶ X", "g : W ⟶ Y"], "goal": "lift (F.map f) (F.map g) ≫ (F.mapBiprod X Y).inv = F.map (lift f g)"}}
+{"state": {"context": ["n : ℕ", "NeZero n", "h : ¬1 = Fin.last (n + 1) := ⋯"], "goal": "Fin.castPred 1 h = 1"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "CommRing R", "i : ℤ"], "goal": "WittVector.ghostFun ↑i = ↑i"}}
+{"state": {"context": ["σ : Type u_1", "R✝ : Type u_2", "R : Type u_3", "inst✝² : CommSemiring R", "ι : Type u_4", "inst✝¹ : DecidableEq ι", "inst✝ : DecidableEq σ", "f : MvPowerSeries σ R", "m : ℕ", "hf : (constantCoeff σ R) f ^ m = 0", "d : σ →₀ ℕ", "n : ℕ", "hn : m + d.degree ≤ n", "k : ℕ →₀ σ →₀ ℕ", "hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n", "s : Finset ℕ := Finset.filter (fun i => k i = 0) (range n)", "hs_def : s = Finset.filter (fun i => k i = 0) (range n)", "hs : s ⊆ range n", "hs' : ∀ i ∈ s, (coeff R (k i)) f = (constantCoeff σ R) f", "hs'' : ∀ i ∈ s, k i = 0"], "goal": "∏ i ∈ range n, (coeff R (k i)) f = 0"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "CommSemiring R", "AddCommMonoid A", "AddCommMonoid B", "Module R A", "Module R B", "CoalgebraStruct R A", "CoalgebraStruct R B", "e : A ≃ₗc[R] B", "x x' : A"], "goal": "x = x' → e x = e x'"}}
+{"state": {"context": ["α : Type u_1", "m m₂ m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "hm : m ≤ m0", "hm₂ : m₂ ≤ m", "MeasureTheory.SigmaFinite (μ.trim ⋯)"], "goal": "MeasureTheory.SigmaFinite (μ.trim hm)"}}
+{"state": {"context": ["X : Type u_1", "X' : Type u_2", "Y : Type u_3", "Y' : Type u_4", "Z : Type u_5", "Z' : Type u_6", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : TopologicalSpace X'", "inst✝³ : TopologicalSpace Y", "inst✝² : TopologicalSpace Y'", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace Z'", "e : PartialHomeomorph X Y", "e' : PartialHomeomorph Y Z", "s : Set X", "hs : IsOpen s", "s' : Set X", "hs' : IsOpen s'"], "goal": "(ofSet s hs).trans (ofSet s' hs') = ofSet (s ∩ s') ⋯"}}
+{"state": {"context": ["α β : Type u", "ι : Type v", "f : ι → Cardinal.{w}", "hf : BddAbove (range f)", "t : Cardinal.{max u w}"], "goal": "lift.{u, w} (iSup f) ≤ t ↔ ∀ (i : ι), lift.{u, w} (f i) ≤ t"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "Algebra R A"], "goal": "↑(ContinuousAlgHom.id R A) = AlgHom.id R A"}}
+{"state": {"context": ["α : Type u_1"], "goal": "ofLex.symm = toLex"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : Type u_5", "x : ι → A", "CommRing R", "CommRing A", "Algebra R A", "hx : AlgebraicIndependent R x"], "goal": "(MvPolynomial.aeval x).comp hx.repr = (Algebra.adjoin R (Set.range x)).val"}}
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+{"state": {"context": ["T : Type u", "inst✝² : Category.{v, u} T", "C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "X✝ Y✝ : Arrow C", "f : X✝ ⟶ Y✝", "w : f.left ≫ Y✝.hom = X✝.hom ≫ f.right := f.w"], "goal": "(𝟭 D).map (F.map f.left) ≫ ((fun a => { left := F.obj a.left, right := F.obj a.right, hom := F.map a.hom }) Y✝).hom = ((fun a => { left := F.obj a.left, right := F.obj a.right, hom := F.map a.hom }) X✝).hom ≫ (𝟭 D).map (F.map f.right)"}}
+{"state": {"context": ["A : Type u_1", "Add A", "e₁ e₂ : AddAut A"], "goal": "⇑(e₁ * e₂) = ⇑e₁ ∘ ⇑e₂"}}
+{"state": {"context": ["β : Type u_1", "α : Type u_2", "A : List β → Set α", "x : ℕ → β", "hx : x ∈ (inducedMap A).fst", "y : ℕ → β", "hy : y ∈ (inducedMap A).fst", "hxy : (inducedMap A).snd ⟨x, hx⟩ = (inducedMap A).snd ⟨y, hy⟩", "n : ℕ", "ih : res x n = res y n", "hA : ¬x n = y n"], "goal": "∃ x x_1 x_2, ¬x_1 = x_2 ∧ ¬_root_.Disjoint (A (x_1 :: x)) (A (x_2 :: x))"}}
+{"state": {"context": ["α : Type u_1", "a : αᵒᵈ"], "goal": "OrderDual.toDual (OrderDual.ofDual a) = a"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "σ : Type v", "M : Type w", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I : Ideal (MvPolynomial σ R)", "p : MvPolynomial σ R", "hcoe : ∀ (m : σ →₀ ℕ), coeff m p ∈ Ideal.comap C I"], "goal": "p ∈ I"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "M : Type w", "L.Structure M", "α : Type u'", "L'.Structure M", "φ : L →ᴸ L'", "φ.IsExpansionOn M", "t : L.Term α", "v : α → M"], "goal": "FirstOrder.Language.Term.realize v (φ.onTerm t) = FirstOrder.Language.Term.realize v t"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "N : Type u_4", "inst✝⁷ : CommRing R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup N", "inst✝³ : Module R N", "inst✝² : _root_.Finite ι", "P : RootSystem ι R M N", "inst✝¹ : CharZero R", "inst✝ : NoZeroSMulDivisors R M", "P₁ P₂ : RootSystem ι R M N", "he : P₁.toLin = P₂.toLin", "hr : P₁.root = P₂.root", "i : ι"], "goal": "MapsTo (⇑(preReflection (P₁.root i) (P₁.toLin.flip (P₂.coroot i)))) (range ⇑P₁.root) (range ⇑P₁.root)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a✝ a₁ a₂ : α", "b✝ b₁ b₂ : β", "inst✝³ : LinearOrderedRing α", "inst✝² : LinearOrderedAddCommGroup β", "inst✝¹ : Module α β", "inst✝ : PosSMulStrictMono α β", "a : α", "b : β", "hab : 0 ≤ a • b"], "goal": "(0 ≤ a → b < 0) → ¬(a ≤ 0 → 0 < b)"}}
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+{"state": {"context": ["𝕜 : Type u_1", "inst✝² : RCLike 𝕜", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace 𝕜 E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "μ : 𝕜", "hμ : HasEigenvalue T μ", "v : E", "hv₁ : T v = μ • v", "hv₂ : v ≠ 0"], "goal": "(starRingEnd 𝕜) μ = μ"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsProbabilityMeasure μ", "ι : Type u_2", "β : Type u_3", "LinearOrder ι", "mβ : MeasurableSpace β", "NormedAddCommGroup β", "BorelSpace β", "f : ι → Ω → β", "i j : ι", "SecondCountableTopology β", "CompleteSpace β", "NormedSpace ℝ β", "hf : ∀ (i : ι), MeasureTheory.StronglyMeasurable (f i)", "hfi : ProbabilityTheory.iIndepFun (fun x => mβ) f μ", "hij : i < j"], "goal": "μ[f j|↑(MeasureTheory.Filtration.natural f hf) i] =ᵐ[μ] fun x => ∫ (x : Ω), f j x ∂μ"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{u_2, u_1} C", "inst✝² : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "inst✝¹ : S₁.HasRightHomology", "inst✝ : S₂.HasRightHomology", "h₀ : S₁.X₁ ⟶ S₂.X₁", "h₀_f : h₀ ≫ S₂.f = 0", "h₁ : S₁.X₂ ⟶ S₂.X₁", "h₂ : S₁.X₃ ⟶ S₂.X₂", "h₃ : S₁.X₃ ⟶ S₂.X₃", "g_h₃ : S₁.g ≫ h₃ = 0"], "goal": "rightHomologyMap (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃) = 0"}}
+{"state": {"context": ["s : Set Ordinal.{u}", "a✝ : Ordinal.{u}", "a : Ordinal.{u_1}"], "goal": "𝓝[≤] a = 𝓝 a"}}
+{"state": {"context": ["R S : CommRingCat", "φ : R ⟶ S"], "goal": "AlgebraicGeometry.Spec.preimage (AlgebraicGeometry.Spec.map φ) = φ"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "a : α", "hf : f.NeBot"], "goal": "f ≤ pure a ↔ f = pure a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace ℝ F", "inst✝⁶ : NormedAddCommGroup F'", "inst✝⁵ : NormedSpace ℝ F'", "inst✝⁴ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "T T' T'' : Set α → E →L[ℝ] F", "C C' C'' : ℝ", "hT : DominatedFinMeasAdditive μ T C", "hT' : DominatedFinMeasAdditive μ T' C'", "f✝ : ↥(Lp E 1 μ)", "f : ↥(simpleFunc E 1 μ)"], "goal": "((setToL1 hT + setToL1 hT').comp (coeToLp α E ℝ)) f = (setToL1SCLM α E μ ⋯) f"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommMonoid R", "R' : Type u_2", "inst✝ : CommMonoidWithZero R'", "χ : MulChar R R'", "a✝ : Rˣ"], "goal": "(χ * 1) ↑a✝ = χ ↑a✝"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "f : ArithmeticFunction R", "hf : f.IsMultiplicative", "x y : ℕ", "hx : ¬x = 0", "hy : ¬y = 0", "hgcd_ne_zero : x.gcd y ≠ 0", "hlcm_ne_zero : x.lcm y ≠ 0"], "goal": "f (x.lcm y) * f (x.gcd y) = f x * f y"}}
+{"state": {"context": ["x x' : SetTheory.PGame"], "goal": "x.insertRight x' ≤ x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "F : Type u_5", "𝕜 : Type u_6", "inst✝² : MeasurableSpace α", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedAddCommGroup F", "p : ℝ≥0∞", "μ : Measure α"], "goal": "↑(toSimpleFunc 0) =ᶠ[ae μ] 0"}}
+{"state": {"context": ["R : Type u_3", "A : Type u_4", "Semifield R", "Ring A", "Algebra R A", "a : A", "x : R"], "goal": "x ∈ quasispectrum R a ↔ x = 0 ∨ x ∈ spectrum R a"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "S₂ : Type w", "σ : Type u_1", "CommSemiring R", "CommSemiring S₁", "f : R →+* S₁", "g : σ → S₁", "s : Finset S₂", "p : S₂ → MvPolynomial σ R"], "goal": "MvPolynomial.eval₂ f g (∑ x ∈ s, p x) = ∑ x ∈ s, MvPolynomial.eval₂ f g (p x)"}}
+{"state": {"context": ["𝕜 : Type u_3", "E : Type u_7", "𝕜' : Type u_13", "NormedField 𝕜", "SeminormedRing 𝕜'", "NormedAlgebra 𝕜 𝕜'", "NormOneClass 𝕜'", "AddCommGroup E", "Module 𝕜' E", "SMul 𝕜 E", "IsScalarTower 𝕜 𝕜' E", "p : Seminorm 𝕜' E"], "goal": "(Seminorm.restrictScalars 𝕜 p).closedBall = p.closedBall"}}
+{"state": {"context": ["n : ℕ", "z : ℂ", "hz : 1 + z ∈ slitPlane", "hz' : -z ≠ 1"], "goal": "(-z) ^ n * (1 + z)⁻¹ = (1 + z)⁻¹ - ∑ j ∈ Finset.range n, (-1) ^ j * z ^ j"}}
+{"state": {"context": [], "goal": "UniformEmbedding Int.cast"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "f : ↥(simpleFunc G 1 μ)"], "goal": "‖f‖ = ∑ x ∈ (toSimpleFunc f).range, (μ (↑(toSimpleFunc f) ⁻¹' {x})).toReal * ‖x‖"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "μ : MeasureTheory.Measure ℝ", "f : ℝ → E", "MeasureTheory.NoAtoms μ", "h_int : MeasureTheory.Integrable f μ", "a : ℝ"], "goal": "Continuous fun b => ∫ (x : ℝ) in a..b, f x ∂μ"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e e' : PartialHomeomorph M H", "f f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s t : Set M", "x : M", "Q : (H → H) → Set H → H → Prop", "hG : G.LocalInvariantProp G' P", "mono : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, t ⊆ s → P f s x → P f t x", "h : LiftProp P g"], "goal": "LiftPropOn P g s"}}
+{"state": {"context": ["m k n : Nat"], "goal": "m + n ≤ k + n ↔ m ≤ k"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : α → F", "C : ℝ≥0", "hfC : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ C", "hμ : NeZero μ", "hp : ¬p = 0", "this : ∀ᵐ (x : α) ∂μ, ‖f x‖₊ ≤ ‖↑C‖₊"], "goal": "eLpNorm f p μ ≤ C • μ Set.univ ^ p.toReal⁻¹"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End R M", "μ : R", "hμ : f.HasEigenvalue μ"], "goal": "μ ∈ spectrum R f"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "m0 : MeasurableSpace α", "μ : Measure α", "v : VitaliFamily μ", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : SecondCountableTopology α", "inst✝² : BorelSpace α", "inst✝¹ : IsLocallyFiniteMeasure μ", "ρ : Measure α", "inst✝ : IsLocallyFiniteMeasure ρ", "hρ : ρ ≪ μ", "w : Set ℝ≥0∞", "w_count : w.Countable", "w_dense : Dense w", "left✝ : 0 ∉ w", "w_top : ⊤ ∉ w", "I : ∀ x ∈ w, x ≠ ⊤", "A : ∀ c ∈ w, ∀ d ∈ w, c < d → ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)"], "goal": "∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)"}}
+{"state": {"context": ["a : ℝ"], "goal": "ContinuousOn (fun x => jacobiTheta₂' (↑a) (I * ↑x) / (-2 * ↑π)) (Ioi 0)"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_6", "β' : Type u_7", "_mΩ : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "f : Ω → β", "g : Ω → β'", "_mβ : MeasurableSpace β", "_mβ' : MeasurableSpace β'", "Neg β", "MeasurableNeg β", "hfg : ProbabilityTheory.IndepFun f g μ"], "goal": "ProbabilityTheory.IndepFun (-f) g μ"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "n : ℤ"], "goal": "U R (n + 1) = 2 * X * U R n - U R (n - 1)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "G : Type u_5", "H : Type u_6", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "f : E → F", "x : E"], "goal": "AnalyticWithinAt 𝕜 f univ x ↔ AnalyticAt 𝕜 f x"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "cut : α → Ordering", "f : α → α", "t : RBSet α cmp"], "goal": "(∀ {x : α}, (zoom cut t.val).fst.root? = some x → cmpEq cmp (f x) x) → OnRoot (fun x => cmpEq cmp (f x) x) (zoom cut t.val).fst"}}
+{"state": {"context": ["α : Type u_1", "CompleteLattice α", "AddGroup α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "s t : Set α"], "goal": "sSup (s - t) = sSup s - sInf t"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "c : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "a : ℝ", "h0 : 0 < rexp a", "b : ℝ", "hle : a ≤ b", "hd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z", "A : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)", "hc : ContinuousOn f A", "R : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]", "g : ℂ → ℂ := fun x => c + cexp x", "hdg : Differentiable ℂ g"], "goal": "∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp b) θ) = ∫ (θ : ℝ) in 0 ..2 * π, I • f (circleMap c (rexp a) θ)"}}
+{"state": {"context": ["G : Type u_1", "GroupWithZero G", "a : G", "ha : a ≠ 0"], "goal": "⇑(Equiv.symm (Equiv.mulRight₀ a ha)) = fun x => x * a⁻¹"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "x y : α"], "goal": "Metric.diam {x, y} = dist x y"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "A : Type u_3", "T : Type u_4", "B : Type u_5", "ι : Type u_6", "inst✝¹¹ : SemilatticeSup B", "inst✝¹⁰ : OrderBot B", "inst✝⁹ : SemilatticeInf T", "inst✝⁸ : OrderTop T", "inst✝⁷ : CommSemiring R", "inst✝⁶ : AddCommMonoid A", "inst✝⁵ : AddCommMonoid B", "inst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝² : AddCommMonoid T", "inst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "degb : A → B", "degt : A → T", "degb0 : degb 0 ≤ 0", "degbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b", "a✝ : List R[A]"], "goal": "(Multiset.prod (Quot.mk Setoid.r a✝)).support.sup degb ≤ (Multiset.map (fun f => f.support.sup degb) (Quot.mk Setoid.r a✝)).sum"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E", "x : E", "s t : Set E", "hs : StarConvex 𝕜 x s", "ht : StarConvex 𝕜 x t"], "goal": "StarConvex 𝕜 x (s ∪ t)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "μ : Measure α", "f✝ f : α → ℝ", "M : ℝ", "f_intble : Integrable f μ", "f_nn : 0 ≤ᶠ[ae μ] f", "f_bdd : f ≤ᶠ[ae μ] fun x => M"], "goal": "∫ (ω : α), f ω ∂μ = ∫ (t : ℝ) in Ioc 0 M, (μ {a | t ≤ f a}).toReal"}}
+{"state": {"context": [], "goal": "(𝓝 ⊤).HasBasis (fun x => True) fun x => Set.Ioi ↑x"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "HasContDiffBump E", "MeasurableSpace E", "c : E", "f : ContDiffBump c", "μ : MeasureTheory.Measure E", "x : E"], "goal": "0 ≤ f.normed μ x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : GeneralizedCoheytingAlgebra α", "a✝ b✝ c d a b : α", "h : a ≤ b"], "goal": "a ∆ b = b \\ a"}}
+{"state": {"context": ["r : ℝ≥0", "b : ℝ≥0∞"], "goal": "Filter.Tendsto (fun b => ↑r - b) (𝓝 b) (𝓝 (↑r - b))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "TopologicalSpace H", "TopologicalSpace M", "I : ModelWithCorners 𝕜 E H", "ChartedSpace H M", "I.Boundaryless", "x : M"], "goal": "IsOpen (extChartAt I x).target"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "d : D", "X : (CategoryTheory.CostructuredArrow F.op (Opposite.op d))ᵒᵖ"], "goal": "(CategoryTheory.CostructuredArrow.toStructuredArrow' F d).obj X =\n  CategoryTheory.StructuredArrow.mk (Opposite.unop X).hom.unop"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.ConcreteCategory C", "X₁ X₂ S : C", "f₁ : X₁ ⟶ S", "f₂ : X₂ ⟶ S", "CategoryTheory.Limits.HasPullback f₁ f₂", "CategoryTheory.Limits.PreservesLimit (CategoryTheory.Limits.cospan f₁ f₂) (CategoryTheory.forget C)", "x₁ : (CategoryTheory.forget C).obj X₁", "x₂ : (CategoryTheory.forget C).obj X₂", "h : f₁ x₁ = f₂ x₂"], "goal": "(CategoryTheory.Limits.pullback.fst f₁ f₂) (CategoryTheory.Limits.Concrete.pullbackMk f₁ f₂ x₁ x₂ h) = x₁"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : InnerProductSpace ℝ E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace ℝ F", "φ : E ≃ₗᵢ[ℝ] F", "n : ℕ", "_i : Fact (finrank ℝ E = n + 1)", "o : Orientation ℝ E (Fin (n + 1))", "inst✝ : Fact (finrank ℝ F = n + 1)", "x : Fin (n + 1) → F", "e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o", "he : e.toBasis.orientation = o"], "goal": "((map (Fin (n + 1)) φ.toLinearEquiv) o).volumeForm x = o.volumeForm (⇑φ.symm ∘ x)"}}
+{"state": {"context": ["R S : Type u", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "M : Submonoid R", "N : Submonoid S", "R' S' : Type u", "inst✝³ : CommRing R'", "inst✝² : CommRing S'", "f : R →+* S", "inst✝¹ : Algebra R R'", "inst✝ : Algebra S S'", "I J : Ideal R", "h : ∀ (P : Ideal R) (hP : P.IsMaximal), map (algebraMap R (Localization.AtPrime P)) I ≤ map (algebraMap R (Localization.AtPrime P)) J", "x : R", "hx : x ∈ I"], "goal": "Submodule.colon J (span {x}) = ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "iso : E ≃L[𝕜] F", "f : F → G", "s : Set F", "x : E", "f' : E →L[𝕜] G"], "goal": "HasFDerivWithinAt (f ∘ ⇑iso) f' (⇑iso ⁻¹' s) x ↔ HasFDerivWithinAt f (f'.comp ↑iso.symm) s (iso x)"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "NonAssocSemiring α", "Preorder α", "NonAssocSemiring β", "Preorder β", "f : α ≃+*o β"], "goal": "⇑↑f = ⇑f"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "hf : f.NeBot", "s : Set α", "hs : s ∈ f"], "goal": "s.Nonempty"}}
+{"state": {"context": ["ι : Sort u_1", "V : Type u", "G : SimpleGraph V", "a b c u v w : V", "e : Sym2 V", "G₁ G₂ : SimpleGraph V"], "goal": "G.Adj a b ↔ ∃ e, ↑e = s(a, b)"}}
+{"state": {"context": ["a : Int"], "goal": "∃ n, a = ↑n ∨ a = -↑n"}}
+{"state": {"context": ["α : Type u_1", "Max α", "n : Nat", "a : α", "w : max a a = a", "h : 0 < n"], "goal": "(List.replicate n a).maximum? = some a"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{u₂, u₁} C", "inst✝² : PreGaloisCategory C", "F : C ⥤ FintypeCat", "inst✝¹ : FiberFunctor F", "A X : C", "inst✝ : IsConnected A", "a : ↑(F.obj A)", "f g : A ⟶ X", "h : F.map f a = F.map g a", "this : IsIso (equalizer.ι f g)"], "goal": "f = g"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "DivisionRing k", "AddCommGroup V", "Module k V", "FiniteDimensional k V", "n : ℕ", "T : Affine.Simplex k V n", "hrank : FiniteDimensional.finrank k V = n"], "goal": "affineSpan k (Set.range T.points) = ⊤"}}
+{"state": {"context": ["α : Type u_1", "s t : Set α"], "goal": "Set.up s < Set.up t ↔ s ⊂ t"}}
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+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "s : Set ↑(X.presheaf.obj (Opposite.op ⊤))"], "goal": "⇑(AlgebraicGeometry.ΓSpec.adjunction.unit.app X).val.base ⁻¹' PrimeSpectrum.zeroLocus s = X.zeroLocus s"}}
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+{"state": {"context": ["P : Type u_1", "Preorder P", "F : Order.PFilter P"], "goal": "DirectedOn (fun x x_1 => x ≥ x_1) ↑F"}}
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+{"state": {"context": ["α : Type u_1", "Monoid α", "x y z : α"], "goal": "x ~ᵤ y → y ~ᵤ z → x ~ᵤ z"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "x✝ y✝ x y : V", "hx : x ≠ 0", "hy : y ≠ 0"], "goal": "⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ ↔ angle x y = 0"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "AddGroup α", "e : α", "x : Finset α × Finset α"], "goal": "(Finset.addETransformLeft e x).2 = x.2 ∪ (-e +ᵥ x.2)"}}
+{"state": {"context": ["R : Type u_1", "x y : R", "n m p : ℕ", "inst✝ : Ring R", "hp : n + m ≤ p + 1", "h_comm : Commute x y", "h : (x - y) ^ n = 0"], "goal": "x ^ m ∣ y ^ p"}}
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+{"state": {"context": ["R : Type u_1", "S : Type u_2", "Semiring R", "Semiring S", "f : R →+* S", "x : S", "p : R[X]"], "goal": "Polynomial.eval₂ f x p = p.sum fun e a => f a * x ^ e"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "inst✝³ : Fintype n", "inst✝² : Fintype o", "inst✝¹ : CommRing R", "inst✝ : StrongRankCondition R", "A : Matrix m n R", "B : Matrix n o R"], "goal": "(A * B).rank ≤ A.rank"}}
+{"state": {"context": ["x : ℝ", "hx : 0 ≤ x", "y : ℝ"], "goal": "√(x * y) = √x * √y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2"], "goal": "Function.Injective Option.map"}}
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+{"state": {"context": ["m n : ℕ∞"], "goal": "↑m ≤ ↑n ↔ m ≤ n"}}
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+{"state": {"context": [], "goal": "LinearMap.det Complex.conjAe.toLinearMap = -1"}}
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+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "CommRing S", "Algebra R S", "Rₘ : Type u_3", "Sₘ : Type u_4", "CommRing Rₘ", "Algebra R Rₘ", "CommRing Sₘ", "Algebra S Sₘ", "M : Submonoid R", "IsLocalization M Rₘ", "IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ", "Algebra Rₘ Sₘ", "Algebra R Sₘ", "IsScalarTower R Rₘ Sₘ", "IsScalarTower R S Sₘ", "ι : Type u_5", "Fintype ι", "DecidableEq ι", "b : Basis ι R S", "a : S"], "goal": "(algebraMap R Rₘ).mapMatrix ((Algebra.leftMulMatrix b) a) =\n  (Algebra.leftMulMatrix (Basis.localizationLocalization Rₘ M Sₘ b)) ((algebraMap S Sₘ) a)"}}
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+{"state": {"context": ["B : Type u", "F : Type v", "E : B → Type w₁", "B' : Type w₂", "TopologicalSpace B'", "TopologicalSpace (Bundle.TotalSpace F E)", "f : B' → B"], "goal": "Continuous Bundle.TotalSpace.proj"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : Sym2 α", "f : α → β", "r : α → α → Prop", "s : Set (Sym2 α)", "x y : α"], "goal": "ToRel s x y → ToRel s y x"}}
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+{"state": {"context": ["α : Type u", "x y : List α"], "goal": "y = x ++ y ↔ x = []"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ"], "goal": "cosZeta a s = (cosZeta a s + I * sinZeta a s + (cosZeta a s + I * -sinZeta a s)) / 2"}}
+{"state": {"context": ["n a b : ℤ"], "goal": "a ≡ b [ZMOD n] ↔ n ∣ b - a"}}
+{"state": {"context": ["f : ℂ → ℂ", "c : ℂ", "R : ℝ", "hd : DifferentiableOn ℂ f (Metric.ball c R)", "h_maps : Set.MapsTo f (Metric.ball c R) (Metric.ball c R)", "hc : f c = c", "h₀ : 0 < R"], "goal": "Complex.abs (deriv f c) ≤ 1"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Monoid (G i)", "inst✝ : Monoid H", "φ : (i : ι) → H →* G i", "i : ι", "x : H"], "goal": "(of i) ((φ i) x) = (base φ) x"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "x : α", "z : ℤ"], "goal": "|x - ↑(round x)| ≤ |x - ↑z|"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "CommSemiring ι", "Module ι (Additive ℤˣ)", "DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "CommRing R", "(i : ι) → AddCommGroup (𝒜 i)", "(i : ι) → AddCommGroup (ℬ i)", "(i : ι) → Module R (𝒜 i)", "(i : ι) → Module R (ℬ i)", "DirectSum.GRing 𝒜", "DirectSum.GRing ℬ", "DirectSum.GAlgebra R 𝒜", "DirectSum.GAlgebra R ℬ"], "goal": "(TensorProduct.gradedComm R 𝒜 ℬ) 1 = 1"}}
+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "ι✝ : Type u_4", "inst✝⁸ : Countable ι✝", "α : Type u_5", "β : Type u_6", "ι : Type u_7", "inst✝⁷ : TopologicalSpace ι", "inst✝⁶ : MetrizableSpace ι", "inst✝⁵ : MeasurableSpace ι", "inst✝⁴ : SecondCountableTopology ι", "inst✝³ : OpensMeasurableSpace ι", "inst✝² : TopologicalSpace β", "inst✝¹ : PseudoMetrizableSpace β", "inst✝ : MeasurableSpace α", "u : ι → α → β", "hu_cont : ∀ (x : α), Continuous fun i => u i x", "h : ∀ (i : ι), StronglyMeasurable (u i)", "this✝¹ : MeasurableSpace β := borel β", "this✝ : BorelSpace β", "t_sf : ℕ → ι →ₛ ι", "ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))", "U : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.1) p.2", "h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)", "n : ℕ", "h_str_meas : StronglyMeasurable fun p => u (↑p.1) p.2"], "goal": "StronglyMeasurable (U n)"}}
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+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I✝ J K L I : Ideal R", "hI : I.IsPrime", "s : Finset R"], "goal": "∏ x ∈ s, x ∈ I ↔ ∃ p ∈ s, p ∈ I"}}
+{"state": {"context": ["M : Type u_1", "Neg M", "a : M"], "goal": "DomAddAct.mk (-a) = -DomAddAct.mk a"}}
+{"state": {"context": ["R₁ : Type u_1", "Semiring R₁", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "Module R₁ M₁", "p : Submodule R₁ M₁"], "goal": "↑p.subtypeL = p.subtype"}}
+{"state": {"context": ["α β : Frm", "e : ↑α ≃o ↑β", "a : ↑α"], "goal": "(Frm.Iso.mk e).hom a = e a"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "k : Type u_1", "inst✝² : Field k", "inst✝¹ : CharP k p", "inst✝ : IsAlgClosed k", "a₁ a₂ : 𝕎 k", "ha₁ : a₁.coeff 0 ≠ 0", "ha₂ : a₂.coeff 0 ≠ 0", "h : frobeniusRotation p ha₁ ha₂ = 0"], "goal": "solution p a₁ a₂ = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : Monoid R", "S : Submonoid R", "inst✝² : OreSet S", "X : Type u_2", "inst✝¹ : AddMonoid X", "inst✝ : DistribMulAction R X", "r₁ : X", "s₁ : ↥S", "r₂ : X", "s₂ : ↥S", "rb sb : R", "hb : sb * ↑s₁ = rb * ↑s₂", "h : sb * ↑s₁ ∈ S", "ra : R", "sa : ↥S", "ha : ↑sa * ↑s₁ = ra * ↑s₂", "rc : R", "sc : ↥S", "hc : ↑sc * sb = rc * ↑sa", "this : ↑sc * rb * ↑s₂ = rc * ra * ↑s₂", "sd : ↥S", "hd : ↑sd * (↑sc * rb) = ↑sd * (rc * ra)"], "goal": "∃ u v, u • (sb • r₁ + rb • r₂) = v • (sa • r₁ + ra • r₂) ∧ ↑u * ↑⟨sb * ↑s₁, h⟩ = v * ↑(sa * s₁)"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "LinearOrderedField 𝕜", "TopologicalSpace 𝕜", "OrderTopology 𝕜", "l : Filter α", "f : α → 𝕜", "h : Filter.Tendsto f l Filter.atTop"], "goal": "Filter.Tendsto f⁻¹ l (𝓝 0)"}}
+{"state": {"context": ["M : Type u", "MulOneClass M", "P : Prop", "Decidable P", "a b : M"], "goal": "(if P then 1 else a * b) = (if P then 1 else a) * if P then 1 else b"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "I : Type u_3", "J : Type u_4", "inst✝⁷ : Category.{u_6, u_1} C", "inst✝⁶ : Category.{u_5, u_2} D", "inst✝⁵ : Zero I", "inst✝⁴ : DecidableEq I", "inst✝³ : HasInitial C", "F : C ⥤ D ⥤ D", "X : C", "e : F.obj X ≅ 𝟭 D", "inst✝² : (Y : D) → PreservesColimit (Functor.empty C) (F.flip.obj Y)", "p : I × J → J", "hp : ∀ (j : J), p (0, j) = j", "Y Y' : GradedObject J D", "φ : Y ⟶ Y'", "inst✝¹ : (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y).HasMap p", "inst✝ : (((mapBifunctor F I J).obj ((single₀ I).obj X)).obj Y').HasMap p", "j : J"], "goal": "φ j ≫ e.inv.app (Y' j) ≫ (F.map (singleObjApplyIso 0 X).inv).app (Y' j) ≫ ιMapBifunctorMapObj F p ((single₀ I).obj X) Y' 0 j j ⋯ = e.inv.app (Y j) ≫ (F.map (singleObjApplyIso 0 X).inv).app (Y j) ≫ (F.map (𝟙 ((single₀ I).obj X 0))).app (Y j) ≫ (F.obj ((single₀ I).obj X 0)).map (φ j) ≫ ιMapBifunctorMapObj F p ((single₀ I).obj X) Y' 0 j j ⋯"}}
+{"state": {"context": ["z x y p : ℝ", "h : 1 ≤ p"], "goal": "ContinuousAt (fun x => x ^ p) 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : SuccOrder α", "inst✝ : PredOrder α", "a b i : α", "n : ℕ", "hn : ¬IsMax (succ^[n - 1] i) → pred^[n] (succ^[n] i) = i", "hin : ¬IsMax (succ^[n] i)", "h_not_max : ¬IsMax (succ^[n - 1] i)"], "goal": "pred^[n + 1] (succ^[n + 1] i) = i"}}
+{"state": {"context": [], "goal": "Nat.factorial 2 = 2"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "LE α", "f : ι → LowerSet α"], "goal": "↑(⨅ i, f i) = ⋂ i, ↑(f i)"}}
+{"state": {"context": ["ξ : ℝ", "hξ : Irrational ξ", "q : ℚ"], "goal": "∃ q', |ξ - ↑q'| < 1 / ↑q'.den ^ 2 ∧ |ξ - ↑q'| < |ξ - ↑q|"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "NormedLinearOrderedField β", "u v : α → β", "l : Filter α", "OrderTopology β", "huv : u ~[l] v", "hu : Filter.Tendsto u l Filter.atBot"], "goal": "Filter.Tendsto v l Filter.atBot"}}
+{"state": {"context": ["ι : Type uι", "inst✝⁵ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝¹ : SeminormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)", "G : Type (max uι u𝕜 uE)", "x✝¹ : SeminormedAddCommGroup G", "x✝ : NormedSpace 𝕜 G", "hp : p = (normSeminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G →L[𝕜] G)).comp (toDualContinuousMultilinearMap G)", "x : ⨂[𝕜] (i : ι), E i"], "goal": "‖(toDualContinuousMultilinearMap G) x‖ ≤ projectiveSeminorm x"}}
+{"state": {"context": ["a b c x : ℕ"], "goal": "x ∈ Icc a.succ b ↔ x ∈ Ioc a b"}}
+{"state": {"context": ["𝕜 : Type u_1", "RCLike 𝕜", "E : Type u_2", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "T : E →L[𝕜] E", "r : ℝ", "hr : 0 < r"], "goal": "⨆ x, T.rayleighQuotient ↑x = ⨆ x, T.rayleighQuotient ↑x"}}
+{"state": {"context": ["α : Type u_1", "E' : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "m m0 : MeasurableSpace α", "μ : Measure α", "inst✝⁸ : RCLike 𝕜", "inst✝⁷ : NormedAddCommGroup E'", "inst✝⁶ : InnerProductSpace 𝕜 E'", "inst✝⁵ : CompleteSpace E'", "inst✝⁴ : NormedSpace ℝ E'", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "hm : m ≤ m0", "f g : ↥(Lp E' p μ)", "hp_ne_zero : p ≠ 0", "hp_ne_top : p ≠ ⊤", "hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (↑↑f) s μ", "hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (↑↑g) s μ", "hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑f x ∂μ = ∫ (x : α) in s, ↑↑g x ∂μ", "hf_meas : AEStronglyMeasurable' m (↑↑f) μ", "hg_meas : AEStronglyMeasurable' m (↑↑g) μ", "hfg' : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, ↑↑(f - g) x ∂μ = 0", "hfg_int : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (↑↑(f - g)) s μ"], "goal": "↑↑(f - g) =ᶠ[ae μ] 0"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "Fact (FiniteDimensional.finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "((-o).kahler x) y = (starRingEnd ℂ) ((o.kahler x) y)"}}
+{"state": {"context": ["R : Type u_1", "S : Type ?u.189967", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing S", "inst✝⁸ : Algebra R S", "inst✝⁷ : LocalRing R", "M : Type u_2", "N : Type ?u.190432", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R M", "inst✝³ : Module R N", "P : Type ?u.191572", "inst✝² : AddCommGroup P", "inst✝¹ : Module R P", "f✝ : M →ₗ[R] N", "g : N →ₗ[R] P", "hg : Surjective ⇑g", "h : Exact ⇑f✝ ⇑g", "inst✝ : FinitePresentation R M", "H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪))", "I : Type (max u_2 u_1) := Free.ChooseBasisIndex k (k ⊗[R] M)", "b : Basis I k (k ⊗[R] M) := Free.chooseBasis k (k ⊗[R] M)", "this✝² : IsNoetherian k (k ⊗[R] (I →₀ R)) := isNoetherian_of_isNoetherianRing_of_finite k (k ⊗[R] (I →₀ R))", "f : k ⊗[R] M → M", "hf : ∀ (b : k ⊗[R] M), ((TensorProduct.mk R k M) 1) (f b) = b", "i : (I →₀ R) →ₗ[R] M := Finsupp.total I M R (f ∘ ⇑b)", "hi : Surjective ⇑i", "this✝¹ : Finite R ↥(LinearMap.ker i)", "this✝ : Surjective ⇑(LinearMap.baseChange k i)", "hi' : Function.Bijective ⇑(LinearMap.baseChange k i)", "this : ∀ {g₁ : ↥(LinearMap.ker i) →ₗ[R] I →₀ R} {g₂ : (I →₀ R) →ₗ[R] M}, Exact ⇑(Submodule.subtype 𝔪) ⇑(Submodule.mkQ 𝔪) → Surjective ⇑(Submodule.mkQ 𝔪) → Exact ⇑g₁ ⇑g₂ → Surjective ⇑g₂ → Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype 𝔪)) → Function.Injective ⇑(LinearMap.lTensor R g₁) → Function.Injective ⇑(LinearMap.lTensor (R ⧸ 𝔪) g₁)", "x : k ⊗[R] ↥(LinearMap.ker i)"], "goal": "(LinearMap.baseChange k i) ((LinearMap.lTensor (R ⧸ 𝔪) (LinearMap.ker i).subtype) x) = (LinearMap.baseChange k i) ((LinearMap.lTensor (R ⧸ 𝔪) (LinearMap.ker i).subtype) 0)"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁵ : DivisionRing K", "inst✝⁴ : AddCommGroup V", "inst✝³ : Module K V", "V₂ : Type v'", "inst✝² : AddCommGroup V₂", "inst✝¹ : Module K V₂", "inst✝ : FiniteDimensional K V", "S : Submodule K V", "h : finrank K ↥S = finrank K V", "this✝¹ : IsNoetherian K V", "bS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S", "bS_eq : bS = Basis.ofVectorSpace K ↥S", "this✝ : LinearIndependent (ι := { x // x ∈ Subtype.val '' Basis.ofVectorSpaceIndex K ↥S }) K Subtype.val", "b : Basis (↑(this✝.extend ⋯)) K V := Basis.extend this✝", "b_eq : b = Basis.extend this✝", "i1 : Fintype ↑(this✝.extend ⋯) := ⋯.fintype", "i2 : Fintype ↑(Subtype.val '' Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype", "i3 : Fintype ↑(Basis.ofVectorSpaceIndex K ↥S) := ⋯.fintype", "this : Subtype.val '' Basis.ofVectorSpaceIndex K ↥S = this✝.extend ⋯"], "goal": "S = Submodule.map S.subtype (span K (Basis.ofVectorSpaceIndex K ↥S))"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "l : Filter α", "CountableInterFilter l"], "goal": "m ≤ EventuallyMeasurableSpace m l"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero R", "inst✝³ : UniqueFactorizationMonoid R", "inst✝² : NormalizationMonoid R", "inst✝¹ : DecidableRel Dvd.dvd", "inst✝ : DecidableEq R", "p x : R", "hp : Irreducible p", "hnorm : normalize p = p", "n : ℕ", "hle : p ^ n ∣ x", "hlt : ¬p ^ (n + 1) ∣ x", "this : DecidableRel fun x x_1 => x ∣ x_1 := fun x x_1 => Classical.propDecidable ((fun x x_2 => x ∣ x_2) x x_1)", "hx0 : ¬x = 0"], "goal": "↑(count p (normalizedFactors x)) = ↑n"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s : Set P", "p : Fin 3 → P", "hps : Set.range p ⊆ s", "hpi : Function.Injective p", "v : V", "hv0 : v ≠ 0", "c : P", "r : ℝ", "hs : ∀ p ∈ s, dist p c = r", "hs' : ∀ (i : Fin 3), dist (p i) c = r", "f : Fin 3 → ℝ", "hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0", "hf0 : f 0 = 0", "hfi : Function.Injective f", "hsd : ∀ (i : Fin 3), f i = 0 ∨ f i = -2 * ⟪v, p 0 -ᵥ c⟫_ℝ / ⟪v, v⟫_ℝ"], "goal": "False"}}
+{"state": {"context": ["k : Type u_1", "V₁ : Type u_2", "P₁ : Type u_3", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "S₁ S₂ : AffineSubspace k P₁", "Nonempty ↥S₁", "Nonempty ↥S₂", "h : S₁ = S₂", "x : ↥S₁"], "goal": "↑((AffineEquiv.ofEq S₁ S₂ h) x) = ↑x"}}
+{"state": {"context": ["R : Type u", "Mul R", "StarMul R", "x y : R"], "goal": "star (star x * y) = star y * x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "self : S₁.Hom S₂"], "goal": "CategoryTheory.CategoryStruct.comp self.τ₁ S₂.f = CategoryTheory.CategoryStruct.comp S₁.f self.τ₂"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_3", "X : Type u_5", "TopologicalSpace X", "TopologicalSpace M", "Monoid M", "ContinuousMul M", "f : ι → X → M", "l : List ι", "h : ∀ i ∈ l, Continuous (f i)"], "goal": "Continuous fun a => (List.map (fun i => f i a) l).prod"}}
+{"state": {"context": ["ι : Type u_1", "M : ι → Type u_2", "(i : ι) → Monoid (M i)", "C : Monoid.CoprodI M → Prop", "m : Monoid.CoprodI M", "one : C 1", "mul : ∀ {i : ι} (m : M i) (x : Monoid.CoprodI M), C x → C (Monoid.CoprodI.of m * x)"], "goal": "C m"}}
+{"state": {"context": ["β : Type u_2", "Ω : Type u_3", "mβ : MeasurableSpace β", "MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "ρ : MeasureTheory.Measure (β × Ω)", "MeasureTheory.IsFiniteMeasure ρ", "E : Type u_4", "f : β × Ω → E", "NormedAddCommGroup E", "NormedSpace ℝ E", "s : Set β", "hs : MeasurableSet s", "t : Set Ω", "ht : MeasurableSet t", "hf : MeasureTheory.IntegrableOn f (s ×ˢ t) ρ"], "goal": "∫ (b : β) in s, ∫ (ω : Ω) in t, f (b, ω) ∂ρ.condKernel b ∂ρ.fst = ∫ (x : β × Ω) in s ×ˢ t, f x ∂ρ"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : CompleteSpace E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "μ ν : Measure α", "s t N : Set α", "f : α → ℝ≥0∞", "hμ : μ ≠ 0", "hint : ∫⁻ (a : α), f a ∂μ ≠ ⊤", "hN : μ N = 0"], "goal": "∃ x ∉ N, ⨍⁻ (a : α), f a ∂μ ≤ f x"}}
+{"state": {"context": ["a b c : Int", "h : c ≠ 0"], "goal": "a * c = b * c ↔ a = b"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "iso : E ≃ₗᵢ[𝕜] F", "f : G → E", "x : G", "f' : G →L[𝕜] F"], "goal": "HasFDerivAt (⇑iso ∘ f) f' x ↔\n  HasFDerivAt f ((↑{ toLinearEquiv := iso.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).comp f')\n    x"}}
+{"state": {"context": ["R : Type u", "a : R", "Semiring R", "ha : a ≠ 0"], "goal": "(Polynomial.C a).degree = 0"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_6, u_1} C", "CategoryTheory.Category.{u_7, u_2} D", "A : Type u_5", "AddMonoid A", "CategoryTheory.HasShift D A", "F G : CategoryTheory.SingleFunctors C D A", "iso : (a : A) → F.functor a ≅ G.functor a", "comm :\n  ∀ (n a a' : A) (ha' : n + a = a'),\n    (F.shiftIso n a a' ha').hom ≫ (iso a).hom =\n      CategoryTheory.whiskerRight (iso a').hom (CategoryTheory.shiftFunctor D n) ≫ (G.shiftIso n a a' ha').hom", "a : A"], "goal": "(CategoryTheory.SingleFunctors.isoMk iso comm).hom.hom a = (iso a).hom"}}
+{"state": {"context": ["α : Type u", "f : α → α", "m n : ℕ"], "goal": "f^[m + n] = f^[m] ∘ f^[n]"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F", "s : Set E", "s_conv : Convex ℝ s", "hne : (interior s).Nonempty", "f : E → F", "f' : E → E →L[ℝ] F", "f'' : E →L[ℝ] E →L[ℝ] F", "hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x", "x : E", "xs : x ∈ s", "hx : HasFDerivWithinAt f' f'' (interior s) x", "v w : E"], "goal": "(f'' v) w = (f'' w) v"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "SeminormedAddCommGroup E", "NormedSpace 𝕜 E"], "goal": "(NormedSpace.dualPairing 𝕜 E).SeparatingLeft"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "δ : ℝ", "s : Set ℝ", "hsδ : s ⊆ Set.Ioi δ", "hs : ∀ (ε : ℝ), δ < ε → (s ∩ Set.Ioc δ ε).Nonempty", "E : Set α"], "goal": "Metric.cthickening δ E = ⋂ ε ∈ s, Metric.cthickening ε E"}}
+{"state": {"context": ["α : Type u", "GroupWithZero α", "a : α", "Invertible a"], "goal": "a⁻¹ * a = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace γ", "inst✝³ : MeasurableSpace δ", "μ : Measure α", "ν : Measure β", "f : α → γ", "g : β → γ", "inst✝² : TopologicalSpace γ", "inst✝¹ : MetrizableSpace γ", "inst✝ : BorelSpace γ", "h : IdentDistrib f g μ ν", "hf : AEStronglyMeasurable f μ"], "goal": "∃ t, IsSeparable t ∧ ∀ᵐ (x : β) ∂ν, g x ∈ t"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "AddCommMonoid α", "AddCommMonoid β", "A₁ A₂ : Set α", "B₁ B₂ : Set β", "f : α → β", "n : ℕ", "hA : A₁ ⊆ A₂", "hf : IsAddFreimanIso n A₂ B₂ f", "hf' : Set.BijOn f A₁ B₁"], "goal": "IsAddFreimanIso n A₁ B₁ f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "y : β", "o : Option α"], "goal": "y ∈ Option.map f o ↔ ∃ x, x ∈ o ∧ f x = y"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "P : Type u_6", "Q : Type u_7", "AddCommMonoid P", "AddCommMonoid Q", "Module R P", "Module R Q", "p : Submodule R P", "q : Submodule R Q"], "goal": "LinearMap.range (TensorProduct.mapIncl p q) = Submodule.span R (Set.image2 (fun x x_1 => x ⊗ₜ[R] x_1) ↑p ↑q)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "ι : Type u_4", "f : X → Y", "g : Y → Z", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z"], "goal": "IsClosedMap f ↔ ∀ (s : Set X) (y : Y), MapClusterPt y (𝓟 s) f → ∃ x, f x = y ∧ ClusterPt x (𝓟 s)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : LinearOrderedSemifield α", "l : Filter β", "f : β → α", "r c : α", "n : ℕ", "hr : 0 < r", "hf : Tendsto f l atTop"], "goal": "Tendsto (fun x => f x / r) l atTop"}}
+{"state": {"context": ["b : ℕ", "b_ge_two : 2 ≤ b", "p : ℕ", "p_prime : Prime p", "p_gt_two : 2 < p", "A : ℕ := (b ^ p - 1) / (b - 1)", "B : ℕ := (b ^ p + 1) / (b + 1)", "AB_dvd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1", "this : 2 ≤ 2 * p - 2"], "goal": "p < b ^ (2 * p - 2)"}}
+{"state": {"context": ["α : Type u_1", "Neg α", "x : α"], "goal": "AddOpposite.op (-x) = -AddOpposite.op x"}}
+{"state": {"context": ["K : Type u", "DivisionRing K", "s : Set K", "t : Subfield K"], "goal": "Subfield.closure s ≤ t ↔ s ⊆ ↑t"}}
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+{"state": {"context": ["R : Type u_1", "inst✝⁶ : Ring R", "E : Type u_2", "inst���⁵ : AddCommGroup E", "inst✝⁴ : Module R E", "F : Type u_3", "inst✝³ : AddCommGroup F", "inst✝² : Module R F", "G : Type u_4", "inst✝¹ : AddCommGroup G", "inst✝ : Module R G", "f : E →ₗ.[R] F", "hf : LinearMap.ker f.toFun = ⊥", "x : F × E", "hv : x ∈ Submodule.map (LinearEquiv.prodComm R E F) f.graph", "hv' : x.1 = 0"], "goal": "x.2 = 0"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "p : ι → Prop", "s : ι → Set α", "𝓕 : Filter ι", "a : α"], "goal": "bliminf s cofinite p = {x | {n | p n ∧ x ∉ s n}.Finite}"}}
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+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "S : Type u_4", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Ring B", "inst✝² : CommRing S", "inst✝¹ : Algebra R A", "inst✝ : Algebra R B", "f✝ f : R →+* S", "p : R[X]", "x : S"], "goal": "(normalizeScaleRoots p).degree = p.degree"}}
+{"state": {"context": ["n : ℕ", "f : Mathlib.Vector ℕ n →. ℕ", "hf : Nat.Partrec' f"], "goal": "Nat.Partrec' fun v => f v.tail"}}
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+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y", "s : Set X"], "goal": "⇑h '' sᶜ = (⇑h '' s)ᶜ"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "inst✝² : Group G", "inst✝¹ : MulAction G α", "inst✝ : DecidableEq α", "S : Set (Equiv.Perm α)", "hS : ∀ f ∈ S, f.IsSwap", "f : Equiv.Perm α", "hf : ∀ (x : α), f x ∈ orbit (↥(closure S)) x", "supp : Set α := (fixedBy α f)ᶜ", "fin : supp.Finite", "supp_eq : supp = (fixedBy α f)ᶜ"], "goal": "f ∈ closure S"}}
+{"state": {"context": ["R : Type uR", "A₁ : Type uA₁", "A₂ : Type uA₂", "CommSemiring R", "CommSemiring A₁", "CommSemiring A₂", "Algebra R A₁", "Algebra R A₂", "e : A₁ ≃ₐ[R] A₂", "α : Type u_1", "Zero α", "ι : Type u_2", "f : ι →₀ α", "g : ι → α → A₁"], "goal": "e (f.prod g) = f.prod fun i a => e (g i a)"}}
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+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "Disjoint (Set.diagonal α) s.offDiag"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "i j : ℕ", "a : Fin i → M", "b : Fin j → M"], "goal": "(List.ofFn fun i_1 => (TensorAlgebra.ι R) (Fin.append a b i_1)).prod = (List.ofFn fun i => (TensorAlgebra.ι R) (a i)).prod * (List.ofFn fun i => (TensorAlgebra.ι R) (b i)).prod"}}
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+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "inst✝ : CommMonoid β", "r : Finset (α × γ)", "s : Finset γ", "t : γ → Finset α", "h : ∀ (p : α × γ), p ∈ r ↔ p.2 ∈ s ∧ p.1 ∈ t p.2", "f : α × γ → β"], "goal": "∏ p ∈ r, f p = ∏ x ∈ s.sigma t, f (x.snd, x.fst)"}}
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+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_3", "i : ι"], "goal": "Set.range (Sigma.mk i) = Sigma.fst ⁻¹' {i}"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "A : Type u_5", "CommRing R", "CommRing A", "Algebra R A", "e : ι ≃ ι'", "f : ι' → A", "g : ι → A", "h : f ∘ ⇑e = g"], "goal": "AlgebraicIndependent R g ↔ AlgebraicIndependent R f"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "v : K", "n : ℕ"], "goal": "(of v).convs' (n + 1) = ↑⌊v⌋ + 1 / (of (fract v)⁻¹).convs' n"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : CompleteLattice α", "u : ℕ → α", "n : ℕ"], "goal": "⨆ k, ⨆ (_ : k < n + 1), u k = u 0 ⊔ ⨆ k, ⨆ (_ : k < n), u (k + 1)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "r✝ r : α → α → Prop", "inst✝¹ : IsRefl α r", "inst✝ : IsTrans α r", "s : Set α", "h : s.PartiallyWellOrderedOn r"], "goal": "{l | ∀ x ∈ l, x ∈ s}.PartiallyWellOrderedOn (List.SublistForall₂ r)"}}
+{"state": {"context": ["α : Type u", "l : List α"], "goal": "l.split.1.length ≤ l.length"}}
+{"state": {"context": ["m : ℤ", "n : ℕ", "hn : Even n", "hm : m ∉ Ico 0 ↑n", "a : ℕ", "ha : a ∈ Finset.range n", "h : m - ↑a = 0"], "goal": "m ∈ Ico 0 ↑n"}}
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+{"state": {"context": ["n p k : ℕ", "hp : Prime p", "hk : k ≠ 0"], "goal": "(p ^ k).minFac = p"}}
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+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝⁵ : CommRing P", "inst✝⁴ : Algebra R P", "loc : IsLocalization S P", "R₁ : Type u_3", "inst✝³ : CommRing R₁", "K : Type u_4", "inst✝² : Field K", "inst✝¹ : Algebra R₁ K", "inst✝ : IsFractionRing R₁ K", "I J : Ideal R₁", "x y : R₁", "hy : y ∈ R₁⁰", "this : spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩) * spanSingleton R₁⁰ ((algebraMap R₁ K) y) = 1", "y' : (FractionalIdeal R₁⁰ K)ˣ := Units.mkOfMulEqOne (spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩)) (spanSingleton R₁⁰ ((algebraMap R₁ K) y)) this", "coe_y' : ↑y' = spanSingleton R₁⁰ (mk' K 1 ⟨y, hy⟩)"], "goal": "spanSingleton R₁⁰ (mk' K x ⟨y, hy⟩) * ↑I = ↑J ↔ Ideal.span {x} * I = Ideal.span {y} * J"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f✝ g✝ f g : Perm α", "x : α"], "goal": "x ∈ (f * g).support → x ∈ f.support ∪ g.support"}}
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+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "n : ℕ", "f : 𝕜 → F", "a : 𝕜"], "goal": "iteratedDeriv n (fun x => f (-x)) a = (-1) ^ n • iteratedDeriv n f (-a)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "MeasurableSpace α", "NormedAddCommGroup E", "NormedAddCommGroup F", "f : α → E", "g : α → F", "μ : MeasureTheory.Measure α", "TopologicalSpace α", "SecondCountableTopology α", "(Filter.cocompact α).IsMeasurablyGenerated", "hf : MeasureTheory.LocallyIntegrable f μ", "ho : f =O[Filter.cocompact α] g", "hg : MeasureTheory.IntegrableAtFilter g (Filter.cocompact α) μ"], "goal": "MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁵ : Ring K", "inst✝⁴ : StrongRankCondition K", "inst✝³ : AddCommGroup V", "inst✝² : Module K V", "inst✝¹ : Module.Free K V", "inst✝ : Module.Finite K V"], "goal": "finrank K V ≤ 1 ↔ ∃ v, ∀ (w : V), ∃ c, c • v = w"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "PartialOrder G", "PartialOrder P", "SMul G P", "IsOrderedCancelSMul G P", "s : Set G", "t : Set P", "hs : s.IsPWO", "ht : t.IsPWO"], "goal": "{a | (Finset.SMulAntidiagonal hs ht a).Nonempty} ⊆ s • t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : PseudoEMetricSpace α", "s : Set ℝ", "h : Bornology.IsBounded s"], "goal": "0 ≤ sSup s - sInf s"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDomain R", "IsPrincipalIdealRing R", "x y : R", "nonzero : ¬(x = 0 ∧ y = 0)", "H : ∀ (z : R), Prime z → z ∣ x → ¬z ∣ y"], "goal": "IsCoprime x y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : TopologicalSpace α", "inst✝ : TopologicalSpace β", "f : α → β", "s✝ : Set β", "ι : Type u_3", "U : ι → Opens β", "hU : iSup U = ⊤", "H : ∀ (i : ι), IsClosedMap ((U i).carrier.restrictPreimage f)", "s : Set α", "hs : IsClosed s"], "goal": "IsClosed (f '' s)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : 𝕜 → F", "f' : F", "x : 𝕜", "L : Filter 𝕜", "hf : HasDerivAtFilter f f' x L", "c : F"], "goal": "HasDerivAtFilter (fun x => f x - c) f' x L"}}
+{"state": {"context": ["s n : Nat"], "goal": "List.range' s (n + 1) = List.range' s n ++ [s + n]"}}
+{"state": {"context": ["D : TopCat.GlueData", "i : D.J", "U : TopologicalSpace.Opens ↑(D.U i)"], "goal": "IsOpen (⇑(D.ι i) '' ↑U)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v"], "goal": "#((a : α) × (β a → WType β)) = sum fun a => #(WType β) ^ lift.{u, v} #(β a)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "h : HasFPowerSeriesAt 0 p x", "n : ℕ"], "goal": "∀ (y : E), ((p n) fun x => y) = 0"}}
+{"state": {"context": ["R : Type u_2", "CommRing R", "r : R", "M : Type u_1", "M' : Type u_3", "AddCommGroup M", "Module R M", "AddCommGroup M'", "Module R M'", "f : M →ₗ[R] M'"], "goal": "(QuotSMulTop.map r) f ∘ₗ (r • ⊤).mkQ = (r • ⊤).mkQ ∘ₗ f"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Add α", "s t : Finset α"], "goal": "(s + t).Nonempty → t.Nonempty"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n : ℕ", "a a' : α", "i : Fin n", "v : Vector α n"], "goal": "v.toList.get (Fin.cast ⋯ i) ∈ v.toList"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "n : ℤ", "z₁ z₂ : CochainComplex.HomComplex.Cochain F G n", "p q : ℤ", "hpq : p + n = q"], "goal": "(z₁ - z₂).v p q hpq = z₁.v p q hpq - z₂.v p q hpq"}}
+{"state": {"context": ["R₁ : Type u_1", "Semiring R₁", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "M₃ : Type u_7", "TopologicalSpace M₃", "AddCommMonoid M₃", "Module R₁ M₁", "Module R₁ M₂", "Module R₁ M₃", "ContinuousAdd M₃", "f₁ : M₁ →L[R₁] M₃", "f₂ : M₂ →L[R₁] M₃", "x : M₁ × M₂"], "goal": "(f₁.coprod f₂) x = f₁ x.1 + f₂ x.2"}}
+{"state": {"context": ["α : Type u_2", "SemilatticeInf α", "s : Set α", "hs : InfClosed s"], "goal": "DirectedOn (fun x x_1 => x ≥ x_1) s"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s t : Set α", "hs : MeasurableSet s", "ht : MeasurableSet t"], "goal": "μ (symmDiff s t) = μ (s \\ t) + μ (t \\ s)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "n : ℕ", "F G : CategoryTheory.ComposableArrows C (n + 1)", "f g : F ⟶ G", "h₀ : CategoryTheory.ComposableArrows.app' f 0 ⋯ = CategoryTheory.ComposableArrows.app' g 0 ⋯", "h₁ : CategoryTheory.ComposableArrows.δ₀Functor.map f = CategoryTheory.ComposableArrows.δ₀Functor.map g"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n n' m✝ : ℕ", "s : Sym α n", "a✝ b : α", "inst✝ : DecidableEq α", "a : α", "m : (i : Fin (n + 1)) × Sym α (n - ↑i)", "h : a ∉ m.snd"], "goal": "∀ (a_1 : α), a_1 ∉ filter (fun x => a ≠ x) ↑(replicate (↑m.fst) a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝ : SigmaFinite μ", "s : Set α", "hμs : μ s ≠ ⊤", "t : Set α := toMeasurable μ s", "ht : t = toMeasurable μ s"], "goal": "∫ (x : α) in s, (μ.rnDeriv ν x).toReal ∂ν ≤ (μ s).toReal"}}
+{"state": {"context": ["w x✝ y z✝ : ℝ", "x : ℝ≥0", "z : ℝ", "hx : x ≤ 1", "h_one_le : 1 ≤ z", "h : 0 < x"], "goal": "x ^ z ≤ x ^ 1"}}
+{"state": {"context": ["a b : ℤ", "h : a < b", "P : Prop", "h' : ∀ (n : ℕ), a + ↑n.succ = b → P"], "goal": "P"}}
+{"state": {"context": ["R : Type u"], "goal": "Function.Injective (@NonAssocRing.toNonUnitalNonAssocRing R)"}}
+{"state": {"context": ["a : ℝ"], "goal": "frontier {z | z.re < a} = {z | z.re = a}"}}
+{"state": {"context": ["X : Type u_3", "Y : Type u_4", "TopologicalSpace X", "f : X → Y", "h : ∀ (x : X), (↑f).IsConstant"], "goal": "IsLocallyConstant f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "f : α → δ", "g : δ → β", "hg : AEStronglyMeasurable g (Measure.map f μ)", "hf : AEMeasurable f μ"], "goal": "Memℒp g 1 (Measure.map f μ) ↔ Memℒp (g ∘ f) 1 μ"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝¹ : MeasurableSpace α", "inst✝ : StandardBorelSpace α", "s : Set α", "hs : MeasurableSet s", "this : UpgradedStandardBorel α := upgradeStandardBorel α", "t : TopologicalSpace α", "hle : t ≤ UpgradedStandardBorel.toTopologicalSpace", "ht : PolishSpace α", "s_clopen : IsClosed s ∧ IsOpen s"], "goal": "PolishSpace α"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "setOf s = s"}}
+{"state": {"context": ["Ω : Type u_1", "Ω' : Type u_2", "MeasurableSpace Ω", "MeasurableSpace Ω'", "ν : MeasureTheory.ProbabilityMeasure Ω", "f : Ω → Ω'", "hf : AEMeasurable f ↑ν"], "goal": "↑(ν.map hf) = MeasureTheory.Measure.map f ↑ν"}}
+{"state": {"context": ["α : Type u", "Group α", "LT α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b : α"], "goal": "a / b < 1 ↔ a < b"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "l : List α", "h : List.Sorted r l"], "goal": "List.Sorted r l.tail"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "C₃ : Type u_3", "C₄ : Type u_4", "C₁₂ : Type u_5", "C₂₃ : Type u_6", "inst✝⁶ : Category.{?u.66927, u_1} C₁", "inst✝⁵ : Category.{?u.66931, u_2} C₂", "inst✝⁴ : Category.{?u.66935, u_3} C₃", "inst✝³ : Category.{?u.66939, u_4} C₄", "inst✝² : Category.{?u.66943, u_5} C₁₂", "inst✝¹ : Category.{?u.66947, u_6} C₂₃", "F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄", "I₁ : Type u_7", "I₂ : Type u_8", "I₃ : Type u_9", "J : Type u_10", "p : I₁ × I₂ × I₃ → J", "X₁ : GradedObject I₁ C₁", "inst✝ : ∀ (X₂ : GradedObject I₂ C₂) (X₃ : GradedObject I₃ C₃), ((((mapTrifunctor F I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap p", "X₂ Y₂ Z₂ : GradedObject I₂ C₂", "φ : X₂ ⟶ Y₂", "ψ : Y₂ ⟶ Z₂", "X₃ : GradedObject I₃ C₃", "j : J", "i₁ : I₁", "i₂ : I₂", "i₃ : I₃", "h✝ : p (i₁, i₂, i₃) = j"], "goal": "ιMapTrifunctorMapObj F p X₁ X₂ X₃ i₁ i₂ i₃ j h✝ ≫ { app := fun X₃ => mapTrifunctorMapMap F p (𝟙 X₁) (φ ≫ ψ) (𝟙 X₃), naturality := ⋯ }.app X₃ j = ιMapTrifunctorMapObj F p X₁ X₂ X₃ i₁ i₂ i₃ j h✝ ≫ ({ app := fun X₃ => mapTrifunctorMapMap F p (𝟙 X₁) φ (𝟙 X₃), naturality := ⋯ } ≫ { app := fun X₃ => mapTrifunctorMapMap F p (𝟙 X₁) ψ (𝟙 X₃), naturality := ⋯ }).app X₃ j"}}
+{"state": {"context": ["G : Type u_1", "Group G", "X : Type u_2", "MulAction G X", "B : Set X", "H : Subgroup G", "hB : MulAction.IsBlock (↥H) B", "g : G"], "goal": "MulAction.IsBlock (↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)) (g • B)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LE α", "S : Set (UpperSet α)", "s t : UpperSet α", "a : α", "f : (i : ι) → κ i → UpperSet α"], "goal": "↑(⨅ i, ⨅ j, f i j) = ⋃ i, ⋃ j, ↑(f i j)"}}
+{"state": {"context": ["α : Type u_5", "β : Type u_6", "TopologicalSpace α", "CommMonoid α", "T2Space α", "ContinuousMul α", "DecidableEq β", "f : β → α", "a a' : α", "hf : HasProd f a", "b : β", "x : α", "hf' : HasProd (Function.update f b x) a'"], "goal": "a * x = a' * f b"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n : ℕ", "ℱ : MeasureTheory.Filtration ℕ m0", "hf : MeasureTheory.Adapted ℱ f"], "goal": "MeasureTheory.IsStoppingTime ℱ (MeasureTheory.upperCrossingTime a b f N n)"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "R : Type u_3", "S : Type u_4", "inst✝² inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "φ ψ : MvPolynomial σ R", "m n : ℕ", "hφ : φ.IsHomogeneous n", "d : σ →₀ ℕ", "hd : coeff d φ ≠ 0"], "goal": "(d.sum fun x e => e) ≤ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p : Set γ", "f f₁ f₂ f₃ : α → β", "g g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a : α", "b : β", "h : InjOn f s", "h₁ : s₁ ⊆ s", "h₂ : s₂ ⊆ s"], "goal": "f '' s₁ ⊂ f '' s₂ ↔ s₁ ⊂ s₂"}}
+{"state": {"context": ["α : Type u_1", "StrictOrderedSemiring α", "t : ↑(Set.Ioc 0 1)"], "goal": "t ≤ 1"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "x y : αᵃᵒᵖ"], "goal": "edist (AddOpposite.unop x) (AddOpposite.unop y) = edist x y"}}
+{"state": {"context": ["E : Type u_4", "F : Type u_5", "AddGroup E", "AddGroup F", "p : AddGroupSeminorm E", "f : F →+ E"], "goal": "⇑(p.comp f) = ⇑p ∘ ⇑f"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝²³ : NormedAddCommGroup E", "inst✝²² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝²¹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝²⁰ : TopologicalSpace M", "inst✝¹⁹ : ChartedSpace H M", "Is : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹⁸ : NormedAddCommGroup E'", "inst✝¹⁷ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁶ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹⁵ : TopologicalSpace M'", "inst✝¹⁴ : ChartedSpace H' M'", "I's : SmoothManifoldWithCorners I' M'", "F : Type u_8", "inst✝¹³ : NormedAddCommGroup F", "inst✝¹² : NormedSpace 𝕜 F", "G : Type u_9", "inst✝¹¹ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N✝ : Type u_10", "inst✝¹⁰ : TopologicalSpace N✝", "inst✝⁹ : ChartedSpace G N✝", "Js : SmoothManifoldWithCorners J N✝", "F' : Type u_11", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace 𝕜 F'", "G' : Type u_12", "inst✝⁶ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_13", "inst✝⁵ : TopologicalSpace N'", "inst✝⁴ : ChartedSpace G' N'", "J's : SmoothManifoldWithCorners J' N'", "F₁ : Type u_14", "inst✝³ : NormedAddCommGroup F₁", "inst✝² : NormedSpace 𝕜 F₁", "F₂ : Type u_15", "inst✝¹ : NormedAddCommGroup F₂", "inst✝ : NormedSpace 𝕜 F₂", "f f₁ : M → M'", "s s₁ t : Set M", "x✝ : M", "m n : ℕ∞", "x : M", "v : TangentSpace I x", "N : ↑I.symm ⁻¹' (chartAt H x).target ∈ 𝓝 (↑I (↑(chartAt H x) x))", "A : MDifferentiableAt I I.tangent (fun x => { proj := x, snd := 0 }) x", "B : (fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))) v = (v, 0)"], "goal": "fderivWithin 𝕜 (fun x_1 => (↑I (↑(ChartedSpace.chartAt x) (↑(ChartedSpace.chartAt x).symm (↑I.symm x_1))), 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x)) = fderivWithin 𝕜 (fun x' => (x', 0)) (range ↑I ∩ ↑I.symm ⁻¹' (chartAt H x).target) (↑I (↑(chartAt H x) x))"}}
+{"state": {"context": ["α : Type u_1"], "goal": "Monotone Cycle.Chain"}}
+{"state": {"context": ["R : Type u", "inst✝³ : CommRing R", "n G : Type v", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "α β : Type v", "inst✝ : DecidableEq α", "M : Matrix n n R", "k : ℕ", "h : Fintype.card n - 1 ≤ k"], "goal": "↑(Fintype.card n - 1) ≤ ↑k"}}
+{"state": {"context": ["R : Type u₁", "L : Type u₂", "L₂ : Type u₃", "M✝ : Type u₄", "inst✝⁸ : CommRing R", "inst✝⁷ : LieRing L", "inst✝⁶ : LieAlgebra R L", "inst✝⁵ : LieRing L₂", "inst✝⁴ : LieAlgebra R L₂", "inst✝³ : AddCommGroup M✝", "inst✝² : Module R M✝", "inst✝¹ : LieRingModule L M✝", "inst✝ : LieModule R L M✝", "K : LieSubalgebra R L", "hK₁ : IsEngelian R ↥K", "hK₂ : K < K.normalizer", "x : L", "hx₁ : x ∈ K.normalizer", "hx₂ : x ∉ K", "K' : LieSubalgebra R L := let __src := Submodule.span R {x} ⊔ K.toSubmodule; { toSubmodule := __src, lie_mem' := ⋯ }", "hxK' : x ∈ K'", "hKK' : K ≤ K'", "hK' : K' ≤ K.normalizer", "M : Type u₄", "_i1 : AddCommGroup M", "_i2 : Module R M", "_i3 : LieRingModule (↥K') M", "_i4 : LieModule R (↥K') M", "h : ∀ (x : ↥K'), _root_.IsNilpotent ((toEnd R (↥K') M) x)", "I : LieIdeal R ↥K'", "hI₁ : lieIdealSubalgebra R (↥K') I = LieSubalgebra.ofLe hKK'", "hI₂ : Submodule.span R {⟨x, hxK'⟩} ⊔ ↑I = ⊤"], "goal": "LieModule.IsNilpotent R (↥K') M"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace ℝ G", "x y✝ : E", "s t✝ : Set E", "t : Set F", "y : F", "hs : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 s x)) ∧ x ∈ closure s", "ht : Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 t y)) ∧ y ∈ closure t", "this : ∀ ⦃x_1 : E × F⦄, x_1 ∈ (Submodule.span 𝕜 (tangentConeAt 𝕜 s x)).prod (Submodule.span 𝕜 (tangentConeAt 𝕜 t y)) → x_1 ∈ Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y))"], "goal": "Dense ↑(Submodule.span 𝕜 (tangentConeAt 𝕜 (s ×ˢ t) (x, y)))"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "TopologicalSpace β", "Zero β", "f : α →C₀ β", "f' : α → β", "h : f' = ⇑f"], "goal": "⇑(f.copy f' h) = f'"}}
+{"state": {"context": ["n : ℕ"], "goal": "(n + 1)‼ ≤ (n + 1)‼ * n‼"}}
+{"state": {"context": ["V : Type u_1", "W : Type u_2", "V₁ : Type u_3", "V₂ : Type u_4", "V₃ : Type u_5", "inst✝⁷ : SeminormedAddCommGroup V", "inst✝⁶ : SeminormedAddCommGroup W", "inst✝⁵ : SeminormedAddCommGroup V₁", "inst✝⁴ : SeminormedAddCommGroup V₂", "inst✝³ : SeminormedAddCommGroup V₃", "f : NormedAddGroupHom V W", "W₁ : Type u_6", "W₂ : Type u_7", "W₃ : Type u_8", "inst✝² : SeminormedAddCommGroup W₁", "inst✝¹ : SeminormedAddCommGroup W₂", "inst✝ : SeminormedAddCommGroup W₃", "g : NormedAddGroupHom V W", "f₁ g₁ : NormedAddGroupHom V₁ W₁", "f₂ g₂ : NormedAddGroupHom V₂ W₂", "f₃ g₃ : NormedAddGroupHom V₃ W₃", "φ : NormedAddGroupHom V₁ V", "h : f.comp φ = g.comp φ", "C : ℝ", "_C_pos : 0 < C", "hC : ∀ (x : V₁), ‖φ x‖ ≤ C * ‖x‖"], "goal": "∃ C, ∀ (v : V₁), ‖(fun v => ⟨φ v, ⋯⟩) v‖ ≤ C * ‖v‖"}}
+{"state": {"context": ["x y z : SetTheory.PGame"], "goal": "⟦x * (y + z)⟧ = ⟦x * y⟧ + ⟦x * z⟧"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "M₂ : Type w₁", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "N N' : LieSubmodule R L M", "I J : LieIdeal R L", "N₂ : LieSubmodule R L M₂", "h₁ : I ≤ J", "h₂ : N ≤ N'", "m : M", "h : ∀ (N_1 : LieSubmodule R L M), {m | ∃ x n, ⁅↑x, ↑n⁆ = m} ⊆ ↑N_1 → m ∈ N_1"], "goal": "m ∈ ⁅J, N'⁆"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "SemilatticeSup α", "u : Finset α", "s t : Set α"], "goal": "↑u ⊆ s ⊻ t → ∃ s' t', ���s' ⊆ s ∧ ↑t' ⊆ t ∧ u ⊆ s' ⊻ t'"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X", "hs : IsLindelof s", "U : (x : X) → x ∈ s → Set X", "hU : ∀ (x : X) (hx : x ∈ s), U x hx ∈ 𝓝 x"], "goal": "∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, U ↑x ⋯"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddMonoid M", "AddMonoid N"], "goal": "AddMonoid.Coprod.fst.prod AddMonoid.Coprod.snd = AddMonoid.Coprod.toSum"}}
+{"state": {"context": ["p : ℕ", "hp : Nat.Prime p", "s : ℂ", "hs : 1 < s.re", "H : 1 = ↑p ^ (-s)", "this : ‖↑p ^ (-s)‖ ≤ 1 / 2"], "goal": "False"}}
+{"state": {"context": ["n : ℕ"], "goal": "1 ≤ n ↔ n ≠ 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z✝ : Type u_3", "Z : Type u_4", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "f : X ≃ Y", "hf : Inducing ⇑f"], "goal": "Continuous (⇑f ∘ f.invFun)"}}
+{"state": {"context": ["M : Type u_3", "A : Type u_4", "AddCommMonoid M", "s : Set M", "hs : IsAddSubmonoid s", "f : A → M", "t : Finset A"], "goal": "(∀ b ∈ t, f b ∈ s) → ∑ b ∈ t, f b ∈ s"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "ι : Type u_3", "α : Type u_4", "inst✝ : SeminormedAddCommGroup α", "f : ℕ → α", "r : ℝ", "hr : 1 < r", "hf : ∀ (a : ℕ), ∃ b ≥ a, ‖f b‖ ≠ 0", "N₀ : ℕ", "hN₀ : ∀ b ≥ N₀, r * ‖f b‖ ≤ ‖f (b + 1)‖", "N : ℕ", "hNN₀ : N₀ ≤ N", "hN : ‖f N‖ ≠ 0", "h' : Tendsto (fun n => f (n + N)) atTop (𝓝 0)"], "goal": "0 < ((fun a => ‖a‖) ∘ fun n => f (n + N)) 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "ι : Type u_4", "π : ι → Type u_5", "inst✝⁶ : OrderedSemiring 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : AddCommGroup F", "inst✝³ : (i : ι) → AddCommGroup (π i)", "inst✝² : Module 𝕜 E", "inst✝¹ : Module 𝕜 F", "inst✝ : (i : ι) → Module 𝕜 (π i)", "A B : Set E", "x✝ : E", "s : (i : ι) → Set (π i)", "x : (i : ι) → π i", "hx : ∀ (i : ι), x i ∈ s i", "h : ∀ (x₁ : (i : ι) → π i), (∀ (i : ι), x₁ i ∈ s i) → ∀ (x₂ : (i : ι) → π i), (∀ (i : ι), x₂ i ∈ s i) → x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x", "i : ι"], "goal": "∀ x₁ ∈ s i, ∀ x₂ ∈ s i, x i ∈ openSegment 𝕜 x₁ x₂ → x₁ = x i ∧ x₂ = x i"}}
+{"state": {"context": ["n : Nat"], "goal": "n + 1 ≠ n"}}
+{"state": {"context": ["S : Type u_1", "R : Type u_3", "M : Type u_4", "N : Type u_5", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "Q : QuadraticMap R M N", "CommSemiring S", "Algebra S R", "Module S M", "IsScalarTower S R M", "Module S N", "IsScalarTower S R N", "a : S", "x y : M"], "goal": "QuadraticMap.polar (⇑Q) x (a • y) = a • QuadraticMap.polar (⇑Q) x y"}}
+{"state": {"context": ["α✝ : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f f₁ f₂ : Filter α✝", "g g₁ g₂ : Filter β", "m : α✝ → β", "m' : β → γ", "s✝ : Set α✝", "t✝ : Set β", "α : Type u_2", "s : Set α", "t : Set ↑s"], "goal": "𝓟 t = comap Subtype.val (𝓟 (Subtype.val '' t))"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "κ : ι → Sort u_5", "CompleteLattice α", "f : (i : ι) → κ i → α"], "goal": "UpperSet.Ici (⨆ i, ⨆ j, f i j) = ⨆ i, ⨆ j, UpperSet.Ici (f i j)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝⁷ : Preorder α", "inst✝⁶ : Preorder β", "inst✝⁵ : Preorder γ", "inst✝⁴ : Preorder δ", "inst✝³ : MulZeroOneClass α", "inst✝² : MulZeroOneClass β", "inst✝¹ : MulZeroOneClass γ", "inst✝ : MulZeroOneClass δ", "f✝ g✝ f g : α →*₀o β", "h : f.toMonoidWithZeroHom = g.toMonoidWithZeroHom"], "goal": "∀ (a : α), f a = g a"}}
+{"state": {"context": ["num : Int", "den g : Nat", "den_nz : den / g ≠ 0", "reduced : (num.div ↑g).natAbs.Coprime (den / g)", "hn : ↑g ∣ num", "hd : g ∣ den"], "goal": "maybeNormalize num den g den_nz reduced = normalize num den ⋯"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "V : Type u_2", "W : Type u_3", "NormedAddCommGroup V", "NormedSpace ℝ V", "NormedAddCommGroup W", "NormedSpace ℝ W", "SecondCountableTopologyEither V (W →L[ℝ] ℝ)", "MeasurableSpace V", "BorelSpace V", "L : V →L[ℝ] W →L[ℝ] ℝ", "f : V → E", "μ : MeasureTheory.Measure V", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.AEStronglyMeasurable (fun v => VectorFourier.fourierSMulRight L f v) μ"}}
+{"state": {"context": ["α : Type u_1", "AddZeroClass α", "LT α", "ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a b : α", "h : a + b < a"], "goal": "b < 0"}}
+{"state": {"context": ["o₁ o₂ : Ordering"], "goal": "(o₁.then o₂).swap = o₁.swap.then o₂.swap"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Category.{?u.65549, u_2} D", "W : MorphismProperty C", "inst✝ : W.HasLeftCalculusOfFractions", "X Y Y' : C", "f : X ⟶ Y'", "s : Y ⟶ Y'", "hs : W s"], "goal": "(Q W).map f ≫ Qinv s hs = homMk (mk f s hs)"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f✝ f₀ f₁ g : 𝕜 → F", "f'✝ f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "f : 𝕜 → F", "f' : F", "x₀ : 𝕜", "hf : HasDerivAt f f' x₀", "C : ℝ≥0", "hlip : LipschitzWith C f"], "goal": "‖f'‖ ≤ ↑C"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : DistribLattice α", "x y z a b c : α"], "goal": "(a ⊔ a ⊓ b) ⊓ (a ⊓ b ⊔ c) = (a ⊓ b ⊔ a) ⊓ (a ⊓ b ⊔ c)"}}
+{"state": {"context": ["α : Type u_1", "G₀ : Type u_3", "Zero G₀", "Inv G₀", "TopologicalSpace G₀", "HasContinuousInv₀ G₀", "f : α → G₀", "TopologicalSpace α", "hf : Continuous f", "h0 : ∀ (x : α), f x ≠ 0"], "goal": "Continuous fun x => (f x)⁻¹"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V"], "goal": "o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ o.oangle x (-y) = 0"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "AddGroup G", "AddAction G α", "MeasurableSpace α", "s : Set α", "μ : MeasureTheory.Measure α", "MeasurableSpace G", "MeasurableVAdd G α", "MeasureTheory.VAddInvariantMeasure G α μ", "Countable G", "h : MeasureTheory.IsAddFundamentalDomain G s μ", "t : Set α", "ht : ∀ (g : G), g +ᵥ t = t", "hts : μ (t ∩ s) = 0"], "goal": "μ t = 0"}}
+{"state": {"context": ["R : Type r", "CommRing R", "W : WeierstrassCurve R"], "goal": "W.ΨSq 2 = W.Ψ₂Sq"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "N : Type u_2", "CommMonoid N", "f : S.LocalizationMap N", "a₁ b₁ : M", "a₂ b₂ : ↥S"], "goal": "f.mk' a₁ a₂ = f.mk' b₁ b₂ ↔ (Localization.r S) (a₁, a₂) (b₁, b₂)"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝ : MeasurableSpace Ω", "μ : FiniteMeasure Ω"], "goal": "μ.mass ≠ 0 ↔ μ ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : Fintype α", "s t : Finset α"], "goal": "Codisjoint s t ↔ ∀ ⦃a : α⦄, a ∉ s → a ∈ t"}}
+{"state": {"context": ["L : Language", "T : L.Theory", "α : Type w", "n : ℕ", "φ ψ φ' ψ' : L.BoundedFormula α n", "h : T.SemanticallyEquivalent φ ψ", "h' : T.SemanticallyEquivalent φ' ψ'"], "goal": "T.SemanticallyEquivalent (φ ⟹ φ') (ψ ⟹ ψ')"}}
+{"state": {"context": ["𝕜 : Type u_1", "F : Type u_2", "RCLike 𝕜", "NormedAddCommGroup F", "InnerProductSpace 𝕜 F", "CompleteSpace F", "f : F → 𝕜", "f' x : F", "s t : Set F", "y : F", "h : s =ᶠ[𝓝[{y}ᶜ] x] t"], "goal": "HasGradientWithinAt f f' s x ↔ HasGradientWithinAt f f' t x"}}
+{"state": {"context": ["x : ℝ", "n : ℕ"], "goal": "x ^ ↑n = x ^ n"}}
+{"state": {"context": ["α : Type u", "Semiring α", "I : Ideal α"], "goal": "¬I.IsPrime ↔ I = ⊤ ∨ ∃ x, ∃ (_ : x ∉ I), ∃ y, ∃ (_ : y ∉ I), x * y ∈ I"}}
+{"state": {"context": ["R : Type u", "S : Type v", "NonAssocSemiring R", "NonAssocSemiring S", "g : S → R", "f : R →+* S", "h : Function.LeftInverse g ⇑f", "x : R"], "goal": "↑((RingEquiv.ofLeftInverseS h) x) = f x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "M : Type u_3", "inst✝² : AddCommMonoid M", "inst✝¹ : TopologicalSpace M", "inst✝ : T2Space M", "v : VectorMeasure α M", "f : ℕ → Set α", "A B : Set α", "hA : MeasurableSet A", "hB : MeasurableSet B", "h : A ⊆ B"], "goal": "↑v A + ↑v (B \\ A) = ↑v B"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t : Set α", "inst✝ : Countable ι", "s : ι → Set α"], "goal": "μ.restrict (⋃ i, s i) = ν.restrict (⋃ i, s i) ↔ ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)"}}
+{"state": {"context": ["G : Type u_3", "Group G", "b : G"], "goal": "Function.Injective fun a => a / b"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "inst✝ : LinearOrderedField α"], "goal": "(∀ (x : α), ∃ n, x < ↑n) → Archimedean α"}}
+{"state": {"context": ["F : Type u_1", "inst✝ : NormedAddCommGroup F", "g : ℝ → F", "R : ℝ", "hR : R ≠ 0"], "goal": "Integrable (fun x => g (R * x)) volume ↔ Integrable g volume"}}
+{"state": {"context": ["α : Type u_1"], "goal": "#[].data = []"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "a b : G₀", "h : Commute a b"], "goal": "Commute a b⁻¹"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "CommRing R", "CommRing S", "CommRing T", "Algebra R S", "Algebra R T", "Algebra S T", "IsScalarTower R S T", "Algebra.EssFiniteType R S"], "goal": "Algebra.EssFiniteType R T ↔ Algebra.EssFiniteType S T"}}
+{"state": {"context": ["α : Type u_1", "Group α", "s : Subgroup α", "x y : α"], "goal": "x • ↑s = y • ↑s ↔ x⁻¹ * y ∈ s"}}
+{"state": {"context": ["f g : MulRingNorm ℚ", "hf_nontriv : f ≠ 1", "bdd : ∀ (n : ℕ), f ↑n ≤ 1", "p : ℕ", "hp0 : 0 < f ↑p", "hp1 : f ↑p < 1", "hmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m"], "goal": "Nat.Prime p"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α✝", "inst✝¹ : MeasurableSpace β", "μ✝ ν✝ ν₁ ν₂ : Measure α✝", "s✝ t : Set α✝", "α : Type u_5", "m₀ : MeasurableSpace α", "μ ν : Measure α", "inst✝ : IsFiniteMeasure μ", "C : Set (Set α)", "hμν : ∀ s ∈ C, μ s = ν s", "m : MeasurableSpace α", "h : m ≤ m₀", "hA : m = generateFrom C", "hC : IsPiSystem C", "h_univ : μ univ = ν univ", "s : Set α", "hs : MeasurableSet s", "this : IsFiniteMeasure ν"], "goal": "∀ (t : Set α), MeasurableSet t → μ t = ν t → μ tᶜ = ν tᶜ"}}
+{"state": {"context": ["F : Type w", "R : Type u", "S : Type v", "NonUnitalNonAssocRing R", "NonUnitalNonAssocRing S", "FunLike F R S", "NonUnitalRingHomClass F R S", "s : NonUnitalSubring S", "f : F"], "goal": "↑(NonUnitalSubring.comap f s) = ⇑f ⁻¹' ↑s"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "Group G", "MulAction G α", "s : Set α", "g : G", "s_ss_fixedBy : s ⊆ MulAction.fixedBy α g"], "goal": "s ∈ MulAction.fixedBy (Set α) g"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f : ℂ → E", "z : ℂ", "x : ℝ"], "goal": "∀ x_1 ∈ norm ∘ f '' (re ⁻¹' {x}), 0 ≤ x_1"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "z : ℤ", "a✝ a b : α", "h : a - ↑⌊a⌋ = b - ↑⌊b⌋"], "goal": "a - b = ↑(⌊a⌋ - ⌊b⌋)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H K : AddSubgroup G"], "goal": "(H ⊓ K).addSubgroupOf K = H.addSubgroupOf K"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "P : (H → H') → Set H → H → Prop", "s✝ t u : Set H", "x✝ : H", "hG : G.LocalInvariantProp G' P", "s : Set H", "x : H", "f : H → H'", "e : PartialHomeomorph H H", "he : e ∈ G", "hxe : x ∈ e.source", "h : P (f ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x)"], "goal": "P f s x"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "I : Ideal R"], "goal": "⊥ * I = ⊥"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l l' : Filter α", "f g : α → β", "h₁ : f =ᶠ[l] g", "h₂ : l' ≤ l"], "goal": "f =ᶠ[l'] g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K : ℝ≥0", "f : α → β", "hf : AntilipschitzWith K f"], "goal": "comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace E", "inst✝ : TopologicalSpace X", "f : E → X", "s : Set X", "hf : IsCoveringMap f", "e₁ e₂ : E", "he : f e₁ = f e₂", "hne : e₁ ≠ e₂", "left✝ : DiscreteTopology ↑(f ⁻¹' {f e₁})", "t : Trivialization (↑(f ⁻¹' {f e₁})) f", "he₁ : e₁ ∈ t.source", "he₂ : e₂ ∈ t.source", "x : E", "h₁ : x ∈ t.source ∩ Prod.snd ∘ ↑t ⁻¹' {(↑t e₁).2}", "h₂ : x ∈ t.source ∩ Prod.snd ∘ ↑t ⁻¹' {(↑t e₂).2}"], "goal": "↑t.toPartialHomeomorph e₁ = ↑t.toPartialHomeomorph e₂"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "F : MonoidalFunctor C D", "X : C"], "goal": "F.map (ρ_ X).hom = inv (F.μ X (𝟙_ C)) ≫ F.obj X ◁ inv F.ε ≫ (ρ_ (F.obj X)).hom"}}
+{"state": {"context": ["α β : Type u", "Infinite α", "le : #β ≤ #α"], "goal": "#↑{f | Function.Surjective f} = #(α → β)"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "k : Type u_3", "S : Type u_4", "M : Type u_5", "M₂ : Type u_6", "M₃ : Type u_7", "ι : Type u_8", "inst✝² : Semiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "r s : R", "x y : M", "zero_eq_one : 0 = 1"], "goal": "x = 0"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M", "x y : M", "hx : x ∈ S", "hy : y ∈ S"], "goal": "⟨x, hx⟩ + ⟨y, hy⟩ = ⟨x + y, ⋯⟩"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : PartialEquiv α β", "s : Set α", "t : Set β"], "goal": "e.symm.IsImage t s ↔ e.IsImage s t"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "p : α → Prop", "DecidablePred p", "h : (Finset.filter p s).card = s.card", "x : α", "hx : x ∈ s"], "goal": "p x"}}
+{"state": {"context": ["x : ℝ≥0", "z : ℝ", "hx : 1 < x", "hz : 0 < z"], "goal": "1 < x ^ z"}}
+{"state": {"context": [], "goal": "Bipointed.swap ⋙ bipointedToPointedSnd = bipointedToPointedFst"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x : X", "M₀ : Type u_4", "Zero M₀", "f : X → M₀"], "goal": "f =ᶠ[𝓝 x] 0 ↔ x ∉ closure (Function.support f)"}}
+{"state": {"context": ["l : Filter ℂ", "hl : IsExpCmpFilter l", "a z : ℂ", "hz : z ≠ 0"], "goal": "(fun z => abs z ^ a.re) z = (fun z => Real.exp (a.re * Real.log (abs z))) z"}}
+{"state": {"context": ["α : Type u", "s t : Set α", "hs : s.Infinite", "ht : t.Finite"], "goal": "∃ a ∈ s, a ∉ t"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝⁴ : Ring k", "inst✝³ : AddCommGroup V", "inst✝² : Module k V", "inst✝¹ : AffineSpace V P", "ι : Type u_4", "A : Set P", "inst✝ : Nonempty ↑A", "x : P", "hx : x ∈ ↑(affineSpan k A)", "c : k", "u : P", "hu : u ∈ affineSpan k A", "v : P", "hv : v ∈ affineSpan k A", "w : P", "hw : w ∈ affineSpan k A"], "goal": "⟨u, hu⟩ ∈ ↑(affineSpan k (Subtype.val ⁻¹' A)) → ⟨v, hv⟩ ∈ ↑(affineSpan k (Subtype.val ⁻¹' A)) → ⟨w, hw⟩ ∈ ↑(affineSpan k (Subtype.val ⁻¹' A)) → ⟨c • (u -ᵥ v) +ᵥ w, ⋯⟩ ∈ ↑(affineSpan k (Subtype.val ⁻¹' A))"}}
+{"state": {"context": ["z : ℂ", "hz : ‖z‖ < 1", "n : ℕ"], "goal": "-(z ^ n / ↑n) = (-1) ^ (n + 1) * (-z) ^ n / ↑n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : PseudoMetricSpace α", "π : β → Type u_3", "inst✝² : Fintype β", "inst✝¹ : (b : β) → PseudoMetricSpace (π b)", "inst✝ : Nonempty β", "a b : α"], "goal": "(dist (fun x => a) fun x => b) = dist a b"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "H : Type u_3", "inst✝² : Category.{u_4, u_1} C", "inst✝¹ : Category.{u_5, u_2} D", "inst✝ : Category.{u_6, u_3} H", "L : C ⥤ D", "F : C ⥤ H", "E E' : L.RightExtension F", "h✝ h : E.IsPointwiseRightKanExtension", "G : L.RightExtension F", "f₁ f₂ : G ⟶ E", "Y : D", "X : StructuredArrow Y L"], "goal": "f₁.left.app Y ≫ (E.coneAt Y).π.app X = f₂.left.app Y ≫ (E.coneAt Y).π.app X"}}
+{"state": {"context": ["α : Type u_1", "α₁ : Type u_2", "α₂ : Type u_3", "β : α → Type u_4", "β₁ : α₁ → Type u_5", "β₂ : α₂ → Type u_6", "f₁ : α₁ → α₂", "f₂ : (a : α₁) → β₁ a → β₂ (f₁ a)", "h₁ : Injective f₁", "h₂ : ∀ (a : α₁), Injective (f₂ a)", "i : α₁", "x : β₁ i", "j : α₁", "y : β₁ j", "h : Sigma.map f₁ f₂ ⟨i, x⟩ = Sigma.map f₁ f₂ ⟨j, y⟩"], "goal": "⟨i, x⟩ = ⟨j, y⟩"}}
+{"state": {"context": ["M : Type u_1", "OrderedCancelAddCommMonoid M", "ExistsAddOfLE M", "a b c : M"], "goal": "(fun x => x + a) '' Set.Ioc b c = Set.Ioc (b + a) (c + a)"}}
+{"state": {"context": ["a b n : ℕ", "hn : n ≠ 0"], "goal": "a ^ n ≤ b ^ n ↔ a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "SuccOrder α", "PredOrder α", "NoMaxOrder α", "a : α"], "goal": "Order.pred (Order.succ a) = a"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "N : Type w'", "L.Structure M", "L.Structure N", "P : Type u_1", "L.Structure P", "Q : Type u_2", "L.Structure Q", "f : L.Hom M N", "g : L.Hom N P", "h : L.Hom P Q"], "goal": "(h.comp g).comp f = h.comp (g.comp f)"}}
+{"state": {"context": ["J : Type v'", "inst✝³ : Category.{u', v'} J", "C : Type u", "inst✝² : Category.{v, u} C", "W✝ X✝ Y Z✝ : C", "f✝ : W✝ ⟶ X✝", "g : W✝ ⟶ Y", "h✝ : X✝ ⟶ Z✝", "i✝ : Y ⟶ Z✝", "W E X Z : C", "c : BinaryCofan W E", "inst✝¹ : FinitaryExtensive C", "inst✝ : HasPullbacks C", "hc : IsColimit c", "f : W ⟶ X", "h : X ⟶ Z", "i : c.pt ⟶ Z", "H : IsPushout f c.inl h i", "hc₁ : IsColimit (BinaryCofan.mk (c.inr ≫ i) h)", "W' X' Y' Z' : C", "f' : W' ⟶ X'", "g' : W' ⟶ Y'", "h' : X' ⟶ Z'", "i' : Y' ⟶ Z'", "αW : W' ⟶ W", "αX : X' ⟶ X", "αY : Y' ⟶ ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }", "αZ : Z' ⟶ Z", "hf : IsPullback f' αW αX f", "hg : IsPullback g' αW αY c.inl", "hh : CommSq h' αX αZ h", "hi : CommSq i' αY αZ i", "w : CommSq f' g' h' i'", "hc₂ : IsColimit (BinaryCofan.mk g' (pullback.fst αY c.inr))"], "goal": "Nonempty (IsColimit (BinaryCofan.mk ((BinaryCofan.mk g' (pullback.fst αY c.inr)).inr ≫ i') h')) ↔ IsPullback h' αX αZ h ∧ IsPullback i' αY αZ i"}}
+{"state": {"context": ["K R : Type u", "V V₁ V₂ V₃ : Type v", "V' V'₁ : Type v'", "V'' : Type v''", "ι✝ : Type w", "ι' : Type w'", "η : Type u₁'", "φ : η → Type u_1", "inst✝ : DivisionRing K", "ι : Type v", "hι : Infinite ι", "ιK : Type (max u v)", "bK : Basis ιK K (ι → K)", "e : ℕ ↪ ι", "this : lift.{max u v, u} (Module.rank K (ℕ → K)) ≤ lift.{u, max u v} (Module.rank K (ι → K))"], "goal": "Module.rank K (ι → K) = #(ι → K)"}}
+{"state": {"context": ["α : Type u_1", "G G' : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "s t u : Finset α", "inst✝ : DecidableEq α", "dst : 2 * ε ≤ ↑(G.edgeDensity s t)", "dsu : 2 * ε ≤ ↑(G.edgeDensity s u)", "dtu : 2 * ε ≤ ↑(G.edgeDensity t u)", "utu : G.IsUniform ε t u", "x : α", "hxY : ↑t.card * (↑(G.edgeDensity s t) - ε) ≤ ↑(filter (G.Adj x) t).card", "hsu : ↑u.card * (↑(G.edgeDensity s u) - ε) ≤ ↑(filter (G.Adj x) u).card", "hY : ↑t.card * ε ≤ ↑(filter (G.Adj x) t).card", "hZ : ↑u.card * ε ≤ ↑(filter (G.Adj x) u).card", "this✝ : |↑(G.edgeDensity (filter (G.Adj x) t) (filter (G.Adj x) u)) - ↑(G.edgeDensity t u)| < ε", "this : ε ≤ ↑(↑(G.interedges (filter (G.Adj x) t) (filter (G.Adj x) u)).card / (↑(filter (G.Adj x) t).card * ↑(filter (G.Adj x) u).card))"], "goal": "ε ^ 3 * ↑t.card * ↑u.card ≤ ↑(filter (fun x => match x with | (y, z) => G.Adj y z) (filter (G.Adj x) t ×ˢ filter (G.Adj x) u)).card"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "x y z : α", "ε ε₁ ε₂ : ℝ≥0∞", "s✝ t s : Set α", "hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε"], "goal": "∃ t ⊆ s, t.Countable ∧ s ⊆ closure t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedAddCommGroup β", "p : ℝ≥0∞", "inst✝ : IsFiniteMeasure μ", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "f : ℕ → α → β", "g : α → β", "hf : ∀ (n : ℕ), StronglyMeasurable (f n)", "hg : StronglyMeasurable g", "hg' : Memℒp g p μ", "hui : UnifIntegrable f p μ", "hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))", "ε : ℝ≥0∞", "hε : ε > 0", "h : ε < ⊤", "hμ : ¬μ = 0", "hε' : 0 < ε.toReal / 3"], "goal": "∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "inst✝⁴ : MeasurableSpace G", "inst✝³ : MeasurableSpace α", "inst✝² : CommGroup G", "inst✝¹ : MeasurableMul₂ G", "inst✝ : MeasurableInv G", "μ : Measure α", "f g : α → G", "hf : AEMeasurable f μ", "h : AEMeasurable (f * g) μ"], "goal": "g = f * g * f⁻¹"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : 𝕜", "ι : Type u_1", "u : Finset ι", "A : ι → 𝕜 → F", "h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x"], "goal": "deriv (fun y => ∑ i ∈ u, A i y) x = ∑ i ∈ u, deriv (A i) x"}}
+{"state": {"context": ["k : Type u_1", "inst✝³ : Field k", "K : Type u_2", "inst✝² : Field K", "F : Type u_3", "inst✝¹ : Field F", "inst✝ : NumberField K", "f : k ≃+* K", "w : InfinitePlace K"], "goal": "(w.comap ↑f).IsReal ↔ w.IsReal"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : Option α → β"], "goal": "Set.range f = insert (f none) (Set.range (f ∘ some))"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "β : Type u_4", "Preorder α", "Preorder β", "Subsingleton ι", "f : ι → α", "g : ι → β", "s : Set ι"], "goal": "AntivaryOn f g s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "self : S.ShortExact"], "goal": "S.Exact"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "e : S₁ ≅ S₂", "S₁.HasLeftHomology", "S₂.HasLeftHomology"], "goal": "(CategoryTheory.ShortComplex.leftHomologyMapIso e).hom = CategoryTheory.ShortComplex.leftHomologyMap e.hom"}}
+{"state": {"context": ["β : Type u_1", "o : Ordinal.{u_1}", "c : Cardinal.{u_1}", "ho : o.card ≤ c", "hc : ℵ₀ ≤ c", "A : Ordinal.{u_1} → Set β", "hA : ∀ j < o, #↑(A j) ≤ c"], "goal": "#(Quotient.out o).α * ⨆ i, #↑(((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm) i) ≤ c * c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "f : α → β", "hf : MeasurableEmbedding f", "g : α → γ", "hg : Measurable g", "hne : β → Nonempty γ"], "goal": "∃ g', Measurable g' ∧ g' ∘ f = g"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁶ : Category.{u_3, u_1} C", "inst✝⁵ : Category.{?u.24208, u_2} D", "inst✝⁴ : ConcreteCategory D", "inst✝³ : Preregular C", "inst✝² : FinitaryPreExtensive C", "F G : Cᵒᵖ ⥤ Type w", "f : F ⟶ G", "inst✝¹ : PreservesFiniteProducts F", "inst✝ : PreservesFiniteProducts G", "U✝ : C", "y : (forget (Type w)).obj (G.obj (op U✝))", "α : Type", "w✝ : Finite α", "Z : α → C", "π : (a : α) → Z a ⟶ U✝", "h : EffectiveEpiFamily Z π", "h' : ∀ (a : α), (Presheaf.imageSieve f y).arrows (π a)", "x : (a : α) → F.obj (op (Z a)) := fun a => Exists.choose ⋯", "x✝ : Fintype α := Fintype.ofFinite α", "i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (∐ Z)) := (Types.productIso fun a => F.obj (op (Z a))).symm ≪≫ (PreservesProduct.iso F fun a => op (Z a)).symm ≪≫ F.mapIso (opCoproductIsoProduct Z).symm", "this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G", "a : α"], "goal": "((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app (op { as := a })) ((f.app (op (∐ Z))) (i'.hom x)) = ((G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).π.app (op { as := a })) ((G.map (Sigma.desc π).op) y)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_4, u_1} C", "inst✝³ : Category.{u_3, u_2} D", "inst✝² : Preadditive C", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "inst✝ : W.HasLeftCalculusOfFractions", "X Y Z : C", "f₁ f₂ : L.obj X ⟶ L.obj Y", "α : W.LeftFraction₂ X Y", "h₁ : f₁ = α.fst.map L ⋯", "h₂ : f₂ = α.snd.map L ⋯"], "goal": "add' W f₁ f₂ = add' W f₂ f₁"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r"], "goal": "(deriv fun x => 1 - ε x) =ᶠ[atTop] -deriv ε"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "a : Products I", "ha : Products.isGood C a", "b : Products I", "hb : Products.isGood C b", "h : Products.eval C a = Products.eval C b", "h' : b < a"], "goal": "⟨a, ha⟩ = ⟨b, hb⟩"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "CategoryTheory.Category.{v, u} C", "AddMonoid A", "CategoryTheory.HasShift C A", "a : A", "X : C"], "goal": "(CategoryTheory.shiftFunctorAdd' C a 0 a ⋯).hom.app X =\n  (CategoryTheory.shiftFunctorZero C A).inv.app ((CategoryTheory.shiftFunctor C a).obj X)"}}
+{"state": {"context": ["α : Type u_2", "BooleanAlgebra α", "a b : α"], "goal": "a ⇔ b ≤ a ↔ Codisjoint a b"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "MeasurableSpace α", "NormedAddCommGroup E", "f : α → E", "s : Set α", "μ : MeasureTheory.Measure α", "h1s : Function.support f ⊆ s"], "goal": "MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder α", "Preorder β", "f : α → β", "s t : Set α", "c : α", "h₁ : StrictMonoOn f s", "h₂ : StrictMonoOn f t", "hs : IsGreatest s c", "ht : IsLeast t c"], "goal": "StrictMonoOn f (s ∪ t)"}}
+{"state": {"context": ["a b c : Prop", "h : a ↔ b"], "goal": "a ∧ c ↔ b ∧ c"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "β : Sort u_1", "f g : β → C", "z : (b : β) → f b ⟶ g b", "j j' : β", "w : j = j'"], "goal": "CategoryTheory.CategoryStruct.comp (z j) (CategoryTheory.eqToHom ⋯) =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom ⋯) (z j')"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y : C", "f✝ : X ⟶ Y", "f g : Arrow C", "F : MonoFactorisation f.hom", "sq : f ⟶ g", "inst✝ : IsIso sq"], "goal": "(inv sq.left ≫ F.e) ≫ F.m ≫ sq.right = g.hom"}}
+{"state": {"context": ["α : Type u", "CompleteLattice α", "f : α →o α", "x : α", "hx : f x ≤ x", "y : ↑(Function.fixedPoints ⇑f)"], "goal": "y ≤ f.prevFixed x hx ↔ ↑y ≤ x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : PseudoEMetricSpace α", "inst✝¹ : PseudoEMetricSpace β", "inst✝ : PseudoEMetricSpace γ", "K : ℝ≥0", "f : α → β", "x y : α"], "goal": "edist x y ≤ ↑1 * edist (id x) (id y)"}}
+{"state": {"context": ["R : Type u_2", "Semiring R", "f : RingInvo R", "a : R"], "goal": "↑f a = f a"}}
+{"state": {"context": ["R : Type u", "S : Type v", "F : Type u_1", "Ring R", "Ring S", "FunLike F R S", "RingHomClass F R S", "f : F", "hf : Function.Surjective ⇑f", "I J : Ideal S"], "goal": "Ideal.comap f I ≤ Ideal.comap f J ↔ I ≤ J"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "inst✝ : SFinite ν", "s : Set (α × β)", "hs : MeasurableSet s", "h2s : (μ.prod ν) s ≠ ⊤"], "goal": "∫⁻ (a : α), ENNReal.ofReal (ν (Prod.mk a ⁻¹' s)).toReal ∂μ < ⊤"}}
+{"state": {"context": ["α : Type u", "β : Type v", "GroupWithZero α", "MulAction α β", "c : α", "hc : c ≠ 0", "x : β"], "goal": "c⁻¹ • c • x = x"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "P₂ : Type u_3", "V₁ : Type u_6", "V₂ : Type u_7", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AffineSpace V₂ P₂", "f : P₁ ≃ᵃ[k] P₂", "s : Set P₂"], "goal": "⇑f.symm '' s = ⇑f ⁻¹' s"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "Group α", "a : α", "ha : 1 < a⁺ᵐ "], "goal": "a⁺ᵐ = a"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "CategoryTheory.Category.{v₁, u₂} 𝒳", "CategoryTheory.Category.{v₂, u₁} 𝒮", "p : 𝒳 ⥤ 𝒮", "a b : 𝒳", "hab : a = b", "S : 𝒮", "hS : p.obj b = S"], "goal": "p.IsHomLift (𝟙 S) (CategoryTheory.eqToHom hab)"}}
+{"state": {"context": ["R : Type u_1", "M M₁ M₂ M₃ : Type u", "M' : Type v", "inst✝¹² : Ring R", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : AddCommGroup M₁", "inst✝⁹ : AddCommGroup M₂", "inst✝⁸ : AddCommGroup M₃", "inst✝⁷ : AddCommGroup M'", "inst✝⁶ : Module R M", "inst✝⁵ : Module R M₁", "inst✝⁴ : Module R M₂", "inst✝³ : Module R M₃", "inst✝² : Module R M'", "inst✝¹ : HasRankNullity.{u, u_1} R", "inst✝ : StrongRankCondition R", "s : Set M", "hs : LinearIndependent (ι := ↑s) R Subtype.val"], "goal": "∃ t, s ⊆ t ∧ #↑t = Module.rank R M ∧ LinearIndependent (ι := ↑t) R Subtype.val"}}
+{"state": {"context": ["f : ℕ → ℕ"], "goal": "(Nat.Primrec' fun v => f v.head) ↔ Primrec f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Preorder α", "t₁ t₂ : Ordnode α", "o₁ : WithBot α", "o₂ : WithTop α", "x : α", "h₁ : t₁.Bounded o₁ ↑x", "h₂ : t₂.Bounded (↑x) o₂", "y : α", "yx : y < x"], "goal": "All (fun z => y < z) t₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s s' : Set α", "x✝ : α", "p : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "x : α"], "goal": "UniformContinuous fun x_1 => toFun x_1 x"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "HasOuterApproxClosed X", "F : Set X", "hF : IsClosed F"], "goal": "Filter.Tendsto (fun n x => (hF.apprSeq n) x) Filter.atTop (𝓝 (F.indicator fun x => 1))"}}
+{"state": {"context": ["R : Type u_1", "Semiring R"], "goal": "LaurentPolynomial.degree 0 = ⊥"}}
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+{"state": {"context": [], "goal": "Measurable Complex.im"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "WeaklyLocallyCompactSpace X", "x : X"], "goal": "Disjoint (𝓝 x) (Filter.cocompact X)"}}
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+{"state": {"context": ["P : TopCat → Prop", "X Y : CompHausLike P", "f : X ⟶ Y", "hf : Function.Surjective ⇑f"], "goal": "CategoryTheory.Epi f"}}
+{"state": {"context": ["x : ℂ"], "goal": "Complex.cos (2 * ↑Real.pi - x) = Complex.cos x"}}
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+{"state": {"context": ["M : Type u_2", "N : Type u_4", "P : Type u_6", "AddGroup M", "AddGroup N", "AddGroup P", "f : M →+ N", "g : N →+ P", "hfg : Function.Exact ⇑f ⇑g"], "goal": "g.ker = f.range"}}
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+{"state": {"context": ["R : Type u_2", "Ring R", "S₁ S₂ : Subring Rᵐᵒᵖ"], "goal": "(S₁ ⊔ S₂).unop = S₁.unop ⊔ S₂.unop"}}
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+{"state": {"context": ["M : Type u_1", "Monoid M", "R : Type u_10", "Semiring R", "MulSemiringAction M R", "x : R"], "goal": "(MulSemiringActionHom.id M) x = x"}}
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+{"state": {"context": ["d : ℤ", "a : Solution₁ d", "hax : 0 < a.x", "hay : 0 < a.y", "n : ℕ", "hn : 0 < n"], "goal": "0 < (a ^ n.pred.succ).y"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "inst✝¹¹ : CommRing R", "inst✝¹⁰ : LieRing L", "inst✝⁹ : LieAlgebra R L", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "inst✝⁶ : LieRingModule L M", "inst✝⁵ : LieModule R L M", "inst✝⁴ : IsDomain R", "inst✝³ : IsPrincipalIdealRing R", "inst✝² : Module.Free R M", "inst✝¹ : Module.Finite R M", "inst✝ : LieAlgebra.IsNilpotent R L", "χ : L → R", "x : L", "n : Module.End R ↥↑(weightSpace M χ) := (toEnd R L ↥↑(weightSpace M χ)) x - χ x • LinearMap.id", "h₁ : (toEnd R L ↥↑(weightSpace M χ)) x = n + χ x • LinearMap.id"], "goal": "(LinearMap.trace R ↥↑(weightSpace M χ) ∘ₗ ↑(toEnd R L ↥↑(weightSpace M χ))) x = (finrank R ↥↑(weightSpace M χ) • χ) x"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y : Mon_ C", "M N : Bimod X Y", "f : M ⟶ N"], "goal": "X.X ◁ f.hom ≫ colimit.ι (parallelPair (X.mul ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one = X.X ◁ f.hom ≫ (λ_ (X.X ⊗ N.X)).inv ≫ (((α_ (𝟙_ C) X.X N.X).inv ≫ X.one ▷ X.X ▷ N.X) ≫ X.mul ▷ N.X) ≫ coequalizer.π (X.mul ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)"}}
+{"state": {"context": [], "goal": "IsGδ {x | (fun x => x ∈ ⋂ n, ⋃ a, ⋃ b, ⋃ (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n)) x}"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "CategoryTheory.Limits.HasColimit F", "E : K ⥤ J", "CategoryTheory.Limits.HasColimit (E ⋙ F)", "c : CategoryTheory.Limits.Cocone F"], "goal": "CategoryTheory.Limits.colimit.pre F E ≫ CategoryTheory.Limits.colimit.desc F c =\n  CategoryTheory.Limits.colimit.desc (E ⋙ F) (CategoryTheory.Limits.Cocone.whisker E c)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "J : CategoryTheory.GrothendieckTopology C", "R : CategoryTheory.Sheaf J RingCat", "M : SheafOfModules R"], "goal": "((SheafOfModules.toSheaf R).obj M).val = M.val.presheaf"}}
+{"state": {"context": ["G : Type w", "H : Type x", "TopologicalSpace G", "Group G", "TopologicalGroup G", "Group H", "TopologicalSpace H", "TopologicalGroup H", "f : G →* H", "hf : Continuous ⇑f", "hf' : DenseRange ⇑f", "s : Subgroup G", "hs : s.topologicalClosure = ⊤"], "goal": "(Subgroup.map f s).topologicalClosure = ⊤"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Semiring R", "AddCommMonoid M", "Module R M", "p q : Submodule R M", "h : ∀ (x : M), x ∈ p ↔ x ∈ q"], "goal": "p = q"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a b c : α"], "goal": "a ∈ Ι b c ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b"}}
+{"state": {"context": ["α : Type u_6", "n : Type u_7", "Zero α", "One α", "DecidableEq n", "AddGroup n"], "goal": "Matrix.circulant (Pi.single 0 1) = 1"}}
+{"state": {"context": ["b : Bool"], "goal": "SNum.bit b (SNum.zero b) = SNum.zero b"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "m : MeasurableSpace X", "μ : MeasureTheory.Measure X", "μ.IsOpenPosMeasure", "U : Set X", "hU : IsOpen U"], "goal": "0 < μ U ↔ U.Nonempty"}}
+{"state": {"context": ["E : Type u_4", "AddGroup E", "p q : NonarchAddGroupNorm E"], "goal": "⇑(p ⊔ q) = ⇑p ⊔ ⇑q"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "E : Type u_4", "F : Type u_6", "ι : Type u_11", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜₂", "σ₁₂ : 𝕜 →+* 𝕜₂", "RingHomIsometric σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup F", "NormedSpace 𝕜 E", "NormedSpace 𝕜₂ F", "f : ι → E →SL[σ₁₂] F"], "goal": "[EquicontinuousAt (DFunLike.coe ∘ f) 0, Equicontinuous (DFunLike.coe ∘ f), UniformEquicontinuous (DFunLike.coe ∘ f),\n    ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖, ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖,\n    ∃ C, ∀ (i : ι), ‖f i‖ ≤ C, ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C, BddAbove (Set.range fun x => ‖f x‖),\n    ⨆ i, ↑‖f i‖₊ < ⊤].TFAE"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "p : ℕ", "inst✝ : ExpChar R p", "h : Function.Injective ⇑(frobenius R p)", "x : R"], "goal": "(∃ n, x ^ p ^ n = 0) → x = 0"}}
+{"state": {"context": ["α : Type u", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "DenselyOrdered α", "δ : Type u_1", "LinearOrder δ", "TopologicalSpace δ", "OrderClosedTopology δ", "a b : α", "hab : a ≤ b", "f : α → δ", "hf : ContinuousOn f (Set.Icc a b)"], "goal": "Set.Ioo (f b) (f a) ⊆ f '' Set.Ioo a b"}}
+{"state": {"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "s : Finset I", "x y : ↑(π C fun x => x ∈ s)", "a : I", "ha : ↑y a = true", "hx : ↑x a = false"], "goal": "a ∈ s ∧ (if ↑x a = true then e (π C fun x => x ∈ s) a else 1 - e (π C fun x => x ∈ s) a) = 1 - e (π C fun x => x ∈ s) a"}}
+{"state": {"context": ["G : PGame", "inst✝ : G.Impartial"], "goal": "G ≈ 0 ∨ G ‖ 0"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝ : MonoidWithZero M₀", "a✝ : Nontrivial M₀"], "goal": "inverse 0 = 0"}}
+{"state": {"context": ["F : Type u", "Field F", "f : F[X]", "hf : Irreducible f"], "goal": "f.Separable ↔ Polynomial.derivative f ≠ 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u'", "CategoryTheory.Category.{v', u'} D", "W : CategoryTheory.MorphismProperty C", "L : CategoryTheory.Functor C D", "h : W.IsInvertedBy L"], "goal": "W.op.IsInvertedBy L.op"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "G : SimpleGraph α", "DecidableRel G.Adj", "P : Finpartition Finset.univ", "ε : ℝ", "Nonempty α", "hε : 0 < ε", "hP : P.IsEquipartition", "hPε : P.IsUniform G (ε / 8)", "hP' : 4 / ε ≤ ↑P.parts.card"], "goal": "2 * (↑G.edgeFinset.card - ↑(SimpleGraph.regularityReduced P G (ε / 8) (ε / 4)).edgeFinset.card) <\n  2 * ε * ↑(Fintype.card α ^ 2)"}}
+{"state": {"context": ["F : Type u_1", "inst✝⁶ : Field F", "E : Type u_2", "inst✝⁵ : Field E", "inst✝⁴ : Algebra F E", "α : E", "K : Type u", "inst✝³ : Field K", "inst✝² : Algebra F K", "L : Type u_3", "inst✝¹ : Field L", "inst✝ : Algebra K L", "S : Set L", "hS : ∀ x ∈ S, IsIntegral K x"], "goal": "Algebra.IsAlgebraic K ↥(adjoin K S)"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁹ : CommRing R", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "inst✝⁶ : Field K", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module K V", "inst✝³ : NoZeroSMulDivisors R M", "L : Type u_1", "F : Type u_2", "inst✝² : Add L", "inst✝¹ : FunLike F L (End R M)", "inst✝ : AddHomClass F L (End R M)", "f : F", "μ : L → R", "h_ne : ⨅ x, ⨆ k, ((f x).genEigenspace (μ x)) k ≠ ⊥", "h : ∀ (x y : L), Commute (f x) (f y)", "x y : L", "contra : ¬μ (x + y) = μ x + μ y", "g : L → Submodule R M := fun x => ⨆ k, ((f x).genEigenspace (μ x)) k", "this : ⨅ x, g x ≤ g x ⊓ g y ⊓ g (x + y)"], "goal": "Disjoint (⨆ k, ((f x + f y).genEigenspace (μ x + μ y)) k) (⨆ k, ((f x + f y).genEigenspace (μ (x + y))) k)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddZeroClass M", "AddZeroClass N", "s : AddSubmonoid M"], "goal": "AddSubmonoid.map (AddMonoidHom.inl M N) s = s.prod ⊥"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "Fintype ι", "Fintype ι'", "π : ι ⊕ ι' → Type u_4", "m : (i : ι ⊕ ι') → MeasurableSpace (π i)", "μ : (i : ι ⊕ ι') → MeasureTheory.Measure (π i)", "∀ (i : ι ⊕ ι'), MeasureTheory.SigmaFinite (μ i)"], "goal": "MeasureTheory.MeasurePreserving (⇑(MeasurableEquiv.sumPiEquivProdPi π).symm)\n  ((MeasureTheory.Measure.pi fun i => μ (Sum.inl i)).prod (MeasureTheory.Measure.pi fun i => μ (Sum.inr i)))\n  (MeasureTheory.Measure.pi μ)"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "a b : Associates.FactorSet α"], "goal": "(a + b).prod = a.prod * b.prod"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "a₁ a₂ : Fin (n + 2)", "ha₂ : ⟨a₁, a₂⟩ ∉ {⟨1, 0⟩}", "ha₁ : a₂ < a₁", "H : 0 < a₁"], "goal": "(if (swap 0 1) a₁ ≤ (swap 0 1) a₂ then -1 else 1) = 1"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "ι : Type u_3", "t : Finset ι", "φ : ι → MvPolynomial σ R", "DecidableEq σ", "h : Pairwise (Disjoint on fun i => (φ i).vars)"], "goal": "(∑ i ∈ t, φ i).vars = t.biUnion fun i => (φ i).vars"}}
+{"state": {"context": ["α : Type u_2", "AddZeroClass α"], "goal": "⇑Set.singletonAddMonoidHom = singleton"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u", "x : α (Fin.last n)", "p : (i : Fin n) → α i.castSucc", "i : Fin n", "y : α i.castSucc"], "goal": "Fin.snoc (Function.update p i y) x = Function.update (Fin.snoc p x) i.castSucc y"}}
+{"state": {"context": ["E : Type u", "NormedAddCommGroup E", "NormedSpace ℂ E", "F : Type v", "NormedAddCommGroup F", "NormedSpace ℂ F", "f : E → F", "U : Set E", "c : E", "hc : IsPreconnected U", "ho : IsOpen U", "hd : DifferentiableOn ℂ f U", "hcU : c ∈ U", "hm : IsMaxOn (norm ∘ f) U c"], "goal": "Set.EqOn (norm ∘ f) (Function.const E ‖f c‖) U"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "TopologicalSpace α", "Zero β", "TopologicalSpace β", "G : Type u_5", "FunLike G α β", "ContinuousMapClass G α β", "CompactSpace α"], "goal": "CompactlySupportedContinuousMapClass G α β"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "MeasurableSpace α", "NormedAddCommGroup E", "f : α → E", "s : Set α", "μ ν : MeasureTheory.Measure α", "h : MeasureTheory.IntegrableOn f s ν", "hμ : μ ≤ ν"], "goal": "MeasureTheory.IntegrableOn f s μ"}}
+{"state": {"context": ["E : Type uE", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : FiniteDimensional ℝ E", "H : Type uH", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "c : M", "f : SmoothBumpFunction I c", "x : M", "h : ↑f x = 0"], "goal": "0 ∈ Icc 0 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : AddCommSemigroup α", "inst✝⁴ : PartialOrder α", "inst✝³ : ExistsAddOfLE α", "inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝¹ : Sub α", "inst✝ : OrderedSub α", "a b c d : α", "hb : AddLECancellable b", "h : b ≤ a"], "goal": "a - b + c = a + c - b"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_3", "α : Type u_7", "tX : TopologicalSpace X", "uα : UniformSpace α", "l : Filter ι", "F : ι → X → α", "f : X → α", "hF : Equicontinuous F", "hf : Continuous f"], "goal": "IsClosed {x | Filter.Tendsto (fun x_1 => F x_1 x) l (nhds (f x))}"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M N P : Bimod X Y", "f : M ⟶ N", "g : N ⟶ P", "Q : Bimod Y Z"], "goal": "coequalizer.π (M.actRight ▷ Q.X) ((α_ M.X Y.X Q.X).hom ≫ M.X ◁ Q.actLeft) ≫ (whiskerRight (f ≫ g) Q).hom = coequalizer.π (M.actRight ▷ Q.X) ((α_ M.X Y.X Q.X).hom ≫ M.X ◁ Q.actLeft) ≫ (whiskerRight f Q ≫ whiskerRight g Q).hom"}}
+{"state": {"context": ["C : Type u_1", "A : Type u_2", "CategoryTheory.Category.{u_5, u_1} C", "CategoryTheory.Category.{u_6, u_2} A", "F : C ⥤ A", "M : Type u_3", "AddMonoid M", "CategoryTheory.HasShift C M", "F.ShiftSequence M", "X Y : C", "n a a' : M", "ha' : n + a = a'", "f : X ⟶ Y"], "goal": "(F.shift a).map ((CategoryTheory.shiftFunctor C n).map f) ≫ (F.shiftIso n a a' ha').hom.app Y =\n  (F.shiftIso n a a' ha').hom.app X ≫ (F.shift a').map f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "hab : a ≤ b", "hcd : c ≤ d", "h₁ : a < d", "h₂ : c < b"], "goal": "Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "NormalSpace X", "s t : Set X", "hs : IsClosed s", "ht : IsClosed t", "hd : Disjoint s t", "a b : ℝ", "hle : a ≤ b"], "goal": "∃ f, Set.EqOn (⇑f) (Function.const X a) s ∧ Set.EqOn (⇑f) (Function.const X b) t ∧ ∀ (x : X), f x ∈ Set.Icc a b"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "hl : l.Subsingleton"], "goal": "l.IsCountablyGenerated"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "NormedField 𝕜₁", "NormedField 𝕜₂", "σ : 𝕜₁ →+* 𝕜₂", "E : Type u_3", "F : Type u_4", "AddCommGroup E", "Module 𝕜₁ E", "TopologicalSpace E", "AddCommGroup F", "Module 𝕜₂ F", "UniformSpace F", "UniformAddGroup F", "𝔖 : Set (Set E)"], "goal": "UniformEmbedding DFunLike.coe"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "T0Space X", "h : X ≃ₜ Y"], "goal": "T0Space Y"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "ι' : Sort u_5", "l l' : Filter α", "p : ι → Prop", "s : ι → Set α", "p' : ι' → Prop", "s' : ι' → Set α", "hl : l.HasBasis p s", "hl' : l'.HasBasis p' s'"], "goal": "(l ⊓ l').HasBasis (fun i => p i.fst ∧ p' i.snd) fun i => s i.fst ∩ s' i.snd"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "P : Finpartition Finset.univ", "G : SimpleGraph α", "DecidableRel G.Adj", "ε : ℝ", "Nonempty α", "hP : P.IsEquipartition", "hP₇ : 7 ≤ P.parts.card", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPG : ¬P.IsUniform G ε", "hε₀ : 0 ≤ ε", "hε₁ : ε ≤ 1"], "goal": "↑(P.energy G) + ε ^ 5 / 4 ≤ ↑((SzemerediRegularity.increment hP G ε).energy G)"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "M : MvPolynomial σ R → Prop", "p : MvPolynomial σ R", "h_C : ∀ (a : R), M (MvPolynomial.C a)", "h_add_weak :\n  ∀ (a : σ →₀ ℕ) (b : R) (f : (σ →₀ ℕ) →₀ R),\n    a ∉ f.support →\n      b ≠ 0 →\n        M f →\n          M ((MvPolynomial.monomial a) b) →\n            M\n              ((let_fun this := (MvPolynomial.monomial a) b;\n                this) +\n                f)", "h_X : ∀ (p : MvPolynomial σ R) (n : σ), M p → M (p * MvPolynomial.X n)"], "goal": "M p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : MeasurableSpace α", "inst✝¹ : TopologicalSpace α", "μ : Measure α", "inst✝ : μ.WeaklyRegular", "x : ℝ≥0∞", "hx : x ≠ ⊤", "this : (x • μ).OuterRegular"], "goal": "(x • μ).WeaklyRegular"}}
+{"state": {"context": ["α : Type u_2", "CancelCommMonoidWithZero α", "self : GCDMonoid α", "a b : α"], "goal": "Associated (gcd a b * lcm a b) (a * b)"}}
+{"state": {"context": ["G : Type u_1", "Group G", "p : ℕ", "P : Sylow p G", "hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P", "Fact (Nat.Prime p)", "Finite (Sylow p G)", "(↑P).FiniteIndex"], "goal": "⇑((MonoidHom.transferSylow P hP).restrict ↑P) = fun x => x ^ (↑P).index"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : TopologicalSpace α", "β : Type u_2", "inst✝² : Preorder β", "f✝ g : α → β", "x : α", "s t : Set α", "y z : β", "ι : Sort u_3", "δ : Type u_4", "δ' : Type u_5", "inst✝¹ : CompleteLinearOrder δ", "inst✝ : ConditionallyCompleteLinearOrder δ'", "f : ι → α → δ'", "bdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i => f i y)", "h : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x", "h✝ : IsEmpty ι"], "goal": "LowerSemicontinuousWithinAt (fun x' => ⨆ i, f i x') s x"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : CompleteSpace E", "f : ℝ → E", "c ca cb : E", "l l' la la' lb lb' : Filter ℝ", "lt : Filter ι", "a b z : ℝ", "u v ua ub va vb : ι → ℝ", "inst✝¹ : FTCFilter a la la'", "inst✝ : FTCFilter b lb lb'", "hab : IntervalIntegrable f volume a b", "hmeas : StronglyMeasurableAtFilter f la' volume", "hf : Tendsto f (la' ⊓ ae volume) (𝓝 c)", "hu : Tendsto u lt la", "hv : Tendsto v lt la"], "goal": "(fun t => ((∫ (x : ℝ) in v t..b, f x) - ∫ (x : ℝ) in u t..b, f x) + (v t - u t) • c) =o[lt] (v - u)"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝¹ : HasCoequalizers C", "R S T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)"], "goal": "(α_ (𝟙_ C) P.X Q.X).inv ≫ (λ_ P.X).hom ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) = (λ_ (P.X ⊗ Q.X)).hom ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y : F", "h : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫_ℝ + ‖y‖ ^ 2"], "goal": "‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_5", "Semiring R", "α' : Type u_7", "M' : Type u_8", "AddCommMonoid M'", "Module R M'", "v' : α' → M'", "f : α → α'"], "goal": "Finsupp.total α' M' R v' ∘ₗ Finsupp.lmapDomain R R f = Finsupp.total α M' R (v' ∘ f)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "p q : Multiset (Associates α)", "s t : Multiset α", "hs : ∀ a ∈ Multiset.map (Quot.mk Setoid.r) t, Irreducible a", "ht : ∀ a ∈ Multiset.map (Quot.mk Setoid.r) s, Irreducible a", "eq : (Multiset.map (Quot.mk Setoid.r) t).prod = (Multiset.map (Quot.mk Setoid.r) s).prod"], "goal": "Multiset.map (Quot.mk Setoid.r) t = Multiset.map (Quot.mk Setoid.r) s"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "∀ {x : Fintype G} {p : ℕ} [hp : Fact (Nat.Prime p)], Fintype.card G = p → ∀ {g : G}, g ≠ 1 → Submonoid.powers g = ⊤"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ", "hf : 0 ≤ᵐ[μ] f", "hfm : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "∫ (a : α), f a ∂μ = (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal"}}
+{"state": {"context": ["R : Type u", "CommRing R", "s f : R"], "goal": "(AlgebraicGeometry.StructureSheaf.toBasicOpen R s) ((algebraMap R (Localization.Away s)) f) =\n  AlgebraicGeometry.StructureSheaf.const R f 1 (PrimeSpectrum.basicOpen s) ⋯"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SeminormedAddCommGroup α", "SeminormedAddCommGroup β", "x y : WithLp 2 (α × β)"], "goal": "nndist x y = NNReal.sqrt (nndist x.1 y.1 ^ 2 + nndist x.2 y.2 ^ 2)"}}
+{"state": {"context": ["n : Type u_1", "p : Type u_2", "q : Type u_3", "l : Type u_4", "R : Type u₂", "inst✝⁶ : DecidableEq n", "inst✝⁵ : DecidableEq p", "inst✝⁴ : DecidableEq q", "inst✝³ : DecidableEq l", "inst✝² : CommRing R", "inst✝¹ : Fintype n", "inst✝ : Nontrivial R", "h : 1 < Fintype.card n", "j i : n", "hij : j ≠ i"], "goal": "¬IsLieAbelian ↥(sl n R)"}}
+{"state": {"context": ["n : ℕ", "i : ZMod n"], "goal": "i.cast = i"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Preadditive C", "α : Type u_3", "AddRightCancelSemigroup α", "One α", "K : HomologicalComplex (CategoryTheory.Idempotents.Karoubi C) (ComplexShape.up α)", "n : α"], "goal": "((CategoryTheory.Idempotents.karoubiCochainComplexEquivalence C α).inverse.obj K).X.X n = (K.X n).X"}}
+{"state": {"context": ["s : List Char", "i b e n : Nat"], "goal": "∀ (c : Char) (cs : List Char), go₂ (c :: cs) { byteIdx := b + n } { byteIdx := e + n } = go₂ (c :: cs) { byteIdx := b } { byteIdx := e }"}}
+{"state": {"context": ["X : Scheme", "𝒰 : X.OpenCover", "x y z : 𝒰.J"], "goal": "(fun x y z => 𝒰.gluedCoverT' x y z) x y z ≫ pullback.snd ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y z) ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y x) = pullback.fst ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x y) ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x z) ≫ (fun x y => (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom) x y"}}
+{"state": {"context": ["ι : Type u_2", "α : Type u_3", "β : Type u_4", "AddCommMonoid α", "AddCommMonoid β", "s : Multiset ι", "r : α → β → Prop", "f : ι → α", "g : ι → β", "h₁ : r 0 0", "h₂ : ∀ ⦃a : ι⦄ ⦃b : α⦄ ⦃c : β⦄, r b c → r (f a + b) (g a + c)"], "goal": "r (Multiset.map f s).sum (Multiset.map g s).sum"}}
+{"state": {"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝⁴ : Category.{max v u, w} D", "inst✝³ : ConcreteCategory D", "inst✝² : PreservesLimits (forget D)", "inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "X : C", "P : Cᵒᵖ ⥤ D", "x : (forget D).obj (P.obj (op X))"], "goal": "(Meq.equiv P ⊤) ((Multiequalizer.lift (⊤.index P) (P.obj (op X)) (fun I => P.map I.f.op) ⋯) x) = (Meq.equiv P ⊤) ((Meq.equiv P ⊤).symm (Meq.mk ⊤ x))"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "C : Type uC", "D : Type uD", "E : Type uE", "F : Type uF", "M : Type uM", "ι : Type uι", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "b : Basis ι R M", "a : A", "i : ι"], "goal": "(basis A b).repr.symm (Finsupp.single i a) = a ⊗ₜ[R] b.repr.symm (Finsupp.single i 1)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : IsDomain B", "inst✝² : Algebra A B", "K : Type u_6", "inst✝¹ : Field K", "inst✝ : Algebra A S", "pb : PowerBasis A S", "p : A[X]", "ne_zero : p ≠ 0", "root : (aeval pb.gen) p = 0", "hlt : p.natDegree < pb.dim"], "goal": "∑ i : Fin pb.dim, (monomial ↑i) (p.coeff ↑i) = 0"}}
+{"state": {"context": ["K : Type u", "Field K", "L : Type u_1", "J : Type u_2", "Field L", "Field J", "A : ValuationSubring J", "g : L →+* J", "f : K →+* L"], "goal": "(A.comap g).comap f = A.comap (g.comp f)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "p q : R[X]", "r : R"], "goal": "(p.comp q).IsRoot r ↔ p.IsRoot (Polynomial.eval r q)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrderedField α", "inst✝¹ : Ring β", "abv : β → α", "inst✝ : IsAbsoluteValue abv", "f : ℕ → β", "g : CauSeq β abv", "h : ∀ ε > 0, ∃ i, ∀ j ≥ i, abv (f j - ↑g j) < ε", "ε : α", "ε0 : ε > 0", "i : ℕ", "hi : ∀ j ≥ i, abv (f j - ↑g j) < ε / 2 / 2 ∧ ∀ k ≥ j, abv (↑g k - ↑g j) < ε / 2", "j : ℕ", "ij : j ≥ i"], "goal": "abv (f j - f i) < ε"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a b : α", "h : a < b"], "goal": "IsGreatest (Set.Ioc a b) b"}}
+{"state": {"context": ["F : Type u", "K✝¹ : Type v", "L : Type w", "inst✝³ : Field K✝¹", "inst✝² : Field L", "inst✝¹ : Field F", "n✝ : ℕ", "K✝ : Type u", "inst✝ : Field K✝", "n : ℕ", "ih : (fun n => ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Splits (algebraMap K (SplittingFieldAux n f)) f) n", "K : Type u", "x✝ : Field K", "f : K[X]", "hf : f.natDegree = n.succ"], "goal": "Splits (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)) ((X - C (AdjoinRoot.root f.factor)) * f.removeFactor)"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "inst✝ : Nontrivial R", "ι : Type u_1", "f : ι → R", "s : Finset ι", "x y : ι", "hfs : ((X - C (f y)) ^ 2 * ∏ x ∈ (s.erase x).erase y, (X - C (f x))).Separable", "hx : x ∈ s", "hy : y ∈ s", "hfxy : f x = f y", "hxy : ¬x = y"], "goal": "False"}}
+{"state": {"context": ["k : Type u", "inst✝¹ : Field k", "inst✝ : IsAlgClosed k", "x : k"], "goal": "∃ z, x = z * z"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : CompleteSpace E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "μ ν : Measure α", "s t : Set α", "f g : α → ℝ≥0∞", "hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤", "hμ : μ ≠ 0"], "goal": "⨍⁻ (x : α), f x ∂μ < ⊤"}}
+{"state": {"context": ["R : Type u", "CommRing R", "M N P : Type u", "AddCommGroup M", "AddCommGroup N", "AddCommGroup P", "Module R M", "Module R N", "Module R P", "Coalgebra R M", "Coalgebra R N", "Coalgebra R P"], "goal": "Coalgebra.counit = Coalgebra.counit"}}
+{"state": {"context": ["a x : ℝ"], "goal": "↑(HurwitzZeta.oddKernel (↑a) x) = HurwitzZeta.jacobiTheta₂'' (↑a) (Complex.I * ↑x)"}}
+{"state": {"context": ["F : Type u_2", "E : Type u_3", "CommRing F", "Ring E", "Algebra F E"], "goal": "Module.rank F ↥⊤ = Module.rank F E"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E"], "goal": "↑𝓘(𝕜, E) = id"}}
+{"state": {"context": ["α : Type u_1", "Neg α", "Small.{u_2, u_1} α", "x : α"], "goal": "(equivShrink α) (-x) = -(equivShrink α) x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D"], "goal": "∀ {X Y : CategoryTheory.Functor C D} (τ : X ⟶ Y) (f : CategoryTheory.Arrow C),\n  (((CategoryTheory.Functor.mapArrowFunctor C D).map τ).app f).right = τ.app f.right"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : AddCommMonoid M", "inst✝² : AddCommMonoid N", "inst✝¹ : Module R M", "inst✝ : Module R N", "M₁ M₂ : Submodule R M", "N₁ N₂ : Submodule R N", "s : Set (M ⊗[R] N)", "hs : s.Finite", "w✝ : Submodule R M", "N' : Submodule R N", "left✝ : Module.Finite R ↥w✝", "hfin : Module.Finite R ↥N'", "h : s ⊆ ↑(LinearMap.range (mapIncl w✝ N'))"], "goal": "∃ N', Module.Finite R ↥N' ∧ s ⊆ ↑(LinearMap.range (LinearMap.lTensor M N'.subtype))"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : C", "f : X ⟶ Y", "Z : C"], "goal": "(CategoryTheory.MonoidalCategory.whiskerRight f Z).mop =\n  CategoryTheory.MonoidalCategory.whiskerLeft { unmop := Z } f.mop"}}
+{"state": {"context": ["α : Type u_1", "p : Multiset α → Prop", "S : Multiset α", "h₁ : p 0", "h₂ : ∀ {a : α} {s : Multiset α}, a ∈ S → s ⊆ S → p s → p (insert a s)"], "goal": "p S"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "K : Type u_2", "inst✝⁴ : Field K", "inst✝³ : Algebra R K", "inst✝² : IsFractionRing R K", "inst✝¹ : IsDomain R", "inst✝ : IsIntegrallyClosed R", "p q : R[X]", "hp : p.Monic", "hq : q.Monic", "r : K[X]", "hr : Polynomial.map (algebraMap R K) p = Polynomial.map (algebraMap R K) q * r", "d' : R[X]", "hr' : Polynomial.map (algebraMap R K) d' = r"], "goal": "q ∣ p"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_4", "E : B → Type u_6", "NontriviallyNormedField 𝕜", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "TopologicalSpace (Bundle.TotalSpace F E)", "(x : B) → TopologicalSpace (E x)", "EB : Type u_7", "NormedAddCommGroup EB", "NormedSpace 𝕜 EB", "HB : Type u_8", "TopologicalSpace HB", "IB : ModelWithCorners 𝕜 EB HB", "TopologicalSpace B", "ChartedSpace HB B", "FiberBundle F E", "x : Bundle.TotalSpace F E", "y : EB × F", "hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target"], "goal": "writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) x (↑(trivializationAt F E x.proj)) y = y"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "a₃ b c : R", "a₁ : R", "a₂ : ℕ"], "goal": "a₃ * b = c → a₁ ^ a₂ * a₃ * b = a₁ ^ a₂ * c"}}
+{"state": {"context": ["R : Type u", "Ring R", "s : Subring R", "x y : R"], "goal": "x ∈ s → y ∈ s → x * y ∈ s"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "IsModularLattice α", "x : α", "y : α", "z : α", "h : x ≤ z"], "goal": "(x ⊔ y) ⊓ z = x ⊔ y ⊓ z"}}
+{"state": {"context": ["n : Int", "d : Nat", "z : d ≠ 0", "n' : Int", "d' : Nat", "z' : d' ≠ 0", "c : n'.natAbs.Coprime d'", "h : Rat.normalize n d z = { num := n', den := d', den_nz := z', reduced := c }"], "goal": "∃ m, m ≠ 0 ∧ n = n' * ↑m ∧ d = d' * m"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "h₁ : α = α := ⋯", "h₂ : HEq r r := ⋯"], "goal": "RelIso.cast h₁ h₂ = RelIso.refl r"}}
+{"state": {"context": ["x y : ℝ≥0"], "goal": "↑x ≠ ↑y ↔ x ≠ y"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "l : Filter α", "p : ι → Prop", "s : ι → Set α", "t : Set α", "i : ι", "hl : l.HasBasis p s", "hi : p i", "ht : s i ⊆ t"], "goal": "t ∈ l"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "SmoothManifoldWithCorners I M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "SmoothManifoldWithCorners I' M'", "s : Set M", "hs : UniqueMDiffOn I s", "e : PartialHomeomorph M M'", "he : PartialHomeomorph.MDifferentiable I I' e"], "goal": "UniqueMDiffOn I' (e.target ∩ ↑e.symm ⁻¹' s)"}}
+{"state": {"context": ["Δ₁ Δ₂ : SimplexCategoryᵒᵖ", "A : SimplicialObject.Splitting.IndexSet Δ₁", "p : Δ₁ ⟶ Δ₂", "CategoryTheory.Epi p.unop"], "goal": "(A.epiComp p).fst = A.fst"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n ν : ℕ", "h : ν ≤ n"], "goal": "bernsteinPolynomial R n ν = (bernsteinPolynomial R n (n - ν)).comp (1 - X)"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "CommRing R", "x : WittVector p R"], "goal": "WittVector.frobenius (WittVector.verschiebung x) = x * ↑p"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q r : ℚ", "hq : q ≠ 0", "hr : r ≠ 0"], "goal": "padicValRat p (q / r) = padicValRat p q - padicValRat p r"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "M : Type u_3", "R : Type u_4", "A : Type u_5", "inst✝¹ : Monoid M", "inst✝ : AddMonoid A", "s✝ t u : Set M", "s : Submonoid M", "x : M"], "goal": "x ∈ ↑s * ↑s → x ∈ ↑s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Ring S", "inst✝⁸ : Algebra R S", "K : Type u_4", "L : Type u_5", "F : Type u_6", "inst✝⁷ : Field K", "inst✝⁶ : Field L", "inst✝⁵ : Field F", "inst✝⁴ : Algebra K L", "inst✝³ : Algebra K F", "ι : Type w", "E : Type u_7", "inst✝² : Field E", "inst✝¹ : Algebra K E", "inst✝ : Algebra R F", "pb : PowerBasis R S", "hE : Splits (algebraMap R F) (minpoly R pb.gen)", "hfx : IsSeparable R pb.gen", "this : DecidableEq F := Classical.decEq F"], "goal": "(algebraMap R F) ((norm R) pb.gen) = ∏ σ : S →ₐ[R] F, σ pb.gen"}}
+{"state": {"context": ["p : ℝ≥0∞", "α : Type u_2", "β : Type u_3", "x : α × β"], "goal": "((WithLp.equiv p (α × β)).symm x).1 = x.1"}}
+{"state": {"context": ["α : Type u", "CancelMonoid α", "a : α"], "goal": "IsConj a 1 ↔ a = 1"}}
+{"state": {"context": ["z✝ w✝ : ℍ", "r R : ℝ", "z w : ℍ"], "goal": "Real.cosh (dist z w) = ((z.re - w.re) ^ 2 + z.im ^ 2 + w.im ^ 2) / (2 * z.im * w.im)"}}
+{"state": {"context": ["C : Type u_1", "ι : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j : ι", "hij : c.Rel i j", "CategoryTheory.CategoryWithHomology C"], "goal": "(HomologicalComplex.HomologySequence.composableArrows₃ K i j).Exact"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "h : λ ^ (3 * S.multiplicity - 2 + 1 + 1) * λ ∣ (λ ^ S.multiplicity * FermatLastTheoremForThreeGen.Solution.w S) ^ 3"], "goal": "False"}}
+{"state": {"context": ["F : Type u_1", "Field F", "x y : F", "hxy : x ≠ y"], "goal": "Lagrange.basisDivisor x y + Lagrange.basisDivisor y x = 1"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α"], "goal": "(𝓤 α).HasBasis (fun x => True) fun n => {p | edist p.1 p.2 < 2⁻¹ ^ n}"}}
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+{"state": {"context": ["a : Int"], "goal": "a < 0 → ∃ n, a = Int.negSucc n"}}
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+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X", "h : Dense s"], "goal": "DenseRange Subtype.val"}}
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+{"state": {"context": ["G : Type u_1", "AddGroup G", "H K : AddSubgroup G", "h : H ≤ K"], "goal": "K.index ∣ H.index"}}
+{"state": {"context": ["α : Type u_2", "SemilatticeInf α", "s : Set α", "hs : IsLowerSet s"], "goal": "InfClosed s"}}
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+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "g : E → F", "c : ℝ≥0", "f : ↥(Lp ℝ p μ)", "a : α", "h : ↑↑(negPart f) a = max (-↑↑f a) 0"], "goal": "↑↑(negPart f) a = -min (↑↑f a) 0"}}
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+{"state": {"context": ["R : Type u_1", "CommSemiring R", "S : Type u_2", "CommSemiring S", "Algebra R S", "x : R", "IsLocalization.Away x S"], "goal": "(algebraMap R S) x * IsLocalization.Away.invSelf x = 1"}}
+{"state": {"context": ["a b : ℤ", "h : a ≤ b"], "goal": "↑(Ioc a b).card = b - a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : Filter α", "f : α → β", "lb : Filter β"], "goal": "(↑f).Tendsto lb ↔ Filter.Tendsto f l lb"}}
+{"state": {"context": ["X : Type u_1", "α : Type u_2", "α' : Type u_3", "β : Type u_4", "γ : Type u_5", "δ : Type u_6", "M : Type u_7", "E : Type u_8", "R : Type u_9", "inst✝⁴ : TopologicalSpace α", "inst✝³ : TopologicalSpace α'", "inst✝² : One β", "inst✝¹ : One γ", "inst✝ : One δ", "g : β → γ", "f : α → β", "f₂ : α → γ", "m : β → γ → δ", "x : α"], "goal": "x ∉ mulTSupport f ↔ f =ᶠ[𝓝 x] 1"}}
+{"state": {"context": ["R : Type u_1", "LinearOrderedSemifield R", "FloorSemiring R", "b : ℕ", "hb : 1 < b", "x : ℤ", "r : R", "hr : 0 < r"], "goal": "r < ↑b ^ x ↔ Int.log b r < x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s s' : Finset α", "t t' : Finset β", "a : α", "b✝ b : β", "x : α", "y : β"], "goal": "(x, y) ∈ s ×ˢ {b} ↔ (x, y) ∈ map { toFun := fun i => (i, b), inj' := ⋯ } s"}}
+{"state": {"context": ["α : Type u_1", "DivisionCommMonoid α", "a b c : α"], "goal": "a / (b * c) = a / b * (1 / c)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "ι : Type v", "M : ι → Type w", "(i : ι) → AddCommMonoid (M i)", "(i : ι) → Module R (M i)", "b : R", "v : DirectSum ι fun i => M i", "i : ι"], "goal": "(b • v) i = b • v i"}}
+{"state": {"context": ["L : Type u_1", "TopologicalSpace L", "ι : Type u_3", "α : Type u_4", "s : Finset ι", "f : ι → α → L", "l : Filter α", "g : ι → L", "SemilatticeSup L", "OrderBot L", "ContinuousSup L", "hs : ∀ i ∈ s, Filter.Tendsto (f i) l (𝓝 (g i))"], "goal": "Filter.Tendsto (fun a => s.sup fun x => f x a) l (𝓝 (s.sup g))"}}
+{"state": {"context": ["n m : Nat", "h : n ≤ m", "k : Nat"], "goal": "n - k ≤ m - k"}}
+{"state": {"context": [], "goal": "[].TFAE"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "x : R"], "goal": "IsCoprime x 0 ↔ IsUnit x"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "CommRing R", "LieRing L", "LieAlgebra R L", "LieRing L'", "LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I₁ I₂ : LieIdeal R L"], "goal": "LieIdeal.map f ⁅I₁, I₂⁆ ≤ ⁅LieIdeal.map f I₁, LieIdeal.map f I₂⁆"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "M : Type u_3", "N : Type u_6", "Monoid R", "Monoid S", "σ : R → S", "MulAction R M", "MulAction S N", "F : Type u_7", "h : F", "FunLike F M N", "MulActionSemiHomClass F σ M N", "c : R", "t : Set N", "hc : IsUnit c", "hc' : IsUnit (σ c)"], "goal": "⇑h ⁻¹' (σ c • t) = c • ⇑h ⁻¹' t"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "ι : Sort u_3", "p : ι → Prop", "s : ι → Set α", "h : l.HasBasis p s"], "goal": "l.Subsingleton ↔ ∃ i, p i ∧ (s i).Subsingleton"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X Y : SimplicialObject C", "n : ℕ", "P : Type := Fin (n + 2) × Fin (n + 3)", "S : Finset P := Finset.filter (fun ij => ↑ij.2 ≤ ↑ij.1) Finset.univ", "φ : (ij : P) → ij ∈ S → P := fun ij hij => (ij.2.castLT ⋯, ij.1.succ)"], "goal": "∀ b ∈ Sᶜ, ∃ a, ∃ (ha : a ∈ S), φ a ha = b"}}
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+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "TopologicalSpace α", "μ : MeasureTheory.Measure α", "R1Space α", "OpensMeasurableSpace α", "h : μ.InnerRegular"], "goal": "μ.InnerRegularWRT IsClosed IsOpen"}}
+{"state": {"context": ["p : Prop", "Decidable p"], "goal": "p → decide p = true"}}
+{"state": {"context": ["ι : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "Z : Type u_5", "inst✝³ : TopologicalSpace B", "inst✝² : TopologicalSpace F", "proj : Z → B", "inst✝¹ : TopologicalSpace Z", "inst✝ : TopologicalSpace (TotalSpace F E)", "e✝ : Trivialization F proj", "x✝ : Z", "e' : Trivialization F TotalSpace.proj", "x' : TotalSpace F E", "b✝ : B", "y : E b✝", "e : Trivialization F proj", "b : B", "h : b ∈ e.baseSet", "x : F"], "goal": "e.coordChange e b x = x"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "s✝ : Set E", "r : ℝ", "hr : r ≠ 0", "x y : E", "s t : Set E"], "goal": "ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ ⊤"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "AddMonoid α", "n : ℕ", "hn : n ≠ 0"], "goal": "n • ∅ = ∅"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "X : C", "S : J.Cover X", "T : (I : S.Arrow) → J.Cover I.Y", "I : (S.bind T).Arrow"], "goal": "S.sieve.arrows I.fromMiddleHom"}}
+{"state": {"context": ["R : Type u", "a b : R", "m n : ℕ", "inst✝ : Semiring R", "p q : R[X]"], "goal": "p * q = ∑ i ∈ p.support, q.sum fun j a => (monomial (i + j)) (p.coeff i * a)"}}
+{"state": {"context": ["α : Type u_2", "SemilatticeInf α", "OrderTop α"], "goal": "¬InfPrime ⊤"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "x y : FreeCommRing α", "s✝ t : Set α", "i✝ : ℤ", "s : Set α", "i : ℕ", "hi : (↑↑i).IsSupported s"], "goal": "(↑↑i + 1).IsSupported s"}}
+{"state": {"context": ["K : Type uK", "Field K", "V₁ : Type uV₁", "AddCommGroup V₁", "Module K V₁"], "goal": "(Module.dualPairing K V₁).Nondegenerate"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "hf : S.f = 0", "S.HasHomology"], "goal": "CategoryTheory.CategoryStruct.comp (S.asIsoHomologyπ hf).inv S.homologyπ = CategoryTheory.CategoryStruct.id S.homology"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "D₁ : Type u_3", "D₂ : Type u_4", "inst✝⁷ : Category.{u_6, u_1} C₁", "inst✝⁶ : Category.{u_8, u_2} C��", "inst✝⁵ : Category.{u_5, u_3} D₁", "inst✝⁴ : Category.{u_7, u_4} D₂", "G : C₁ ⥤ C₂", "F : C₂ ⥤ C₁", "adj : G ⊣ F", "L₁ : C₁ ⥤ D₁", "W₁ : MorphismProperty C₁", "inst✝³ : L₁.IsLocalization W₁", "L₂ : C₂ ⥤ D₂", "W₂ : MorphismProperty C₂", "inst✝² : L₂.IsLocalization W₂", "G' : D₁ ⥤ D₂", "F' : D₂ ⥤ D₁", "inst✝¹ : CatCommSq G L₁ L₂ G'", "inst✝ : CatCommSq F L₂ L₁ F'", "X₁ : C₁", "this : Lifting L₁ W₁ ((G ⋙ F) ⋙ L₁) (G' ⋙ F') := { iso' := CatCommSq.iso'.symm }"], "goal": "(ε adj L₁ W₁ L₂ G' F').app (L₁.obj X₁) = L₁.map (adj.unit.app X₁) ≫ (CatCommSq.iso F L₂ L₁ F').hom.app (G.obj X₁) ≫ F'.map ((CatCommSq.iso G L₁ L₂ G').hom.app X₁)"}}
+{"state": {"context": ["α : Type u_1", "f : ℕ → ℝ≥0", "k : ℕ", "a : ℝ≥0"], "goal": "HasSum (fun n => ↑(f n)) (↑a + ∑ i ∈ Finset.range k, ↑(f i)) ↔ HasSum f (a + ∑ i ∈ Finset.range k, f i)"}}
+{"state": {"context": ["R : Type u_1", "CommMonoidWithZero R", "χ : DirichletCharacter R 1"], "goal": "χ = 1"}}
+{"state": {"context": ["β : Type v", "π : β → Type u_2", "Fintype β", "(b : β) → PseudoEMetricSpace (π b)", "s : (b : β) → Set (π b)", "c : ℝ≥0∞", "h : ∀ (b : β), EMetric.diam (s b) ≤ c"], "goal": "EMetric.diam (Set.univ.pi s) ≤ c"}}
+{"state": {"context": ["ι₁ : Type u_4", "ι₂ : Type u_5", "β₁ : ι₁ → Type u_6", "β₂ : ι₂ → Type u_7", "(i₁ : ι₁) → UniformSpace (β₁ i₁)", "(i₂ : ι₂) → UniformSpace (β₂ i₂)", "e : ι₁ ≃ ι₂", "F : (i₁ : ι₁) → β₁ i₁ ≃ᵤ β₂ (e i₁)"], "goal": "(UniformEquiv.piCongr e F).toEquiv = (Equiv.piCongrRight fun i => (F i).toEquiv).trans (Equiv.piCongrLeft β₂ e)"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "l : Filter α", "CountableInterFilter l", "β : Type u_2", "MeasurableSpace β", "f g : α → β", "hf : Measurable f", "hgf : g =ᶠ[l] f"], "goal": "EventuallyMeasurable m l g"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "X : Type u_3", "Y : Type u_4", "Z : Type u_5", "inst✝ : TopologicalSpace X", "f g : X → Y", "A : Set X", "x : X", "h : ∀ (x : X), (↑f).IsConstant", "s : Set Y", "a : X", "ha : a ∈ f ⁻¹' s", "b : Y", "hb : f =ᶠ[𝓝 a] fun x => b"], "goal": "f ⁻¹' s ∈ 𝓝 a"}}
+{"state": {"context": ["ι : Type u", "β : ι → Type v", "ι₁ : Type u₁", "DecidableEq ι₁", "β₁ : ι₁ → Type v₁", "(i₁ : ι₁) → Zero (β₁ i₁)", "(i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)", "(i : ι) → AddCommMonoid (β i)", "f : Π₀ (i₁ : ι₁), β₁ i₁", "g : (i₁ : ι₁) → β₁ i₁ → Π₀ (i : ι), β i", "i₂ : ι"], "goal": "(f.sum g) i₂ = f.sum fun i₁ b => (g i₁ b) i₂"}}
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+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X", "hs : IsCompact s", "f : Filter X", "hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f"], "goal": "sᶜ ∈ f"}}
+{"state": {"context": ["α : Type u_1", "f : α → α", "x : α", "m : ℕ", "hm : Function.IsPeriodicPt f m x", "n : ℕ"], "goal": "Function.IsPeriodicPt f (m * n) x"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "Semiring R", "φ : ι → Type u_3", "(i : ι) → AddCommMonoid (φ i)", "(i : ι) → Module R (φ i)", "DecidableEq ι", "i : ι"], "goal": "LinearMap.proj i ∘ₗ LinearMap.stdBasis R φ i = LinearMap.id"}}
+{"state": {"context": ["α : Type u_1", "EMetricSpace α", "s : Set α", "hs : s.Finite"], "goal": "∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y"}}
+{"state": {"context": ["n : Type u_2", "𝕜 : Type u_4", "Fintype n", "RCLike 𝕜", "DecidableEq n", "A : Matrix n n 𝕜", "hA : A.PosSemidef", "x : n → 𝕜"], "goal": "(((Matrix.toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ (Matrix.toLin' A) x = 0"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s t : LowerSet α"], "goal": "↑s ⊂ ↑t ↔ s < t"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x✝ y z : X", "ι : Type u_3", "F : Set X", "p : ι → Prop", "s : X → ι → Set X", "h : ∀ (x : X), (𝓝 x).HasBasis p (s x)", "h' : ∀ (x : X) (i : ι), p i → IsPathConnected (s x i)", "x : X"], "goal": "∀ (i' : Set X), i' ∈ 𝓝 x ∧ IsPathConnected i' → ∃ i, p i ∧ s x i ⊆ id i'"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹² : LinearOrderedCommRing 𝕜", "inst✝¹¹ : LinearOrderedCommRing E", "inst✝¹⁰ : LinearOrderedAddCommGroup F", "inst✝⁹ : Module 𝕜 E", "inst✝⁸ : Module 𝕜 F", "inst✝⁷ : Module E F", "inst✝⁶ : IsScalarTower 𝕜 E F", "inst✝⁵ : SMulCommClass 𝕜 E F", "inst✝⁴ : OrderedSMul 𝕜 E", "inst✝³ : OrderedSMul 𝕜 F", "inst✝² : OrderedSMul E F", "s : Set 𝕜", "f✝ : 𝕜 → E", "g✝ : 𝕜 → F", "inst✝¹ : IsScalarTower 𝕜 E E", "inst✝ : SMulCommClass 𝕜 E E", "f g : 𝕜 → E", "n : ℕ"], "goal": "ConvexOn 𝕜 univ fun x => x ^ (n + n)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Semiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝¹ : Fintype m", "inst✝ : DecidableEq m", "M : Matrix m n R", "this : DecidableEq m := Classical.decEq m"], "goal": "span R (Set.range fun x => Pi.single x 1 ᵥ* M) = span R (Set.range M)"}}
+{"state": {"context": ["a b : Prop"], "goal": "¬(a → b) → ¬¬a"}}
+{"state": {"context": ["α : Type u_2", "G : Type u_4", "TopologicalSpace α", "Group G", "MulAction G α", "ContinuousConstSMul G α", "c : G"], "goal": "IsOpenMap fun x => c • x"}}
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+{"state": {"context": ["n : ℕ", "α : Type u_1", "j : Fin (n + 1)", "op : α → α → α", "g : Fin (n + 1) → α", "k : Fin n", "h : ↑j < ↑k"], "goal": "j.contractNth op g k = g k.succ"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : PseudoEMetricSpace α", "δ : ℝ", "δ_pos : 0 < δ", "E : Set α", "f : α → ℝ≥0 × ℝ≥0∞ := fun x => (1, infEdist x E / ENNReal.ofReal δ)", "sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := fun p => ↑p.1 - p.2"], "goal": "Continuous fun x => (f x).2"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "inst✝ : LinearOrderedField α", "r : ℝ", "ha : 0 ≤ r", "f : ι → ℝ"], "goal": "(⨅ i, f i) * r = ⨅ i, f i * r"}}
+{"state": {"context": ["A : Type u_5", "NormedRing A", "a b : ℝ", "μ : MeasureTheory.Measure ℝ", "f : ℝ → A", "hf : IntervalIntegrable f μ a b", "c : A"], "goal": "IntervalIntegrable (fun x => f x * c) μ a b"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "a✝ a : α", "ha : a ∉ s"], "goal": "s ⋖ cons a s ha"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝¹ : HasAffineProperty P Q", "X Y : Scheme", "f : X ⟶ Y", "inst✝ : IsAffine Y", "this✝ : Q.IsLocal := isLocal_affineProperty P", "this : ∀ (i : (Scheme.openCoverOfIsIso (𝟙 Y)).J), IsAffine ((Scheme.openCoverOfIsIso (𝟙 Y)).obj i)"], "goal": "(∀ (i : ((Scheme.openCoverOfIsIso (𝟙 Y)).pullbackCover f).J), Q ((Scheme.openCoverOfIsIso (𝟙 Y)).pullbackHom f i)) ↔ Q f"}}
+{"state": {"context": ["α : Type u_2", "OrderedAddCommMonoid α", "s t : Multiset α", "h : Multiset.Rel (fun x x_1 => x ≤ x_1) s t"], "goal": "s.sum ≤ t.sum"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β"], "goal": "s ×ˢ t = (Prod.mk '' s).seq t"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "X : Type u_4", "Y : Type u_5", "l₁ l₂ : Filter α"], "goal": "l₁ ⤳ l₂ ↔ l₁ ≤ l₂"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "α : Type v", "β : Type w", "ι : Type u_6", "M : Matrix m n α", "i : m", "j : n", "b : n → α", "c : m → α", "inst✝ : DecidableEq m", "i✝ : n", "j✝ : m"], "goal": "(if j✝ = i then b i✝ else Mᵀ i✝ j✝) = if j✝ = i then b i✝ else M j✝ i✝"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_3", "Fintype ι", "(i : ι) → MeasurableSpace (α i)", "μ : (i : ι) → MeasureTheory.Measure (α i)", "∀ (i : ι), MeasureTheory.SigmaFinite (μ i)", "i : ι"], "goal": "Filter.Tendsto (Function.eval i) (MeasureTheory.ae (MeasureTheory.Measure.pi μ)) (MeasureTheory.ae (μ i))"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_2", "AddCommMonoid M", "Module R M", "A : Type u_3", "Semiring A", "Algebra R A", "f : M →ₗ[R] A", "g : TensorAlgebra R M →ₐ[R] A"], "goal": "g.toLinearMap ∘ₗ TensorAlgebra.ι R = f ↔ g = (TensorAlgebra.lift R) f"}}
+{"state": {"context": ["S : Type u_3", "AddGroup S", "Mul S", "t : RingCon S", "a b c d : S", "h : t a b", "h' : t c d"], "goal": "t (a - c) (b - d)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "f✝ : α → β", "n : ℕ", "f : ℕ → α", "hf : ∀ i < n, f i ∈ s", "f_inj : ∀ i < n, ∀ j < n, f i = f j → i = j", "hs : autoParam s.Finite _auto✝"], "goal": "n ≤ s.ncard"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "inst✝³ : Group G", "inst✝² : MulAction G α", "M : Type u_3", "inst✝¹ : Monoid M", "inst✝ : MulAction M α", "s : Set α", "g : G", "s_subset : (fixedBy α g)ᶜ ⊆ s"], "goal": "s ∈ fixedBy (Set α) g⁻¹"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "bg : β ≃ γ", "e : α ≃ β"], "goal": "((Equiv.refl α).equivCongr bg) e = e.trans bg"}}
+{"state": {"context": ["E : Type u_1", "X : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : TopologicalSpace X", "a✝ b✝ b₀✝ b₁ b₂ : ℝ", "μ : Measure ℝ", "f : ℝ → E", "inst✝ : FirstCountableTopology X", "F : X → ℝ → E", "bound : ℝ → ℝ", "a b a₀ b₀ : ℝ", "x₀ : X", "hF_meas : ∀ (x : X), AEStronglyMeasurable (F x) (μ.restrict (Ι a b))", "h_bound : ∀ᶠ (x : X) in 𝓝 x₀, ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ‖F x t‖ ≤ bound t", "bound_integrable : IntervalIntegrable bound μ a b", "h_cont : ∀ᵐ (t : ℝ) ∂μ.restrict (Ι a b), ContinuousAt (fun x => F x t) x₀", "ha₀ : a₀ ∈ Ioo a b", "hb₀ : b₀ ∈ Ioo a b", "hμb₀ : μ {b₀} = 0", "hsub : ∀ {a₀ b₀ : ℝ}, a₀ ∈ Ioo a b → b₀ ∈ Ioo a b → Ι a₀ b₀ ⊆ Ι a b", "Ioo_nhds : Ioo a b ∈ 𝓝 b₀", "Icc_nhds : Icc a b ∈ 𝓝 b₀"], "goal": "ContinuousAt (fun p => ∫ (t : ℝ) in a₀..p.2, F p.1 t ∂μ) (x₀, b₀)"}}
+{"state": {"context": ["α : Type u", "PartialOrder α", "DecidableRel fun x x_1 => x ≤ x_1", "a b : α", "hab : a ≤ b"], "goal": "a = b ∨ a < b"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹⁰ : MeasurableSpace G", "inst✝⁹ : Group G", "inst✝⁸ : MeasurableMul₂ G", "μ ν : Measure G", "inst✝⁷ : SFinite ν", "inst✝⁶ : SFinite μ", "s : Set G", "inst✝⁵ : MeasurableInv G", "inst✝⁴ : μ.IsMulLeftInvariant", "μ' ν' : Measure G", "inst✝³ : SigmaFinite μ'", "inst✝² : SigmaFinite ν'", "inst✝¹ : μ'.IsMulLeftInvariant", "inst✝ : ν'.IsMulLeftInvariant", "h2s : ν' s ≠ 0", "h3s : ν' s ≠ ⊤"], "goal": "∀ᵐ (x : G) ∂μ', ν' ((fun y => y * x) ⁻¹' s) < ⊤"}}
+{"state": {"context": ["M₀ : Type u_1", "G₀ : Type u_2", "inst✝ : GroupWithZero G₀", "s : Set G₀", "a b : G₀"], "goal": "center G₀ˣ ⊆ Units.val ⁻¹' center G₀"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedAddCommGroup α", "OrderedAddCommGroup β", "f₁ f₂ : ι → α", "g : ι → β", "h₁ : Antivary f₁ g", "h₂ : Monovary f₂ g"], "goal": "Antivary (f₁ - f₂) g"}}
+{"state": {"context": ["M : Type u_1", "Add M", "Preorder M", "CovariantClass M M (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "CovariantClass M M (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "s t : Set M", "hs : BddAbove s", "ht : BddAbove t"], "goal": "BddAbove (s + t)"}}
+{"state": {"context": [], "goal": "Filter.Tendsto Real.Wallis.W Filter.atTop (𝓝 (π / 2))"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝ : PseudoEMetricSpace α", "δ✝ : ℝ", "s : Set α", "x : α", "δ : ℝ", "E : Set α"], "goal": "thickening δ (thickening δ E)ᶜ ⊆ Eᶜ"}}
+{"state": {"context": ["A : Type u₁", "inst✝⁵ : Category.{v₁, u₁} A", "B : Type u₂", "inst✝⁴ : Category.{v₂, u₂} B", "T : Type u₃", "inst✝³ : Category.{v₃, u₃} T", "A' : Type u_1", "B' : Type u_2", "T' : Type u_3", "inst✝² : Category.{?u.25322, u_1} A'", "inst✝¹ : Category.{?u.25326, u_2} B'", "inst✝ : Category.{?u.25330, u_3} T'", "L : A ⥤ T", "R : B ⥤ T", "L₁ L₂ L₃ : A ⥤ T", "R₁ R₂ R₃ : B ⥤ T", "L' : A' ⥤ T'", "R' : B' ⥤ T'", "F₁ : A ⥤ A'", "F₂ : B ⥤ B'", "F : T ⥤ T'", "α : F₁ ⋙ L' ⟶ L ⋙ F", "β : R ⋙ F ⟶ F₂ ⋙ R'", "X Y : Comma L R", "φ : X ⟶ Y"], "goal": "α.app X.left ≫ (L ⋙ F).map φ.left ≫ F.map Y.hom ≫ β.app Y.right = α.app X.left ≫ F.map X.hom ≫ (R ⋙ F).map φ.right ≫ β.app Y.right"}}
+{"state": {"context": ["α : Type u", "Semiring α"], "goal": "Ideal.span 1 = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν ν' : Measure α", "inst✝² : SigmaFinite μ", "inst✝¹ : SigmaFinite ν", "inst✝ : SigmaFinite ν'", "hμν' : μ ⟂ₘ ν'", "hνν' : ν ⟂ₘ ν'", "h_ac : ν ≪ ν + ν'", "h₁ : (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae (ν + ν')] (μ.singularPart ν).rnDeriv (ν + ν') + (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν')", "h₂ : (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν =ᶠ[ae ν] (μ.singularPart ν).rnDeriv ν + (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν"], "goal": "(μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν + ν') =ᶠ[ae ν] (μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)).rnDeriv ν"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P", "h : ∡ p₁ p₂ p₃ = ↑(π / 2)", "hs : (∡ p₃ p₁ p₂).sign = 1"], "goal": "(∡ p₃ p₁ p₂).cos * dist p₁ p₃ = dist p₁ p₂"}}
+{"state": {"context": ["x : SetTheory.PGame"], "goal": "⟦1 * x⟧ = ⟦x⟧"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "s t : Submodule R M", "Module.Finite R ↥t", "hst : s ≤ t"], "goal": "FiniteDimensional.finrank R ↥s ≤ FiniteDimensional.finrank R ↥t"}}
+{"state": {"context": ["n : ℕ", "NeZero n", "a : Fin n"], "goal": "↑↑a = a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝²³ : NormedAddCommGroup E", "inst✝²² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝²¹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝²⁰ : TopologicalSpace M", "inst✝¹⁹ : ChartedSpace H M", "Is : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹⁸ : NormedAddCommGroup E'", "inst✝¹⁷ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁶ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹⁵ : TopologicalSpace M'", "inst✝¹⁴ : ChartedSpace H' M'", "I's : SmoothManifoldWithCorners I' M'", "F : Type u_8", "inst✝¹³ : NormedAddCommGroup F", "inst✝¹² : NormedSpace 𝕜 F", "G : Type u_9", "inst✝¹¹ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N : Type u_10", "inst✝¹⁰ : TopologicalSpace N", "inst✝⁹ : ChartedSpace G N", "Js : SmoothManifoldWithCorners J N", "F' : Type u_11", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace 𝕜 F'", "G' : Type u_12", "inst✝⁶ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_13", "inst✝⁵ : TopologicalSpace N'", "inst✝⁴ : ChartedSpace G' N'", "J's : SmoothManifoldWithCorners J' N'", "F₁ : Type u_14", "inst✝³ : NormedAddCommGroup F₁", "inst✝² : NormedSpace 𝕜 F₁", "F₂ : Type u_15", "inst✝¹ : NormedAddCommGroup F₂", "inst✝ : NormedSpace 𝕜 F₂", "f✝ f₁ : M → M'", "s✝ s₁ t : Set M", "x : M", "m n : ℕ∞", "f : H → H'", "s : Set H", "hf : ContMDiffOn I I' n f s", "hmn : m + 1 ≤ n", "hs : UniqueMDiffOn I s", "m_le_n : m ≤ n", "one_le_n : 1 ≤ n", "U' : UniqueDiffOn 𝕜 (range ↑I ∩ ↑I.symm ⁻¹' s)", "p : TangentBundle I H", "q : TangentBundle I' H'"], "goal": "ContDiffOn 𝕜 m (fun p => (↑I' (f (↑I.symm p.1)), (mfderivWithin I I' f s (↑I.symm p.1)) p.2)) ((range ↑I ∩ ↑I.symm ⁻¹' s) ×ˢ univ)"}}
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+{"state": {"context": ["α✝ : Type u_1", "r : α✝ → α✝ → Prop", "α : Type u_2", "h : ∀ x < #α, 2 ^ x < #α"], "goal": "#{ s // #↑s < (#α).ord.cof } = #α"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_7", "M : Type u_8", "M₂ : Type u_10", "Semiring R", "Semiring S", "AddCommMonoid M", "AddCommMonoid M₂", "module_M : Module R M", "module_S_M₂ : Module S M₂", "σ : R →+* S", "σ' : S →+* R", "re₁ : RingHomInvPair σ σ'", "re₂ : RingHomInvPair σ' σ", "e : M ≃ₛₗ[σ] M₂"], "goal": "⇑e.toEquiv = ⇑e"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "V : ℕ → Set (α × α)", "hV : ∀ (n : ℕ), V n ∈ 𝓤 α", "u : ℕ → α", "a : α", "hu : Tendsto u atTop (𝓝 a)", "n : ℕ", "hn : ∀ b ≥ n, b ∈ u ⁻¹' ball a (Prod.swap ⁻¹' V 0)", "φ : ℕ → ℕ", "φ_mono : StrictMono φ", "hφV : ∀ (n_1 : ℕ), ((u ∘ fun a => a + n) (φ (n_1 + 1)), (u ∘ fun a => a + n) (φ n_1)) ∈ V (n_1 + 1)"], "goal": "∃ φ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧ ∀ (n : ℕ), (u (φ (n + 1)), u (φ n)) ∈ V (n + 1)"}}
+{"state": {"context": ["F : Type v'", "R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝¹⁶ : CommSemiring R", "inst✝¹⁵ : StarRing R", "inst✝¹⁴ : NonUnitalSemiring A", "inst✝¹³ : StarRing A", "inst✝¹² : Module R A", "inst✝¹¹ : IsScalarTower R A A", "inst✝¹⁰ : SMulCommClass R A A", "inst✝⁹ : StarModule R A", "inst✝⁸ : NonUnitalSemiring B", "inst✝⁷ : StarRing B", "inst✝⁶ : Module R B", "inst✝⁵ : IsScalarTower R B B", "inst✝⁴ : SMulCommClass R B B", "inst✝³ : StarModule R B", "inst✝² : FunLike F A B", "inst✝¹ : NonUnitalAlgHomClass F R A B", "inst✝ : NonUnitalStarAlgHomClass F R A B", "s : Set A", "p : (x : A) → x ∈ adjoin R s → Prop", "a : A", "ha : a ∈ adjoin R s", "mem : ∀ (x : A) (hx : x ∈ s), p x ⋯", "add : ∀ (x : A) (hx : x ∈ adjoin R s) (y : A) (hy : y ∈ adjoin R s), p x hx → p y hy → p (x + y) ⋯", "zero : p 0 ⋯", "mul : ∀ (x : A) (hx : x ∈ adjoin R s) (y : A) (hy : y ∈ adjoin R s), p x hx → p y hy → p (x * y) ⋯", "smul : ∀ (r : R) (x : A) (hx : x ∈ adjoin R s), p x hx → p (r • x) ⋯", "star : ∀ (x : A) (hx : x ∈ adjoin R s), p x hx → p (Star.star x) ⋯"], "goal": "p a ha"}}
+{"state": {"context": ["n : ℕ", "m : ℤ", "hnpos : 0 < n", "N : ℤ", "D : ℕ", "P : D ≠ 0", "C : N.natAbs.Coprime D", "hxr : ↑{ num := N, den := D, den_nz := P, reduced := C } ^ n = ↑m", "hv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y"], "goal": "False"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : Matrix.SpecialLinearGroup n R", "m : ℕ"], "goal": "↑(A ^ m) = ↑A ^ m"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "TopologicalSpace α", "TopologicalSpace β", "TopologicalSpace γ", "TopologicalSpace δ", "f : SpectralMap γ δ", "g : SpectralMap β γ", "h : SpectralMap α β"], "goal": "(f.comp g).comp h = f.comp (g.comp h)"}}
+{"state": {"context": ["E : Type u_2", "NormedAddCommGroup E", "v : ℝ → E → E", "tMin t₀ tMax : ℝ", "x₀ : E", "L : ℝ≥0", "R C : ℝ", "self : IsPicardLindelof v tMin t₀ tMax x₀ L R C", "x : E"], "goal": "x ∈ Metric.closedBall x₀ R → ContinuousOn (fun t => v t x) (Set.Icc tMin tMax)"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "σ : Equiv.Perm α", "h : σ.IsThreeCycle"], "goal": "σ.cycleType = {3}"}}
+{"state": {"context": ["m n : ℤ"], "goal": "m ≤ n - 1 ↔ m < n"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "P : {a b : α} → Relation.TransGen r a b → Prop", "a b : α", "h : Relation.TransGen r a b", "base : ∀ {a b : α} (h : r a b), P ⋯", "ih : ∀ {a b c : α} (h₁ : Relation.TransGen r a b) (h₂ : Relation.TransGen r b c), P h₁ → P h₂ → P ⋯"], "goal": "P h"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v✝ : ι → M", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a b : R", "x y : M", "v : ι → M", "hv : ∀ (s : Finset ι) (g : ι → R), (∀ i ∉ s, g i = 0) → ∑ i ∈ s, g i • v i = 0 → ∀ (i : ι), g i = 0", "w : ι → Rˣ", "s : Finset ι", "g : ι → R", "hgs : ∀ i ∉ s, g i = 0", "hsum : ∑ i ∈ s, g i • (w • v) i = 0", "i : ι"], "goal": "g i = 0"}}
+{"state": {"context": ["a b : ℤ"], "goal": "Finset.Icc a b = Finset.map (Nat.castEmbedding.trans (addLeftEmbedding a)) (Finset.range (b + 1 - a).toNat)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v, u₁} C", "D : CategoryTheory.GlueData C"], "goal": "D.diagram.L = (D.J × D.J)"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : E → F", "h'f : Function.HasTemperateGrowth (fderiv ℝ f)", "hf : Differentiable ℝ f", "k : ℕ", "C : ℝ", "h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k"], "goal": "Function.HasTemperateGrowth f"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "M : Type u_2", "AddCommGroup M", "R : Type u_3", "CommRing R", "Module R M", "b : ι → M", "DecidableEq ι", "r : R"], "goal": "Matrix.Represents b ((algebraMap R (Matrix ι ι R)) r) ((algebraMap R (Module.End R M)) r)"}}
+{"state": {"context": ["α : Type u_1", "LE α", "a : WithBot α", "b : WithTop αᵒᵈ"], "goal": "a ≤ WithTop.ofDual b ↔ b ≤ WithBot.toDual a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "s : Set M", "Zero M'"], "goal": "SmoothOn I I' 0 s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "W₁ W₂ : CategoryTheory.MorphismProperty C", "data : W₁.FunctorialFactorizationData W₂", "X Y X' Y' : C", "f : X ⟶ Y", "g : X' ⟶ Y'", "φ : CategoryTheory.Arrow.mk f ⟶ CategoryTheory.Arrow.mk g"], "goal": "CategoryTheory.CategoryStruct.comp (data.mapZ φ) (data.factorizationData g).p =\n  CategoryTheory.CategoryStruct.comp (data.factorizationData f).p φ.right"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_4", "Preorder α", "Preorder β", "u : β → α", "l₁ l₂ : LowerAdjoint u"], "goal": "l₁.toFun = l₂.toFun → l₁ = l₂"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "N : Type u_2", "CommMonoid N", "f : S.LocalizationMap N", "x : M", "y : ↥S"], "goal": "(Localization.mulEquivOfQuotient f).symm (f.mk' x y) = Localization.mk x y"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C"], "goal": "(Comon_.Comon_EquivMon_OpOp C).unitIso =\n  CategoryTheory.NatIso.ofComponents (fun x => CategoryTheory.Iso.refl ((𝟭 (Comon_ C)).obj x)) ⋯"}}
+{"state": {"context": ["R : Type u", "inst✝¹⁰ : CommSemiring R", "A : Type u", "inst✝⁹ : CommSemiring A", "inst✝⁸ : Algebra R A", "B : Type u", "inst✝⁷ : Semiring B", "inst✝⁶ : Algebra R B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsScalarTower R A B", "inst✝³ : FormallyUnramified R A", "inst✝² : FormallyUnramified A B", "C : Type u", "inst✝¹ : CommRing C", "inst✝ : Algebra R C", "I : Ideal C", "hI : I ^ 2 = ⊥", "f₁ f₂ : B →ₐ[R] C", "e : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂", "e' : f₁.comp (IsScalarTower.toAlgHom R A B) = f₂.comp (IsScalarTower.toAlgHom R A B)", "this : Algebra A C := (f₁.comp (IsScalarTower.toAlgHom R A B)).toAlgebra", "F₁ : B →ₐ[A] C := { toRingHom := f₁.toRingHom, commutes' := ⋯ }"], "goal": "f₁ = f₂"}}
+{"state": {"context": ["R : Type u", "inst✝¹⁸ : CommRing R", "S : Type v", "inst✝¹⁷ : CommRing S", "f✝ : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝¹⁶ : Algebra R S", "K : Type u_1", "inst✝¹⁵ : Field K", "inst✝¹⁴ : Algebra R K", "hRK : IsFractionRing R K", "L : Type u_2", "inst✝¹³ : Field L", "inst✝¹² : Algebra S L", "inst✝¹¹ : IsFractionRing S L", "V : Type u_3", "V' : Type u_4", "V'' : Type u_5", "inst✝¹⁰ : AddCommGroup V", "inst✝⁹ : Module R V", "inst✝⁸ : Module K V", "inst✝⁷ : IsScalarTower R K V", "inst✝⁶ : AddCommGroup V'", "inst✝⁵ : Module R V'", "inst✝⁴ : Module S V'", "inst✝³ : IsScalarTower R S V'", "inst✝² : AddCommGroup V''", "inst✝¹ : Module R V''", "inst✝ : IsDedekindDomain R", "hRS : RingHom.ker (algebraMap R S) ≠ ⊤", "f : V'' →ₗ[R] V", "hf : Function.Injective ⇑f", "f' : V'' →ₗ[R] V'", "ι : Type u_6", "b : ι → V''", "s : Finset ι", "g : ι → K", "eq : ∑ i ∈ s, g i • (⇑f ∘ b) i = 0", "j' : ι", "hj's : j' ∈ s", "hj'g : ¬g j' = 0", "a : K", "j : ι", "hjs : j ∈ s", "hgI✝ : a * g j ∉ ↑(RingHom.ker (algebraMap R S))", "g'' : (i : ι) → i ∈ s → R", "hg'' : ∀ (i : ι) (a_1 : i ∈ s), (algebraMap R K) (g'' i a_1) = a * g i", "this : (a : Prop) → Decidable a := Classical.propDecidable", "g' : ι → R := fun i => if h : i ∈ s then g'' i h else 0", "hg' : ∀ i ∈ s, (algebraMap R K) (g' i) = a * g i", "hgI : (algebraMap R S) (g' j) ≠ 0"], "goal": "∑ i ∈ s, (fun i => (algebraMap R S) (g' i)) i • (⇑f' ∘ b) i = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "Z : Type u_5", "Z' : Type u_6", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "TopologicalSpace Z'", "e : PartialHomeomorph X Y", "f' : Y ≃ₜ Z", "f'' : Z ≃ₜ Z'"], "goal": "(e.transHomeomorph f').transHomeomorph f'' = e.transHomeomorph (f'.trans f'')"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "S : CategoryTheory.Sieve X", "s : CategoryTheory.Limits.Cocone S.arrows.diagram"], "goal": "(S.arrows.yonedaFamilyOfElements_fromCocone s).Compatible"}}
+{"state": {"context": ["G : Type u_1", "SeminormedAddCommGroup G", "H : Type u_2", "SeminormedAddCommGroup H"], "goal": "NormedAddGroupHom.completion 0 = 0"}}
+{"state": {"context": ["α✝ : Sort u", "β : Sort v", "γ : Sort w", "α : Type u_1", "inst✝ : DecidableEq α", "a b : α", "s : Set α", "hs : a ∈ s ↔ b ∈ s", "x : α", "hx : x ∈ s"], "goal": "x ∈ ⇑(swap a b) '' s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "MeasurableSpace.CountableOrCountablyGenerated α β", "κ : ProbabilityTheory.Kernel α (β × Ω)", "ProbabilityTheory.IsFiniteKernel κ", "a : α", "s : Set β", "hs : MeasurableSet s", "t : Set Ω", "ht : MeasurableSet t"], "goal": "∫⁻ (b : β) in s, (κ.condKernel (a, b)) t ∂κ.fst a = (κ a) (s ×ˢ t)"}}
+{"state": {"context": ["δ : ℝ", "x : UnitAddCircle", "n : ℕ", "hn : 0 < n"], "goal": "x ∈ approxAddOrderOf UnitAddCircle n δ ↔ ∃ m < n, gcd m n = 1 ∧ ‖x - ↑(↑m / ↑n)‖ < δ"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝¹ : UniformSpace α", "inst✝ : UniformSpace β"], "goal": "𝓤 (α × β) = map (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) (𝓤 α ×ˢ 𝓤 β)"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "inst✝¹ : MeasurableSpace α", "μ : Measure α", "s t : Set α", "f : α → β", "inst✝ : Zero β", "hs : MeasurableSet s"], "goal": "s.indicator f =ᶠ[ae (μ.restrict s)] f"}}
+{"state": {"context": ["G : Type u_1", "inst✝³ : Group G", "inst✝² : TopologicalSpace G", "inst✝¹ : TopologicalGroup G", "inst✝ : CompactSpace G"], "goal": "UniformContinuous fun p => p.1 / p.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : LinearOrderedField α", "inst✝² : Ring β", "abv : β → α", "inst✝¹ : IsAbsoluteValue abv", "f g : ℕ → β", "a✝ : ℕ → α", "inst✝ : Archimedean α", "a x : α", "hx1 : |x| < 1"], "goal": "IsCauSeq abs fun m => ∑ n ∈ range m, a * x ^ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "l : Filter α", "f : l.Germ β", "g : l.Germ γ", "h : l.Germ δ", "p : l.Germ β → l.Germ γ → l.Germ δ → Prop", "H : ∀ (f : α → β) (g : α → γ) (h : α → δ), p ↑f ↑g ↑h"], "goal": "p f g h"}}
+{"state": {"context": ["F : Type u", "E : Type v", "Field F", "Field E", "Algebra F E", "Algebra.IsSeparable F E", "S : Set E", "q : ℕ", "ExpChar F q"], "goal": "IntermediateField.adjoin F S = IntermediateField.adjoin F ((fun x => x ^ q) '' S)"}}
+{"state": {"context": ["M : Type w", "A : Set M", "L : FirstOrder.Language", "L.Structure M", "α : Type u₁", "f g : Set (α → M)", "hf : A.Definable L f", "hg : A.Definable L g"], "goal": "A.Definable L (f ∪ g)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℂ E"], "goal": "finrank ℝ E = 2 * finrank ℂ E"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommSemiring R", "M✝ : Submonoid R", "S : Type u_2", "inst✝⁸ : CommSemiring S", "inst✝⁷ : Algebra R S", "P : Type u_3", "inst✝⁶ : CommSemiring P", "N✝ : Submonoid S", "T : Type u_4", "inst✝⁵ : CommSemiring T", "inst✝⁴ : Algebra R T", "M N : Submonoid R", "h : M ≤ N", "inst✝³ : IsLocalization M S", "inst✝² : IsLocalization N T", "inst✝¹ : Algebra S T", "inst✝ : IsScalarTower R S T", "y : T", "x : R", "s : ↥N", "e : y * (algebraMap R T) ↑(x, s).2 = (algebraMap R T) (x, s).1"], "goal": "∃ x, y * (algebraMap S T) ↑x.2 = (algebraMap S T) x.1"}}
+{"state": {"context": ["P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop", "hP₁ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), P ⇑f", "hP₂ :\n  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)\n    (g : β → γ), P f → P g → P (g ∘ f)"], "goal": "(AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso"}}
+{"state": {"context": ["G : Type u_1", "Group G"], "goal": "upperCentralSeries G 0 = ⊥"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_3", "inst✝ : StrictOrderedSemiring R", "a x y : R", "n m : ℕ", "h₀ : 0 < a", "h₁ : a < 1", "hn : 1 < n"], "goal": "a ^ n < a"}}
+{"state": {"context": ["C : Type u₁", "W : Type (v + 1)", "CategoryTheory.Category.{v, v + 1} W", "CategoryTheory.MonoidalCategory W", "CategoryTheory.EnrichedCategory W C", "X : CategoryTheory.ForgetEnrichment W C"], "goal": "CategoryTheory.ForgetEnrichment.of W (CategoryTheory.ForgetEnrichment.to W X) = X"}}
+{"state": {"context": [], "goal": "2 * arctan 3⁻¹ + arctan 7⁻¹ = π / 4"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "MetricSpace X", "K U : ι → Set X", "hK : ∀ (i : ι), IsClosed (K i)", "hU : ∀ (i : ι), IsOpen (U i)", "hKU : ∀ (i : ι), K i ⊆ U i", "hfin : LocallyFinite K"], "goal": "∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x ↑(δ x) ⊆ U i"}}
+{"state": {"context": ["p : ℕ", "idx : Type u_2", "hp : Fact (Nat.Prime p)", "Φ : MvPolynomial idx ℚ", "h : MvPolynomial.constantCoeff Φ = 0", "n : ℕ"], "goal": "MvPolynomial.constantCoeff (wittStructureRat p Φ n) = 0"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve R", "Nontrivial R", "n : ℤ", "hn : n ≠ 0"], "goal": "0 < (W.Φ n).natDegree"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝² : LinearOrder E", "inst✝¹ : Zero E", "inst✝ : MeasurableSpace α", "f : α →ₛ ℝ", "a : α"], "goal": "↑(map (fun b => max b 0) f - map (fun b => max b 0) (-f)) a = ↑f a"}}
+{"state": {"context": ["R : Type u", "Semiring R", "n : ℕ", "r : R"], "goal": "(Polynomial.monomial n) r * Polynomial.X = (Polynomial.monomial (n + 1)) r"}}
+{"state": {"context": ["n : Type u_1", "p : Type u_2", "R : Type u₂", "𝕜 : Type u_3", "inst✝³ : Field 𝕜", "inst✝² : DecidableEq n", "inst✝¹ : DecidableEq p", "inst✝ : CommRing R", "r : ℕ", "M : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜", "hM : M (inr ()) (inr ()) ≠ 0", "i : Fin r"], "goal": "∀ k ≤ r, ((List.drop k (listTransvecCol M)).prod * M) (inl i) (inr ()) = if k ≤ ↑i then 0 else M (inl i) (inr ())"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(π / 2)"], "goal": "(o.oangle (x - y) x).sin = ‖y‖ / ‖x - y‖"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D"], "goal": "∀ {X Y : C × D} (f : X ⟶ Y), (CategoryTheory.Prod.braiding C D).functor.map f = (f.2, f.1)"}}
+{"state": {"context": ["T : Type w", "CommRing T", "p : Polynomial T", "S : Type u_1", "CommRing S", "IsDomain S", "Algebra T S", "NoZeroSMulDivisors T S", "a : S"], "goal": "a ∈ p.rootSet S ↔ p ≠ 0 ∧ (Polynomial.aeval a) p = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "K : Type u_3", "inst✝² : Field K", "inst✝¹ : Module K M", "f g : End K M", "inst✝ : FiniteDimensional K M", "hf : f.IsSemisimple", "a : End K M", "ha : a ∈ (aeval f).range"], "goal": "a.IsSemisimple"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "E : Type u_3", "H : Type u_4", "hom : Type u_5", "inst✝¹⁰ : NormedField 𝕜", "inst✝⁹ : AddCommGroup H", "inst✝⁸ : Module 𝕜 H", "inst✝⁷ : AddCommGroup E", "inst✝⁶ : Module 𝕜 E", "inst✝⁵ : TopologicalSpace H", "inst✝⁴ : UniformSpace E", "inst✝³ : UniformAddGroup E", "inst✝² : ContinuousSMul 𝕜 E", "𝔖 : Set (Set α)", "inst✝¹ : FunLike hom H (α → E)", "inst✝ : LinearMapClass hom 𝕜 H (α → E)", "φ : hom", "hφ : Inducing (⇑ofFun ∘ ⇑φ)", "h : ∀ (u : H), Bornology.IsVonNBounded 𝕜 (Set.range (φ u))", "this : TopologicalAddGroup H", "hb : (𝓝 0).HasBasis (fun x => x ∈ 𝓝 0) fun V => {u | ∀ (x : α), φ u x ∈ V}"], "goal": "ContinuousSMul 𝕜 H"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "OrderTop α", "OrderBot β", "LocallyFiniteOrder α", "LocallyFiniteOrder β", "a : α", "b : β"], "goal": "Finset.Ioo (Sum.inlₗ a) (Sum.inrₗ b) = Finset.map toLex.toEmbedding ((Finset.Ioi a).disjSum (Finset.Iio b))"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : WeierstrassCurve R", "h : 3 ≠ 0"], "goal": "W.Ψ₃.coeff 4 ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹ : NormedLinearOrderedField 𝕜", "inst✝ : OrderTopology 𝕜", "p q : ℕ", "hpq : p < q"], "goal": "∀ᶠ (x : 𝕜) in atTop, x ^ q = 0 → x ^ p = 0"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_4", "γ : Type u_6", "l : Type u_8", "m : Type u_9", "n : Type u_10", "p : Type u_11", "f : α → β → γ", "A : Matrix l m α", "B : Matrix n p β", "i : l × n", "j : m × p"], "goal": "Matrix.kroneckerMap f A B i j = f (A i.1 j.1) (B i.2 j.2)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "s : Multiset α"], "goal": "{x | x ∈ s}.Finite"}}
+{"state": {"context": ["β : Type u_5", "CommMonoid β", "a u : β", "hu : IsUnit u"], "goal": "u * a ~ᵤ a"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ : ℕ", "χ : DirichletCharacter R n✝", "n : ℕ"], "goal": "1 ∣ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : MeasurableSpace α", "inst✝¹ : TopologicalSpace α", "μ : Measure α", "inst✝ : μ.WeaklyRegular", "x : ℝ≥0∞", "hx : x ≠ ⊤"], "goal": "(x • μ).WeaklyRegular"}}
+{"state": {"context": ["α β γ : Type u", "S S' : Set (α → ℕ)", "l : List (Set (α → ℕ))", "d : List.Forall Dioph l"], "goal": "∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S => S v) l ↔ ∃ t, List.Forall (fun p => p (v ⊗ t) = 0) pl"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{u₂, u₁} C", "CategoryTheory.GaloisCategory C", "F : CategoryTheory.Functor C FintypeCat", "CategoryTheory.PreGaloisCategory.FiberFunctor F", "X A : C", "u : A ⟶ CategoryTheory.PreGaloisCategory.selfProd F X", "a : ↑(F.obj A)", "h : F.map u a = CategoryTheory.PreGaloisCategory.mkSelfProdFib F X", "x : ↑(F.obj X)"], "goal": "F.map (CategoryTheory.PreGaloisCategory.selfProdProj u x) a = x"}}
+{"state": {"context": ["α : Type u", "Fintype α", "Fintype (Shrink.{v, u} α)"], "goal": "Fintype.card (Shrink.{v, u} α) = Fintype.card α"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : PartialOrder α", "s : Set α", "a : α"], "goal": "(∀ ⦃b : α⦄, b ≤ a → a ∈ s → b ∈ s) ↔ ∀ ⦃b : α⦄, b < a → a ∈ s → b ∈ s"}}
+{"state": {"context": ["a✝ b c✝ d m n k : ℕ", "p q : ℕ → Prop", "a c x✝ : ℕ", "dvd : c + 1 ∣ a + 1", "dvd2 : a + 1 ∣ x✝"], "goal": "x✝ / ((a + 1) / (c + 1)) / (c + 1) = x✝ / (a + 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x : α", "inst✝¹ : Ring α", "inst✝ : SubtractionMonoid β", "h : Antiperiodic f c", "hi : f 0 = 0", "n : ℕ"], "goal": "f (↑(Int.negSucc n) * c) = 0"}}
+{"state": {"context": ["z : ℂ"], "goal": "(norm ℝ) z = normSq z"}}
+{"state": {"context": ["R : Type u_1", "G : Type u_3", "AddCommGroup G", "CommRing R", "Nontrivial R"], "goal": "Algebra.FiniteType R (AddMonoidAlgebra R G) ↔ AddGroup.FG G"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R", "M : Type u_2", "AddCommGroup M", "Module R M", "N : Type u_3", "AddCommGroup N", "Module R N"], "goal": "AdicCompletion.map I 0 = 0"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "ι : Type u_5", "κ : Type u_6", "inst✝ : LinearOrder α", "s : Finset α", "H : s.Nonempty", "x : α"], "goal": "s.max = ↑(s.max' H)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Field K", "inst✝² : IsDedekindDomain A", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "P I : Ideal A", "P_prime : P.IsPrime", "hP : P ≠ ⊥", "i : ℕ", "hlt : P ^ (i + 1) < I", "hle : I ≤ P ^ i"], "goal": "I = P ^ i"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j k : ι", "hi : c.prev j = i", "hk : c.next j = k", "K.HasHomology j", "(K.sc' i j k).HasHomology"], "goal": "(K.opcyclesIsoSc' i j k hi hk).inv ≫ K.fromOpcycles j k = (K.sc' i j k).fromOpcycles"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : Inducing f", "s : Set X"], "goal": "closure s = f ⁻¹' closure (f '' s)"}}
+{"state": {"context": ["K : Type u_1", "inst✝ : Field K", "x : (K →+* ℂ) → ℂ", "φ : K →+* ℂ", "hx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))"], "goal": "(starRingEnd ℂ) (0 φ) = 0 (ComplexEmbedding.conjugate φ)"}}
+{"state": {"context": ["n : ZNum"], "goal": "ZNum.ofInt' ↑n = n"}}
+{"state": {"context": [], "goal": "Nat.primeFactors 0 = ∅"}}
+{"state": {"context": ["n : ℕ", "K : Type u_1", "inst✝³ : CommRing K", "μ : K", "h : IsPrimitiveRoot μ n", "inst✝² : IsDomain K", "inst✝¹ : CharZero K", "inst✝ : DecidableEq K", "hn : ¬n = 0", "hpos : 0 < n", "m : ℕ", "left✝ : m < n", "hcop : m.Coprime n", "hx : μ ^ m ∈ primitiveRoots n K"], "goal": "μ ^ m ∈ (map (Int.castRingHom K) (minpoly ℤ μ)).roots"}}
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+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "f : α → β"], "goal": "MonotoneOn f Set.univ ↔ Monotone f"}}
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+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G", "x : G"], "goal": "x⁻¹ ∈ H ↔ x ∈ H"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "I : Ideal R", "hi : I.IsPrime"], "goal": "I.IsPrimary"}}
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+{"state": {"context": [], "goal": "Xor' True = Not"}}
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+{"state": {"context": ["i : Nat", "xs : Lean.Omega.IntList", "h : List.length xs ≤ i"], "goal": "xs.get i = 0"}}
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+{"state": {"context": ["α : Type u_1", "UniformSpace α", "AddGroup α", "UniformAddGroup α", "n : ℤ"], "goal": "UniformContinuous fun x => n • x"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : Set P", "p1 p2 : P", "hp1 : p1 ∈ s", "hp2 : p2 ∈ s"], "goal": "p1 -ᵥ p2 ∈ vectorSpan k s"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "H✝ : Subgroup G", "inst✝ : H✝.Normal", "H : Subgroup G", "hG : Group.IsNilpotent G"], "goal": "Group.nilpotencyClass ↥H ≤ Group.nilpotencyClass G"}}
+{"state": {"context": ["m n : ℕ", "α : Type u_1", "xs : Fin m → α", "ys : Fin n → α"], "goal": "Fin.append xs ys ∘ Fin.rev = Fin.append (ys ∘ Fin.rev) (xs ∘ Fin.rev) ∘ Fin.cast ⋯"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "s : Finset α"], "goal": "Finset.filter p s = s ↔ ∀ x ∈ s, p x"}}
+{"state": {"context": ["G : Type u_10", "InvolutiveNeg G"], "goal": "Equiv.symm (Equiv.neg G) = Equiv.neg G"}}
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+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "TopologicalSpace H", "TopologicalSpace M", "ChartedSpace H M", "TopologicalSpace H'", "TopologicalSpace M'", "ChartedSpace H' M'", "P : (H → H') → Set H → H → Prop", "g : M → M'", "s t : Set M", "x : M", "mono_of_mem : ∀ ⦃s : Set H⦄ ⦃x : H⦄ ⦃t : Set H⦄ ⦃f : H → H'⦄, s ∈ 𝓝[t] x → P f s x → P f t x", "h : ChartedSpace.LiftPropWithinAt P g s x", "hst : s ∈ 𝓝[t] x"], "goal": "ChartedSpace.LiftPropWithinAt P g t x"}}
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+{"state": {"context": ["R : Type u", "inst✝¹ : CommSemiring R", "S : Type v", "inst✝ : CommSemiring S", "p✝ q✝ : ℕ", "p q : R[X]", "hsep : p.Separable", "hq : 1 < multiplicity q p", "h : { Dom := Part.Dom 1, get := fun x => 2 } ≤ multiplicity q p"], "goal": "↑2 ≤ multiplicity q p"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "M : Type u_1", "inst✝¹ : Category.{?u.113036, u_1} M", "inst✝ : MonoidalCategory M", "F : MonoidalFunctor M (C ⥤ C)", "m n : M", "h₁ : m ⊗ n ≅ 𝟙_ M", "h₂ : n ⊗ m ≅ 𝟙_ M", "H : h₁.hom ▷ m ≫ (λ_ m).hom = (α_ m n m).hom ≫ m ◁ h₂.hom ≫ (ρ_ m).hom", "X : C"], "goal": "(F.obj m).map ((unitOfTensorIsoUnit F m n h₁).inv.app X) ≫ (F.μ n m).app ((F.obj m).obj X) ≫ (F.map h₂.hom).app ((F.obj m).obj X) ≫ F.εIso.inv.app ((F.obj m).obj X) = 𝟙 ((F.obj m).obj X)"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "R₁ R₂ : ComposableArrows C 4", "hR₁ : R₁.Exact", "hR₂ : R₂.Exact", "φ : R₁ ⟶ R₂", "h₀ : Epi (φ.app 0)", "h₁ : IsIso (φ.app 1)", "h₂ : IsIso (φ.app ⟨3, ⋯⟩)", "h₃ : Mono (φ.app ⟨4, ⋯⟩)", "this : Mono (app' φ 2 ⋯)"], "goal": "IsIso (app' φ 2 ⋯)"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s : Set α"], "goal": "minimals r (minimals r s) = minimals r s"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "inst✝¹ : Fintype F", "inst✝ : DecidableEq F", "hF : ringChar F ≠ 2", "x✝ : ∃ a, (quadraticChar F) a = -1", "a : F", "ha : (quadraticChar F) a = -1"], "goal": "IsUnit a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m m0 : MeasurableSpace α", "μ : Measure α", "s✝ t : Set α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "p : ℝ≥0∞", "f✝ f g : α → ℝ", "hf : Integrable f μ", "hg : Integrable g μ", "hf_le : ∀ (s : Set α), MeasurableSet s �� μ s < ⊤ → ∫ (x : α) in s, f x ∂μ ≤ ∫ (x : α) in s, g x ∂μ", "s : Set α", "hs : MeasurableSet s"], "goal": "μ s < ⊤ → 0 ≤ ∫ (x : α) in s, (g - f) x ∂μ"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs : s ≠ 0", "hs' : s ≠ 1 ∨ a ≠ 0"], "goal": "DifferentiableAt ℂ (HurwitzZeta.completedCosZeta a) s"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Ring R", "inst✝¹ : IsDomain R", "S : Submonoid R", "inst✝ : StrongRankCondition R", "h : ¬Nonempty (OreLocalization.OreSet R⁰)", "this : ℵ₀ ≤ 1"], "goal": "False"}}
+{"state": {"context": ["X₁ : Type u₁", "inst✝⁵ : TopologicalSpace X₁", "X : Type u", "inst✝⁴ : TopologicalSpace X", "inst✝³ : NormalSpace X", "s : Set X", "hs : IsClosed s", "e : X₁ → X", "he : ClosedEmbedding e", "Y✝ : Type v", "inst✝² : TopologicalSpace Y✝", "inst✝¹ : TietzeExtension Y✝", "Y : Type v", "inst✝ : TopologicalSpace Y", "f : C(X₁, Y)", "t : Set Y", "hf : ∀ (x : X₁), f x ∈ t", "ht : TietzeExtension ↑t", "g : C(X, ↑t)", "hg : g.comp { toFun := e, continuous_toFun := ⋯ } = { toFun := Set.codRestrict (⇑f) t hf, continuous_toFun := ⋯ }"], "goal": "∃ g, (∀ (x : X), g x ∈ t) ∧ g.comp { toFun := e, continuous_toFun := ⋯ } = f"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "X : Type u_2", "inst✝¹ : MulAction G X", "N : Subgroup G", "inst✝ : N.Normal", "a : X"], "goal": "∀ (g : G), g • orbit (↥N) a = orbit (↥N) a ∨ Disjoint (g • orbit (↥N) a) (orbit (↥N) a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "δ : Type u_7", "s : Set α", "t : Set β", "f : α → β' → γ", "g : β → β'", "f' : β → α → δ", "g' : δ → γ", "h_right_anticomm : ∀ (a : α) (b : β), f a (g b) = g' (f' b a)"], "goal": "Set.image2 f s (g '' t) = g' '' Set.image2 f' t s"}}
+{"state": {"context": ["E : Type u_1", "β : Type u_2", "AddCommGroup E", "TopologicalSpace E", "Module ℝ E", "TopologicalAddGroup E", "ContinuousSMul ℝ E", "OrderedAddCommGroup β", "Module ℝ β", "OrderedSMul ℝ β", "f : E → β", "a : E", "h_local_max : IsLocalMax f a", "h_conc : ConcaveOn ℝ Set.univ f", "x : E"], "goal": "f x ≤ f a"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_2", "F : Type u_3", "CommRing R", "AddCommGroup E", "AddCommGroup F", "Module R E", "Module R F", "TopologicalSpace E", "TopologicalSpace F", "ContinuousAdd E", "ContinuousAdd F", "TopologicalSpace R", "ContinuousSMul R E", "ContinuousSMul R F", "f : E →ₗ.[R] F"], "goal": "f.closure.HasCore f.domain"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "f g : M →ₗ[R] M", "h : Commute f g", "p : Submodule R M", "hf : Set.MapsTo ⇑f ↑p ↑p", "hg : Set.MapsTo ⇑g ↑p ↑p"], "goal": "Commute (f.restrict hf) (g.restrict hg)"}}
+{"state": {"context": ["G : Type u_2", "S : Type u_4", "Group G", "SetLike S G", "SubgroupClass S G", "s : S", "a : G", "ha : a ∈ s"], "goal": "MulOpposite.op a • ↑s = ↑s"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "DecidableEq α", "s : Sym α n.succ", "a : α", "h : a ∈ s"], "goal": "a ::ₛ s.erase a h = s"}}
+{"state": {"context": ["x : SetTheory.PGame"], "goal": "SetTheory.PGame.LF 0 x ↔ ∃ i, 0 ≤ x.moveLeft i"}}
+{"state": {"context": ["f : ℕ → ℂ", "s : ℂ", "h : abscissaOfAbsConv f < ↑s.re", "x : ℝ", "hxs : x < s.re", "hf : LSeriesSummable f ↑x", "y : ℝ", "hxy : x < y", "hys : y < s.re", "S : Set ℂ := {z | y < z.re}"], "goal": "LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "T2Space X", "x : X", "s : Set X", "h : x ∈ closure s"], "goal": "lim (𝓝[s] x) = x"}}
+{"state": {"context": ["x : Fin 2 → ℤ", "hx : x ≠ 0"], "goal": "1 ≤ (↑(x 0) / ‖x‖) ^ 2 ∨ 1 ≤ (↑(x 1) / ‖x‖) ^ 2"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "x y : α"], "goal": "x ≤ y → ⌊x⌋₊ ≤ ⌊y⌋₊"}}
+{"state": {"context": ["Fq : Type u_1", "inst✝¹ : Fintype Fq", "inst✝ : Ring Fq", "d m : ℕ", "hm : Fintype.card Fq ^ d ≤ m", "b : Fq[X]", "A : Fin m.succ → Fq[X]", "hA : ∀ (i : Fin m.succ), (A i).degree < b.degree", "hb : b ≠ 0", "f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff (b.natDegree - ↑j.succ)", "this : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ)", "i₀ i₁ : Fin m.succ", "i_ne : i₀ ≠ i₁", "i_eq : f i₀ = f i₁", "j : ℕ", "hj : b.natDegree - d ≤ j"], "goal": "(A i₁ - A i₀).coeff j = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "β' : Type u_3", "γ : Type u_4", "l₁ : List α", "l₂ : List β", "f : β → β'", "g : α → β' → γ"], "goal": "List.zipWith g l₁ (List.map f l₂) = List.zipWith (fun a b => g a (f b)) l₁ l₂"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "x : ↑{0}ᶜ"], "goal": "↑((homeomorphUnitSphereProd E) x).2 = ‖↑x‖"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End R M", "p : Submodule R M", "hfp : ∀ x ∈ p, f x ∈ p", "μ : R", "hμp : Disjoint (f.eigenspace μ) p", "x : ↥p", "hx : x ∈ eigenspace (LinearMap.restrict f hfp) μ"], "goal": "x ∈ ⊥"}}
+{"state": {"context": ["D : GlueData", "α : Type u", "inst✝ : TopologicalSpace α", "J : Type u", "U : J → Opens α", "s : Set ↑(ofOpenSubsets U).glued", "hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑((ofOpenSubsets U).ι i) ⁻¹' s)", "i : (ofOpenSubsets U).J", "x : ↑(of α)", "hx' : x ∈ U i", "hx : ((ofOpenSubsets U).ι i) ⟨x, hx'⟩ ∈ s"], "goal": "∃ t ⊆ ⇑(fromOpenSubsetsGlue U) '' s, IsOpen t ∧ (fromOpenSubsetsGlue U) (((ofOpenSubsets U).ι i) ⟨x, hx'⟩) ∈ t"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "x y : R"], "goal": "⇑(Polynomial.evalEvalRingHom x y) = Polynomial.evalEval x y"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "SeminormedAddCommGroup M", "SeminormedAddCommGroup N", "S : AddSubgroup M", "f : NormedAddGroupHom M N", "hf : ∀ x ∈ S, f x = 0", "x : M ⧸ S"], "goal": "‖(QuotientAddGroup.lift S f.toAddMonoidHom hf) x‖ ≤ ‖f‖ * ‖x‖"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x y : X", "γ : Path x y"], "goal": "γ 0 = x"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s✝ s₁ t✝ u : Set E", "f✝ f₁ : E → F", "g✝ : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "s : Set E", "t : Set F", "g : F → G", "f : E → F", "hg : ContDiffOn 𝕜 n g t", "hf : ContDiffOn 𝕜 n f s", "st : s ⊆ f ⁻¹' t", "Eu : Type (max uE uF uG) := ULift.{max uF uG, uE} E", "Fu : Type (max uE uF uG) := ULift.{max uE uG, uF} F", "Gu : Type (max uE uF uG) := ULift.{max uE uF, uG} G", "isoE : Eu ≃L[𝕜] E", "isoF : Fu ≃L[𝕜] F", "isoG : Gu ≃L[𝕜] G", "fu : Eu → Fu := (⇑isoF.symm ∘ f) ∘ ⇑isoE", "fu_diff : ContDiffOn 𝕜 n fu (⇑isoE ⁻¹' s)", "gu : Fu → Gu := (⇑isoG.symm ∘ g) ∘ ⇑isoF"], "goal": "ContDiffOn 𝕜 n (g ∘ f) s"}}
+{"state": {"context": ["E : Type u_4", "F : Type u_5", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F"], "goal": "⇑0 = 0"}}
+{"state": {"context": ["U : Type u_1", "Quiver U", "u v w u' w' : U", "p : Quiver.Path u v", "e : v ⟶ w", "hu : u = u'", "hw : w = w'"], "goal": "Quiver.Path.cast hu hw (p.cons e) = (Quiver.Path.cast hu ⋯ p).cons (Quiver.Hom.cast ⋯ hw e)"}}
+{"state": {"context": ["F : Type u_1", "R : Type u_2", "A : Type u_3", "B : Type u_4", "CommSemiring R", "StarRing R", "Semiring A", "Algebra R A", "StarRing A", "StarModule R A", "Semiring B", "Algebra R B", "StarRing B", "FunLike F A B", "AlgHomClass F R A B", "StarAlgHomClass F R A B", "s : Set A", "h : StarAlgebra.adjoin R s = ⊤", "f g : F", "hs : Set.EqOn (⇑f) (⇑g) s"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "x y : Part α", "z : Part α", "hx : x ≤ z", "hy : y ≤ z"], "goal": "x ≤ y ∨ y ≤ x"}}
+{"state": {"context": ["E : Type u_1", "SeminormedAddCommGroup E", "δ : ℝ", "x y : E"], "goal": "{x} + Metric.ball y δ = Metric.ball (x + y) δ"}}
+{"state": {"context": ["α : Type u_1", "r s : Setoid α", "x y : α"], "goal": "(r ⊓ s).Rel x y ↔ r.Rel x y ∧ s.Rel x y"}}
+{"state": {"context": ["α β : Type u", "c : Cardinal.{?u.231592}", "f : α → β", "s : Set α", "t : Set β", "h : t ⊆ range fun x => f ↑x"], "goal": "#↑{x | f ↑x ∈ t} = #↑((fun x => f ↑x) ⁻¹' t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f : α → β", "hf : Continuous f"], "goal": "Measurable f"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "h : J ⊔ I = ⊤"], "goal": "I * K ⊔ J = K ⊔ J"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : UniqueFactorizationMonoid α", "a : α"], "goal": "a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a"}}
+{"state": {"context": ["m : ℕ", "hm : m ≠ 1", "n : ℕ"], "goal": "n ∈ {n | n ≡ 1 [MOD m]} ↔ n ∈ {n | n % m = 1}"}}
+{"state": {"context": ["I : Type u", "LinearOrder I", "IsWellOrder I fun x x_1 => x < x_1"], "goal": "Profinite.NobelingProof.P I 0"}}
+{"state": {"context": [], "goal": "Cardinal.univ.{u, v}.ord = Ordinal.univ.{u, v}"}}
+{"state": {"context": ["α : Type u_2", "InvolutiveNeg α", "a : α"], "goal": "-{a} = {-a}"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f✝ f : α → β", "hf : Bijective f", "s : Set β", "t : Set α"], "goal": "f ⁻¹' s = t ↔ s = f '' t"}}
+{"state": {"context": ["x✝ y✝ z : ℝ", "n : ℕ", "x y : ℝ", "h : 0 ≤ x", "h₁ : x ≤ y", "h₁' : x < y", "h₂ : 0 ≤ 0"], "goal": "x ^ 0 ≤ y ^ 0"}}
+{"state": {"context": ["G : Type u_1", "Group G", "X : Type u_2", "MulAction G X", "C : SubMulAction G X", "B : Set X", "hB : MulAction.IsBlock G B"], "goal": "MulAction.IsBlock G (Subtype.val ⁻¹' B)"}}
+{"state": {"context": ["k : Type u", "inst✝¹¹ : CommRing k", "G : Type u", "inst✝¹⁰ : Group G", "V : Type v", "inst✝⁹ : AddCommGroup V", "inst✝⁸ : Module k V", "inst✝⁷ : Module (MonoidAlgebra k G) V", "inst✝⁶ : IsScalarTower k (MonoidAlgebra k G) V", "W : Type w", "inst✝⁵ : AddCommGroup W", "inst✝⁴ : Module k W", "inst✝³ : Module (MonoidAlgebra k G) W", "inst✝² : IsScalarTower k (MonoidAlgebra k G) W", "π : W →ₗ[k] V", "i : V →ₗ[MonoidAlgebra k G] W", "h : ∀ (v : V), π (i v) = v", "inst✝¹ : Fintype G", "inst✝ : Invertible ↑(Fintype.card G)", "v : W"], "goal": "(equivariantProjection G π) v = ⅟↑(Fintype.card G) • ∑ g : G, (π.conjugate g) v"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "R : Type u_6", "NormedRing R", "f : α → R", "hf : MeasureTheory.Memℒp f p μ", "c : R"], "goal": "MeasureTheory.Memℒp (fun x => c * f x) p μ"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁴ : CommRing R", "inst✝³ : CommRing A", "inst✝² : Field K", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "h : IsDedekindDomain A", "I : Ideal A", "hI0 : I ≠ ⊥", "hJ0 : ¬↑I * (↑I)⁻¹ = 0", "x : K", "hx : x ∈ (fun a => ↑a) (↑I * (↑I)⁻¹)⁻¹", "x_mul_mem : ∀ b ∈ (↑I)⁻¹, x * b ∈ (↑I)⁻¹", "y : K", "hy : y ∈ Subalgebra.toSubmodule (Polynomial.aeval x).range"], "goal": "y ∈ ↑(↑I)⁻¹"}}
+{"state": {"context": ["ι : Sort u_1", "G : Type u_2", "Group G", "S : ι → Subgroup Gᵐᵒᵖ"], "goal": "(iInf S).unop = ⨅ i, (S i).unop"}}
+{"state": {"context": ["η : Type u_1", "α : Type u_2", "ι✝ : Type u_3", "κ✝ : Type u_4", "M : Type u_5", "κ : Type u_6", "inst✝¹ : AddCommMonoid M", "S : Finset M", "inst✝ : Finite κ", "C : M → κ", "ι : Type", "_inst : Fintype ι", "hι : ∀ (C : (ι → { x // x ∈ S }) → κ), ∃ l, Line.IsMono C l"], "goal": "∃ a > 0, ∃ b c, ∀ s ∈ S, C (a • s + b) = c"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s✝ : σ →₀ ℕ", "inst✝ : CommSemiring R", "p q : MvPolynomial σ R", "ι : Type u_3", "s : Finset ι", "f : ι → MvPolynomial σ R"], "goal": "(s.sum f).totalDegree ≤ s.sup fun i => (f i).totalDegree"}}
+{"state": {"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝⁴ : Category.{max v u, w} D", "inst✝³ : ConcreteCategory D", "inst✝² : PreservesLimits (forget D)", "inst✝¹ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "X : C", "P : Cᵒᵖ ⥤ D", "S : J.Cover X", "x : (forget D).obj (P.obj (op X))"], "goal": "(⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P X) (op ⊤)) x = (colimit.ι (J.diagram P X) (op S)) ((Meq.equiv P S).symm (Meq.mk S x))"}}
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+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "p : Equiv.Perm α", "x : α", "k : ℕ"], "goal": "p.toList ((p ^ k) x) = (p.toList x).rotate k"}}
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+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁶ : Ring R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'", "inst✝³ : AddCommGroup M''", "inst✝² : Module R M", "inst✝¹ : Module R M'", "inst✝ : Module R M''", "a✝ b : R", "x y : M", "hv✝ hv : LinearIndependent R v", "i : ι", "a : R", "l : ι →₀ R", "hl : Finsupp.single i a ∈ Finsupp.supported R R (univ \\ {i})", "e : (Finsupp.total ι M R v) l = a • v i", "hn : ¬a = 0"], "goal": "False"}}
+{"state": {"context": ["a b c : ℕ"], "goal": "a < c - b ↔ b + a < c"}}
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+{"state": {"context": ["R✝ : Type u", "S : Type v", "a b : R✝", "m n✝ : ℕ", "ι : Type y", "inst✝² : Ring R✝", "p : R✝[X]", "R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : Nontrivial R", "n : ℕ", "h : IsUnit (X ^ n - 1)", "hn : n ≠ 0"], "goal": "False"}}
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+{"state": {"context": ["F : Type u_1", "M : Type u_2", "N : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝¹² : NonUnitalNonAssocSemiring α", "inst✝¹¹ : Monoid M", "inst✝¹⁰ : Monoid N", "inst✝⁹ : Semiring R", "inst✝⁸ : DistribMulAction M α", "inst✝⁷ : SMulCommClass M α α", "inst✝⁶ : IsScalarTower M α α", "inst✝⁵ : DistribMulAction N α", "inst✝⁴ : SMulCommClass N α α", "inst✝³ : IsScalarTower N α α", "inst✝² : Module R α", "inst✝¹ : SMulCommClass R α α", "inst✝ : IsScalarTower R α α", "T S : CentroidHom α", "a b : α"], "goal": "(⇑T ∘ ⇑S) (a * b) = (⇑S ∘ ⇑T) (a * b)"}}
+{"state": {"context": ["V : Type u_5", "K : Type u_6", "Field K", "AddCommGroup V", "Module K V", "FiniteDimensional K V", "B : LinearMap.BilinForm K V", "W : Submodule K V", "hB₀ : B.IsRefl"], "goal": "IsCompl W (B.orthogonal W) ↔ Disjoint W (B.orthogonal W)"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "inst✝¹ : Group G", "hp : Fact (Nat.Prime p)", "P : Sylow p G", "inst✝ : (↑P).Normal", "fP : (↑P).FiniteIndex", "h : p ∣ Nat.card (G ⧸ ↑P)", "x : G ⧸ ↑P", "hx : orderOf x = p"], "goal": "False"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m' mΩ : MeasurableSpace Ω", "inst✝² : StandardBorelSpace Ω", "inst✝¹ : Nonempty Ω", "hm' : m' ≤ mΩ", "s t : Set Ω", "hs : MeasurableSet s", "ht : MeasurableSet t", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ"], "goal": "CondIndepSet m' hm' s t μ ↔ μ[(s ∩ t).indicator fun ω => 1|m'] =ᶠ[ae μ] μ[s.indicator fun ω => 1|m'] * μ[t.indicator fun ω => 1|m']"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "a : ℕ", "ha0 : ↑a ≠ 0"], "goal": "↑↑a ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop"], "goal": "type (Sum.Lex EmptyRelation fun x x_1 => x < x_1) ≤ type fun x x_1 => x < x_1"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "β : ι → Type u_4", "mβ : (i : ι) → MeasurableSpace (β i)", "_mα : MeasurableSpace α", "m : ι → MeasurableSpace Ω", "_mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "π : ι → Set (Set Ω)", "s : ι → Set Ω", "S : Finset ι", "f : (x : ι) → Ω → β x", "inst✝ : Fintype ι", "h : iIndepSets π κ μ", "hs : ∀ (i : ι), s i ∈ π i"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (⋂ i, s i) = ∏ i : ι, (κ a) (s i)"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "R : α → β → Prop", "h : Relator.BiTotal R"], "goal": "((R ⇒ Iff) ⇒ Iff) (fun p => ∀ (i : α), p i) fun q => ∀ (i : β), q i"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "M : Type u_3", "AddCommGroup M", "TopologicalSpace M", "TopologicalAddGroup M", "v : MeasureTheory.VectorMeasure α M", "i : Set α"], "goal": "↑(-v) i = -↑v i"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "m0 : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s✝ s' t✝ : Set α", "inst✝ : Countable ι", "s : ι → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) s", "t : Set α", "ht : MeasurableSet t"], "goal": "Directed (fun x x_1 => x ⊆ x_1) fun i => t ∩ s i"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : ↥(MeasureTheory.Lp E p μ)"], "goal": "f = 0 ↔ ↑↑f =ᵐ[μ] 0"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "CommRing A", "Field K", "CommRing B", "Field L", "Algebra A K", "Algebra B L", "Algebra A B", "Algebra K L", "Algebra A L", "IsScalarTower A K L", "IsScalarTower A B L", "IsDomain A", "IsFractionRing A K", "IsIntegralClosure B A L", "IsFractionRing B L", "FiniteDimensional K L", "Algebra.IsSeparable K L", "IsIntegrallyClosed A", "IsDedekindDomain B", "I J : FractionalIdeal B⁰ L", "hI : I ≠ 0", "hIJ : I * J ≤ 1"], "goal": "J ≤ FractionalIdeal.dual A K I"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝⁸ : CommRing R", "inst✝⁷ : LieRing L", "inst✝⁶ : LieAlgebra R L", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : LieRingModule L M", "inst✝² : LieModule R L M", "inst✝¹ : CommRing A", "inst✝ : Algebra R A", "N✝ : LieSubmodule R L M", "I : LieIdeal R L", "N : LieSubmodule R L M", "s : Set (A ⊗[R] M) := {m | ∃ x ∈ I, ∃ n ∈ N, 1 ⊗ₜ[R] ⁅x, n⁆ = m}", "this : ⇑((TensorProduct.mk R A M) 1) '' {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m} = s"], "goal": "s ⊆ {m | ∃ x ∈ baseChange A I, ∃ n ∈ baseChange A N, ⁅x, n⁆ = m}"}}
+{"state": {"context": ["C : Type u_1", "inst✝⁵ : Category.{u_6, u_1} C", "D : Type u_2", "inst✝⁴ : Category.{u_5, u_2} D", "E : Type u_3", "inst✝³ : Category.{?u.50538, u_3} E", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "L : GrothendieckTopology E", "A : Type u_4", "inst✝² : Category.{?u.50590, u_4} A", "G : C ⥤ D", "inst✝¹ : G.IsCoverDense K", "inst✝ : G.IsLocallyFull K", "ℱ : Dᵒᵖ ⥤ Type v", "ℱ' : SheafOfTypes K", "α : G.op ⋙ ℱ ⟶ G.op ⋙ ℱ'.val", "X : D", "Y : C", "f : op X ⟶ op (G.obj Y)", "a✝ : ℱ.obj (op X)"], "goal": "(appHom α X ≫ ℱ'.val.map f) a✝ = (ℱ.map f ≫ α.app (op Y)) a✝"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "i j : Fin 2"], "goal": "(basisOneI.repr (![1, I] i)) j = (Finsupp.single i 1) j"}}
+{"state": {"context": ["M : Type u_4", "AddMonoid M", "l : List M"], "goal": "(l.get? 0).getD 0 + l.tail.sum = l.sum"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ : MeasurableSpace Ω", "μ : Measure Ω", "f g : Ω → ℝ≥0∞", "X Y : Ω → ℝ", "hXY : IndepFun X Y μ", "hX : Integrable X μ", "hY : Integrable Y μ", "pos : ℝ → ℝ := fun x => max x 0", "neg : ℝ → ℝ := fun x => max (-x) 0", "posm : Measurable pos", "negm : Measurable neg", "Xp : Ω → ℝ := pos ∘ X", "Xm : Ω → ℝ := neg ∘ X", "Yp : Ω → ℝ := pos ∘ Y", "Ym : Ω → ℝ := neg ∘ Y", "hXpm : X = Xp - Xm", "hYpm : Y = Yp - Ym", "hp1 : 0 ≤ Xm", "hp2 : 0 ≤ Xp", "hp3 : 0 ≤ Ym", "hp4 : 0 ≤ Yp", "hm1 : AEMeasurable Xm μ", "hm2 : AEMeasurable Xp μ", "hm3 : AEMeasurable Ym μ", "hm4 : AEMeasurable Yp μ", "hv1 : Integrable Xm μ", "hv2 : Integrable Xp μ", "hv3 : Integrable Ym μ", "hv4 : Integrable Yp μ", "hi1 : IndepFun Xm Ym μ", "hi2 : IndepFun Xp Ym μ", "hi3 : IndepFun Xm Yp μ", "hi4 : IndepFun Xp Yp μ", "hl1 : Integrable (Xm * Ym) μ", "hl2 : Integrable (Xp * Ym) μ", "hl3 : Integrable (Xm * Yp) μ"], "goal": "integral μ (X * Y) = integral μ X * integral μ Y"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "ι : Type u_2", "DecidableEq ι", "Preorder ι", "G : ι → Type u_3", "(i : ι) → AddCommGroup (G i)", "(i : ι) → Module R (G i)", "f : (i j : ι) → i ≤ j → G i →ₗ[R] G j", "M : Type u_4", "AddCommGroup M", "Module R M", "i : ι", "g : G i", "m : M"], "goal": "(TensorProduct.fromDirectLimit f M)\n    ((Module.DirectLimit.of R ι (fun x => G x ⊗[R] M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)) =\n  (Module.DirectLimit.of R ι G f i) g ⊗ₜ[R] m"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "s : σ", "n : ℕ", "φ : MvPowerSeries σ R"], "goal": "MvPowerSeries.X s ^ n ∣ φ ↔ ∀ (m : σ →₀ ℕ), m s < n → (MvPowerSeries.coeff R m) φ = 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α"], "goal": "toIcoDiv hp a b • p - b = -toIcoMod hp a b"}}
+{"state": {"context": [], "goal": "↑ArithmeticFunction.zeta * ArithmeticFunction.moebius = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "inst✝² : MetricSpace β", "μ : Measure α", "n✝ : ℕ", "i j : ι", "s : Set α", "ε : ℝ", "f : ι → α → β", "g : α → β", "inst✝¹ : SemilatticeSup ι", "inst✝ : Nonempty ι", "n : ℕ", "hfg : μ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (f n a) (g a) < ε)} = 0", "x : α"], "goal": "(x ∈ s ∧ ∀ (i : ι), ∃ k ≥ i, 1 / (↑n + 1) < dist (f k x) (g x)) → x ∈ {a | ¬(a ∈ s → ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (f n a) (g a) < ε)}"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι"], "goal": "BoxIntegral.Box.Icc I = Set.univ.pi fun i => Set.Icc (I.lower i) (I.upper i)"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "MeasurableSpace X", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : X → E", "μ : MeasureTheory.Measure X", "PartialOrder X", "x y : X", "hy : μ {y} = 0"], "goal": "∫ (t : X) in Set.Icc x y, f t ∂μ = ∫ (t : X) in Set.Ico x y, f t ∂μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : NormedRing A", "inst✝¹ : NormedAlgebra 𝕜 A", "inst✝ : CompleteSpace A", "a : A", "n : ℕ", "k : 𝕜", "hk : k ∈ σ a", "pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))"], "goal": "↑‖k‖₊ ≤ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "N : Matroid β", "f : α → β"], "goal": "(N.comap f).E = f ⁻¹' N.E"}}
+{"state": {"context": ["G : Type u_2", "AddCommGroup G", "TopologicalSpace G", "TopologicalAddGroup G", "f : ℕ → G", "k : ℕ"], "goal": "(Summable fun n => f (n + k)) ↔ Summable f"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "f : α → ι", "g : (Perm α)ᵈᵐᵃ"], "goal": "g • f = f ↔ f ∘ ⇑(mk.symm g) = f"}}
+{"state": {"context": ["ι : Type u", "β : ι → Type v", "DecidableEq ι", "(i : ι) → Zero (β i)", "i : ι"], "goal": "Function.Injective (DFinsupp.single i)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "inst✝ : Monoid α", "n : ℕ", "a m : α"], "goal": "IsSquare m ↔ ∃ c, m = c ^ 2"}}
+{"state": {"context": ["ι : Type u_1", "inst✝⁴ : Fintype ι", "M : Type u_2", "inst✝³ : AddCommGroup M", "R : Type u_3", "inst✝² : CommRing R", "inst✝¹ : Module R M", "I : Ideal R", "b : ι → M", "hb : Submodule.span R (Set.range b) = ⊤", "inst✝ : DecidableEq ι", "A A' : Matrix ι ι R", "f f' : Module.End R M", "h : Represents b A f", "h' : Represents b A' f'"], "goal": "((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.total R R) b)) (algEquivMatrix'.symm A * algEquivMatrix'.symm A') = (PiToModule.fromEnd R b) (f * f')"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "α : Type u_3", "inst✝³ : Fintype n", "inst✝² : DecidableEq n", "inst✝¹ : Semiring α", "A : Matrix m n α", "B : Matrix n n α", "inst✝ : Invertible B"], "goal": "A * ⅟B * B = A"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_3", "V : Type u_4", "PartialOrder Γ", "AddCommMonoid V", "SMul R V", "x y : HahnSeries Γ V"], "goal": "(HahnModule.of R) (x + y) = (HahnModule.of R) x + (HahnModule.of R) y"}}
+{"state": {"context": ["b : Ordinal.{u_1}", "IH : ∀ k < b, ∀ ⦃a : Ordinal.{u_1}⦄, a < k → beth a < beth k", "a : Ordinal.{u_1}", "h : a < b"], "goal": "beth a < beth b"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_5", "ι : Type u_8", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "Nonempty ι", "TopologicalSpace E", "p : SeminormFamily 𝕜 E ι", "hp : WithSeminorms p"], "goal": "(𝓝 0).HasBasis (fun s => s ∈ p.basisSets) id"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "AddCommSemigroup α", "Sub α", "OrderedSub α", "a b : α", "hb : AddLECancellable b"], "goal": "a + b - b = a"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "Field K", "Field L", "Algebra K L", "self : IntermediateField K L", "x : L"], "goal": "x ∈ self.carrier → x⁻¹ ∈ self.carrier"}}
+{"state": {"context": ["X Y : Grp", "f : X ⟶ Y"], "goal": "Grp.uliftFunctor.map f = Grp.ofHom (MulEquiv.ulift.symm.toMonoidHom.comp (MonoidHom.comp f MulEquiv.ulift.toMonoidHom))"}}
+{"state": {"context": ["S : Type u_2", "Ring S", "A : Type u_4", "CommRing A", "Algebra A S", "pb : PowerBasis A S"], "goal": "IsIntegral A pb.gen"}}
+{"state": {"context": ["K : Type u", "Field K", "x : RatFunc K"], "goal": "IsCoprime x.num x.denom"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "h : f.NeBot"], "goal": "∃ u, ↑u ≤ f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : TopologicalSpace α", "inst✝¹ : LinearOrder α", "inst✝ : OrderTopology α", "a : α"], "goal": "𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "J : Type u₂", "inst✝ : Category.{v₂, u₂} J", "F : Jᵒᵖ ⥤ C", "c : Cone F.rightOp", "hc : IsLimit c", "s : Cocone F", "m : (coconeOfConeRightOp c).pt ⟶ s.pt", "w : ∀ (j : Jᵒᵖ), (coconeOfConeRightOp c).ι.app j ≫ m = s.ι.app j", "j : J"], "goal": "(m.op ≫ c.π.app j).unop = (((fun s => (hc.lift (coneRightOpOfCocone s)).unop) s).op ≫ c.π.app j).unop"}}
+{"state": {"context": ["G : Type u_1", "AddLeftCancelMonoid G", "x : G", "hx : IsOfFinAddOrder x", "n : ℕ", "hn : ∃ m, m • x = n • x"], "goal": "(finEquivMultiples x hx).symm ⟨n • x, hn⟩ = ⟨n % addOrderOf x, ⋯⟩"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w : W", "hw : w ≠ 1"], "goal": "∃ i, cs.IsLeftDescent w i"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "inst✝¹ : MeasurableSpace E", "m : MeasurableSpace Ω", "ℙ : Measure Ω", "μ : Measure E", "inst✝ : IsFiniteMeasure ℙ", "X : Ω → ℝ", "f : ℝ → ℝ", "g : ℝ → ℝ≥0∞", "hg : pdf X ℙ volume =ᶠ[ae volume] g", "hgi : ∫⁻ (x : ℝ), ↑‖f x‖₊ * g x ≠ ⊤", "this : (fun x => ↑‖f x‖₊ * g x) =ᶠ[ae volume] fun x => ↑‖f x * (pdf X ℙ volume x).toReal‖₊"], "goal": "∫⁻ (a : ℝ), ↑‖f a * (pdf X ℙ volume a).toReal‖₊ < ⊤"}}
+{"state": {"context": ["α : Type u", "M : Type u_1", "N : Type u_2", "inst✝ : DivisionCommMonoid α", "a b c d : α", "hb : IsUnit b", "hd : IsUnit d"], "goal": "a / b = c / d ↔ a * d = c * b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x : Fin n → ℕ"], "goal": "0 < 1 - 2 / rexp 1"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : MulZeroClass R", "a b : R"], "goal": "¬IsLeftRegular 0 ↔ Nontrivial R"}}
+{"state": {"context": ["x : SetTheory.PGame", "xl' xr' : Type u_1", "el : xl' ≃ x.LeftMoves", "er : xr' ≃ x.RightMoves", "j : x.RightMoves"], "goal": "(SetTheory.PGame.relabel el er).moveRight (er.symm j) = x.moveRight j"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "AddZeroClass M", "AddZeroClass N", "f : M →+ N"], "goal": "(↑f).toFun = ⇑f"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f g : α → β", "inst✝ : CommMonoid β", "s : Finset ι", "t : ι → Finset α", "h : (↑s).PairwiseDisjoint t"], "goal": "1 = Multiset.foldr (fun x x_1 => x * x_1) ⋯ 1 (Multiset.map (fun i => 1) s.val)"}}
+{"state": {"context": ["α✝ : Type u", "β : Type u_1", "γ : Type u_2", "r : α✝ → α✝ → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "α : Type u"], "goal": "Small.{v, u} α ↔ ∃ c', lift.{v + 1, u} #α = lift.{max u (v + 1), v} c'"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "A : Type u_4", "B : Type u_5", "C : Type u_6", "inst✝⁴ : AddCommMonoid A", "inst✝³ : AddCommMonoid B", "inst✝² : AddCommMonoid C", "t : ι → A → C", "h0 : ∀ (i : ι), t i 0 = 0", "h1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y", "s : Finset α", "f✝ : α → ι →₀ A", "i : ι", "g : ι →₀ A", "k : ι → A → γ → B", "x : γ", "β : Type u_7", "M : Type u_8", "M' : Type u_9", "N : Type u_10", "P : Type u_11", "G : Type u_12", "H : Type u_13", "R : Type u_14", "S : Type u_15", "inst✝¹ : Fintype α", "inst✝ : AddCommMonoid M", "f : α →₀ M"], "goal": "∑ a : α, single a (f a) = f"}}
+{"state": {"context": ["α : Type u_1", "AddGroup α", "LinearOrder α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "DenselyOrdered α", "a b : α"], "goal": "a ≤ b ↔ ∀ (ε : α), 0 < ε → a ≤ b + ε"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type x", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq β", "inst✝ : DecidableEq γ", "b : β", "c : γ", "f : δ ≃. β", "x✝ : δ", "a✝ : γ", "h : ∀ (a : δ), ¬f a = some b"], "goal": "a✝ ∈ (f.trans (single b c)) x✝ ↔ a✝ ∈ ⊥ x✝"}}
+{"state": {"context": ["n : ℕ", "i : Fin (n + 1)", "h₀ : 0 < i", "hₙ : i < Fin.last n", "j : Fin n"], "goal": "{i, j.castSucc, j.succ}.card ≤ n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "c : CategoryTheory.Limits.BinaryCofan X Y"], "goal": "CategoryTheory.IsVanKampenColimit c ↔\n  ∀ {X' Y' : C} (c' : CategoryTheory.Limits.BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt),\n    CategoryTheory.CategoryStruct.comp αX c.inl = CategoryTheory.CategoryStruct.comp c'.inl f →\n      CategoryTheory.CategoryStruct.comp αY c.inr = CategoryTheory.CategoryStruct.comp c'.inr f →\n        (Nonempty (CategoryTheory.Limits.IsColimit c') ↔\n          CategoryTheory.IsPullback c'.inl αX f c.inl ∧ CategoryTheory.IsPullback c'.inr αY f c.inr)"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "ι₁ : Type u_4", "ι₂ : Type u_5", "inst✝⁷ : CommRing R", "inst✝⁶ : AddCommGroup M₁", "inst✝⁵ : AddCommGroup M₂", "inst✝⁴ : Module R M₁", "inst✝³ : Module R M₂", "inst✝² : Fintype ι₁", "inst✝¹ : Finite ι₂", "inst✝ : DecidableEq ι₁", "b₁ : Basis ι₁ R M₁", "b₂ : Basis ι₂ R M₂"], "goal": "(fun i => ((toMatrix b₁ b₂) 0).toMvPolynomial i) = 0"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "T₁ T₂ : Triangle C", "hT₁ : T₁ ∈ distinguishedTriangles", "hT₂ : T₂ ∈ distinguishedTriangles", "a✝ : T₁.obj₁ ⟶ T₂.obj₁", "c : T₁.obj₃ ⟶ T₂.obj₃", "comm : T₁.mor₃ ≫ (shiftFunctor C 1).map a✝ = c ≫ T₂.mor₃", "a : T₁.invRotate.obj₃ ⟶ T₂.invRotate.obj₃", "ha₁ : T₁.invRotate.mor₂ ≫ a = a✝ ≫ T₂.invRotate.mor₂", "ha₂ : T₁.invRotate.mor₃ ≫ (shiftFunctor C 1).map ((shiftFunctor C (-1)).map c) = a ≫ T₂.invRotate.mor₃"], "goal": "∃ b, T₁.mor₁ ≫ b = a✝ ≫ T₂.mor₁ ∧ T₁.mor₂ ≫ c = b ≫ T₂.mor₂"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "J : Type u_4", "inst✝¹⁵ : Category.{u_7, u_1} C", "inst✝¹⁴ : Category.{u_6, u_2} D", "inst✝¹³ : Category.{?u.106857, u_3} E", "inst✝¹² : Category.{?u.106861, u_4} J", "F₁ F₂ F₃ : C ⥤ D", "τ : F₁ ⟶ F₂", "τ' : F₂ ⟶ F₃", "e : F₁ ≅ F₂", "G G' : D ⥤ E", "τ'' : G ⟶ G'", "H : E ⥤ J", "A : Type u_5", "inst✝¹¹ : AddMonoid A", "inst✝¹⁰ : HasShift C A", "inst✝⁹ : HasShift D A", "inst✝⁸ : HasShift E A", "inst✝⁷ : HasShift J A", "inst✝⁶ : F₁.CommShift A", "inst✝⁵ : F₂.CommShift A", "inst✝⁴ : F₃.CommShift A", "inst✝³ : G.CommShift A", "inst✝² : G'.CommShift A", "inst✝¹ : H.CommShift A", "inst✝ : CommShift τ A", "a : A", "X : C"], "goal": "(shiftFunctor D a).map (τ.app X) = (F₁.commShiftIso a).inv.app X ≫ τ.app ((shiftFunctor C a).obj X) ≫ (F₂.commShiftIso a).hom.app X"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : GeneralizedCoheytingAlgebra α", "a b c d : α"], "goal": "a ∆ b \\ c = a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : LinearOrderedField α", "inst✝¹ : Ring β", "abv : β → α", "inst✝ : IsAbsoluteValue abv", "f g : ℕ → β", "a : ℕ → α", "n : ℕ", "hm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m", "hg : IsCauSeq abs fun n => ∑ i ∈ range n, a i", "ε : α", "ε0 : ε > 0", "i : ℕ", "hi : ∀ j ≥ i, |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2", "j : ℕ", "ji : j ≥ max n i", "hi₁ : |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| < ε / 2", "hi₂ : |(fun n => ∑ i ∈ range n, a i) (max n i) - (fun n => ∑ i ∈ range n, a i) i| < ε / 2", "sub_le : |∑ k ∈ range j, a k - ∑ k ∈ range (max n i), a k| ≤ |∑ k ∈ range j, a k - ∑ k ∈ range i, a k| + |∑ k ∈ range i, a k - ∑ k ∈ range (max n i), a k|", "this : |(fun n => ∑ i ∈ range n, a i) j - (fun n => ∑ i ∈ range n, a i) i| + |(fun n => ∑ i ∈ range n, a i) i - ∑ k ∈ range (max n i), a k| < ε"], "goal": "abv ((fun n => ∑ i ∈ range n, f i) j - (fun n => ∑ i ∈ range n, f i) (max n i)) ≤ ∑ k ∈ range j, a k - ∑ k ∈ range (max n i), a k"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "σ : Type u_5", "inst✝⁴ : Primcodable α", "inst✝³ : Primcodable β", "inst✝² : Primcodable γ", "inst✝¹ : Primcodable δ", "inst✝ : Primcodable σ", "f : α → β", "g : α → ℕ × β → β", "hf : Primrec f", "hg : Primrec₂ g", "n : ℕ", "a : α"], "goal": "Nat.rec 0 (fun n_1 n_ih => Nat.rec (encode (Option.map f (decode n_1))) (fun y IH => encode (Option.map (fun p => g p.1 p.2) ((decode n_1).bind fun a => Option.map (Prod.mk a ∘ Prod.mk y) (decode IH.pred)))) (Nat.unpair n).2) (encode (some a)) = encode ((some a).bind fun a => some (Nat.rec (f a) (fun n IH => g a (n, IH)) (Nat.unpair n).2))"}}
+{"state": {"context": ["R : Type u_1", "S : Type ?u.268667", "inst✝¹² : CommRing R", "inst✝¹¹ : CommRing S", "inst✝¹⁰ : Algebra R S", "inst✝⁹ : LocalRing R", "M : Type u_2", "N : Type u_3", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : AddCommGroup N", "inst✝⁶ : Module R M", "inst✝⁵ : Module R N", "P : Type u_4", "inst✝⁴ : AddCommGroup P", "inst✝³ : Module R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "hg : Surjective ⇑g", "h : Exact ⇑f ⇑g", "inst✝² : Finite R M", "inst✝¹ : Finite R N", "inst✝ : Free R N", "hf : Function.Injective ⇑(LinearMap.lTensor k f)", "this : FinitePresentation R P"], "goal": "Free R P"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜'", "σ : 𝕜 →+* 𝕜'", "σ' : 𝕜' →+* 𝕜", "RingHomInvPair σ σ'", "RingHomIsometric σ", "RingHomIsometric σ'", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_4", "NormedAddCommGroup F", "NormedSpace 𝕜' F", "f : E →SL[σ] F", "CompleteSpace F", "CompleteSpace E", "hsurj : Function.Surjective ⇑f", "s : Set F"], "goal": "frontier (⇑f ⁻¹' s) = ⇑f ⁻¹' frontier s"}}
+{"state": {"context": ["a b c d m n i : ℤ"], "goal": "-i - i = 2 * -i"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "C : Set (Set α)", "self : μ.FiniteSpanningSetsIn C"], "goal": "⋃ i, self.set i = Set.univ"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : GeneralizedBooleanAlgebra α", "inst✝² : DecidableRel Disjoint", "inst✝¹ : DecidableRel fun x x_1 => x ≤ x_1", "s✝ : Finset α", "u✝ v a b : α", "inst✝ : DecidableEq α", "u : α", "s : Finset α"], "goal": "filter (fun x => compress u u x ∈ s) s ∪ filter (fun x => x ∉ s) (image (compress u u) s) = s"}}
+{"state": {"context": ["α : Type u", "s t : Set α", "x : α", "h1 : x ∈ s", "h2 : ¬x ∈ t"], "goal": "x ∈ s \\ t"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "p p' : Submodule R M", "h : p ≤ p'"], "goal": "Submodule.map p'.subtype (LinearMap.range (Submodule.inclusion h)) = p"}}
+{"state": {"context": ["n : ℕ", "a : Fin n", "b : Fin (n + 1)"], "goal": "a.succ.predAbove b.succ = (a.predAbove b).succ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "μ : MeasureTheory.Measure α", "κ : ProbabilityTheory.Kernel α β", "MeasureTheory.SFinite μ", "ProbabilityTheory.IsSFiniteKernel κ", "s : Set (α × β)", "hs : MeasurableSet s"], "goal": "(μ ⊗ₘ κ) s = ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' s) ∂μ"}}
+{"state": {"context": ["X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "s : Set ↑Γ(X, ⊤)", "hs : Ideal.span s = ⊤", "hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)"], "goal": "∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "ϕ : L →ᴸ L'", "M : Type u_1", "L.Structure M", "L'.Structure M", "ϕ.IsExpansionOn M", "n : ℕ", "R : L.Relations n", "x : Fin n → M"], "goal": "FirstOrder.Language.Structure.RelMap (ϕ.onRelation R) x = FirstOrder.Language.Structure.RelMap R x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "OmegaCompletePartialOrder α", "OmegaCompletePartialOrder β", "f : α →o β"], "goal": "OmegaCompletePartialOrder.Continuous' ⇑f ↔ OmegaCompletePartialOrder.Continuous f"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "φ ψ : PowerSeries R"], "goal": "(φ * ψ).order = φ.order + ψ.order"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : LocallyFiniteOrderBot α", "inst✝ : DecidableEq α", "b a✝ : α"], "goal": "a✝ ∈ (Iic b).erase b ↔ a✝ ∈ Iio b"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "n : σ →₀ ℕ", "φ : MvPowerSeries σ R", "a : R"], "goal": "(MvPowerSeries.coeff R n) ((MvPowerSeries.C σ R) a * φ) = a * (MvPowerSeries.coeff R n) φ"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "f : E →ₛₗᵢ[σ₁₂] E₂", "x : E"], "goal": "f (-x) = -f x"}}
+{"state": {"context": ["z : ℂ", "hz : ‖z‖ < 1"], "goal": "log (1 + z) - z = log (1 + z) - logTaylor (1 + 1) z"}}
+{"state": {"context": ["a b : Int"], "goal": "-a ∣ b ↔ a ∣ b"}}
+{"state": {"context": ["l m r : List Char", "s : Substring"], "goal": "Substring.ValidFor l m r s → ∀ (n : Nat), Substring.ValidFor l (List.take n m) (List.drop n m ++ r) (s.take n)"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "x : M"], "goal": "2 • x = x + x"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "TopologicalSpace.SeparableSpace α", "PartialOrder α"], "goal": "∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∈ s) ∧ ∀ (x : α), IsTop x → x ∈ s"}}
+{"state": {"context": ["α : Type u_1", "E✝ : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝³ : NormedAddCommGroup E✝", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedAddCommGroup G", "E : Type u_5", "inst✝ : NormedAddCommGroup E", "f : ℕ → α → E", "p : ℝ", "hp_pos : 0 < p", "hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ", "f_lim : α → E", "h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))", "h_pow_liminf : liminf (fun n => eLpNorm' (f n) p μ) atTop ^ p = liminf (fun n => eLpNorm' (f n) p μ ^ p) atTop"], "goal": "liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ p ∂μ) atTop = liminf (fun n => eLpNorm' (f n) p μ ^ p) atTop"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "T : Set α → E →L[ℝ] F", "h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0", "h_add : FinMeasAdditive μ T", "f g : α →ₛ E", "hf : Integrable (↑f) μ", "h : ↑f =ᶠ[ae μ] ↑g", "x y : E", "hxy : x ≠ y"], "goal": "μ (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0"}}
+{"state": {"context": ["α : Type u", "x : α"], "goal": "(FreeAddSemigroup.of x).tail = []"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝⁹ : CommSemiring ι", "inst✝⁸ : Module ι (Additive ℤˣ)", "inst✝⁷ : DecidableEq ι", "inst✝⁶ : CommRing R", "inst✝⁵ : Ring A", "inst✝⁴ : Ring B", "inst✝³ : Algebra R A", "inst✝² : Algebra R B", "𝒜 : ι → Submodule R A", "ℬ : ι → Submodule R B", "inst✝¹ : GradedAlgebra 𝒜", "inst✝ : GradedAlgebra ℬ"], "goal": "∀ (m : R) (x : B), { toFun := fun b => 1 ᵍ⊗ₜ[R] b, map_add' := ⋯ }.toFun (m • x) = (RingHom.id R) m • { toFun := fun b => 1 ᵍ⊗ₜ[R] b, map_add' := ⋯ }.toFun x"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_5"], "goal": "∀ {x : MeasurableSpace α} (f : α → ℝ≥0∞) (μ : ι → MeasureTheory.Measure α),\n  HasSum (fun i => ∫⁻ (a : α), f a ∂μ i) (∫⁻ (a : α), f a ∂MeasureTheory.Measure.sum μ)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_3", "N : Type u_4", "ι : Type u_5", "CommSemiring R", "AddCommMonoid M", "AddCommMonoid N", "Module R M", "Module R N", "TopologicalSpace R", "TopologicalSpace M", "TopologicalSpace N", "ContinuousSMul R N", "f : M [⋀^ι]→L[R] R", "z : N"], "goal": "∀ (a : (i : ι) → (fun x => M) i), (f.smulRight z) a = f a • z"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "k : ℕ", "𝒜 : Finset (Finset α)"], "goal": "𝒜 # k ⊆ Finset.falling k 𝒜"}}
+{"state": {"context": [], "goal": "SetTheory.PGame.star.RightMoves = PUnit.{u_1 + 1}"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "x y : α", "inst✝¹ : DecidableRel f.SameCycle", "inst✝ : DecidableRel g.SameCycle", "h : f.SameCycle x y", "z : α"], "goal": "(f.cycleOf x) z = (f.cycleOf y) z"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁶ : NormedAddCommGroup E", "inst✝¹⁵ : NormedAddCommGroup E'", "inst✝¹⁴ : NormedAddCommGroup E''", "inst✝¹³ : NormedAddCommGroup F", "f✝ f' : G → E", "g✝ g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝¹² : RCLike 𝕜", "inst✝¹¹ : NormedSpace 𝕜 E", "inst✝¹⁰ : NormedSpace 𝕜 E'", "inst✝⁹ : NormedSpace 𝕜 E''", "inst✝⁸ : NormedSpace ℝ F", "inst✝⁷ : NormedSpace 𝕜 F", "inst✝⁶ : CompleteSpace F", "inst✝⁵ : MeasurableSpace G", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : BorelSpace G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup P", "inst✝ : NormedSpace 𝕜 P", "μ : Measure G", "L✝ : E →L[𝕜] E' →L[𝕜] F", "n : ℕ∞", "L : E →L[𝕜] E' →L[𝕜] F", "s : Set P", "v : P → G", "hv : ContDiffOn 𝕜 n v s", "f : G → E", "g : P → G → E'", "k : Set G", "hs : IsOpen s", "hk : IsCompact k", "hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0", "hf : LocallyIntegrable f μ", "hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)", "x : P", "hx : x ∈ s"], "goal": "x ∈ (fun x => (_root_.id x, v x)) ⁻¹' s ×ˢ univ"}}
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+{"state": {"context": ["f : ℕ → ℂ", "l : ℂ", "h : Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 l)", "z : ℂ", "hz : ‖z‖ < 1", "s : ℕ → ℂ := fun n => ∑ i ∈ range n, f i", "k : Tendsto (fun x => ∑ i ∈ range x, f i - ∑ i ∈ range x, f i * z ^ i) atTop (𝓝 (l - ∑' (b : ℕ), f b * z ^ b))"], "goal": "Tendsto (fun n => (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range (i + 1), f j) * z ^ i) atTop (𝓝 (l - ∑' (n : ℕ), f n * z ^ n))"}}
+{"state": {"context": ["K : Type u", "CommRing K", "p : ℕ", "Fact (Nat.Prime p)", "CharP K p", "x y : ℕ"], "goal": "↑x = ↑y ↔ ↑x = ↑y"}}
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+{"state": {"context": ["m n : ℕ+", "k l : ℕ", "h : m.Coprime n"], "goal": "(↑m ^ k).Coprime (↑n ^ l)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s✝ t u v : Finset α", "a✝ b : α", "s : Finset α", "a : α"], "goal": "s.erase a = ∅ ↔ s = ∅ ∨ s = {a}"}}
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+{"state": {"context": ["E : Type u_1", "SeminormedCommGroup E", "δ : ℝ", "s : Set E"], "goal": "s / Metric.ball 1 δ = Metric.thickening δ s"}}
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+{"state": {"context": ["α : Type u", "p q : α → Prop"], "goal": "{x | x ∈ {y | p y} ∧ q x} = {x | p x ∧ q x}"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π : BoxIntegral.Prepartition I", "h : π.IsPartition"], "goal": "π.iUnion = ↑I"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "c : R", "f : RingSeminorm R", "hf1 : f 1 ≤ 1", "hc : f c ≠ 0", "hpm : IsPowMul ⇑f", "x : R"], "goal": "∀ (b : ℕ), 1 ≤ b → seminormFromConst_seq c f x b ≤ f x"}}
+{"state": {"context": ["X : Type u_1", "MetricSpace X", "CompleteSpace X", "x : X", "ε : ℝ", "ε_pos : 0 < ε", "ϕ : X → ℝ", "cont : Continuous ϕ", "nonneg : ∀ (y : X), 0 ≤ ϕ y"], "goal": "∃ ε' > 0, ∃ x', ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ (y : X), d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x'"}}
+{"state": {"context": ["α : Type u", "c d : Cardinal.{u}"], "goal": "∀ (i i_1 : ℕ), i < i_1 → i < i_1"}}
+{"state": {"context": ["X : TopCat"], "goal": "∀ (Z B : LightProfinite) (π : Z ⟶ B) [inst : EffectiveEpi π], QuotientMap ⇑(LightProfinite.toTopCat.map π)"}}
+{"state": {"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X : Scheme", "U : X.Opens", "r : ↑Γ(X, U)"], "goal": "U.ι ''ᵁ (↑U).basicOpen (U.topIso.inv r) = X.basicOpen r"}}
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+{"state": {"context": [], "goal": "Ordinal.type EmptyRelation = 0"}}
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+{"state": {"context": ["α : Type u", "L₁ L₂ L₃ : List (α × Bool)"], "goal": "FreeAddGroup.Red.Step L₁ L₂ → FreeAddGroup.Red.Step (L₁ ++ L₃) (L₂ ++ L₃)"}}
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+{"state": {"context": ["σ R : Type u", "inst✝ : CommSemiring R"], "goal": "max (max (lift.{u, u} #R) (lift.{u, u} #σ)) ℵ₀ ≤ max (max #R #σ) ℵ₀"}}
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+{"state": {"context": ["x y z : ℤ", "h : PythagoreanTriple x y z"], "goal": "x * x + y * y = z * z"}}
+{"state": {"context": ["S : Type u_1", "Semiring S", "l : List S[X]", "n : ℕ", "hl : ∀ p ∈ l, p.natDegree ≤ n"], "goal": "l.prod.coeff (l.length * n) = (List.map (fun p => p.coeff n) l).prod"}}
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+{"state": {"context": ["R : Type u", "CommRing R", "x y z : R"], "goal": "IsCoprime (z * y + x) y ↔ IsCoprime x y"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "α : Type u_2", "inst✝ : Mul α", "J : Subgroup G", "g : G", "H K : Subgroup G", "a b : G", "hb : b ∈ doset a ↑H ↑K"], "goal": "doset b ↑H ↑K = doset a ↑H ↑K"}}
+{"state": {"context": ["n : ℕ", "x : ℝ"], "goal": "Irrational (↑n - x) ↔ Irrational x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : T1Space X", "s : Set X", "h : IsPreconnected s", "hs : s.Nontrivial", "hf : s.Finite", "this : DiscreteTopology ↑s"], "goal": "Subsingleton ↑s"}}
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+{"state": {"context": ["R : Type u_1", "S : Type u_2", "n : Type u_3", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : Fintype n", "inst✝ : DecidableEq n", "f : R →+* S"], "goal": "((optionEquivLeft R (n × n)).symm (univ R n)).IsHomogeneous (Fintype.card n)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "s : Multiset R"], "goal": "(map (fun x => X + C (-x)) s).prod = ∑ j ∈ Finset.range (card s + 1), (-1) ^ j * (C (s.esymm j) * X ^ (card s - j))"}}
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+{"state": {"context": ["M : Type u_1", "Monoid M", "MeasurableSpace M", "MeasurableMul₂ M", "μ ν ρ : MeasureTheory.Measure M", "MeasureTheory.SFinite μ", "MeasureTheory.SFinite ν", "MeasureTheory.SFinite ρ"], "goal": "μ.mconv (ν + ρ) = μ.mconv ν + μ.mconv ρ"}}
+{"state": {"context": ["n k p : ℕ", "inst✝ : Fact (Nat.Prime p)", "a : ℕ", "ha₁ : n < p ^ a", "ha₂ : k < p ^ a"], "goal": "n.choose k ≡ ∏ i ∈ range a, (n / p ^ i % p).choose (k / p ^ i % p) [MOD p]"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : Zero R", "a b : Γ", "r : R", "x : HahnSeries Γ R", "i : Γ", "hx : x ≠ 0", "hi : i ∈ x.support"], "goal": "¬i < ⋯.min ⋯"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p₁ p₂ p₃ : P"], "goal": "EuclideanGeometry.angle p₁ p₂ p₃ = 0 ↔ p₁ = p₃ ∧ p₁ ≠ p₂ ∨ Sbtw ℝ p₂ p₁ p₃ ∨ Sbtw ℝ p₂ p₃ p₁"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X"], "goal": "(⨅ s, ⨅ (_ : IsClosed s ∧ IsLindelof s), 𝓟 sᶜ).HasBasis (fun s => IsClosed s ∧ IsLindelof s) compl"}}
+{"state": {"context": ["α : Type u_1", "CancelMonoidWithZero α", "a b c : α", "ha : a ≠ 0"], "goal": "a * b ∣ a * c ↔ b ∣ c"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "n : ℕ", "hn : 0 < n", "p : FormalMultilinearSeries 𝕜 E F", "q : FormalMultilinearSeries 𝕜 F E", "v : Fin n → F", "A : univ = {c | 1 < c.length}.toFinset ∪ {Composition.single n hn}", "B : Disjoint {c | 1 < c.length}.toFinset {Composition.single n hn}", "C : (p (Composition.single n hn).length) (q.applyComposition (Composition.single n hn) v) = (p 1) fun x => (q n) v"], "goal": "(p.comp q n) v = ∑ c ∈ {c | 1 < c.length}.toFinset, (p c.length) (q.applyComposition c v) + (p 1) fun x => (q n) v"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H K : Subgroup G", "h1 : Disjoint H K", "h2 : ↑H * ↑K = Set.univ"], "goal": "H.IsComplement' K"}}
+{"state": {"context": ["R : Type u", "inst✝⁵ : OrderedRing R", "inst✝⁴ : StarRing R", "inst✝³ : StarOrderedRing R", "inst✝² : Algebra ℝ R", "inst✝¹ : OrderedSMul ℝ R", "inst✝ : StarModule ℝ R", "A₀ A₁ B₀ B₁ : R", "T : IsCHSHTuple A₀ A₁ B₀ B₁", "M : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x", "P : R := (√2)⁻¹ • (A₁ + A₀) - B₀", "Q : R := (√2)⁻¹ • (A₁ - A₀) + B₁", "w : √2 ^ 3 • 1 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁ = (√2)⁻¹ • (P ^ 2 + Q ^ 2)", "pos : 0 ≤ (√2)⁻¹ • (P ^ 2 + Q ^ 2)"], "goal": "0 ≤ √2 ^ 3 • 1 - (A₀ * B₀ + A₀ * B₁ + A₁ * B₀ - A₁ * B₁)"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0"], "goal": "w₁ + w₂ + w₃ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "RCLike 𝕂", "NormedCommRing 𝔸", "NormedAlgebra 𝕂 𝔸", "CompleteSpace 𝔸", "ι : Type u_3", "s : Finset ι", "f : ι → 𝔸"], "goal": "NormedSpace.exp 𝕂 (∑ i ∈ s, f i) = ∏ i ∈ s, NormedSpace.exp 𝕂 (f i)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "t : CategoryTheory.Limits.Fork f g", "lift : (s : CategoryTheory.Limits.Fork f g) → s.pt ⟶ t.pt", "fac : ∀ (s : CategoryTheory.Limits.Fork f g), CategoryTheory.CategoryStruct.comp (lift s) t.ι = s.ι", "uniq :\n  ∀ (s : CategoryTheory.Limits.Fork f g) (m : s.pt ⟶ t.pt), CategoryTheory.CategoryStruct.comp m t.ι = s.ι → m = lift s", "s : CategoryTheory.Limits.Fork f g"], "goal": "(CategoryTheory.Limits.Fork.IsLimit.mk t lift fac uniq).lift s = lift s"}}
+{"state": {"context": ["α : Type u_1", "Monoid α", "DecidableRel fun x x_1 => x ∣ x_1", "a b : α"], "goal": "multiplicity a b = 0 ↔ ¬a ∣ b"}}
+{"state": {"context": ["F : Type w", "R : Type u", "S : Type v", "NonUnitalNonAssocRing R", "NonUnitalNonAssocRing S", "FunLike F R S", "NonUnitalRingHomClass F R S", "f : F", "s : NonUnitalSubring R"], "goal": "↑(NonUnitalSubring.map f s) = ⇑f '' ↑s"}}
+{"state": {"context": ["Γ✝ : Type u_1", "R : Type u_2", "inst✝³ : PartialOrder Γ✝", "inst✝² : AddMonoid R", "Γ : Type u_3", "inst✝¹ : Zero Γ", "inst✝ : LinearOrder Γ", "x y : HahnSeries Γ R", "hxy : x + y ≠ 0", "hx : ¬x = 0", "hy : ¬y = 0"], "goal": "min (⋯.min ⋯) (⋯.min ⋯) ≤ ⋯.min ⋯"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "f : ι → α → β", "μ : MeasureTheory.Measure α", "p : α → (ι → β) → Prop", "CompleteLattice β", "Countable ι", "hf : ∀ (i : ι), AEMeasurable (f i) μ", "hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x"], "goal": "⨆ n, aeSeq hf p n =ᵐ[μ] ⨆ n, f n"}}
+{"state": {"context": ["p m n : ℕ", "pp : Nat.Prime p", "h : p ∣ m ^ n"], "goal": "p ∣ m"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "σ α β : Type u_1", "Monad m", "x : StateT σ m α", "f : α → StateT σ m β", "s : σ"], "goal": "(x >>= f).run s = do\n  let p ← x.run s\n  (f p.fst).run p.snd"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommRing R", "CommRing S", "Algebra R S", "g : R[X]", "pb : PowerBasis R S", "h₁ : (Polynomial.aeval (AdjoinRoot.root g)) (minpoly R pb.gen) = 0", "h₂ : (Polynomial.aeval pb.gen) g = 0"], "goal": "↑(AdjoinRoot.equiv' g pb h₁ h₂) = AdjoinRoot.liftHom g pb.gen h₂"}}
+{"state": {"context": ["a b c : Ordinal.{u_1}", "ba : b + a = a", "l : c.IsLimit", "IH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b", "c' : Ordinal.{u_1}", "h : c' < c"], "goal": "Ordinal.{u_1}"}}
+{"state": {"context": ["C✝ : Type u", "𝒞 : Category.{v, u} C✝", "inst✝² : MonoidalCategory C✝", "C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : MonoidalCategory C", "U V W X✝ Y✝ Z✝ X Y Z Z' : C", "h : Z ⟶ Z'"], "goal": "(𝟙 (X ⊗ Y) ⊗ h) ≫ (α_ X Y Z').hom = (α_ X Y Z).hom ≫ (𝟙 X ⊗ 𝟙 Y ⊗ h)"}}
+{"state": {"context": ["R : Type u", "Ring R", "X₁ X₂ X₃ : ModuleCat R", "f : X₁ ⟶ X₂", "g : X₂ ⟶ X₃", "hfg : LinearMap.range f ≤ LinearMap.ker g"], "goal": "(CategoryTheory.ShortComplex.moduleCatMkOfKerLERange f g hfg).X₃ = X₃"}}
+{"state": {"context": ["Γ₀ : Type u_2", "LinearOrderedCommGroupWithZero Γ₀"], "goal": "𝓝 0 = ⨅ γ, ⨅ (_ : γ ≠ 0), 𝓟 (Set.Iio γ)"}}
+{"state": {"context": ["A : Type u_1", "inst✝ : SeminormedGroup A", "n : ℕ", "δ : ℝ", "a : A"], "goal": "a ∈ approxOrderOf A n δ ↔ ∃ b, orderOf b = n ∧ a ∈ ball b δ"}}
+{"state": {"context": ["α : Type u", "inst✝² : UniformSpace α", "β : Type v", "γ : Type w", "inst✝¹ : UniformSpace β", "inst✝ : UniformSpace γ", "f g : CauchyFilter α", "a b : α", "ha : ↑f ≤ 𝓝 a", "hb : ↑g ≤ 𝓝 b"], "goal": "Inseparable a b ↔ Inseparable f g"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "NormedAddCommGroup F", "g : E → F", "c : ℝ≥0", "Fact (1 ≤ p)", "hg : LipschitzWith c g", "g0 : g 0 = 0"], "goal": "LipschitzWith c (hg.compLp g0)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "l✝ : List α", "x✝ x y x' y' : α", "l l' : List α", "hd : (x :: y :: l).Nodup", "hd' : (x' :: y' :: l').Nodup", "h : (x :: y :: l).formPerm = (x' :: y' :: l').formPerm"], "goal": "(x :: y :: l) ~r (x' :: y' :: l')"}}
+{"state": {"context": ["ι✝ : Type u_1", "κ✝ : Type u_2", "α✝ : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α✝", "a : α✝", "f✝ g✝ : α✝ → β", "inst✝¹ : CommMonoid β", "ι : Type u_6", "κ : Type u_7", "α : Type u_8", "inst✝ : CommMonoid α", "s : Finset ι", "t : Finset κ", "f : ι → α", "g : κ → α", "e : ι → κ", "he : Set.InjOn e ↑s", "hest : Set.MapsTo e ↑s ↑t", "h' : ∀ i ∈ t, i ∉ e '' ↑s → g i = 1", "h : ∀ i ∈ s, f i = g (e i)"], "goal": "∏ i ∈ s, f i = ∏ j ∈ t, g j"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "Group G", "MulAction G α", "H : Subgroup G", "a b : α", "h : ⟦a⟧ = ⟦b⟧"], "goal": "⟦a⟧ = ⟦b⟧"}}
+{"state": {"context": ["X : Type u_3", "Z : Type u_5", "β : Type u_6", "MeasurableSpace X", "StandardBorelSpace X", "MeasurableSpace β", "MeasurableSpace Z", "MeasurableSpace.CountablySeparated Z", "f : X → Z", "hf : Measurable f", "hsurj : Function.Surjective f", "g : Z → β"], "goal": "Measurable (g ∘ f) ↔ Measurable g"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{?u.46503, u_1} C", "inst✝³ : Category.{u_3, u_2} D", "G : C ⥤ D", "A : Type w", "inst✝² : Category.{w', w} A", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "inst✝¹ : G.IsCocontinuous J K", "inst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F", "F : Cᵒᵖ ⥤ A", "hF : Presheaf.IsSheaf J F", "R : Dᵒᵖ ⥤ A", "α : G.op ⋙ R ⟶ F", "hR : (Functor.RightExtension.mk R α).IsPointwiseRightKanExtension", "X : D", "S : K.Cover X", "s : Multifork (S.index R)", "W✝ : A", "f g : W✝ ⟶ R.obj (op X)", "h : ∀ (i : S.Arrow), f ≫ R.map i.f.op = g ≫ R.map i.f.op", "j : StructuredArrow (op X) G.op", "W : C", "i : W ⟶ unop j.right", "hi : (GrothendieckTopology.Cover.sieve ⟨Sieve.functorPullback G (Sieve.pullback j.hom.unop ↑S), ⋯⟩).arrows i", "eq : f ≫ R.map (j.hom ≫ (G.map i).op) = g ≫ R.map (j.hom ≫ (G.map i).op)"], "goal": "(f ≫ R.map j.hom ≫ α.app j.right) ≫ F.map i.op = (g ≫ R.map j.hom ≫ α.app j.right) ≫ F.map i.op"}}
+{"state": {"context": ["α : Type u_1", "AddGroupWithOne α", "n : ZNum"], "goal": "↑n.bitm1 = ↑n + ↑n - 1"}}
+{"state": {"context": ["ι : Type u_1", "I J J₁ J₂ : Box ι", "π π₁ π₂ : Prepartition I", "x : ι → ℝ"], "goal": "x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J"}}
+{"state": {"context": ["α : Type u_1", "p q : Perm α", "n : ℤ", "x : α", "hx : ¬(p ^ n) x = x", "H : p x = x"], "goal": "False"}}
+{"state": {"context": ["x : ℤ[i]"], "goal": "{ re := ↑(star x).re, im := ↑(star x).im } = (starRingEnd ℂ) { re := ↑x.re, im := ↑x.im }"}}
+{"state": {"context": ["S : Type u_1", "inst✝⁷ : CommRing S", "inst✝⁶ : IsDomain S", "inst✝⁵ : Nontrivial S", "inst✝⁴ : IsDedekindDomain S", "inst✝³ : Module.Free ℤ S", "inst✝² : Module.Finite ℤ S", "I : Ideal S", "E : Type u_2", "inst✝¹ : EquivLike E S ↥I", "inst✝ : AddEquivClass E S ↥I", "e : E", "hI : ¬I = ⊥", "ι : Type u_1 := Module.Free.ChooseBasisIndex ℤ S", "b : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := Module.Free.chooseBasis ℤ S", "h✝ : Nonempty ι", "this✝¹ : Fintype (S ⧸ I) := I.fintypeQuotientOfFreeOfNeBot hI", "this✝ : DecidableEq ι := Classical.decEq ι", "a : Module.Free.ChooseBasisIndex ℤ S → ℤ := smithCoeffs b I hI", "b' : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ S := ringBasis b I hI", "ab : Basis (Module.Free.ChooseBasisIndex ℤ S) ℤ ↥I := selfBasis b I hI", "ab_eq : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), ↑((selfBasis b I hI) i) = smithCoeffs b I hI i • (ringBasis b I hI) i", "e' : S ≃ₗ[ℤ] ↥I := b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))", "f : S →ₗ[ℤ] S := ↑ℤ (Submodule.subtype I) ∘ₗ ↑e'", "f_apply : ∀ (x : S), f x = ↑((b'.equiv ab (Equiv.refl (Module.Free.ChooseBasisIndex ℤ S))) x) := fun x => rfl", "ha : ∀ (i : Module.Free.ChooseBasisIndex ℤ S), f (b' i) = a i • b' i", "this : DecidableEq ι := Classical.decEq ι"], "goal": "((LinearMap.toMatrix b' b') f).det.natAbs = (Matrix.diagonal a).det.natAbs"}}
+{"state": {"context": ["K : Type u", "Field K", "G : Type u_1", "Group G", "MulSemiringAction G K", "g : G", "S : ValuationSubring K", "x : K"], "goal": "x ∈ g • S ↔ g⁻¹ • x ∈ S"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "R : Type u_5", "inst✝⁶ : MeasurableSpace X", "inst✝⁵ : TopologicalSpace X", "inst✝⁴ : MeasurableSpace Y", "inst✝³ : TopologicalSpace Y", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "f✝ g : X → E", "μ : Measure X", "s : Set X", "inst✝ : IsLocallyFiniteMeasure μ", "f : X → E", "p : ℝ≥0∞", "hf : Memℒp f p μ", "hp : 1 ≤ p", "x : X"], "goal": "IntegrableAtFilter f (𝓝 x) μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α", "M : Type u_3", "inst✝² : AddCommMonoid M", "inst✝¹ : TopologicalSpace M", "inst✝ : OrderedAddCommMonoid M", "s : SignedMeasure α", "i j : Set α", "hi : ¬restrict s i ≤ restrict 0 i"], "goal": "MeasureTheory.SignedMeasure.ExistsOneDivLT s i (MeasureTheory.SignedMeasure.findExistsOneDivLT s i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "E : Type u_4", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝⁶ : NormedAddCommGroup E", "κ : Kernel α β", "inst✝⁵ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝⁴ : IsSFiniteKernel η", "a : α", "inst✝³ : NormedSpace ℝ E", "E' : Type u_5", "inst✝² : NormedAddCommGroup E'", "inst✝¹ : CompleteSpace E'", "inst✝ : NormedSpace ℝ E'", "g : ↥(Lp E 1 ((κ ⊗ₖ η) a))"], "goal": "Tendsto (fun i => ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a) (𝓝 g) (𝓝 0)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "P : CategoryTheory.MorphismProperty Cᵒᵖ", "hP : P.StableUnderBaseChange"], "goal": "P.unop.StableUnderCobaseChange"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "MeasureTheory.SFinite ν", "MeasureTheory.SFinite μ", "L : Type u_8", "RCLike L", "f : α → L", "g : β → L"], "goal": "∫ (z : α × β), f z.1 * g z.2 ∂μ.prod ν = (∫ (x : α), f x ∂μ) * ∫ (y : β), g y ∂ν"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "P Q : C", "f : P ⟶ Q", "x y : CategoryTheory.Abelian.Pseudoelement P"], "goal": "CategoryTheory.Abelian.Pseudoelement.pseudoApply f x = CategoryTheory.Abelian.Pseudoelement.pseudoApply f y →\n  ∃ z,\n    CategoryTheory.Abelian.Pseudoelement.pseudoApply f z = 0 ∧\n      ∀ (R : C) (g : P ⟶ R),\n        CategoryTheory.Abelian.Pseudoelement.pseudoApply g y = 0 →\n          CategoryTheory.Abelian.Pseudoelement.pseudoApply g z = CategoryTheory.Abelian.Pseudoelement.pseudoApply g x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : β →. γ", "g : α →. β", "a : α"], "goal": "f.comp g a = (g a).bind f"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "δ : ℝ", "s : Set α", "x : α"], "goal": "x ∈ Metric.cthickening δ s ↔ EMetric.infEdist x s ≤ ENNReal.ofReal δ"}}
+{"state": {"context": ["R : Type u", "inst✝ : Ring R", "S : ShortComplex (ModuleCat R)", "X₁ X₂ X₃ : ModuleCat R", "f : X₁ ⟶ X₂", "g : X₂ ⟶ X₃", "hfg : LinearMap.range f = LinearMap.ker g"], "goal": "(moduleCatMkOfKerLERange f g ⋯).Exact"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "b : E × F → G", "u : Set (E × F)", "h : IsBoundedBilinearMap 𝕜 b"], "goal": "DifferentiableOn 𝕜 b u"}}
+{"state": {"context": ["α : Type u_1", "M : IndepMatroid α"], "goal": "M.matroid.E = M.E"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "f : X → Y", "s : Set X", "x : X", "y : Y", "t : Set Y", "h : MapsTo f s t", "hc : Continuous f"], "goal": "MapsTo f (closure s) (closure t)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "T : Set α → F →L[ℝ] F'", "h_add : FinMeasAdditive μ T", "f : α →ₛ G", "hf : Integrable (↑f) μ", "g : G → F", "hg : g 0 = 0", "T_empty : T ∅ = 0", "hfp : ∀ x ∈ f.range, x ≠ 0 → μ (↑f ⁻¹' {x}) ≠ ⊤"], "goal": "∑ x ∈ Finset.image g f.range, (T (↑(map g f) ⁻¹' {x})) x = ∑ x ∈ f.range, (T (↑f ⁻¹' {x})) (g x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoEMetricSpace α", "PseudoEMetricSpace β", "K : ℝ≥0", "s t : Set α", "f : α → β", "hf : LipschitzOnWith K f t", "h : s ⊆ t"], "goal": "LipschitzOnWith K f s"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ ν : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : α → E", "h : MeasureTheory.Memℒp f p (μ + ν)"], "goal": "MeasureTheory.Memℒp f p μ"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝² : HasCoequalizers C", "R S : Mon_ C", "P : Bimod R S", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)"], "goal": "(α_ P.X S.X S.X).hom ≫ (α_ P.X S.X S.X).inv ≫ P.actRight ▷ S.X ≫ P.actRight = coequalizer.π (P.actRight ▷ S.X) ((α_ P.X S.X S.X).hom ≫ P.X ◁ S.mul) ▷ S.X ≫ coequalizer.desc P.actRight ⋯ ▷ S.X ≫ P.actRight"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Computation α", "f : α → Computation β", "g : β → Computation γ"], "goal": "((s.bind f).bind g = s.bind fun x => (f x).bind g) ∨ ∃ s_1, (s.bind f).bind g = (s_1.bind f).bind g ∧ (s.bind fun x => (f x).bind g) = s_1.bind fun x => (f x).bind g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : MeasurableSpace α", "inst✝ : MeasurableSpace β", "m : Measure (Measure α)", "f : α → ℝ≥0∞", "hf : Measurable f"], "goal": "⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * ∫⁻ (μ : Measure α), μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m = ∫⁻ (μ : Measure α), ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * μ (↑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "ContinuousSMul 𝕜 E", "U : Set E", "hU : IsClosed U"], "goal": "IsClosed (balancedCore 𝕜 U)"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "AddGroup G", "AddAction G α", "s : Set α"], "goal": "s \\ MeasureTheory.addFundamentalFrontier G s = MeasureTheory.addFundamentalInterior G s"}}
+{"state": {"context": [], "goal": "↑1 = 1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "X Y : C", "t : CategoryTheory.Limits.BinaryBicone X Y", "ht : CategoryTheory.Limits.IsColimit t.toCocone"], "goal": "t.fst = ht.desc (CategoryTheory.Limits.BinaryCofan.mk (𝟙 X) 0)"}}
+{"state": {"context": ["α : Type u_1", "mα : MeasurableSpace α", "μ : Measure α", "s t✝ : Set α", "inst✝ : SFinite μ", "t : Set α", "ht : MeasurableSet t"], "goal": "(μ.restrict μ.sigmaFiniteSetᶜ) t = (⊤ • μ.toFinite.restrict μ.sigmaFiniteSetᶜ) t"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End R M", "μ : R", "h : f.HasEigenvalue μ", "n : ℕ"], "goal": "∃ x ∈ (f ^ n).eigenspace (μ ^ n), x ≠ 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "K : CochainComplex C ℤ", "a b ab : ℤ", "h : a + b = ab", "n : ℤ"], "goal": "((CategoryTheory.shiftFunctorAdd' (CochainComplex C ℤ) a b ab h).hom.app K).f n = (HomologicalComplex.XIsoOfEq K ⋯).hom"}}
+{"state": {"context": ["M : Type u_1", "AddMonoid M", "MeasurableSpace M", "μ : MeasureTheory.Measure M"], "goal": "μ.conv 0 = 0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : ClosedEmbedding f", "s : Set X"], "goal": "closure (f '' s) = f '' closure s"}}
+{"state": {"context": ["n' : Type u_1", "DecidableEq n'", "Fintype n'", "R : Type u_2", "CommRing R", "A : Matrix n' n' R", "h : IsUnit A.det", "n : ℤ"], "goal": "IsUnit (A ^ n).det"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "Preorder α", "Preorder β", "Preorder γ", "g : β → γ", "f : α → β", "hg : StrictMono g", "hf : StrictAnti f"], "goal": "StrictAnti (g ∘ f)"}}
+{"state": {"context": ["n : ℕ", "NeZero n", "i : ZMod (2 * n)"], "goal": "orderOf (QuaternionGroup.a i) = 2 * n / (2 * n).gcd i.val"}}
+{"state": {"context": ["I : Type w₁", "C : I → Type u₁", "(i : I) → CategoryTheory.Category.{v₁, u₁} (C i)", "i : I", "X Y Z : C i", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "(CategoryTheory.Sigma.SigmaHom.mk f).comp (CategoryTheory.Sigma.SigmaHom.mk g) =\n  CategoryTheory.Sigma.SigmaHom.mk (CategoryTheory.CategoryStruct.comp f g)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : ι → Type u_4", "CommSemiring R", "(i : ι) → Semiring (A i)", "(i : ι) → Algebra R (A i)", "S : Type u_5", "Semiring S", "Algebra R S", "f : MultilinearMap R A S", "one : f 1 = 1", "mul : ∀ (x y : (i : ι) → A i), f (x * y) = f x * f y", "a : ⨂[R] (i : ι), A i"], "goal": "(PiTensorProduct.liftAlgHom f one mul) a = (PiTensorProduct.lift f) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2"], "goal": "Infinite (α × β) ↔ Infinite α ∧ Nonempty β ∨ Nonempty α ∧ Infinite β"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E", "s : Set E", "h : IsBoundedLinearMap 𝕜 f"], "goal": "HasFDerivWithinAt f h.toContinuousLinearMap s x"}}
+{"state": {"context": ["R : Type u", "M : Type v", "AddCommGroup M", "Ring R", "Module R M", "S : Submodule R M", "S.IsPrincipal"], "goal": "Submodule.span R {Submodule.IsPrincipal.generator S} = S"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "inst✝⁴ : Field K", "inst✝³ : Field L", "inst✝² : Field L'", "inst✝¹ : Algebra K L", "inst✝ : Algebra K L'", "S F E✝ E'✝ : IntermediateField K L", "h : F ≤ E✝", "h' : F ≤ E'✝", "x : L", "E E' : { E // F ≤ E }"], "goal": "extendScalars ⋯ ≤ extendScalars ⋯ ↔ E ≤ E'"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q r : ℚ", "hqr : q + r ≠ 0", "h : padicValRat p q ≤ padicValRat p r", "hq : ¬q = 0", "hr : ¬r = 0", "hqn : q.num ≠ 0", "hqd : ↑q.den ≠ 0", "hrn : r.num ≠ 0", "hrd : ↑r.den ≠ 0", "hqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)"], "goal": "padicValRat p q ≤ padicValRat p (q + r)"}}
+{"state": {"context": ["ι : Type u_1", "inst✝³ : DecidableEq ι", "A : ι → Type u_2", "inst✝² : (i : ι) → AddCommMonoid (A i)", "inst✝¹ : AddMonoid ι", "inst✝ : GSemiring A", "a : A 0", "n : ℕ"], "goal": "(of A 0) (a ^ (n + 1)) = (of A 0) a ^ (n + 1)"}}
+{"state": {"context": ["E : Type u_1", "V : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℂ E", "f : V → E", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : MeasurableSpace V", "inst✝³ : BorelSpace V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : FiniteDimensional ℝ V", "inst✝ : CompleteSpace E", "hf1 : Continuous f", "hf2 : HasCompactSupport f", "ε : ℝ", "hε : ε > 0", "R : ℝ", "hR_bd : ∀ (x : V), R ≤ ‖x‖ → f x = 0", "A : Set V := {v | ‖v‖ ≤ R + 1}", "mA : MeasurableSet A", "B : ℝ≥0", "hB_pos : 0 < B", "hB_vol : volume A ≤ ↑B", "δ : ℝ", "hδ1 : δ > 0", "hδ2 : ∀ {a b : V}, dist a b < δ → dist (f a) (f b) < ε / ↑B", "w : V", "hw_bd : 1 / 2 + 1 / (2 * δ) ≤ ‖w‖", "hw_ne : w ≠ 0", "hw'_nm : ‖i w‖ = 1 / (2 * ‖w‖)", "this : ‖1 / 2‖ = 2⁻¹"], "goal": "∫ (a : V), ‖f a - f (a + (1 / (2 * ‖w‖ ^ 2)) • w)‖ < 2 * ε"}}
+{"state": {"context": ["n : ℕ", "h : 0 < n", "NeZero n"], "goal": "Fin.ofNat' 1 h = 1"}}
+{"state": {"context": ["F : Type u", "Field F", "DecidableEq F", "h2 : 2 = 0"], "goal": "EllipticCurve.ofJ 0 = EllipticCurve.ofJ0 F"}}
+{"state": {"context": ["R : Type u", "Semiring R", "f : R[X]", "n : ℕ"], "goal": "Polynomial.X ^ n ∣ f ↔ ∀ d < n, f.coeff d = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ✝ : Type w", "ι : Sort x", "γ : Type u_1", "inst✝¹ : Preorder β", "inst✝ : Preorder γ", "f : α → β", "g : β → γ", "hf : BddAbove (range f)", "hg : Monotone g"], "goal": "BddAbove (range (g ∘ f))"}}
+{"state": {"context": ["R : Type u_1", "M : Type v₁", "N : Type v₂", "S : Type v₃", "inst✝³³ : AddCommMonoid M", "inst✝³² : AddCommMonoid N", "inst✝³¹ : CommSemiring R", "inst✝³⁰ : CommSemiring S", "inst✝²⁹ : Algebra R S", "inst✝²⁸ : Module R M", "inst✝²⁷ : Module R N", "inst✝²⁶ : Module S N", "inst✝²⁵ : IsScalarTower R S N", "f✝ : M →ₗ[R] N", "h : IsBaseChange S f✝", "P : Type u_2", "Q : Type u_3", "inst✝²⁴ : AddCommMonoid P", "inst✝²³ : Module R P", "inst✝²² : AddCommMonoid Q", "inst✝²¹ : Module S Q", "T : Type u_4", "O : Type u_5", "inst✝²⁰ : CommSemiring T", "inst✝¹⁹ : Algebra R T", "inst✝¹⁸ : Algebra S T", "inst✝¹⁷ : IsScalarTower R S T", "inst✝¹⁶ : AddCommMonoid O", "inst✝¹⁵ : Module R O", "inst✝¹⁴ : Module S O", "inst✝¹³ : Module T O", "inst✝¹² : IsScalarTower S T O", "inst✝¹¹ : IsScalarTower R S O", "inst✝¹⁰ : IsScalarTower R T O", "R' : Type u_6", "S' : Type u_7", "inst✝⁹ : CommSemiring R'", "inst✝⁸ : CommSemiring S'", "inst✝⁷ : Algebra R R'", "inst✝⁶ : Algebra S S'", "inst✝⁵ : Algebra R' S'", "inst✝⁴ : Algebra R S'", "inst✝³ : IsScalarTower R R' S'", "inst✝² : IsScalarTower R S S'", "H : IsPushout R S R' S'", "A : Type u_8", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "f : S →ₐ[R] A", "g : R' →ₐ[R] A", "hf : ∀ (x : S) (y : R'), f x * g y = g y * f x", "this✝² : Module S A := Module.compHom A f.toRingHom", "this✝¹ : IsScalarTower R S A", "this✝ : IsScalarTower S A A", "this : ∀ (x : R'), (⋯.lift g.toLinearMap) ((algebraMap R' S') x) = g x"], "goal": "S' →ₐ[R] A"}}
+{"state": {"context": ["α : Type u_1", "s t u : Multiset α"], "goal": "s + t ≤ s + u ↔ t ≤ u"}}
+{"state": {"context": ["R : Type u", "ι : Type w", "s : Finset ι", "inst✝ : CommSemiring R", "f : ι → R[X]", "t✝ : Multiset R[X]", "a : R[X]", "t : Multiset R[X]", "ih : (Multiset.map leadingCoeff t).prod ≠ 0 → t.prod.leadingCoeff = (Multiset.map leadingCoeff t).prod", "h : a.leadingCoeff * (Multiset.map leadingCoeff t).prod ≠ 0"], "goal": "a.leadingCoeff * t.prod.leadingCoeff ≠ 0"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "α : Type u_12", "Zero α"], "goal": "Matrix.fromBlocks 0 0 0 0 = 0"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_3", "σ : Type u_4", "DecidableEq ι", "AddCommMonoid M", "SetLike σ M", "AddSubmonoidClass σ M", "ℳ : ι → σ", "DirectSum.Decomposition ℳ", "x : M", "i j : ι", "hx : x ∈ ℳ i", "hij : i ≠ j"], "goal": "↑(((DirectSum.decompose ℳ) x) j) = 0"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "S : StarSubalgebra R A"], "goal": "Set.range ⇑(algebraMap R A) ≤ ↑S"}}
+{"state": {"context": ["L : Language", "T : L.Theory", "α : Type w", "M : Type w'", "inst✝² : L.Structure M", "inst✝¹ : Nonempty M", "inst✝ : M ⊨ T", "p : T.CompleteType α"], "goal": "∃ M, p ∈ T.realizedTypes (↑M) α"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{u_4, u_1} C", "A : Type u_2", "B : Type u_3", "inst✝² : AddMonoid A", "inst✝¹ : AddMonoid B", "φ : A →+ B", "inst✝ : HasShift C B", "X : PullbackShift C φ", "a₁ a₂ a₃ : A", "h : a₁ + a₂ = a₃", "b₁ b₂ b₃ : B", "h₁ : b₁ = φ a₁", "h₂ : b₂ = φ a₂", "h₃ : b₃ = φ a₃"], "goal": "(shiftFunctorZero (PullbackShift C φ) A).hom.app X = (pullbackShiftIso C φ 0 0 ⋯).hom.app X ≫ (shiftFunctorZero C B).hom.app X"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X : C", "hX : CategoryTheory.Limits.IsTerminal X"], "goal": "CategoryTheory.Limits.IsZero X"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "𝕜 : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝² : RCLike 𝕜", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "t : Set E", "f : α → E", "hf : ∀ (c : Dual 𝕜 E), (fun x => c (f x)) =ᶠ[ae μ] 0", "h't : ∀ᵐ (x : α) ∂μ, f x ∈ t", "d : Set E", "d_count : d.Countable", "hd : t ⊆ closure d", "this : Encodable ↑d", "s : ↑d → E →L[𝕜] 𝕜", "hs : ∀ (x : ↑d), ‖s x‖ ≤ 1 ∧ (s x) ↑x = ↑‖↑x‖", "A : ∀ a ∈ t, (∀ (x : ↑d), (s x) a = 0) → a = 0", "hfs : ∀ (y : ↑d), ∀ᵐ (x : α) ∂μ, (s y) (f x) = 0", "hf' : ∀ᵐ (x : α) ∂μ, ∀ (y : ↑d), (s y) (f x) = 0"], "goal": "f =ᶠ[ae μ] 0"}}
+{"state": {"context": ["α : Type u_1", "Ring α", "n : ℕ", "a a' : α", "h : Mathlib.Meta.NormNum.IsInt a (Int.ofNat n)", "e : ↑n = a'"], "goal": "a = a'"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "S : Type u_10", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Nontrivial S", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Algebra R S", "inst✝³ : Module S M", "inst✝² : Module R M", "inst✝¹ : IsScalarTower R S M", "inst✝ : NoZeroSMulDivisors R S", "b : Basis ι S M", "m : ↥(span R (range ⇑b))", "i : ι"], "goal": "(algebraMap R S) (((restrictScalars R b).repr m) i) = (b.repr ↑m) i"}}
+{"state": {"context": ["R : Type u_1", "Rack R", "x y : R"], "goal": "MulOpposite.op x ◃⁻¹ MulOpposite.op y = MulOpposite.op (x ◃ y)"}}
+{"state": {"context": ["α : Type u_1", "σ : Type u_2", "inst✝¹ : Primcodable α", "inst✝ : Primcodable σ", "c : α → Code", "hc : Primrec c", "z : α → σ", "hz : Primrec z", "s : α → σ", "hs : Primrec s", "l : α → σ", "hl : Primrec l", "r : α → σ", "hr : Primrec r", "pr : α → Code × Code × σ × σ → σ", "hpr : Primrec₂ pr", "co : α → Code × Code × σ × σ → σ", "hco : Primrec₂ co", "pc : α → Code × Code × σ × σ → σ", "hpc : Primrec₂ pc", "rf : α → Code × σ → σ", "hrf : Primrec₂ rf", "PR✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg => pr a (cf, cg, hf, hg)", "CO✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg => co a (cf, cg, hf, hg)", "PC✝ : α → Code → Code → σ → σ → σ := fun a cf cg hf hg => pc a (cf, cg, hf, hg)", "RF✝ : α → Code → σ → σ := fun a cf hf => rf a (cf, hf)", "F : α → Code → σ := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR✝ a) (CO✝ a) (PC✝ a) (RF✝ a)", "G₁ : (α × List σ) × ℕ × ℕ → Option σ := fun p => (p.1.2.get? p.2.2).bind fun s => (p.1.2.get? (unpair p.2.2).1).bind fun s₁ => Option.map (fun s₂ => bif p.2.1.bodd then bif p.2.1.div2.bodd then rf p.1.1 (ofNat Code p.2.2, s) else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂) else bif p.2.1.div2.bodd then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂) else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s₁, s₂)) (p.1.2.get? (unpair p.2.2).2)", "this✝ : Primrec G₁", "G : α → List σ → Option σ := fun a IH => Nat.casesOn IH.length (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G₁ ((a, IH), n, n.div2.div2)", "this : Primrec₂ G"], "goal": "Primrec fun a => F a (c a)"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "N : Type u_4", "CommRing R", "AddCommGroup M₁", "AddCommGroup M₂", "AddCommGroup N", "Module R M₁", "Module R M₂", "Module R N", "Q₁ : QuadraticForm R M₁", "Q₂ : QuadraticForm R M₂", "Qₙ : QuadraticForm R N", "f₁ : Q₁ →qᵢ Qₙ", "f₂ : Q₂ →qᵢ Qₙ", "hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)", "m₁ : CliffordAlgebra Q₁", "m₂ : CliffordAlgebra Q₂", "i₁ : ZMod 2", "hm₁ : m₁ ∈ CliffordAlgebra.evenOdd Q₁ i₁", "hm₂ : m₂ ∈ CliffordAlgebra.evenOdd Q₂ 0"], "goal": "Commute ((CliffordAlgebra.map f₁) m₁) ((CliffordAlgebra.map f₂) m₂)"}}
+{"state": {"context": ["ι✝ : Type u", "α : Type v", "inst✝¹ : DecidableEq α", "t✝ : ι✝ → Finset α", "inst✝ : Fintype ι✝", "ι : Type u", "t : ι → Finset α", "s : Finset ι", "hus : s.card = (s.biUnion t).card", "ht : ∀ (s : Finset ι), s.card ≤ (s.biUnion t).card", "s' : Finset ↑(↑s)ᶜ", "this : DecidableEq ι"], "goal": "s'.card ≤ (s'.biUnion fun x' => t ↑x' \\ s.biUnion t).card"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w : ι → k", "p : ι → P", "b : P", "c : k"], "goal": "(s.weightedVSubOfPoint p b) (c • w) = c • (s.weightedVSubOfPoint p b) w"}}
+{"state": {"context": ["M : Type u", "inst✝⁴ : Monoid M", "G : Type u", "inst✝³ : Group G", "F : Type v", "inst✝² : Field F", "inst✝¹ : MulSemiringAction M F", "inst✝ : MulSemiringAction G F", "m : M", "s✝ : Finset F", "this : IsEmpty ↑↑∅", "a : F", "s : Finset F", "has : a ∉ s", "ih : (LinearIndependent ↥(subfield G F) fun (i : ↑↑s) => ↑i) → LinearIndependent F fun (i : ↑↑s) => (toFun G F) ↑i", "hs : (LinearIndependent ↥(subfield G F) fun (b : ↑↑s) => ↑b) ∧ a ∉ Submodule.span ↥(subfield G F) ↑s", "l : F →₀ F", "hl : l ∈ Finsupp.supported F F ↑s", "hla : ∑ i ∈ s, l i • (toFun G F) i = (toFun G F) a"], "goal": "False"}}
+{"state": {"context": ["G : Type u_1", "Group G", "p : ℕ", "P : Sylow p G", "hP : (↑P).normalizer ≤ Subgroup.centralizer ↑P", "Fact (Nat.Prime p)", "Finite (Sylow p G)", "(↑P).FiniteIndex", "g : G", "hg : g ∈ P"], "goal": "(MonoidHom.transferSylow P hP) g = ⟨g ^ (↑P).index, ⋯⟩"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "inst✝ : ConditionallyCompleteLinearOrderBot α", "s : Set (WithTop α)", "hs : s = ∅"], "goal": "IsLUB s (sSup s)"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E", "S : IntermediateField F E", "h : S.FG"], "goal": "S.FG"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : LinearOrder α", "inst✝² : ConditionallyCompleteLinearOrder β", "inst✝¹ : TopologicalSpace β", "inst✝ : OrderTopology β", "f : α → β", "hf : Monotone f", "x y : α", "h : x < y", "this✝ : TopologicalSpace α := Preorder.topology α", "this : OrderTopology α", "h' : 𝓝[<] y ≠ ⊥"], "goal": "f x ≤ sSup (f '' Iio y)"}}
+{"state": {"context": ["α : Type u_1", "OrderedAddCommGroup α", "a b c : α"], "goal": "(fun x => x - a) '' Set.Icc b c = Set.Icc (b - a) (c - a)"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p q : ℝ≥0"], "goal": "2⁻¹ + 2⁻¹ = 1"}}
+{"state": {"context": ["α β : Type u", "c : Cardinal.{u_1}"], "goal": "1 ≤ c ↔ c ≠ 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "s : Sphere P", "p₁ p₂ : P", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "h : p₁ ≠ p₂"], "goal": "∡ p₁ s.center p₂ = ↑π - 2 • ∡ s.center p₂ p₁"}}
+{"state": {"context": ["α : Type u", "β : Type v", "e : α ≃ β", "Add β", "Mul β", "a : α"], "goal": "e.ringEquiv a = e a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f : α → β", "G : Type u_6", "Group G", "MulAction G β", "ContinuousConstSMul G β", "c : G"], "goal": "MeasureTheory.AEStronglyMeasurable (fun x => c • f x) μ ↔ MeasureTheory.AEStronglyMeasurable f μ"}}
+{"state": {"context": ["M : Type u_1", "X : Type u_2", "Y : Type u_3", "TopologicalSpace M", "TopologicalSpace X", "TopologicalSpace Y", "SMul M X", "ContinuousSMul M X", "f : Y → M", "g : Y → X", "hf : Continuous f", "hg : Continuous g"], "goal": "Continuous fun x => f x • g x"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "s t : TopologicalSpace.CompactOpens α"], "goal": "↑(s ⊔ t) = ↑s ∪ ↑t"}}
+{"state": {"context": ["R : Type u_1", "n : ℕ", "inst✝ : CommRing R", "a b x y : R", "p : ℕ", "hp : Odd p"], "goal": "↑p ^ 2 ∣ ∑ i ∈ range p, (a + ↑p * b) ^ i * a ^ (p - 1 - i) - ↑p * a ^ (p - 1)"}}
+{"state": {"context": ["F : Type u_1", "inst✝² : Field F", "E : Type u_2", "inst✝¹ : Field E", "inst✝ : Algebra F E", "p : F[X]", "sp : Polynomial.IsSplittingField F E p", "hp : p.Separable", "hFE : FiniteDimensional F E", "this : DecidableEq E := Classical.decEq E", "s : Set E := p.rootSet E", "adjoin_root : IntermediateField.adjoin F s = ⊤", "P : IntermediateField F E → Prop := fun K => Fintype.card (↥K →ₐ[F] E) = finrank F ↥K"], "goal": "P ⊥"}}
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+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r✝ r : ℚ", "h : ‖↑r‖ ≤ 1"], "goal": "¬‖↑r.den‖ < 1"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : (CategoryTheory.ShortComplex C)ᵒᵖ"], "goal": "(CategoryTheory.ShortComplex.opFunctor C).obj S = (Opposite.unop S).op"}}
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+{"state": {"context": ["α : Type u_5", "NonUnitalNonAssocSemiring α", "x y : CentroidHom α"], "goal": "(x * y).toEnd = x.toEnd * y.toEnd"}}
+{"state": {"context": ["V : Type u", "a b : V", "s : Set (SimpleGraph V)", "hs : s.Nonempty"], "goal": "(sInf s).Adj a b ↔ ∀ G ∈ s, G.Adj a b"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "S : CategoryTheory.ShortComplex C", "CategoryTheory.Balanced C", "hS : S.Exact", "h : S.LeftHomologyData", "CategoryTheory.Mono S.f"], "goal": "CategoryTheory.IsIso h.f'"}}
+{"state": {"context": ["C : Type u", "inst✝ : Groupoid C", "S✝ S T : Subgroupoid C", "h : S ≤ T", "s t : ↑S.objs", "x✝¹ x✝ : s ⟶ t", "f : ↑s ⟶ ↑t", "hf : f ∈ S.arrows ↑s ↑t", "g : ↑s ⟶ ↑t", "hg : g ∈ S.arrows ↑s ↑t"], "goal": "⟨f, ⋯⟩ = ⟨g, ⋯⟩ → ⟨f, hf⟩ = ⟨g, hg⟩"}}
+{"state": {"context": ["x x₁✝ x₂✝ x₃ x' y y₁ y₂ y₃ y'✝ : PGame", "h : IH1 x y", "x₁ x₂ y' : PGame", "h₁ : x₁.IsOption (-x)", "h₂ : x₂.IsOption (-x)", "hy : y' = y ∨ y'.IsOption y"], "goal": "P24 x₁ x₂ y'"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace β", "Group α", "MulAction α β", "ContinuousConstSMul α β", "s : Set α", "t : Set β"], "goal": "s • interior t ⊆ interior (s • t)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "Preorder α", "OrderClosedTopology α", "SemilatticeInf β", "f : β → α", "a : α", "hf : Monotone f", "ha : Filter.Tendsto f Filter.atBot (𝓝 a)", "b : β"], "goal": "a ≤ f b"}}
+{"state": {"context": ["ι : Type u_1", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "c : ComplexShape ι", "K L K' L' : HomologicalComplex C c", "φ : K ⟶ L", "φ' : K' ⟶ L'", "e : CategoryTheory.Arrow.mk φ ≅ CategoryTheory.Arrow.mk φ'", "∀ (i : ι), K.HasHomology i", "∀ (i : ι), L.HasHomology i", "∀ (i : ι), K'.HasHomology i", "∀ (i : ι), L'.HasHomology i"], "goal": "QuasiIso φ ↔ QuasiIso φ'"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ dt : ℝ"], "goal": "IsIntegralCurveOn γ v s ↔ IsIntegralCurveOn (γ ∘ fun x => x + dt) v {t | t + dt ∈ s}"}}
+{"state": {"context": ["n : ℕ", "NeZero n", "a : ℕ"], "goal": "Fin.ofNat'' a = ↑a"}}
+{"state": {"context": ["𝕜 : Type u_2", "E : Type u_1", "OrderedSemiring 𝕜", "OrderedAddCommGroup E", "Module 𝕜 E", "OrderedSMul 𝕜 E", "TopologicalSpace E", "OrderClosedTopology E"], "goal": "↑↑(ProperCone.positive 𝕜 E) = ConvexCone.positive 𝕜 E"}}
+{"state": {"context": ["n k : ℕ", "h : k.succ ≤ n.succ"], "goal": "(n.succ - k.succ)! * n.succ.descFactorial k.succ = n.succ !"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "β : ι → Type u_3", "s s₁ s₂ : Set ι", "t✝ t₁ t₂ : (i : ι) → Set (α i)", "i : ι", "inst✝ : DecidableEq ι", "hi : i ∈ s", "f : (j : ι) → α j", "a : α i", "t : (j : ι) → α j → Set (β j)"], "goal": "i ∉ s \\ {i}"}}
+{"state": {"context": ["n d : PosNum"], "goal": "↑(↑n.pred' / ↑d).succ = ↑(↑n.pred' / (↑d.pred' + 1)).succ"}}
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+{"state": {"context": ["A : Type u_1", "B : Type u_2", "Field A", "Ring B", "Algebra A B", "x : B", "IsReduced B"], "goal": "IsRadical (minpoly A x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "β : Type u_4", "ι : Type u_5", "inst✝⁵ : LinearOrderedField 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : LinearOrderedAddCommGroup β", "inst✝² : Module 𝕜 E", "inst✝¹ : Module 𝕜 β", "inst✝ : OrderedSMul 𝕜 β", "s : Set E", "f✝ : E → β", "t : Finset ι", "w : ι → 𝕜", "p : ι → E", "x✝ y✝ z✝ : E", "f : 𝕜 → β", "x y z : 𝕜", "hf : ConcaveOn 𝕜 (Set.Icc x y) f", "hz : z ∈ Set.Icc x y"], "goal": "min (f x) (f y) ≤ f z"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "S : Type u_5", "π : ι → Type u_6", "inst✝³ : ConditionallyCompleteLinearOrder α", "inst✝² : TopologicalSpace α", "inst✝¹ : OrderTopology α", "inst✝ : DenselyOrdered α", "f : Filter β", "u : β → α", "s : Set α", "hs : Dense s", "H : ∀ a ∈ s, ∀ b ∈ s, a < b → ¬((∃ᶠ (n : β) in f, u n < a) ∧ ∃ᶠ (n : β) in f, b < u n)", "h : autoParam (IsBoundedUnder (fun x x_1 => x ≤ x_1) f u) _auto✝", "h' : autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) f u) _auto✝", "hbot : f.NeBot", "hlt : liminf u f < limsup u f"], "goal": "False"}}
+{"state": {"context": [], "goal": "Hyperreal.omega.InfinitePos"}}
+{"state": {"context": ["R : Type u", "inst✝ : AddGroupWithOne R", "m n : ℕ", "h : m ≤ n"], "goal": "↑(n - m) + ↑m = ↑n"}}
+{"state": {"context": ["E : Type u_3", "SeminormedAddCommGroup E", "a₁ a₂ b₁ b₂ : E"], "goal": "nndist (a₁ + a₂) (b₁ + b₂) ≤ nndist a₁ b₁ + nndist a₂ b₂"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "R : Type u_4", "Fintype n", "CommSemiring R", "M : Matrix m n R", "i : m"], "goal": "(M.toMvPolynomial i).IsHomogeneous 1"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Finset α"], "goal": "(s \\ t).card + (s ∩ t).card = s.card"}}
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+{"state": {"context": ["R : Type u", "S₁ : Type v", "CommSemiring R", "a : MvPolynomial S₁ R[X]"], "goal": "(MvPolynomial.optionEquivRight R S₁).symm a =\n  (MvPolynomial.aevalTower (Polynomial.aeval (MvPolynomial.X none)) fun i => MvPolynomial.X (some i)) a"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M", "N : LieSubmodule R L M", "x : M"], "goal": "x ∈ ↑N ↔ x ∈ N"}}
+{"state": {"context": ["F : Type u_10", "M : outParam (Type u_11)", "N : outParam (Type u_12)", "One M", "One N", "FunLike F M N", "self : OneHomClass F M N", "f : F"], "goal": "f 1 = 1"}}
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+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "f g : PadicSeq p", "hfg : ¬f + g ≈ 0", "hf : ¬f ≈ 0", "hg : g ≈ 0", "hfg' : f + g ≈ f", "hcfg : (f + g).norm = f.norm"], "goal": "(f + g).norm ≤ max f.norm g.norm"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "F : Type u_3", "AddCommGroup F", "Module R F", "f : E →ₗ.[R] F", "hf : LinearMap.ker f.toFun = ⊥", "x : ↥f.domain"], "goal": "(↑f x, ↑x) ∈ f.inverse.graph"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDomain R", "NormalizedGCDMonoid R", "p : Polynomial R"], "goal": "p.primPart.content = 1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝ : PseudoMetricSpace α", "x y z : α", "δ ε ε₁ ε₂ : ℝ", "s : Set α", "h : x ∈ ball y ε"], "goal": "∃ ε' < ε, x ∈ ball y ε'"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "Sub α", "A B : Matrix m n α", "i : m", "j : n"], "goal": "(A - B) i j = A i j - B i j"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s✝ t u : Set α", "π : δ → Type u_6", "inst✝³ : MeasurableSpace α", "inst✝² : (a : δ) → MeasurableSpace (π a)", "inst✝¹ : MeasurableSpace γ", "inst✝ : DecidableEq δ", "s : Finset δ", "x : (i : δ) → π i"], "goal": "∀ (a : δ), Measurable fun x_1 => if hi : a ∈ s then x_1 ⟨a, hi⟩ else x a"}}
+{"state": {"context": [], "goal": "∀ (a : ℝ × ℝ), (Complex.equivRealProdLm.symm a).re = a.1"}}
+{"state": {"context": ["α : Type u", "LinearOrder α", "IsWellOrder α fun x x_1 => x < x_1", "l m : List α", "o : Ordinal.{u}", "hl : List.Sorted (fun x x_1 => x > x_1) l", "hm : List.Sorted (fun x x_1 => x > x_1) m", "hmltl : m < l", "hlt : ∀ i ∈ l, Ordinal.typein (fun x x_1 => x < x_1) i < o", "i : α"], "goal": "i ∈ m → Ordinal.typein (fun x x_1 => x < x_1) i < o"}}
+{"state": {"context": ["α : Type u", "a : α", "as : List α"], "goal": "(a :: as).dropLast.length = as.length"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : 𝕜", "L : Filter 𝕜", "ι : Type u_1", "u : Finset ι", "A : ι → 𝕜 → F", "A' : ι → F", "h : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) x L"], "goal": "HasDerivAtFilter (fun y => ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) x L"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "s : Set α", "H : Function.Surjective f"], "goal": "(f '' s)ᶜ ⊆ f '' sᶜ"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "c g' : Perm (Fin 5)", "h : 3 ∈ (c * g').cycleType", "hd : c.Disjoint g'", "left✝ : c.IsCycle", "h3 : c.support.card = 3"], "goal": "(c * (g' * (c * g'))).IsThreeCycle"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "β : Type u_3", "ι : Type u_4", "inst✝ : TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s s₁ s₂ : Set α", "v : Set τ", "hv : v ∈ f"], "goal": "ω f ϕ s = ⋂ u, closure (image2 ϕ (↑u ∩ v) s)"}}
+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "s.encard ≠ 0 ↔ s.Nonempty"}}
+{"state": {"context": ["G : Type u", "H : Type v", "Add G", "Add H", "f : G ≃+ H"], "goal": "TwoUniqueSums G ↔ TwoUniqueSums H"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "Fintype { x // x ∈ s }"], "goal": "Fintype.card { x // x ∈ s } = s.card"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "β : Type u_4", "inst✝⁴ : OrderedRing 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : AddCommGroup F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "s t✝ : Set E", "hs : Convex 𝕜 s", "x y : E", "hx : x ∈ s", "hy : x + y ∈ s", "t : 𝕜", "ht : t ∈ Icc 0 1", "h : x + t • y = (1 - t) • x + t • (x + y)"], "goal": "(1 - t) • x + t • (x + y) ∈ s"}}
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+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "inst✝ : Group G", "A : Rep k G", "h1 : ↑(oneCochainsLequiv A) ≫ ModuleCat.ofHom (dOne A) = (inhomogeneousCochains A).d 1 2 ≫ ↑(twoCochainsLequiv A)", "h2 : ↑(twoCochainsLequiv A) ≫ ModuleCat.ofHom (dTwo A) = (inhomogeneousCochains A).d 2 3 ≫ ↑(threeCochainsLequiv A)"], "goal": "ModuleCat.ofHom (dOne A) ≫ ModuleCat.ofHom (dTwo A) = 0"}}
+{"state": {"context": ["i n : ℕ", "h0 : n ≠ 0", "hi : i.Coprime n"], "goal": "IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I * (↑i / ↑n))) n"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "x₀ x₁ : X", "p₀ p₁ : Path x₀ x₁", "F : p₀.Homotopy p₁", "x : ↑I × ↑I"], "goal": "F.symm x = F (σ x.1, x.2)"}}
+{"state": {"context": ["a : ℤ", "n : ℕ"], "goal": "Int.pred^[n] a = a - ↑n"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "M' : Type u_3", "N' : Type u_4", "MulOneClass M", "MulOneClass N", "MulOneClass M'", "MulOneClass N'", "M'' : Type u_6", "N'' : Type u_7", "MulOneClass M''", "MulOneClass N''", "f' : M' →* M''", "g' : N' →* N''", "f : M →* M'", "g : N →* N'", "x : Monoid.Coprod M N"], "goal": "(Monoid.Coprod.map f' g') ((Monoid.Coprod.map f g) x) = (Monoid.Coprod.map (f'.comp f) (g'.comp g)) x"}}
+{"state": {"context": ["R : Type u", "Ring R", "p q : R[X]"], "goal": "(p /ₘ q).degree ≤ p.degree"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "x y z : α", "ε ε₁ ε₂ : ℝ≥0∞", "s t✝ t : Set α", "xs : x ∈ s", "yt : y ∈ t", "A : ∀ a ∈ s, ∀ b ∈ t, edist a b ≤ diam s + edist x y + diam t", "a : α", "ha : a ∈ s ∪ t", "b : α", "hb : b ∈ s ∪ t", "h'a : a ∈ t", "h'b : b ∈ s"], "goal": "edist a b ≤ diam s + edist x y + diam t"}}
+{"state": {"context": ["B : Type u", "CategoryTheory.Bicategory B", "a b : B", "f g : a ⟶ b", "CategoryTheory.BicategoricalCoherence f g"], "goal": " ⊗𝟙 = ⊗𝟙 ≫ (λ_ g).inv"}}
+{"state": {"context": ["F : Type u_1", "inst✝⁴ : Field F", "inst✝³ : Infinite F", "E : Type u_2", "inst✝² : Field E", "ϕ : F →+* E", "α β : E", "inst✝¹ : Algebra F E", "inst✝ : Algebra.IsSeparable F E", "hα : IsIntegral F α", "hβ : IsIntegral F β", "f : F[X] := minpoly F α", "g : F[X] := minpoly F β", "ιFE : F →+* E := algebraMap F E", "ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField", "c : F", "hc : ∀ α' ∈ (Polynomial.map (ιEE'.comp ιFE) f).roots, ∀ β' ∈ (Polynomial.map (ιEE'.comp ιFE) g).roots, -(α' - ιEE' α) / (β' - ιEE' β) ≠ (ιEE'.comp ιFE) c", "γ : E := α + c • β", "p : (↥F⟮γ⟯)[X] := EuclideanDomain.gcd ((Polynomial.map (algebraMap F ↥F⟮γ⟯) f).comp (C (AdjoinSimple.gen F γ) - C ⟨(algebraMap F E) c, ⋯⟩ * X)) (Polynomial.map (algebraMap F ↥F⟮γ⟯) g)", "h : E[X] := EuclideanDomain.gcd ((Polynomial.map ιFE f).comp (C γ - C (ιFE c) * X)) (Polynomial.map ιFE g)", "map_g_ne_zero : Polynomial.map ιFE g ≠ 0", "h_ne_zero : h ≠ 0", "h_sep : h.Separable", "h_root : eval β h = 0", "h_splits : Splits ιEE' h"], "goal": "Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = C h.leadingCoeff * (X - C β)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν✝ ν μ : Measure α", "inst✝¹ : IsFiniteMeasure ν", "inst✝ : ν.HaveLebesgueDecomposition μ", "r : ℝ≥0", "hr : r ≠ 0"], "goal": "ν.rnDeriv (r • μ) =ᶠ[ae (r • μ)] r⁻¹ • ν.rnDeriv μ"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NormedField 𝕜", "SeminormedAddCommGroup E", "NormedSpace 𝕜 E", "c : 𝕜", "hc : c ≠ 0", "x : E", "r : ℝ"], "goal": "c • Metric.ball x r = Metric.ball (c • x) (‖c‖ * r)"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeSup α", "OrderTop α", "a b c : α"], "goal": "Codisjoint (a ⊔ b) c ↔ Codisjoint (a ⊔ c) b"}}
+{"state": {"context": ["n p : ℕ", "pp : Prime p", "k x : ℕ"], "goal": "x ∈ (p ^ k).divisors ↔ ∃ j ≤ k, x = p ^ j"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : BooleanAlgebra α", "s : Finset α", "hs : (↑s).Intersecting", "x : α", "hx' : x ∈ s", "hx : { toFun := compl, inj' := ⋯ } x ∈ s", "hxc : { toFun := compl, inj' := ⋯ } x ∈ map { toFun := compl, inj' := ⋯ } s"], "goal": "False"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L"], "goal": "IsLieAbelian L ↔ LieAlgebra.center R L = ⊤"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "A : Type u_2", "AddMonoid A", "CategoryTheory.HasShift C A", "X : CategoryTheory.OppositeShift C A", "a b : A"], "goal": "(CategoryTheory.shiftFunctorAdd (CategoryTheory.OppositeShift C A) a b).inv.app X =\n  ((CategoryTheory.shiftFunctorAdd C a b).hom.app (Opposite.unop X)).op"}}
+{"state": {"context": ["p : ℕ"], "goal": "Nat.rec (unpair p).1 (fun y IH => IH - 1) (unpair p).2 = (unpair p).1 - (unpair p).2"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "s.Subsingleton → Subsingleton ↑s"}}
+{"state": {"context": ["α : Type u_1", "l : List (Option α)"], "goal": "l.length = l.reduceOption.length + (List.filter Option.isNone l).length"}}
+{"state": {"context": ["p : ℕ → Prop", "inst✝ : DecidablePred p", "n k : ℕ", "h : k < count p n"], "goal": "count p (nth p k) < count p n"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X : PresheafedSpace C", "x : ↑↑X", "e_1✝ : (X.presheaf.stalk x ⟶ X.presheaf.stalk x) = ((stalkFunctor C x).obj X.presheaf ⟶ (stalkFunctor C x).obj X.presheaf)"], "goal": "(𝟙 X).c ≫ (Pushforward.id X.presheaf).hom = 𝟙 X.presheaf"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimits C", "X : TopCat", "ℱ : TopCat.Presheaf C X", "x : ↑X"], "goal": "(TopCat.Presheaf.stalkFunctor C x).obj ℱ = ℱ.stalk x"}}
+{"state": {"context": ["R : Type u_3", "Γ₀ : Type u_4", "Ring R", "LinearOrderedAddCommMonoidWithTop Γ₀", "f : R → Γ₀", "h0 : f 0 = ⊤", "h1 : f 1 = 0", "hadd : ∀ (x y : R), min (f x) (f y) ≤ f (x + y)", "hmul : ∀ (x y : R), f (x * y) = f x + f y", "r : R"], "goal": "(AddValuation.of f h0 h1 hadd hmul) r = f r"}}
+{"state": {"context": ["α : Type u", "inst✝ : Monoid α", "a b : α", "x✝ : Invertible a"], "goal": "⅟a * (a * b) = b"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a b : α", "hb : 0 < b"], "goal": "1 < a / b ↔ b < a"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "s : AList β"], "goal": "AList.lookup a s = none ↔ a ∉ s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "s : CategoryTheory.Limits.BinaryCofan X Y"], "goal": "s.ι.app { as := CategoryTheory.Limits.WalkingPair.left } = s.inl"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : NormedAddCommGroup β", "u v w : α → β", "l : Filter α", "h : u =O[l] 0"], "goal": "u ~[l] 0"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝¹³ : CommSemiring R", "inst✝¹² : StarRing R", "inst✝¹¹ : TopologicalSpace A", "inst✝¹⁰ : Semiring A", "inst✝⁹ : Algebra R A", "inst✝⁸ : StarRing A", "inst✝⁷ : StarModule R A", "inst✝⁶ : TopologicalSemiring A", "inst✝⁵ : ContinuousStar A", "inst✝⁴ : TopologicalSpace B", "inst✝³ : Semiring B", "inst✝² : Algebra R B", "inst✝¹ : StarRing B", "inst✝ : T2Space B", "S : StarSubalgebra R A", "φ ψ : ↥S.topologicalClosure →⋆ₐ[R] B", "hφ : Continuous ⇑φ", "hψ : Continuous ⇑ψ", "h : φ.comp (inclusion ⋯) = ψ.comp (inclusion ⋯)"], "goal": "⇑φ = ⇑ψ"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝¹ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y✝ y' : F", "c c₁ c₂ : ℝ≥0", "ε ε₁ ε₂ : ℝ", "π₁ π₂ : TaggedPrepartition I", "inst✝ : CompleteSpace F", "h : Integrable I l f vol", "hJ : J ≤ I", "y : F", "hy : Tendsto (integralSum f vol) (l.toFilteriUnion I (Prepartition.single I J hJ)) (𝓝 y)"], "goal": "HasIntegral J l f vol y"}}
+{"state": {"context": ["n : ℕ"], "goal": "(antidiagonal (n + 1)).val = (n + 1, 0) ::ₘ Multiset.map (⇑((Embedding.refl ℕ).prodMap { toFun := Nat.succ, inj' := Nat.succ_injective })) (antidiagonal n).val"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "K : Type u_4", "A : Type u_5", "A' : Type u_6", "A'' : Type u_7", "V : Type u", "V' : Type u_8", "x : ι → A", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing A", "inst✝⁵ : CommRing A'", "inst✝⁴ : CommRing A''", "inst✝³ : Algebra R A", "inst✝² : Algebra R A'", "inst✝¹ : Algebra R A''", "a b : R", "η : Type u_9", "inst✝ : Nonempty η", "s : η → Set A", "hs : Directed (fun x x_1 => x ⊆ x_1) s", "h : ∀ (i : η), AlgebraicIndependent R Subtype.val"], "goal": "AlgebraicIndependent R Subtype.val"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "s✝ : Set α", "t : Set β", "f✝ : Filter α", "g✝ : Filter β", "f : Filter α", "g : Filter β", "inst✝ : g.NeBot", "s : Set α"], "goal": "s ∈ map Prod.fst (f ×ˢ g) ↔ s ∈ f"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "W W' : CategoryTheory.MorphismProperty C", "h : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), W f ↔ W' f"], "goal": "W = W'"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "CommMonoid β", "p : Prop", "Decidable p", "s : Finset α", "f g : α → β"], "goal": "(∏ x ∈ s, if p then f x else g x) = if p then ∏ x ∈ s, f x else ∏ x ∈ s, g x"}}
+{"state": {"context": ["A : Type u_1", "CommRing A", "Algebra ℚ A"], "goal": "PowerSeries.exp A * PowerSeries.evalNegHom (PowerSeries.exp A) = 1"}}
+{"state": {"context": ["α : Type u", "LinearOrder α", "OrderBot α", "a : α"], "goal": "min ⊥ a = ⊥"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "AddCommMonoid R", "s : Finset S", "f : S → Tropical R"], "goal": "Tropical.untrop (∏ i ∈ s, f i) = ∑ i ∈ s, Tropical.untrop (f i)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "r : R"], "goal": "TensorPower.toTensorAlgebra (DirectSum.GAlgebra.toFun r) = (algebraMap R (TensorAlgebra R M)) r"}}
+{"state": {"context": ["n : Type u_2", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : Matrix n n R", "j : n", "c : n → R"], "goal": "(A.updateRow j (∑ k : n, c k • A k)).det = c j • A.det"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "Q : AffineTargetMorphismProperty", "inst✝² : HasAffineProperty P Q", "X Y U✝ V✝ : Scheme", "f : X ⟶ Y", "g : U✝ ⟶ Y", "inst✝¹ : IsAffine U✝", "inst✝ : IsOpenImmersion g", "iV : V✝ ⟶ X", "f' : V✝ ⟶ U✝", "h : IsPullback iV f' f g", "H : P.diagonal f", "this : Q.IsLocal := isLocal_affineProperty P", "U V : Scheme", "f₁ : U ⟶ pullback f g", "f₂ : V ⟶ pullback f g", "hU : IsAffine U", "hV : IsAffine V", "hf₁ : IsOpenImmersion f₁", "hf₂ : IsOpenImmersion f₂"], "goal": "IsAffine (pullback (f₁ ≫ pullback.snd f g) (f₂ ≫ pullback.snd f g))"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "E' : Type u_3", "F : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹⁹ : RCLike 𝕜", "inst✝¹⁸ : NormedAddCommGroup E", "inst✝¹⁷ : InnerProductSpace 𝕜 E", "inst✝¹⁶ : CompleteSpace E", "inst✝¹⁵ : NormedAddCommGroup E'", "inst✝¹⁴ : InnerProductSpace 𝕜 E'", "inst✝¹³ : CompleteSpace E'", "inst✝¹² : NormedSpace ℝ E'", "inst✝¹¹ : NormedAddCommGroup F", "inst✝¹⁰ : NormedSpace 𝕜 F", "inst✝⁹ : NormedAddCommGroup G", "inst✝⁸ : NormedAddCommGroup G'", "inst✝⁷ : NormedSpace ℝ G'", "inst✝⁶ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "E'' : Type u_8", "𝕜' : Type u_9", "inst✝⁵ : RCLike 𝕜'", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : InnerProductSpace 𝕜' E''", "inst✝² : CompleteSpace E''", "inst✝¹ : NormedSpace ℝ E''", "hm : m ≤ m0", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "x : E'", "inst✝ : SigmaFinite (μ.trim hm)"], "goal": "∫⁻ (a : α), ↑‖↑↑↑((condexpL2 E' 𝕜 hm) (indicatorConstLp 2 hs hμs x)) a‖₊ ∂μ ≤ μ s * ↑‖x‖₊"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "f : X → Y", "g : Y → Z", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "hf : Continuous f", "hg : Continuous g", "hgf : QuotientMap (g ∘ f)"], "goal": "QuotientMap g"}}
+{"state": {"context": ["α : Type u_1"], "goal": "∀ (a : α), (Equiv.punitProd α).symm a = (PUnit.unit, a)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "a : α", "s : WSeq α"], "goal": "(fun x => a :: x) <$> s.toList = List.cons a <$> Computation.corec (fun x => match Seq.destruct x.2 with | none => Sum.inl x.1.reverse | some (none, s') => Sum.inr (x.1, s') | some (some a, s') => Sum.inr (a :: x.1, s')) ([], s)"}}
+{"state": {"context": ["α : Type u_1", "rα : α → α → Prop", "h : WellFounded rα"], "goal": "WellFounded (Sym2.GameAdd rα)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α", "m : ℤ"], "goal": "toIcoDiv hp a (b + m • p) = toIcoDiv hp a b + m"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "NormedLatticeAddCommGroup β", "f g : BoundedContinuousFunction α β"], "goal": "⇑(f ⊓ g) = ⇑f ⊓ ⇑g"}}
+{"state": {"context": ["α : Type u", "β : Type v", "c : Cardinal.{max u v}", "f : α → β", "hf : ∀ (b : β), Cardinal.lift.{v, u} (Cardinal.mk ↑(f ⁻¹' {b})) ≤ c"], "goal": "Cardinal.lift.{v, u} (Cardinal.mk α) ≤ Cardinal.lift.{u, v} (Cardinal.mk β) * c"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "g : β → γ", "s : Computation α"], "goal": "Computation.map (g ∘ f) s = Computation.map g (Computation.map f s)"}}
+{"state": {"context": ["R : Type u", "inst✝⁸ : Ring R", "ι : Type v", "inst✝⁷ : Preorder ι", "G : ι → Type w", "inst✝⁶ : (i : ι) → AddCommGroup (G i)", "inst✝⁵ : (i : ι) → Module R (G i)", "f : (i j : ι) → i ≤ j → G i →ₗ[R] G j", "inst✝⁴ : DecidableEq ι", "P : Type u₁", "inst✝³ : AddCommGroup P", "inst✝² : Module R P", "g : (i : ι) → G i →ₗ[R] P", "Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x", "inst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)", "inst✝ : (i : ι) → (k : G i) → Decidable (k ≠ 0)", "x : DirectSum ι G", "i j : ι", "hij : i ≤ j", "hx : ∀ k ∈ DFinsupp.support x, k ≤ i", "k : ι", "hk : k ∈ DFinsupp.support x"], "goal": "(DirectSum.toModule R ι (G j) fun k => totalize G f k j) (DFinsupp.single k (x k)) = (f i j hij) ((DirectSum.toModule R ι (G i) fun k => totalize G f k i) (DFinsupp.single k (x k)))"}}
+{"state": {"context": ["m n a b d : ℤ", "h : a ≡ b [ZMOD n]"], "goal": "0 * a ≡ 0 * b [ZMOD 0 * n]"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "Γ₀ : Type u_2", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "hv : Valued K Γ₀", "γ : Γ₀ˣ", "x : hat K", "γ₀ : Γ₀ := extensionValuation x", "h : γ₀ ≠ 0"], "goal": "x ∈ closure (↑K '' {x | v x < ↑γ}) ↔ γ₀ < ↑γ"}}
+{"state": {"context": ["α : Type u_1", "X : Type u_2", "Y : Type u_3", "Z : Type u_4", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "S : DiscreteQuotient X", "x : X"], "goal": "IsClopen (setOf (S.Rel x))"}}
+{"state": {"context": ["y z : PartENat", "hz : z ≠ ⊤", "m : ℕ", "h : ↑m < y"], "goal": "↑m + z < y + z"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.ConcreteCategory C", "(CategoryTheory.forget C).PreservesMonomorphisms"], "goal": "CategoryTheory.MorphismProperty.injective C = CategoryTheory.MorphismProperty.monomorphisms C"}}
+{"state": {"context": ["x y : ℂ"], "goal": "x = y ↔ Complex.abs x = Complex.abs y ∧ x.arg = y.arg"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "A : Type u_2", "Semiring A", "TopologicalSpace A", "Algebra R A"], "goal": "1 = ContinuousAlgHom.id R A"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "p : ℝ≥0∞", "hp_ne_top : p ≠ ⊤", "f : α → F", "c : ℝ≥0∞"], "goal": "MeasureTheory.eLpNorm f p (c • μ) = c ^ (1 / p).toReal • MeasureTheory.eLpNorm f p μ"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "inst✝¹⁵ : Field R", "inst✝¹⁴ : AddCommGroup V", "inst✝¹³ : TopologicalSpace R", "inst✝¹² : TopologicalSpace V", "inst✝¹¹ : TopologicalRing R", "inst✝¹⁰ : TopologicalAddGroup V", "inst✝⁹ : Module R V", "inst✝⁸ : SeparatingDual R V", "𝕜 : Type u_3", "E : Type u_4", "F : Type u_5", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : SeparatingDual 𝕜 E", "inst✝¹ : Nontrivial E", "inst✝ : CompleteSpace (E →L[𝕜] F)", "f : ℕ → F", "hf : CauchySeq f", "v : E", "hv : v ≠ 0", "φ : E →L[𝕜] 𝕜", "hφ : φ v = 1", "g : ℕ → E →L[𝕜] F := fun n => ((ContinuousLinearMap.smulRightL 𝕜 E F) φ) (f n)", "this✝ : CauchySeq g", "a : E →L[𝕜] F", "ha : Tendsto g atTop (𝓝 a)", "this : Tendsto (fun n => (g n) v) atTop (𝓝 (a v))"], "goal": "Tendsto f atTop (𝓝 (a v))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝ : MeasurableSpace β", "μ✝ ν ν₁ ν₂ : Measure α", "s t : Set α", "μ : Measure α", "f : α → β", "hf : AEMeasurable f μ", "h : SigmaFinite (Measure.map f μ)", "n : ℕ"], "goal": "μ ((fun n => f ⁻¹' spanningSets (Measure.map f μ) n) n) < ⊤"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E"], "goal": "Continuous { toFun := fun x => (⟨‖↑x‖⁻¹ • ↑x, ⋯⟩, ⟨‖↑x‖, ⋯⟩), invFun := fun x => ⟨↑x.2 • ↑x.1, ⋯⟩, left_inv := ⋯, right_inv := ⋯ }.toFun"}}
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+{"state": {"context": ["G : Type u_1", "SubNegMonoid G", "n : ℤ", "x : G"], "goal": "SubNegMonoid.zsmul n x = n • x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "f : Filter β", "p q : β → Prop", "u : β → α", "h : ∀ (x : β), p x → q x"], "goal": "Filter.blimsup u f p ≤ Filter.blimsup u f q"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ₁ μ₂ ν ν' : MeasureTheory.Measure α", "h1 : μ₁.AbsolutelyContinuous ν", "h2 : μ₂.AbsolutelyContinuous ν'"], "goal": "(μ₁ + μ₂).AbsolutelyContinuous (ν + ν')"}}
+{"state": {"context": ["K : Type u_3", "DivisionMonoid K", "HasDistribNeg K", "a b : K"], "goal": "-b / a = -(b / a)"}}
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+{"state": {"context": [], "goal": "[].IsZeckendorfRep"}}
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+{"state": {"context": ["α : Type u_1", "s : Set α", "T : Set (Set α)"], "goal": "⋂₀ insert s T = s ∩ ⋂₀ T"}}
+{"state": {"context": ["r p : ℝ"], "goal": "r.toNNReal ≤ p.toNNReal ↔ r ≤ p ∨ r ≤ 0"}}
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+{"state": {"context": ["this : Tendsto (fun x => -(eulerMascheroniSeq' ∘ fun a => a + 1) x) atTop (𝓝 (-γ))"], "goal": "Tendsto (fun N => -↑(harmonic (N + 1)) + log (↑N + 1)) atTop (𝓝 (-γ))"}}
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+{"state": {"context": ["o : Ordinal.{u_3}"], "goal": "o ≤ 0 ↔ o = 0"}}
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+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "z : ℤ", "hz : z = 0"], "goal": "padicNorm p ↑z ≤ 1"}}
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+{"state": {"context": ["G : Type u_2", "α : Type u_3", "MeasurableSpace G", "Div G", "m : MeasurableSpace α", "f : α → G", "MeasurableDiv G", "hf : Measurable f", "c : G"], "goal": "Measurable fun x => f x / c"}}
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+{"state": {"context": ["P : Type u_1", "Preorder P", "IF : Order.Ideal.PrimePair P"], "goal": "(↑IF.I)ᶜ = ↑IF.F"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : CommRing R", "inst✝³ : AddCommGroup M", "inst✝² : AddCommGroup N", "inst✝¹ : Module R M", "inst✝ : Module R N", "Q : QuadraticForm R M", "P : CliffordAlgebra Q → Prop", "algebraMap : ∀ (r : R), P ((_root_.algebraMap R (CliffordAlgebra Q)) r)", "add : ∀ (x y : CliffordAlgebra Q), P x → P y → P (x + y)", "ι_mul : ∀ (x : CliffordAlgebra Q) (m : M), P x → P ((ι Q) m * x)"], "goal": "∀ (x : CliffordAlgebra Q), P (reverse x)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "x : α × β"], "goal": "x.swap.swap = x"}}
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+{"state": {"context": ["L : FirstOrder.Language", "M : Type u_1", "L.Structure M"], "goal": "⊥.FG"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort u_6", "s t u : Set α", "m : MeasurableSpace α", "s₁ s₂ : Set α", "h₁ : MeasurableSet s₁", "h₂ : MeasurableSet s₂"], "goal": "MeasurableSet (s₁ ∪ s₂)"}}
+{"state": {"context": ["n : ℕ"], "goal": "n % 2 = 1 ↔ n % 4 = 1 ∨ n % 4 = 3"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P Q : Fin 3 → F", "hP : W.Equation P", "hQ : W.Equation Q", "hPz : P z ≠ 0", "hQz : Q z ≠ 0", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3", "hZ : ∀ {n : ℕ}, IsUnit (W.dblZ P ^ n)"], "goal": "W.dblXYZ P = W.dblZ P • ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)), 1]"}}
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+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "𝒢 : Filtration ℕ m0", "f : ℕ → Ω → ℝ", "τ π : Ω → ℕ", "inst✝ : SigmaFiniteFiltration μ 𝒢", "hf : Submartingale f 𝒢 μ", "hτ : IsStoppingTime 𝒢 τ", "hπ : IsStoppingTime 𝒢 π", "hle : τ ≤ π", "N : ℕ", "hbdd : ∀ (ω : Ω), π ω ≤ N"], "goal": "∫ (x : Ω), stoppedValue f τ x ∂μ ≤ ∫ (x : Ω), stoppedValue f π x ∂μ"}}
+{"state": {"context": ["α : Type u", "Preorder α", "s : Set α"], "goal": "BddBelow s ↔ ∃ a, s ⊆ Set.Ici a"}}
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+{"state": {"context": ["M : Type u_1", "N : Type u_2", "inst✝¹ : SeminormedAddCommGroup M", "inst✝ : SeminormedAddCommGroup N", "S : AddSubgroup M", "m : M", "ε : ℝ", "hε : 0 < ε", "n : M", "hn : -m + n ∈ S", "hn' : ‖n‖ < ‖(mk' S) m‖ + ε"], "goal": "∃ s ∈ S, ‖m + s‖ < ‖(mk' S) m‖ + ε"}}
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+{"state": {"context": ["m n a b c d : ℕ", "h₁ : c ≡ d [MOD n]", "h₂ : a + c ≡ b + d [MOD n]"], "goal": "a ≡ b [MOD n]"}}
+{"state": {"context": ["α : Type u_2", "CommSemiring α", "ι : Type u_6", "Fintype ι", "a b : α"], "goal": "∑ s : Finset ι, a ^ s.card * b ^ (Fintype.card ι - s.card) = (a + b) ^ Fintype.card ι"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "p : Equiv.Perm α", "x : α"], "goal": "(p.toList x).Nodup"}}
+{"state": {"context": [], "goal": "FermatLastTheoremFor 3"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "CommRing R", "n : ℕ", "hn : 0 < n"], "goal": "WittVector.coeff 1 n = 0"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M"], "goal": "Subsingleton (LieSubmodule R L M) ↔ Subsingleton M"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y P : C", "f : X ⟶ Y", "π : Y ⟶ P", "w : f ≫ π = 0"], "goal": "CategoryTheory.Limits.Cofork.π (CategoryTheory.Limits.CokernelCofork.ofπ π w) = π"}}
+{"state": {"context": ["m n✝ n : ℕ", "a : ZMod n"], "goal": "2 * a = 0 ↔ a = 0 ∨ 2 * a.val = n"}}
+{"state": {"context": ["x✝¹ y✝ a : ℝ", "ha : 0 < a", "x : ℝ", "hex : rexp (1 / a) ≤ x", "y : ℝ", "x✝ : rexp (1 / a) ≤ y", "hxy : x ≤ y"], "goal": "log y / y ^ a ≤ log x / x ^ a"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "s : Set E", "f : E → E", "f' : E → E →L[ℝ] E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hs : MeasurableSet s", "h's : μ s ≠ ⊤", "hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x", "this : Tendsto (fun ε => ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ + 2 * ↑ε * μ s) (𝓝[>] 0) (𝓝 (∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ))"], "goal": "μ (f '' s) ≤ ∫⁻ (x : E) in s, ENNReal.ofReal |(f' x).det| ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝¹ ν✝¹ μ✝ ν✝ : Measure α", "f : α → ℝ≥0∞", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "hf : AEMeasurable f ν", "hf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0", "hf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ⊤", "μ' : Measure α := ν.withDensity (μ.rnDeriv ν)", "h₁ : μ'.rnDeriv (ν.withDensity f) =ᶠ[ae ν] μ.rnDeriv (ν.withDensity f)", "h₂ : μ.rnDeriv ν =ᶠ[ae ν] μ'.rnDeriv ν", "this : SigmaFinite μ'", "hμ' : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) =ᶠ[ae ν] fun x => (f x)⁻¹ * (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x", "x : α", "hx₁ : μ'.rnDeriv (ν.withDensity f) x = μ.rnDeriv (ν.withDensity f) x", "hx₂ : μ.rnDeriv ν x = μ'.rnDeriv ν x", "hx_eq : (ν.withDensity (μ.rnDeriv ν)).rnDeriv (ν.withDensity f) x = (f x)⁻¹ * (ν.withDensity (μ.rnDeriv ν)).rnDeriv ν x"], "goal": "μ.rnDeriv (ν.withDensity f) x = (f x)⁻¹ * μ.rnDeriv ν x"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s t : Set α", "inst✝ : LinearOrder α", "a : α"], "goal": "Bounded (fun x x_1 => x ≤ x_1) (s ∩ {b | a < b}) ↔ Bounded (fun x x_1 => x ≤ x_1) s"}}
+{"state": {"context": ["X✝ Y✝ Z✝ : Stonean", "f✝ : X✝ ⟶ Z✝", "i✝ : Y✝ ⟶ Z✝", "hi✝ : OpenEmbedding ⇑i✝", "X Y Z : Stonean", "f : X ⟶ Z", "i : Y ⟶ Z", "hi : OpenEmbedding ⇑i", "x : (forget Stonean).obj X", "h : x ∈ ⇑f ⁻¹' Set.range ⇑i", "y : (forget Stonean).obj Y", "hy : i y = f x"], "goal": "(fst f hi ≫ f) ⟨x, h⟩ = (snd f hi ≫ i) ⟨x, h⟩"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a : α", "ha : IsWellOrderLimitElement a", "IsWellOrder α fun x x_1 => x < x_1", "b : α", "hb : b < a"], "goal": "wellOrderSucc b < a"}}
+{"state": {"context": ["a b c d : List Turing.PartrecToTM2.Γ'"], "goal": "Turing.PartrecToTM2.K'.elim a b c d Turing.PartrecToTM2.K'.main = a"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "L : Type u_3", "M : Type u_4", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "inst✝⁶ : LieAlgebra.IsNilpotent R L", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : LieRingModule L M", "inst✝² : LieModule R L M", "inst✝¹ : IsNoetherian R M", "inst✝ : IsArtinian R M", "x : L", "k : ℕ", "m : M", "F : LieSubmodule R L M := posFittingComp R L M", "x✝ : M ⧸ F", "hk : ((toEnd R L M) x ^ k) m ∈ F", "this : (((toEnd R L (M ⧸ F)) x ^ k) ∘ₗ ↑(LieSubmodule.Quotient.mk' F)) m = (↑(LieSubmodule.Quotient.mk' F) ∘ₗ (toEnd R L M) x ^ k) m"], "goal": "((toEnd R L (M ⧸ F)) x ^ k) (LieSubmodule.Quotient.mk m) = LieSubmodule.Quotient.mk (((toEnd R L M) x ^ k) m)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "f : M → M'", "x : M", "s : Set M", "Is : SmoothManifoldWithCorners I M", "I's : SmoothManifoldWithCorners I' M'", "f' f₁' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)", "h : HasMFDerivWithinAt I I' f s x f'", "h' : f' = f₁'"], "goal": "HasMFDerivWithinAt I I' f s x f₁'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : CompleteLattice α", "f : ι → α", "c : α"], "goal": "c ∈ Iic (⨅ i, f i) ↔ c ∈ ⨅ i, Iic (f i)"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "F G K L : CochainComplex C ℤ", "n m : ℤ", "φ : F ⟶ G", "p' p q : ℤ", "hpq : p + 0 = q"], "goal": "F.d p' p ≫ (ofHom φ).v p q hpq = F.d p' q ≫ φ.f q"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq β", "l : List α", "f : α → β"], "goal": "Finset.image (fun x => f (l.get x)) Finset.univ = (List.map f l).toFinset"}}
+{"state": {"context": [], "goal": "Real.zero = { cauchy := 0 }"}}
+{"state": {"context": ["A✝ : Type u_1", "inst✝⁹ : AddCommGroup A✝", "inst✝⁸ : Module ℂ A✝", "inst✝⁷ : StarAddMonoid A✝", "inst✝⁶ : StarModule ℂ A✝", "A : Type u_2", "inst✝⁵ : NonUnitalRing A", "inst✝⁴ : StarRing A", "inst✝³ : Module ℂ A", "inst✝² : IsScalarTower ℂ A A", "inst✝¹ : SMulCommClass ℂ A A", "inst✝ : StarModule ℂ A", "a : A", "a_eq : a = ↑(ℜ a) + I • ↑(ℑ a)"], "goal": "star (↑(ℜ a) + I • ↑(ℑ a)) * (↑(ℜ a) + I • ↑(ℑ a)) + (↑(ℜ a) + I • ↑(ℑ a)) * star (↑(ℜ a) + I • ↑(ℑ a)) = 2 • (↑(ℜ a) * ↑(ℜ a) + ↑(ℑ a) * ↑(ℑ a))"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : CommRing A", "inst✝ : Algebra ℚ A", "a b : A", "n x : ℕ", "hx : x ∈ Finset.range n.succ"], "goal": "↑(n.choose x) / ↑n ! = 1 / (↑x ! * ↑(n - x)!)"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝² : Category.{max v u, w} D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "P : Cᵒᵖ ⥤ D", "inst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "hP : ∀ (X : C) (S : J.Cover X), IsIso (S.toMultiequalizer P)", "X : Cᵒᵖ"], "goal": "IsIso (colimit.ι (J.diagram P (unop X)) (op ⊤))"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "s1 s2 : AffineSubspace k P", "hd : s1.direction = s2.direction", "hn : (↑s1 ∩ ↑s2).Nonempty", "p : P", "hq1 : hn.some ∈ ↑s1", "hq2 : hn.some ∈ ↑s2"], "goal": "p ∈ s1 ↔ p ∈ s2"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : MeasurableSpace E", "m μ : Measure E", "Ω : Type u_2", "x✝ : MeasurableSpace Ω", "ℙ : Measure Ω", "X : Ω → E", "s : Set E", "hns : μ s ≠ 0", "hnt : μ s ≠ ⊤", "hu : IsUniform X s ℙ μ", "t : Set E := toMeasurable μ s"], "goal": "Measure.map X ℙ = μ.withDensity (t.indicator ((μ t)⁻¹ • 1))"}}
+{"state": {"context": ["K : Type u", "CommRing K", "p : ℕ", "Fact (Nat.Prime p)", "CharP K p"], "goal": "1 = PerfectClosure.mk K p (0, 1)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "m✝ m : OuterMeasure α", "m0 m1 : MeasurableSpace α", "h : m0 ≤ m1"], "goal": "∀ (s : Set α), MeasurableSet s → m.trim s ≤ m s"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝⁷ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "E' : Type u_5", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝² : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹ : TopologicalSpace M'", "inst✝ : ChartedSpace H' M'", "s : Set H", "x : H", "f : H → H'", "e' : PartialHomeomorph H' H'", "he' : e' ∈ contDiffGroupoid ⊤ I'", "hs : s ⊆ f ⁻¹' e'.source", "hx : f x ∈ e'.source", "h : DifferentiableWithinAt 𝕜 (↑I' ∘ f ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)", "A : (↑I' ∘ f ∘ ↑I.symm) (↑I x) ∈ ↑I'.symm ⁻¹' e'.source ∩ range ↑I'"], "goal": "DifferentiableWithinAt 𝕜 (↑I' ∘ (↑e' ∘ f) ∘ ↑I.symm) (↑I.symm ⁻¹' s ∩ range ↑I) (↑I x)"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "T1Space X", "a b : X"], "goal": "pure a ≤ 𝓝 b ↔ a = b"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "(s ∩ t).Nonempty ↔ ∃ x ∈ s, x ∈ t"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : QuotientMap f", "s : Set Y"], "goal": "IsOpen (f ⁻¹' s) ↔ IsOpen s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹⁴ : CommRing R", "inst✝¹³ : Ring S", "inst✝¹² : Algebra R S", "K : Type u_4", "L : Type u_5", "F : Type u_6", "inst✝¹¹ : Field K", "inst✝¹⁰ : Field L", "inst✝⁹ : Field F", "inst✝⁸ : Algebra K L", "inst✝⁷ : Algebra K F", "ι : Type w", "E : Type u_7", "inst✝⁶ : Field E", "inst✝⁵ : Algebra K E", "inst✝⁴ : Algebra L F", "inst✝³ : IsScalarTower K L F", "inst✝² : IsAlgClosed E", "inst✝¹ : Algebra.IsSeparable K F", "inst✝ : FiniteDimensional K F", "pb : PowerBasis K L", "this✝ : FiniteDimensional L F", "this : Algebra.IsSeparable L F"], "goal": "∏ σ : F →ₐ[K] E, σ ((algebraMap L F) pb.gen) = (∏ σ : L →ₐ[K] E, σ pb.gen) ^ finrank L F"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "μ : M → N → N", "LinearOrder N"], "goal": "(Covariant M N μ fun x x_1 => x < x_1) ↔ Contravariant M N μ fun x x_1 => x ≤ x_1"}}
+{"state": {"context": ["x y : ℕ", "h : y % x = 0"], "goal": "x.gcd y = x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "K L : CochainComplex C ℤ", "n : ℤ", "γ₁ γ₂ : CochainComplex.HomComplex.Cochain K L n", "a n' : ℤ", "hn' : n' + a = n"], "goal": "(γ₁ + γ₂).rightShift a n' hn' = γ₁.rightShift a n' hn' + γ₂.rightShift a n' hn'"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "s t : Set α", "h : s ⊆ t"], "goal": "Measurable (Set.inclusion h)"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "M : Type u_4", "N : Type u_5", "AddCommMonoid M", "TopologicalSpace M", "AddCommMonoid N", "TopologicalSpace N", "v : MeasureTheory.VectorMeasure α M", "w : MeasureTheory.VectorMeasure α N", "s : Set α", "hs : MeasurableSet s", "h₁ : ∀ t ⊆ s, MeasurableSet t → ↑v t = 0", "h₂ : ∀ t ⊆ sᶜ, MeasurableSet t → ↑w t = 0"], "goal": "v ⟂ᵥ w"}}
+{"state": {"context": ["n : Type u", "inst✝² : DecidableEq n", "inst✝¹ : Fintype n", "R : Type v", "inst✝ : CommRing R", "A : GL n R"], "goal": "(↑A).det * (↑A⁻¹).det = 1"}}
+{"state": {"context": ["α : Type u_1", "M✝ : Matroid α", "R✝ I✝ J X Y : Set α", "M : Matroid α", "R I : Set α", "hI : M.Indep I", "hIY : I ⊆ R", "hIn : ¬M.Basis I (R ∩ M.E)", "B' : Set α", "hB' : M.Base B'", "hI' : M.Basis (B' ∩ (R ∩ M.E)) (R ∩ M.E)"], "goal": "∃ x ∈ (B' ∩ (R ∩ M.E)) \\ I, (fun I => M.Indep I ∧ I ⊆ R) (insert x I)"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "I : (FractionalIdeal (𝓞 K)⁰ K)ˣ"], "goal": "2 ^ finrank ℝ (({ w // w.IsReal } → ℝ) × ({ w // w.IsComplex } → ℂ)) ≠ 0"}}
+{"state": {"context": ["V : Type u_4", "NormedAddCommGroup V", "InnerProductSpace ℂ V", "T : V →ₗ[ℂ] V", "x y : V"], "goal": "⟪T x, y⟫_ℂ =\n  (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ - Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ +\n      Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) /\n    4"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_4", "γ : Type u_6", "γ' : Type u_7", "l : Type u_8", "m : Type u_9", "n : Type u_10", "p : Type u_11", "f : α → β → γ", "g : γ → γ'", "A : Matrix l m α", "B : Matrix n p β"], "goal": "(Matrix.kroneckerMap f A B).map g = Matrix.kroneckerMap (fun a b => g (f a b)) A B"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "Nontrivial M", "Module.Finite R M", "r : R", "h : r ∈ (Module.annihilator R M).jacobson"], "goal": "⊤ ≠ r • ⊤"}}
+{"state": {"context": ["X✝ Y✝ : Scheme", "f✝ : X✝ ⟶ Y✝", "X Y : Scheme", "f : X ⟶ Y", "hY : QuasiSeparated (terminal.from Y)", "inst✝ : QuasiSeparated f"], "goal": "QuasiSeparated (f ≫ terminal.from Y)"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "inst✝² : Semiring α", "inst✝¹ : PartialOrder α", "a b c d : α", "inst✝ : MulPosStrictMono α", "hn : 0 < a", "hm : 1 < b"], "goal": "a = 1 * a"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "E : Type u_4", "F : Type u_6", "Fₗ : Type u_7", "NormedAddCommGroup E", "NormedAddCommGroup F", "NormedAddCommGroup Fₗ", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜₂", "NormedSpace 𝕜 E", "NormedSpace 𝕜₂ F", "NormedSpace 𝕜 Fₗ", "σ₁₂ : 𝕜 →+* 𝕜₂", "f : E →SL[σ₁₂] F", "CompleteSpace F", "e : E →L[𝕜] Fₗ", "h_dense : DenseRange ⇑e", "h_e : UniformInducing ⇑e", "g : Fₗ →SL[σ₁₂] F", "H : g.comp e = f"], "goal": "f.extend e h_dense h_e = g"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "TopologicalSpace α", "TopologicalSpace β", "TopologicalSpace γ", "f : SpectralMap β γ", "g : SpectralMap α β", "a : α"], "goal": "(f.comp g) a = f (g a)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "ι : Type u_4", "f : X → Y", "g : Y → Z", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "f_closed : IsClosedMap f", "f_cont : Continuous f", "F : Filter X"], "goal": "F.lift' (closure ∘ image f) = F.lift' (image f ∘ closure)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "M' : Type u_3", "N' : Type u_4", "AddZeroClass M", "AddZeroClass N", "AddZeroClass M'", "AddZeroClass N'", "e : M ≃+ N", "e' : M' ≃+ N'"], "goal": "⇑(AddMonoid.MulEquiv.coprodCongr e e').symm = ⇑(AddMonoid.Coprod.map ↑e.symm ↑e'.symm)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "G : Type u_3", "R : Type u_4", "S : Type u_5", "F : Type u_6", "inst✝² : CommMonoid M", "inst✝¹ : CommMonoid N", "inst✝ : DivisionCommMonoid G", "k l : ℕ", "ζ : M", "f : F", "h✝ h : IsPrimitiveRoot ζ k", "h0 : 0 < k"], "goal": "IsUnit ζ"}}
+{"state": {"context": ["𝕜 : Type u_1", "M : Type u_2", "α : Type u_3", "G : Type u_4", "E : Type u_5", "F : Type u_6", "inst✝⁷ : MeasurableSpace G", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "inst✝³ : NormedAddCommGroup F", "μ : Measure G", "f : G → E", "g : G", "inst✝² : Group G", "inst✝¹ : MeasurableMul G", "inst✝ : μ.IsMulRightInvariant", "hf' : ∀ (x : G), f (x * g) = -f x"], "goal": "∫ (x : G), f x ∂μ = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : CommRing R", "M : Type u_2", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "M' : Type u_3", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : Module R M'", "ι : Type u_4", "inst✝³ : DecidableEq ι", "inst✝² : Fintype ι", "e✝ : Basis ι R M", "N : Type u_5", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "f : N →ₗ[R] M", "e e' : M ≃ₗ[R] N", "x : M"], "goal": "(f ∘ₗ ↑e) x = (f ∘ₗ ↑e') ((e ≪≫ₗ e'.symm) x)"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "x : 𝕜", "𝕜' : Type u_2", "NontriviallyNormedField 𝕜'", "NormedAlgebra 𝕜 𝕜'", "NormedSpace 𝕜' F", "IsScalarTower 𝕜 𝕜' F", "c : 𝕜 → 𝕜'", "hc : DifferentiableAt 𝕜 c x", "f : F"], "goal": "deriv (fun y => c y • f) x = deriv c x • f"}}
+{"state": {"context": ["R : Type u", "inst✝² : CommRing R", "𝓟 : Ideal R", "f : R[X]", "hf : f.IsWeaklyEisensteinAt 𝓟", "S : Type v", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "x : S", "hx : (aeval x) f = 0", "hmo : f.Monic"], "goal": "∀ (i : ℕ), (Polynomial.map (algebraMap R S) f).natDegree ≤ i → x ^ i ∈ Ideal.map (algebraMap R S) 𝓟"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α ��� Prop", "f : β → α", "hr : Pairwise (r on f)", "hf : Function.Surjective f"], "goal": "Pairwise r"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : E → ℝ", "f' : E →L[ℝ] ℝ", "x : E", "s : Set E", "x✝ : ℝ"], "goal": "logDeriv cosh x✝ = tanh x✝"}}
+{"state": {"context": ["ι : Sort u_1", "κ : ι → Sort u_2", "α : Type u_3", "β : Type u_4", "inst✝³ : TopologicalSpace α", "inst✝² : TopologicalSpace β", "s t : Set α", "a x y : α", "inst✝¹ : AlexandrovDiscrete α", "inst✝ : AlexandrovDiscrete β", "f : β → α", "h : Inducing f", "S : Set (Set β)", "U : (s : Set β) → s ∈ S → Set α", "hU : ∀ (s : Set β) (a : s ∈ S), IsOpen (U s a)", "htU : ∀ (s : Set β) (a : s ∈ S), f ⁻¹' U s a = s"], "goal": "∃ t, IsOpen t ∧ f ⁻¹' t = ⋂₀ S"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝ : Field F", "W : Jacobian F", "P : Fin 3 → R", "hPz : P z = 0"], "goal": "W'.Nonsingular P ↔ W'.Equation P ∧ (3 * P x ^ 2 ≠ 0 ∨ 2 * P y ≠ 0 ∨ W'.a₁ * P x * P y ≠ 0)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "r : R", "rs : List R"], "goal": "Ideal.ofList (r :: rs) = Ideal.span {r} ⊔ Ideal.ofList rs"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "Ring k", "MulOneClass G", "z : ℤ"], "goal": "↑z = MonoidAlgebra.single 1 ↑z"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "(∀ (f g : R[X]), f = g) ↔ ∀ (a b : R), a = b"}}
+{"state": {"context": ["K : Type u", "CommRing K", "p : ℕ", "Fact (Nat.Prime p)", "CharP K p", "x1 x2 y : ℕ × K", "H : PerfectClosure.R K p x1 x2"], "goal": "PerfectClosure.mk K p (x1.1 + y.1, (⇑(frobenius K p))^[y.1] x1.2 + (⇑(frobenius K p))^[x1.1] y.2) =\n  PerfectClosure.mk K p (x2.1 + y.1, (⇑(frobenius K p))^[y.1] x2.2 + (⇑(frobenius K p))^[x2.1] y.2)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "k : ℕ"], "goal": "((⇑Polynomial.derivative)^[k] p).natDegree ≤ p.natDegree - k"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommMonoid α", "a : α"], "goal": "a + a ≤ 0 ↔ a ≤ 0"}}
+{"state": {"context": ["R : Type u", "inst✝ : AddGroupWithOne R", "n : ℕ", "x✝ : 0 < n + 1"], "goal": "↑(n + 1 - 1) = ↑n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Semiring R", "p q : R[X]"], "goal": "p.divX.coeff n = p.coeff (n + 1)"}}
+{"state": {"context": ["α : Type u_2", "A : Set α", "D E : Set ↑A"], "goal": "Subtype.val '' (D ∩ E) = Subtype.val '' D ∩ Subtype.val '' E"}}
+{"state": {"context": ["R : Type u", "inst✝¹¹ : Ring R", "P : Type v", "inst✝¹⁰ : AddCommGroup P", "inst✝⁹ : Module R P", "R₀ : Type ?u.49012", "M : Type u_1", "N : Type ?u.52304", "inst✝⁸ : CommRing R₀", "inst✝⁷ : Algebra R₀ R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R₀ M", "inst✝⁴ : Module R M", "inst✝³ : IsScalarTower R₀ R M", "inst✝² : AddCommGroup N", "inst✝¹ : Module R₀ N", "inst✝ : Projective R M", "i : P →ₗ[R] M", "s : M →ₗ[R] P", "H : s ∘ₗ i = LinearMap.id"], "goal": "Projective R P"}}
+{"state": {"context": ["x y : Class", "H : ∀ z ∈ x, y ∈ z", "z : ZFSet", "hz : z ∈ x"], "goal": "∃ x_1, ↑x_1 = y ∧ ∀ (z : ZFSet), x z → x_1 ∈ z"}}
+{"state": {"context": ["k n : ℕ", "hk0 : k ≠ 0", "hk1 : 1 < k", "b : ℕ := k * n !", "hgt : 1 < (eval (↑b) (cyclotomic k ℤ)).natAbs", "p : ℕ := (eval (↑b) (cyclotomic k ℤ)).natAbs.minFac", "hprime : Fact (Prime p)"], "goal": "∃ p, Prime p ∧ n < p ∧ p ≡ 1 [MOD k]"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "Subsingleton α", "f : α → β"], "goal": "Antitone f"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "Star A", "S : NonUnitalStarSubalgebra R A", "s : Set A", "hs : s = ↑S"], "goal": "↑(S.copy s hs) = s"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "ι : Type u_4", "f : X → Y", "g : Y → Z", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "h₁ : Continuous f", "h₂ : Injective f", "h₃ : IsOpenMap f"], "goal": "OpenEmbedding f"}}
+{"state": {"context": ["ι✝ : Type u_1", "inst✝¹⁰ : Fintype ι✝", "A : ι✝ → Type u_2", "inst✝⁹ : (i : ι✝) → MeasurableSpace (A i)", "μ✝ : (i : ι✝) → Measure (A i)", "inst✝⁸ : ∀ (i : ι✝), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "u : E → F", "hu : ContDiff ℝ 1 u", "h2u : HasCompactSupport u", "p : ℝ", "hp✝ : (↑(finrank ℝ E)).IsConjExponent p", "C : ℝ≥0 := lintegralPowLePowLIntegralFDerivConst μ p", "ι : Type := Fin (finrank ℝ E)", "hιcard : #ι = finrank ℝ E", "this✝ : finrank ℝ E = finrank ℝ (ι → ℝ)", "e : E ≃L[ℝ] ι → ℝ := ContinuousLinearEquiv.ofFinrankEq this✝", "this : (Measure.map (⇑e.symm) volume).IsAddHaarMeasure", "hp : (↑#ι).IsConjExponent p", "h0p : 0 ≤ p"], "goal": "∫⁻ (x : E), ↑‖u x‖₊ ^ p ∂μ ≤ ↑C * (∫⁻ (x : E), ↑‖fderiv ℝ u x‖₊ ∂μ) ^ p"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "NormedField 𝕜", "AddCommGroup V", "Module 𝕜 V", "e : ENorm 𝕜 V", "x : V"], "goal": "↑e (-x) = ↑e x"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : EMetricSpace α", "cs : CompleteSpace α", "K : ℝ≥0", "f : α → α", "hf : ContractingWith K f", "s : Set α", "hsc : IsComplete s", "hsf : MapsTo f s s", "hfs : ContractingWith K (MapsTo.restrict f s s hsf)", "x : α", "hxs : x ∈ s", "hx : edist x (f x) ≠ ⊤", "t : Set α", "htc : IsComplete t", "htf : MapsTo f t t", "hft : ContractingWith K (MapsTo.restrict f t t htf)", "y : α", "hyt : y ∈ t", "hy : edist y (f y) ≠ ⊤", "hxy : edist x y ≠ ⊤"], "goal": "efixedPoint' f hsc hsf hfs x hxs hx = efixedPoint' f htc htf hft y hyt hy"}}
+{"state": {"context": ["n : ℕ"], "goal": "Fintype.card (CompositionAsSet n) = 2 ^ (n - 1)"}}
+{"state": {"context": ["α : Type u_2", "AddGroup α", "t : Set α", "b : α"], "goal": "(fun x => x + b) '' t = (fun x => x + -b) ⁻¹' t"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "inst✝² : Category.{v₁, u₂} 𝒳", "inst✝¹ : Category.{v₂, u₁} 𝒮", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ≅ b", "inst✝ : p.IsHomLift f φ.hom"], "goal": "(eqToHom ⋯ ≫ (p.mapIso φ).inv ≫ eqToHom ⋯) ≫ f = 𝟙 S"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝ q✝ r p q : R[X]", "hp : p ≠ 0", "hpq : p.degree < q.degree", "hq : ¬q = 0"], "goal": "p.natDegree < q.natDegree"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_5", "ι : Type u_8", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : SeminormFamily 𝕜 E ι", "Nonempty ι", "x : 𝕜", "U : Set E", "hU : U ∈ p.basisSets"], "goal": "∃ V ∈ AddGroupFilterBasis.toFilterBasis.sets, V ⊆ (fun y => x • y) ⁻¹' U"}}
+{"state": {"context": ["α : Type u_1", "a✝ b✝ c✝ d : α", "inst✝¹ : MulZeroClass α", "inst✝ : PartialOrder α", "h✝ : CovariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x ≤ x_1", "a : { x // 0 ≤ x }", "b c : α", "h : b ≤ c", "ha : 0 = ↑a"], "goal": "↑a * b ≤ ↑a * c"}}
+{"state": {"context": ["M : Type u_1", "Add M", "p p' : AddSubsemigroup M", "x : M"], "goal": "x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'"}}
+{"state": {"context": ["K : Type u", "Field K", "L : Type u_3", "Field L", "Algebra K L", "x : L", "FiniteDimensional K L"], "goal": "(minpoly K x).degree ≤ ↑(FiniteDimensional.finrank K L)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "f : X → Y", "TopologicalSpace X", "TopologicalSpace Y", "hf : Inducing f"], "goal": "Continuous f"}}
+{"state": {"context": ["G : Type u", "M : Type v", "α : Type w", "s✝ : Set α", "m : MeasurableSpace α", "inst✝³ : MeasurableSpace G", "inst✝² : Group G", "inst✝¹ : MulAction G α", "μ : Measure α", "inst✝ : SMulInvariantMeasure G α μ", "s : Set α", "c : G"], "goal": "μ (c • s) = 0 ↔ μ s = 0"}}
+{"state": {"context": ["a b : ℝ", "f : C(↑(Set.Icc a b), ℝ)", "ε : ℝ", "pos : 0 < ε", "w : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑(polynomialFunctions (Set.Icc a b)).toSubsemiring", "m : ℝ[X]", "H : ↑(Polynomial.toContinuousMapOnAlgHom (Set.Icc a b)) m ∈ Metric.ball f ε"], "goal": "∃ p, ‖p.toContinuousMapOn (Set.Icc a b) - f‖ < ε"}}
+{"state": {"context": ["R : Type u_1", "a b : R", "inst✝ : Monoid R", "n : ℕ", "rra : IsRightRegular a"], "goal": "Function.Injective (fun x => x * a)^[n]"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "h : b ∈ closedBall a r"], "goal": "‖b / a‖ ≤ r"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "W : CategoryTheory.MorphismProperty C", "W.HasLeftCalculusOfFractions", "X Y Z : C", "z₁ : W.LeftFraction X Y", "z₂ : W.LeftFraction Y Z", "z₃ : W.LeftFraction z₁.Y' z₂.Y'", "h₃ : CategoryTheory.CategoryStruct.comp z₂.f z₃.s = CategoryTheory.CategoryStruct.comp z₁.s z₃.f", "L : CategoryTheory.Functor C D", "L.IsLocalization W"], "goal": "CategoryTheory.CategoryStruct.comp (z₁.map L ⋯) (z₂.map L ⋯) = (z₁.comp₀ z₂ z₃).map L ⋯"}}
+{"state": {"context": ["α : Type u_3", "m : MeasurableSpace α", "μ ν : MeasureTheory.Measure α", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace ℝ E", "μ.HaveLebesgueDecomposition ν", "hμν : μ ≪ ν", "MeasureTheory.SigmaFinite μ", "f : α → E"], "goal": "MeasureTheory.Integrable (fun x => (μ.rnDeriv ν x).toReal • f x) ν ↔ MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["F : Type w", "R : Type u", "S : Type v", "NonUnitalNonAssocRing R", "NonUnitalNonAssocRing S", "FunLike F R S", "NonUnitalRingHomClass F R S", "s t : NonUnitalSubring S", "f : F"], "goal": "NonUnitalSubring.comap f (s ⊓ t) = NonUnitalSubring.comap f s ⊓ NonUnitalSubring.comap f t"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p✝ q✝ p q : R[X]", "h : {x | eval x p = eval x q}.Infinite"], "goal": "{x | (p - q).IsRoot x}.Infinite"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "a : α", "h : s.Nonempty"], "goal": "s ⊆ {a} ↔ s = {a}"}}
+{"state": {"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "hcont✝ : ContinuousWithinAt f (Ici a) a", "hderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Ioi a) volume", "hf : Tendsto f atTop (𝓝 m)", "hcont : ContinuousOn f (Ici a)", "x : ℝ", "hx : x ∈ Ioi a", "h'x : a ≤ id x"], "goal": "IntervalIntegrable f' volume a (id x)"}}
+{"state": {"context": ["α : Type u", "s t : Stream'.WSeq α", "h : s ~ʷ t", "a : α"], "goal": "a ∈ s ↔ a ∈ t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "B : FilterBasis α"], "goal": "B.filter ≤ generate B.sets"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : Finite α", "f : α → β", "e : α ≃ β", "this : Injective (⇑e.symm ∘ f) ↔ Surjective (⇑e.symm ∘ f)", "hinj : Injective f"], "goal": "Surjective f"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "M : Type u_7", "AddMonoid M", "f g : G →+ M", "s : Set G", "h : Set.EqOn (⇑f) (⇑g) s"], "goal": "Set.EqOn ⇑f ⇑g ↑(AddSubgroup.closure s)"}}
+{"state": {"context": ["u : PrimeMultiset", "n : ℕ+", "h : u.prod ∣ n ↔ u ≤ n.factorMultiset"], "goal": "u.prod ∣ n ↔ u ≤ n.factorMultiset"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "MulOneClass M", "MulOneClass N", "f : M ≃* N", "K : Submonoid M", "x : N"], "goal": "x ∈ Submonoid.map f.toMonoidHom K ↔ f.symm x ∈ K"}}
+{"state": {"context": ["M : Type v", "α : Type w", "m : MeasurableSpace α", "MeasurableSpace M", "SMul M α", "MeasurableSMul M α", "c : M", "μ : MeasureTheory.Measure α", "MeasureTheory.SMulInvariantMeasure M α μ"], "goal": "MeasureTheory.MeasurePreserving (fun x => c • x) μ μ"}}
+{"state": {"context": ["α : Type u", "LinearOrder α", "a b c : α", "h : b < c"], "goal": "min a b < c"}}
+{"state": {"context": ["α : Type u_1", "f f' g : α → α", "h : Function.Commute f g", "h' : Function.Commute f' g"], "goal": "Function.Commute (f ∘ f') g"}}
+{"state": {"context": ["α β : Type u", "s : Set α", "t : (a : α) → a ∈ s → Set β", "c : Cardinal.{u}", "h : #↑(⋃ a, ⋃ (h : a ∈ s), t a h) < c", "a : α", "ha : a ∈ s"], "goal": "#↑(t a ha) < c"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "L₁ : CategoryTheory.Functor C D", "R₁ : CategoryTheory.Functor D C", "adj₁ : L₁ ⊣ R₁"], "goal": "(CategoryTheory.conjugateEquiv adj₁ adj₁).symm (CategoryTheory.CategoryStruct.id R₁) =\n  CategoryTheory.CategoryStruct.id L₁"}}
+{"state": {"context": ["x : ℝ", "n : ℕ∞"], "goal": "ContDiffAt ℝ n Real.arcsin x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "γ : Type u_3", "β : Type u_4", "inst✝⁵ : TopologicalSpace γ", "inst✝⁴ : PolishSpace γ", "inst✝³ : TopologicalSpace β", "inst✝² : T2Space β", "inst✝¹ : MeasurableSpace β", "inst✝ : OpensMeasurableSpace β", "f : γ → β", "f_cont : Continuous f", "f_inj : Injective f", "this✝ : UpgradedPolishSpace γ := upgradePolishSpace γ", "b : Set (Set γ)", "b_count : b.Countable", "b_nonempty : ∅ ∉ b", "hb : IsTopologicalBasis b", "this : Encodable ↑b", "A : Type u_3 := { p // Disjoint ↑p.1 ↑p.2 }", "q : A → Set β", "hq1 : ∀ (p : A), f '' ↑(↑p).1 ⊆ q p", "hq2 : ∀ (p : A), Disjoint (f '' ↑(↑p).2) (q p)", "q_meas : ∀ (p : A), MeasurableSet (q p)", "E : ↑b → Set β := fun s => closure (f '' ↑s) ∩ ⋂ t, ⋂ (ht : Disjoint ↑s ↑t), q ⟨(s, t), ht⟩ \\ q ⟨(t, s), ⋯⟩", "u : ℕ → ℝ", "u_anti : StrictAnti u", "u_pos : ∀ (n : ℕ), 0 < u n", "u_lim : Tendsto u atTop (𝓝 0)"], "goal": "MeasurableSet (range f)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "f : X → Y", "hf : IsLocallyConstant f", "x : X"], "goal": "∃ U, IsOpen U ∧ x ∈ U ∧ ∀ x' ∈ U, f x' = f x"}}
+{"state": {"context": ["s : Set Game", "inst✝ : Small.{u, u + 1} ↑s"], "goal": "BddBelow s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "op : β → β → β", "hc : Std.Commutative op", "ha : Std.Associative op", "f : α → β", "b : β", "s : Finset α", "a : α", "inst✝ : LinearOrder β", "c : β"], "goal": "∀ {x y z : β}, x ≥ min y z ↔ x ≥ y ∨ x ≥ z"}}
+{"state": {"context": ["a b : Ordinal.{u_4}", "a0 : 0 < a"], "goal": "b.IsLimit → (a * b).IsLimit"}}
+{"state": {"context": ["β : Type u_1", "α : Type u_2", "A : List β → Set α", "inst✝¹ : PseudoMetricSpace α", "inst✝ : CompleteSpace α", "hdiam : VanishingDiam A", "hanti : ClosureAntitone A", "hnonempty : ∀ (l : List β), (A l).Nonempty", "x : ℕ → β", "u : ℕ → α", "hu : ∀ (n : ℕ), u n ∈ A (res x n)", "umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)", "this : CauchySeq u"], "goal": "x ∈ (inducedMap A).fst"}}
+{"state": {"context": ["α : Type u", "f : Ultrafilter α"], "goal": "Ultrafilter.of ↑f = f"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : Projective R", "P✝ : Fin 3 → R", "u✝ u : Rˣ", "P : Fin 3 → R", "h : W.Nonsingular (u • P)"], "goal": "W.Nonsingular P"}}
+{"state": {"context": ["G : Type u_1", "Group G", "A' A : Subgroup G", "B : Subgroup G", "hA : A' ≤ A", "hN : (A'.subgroupOf A).Normal"], "goal": "((A' ⊓ B).subgroupOf (A ⊓ B)).Normal"}}
+{"state": {"context": ["R : Type u", "n : Type w", "inst✝² : CommRing R", "inst✝¹ : DecidableEq n", "inst✝ : Fintype n", "J A B : Matrix n n R", "hA : Aᵀ * J = J * -A", "hB : Bᵀ * J = J * -B"], "goal": "⁅A, B⁆ᵀ * J = J * -⁅A, B⁆"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "MeasurableSingletonClass α", "a : α → ℝ≥0∞", "a_mble : Measurable a", "c : ℝ≥0∞", "tsum_le_c : ∑' (i : α), a i ≤ c", "ε : ℝ≥0∞", "ε_ne_zero : ε ≠ 0", "ε_ne_top : ε ≠ ⊤"], "goal": "MeasureTheory.Measure.count {i | ε ≤ a i} ≤ c / ε"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ✝ : MeasurableSpace Ω", "μ✝ : Measure Ω", "g : Ω → ℝ≥0∞", "X Y : Ω → ℝ", "Mf mΩ : MeasurableSpace Ω", "μ : Measure Ω", "hMf : Mf ≤ mΩ", "c : ℝ≥0∞", "T : Set Ω", "h_meas_T : MeasurableSet T", "h_ind : IndepSets {s | MeasurableSet s} {T} μ", "h_mul_indicator : ∀ (g : Ω → ℝ≥0∞), Measurable g → Measurable fun a => g a * T.indicator (fun x => c) a"], "goal": "∀ {f : Ω → ℝ≥0∞}, Measurable f → ∫⁻ (ω : Ω), f ω * T.indicator (fun x => c) ω ∂μ = (∫⁻ (ω : Ω), f ω ∂μ) * ∫⁻ (ω : Ω), T.indicator (fun x => c) ω ∂μ"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "f g : Perm (Fin n)"], "goal": "(∏ x ∈ finPairsLT n, if (f * g) x.fst ≤ (f * g) x.snd then -1 else 1) = ∏ x ∈ finPairsLT n, (if f x.fst ≤ f x.snd then -1 else 1) * if g⁻¹ x.fst ≤ g⁻¹ x.snd then -1 else 1"}}
+{"state": {"context": ["ι : Type u_1", "Fintype ι", "I : BoxIntegral.Box ι"], "goal": "(BoxIntegral.Prepartition.splitCenter I).IsPartition"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "a : α", "s : Computation α", "m : a ∈ s"], "goal": "f a ∈ Computation.map f s"}}
+{"state": {"context": ["𝕜 : Type u_2", "E : Type u_5", "CommSemiring 𝕜", "TopologicalSpace 𝕜", "ContinuousAdd 𝕜", "ContinuousConstSMul 𝕜 𝕜", "AddCommMonoid E", "Module 𝕜 E", "TopologicalSpace E"], "goal": "Function.Bijective ⇑(continuousLinearMapToWeakSpace 𝕜 E)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "AddCommMonoid M", "f : α × β → M", "hf : (Function.support f).Finite"], "goal": "∑ᶠ (ab : α × β), f ab = ∑ᶠ (a : α) (b : β), f (a, b)"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "idx : Type u_2", "inst✝ : CommRing R", "hp : Fact (Nat.Prime p)", "σ : Type u_3", "Φ : MvPolynomial idx ℤ", "f : idx → σ", "n : ℕ"], "goal": "(bind₁ fun k => (bind₁ ((fun i => (rename (Prod.mk i)) (W_ ℚ k)) ∘ f)) ((map (Int.castRingHom ℚ)) Φ)) (xInTermsOfW p ℚ n) = (bind₁ fun i => (bind₁ fun i_1 => (rename (Prod.map f id ∘ Prod.mk i_1)) (W_ ℚ i)) ((map (Int.castRingHom ℚ)) Φ)) (xInTermsOfW p ℚ n)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "a : R", "n : ℕ"], "goal": "Mathlib.Meta.NormNum.IsNat a n → a = n.rawCast + 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s s₁✝ s₂✝ t t₁ t₂ u v : Finset α", "a b : α", "s₁ s₂ s₃ : Finset α", "x✝ : α"], "goal": "x✝ ∈ s₁ ∩ (s₂ ∩ s₃) ↔ x✝ ∈ s₂ ∩ (s₁ ∩ s₃)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "T : Set α → E →L[ℝ] F'", "C : ℝ", "hT_norm : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ‖T s‖ ≤ C * (μ s).toReal", "f : α →ₛ E", "hf : Integrable (↑f) μ", "b : E", "hb✝ : b ∈ f.range", "hb : b ≠ 0"], "goal": "‖T (↑f ⁻¹' {b})‖ ≤ C * (μ (↑f ⁻¹' {b})).toReal"}}
+{"state": {"context": ["k : Type u_1", "Field k", "σ : Type u_2", "V : Set (σ → k)"], "goal": "PrimeSpectrum.vanishingIdeal (MvPolynomial.pointToPoint '' V) = MvPolynomial.vanishingIdeal V"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "L' : Type u_3", "Field K", "Field L", "Field L'", "Algebra K L", "Algebra K L'", "Algebra L' L", "IsScalarTower K L' L", "E : IntermediateField L' L"], "goal": "(IntermediateField.restrictScalars K E).toSubalgebra = Subalgebra.restrictScalars K E.toSubalgebra"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "G : CategoryTheory.Functor D C", "adj : CategoryTheory.Adjunction.CoreHomEquiv F G", "X : C", "Y Y' : D", "f : X ⟶ G.obj Y", "g : Y ⟶ Y'"], "goal": "CategoryTheory.CategoryStruct.comp (F.map f) ((adj.homEquiv (G.obj Y) Y').symm (G.map g)) =\n  CategoryTheory.CategoryStruct.comp ((adj.homEquiv X Y).symm f) g"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "r : α → β → Prop", "self : Concept α β r"], "goal": "intentClosure r self.toProd.1 = self.toProd.2"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "p : α → Prop", "f : α → ℝ≥0∞", "hf : Measurable f"], "goal": "(∀ᵐ (x : α) ∂μ.withDensity f, p x) ↔ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, p x"}}
+{"state": {"context": ["α : Sort u_1", "p : α → Prop", "a1 a2 : { x // p x }"], "goal": "↑a1 = ↑a2 → a1 = a2"}}
+{"state": {"context": ["x : ℝ", "h1 : 0 ≤ x", "h2 : x ≤ 1", "n : ℕ", "hn : 0 < n", "h3 : |x| = x", "h4 : |x| ≤ 1", "h' : |rexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial| ≤ x ^ n * (↑n.succ / (↑n.factorial * ↑n))", "h'' : rexp x - ∑ m ∈ range n, x ^ m / ↑m.factorial ≤ x ^ n * (↑n.succ / (↑n.factorial * ↑n))"], "goal": "rexp x ≤ ∑ m ∈ range n, x ^ m / ↑m.factorial + x ^ n * (↑n + 1) / (↑n.factorial * ↑n)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹¹ : CommRing R", "inst✝¹⁰ : LieRing L", "inst✝⁹ : LieAlgebra R L", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "inst✝⁶ : LieRingModule L M", "inst✝⁵ : LieModule R L M", "k : ℕ", "N : LieSubmodule R L M", "M₂ : Type w₁", "inst✝⁴ : AddCommGroup M₂", "inst✝³ : Module R M₂", "inst✝² : LieRingModule L M₂", "inst✝¹ : LieModule R L M₂", "inst✝ : IsNilpotent R L M", "s : Set ℕ := {k | lowerCentralSeries R L M k = ⊥}", "hs : s.Nonempty", "h : lowerCentralSeries R L M 0 = ⊥"], "goal": "0 ∈ s ∨ s = ∅"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : 𝕜 →ᵃ[𝕜] E", "x : 𝕜"], "goal": "deriv (⇑f) x = f.linear 1"}}
+{"state": {"context": ["R : Type v", "M : Type w", "CommRing R", "AddCommGroup M", "Module R M", "f : Module.End R M", "k : ℕ"], "goal": "(f.genEigenspace 0) k = LinearMap.ker (f ^ k)"}}
+{"state": {"context": ["σ✝ : Type u", "R✝ : Type v", "inst✝¹ : CommSemiring R✝", "σ : Type u", "R : Type v", "inst✝ : CommSemiring R"], "goal": "#(MvPolynomial σ R) ≤ max (max (lift.{u, v} #R) (lift.{v, u} #σ)) ℵ₀"}}
+{"state": {"context": ["p : Char → Bool", "l m r : List Char"], "goal": "Substring.takeWhileAux { data := l ++ m ++ r } { byteIdx := String.utf8Len l + String.utf8Len m } p\n    { byteIdx := String.utf8Len l } =\n  { byteIdx := String.utf8Len l + String.utf8Len (List.takeWhile p m) }"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "l✝ : List α", "a✝ m a : α", "l : List α"], "goal": "Option.rec (some a) (fun val => if val < a then some a else some val) (argmax id l) = max (argmax id l) ↑a"}}
+{"state": {"context": ["X✝ Y : Scheme", "f✝ : X✝ ⟶ Y", "Z X : Scheme", "U : X.Opens", "hU : IsCompact U.carrier", "x f : ↑Γ(X, U)", "H : (x |_ X.basicOpen f) ⋯ = 0", "s : Set ↑X.affineOpens", "hs : s.Finite", "e : U.carrier = ⋃ i ∈ s, ↑↑i"], "goal": "∃ n, f ^ n * x = 0"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "AddGroup G", "AddAction G α", "MeasurableSpace α", "s : Set α", "μ : MeasureTheory.Measure α", "MeasurableSpace G", "MeasurableVAdd G α", "MeasureTheory.VAddInvariantMeasure G α μ", "Countable G", "h : MeasureTheory.IsAddFundamentalDomain G s μ", "f : α → ℝ≥0∞"], "goal": "∫⁻ (x : α), f x ∂μ = ∑' (g : G), ∫⁻ (x : α) in s, f (g +ᵥ x) ∂μ"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "IsEmpty α"], "goal": "krullDim α = ⊥"}}
+{"state": {"context": ["n : ℤ"], "goal": "ZMod.χ₈ ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "s : Set E", "CompleteSpace F", "h : AnalyticOn 𝕜 f s", "n : ℕ"], "goal": "AnalyticOn 𝕜 (iteratedFDeriv 𝕜 n f) s"}}
+{"state": {"context": ["R : Type u_1", "CommMonoidWithZero R", "n : ℕ", "χ : DirichletCharacter R n", "m : ℕ", "hm : n ∣ m"], "goal": "MulChar.toUnitHom ((DirichletCharacter.changeLevel hm) χ) = (MulChar.toUnitHom χ).comp (ZMod.unitsMap hm)"}}
+{"state": {"context": ["e : ONote", "n : ℕ+", "a : ONote"], "goal": "(e.oadd n a).NF → a.NFBelow e.repr"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s : Set (β × γ)", "κ : Kernel α β", "η : Kernel (α × β) γ", "inst✝ : IsFiniteKernel η", "a : α", "hκ : IsSFiniteKernel κ"], "goal": "∫⁻ (b : β), (η (a, b)) Set.univ ∂κ a ≤ (κ a) Set.univ * IsFiniteKernel.bound η"}}
+{"state": {"context": ["α : Type u_3", "Ring α", "a : α", "n : ℤ"], "goal": "Commute a (↑n * a)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝¹ ν✝¹ μ✝ ν✝ : Measure α", "f : α → ℝ≥0∞", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "hf : AEMeasurable f ν", "hf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0", "hf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ⊤", "μ' : Measure α := ν.withDensity (μ.rnDeriv ν)"], "goal": "μ.rnDeriv (ν.withDensity f) =ᶠ[ae ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "x : E", "r : ℝ", "hr : 0 < r", "s : ℝ"], "goal": "μ (Metric.ball x (r * s)) = ENNReal.ofReal (r ^ FiniteDimensional.finrank ℝ E) * μ (Metric.ball 0 s)"}}
+{"state": {"context": ["R : Type u_2", "Add R", "Mul R", "Neg R", "One R", "Zero R", "self : FirstOrder.Ring.CompatibleRing R", "x : Fin 0 → R"], "goal": "FirstOrder.Language.Structure.funMap FirstOrder.Ring.oneFunc x = 1"}}
+{"state": {"context": ["E : Type u_1", "MeasurableSpace E", "μ : MeasureTheory.Measure E", "s : Set E"], "goal": "MeasureTheory.pdf.IsUniform id s (ProbabilityTheory.cond μ s) μ"}}
+{"state": {"context": ["a b : ℕ", "h : b ≤ a"], "goal": "a ≡ b [MOD a - b]"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CommGroup α", "inst✝ : DecidableEq α", "A B C✝ : Finset α", "hA : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card", "x : α", "C : Finset α", "a✝ : x ∉ C", "ih : (A * B * C).card * A.card ≤ (A * B).card * (A * C).card", "A' : Finset α := A ∩ (A * C / {x})", "hA' : A' = A ∩ (A * C / {x})", "C' : Finset α := insert x C", "hC' : C' = insert x C", "h₀ : A' * {x} = A * {x} ∩ (A * C)", "h₁ : A * B * C' = A * B * C ∪ (A * B * {x}) \\ (A' * B * {x})", "h₂ : A' * B * {x} ⊆ A * B * {x}", "h₃ : (A * B * C').card ≤ (A * B * C).card + (A * B).card - (A' * B).card"], "goal": "((A * B * C).card + (A * B).card - (A' * B).card) * A.card ≤ (A * B).card * (A * C').card"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Preorder α", "f g : α → α", "h : Commute f g", "hf : Monotone f", "hg : StrictMono g", "x : α", "hx : f x < g x", "n : ℕ", "hn : 0 < n"], "goal": "f^[0] x ≤ g^[0] x"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : GL n R"], "goal": "↑(Matrix.GeneralLinearGroup.det A)⁻¹ = (↑A⁻¹).det"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "RCLike 𝕜", "NormedAddCommGroup E", "NormedAddCommGroup F", "InnerProductSpace 𝕜 E", "InnerProductSpace 𝕜 F", "CompleteSpace E", "CompleteSpace F", "T : E →L[𝕜] E", "hT : IsSelfAdjoint T", "S : E →L[𝕜] F"], "goal": "IsSelfAdjoint (S.comp (T.comp (ContinuousLinearMap.adjoint S)))"}}
+{"state": {"context": ["R : Type u", "A : Type v", "B : Type w", "CommSemiring R", "Semiring A", "Semiring B", "Algebra R A", "Algebra R B", "f : A →ₐ[R] B"], "goal": "⇑↑f = ⇑f"}}
+{"state": {"context": ["F : Type u_1", "R : Type u", "A : Type v", "B : Type w", "inst✝¹⁰ : CommSemiring R", "inst✝⁹ : NonUnitalNonAssocSemiring A", "inst✝⁸ : Module R A", "inst✝⁷ : IsScalarTower R A A", "inst✝⁶ : SMulCommClass R A A", "inst✝⁵ : NonUnitalNonAssocSemiring B", "inst✝⁴ : Module R B", "inst✝³ : IsScalarTower R B B", "inst✝² : SMulCommClass R B B", "inst✝¹ : FunLike F A B", "inst✝ : NonUnitalAlgHomClass F R A B", "S T : NonUnitalSubalgebra R A"], "goal": "∀ {x : A}, x ∈ T → x ∈ S ⊔ T"}}
+{"state": {"context": ["G : Type u_1", "AddLeftCancelMonoid G", "Finite G", "x y : G", "DecidableEq G"], "goal": "y ∈ AddSubmonoid.multiples x ↔ y ∈ Finset.image (fun x_1 => x_1 • x) (Finset.range (addOrderOf x))"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "(i : ι) → Zero (α i)", "r : ι → ι → Prop", "s : (i : ι) → α i → α i → Prop", "hbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0", "hs : ∀ (i : ι), WellFounded (s i)", "IsTrichotomous ι r", "hr : WellFounded (Function.swap r)"], "goal": "WellFounded (DFinsupp.Lex r s)"}}
+{"state": {"context": ["ι : Type u_1", "B : Type u_2", "F : Type u_3", "TopologicalSpace B", "TopologicalSpace F", "Z : FiberBundleCore ι B F", "b : B", "a : F"], "goal": "↑(Z.localTrivAt b) { proj := b, snd := a } = (b, a)"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "μ : MeasureTheory.Measure Ω", "t : ℝ", "c : ℝ"], "goal": "ProbabilityTheory.mgf (fun x => c) μ t = (μ Set.univ).toReal * rexp (t * c)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.MonoidalPreadditive C", "CategoryTheory.Limits.HasFiniteBiproducts C", "J : Type", "Fintype J", "f : J → C", "X : C", "j : J"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.rightDistributor f X).hom\n    (CategoryTheory.Limits.biproduct.π (fun j => CategoryTheory.MonoidalCategory.tensorObj (f j) X) j) =\n  CategoryTheory.MonoidalCategory.whiskerRight (CategoryTheory.Limits.biproduct.π f j) X"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : α → β → γ", "a : α", "b : β"], "goal": "WithBot.map₂ f ↑a ↑b = ↑(f a b)"}}
+{"state": {"context": ["E : Type u_2", "AddCommGroup E", "Module ℝ E", "s : ConvexCone ℝ E", "p : E →ₗ.[ℝ] ℝ", "hp_nonneg : ∀ (x : ↥p.domain), ↑x ∈ s → 0 ≤ ↑p x", "hp_dense : ∀ (y : E), ∃ x, ↑x + y ∈ s"], "goal": "∃ q ≥ p, q.domain = ⊤ ∧ ∀ (x : ↥q.domain), ↑x ∈ s → 0 ≤ ↑q x"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "x y : α", "h : x ≠ y"], "goal": "(Equiv.swap x y).support = {x, y}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "r : ℝ", "hr : 0 ≤ r", "x : E"], "goal": "x +ᵥ r • closedBall 0 1 = closedBall x r"}}
+{"state": {"context": ["z✝ w✝ : ℍ", "r✝ R : ℝ", "z w : ℍ", "r : ℝ", "H : 2 * z.im * w.im ≠ 0"], "goal": "dist ↑z ↑(w.center r) ^ 2 = 2 * z.im * w.im * (Real.cosh (dist z w) - Real.cosh r) + (w.im * Real.sinh r) ^ 2"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : Preorder α", "inst✝ : LocallyFiniteOrder α", "a a₁ a₂ b b₁ b₂ c x : α", "ha : a₂ ≤ a₁", "hb : b₁ ≤ b₂"], "goal": "Icc a₁ b₁ ⊆ Icc a₂ b₂"}}
+{"state": {"context": ["X : Type u", "Y✝ : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y✝", "s t✝ : Set X", "b : ι → Set X", "hb : IsTopologicalBasis (range b)", "U : Set X", "hUc : IsCompact U", "hUo : IsOpen U", "Y : Type u", "f' : Y → ι", "e : U = ⋃ i, (b ∘ f') i", "hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i", "t : Finset Y", "ht : U ⊆ ⋃ i ∈ t, (b ∘ f') i"], "goal": "∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i"}}
+{"state": {"context": ["G : Type u_7", "AddCommGroup G", "A : Type u_8", "DecidableEq A", "l : List A", "f g : A → G", "a : A"], "goal": "(List.map (fun x => if x = a then f x else g x) l).sum = List.count a l • (f a - g a) + (List.map g l).sum"}}
+{"state": {"context": ["α : Type u", "inst✝² : Group α", "inst✝¹ : LE α", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c d : α"], "goal": "a ≤ c / b ↔ a * b ≤ c"}}
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+{"state": {"context": ["R : Type u", "CommRing R", "W' : WeierstrassCurve.Jacobian R", "P : Fin 3 → R"], "goal": "(MvPolynomial.eval P) W'.polynomialX = W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z ^ 2 + W'.a₄ * P z ^ 4)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "I : Type u_1", "X : C", "Y : I → C", "f : (i : I) → Y i ⟶ X", "i : I"], "goal": "(CategoryTheory.Sieve.ofArrows Y f).arrows (f i)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "PredOrder α", "a b : α", "h : a ≤ b"], "goal": "Order.pred a ≤ Order.pred b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : PseudoMetricSpace α", "s : Set α", "hs : s.Nontrivial"], "goal": "s.einfsep < ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "s t✝ u : Finset α", "f : α → β", "n : ℕ", "t : Finset β", "i : (a : α) → a ∈ s → β", "j : (a : β) → a ∈ t → α", "hi : ∀ (a : α) (ha : a ∈ s), i a ha ∈ t", "hj : ∀ (a : β) (ha : a ∈ t), j a ha ∈ s", "left_inv : ∀ (a : α) (ha : a ∈ s), j (i a ha) ⋯ = a", "right_inv : ∀ (a : β) (ha : a ∈ t), i (j a ha) ⋯ = a", "a1 : α", "h1 : a1 ∈ s", "a2 : α", "h2 : a2 ∈ s", "eq : i a1 h1 = i a2 h2"], "goal": "a1 = a2"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝ : Field K"], "goal": "(Polynomial.idealX K).intValuation Polynomial.X = ↑(Multiplicative.ofAdd (-1))"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : Projective R", "P : Fin 3 → R"], "goal": "(eval P) (X 1 ^ 2 * X 2 + C W.a₁ * X 0 * X 1 * X 2 + C W.a₃ * X 1 * X 2 ^ 2 - (X 0 ^ 3 + C W.a₂ * X 0 ^ 2 * X 2 + C W.a₄ * X 0 * X 2 ^ 2 + C W.a₆ * X 2 ^ 3)) = P 1 ^ 2 * P 2 + W.a₁ * P 0 * P 1 * P 2 + W.a₃ * P 1 * P 2 ^ 2 - (P 0 ^ 3 + W.a₂ * P 0 ^ 2 * P 2 + W.a₄ * P 0 * P 2 ^ 2 + W.a₆ * P 2 ^ 3)"}}
+{"state": {"context": ["R : Type u", "S : Type u'", "M : Type v", "N : Type v'", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing S", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : AddCommGroup N", "inst✝⁶ : Module R M", "inst✝⁵ : Module R N", "inst✝⁴ : Algebra R S", "inst✝³ : Module S N", "inst✝² : IsScalarTower R S N", "p : Submonoid R", "inst✝¹ : IsLocalization p S", "f : M →ₗ[R] N", "inst✝ : IsLocalizedModule p f", "hp : p ≤ R⁰", "h✝ : Nontrivial R", "this : Nontrivial S"], "goal": "Cardinal.lift.{v, v'} (Module.rank S N) ≤ Cardinal.lift.{v', v} (Module.rank R M)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0", "ht : IntegrableOn f t μ"], "goal": "∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ"}}
+{"state": {"context": ["R : Type u_3", "A : Type u_4", "CommSemiring R", "Semiring A", "Algebra R A"], "goal": "(Subalgebra.toSubmodule ⊥).FG"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝ : Semiring R", "p : R[X]", "hp : p.Monic", "q : R[X]", "hq : q ≠ 0", "h : ¬p = 1"], "goal": "¬p.degree = ⊥"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : ℂ → E", "c : ℂ", "R : ℝ"], "goal": "(∮ (z : ℂ) in C(c, R), f z) = ∫ (θ : ℝ) in Set.Icc 0 (2 * π), deriv (circleMap c R) θ • f (circleMap c R θ)"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : Scheme", "f : X.Hom Y", "H : IsOpenImmersion f"], "goal": "f ''ᵁ ⊤ = f.opensRange"}}
+{"state": {"context": ["α : Type u", "f : Filter α", "hf : f.NeBot", "x : α"], "goal": "f = pure x ↔ {x} ∈ f"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝² : NontriviallyNormedField 𝕜", "E : Type v", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f : 𝕜 → E", "x : 𝕜"], "goal": "x ∈ support (deriv f) → x ∈ tsupport f"}}
+{"state": {"context": ["X : Type u_2", "TopologicalSpace X", "S : DiscreteQuotient X"], "goal": "Continuous S.proj"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "ProperSpace α"], "goal": "Bornology.cobounded α = Filter.cocompact α"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_8", "N : Type u_10", "Zero M", "CommMonoid N", "f : α →₀ M", "y : α", "g : α → M → N", "hg : ∀ (i : α), g i 0 = 1"], "goal": "g y (f y) * (Finsupp.erase y f).prod g = f.prod g"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "ι : Type u_4", "s : Set (Set E)", "hs : DirectedOn (fun x x_1 => x ⊆ x_1) s", "h : ∀ a ∈ s, Orthonormal 𝕜 fun x => ↑x"], "goal": "Orthonormal 𝕜 fun x => ↑x"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_8", "Semiring R", "AddCommMonoid M", "Module R M", "m : Type u_17", "n : Type u_18", "f : m → n", "hf : Function.Surjective f"], "goal": "Function.Injective ⇑(LinearMap.funLeft R M f)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "p : α → Prop", "f : (a : α) → p a → β", "l₁ l₂ : List α", "h₁ : ∀ (a : α), a ∈ l₁ → p a", "h₂ : ∀ (a : α), a ∈ l₂ → p a"], "goal": "List.pmap f (l₁ ++ l₂) ⋯ = List.pmap f l₁ h₁ ++ List.pmap f l₂ h₂"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "hx : x ≠ 0", "hy : y ≠ 0"], "goal": "(o.oangle x y).cos = Real.cos (InnerProductGeometry.angle x y)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "Zero M", "a x : α", "b : M"], "goal": "(Finsupp.single a b) x = 0 ↔ x = a → b = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "p : R[X]", "hp : p.Monic", "hp2 : 2 ≤ p.natDegree", "hp3 : p.natDegree ≤ 3", "hp0 : p ≠ 0", "hp1 : p ≠ 1", "h : ∀ (a : R), ¬X - C a ∣ p", "q : R[X]", "hq : q.Monic", "hq1 : q.natDegree = 1"], "goal": "¬q ∣ p"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : CommMonoid M", "S : Submonoid M", "N : Type u_2", "inst✝¹ : CommMonoid N", "P : Type u_3", "inst✝ : CommMonoid P", "n : ℕ", "a : M", "b : ↥S"], "goal": "mk a b ^ n = mk (a ^ n) (b ^ n)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "s : Set P", "h₁ : s.Subsingleton", "h₂ : affineSpan k s = ⊤"], "goal": "s = Set.univ"}}
+{"state": {"context": ["n : ℕ+"], "goal": "n.Coprime 1"}}
+{"state": {"context": ["M : Type u_2", "AddMonoid M", "a : M"], "goal": "2 • a = a + a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_8", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "n : ℕ∞", "L : E →L[𝕜] F", "s : Set E"], "goal": "ContMDiffOn 𝓘(𝕜, E) 𝓘(𝕜, F) n (⇑L) s"}}
+{"state": {"context": ["p : Prop"], "goal": "False ∨ p ↔ p"}}
+{"state": {"context": ["n : Type u_1", "𝕜 : Type u_2", "inst✝² : RCLike 𝕜", "inst✝¹ : Fintype n", "inst✝ : DecidableEq n", "A : Matrix n n 𝕜"], "goal": "(spectrum ℝ A).Finite"}}
+{"state": {"context": ["G : Type u_1", "X : Type u_2", "TopologicalSpace G", "TopologicalSpace X", "AddGroup G", "AddAction G X", "self : ProperVAdd G X"], "goal": "IsProperMap fun gx => (gx.1 +ᵥ gx.2, gx.2)"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝³ : LinearOrder ι", "f : Filtration ι m", "τ : Ω → ι", "inst✝² : TopologicalSpace ι", "inst✝¹ : OrderTopology ι", "inst✝ : FirstCountableTopology ι", "hτ : IsStoppingTime f τ", "i i' : ι", "hi'_lub : IsLUB (Set.Iio i) i'", "hi'_eq_i : i' = i"], "goal": "MeasurableSet {ω | τ ω < i}"}}
+{"state": {"context": ["M : Type u_5", "F : Type u_6", "Group M", "FunLike F M M", "MonoidHomClass F M M", "f : F", "n : ℕ", "x y : M"], "goal": "(⇑f)^[n] (x / y) = (⇑f)^[n] x / (⇑f)^[n] y"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝² : Ring k", "inst✝¹ : AddCommGroup V", "inst✝ : Module k V", "S : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "w : ι → k", "p : P", "h : ∑ i ∈ s, w i = 0"], "goal": "(s.weightedVSub fun x => p) w = 0"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Monoid R", "MulAction R M", "p : SubMulAction R M", "m : ↥p"], "goal": "Subtype.val '' MulAction.orbit R m = MulAction.orbit R ↑m"}}
+{"state": {"context": ["a b : Cardinal.{u}", "n m : ℕ", "ha : ℵ₀ ≤ a"], "goal": "¬¬IsUnit a ∨ ¬∀ (a_1 b : Cardinal.{u}), a = a_1 * b → IsUnit a_1 ∨ IsUnit b"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "inst✝¹ : DecidableEq α", "inst✝ : IsIrrefl α r", "s' s : Multiset α"], "goal": "(∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = (s + t).erase a) ↔ ∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.75295, u_1} C", "X : SimplicialObject C", "s : Splitting X", "inst✝ : Preadditive C", "i j : ℕ", "x✝ : (ComplexShape.down ℕ).Rel i j"], "goal": "∀ b ∈ Finset.univ, b ≠ IndexSet.id (op [i]) → ((s.πSummand b ≫ (s.cofan (op [i])).inj b) ≫ K[X].d i j) ≫ s.πSummand (IndexSet.id (op [j])) = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommMonoid R", "R' : Type u_2", "inst✝¹ : CommRing R'", "R'' : Type u_3", "inst✝ : CommRing R''", "χ : MulChar R R'", "f : R' →+* R''", "hf : Function.Injective ⇑f", "a : Rˣ", "ha : χ ↑a ≠ 1"], "goal": "f (χ ↑a) ≠ f 1"}}
+{"state": {"context": ["X : Scheme", "inst✝ : IsIntegral X", "U : X.Opens", "x : ↥U", "y : ↑(X.presheaf.obj (op U))", "hy : (X.presheaf.germ x) y = (X.presheaf.germ x) 0"], "goal": "y = 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a✝ b : R", "n : ℕ", "inst✝ : CommSemiring R", "a p P : R[X]", "h : ∀ (r : R), r • P = 0 → r = 0", "Q : R[X]", "hQ : P * Q = 0", "l : ℕ", "IH : ∀ m > l, P.coeff m • Q = 0", "hl : (P.coeff l • Q).natDegree = Q.natDegree"], "goal": "P.coeff l • Q = 0"}}
+{"state": {"context": ["K : Type u", "CommRing K", "p : ℕ", "Fact (Nat.Prime p)", "CharP K p", "x : PerfectClosure K p", "q : PerfectClosure K p → Prop", "h : ∀ (x : ℕ × K), q (PerfectClosure.mk K p x)"], "goal": "q x"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Add α", "s t : Set α", "hs : s.Finite", "ht : t.Finite", "hf : (s + t).Finite := ⋯"], "goal": "Set.Finite.toFinset hf = hs.toFinset + ht.toFinset"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "SeminormedAddCommGroup E", "s : Set α", "f : α → E", "a : α"], "goal": "‖s.indicator f a‖ = s.indicator (fun a => ‖f a‖) a"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁴ : AddCommGroup E", "inst✝³ : Module ℝ E", "inst✝² : TopologicalSpace E", "inst✝¹ : ContinuousAdd E", "inst✝ : ContinuousSMul ℝ E", "h : 1 < Module.rank ℝ E", "s : Set E", "hs : s.Countable", "this : Nontrivial E", "a : E", "ha : a ∈ sᶜ", "b : E", "hb : b ∈ sᶜ", "hab : a ≠ b", "c : E := 2⁻¹ • (a + b)", "x : E := 2⁻¹ • (b - a)", "Ia : c - x = a", "Ib : c + x = b", "x_ne_zero : x ≠ 0", "y : E", "hy : LinearIndependent ℝ ![x, y]", "A : {t | ([c + x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable", "B : {t | ([c - x-[ℝ]c + t • y] ∩ s).Nonempty}.Countable", "t : ℝ", "ht : t ∈ ({t | ([c + x-[ℝ]c + t • y] ∩ s).Nonempty} ∪ {t | ([c - x-[ℝ]c + t • y] ∩ s).Nonempty})ᶜ"], "goal": "JoinedIn sᶜ a b"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : GlueData C", "inst✝ : HasLimits C", "i j : D.J", "U : Opens ↑↑(D.U i)"], "goal": "D.diagramOverOpenπ U i ≫ (D.U i).presheaf.map ((((eqToIso ⋯).inv ≫ (homOfLE ⋯).op) ≫ (opensFunctor (D.f i i)).op.map (𝟙 ((Opens.map (D.f i i).base).op.obj (op U)) ≫ eqToHom ⋯)) ≫ (eqToHom ⋯).op) ≫ (D.f i j).c.app (op ((Opens.map (D.ι i).base).obj (⋯.functor.obj U))) ≫ (D.V (i, j)).presheaf.map (eqToHom ⋯) = limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (i, j)))"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²¹ : NontriviallyNormedField 𝕜", "EM : Type u_2", "inst✝²⁰ : NormedAddCommGroup EM", "inst✝¹⁹ : NormedSpace 𝕜 EM", "HM : Type u_3", "inst✝¹⁸ : TopologicalSpace HM", "IM : ModelWithCorners 𝕜 EM HM", "E : Type u_4", "inst✝¹⁷ : NormedAddCommGroup E", "inst✝¹⁶ : NormedSpace 𝕜 E", "H : Type u_5", "inst✝¹⁵ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "H' : Type u_6", "inst✝¹⁴ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E H'", "M : Type u", "inst✝¹³ : TopologicalSpace M", "inst✝¹² : ChartedSpace HM M", "N G A A' R : Type u", "inst✝¹¹ : TopologicalSpace N", "inst✝¹⁰ : ChartedSpace H N", "inst✝⁹ : TopologicalSpace G", "inst✝⁸ : ChartedSpace H G", "inst✝⁷ : TopologicalSpace A", "inst✝⁶ : ChartedSpace H A", "inst✝⁵ : TopologicalSpace A'", "inst✝⁴ : ChartedSpace H' A'", "inst✝³ : TopologicalSpace R", "inst✝² : ChartedSpace H R", "inst✝¹ : CommRing R", "inst✝ : SmoothRing I R", "x : M", "y : (smoothSheaf IM I M R).presheaf.stalk x"], "goal": "(eval IM I M R x) ((forgetStalk IM I M R x).inv y) = smoothSheaf.eval IM I R x y"}}
+{"state": {"context": ["α : Type u", "R : Type u_1", "CommSemiring R", "Small.{v, u} α", "Semiring α", "Algebra R α"], "goal": "∀ (a : α), (Shrink.algEquiv α R).symm a = (equivShrink α) a"}}
+{"state": {"context": ["α : Type u", "l : List α", "n : ℕ"], "goal": "l.rotate (n % l.length) = l.rotate n"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "functor : CategoryTheory.Functor C D", "inverse : CategoryTheory.Functor D C", "unit_iso : CategoryTheory.Functor.id C ≅ functor.comp inverse", "counit_iso : inverse.comp functor ≅ CategoryTheory.Functor.id D", "f :\n  ∀ (X : C),\n    CategoryTheory.CategoryStruct.comp (functor.map (unit_iso.hom.app X)) (counit_iso.hom.app (functor.obj X)) =\n      CategoryTheory.CategoryStruct.id (functor.obj X)"], "goal": "{ functor := functor, inverse := inverse, unitIso := unit_iso, counitIso := counit_iso,\n      functor_unitIso_comp := f }.counitInv =\n  counit_iso.inv"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "X✝ Y✝ : C", "inst✝¹ : HasTerminal C", "inst✝ : HasBinaryProducts C", "X Y Z : C"], "goal": "prod.lift (prod.lift prod.fst (prod.snd ≫ prod.fst)) (prod.snd ≫ prod.snd) ≫ prod.lift prod.snd prod.fst ≫ prod.lift (prod.lift prod.fst (prod.snd ≫ prod.fst)) (prod.snd ≫ prod.snd) = prod.map (𝟙 X) (prod.lift prod.snd prod.fst) ≫ prod.lift (prod.lift prod.fst (prod.snd ≫ prod.fst)) (prod.snd ≫ prod.snd) ≫ prod.map (prod.lift prod.snd prod.fst) (𝟙 Y)"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "a b : CauSeq α abs"], "goal": "b ≤ a ⊔ b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing S", "inst✝⁴ : Algebra R S", "x : S", "I : Ideal R", "inst✝³ : IsDomain R", "inst✝² : IsIntegrallyClosed R", "inst✝¹ : IsDedekindDomain S", "inst✝ : NoZeroSMulDivisors R S", "hI : I.IsMaximal", "hI' : I ≠ ⊥", "hx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤", "hx' : _root_.IsIntegral R x", "J : Ideal S", "hJ : J ∈ normalizedFactors (Ideal.map (algebraMap R S) I)", "this : ↑(Multiset.count (normalize J) (normalizedFactors (Ideal.map (algebraMap R S) I))) = multiplicity (↑((normalizedFactorsMapEquivNormalizedFactorsMinPolyMk hI hI' hx hx') ⟨J, hJ⟩)) (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))"], "goal": "Polynomial.map (Ideal.Quotient.mk I) (minpoly R x) ≠ 0"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝⁷ : Field K", "inst✝⁶ : CharZero K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : IsKilling K L", "inst✝² : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝¹ : H.IsCartanSubalgebra", "inst✝ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα✝ : α.IsNonZero", "n : ℕ", "hn : n ≤ chainLength α β", "hα : ¬α.IsZero", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) β)", "x_ne0 : x ≠ 0", "h e f : L", "isSl2 : IsSl2Triple h e f", "he : e ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)"], "goal": "rootSpace H (-(n • ⇑α) + ⇑(chainTop (⇑α) β)) ≠ ⊥"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "f : X ⟶ Y", "s : CategoryTheory.Limits.CokernelCofork f"], "goal": "f ≫ CategoryTheory.Limits.Cofork.π s = 0"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "IsModularLattice α", "x y z : α", "hxy : x < y", "hinf : y ⊓ z ≤ x ⊓ z"], "goal": "x ⊔ z < y ⊔ z"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "CompleteSpace E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "CompleteSpace F", "f : E → F", "f' : E →L[𝕜] F", "a : E", "hf : HasStrictFDerivAt f f' a", "hf' : LinearMap.range f' = ⊤", "hker : (LinearMap.ker f').ClosedComplemented"], "goal": "HasStrictFDerivAt (HasStrictFDerivAt.implicitFunctionOfComplemented f f' hf hf' hker (f a)) (LinearMap.ker f').subtypeL\n  0"}}
+{"state": {"context": ["ι : Type u_1", "M₀ : Type u_4", "MulZeroOneClass M₀", "s : Set ι", "i : ι", "Nontrivial M₀"], "goal": "s.indicator 1 i = 1 ↔ i ∈ s"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "n : ℕ", "r : ℝ≥0∞", "finite : ∀ (m : ℕ), n ≤ m → p m = 0", "pos : 0 < r", "sum_eq : ∀ y ∈ EMetric.ball 0 r, (∑ i ∈ Finset.range n, (p i) fun x => y) = f (x + y)"], "goal": "HasFiniteFPowerSeriesOnBall f p x n r"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "β : Type u_2", "v : Vector α (n + 1)", "f : α → β", "a : α", "v' : Vector α n", "h : v = a ::ᵥ v'"], "goal": "(map f v).head = f v.head"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : Ring R", "E : Type u_2", "inst✝⁸ : AddCommGroup E", "inst✝⁷ : Module R E", "F : Type u_3", "inst✝⁶ : AddCommGroup F", "inst✝⁵ : Module R F", "G : Type u_4", "inst✝⁴ : AddCommGroup G", "inst✝³ : Module R G", "M : Type u_5", "inst✝² : Monoid M", "inst✝¹ : DistribMulAction M F", "inst✝ : SMulCommClass R M F", "y✝ : M", "f : E →ₗ.[R] F", "x y : E", "x' y' : F", "hx : ∃ y, ↑y = (x, x').1 ∧ ↑f y = (x, x').2", "hy : ∃ y_1, ↑y_1 = (y, y').1 ∧ ↑f y_1 = (y, y').2", "hxy : x = y"], "goal": "x' = y'"}}
+{"state": {"context": ["F : Type u", "K✝¹ : Type v", "L : Type w", "inst✝³ : Field K✝¹", "inst✝² : Field L", "inst✝¹ : Field F", "n✝ : ℕ", "K✝ : Type u", "inst✝ : Field K✝", "n : ℕ", "ih : (fun n => ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤) n", "K : Type u", "x✝ : Field K", "f : K[X]", "hfn : f.natDegree = n.succ", "hndf : f.natDegree ≠ 0", "hfn0 : f ≠ 0", "hmf0 : map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)) (X - C (AdjoinRoot.root f.factor)) * map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)) f.removeFactor ≠ 0"], "goal": "Subalgebra.restrictScalars K (Algebra.adjoin ↥(IsScalarTower.toAlgHom K (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)).range ↑(map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor)) f.removeFactor).roots.toFinset) = ⊤"}}
+{"state": {"context": ["α : Type u_6", "β : Type u_7", "f : α → β", "s : Set α"], "goal": "f '' s = ∅ ↔ s = ∅"}}
+{"state": {"context": ["K : Type u_2", "Field K", "NumberField K"], "goal": "Fintype.card { φ // ¬NumberField.ComplexEmbedding.IsReal φ } = 2 * NumberField.InfinitePlace.NrComplexPlaces K"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "NormedAlgebra 𝕜 ℂ", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : E → ℂ", "s : Set E", "n : ℕ∞", "hf : ContDiffOn 𝕜 n f s"], "goal": "ContDiffOn 𝕜 n (fun x => Complex.exp (f x)) s"}}
+{"state": {"context": ["𝕜 : Type u_1", "m : Type u_2", "n : Type u_3", "l : Type u_4", "RCLike 𝕜", "Fintype m", "Fintype n", "DecidableEq n", "Fintype l", "DecidableEq l", "A : Matrix m n 𝕜", "B : Matrix n l 𝕜"], "goal": "‖A * B‖ ≤ ‖A‖ * ‖B‖"}}
+{"state": {"context": ["x y r : ℕ", "hr : y * y + r = x", "hle : r.ble (2 * y) = true"], "goal": "x.sqrt = y"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : EuclideanDomain R", "inst✝ : GCDMonoid R", "p✝ q✝ p q : R", "hq : q ≠ 0", "r : R", "hr : q = GCDMonoid.gcd p q * r"], "goal": "q / GCDMonoid.gcd p q ≠ 0"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y z : V", "hx : x ≠ 0", "hy : y ≠ 0", "hz : z ≠ 0"], "goal": "o.oangle x z - o.oangle x y = o.oangle y z"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "s : Set α", "x : α"], "goal": "(s.ordConnectedComponent x).Nonempty ↔ x ∈ s"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "ι : Type u_3", "hι : Nonempty ι", "s : ι → Set M", "hs : ∀ (i : ι), IsSubmonoid (s i)", "Directed : ∀ (i j : ι), ∃ k, s i ⊆ s k ∧ s j ⊆ s k"], "goal": "IsSubmonoid (⋃ i, s i)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "𝕜 : Type u_4", "E : Type u_5", "inst✝³ : DecidableEq α", "inst✝² : CommMonoid α", "inst✝¹ : CommMonoid β", "inst✝ : DecidableEq β", "A : Finset α", "B : Finset β", "f : α → β", "hf : IsMulFreimanHom 2 (↑A) (↑B) f", "hf' : Set.BijOn f ↑A ↑B", "s : Finset β", "hsB : s ⊆ B", "hcard : s.card = mulRothNumber B", "hs : ThreeGPFree ↑s", "hsA : invFunOn f ↑A '' ↑s ⊆ ↑A", "hfsA : Set.SurjOn f ↑A ↑s"], "goal": "(image (invFunOn f ↑A) s).card ≤ mulRothNumber A"}}
+{"state": {"context": ["n : Type u_1 → Type u_2", "m : Type u_1 → Type u_3", "α ρ : Type u_1", "MonadLiftT n m", "x : n α", "ctx : ρ"], "goal": "(monadLift x).run ctx = monadLift x"}}
+{"state": {"context": ["x y : SetTheory.PGame"], "goal": "¬x ≤ y ↔ y.LF x"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "N : Type u_5", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "Q : QuadraticMap R M N", "y : M"], "goal": "QuadraticMap.polar (⇑Q) y 0 = 0"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "t : Triangle ℝ P", "i₁ i₂ : Fin 3", "h : i₁ ≠ i₂", "hu : {i₁, i₂} ⊆ univ", "i₃ : Fin 3", "hi₃ : univ \\ {i₁, i₂} = {i₃}", "hi₃₁ : i₃ ≠ i₁", "hi₃₂ : i₃ ≠ i₂"], "goal": "-(∑ x : Fin (0 + 2 + 1), (∑ x_1 : Fin (0 + 2 + 1), ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) * ((↑(0 + 1))⁻¹ - if x_1 = i₁ ∨ x_1 = i₂ then 1 else 0) * (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1)) * dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x_1))) + ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) * (-2 / ↑(0 + 1) - -1) * (dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) * dist (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex))) + (∑ x : Fin (0 + 2 + 1), (-2 / ↑(0 + 1) - -1) * ((↑(0 + 1))⁻¹ - if x = i₁ ∨ x = i₂ then 1 else 0) * (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x)) * dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) (pointsWithCircumcenter t (PointsWithCircumcenterIndex.pointIndex x))) + (-2 / ↑(0 + 1) - -1) * (-2 / ↑(0 + 1) - -1) * (dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) * dist (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex) (pointsWithCircumcenter t PointsWithCircumcenterIndex.circumcenterIndex)))) / 2 = circumradius t * circumradius t"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "Module R M", "Module R N", "Module R P", "Module R Q", "g : P ≃ₗ[R] Q", "f : N ≃ₗ[R] P", "y : TensorProduct R N M"], "goal": "(LinearEquiv.rTensor M (f ≪≫ₗ g)) y = (LinearEquiv.rTensor M g) ((LinearEquiv.rTensor M f) y)"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "β : Type u", "u : F (α ::: β) → β", "x y : (P F).W α"], "goal": "∀ (a : (P F).A) (f' : (P F).drop.B a ⟹ α) (f : (P F).last.B a → (P F).W α), WEquiv ((P F).wMk a f' f) y → recF u ((P F).wMk a f' f) = recF u y"}}
+{"state": {"context": ["K : Type u", "V : Type v", "DivisionRing K", "AddCommGroup V", "Module K V", "S₁ S₂ : Submodule K V", "FiniteDimensional K ↥S₂", "hle : S₁ ≤ S₂", "hd : FiniteDimensional.finrank K ↥S₂ ≤ FiniteDimensional.finrank K ↥S₁"], "goal": "S₁ = S₂"}}
+{"state": {"context": ["α : Type u_1", "a : α", "LE α", "x : WithBot α"], "goal": "↑a ≤ x ↔ ∃ b, x = ↑b ∧ a ≤ b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "A : Type w", "B : Type u₁", "M : Type v₁", "inst✝⁶ : CommSemiring R", "inst✝⁵ : CommSemiring S", "inst✝⁴ : CommSemiring A", "inst✝³ : Algebra R S", "inst✝² : Algebra S A", "inst✝¹ : Algebra R A", "inst✝ : IsScalarTower R S A", "t : Set A", "z : A"], "goal": "Set.range ⇑(algebraMap (↥(toAlgHom R S A).range) A) = Set.range ⇑(algebraMap S A)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "H : Subgroup (Perm α)", "d : DecidablePred fun x => x ∈ H", "τ : Perm α", "h0 : Nat.Prime (Fintype.card α)", "h1 : Fintype.card α ∣ Fintype.card ↥H", "h2 : τ ∈ H", "h3 : τ.IsSwap", "this : Fact (Nat.Prime (Fintype.card α))", "σ : ↥H", "hσ : orderOf σ = Fintype.card α", "hσ1 : orderOf ↑σ = Fintype.card α", "hσ2 : (↑σ).IsCycle"], "goal": "H = ⊤"}}
+{"state": {"context": ["ι : Type u_2", "α : Type u_3", "DivisionCommMonoid α", "m : Multiset ι", "f : ι → α", "n : ℤ"], "goal": "(Multiset.map (fun i => f i ^ n) m).prod = (Multiset.map f m).prod ^ n"}}
+{"state": {"context": ["α : Type u", "R : α → α → Prop", "l : List α"], "goal": "List.Pairwise R l → List.Pairwise (List.Lex (Function.swap R)) l.sublists'"}}
+{"state": {"context": ["I : Type u_4", "Fintype I", "DecidableEq I", "R : I → Type u_5", "(i : I) → Ring (R i)"], "goal": "CompleteOrthogonalIdempotents fun x => Pi.single x 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "f g : Filter β", "p : β → Prop", "u : β → α", "h : f ≤ g"], "goal": "Filter.bliminf u g p ≤ Filter.bliminf u f p"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "𝕜 : Type u_2", "Field 𝕜", "IsAlgClosed 𝕜", "CategoryTheory.Linear 𝕜 C", "CategoryTheory.Limits.HasKernels C", "X Y : C", "FiniteDimensional 𝕜 (X ⟶ X)", "CategoryTheory.Simple X", "CategoryTheory.Simple Y"], "goal": "FiniteDimensional.finrank 𝕜 (X ⟶ Y) ≤ 1"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "UniformSpace R", "UniformSpace M"], "goal": "uniformity (tsze R M) =\n  Filter.comap (fun p => (p.1.fst, p.2.fst)) (uniformity R) ⊓ Filter.comap (fun p => (p.1.snd, p.2.snd)) (uniformity M)"}}
+{"state": {"context": ["E : Type u_3", "NormedAddCommGroup E", "CompleteSpace E", "NormedSpace ℝ E", "a b : ℝ", "μ : MeasureTheory.Measure ℝ", "𝕜 : Type u_6", "RCLike 𝕜", "NormedSpace 𝕜 E", "f : ℝ → 𝕜", "c : E"], "goal": "∫ (x : ℝ) in a..b, f x • c ∂μ = (∫ (x : ℝ) in a..b, f x ∂μ) • c"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "inst✝⁶ : Field K", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝³ : H.IsCartanSubalgebra", "inst✝² : IsTriangularizable K (↥H) L", "inst✝¹ : IsKilling K L", "inst✝ : CharZero K", "α : Weight K (↥H) L"], "goal": "IsCompl Weight.ker (span K {coroot α})"}}
+{"state": {"context": ["M : Type u_1", "inst✝⁵ : CancelMonoidWithZero M", "inst✝⁴ : Finite M", "R : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : IsDomain R", "inst✝¹ : GCDMonoid R", "inst✝ : Unique Rˣ", "a b c : R", "n : ℕ", "cp : IsCoprime a b", "h : a * b = c ^ n"], "goal": "GCDMonoid.gcd a b ∣ 1"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "α : Type u_3", "inst✝¹ : LinearOrder α", "inst✝ : NoTopOrder α", "a : α"], "goal": "∃ b, a < b"}}
+{"state": {"context": ["ι✝ : Type u_1", "inst✝¹⁰ : Fintype ι✝", "A : ι✝ → Type u_2", "inst✝⁹ : (i : ι✝) → MeasurableSpace (A i)", "μ✝ : (i : ι✝) → Measure (A i)", "inst✝⁸ : ∀ (i : ι✝), SigmaFinite (μ✝ i)", "F : Type u_3", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ F", "E : Type u_4", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : MeasurableSpace E", "inst✝² : BorelSpace E", "inst✝¹ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "p : ℝ", "ι : Type := Fin (finrank ℝ E)", "this : finrank ℝ E = finrank ℝ (ι → ℝ)"], "goal": "ℝ≥0"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "ι : Type u_4", "f : X → Y", "g : Y → Z", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "hg : Inducing g"], "goal": "Continuous f ↔ Continuous (g ∘ f)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "i : CategoryTheory.IsIso 0"], "goal": "⋯ = ⋯"}}
+{"state": {"context": ["R M : Type u", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I : Ideal R", "f : R[X]", "a : ℕ"], "goal": "f.coeff a ∈ I ^ a ↔ a ∈ f.support → f.coeff a ∈ I ^ a"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "b : CategoryTheory.Limits.Bicone (CategoryTheory.Limits.pairFunction X Y)"], "goal": "(CategoryTheory.Limits.Bicone.toBinaryBiconeFunctor.obj b).snd = b.π CategoryTheory.Limits.WalkingPair.right"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ✝ ν✝ ν μ : Measure α", "inst✝¹ : IsFiniteMeasure ν", "inst✝ : ν.HaveLebesgueDecomposition μ", "r : ℝ≥0", "hr : r ≠ 0"], "goal": "AEMeasurable (ν.rnDeriv (r • μ)) (r • μ)"}}
+{"state": {"context": ["M : Type u_1", "Mul M"], "goal": "↑Subsemigroup.topEquiv = MulMemClass.subtype ⊤"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "Preorder α", "f : ι → Set α"], "goal": "lowerClosure (⋃ i, f i) = ⨆ i, lowerClosure (f i)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasFiniteCoproducts C", "K : ChainComplex C ℕ", "n : ℕ"], "goal": "((AlgebraicTopology.DoldKan.Γ₀.splitting K).cofan (Opposite.op [n])).inj\n      (SimplicialObject.Splitting.IndexSet.id (Opposite.op [n])) ≫\n    AlgebraicTopology.DoldKan.PInfty.f n =\n  ((AlgebraicTopology.DoldKan.Γ₀.splitting K).cofan (Opposite.op [n])).inj\n    (SimplicialObject.Splitting.IndexSet.id (Opposite.op [n]))"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "ι : Type u_4", "Nontrivial k", "p : ι → P", "ha : AffineIndependent k p", "s1 s2 : Set ι", "p0 : P", "hp0s1 : p0 ∈ affineSpan k (p '' s1)", "hp0s2 : p0 ∈ affineSpan k (p '' s2)"], "goal": "∃ i, i ∈ s1 ∩ s2"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Semiring R", "AddCommMonoid M", "Module R M", "self : IsArtinian R M"], "goal": "WellFounded fun x x_1 => x < x_1"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "inst✝ : IsIrrefl α r", "s : Multiset α", "hs : ∀ a ∈ s, Acc (CutExpand r) {a}"], "goal": "Acc (CutExpand r) s"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Inv α", "inst✝ : Small.{u_2, u_1} α", "x : α"], "goal": "(equivShrink α) x⁻¹ = ((equivShrink α) x)⁻¹"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "NormedAddCommGroup F", "NormedSpace ℝ F", "CompleteSpace F", "f : E → F", "φ : E → ℝ", "x₀ : E", "f' : E →L[ℝ] F", "φ' : E →L[ℝ] ℝ", "hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀", "hf' : HasStrictFDerivAt f f' x₀", "hφ' : HasStrictFDerivAt φ φ' x₀"], "goal": "LinearMap.range (f'.prod φ') ≠ ⊤"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "M : Type u_5", "NatCast M", "n : ℕ", "n.AtLeastTwo"], "goal": "↑(OfNat.ofNat n) = OfNat.ofNat n"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedAddCommGroup E''", "inst✝⁷ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedSpace 𝕜 E", "inst✝⁴ : NormedSpace 𝕜 E'", "inst✝³ : NormedSpace 𝕜 E''", "inst✝² : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝¹ : AddGroup G", "inst✝ : TopologicalSpace G", "hcf : HasCompactSupport f", "hf : Continuous f", "x t : G", "s : Set G", "hx : x ∈ s"], "goal": "(-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t = (-tsupport f + s).indicator (fun t => ‖L.flip‖ * ‖g t‖ * ⨆ i, ‖f i‖) t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "b : Ordinal.{u_4}", "h₁ : 0 ∣ b", "h₂ : b ∣ 0"], "goal": "0 = b"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "OrderedCommGroup α", "LinearOrderedCommGroup β", "s : Set ι", "f : ι → α", "g₁ g₂ : ι → β", "h₁ : AntivaryOn f g₁ s", "h₂ : AntivaryOn f g₂ s"], "goal": "AntivaryOn f (g₁ * g₂) s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "π : α → Type u_6", "π' : β → Type u_7", "f : α → β", "hf : Function.Injective f", "g : (a : α) → π a → π' (f a)", "a : α", "l : Filter (π' (f a))"], "goal": "(comap (Sigma.map f g) (map (Sigma.mk (f a)) l)).HasBasis (fun s => s ∈ l) fun i => Sigma.mk a '' (g a ⁻¹' id i)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "A : Type u₃", "CategoryTheory.Category.{v₃, u₃} A", "B : Type u₄", "CategoryTheory.Category.{v₄, u₄} B", "L : D", "R : CategoryTheory.Functor C D", "L' : B", "R' : CategoryTheory.Functor A B", "F : CategoryTheory.Functor C A", "G : CategoryTheory.Functor D B", "α : L' ⟶ G.obj L", "β : R.comp G ⟶ F.comp R'", "X : CategoryTheory.Comma (CategoryTheory.Functor.fromPUnit L) R"], "goal": "((CategoryTheory.StructuredArrow.map₂ α β).obj X).left = X.left"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g : α → β", "c c₁ c₂ x : α", "inst✝² : Add α", "inst✝¹ : DivisionMonoid β", "inst✝ : HasDistribNeg β", "hf : Antiperiodic f c", "hg : Antiperiodic g c"], "goal": "Periodic (f / g) c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "M : Type u_5", "M' : Type u_6", "N : Type u_7", "P : Type u_8", "G : Type u_9", "H : Type u_10", "R : Type u_11", "S : Type u_12", "inst✝¹ : AddMonoid M", "inst✝ : SMulZeroClass R M", "b : R", "g : α →₀ M", "a : α"], "goal": "¬b • g a = 0 → ¬g a = 0"}}
+{"state": {"context": ["X Y Z : TopCat", "f : X ⟶ Y", "g : Y ⟶ Z", "inst✝ : IsIso g"], "goal": "OpenEmbedding ((forget TopCat).map (f ≫ g)) ↔ OpenEmbedding ⇑f"}}
+{"state": {"context": ["s : ℂ", "hs : 1 < s.re", "n : ℕ", "hn : n ≠ 0"], "goal": "term (fun n => if n = 0 then 0 else 1) s n = 1 / ↑n ^ s"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "x y : R", "h : Commute x y", "n : ℕ", "t : ℕ → ℕ → R := fun n m => x ^ m * y ^ (n - m) * ↑(n.choose m)", "h_first : ∀ (n : ℕ), t n 0 = y ^ n"], "goal": "(x + y) ^ n = ∑ m ∈ range (n + 1), t n m"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "s : PullbackCone f g"], "goal": "∀ (b : (𝒰.pullbackCover s.fst).gluedCover.diagram.R), Multicoequalizer.π (𝒰.pullbackCover s.fst).gluedCover.diagram b ≫ (𝒰.pullbackCover s.fst).fromGlued ≫ gluedLift 𝒰 f g s ≫ p1 𝒰 f g = Multicoequalizer.π (𝒰.pullbackCover s.fst).gluedCover.diagram b ≫ (𝒰.pullbackCover s.fst).fromGlued ≫ s.fst"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "Monoid R", "NonUnitalNonAssocSemiring A", "DistribMulAction R A", "Star A", "NonUnitalNonAssocSemiring B", "DistribMulAction R B", "Star B", "F : Type u_6", "FunLike F A B", "NonUnitalAlgHomClass F R A B", "NonUnitalStarAlgHomClass F R A B", "f : F"], "goal": "⇑↑f = ⇑f"}}
+{"state": {"context": ["x y z : ℝ", "n : ℕ", "hx : 1 < x", "hyz : y < z"], "goal": "rexp (log x * y) < rexp (log x * z)"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "inst✝¹ : NormedAddGroup E", "inst✝ : TopologicalSpace α", "f : α → E", "hf : Continuous f", "h : HasCompactSupport f"], "goal": "∃ C, ∀ (x : α), ‖f x‖ ≤ C"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I : Set α", "ι : Type u_2", "Nonempty ι", "X : ι → Set α", "hI : ∀ (i : ι), M.Basis I (X i)"], "goal": "M.Basis I (⋃ i, X i)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedField α", "a b c d : α", "n : ℤ", "x✝ : α"], "goal": "x✝ = x✝ - 0"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "V₁ : Type u_6", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "u : (P₁ →ᵃ[k] P₁)ˣ", "a : P₁"], "goal": "(AffineEquiv.equivUnitsAffineMap.symm u).symm a = ↑u⁻¹ a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁵ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁴ : NormedAddCommGroup E", "inst✝¹³ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝¹² : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝¹¹ : TopologicalSpace M", "inst✝¹⁰ : ChartedSpace H M", "E' : Type u_5", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝⁷ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁶ : TopologicalSpace M'", "inst✝⁵ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁴ : NormedAddCommGroup E''", "inst✝³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝¹ : TopologicalSpace M''", "inst✝ : ChartedSpace H'' M''", "f f₀ f₁ : M → M'", "x : M", "s t : Set M", "g : M' → M''", "u : Set M'"], "goal": "UniqueDiffWithinAt 𝕜 (range ↑I) (↑(extChartAt I x) x)"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "E : Type u_5", "AddGroup G", "AddAction G α", "MeasurableSpace α", "NormedAddCommGroup E", "s : Set α", "μ : MeasureTheory.Measure α", "MeasurableSpace G", "MeasurableVAdd G α", "MeasureTheory.VAddInvariantMeasure G α μ", "Countable G", "NormedSpace ℝ E", "h : MeasureTheory.IsAddFundamentalDomain G s μ", "f : α → E", "hf : MeasureTheory.Integrable f μ"], "goal": "∫ (x : α), f x ∂μ = ∑' (g : G), ∫ (x : α) in s, f (-g +ᵥ x) ∂μ"}}
+{"state": {"context": ["G : Type u_1", "Group G", "a : G"], "goal": "a ∈ IsSubgroup.center G ↔ ∀ (g : G), g * a = a * g"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "↑⟪x, y⟫_ℝ + (o.areaForm x) y • Complex.I = (starRingEnd ℂ) (↑⟪y, x⟫_ℝ + (o.areaForm y) x • Complex.I)"}}
+{"state": {"context": ["α : Type u", "Add α", "β : Type v", "AddSemigroup β", "f g : AddHom (AddMagma.FreeAddSemigroup α) β", "h : f.comp AddMagma.FreeAddSemigroup.of = g.comp AddMagma.FreeAddSemigroup.of"], "goal": "f = g"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "e : C ≌ D"], "goal": "e.toAdjunction.counit = e.counit"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "PartialOrder Γ", "Zero R", "Zero Γ"], "goal": "HahnSeries.order 0 = 0"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : C", "f✝ : X ⟶ Y", "d₁ d₂ d₃ : Factorisation f✝", "f : d₁.Hom d₂", "g : d₂.Hom d₃"], "goal": "(f.h ≫ g.h) ≫ d₃.π = d₁.π"}}
+{"state": {"context": ["x✝ y✝ x y : ℝ", "hy : y ∈ Ico (-(π / 2)) (π / 2)", "hx₁ : -1 ≤ x", "hx₂ : x ≤ 1"], "goal": "arcsin x ≤ y ↔ x ≤ sin y"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "CategoryTheory.Limits.HasColimit F", "E : K ⥤ J", "CategoryTheory.Limits.HasColimit (E ⋙ F)", "k : K"], "goal": "CategoryTheory.Limits.colimit.ι (E ⋙ F) k ≫ CategoryTheory.Limits.colimit.pre F E =\n  CategoryTheory.Limits.colimit.ι F (E.obj k)"}}
+{"state": {"context": [], "goal": "Filter.Tendsto Int.cast Filter.cofinite (Filter.cocompact ℝ)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort y", "s✝ t u : Set α", "inst✝³ : PseudoEMetricSpace α", "inst✝² : MeasurableSpace α", "inst✝¹ : OpensMeasurableSpace α", "inst✝ : MeasurableSpace β", "x : α", "ε : ℝ≥0∞", "μ : Measure α", "s : Set α", "hs : ∃ R > 0, μ (thickening R s) ≠ ⊤", "h's : IsClosed s"], "goal": "s = closure s"}}
+{"state": {"context": ["K : Type u", "inst✝ : Field K", "n : ℕ", "hζ✝ : (primitiveRoots n K).Nonempty", "hn✝ : 0 < n", "a : K", "H : Irreducible (X ^ n - C a)", "this✝ : Fact (Irreducible (X ^ n - C a)) := { out := H }", "this : Algebra K K[n√a] := inferInstance", "hn : 0 < n", "hn' : ¬n = 1", "ζ : K", "hζ : IsPrimitiveRoot ζ n"], "goal": "∀ x ∈ Finset.range n, ∀ y ∈ Finset.range n, (algebraMap K K[n√a]) ζ ^ x * root (X ^ n - C a) = (algebraMap K K[n√a]) ζ ^ y * root (X ^ n - C a) → x = y"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝³ : Ring R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "ι : Type w", "ι' : Type w'", "inst✝ : StrongRankCondition R", "v : Basis ι R M", "this : Nontrivial R"], "goal": "lift.{v, w} #ι = lift.{w, v} (Module.rank R M)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι : Type u_12", "inst✝² : NormedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "p q : Seminorm 𝕜 E", "x : E", "s : Set (Seminorm 𝕜 E)", "hs₁ : BddAbove s", "hs₂ : s.Nonempty"], "goal": "IsLUB s (sSup s)"}}
+{"state": {"context": ["α : Type u_2", "CompleteLattice α", "k : α", "h : CompleteLattice.IsCompactElement k"], "goal": "IsCoatomic ↑(Set.Iic k)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Semiring R", "p q : R[X]"], "goal": "∀ (n : ℕ) (c : R), c ≠ 0 → (divX_hom ((monomial n) c)).natDegree = n - 1"}}
+{"state": {"context": ["α : Type u_2", "Lattice α", "s : Set α"], "goal": "IsSublattice (⇑OrderDual.ofDual ⁻¹' s) ↔ IsSublattice s"}}
+{"state": {"context": ["M : Type u_5", "N : Type u_6", "Zero M", "Zero N"], "goal": "0 = (0, 0)"}}
+{"state": {"context": ["α : Type u", "Semiring α", "LinearOrder α", "a b c : α", "ExistsAddOfLE α", "MulPosMono α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "h : a * c < b * c", "hc : c ≤ 0"], "goal": "b < a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "f : r ↪r s", "y : Set β"], "goal": "maximals r (⇑f ⁻¹' y) = ⇑f ⁻¹' maximals s (y ∩ Set.range ⇑f)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "E : Type u_3", "F : Type u_4", "ι : Type u_5", "ι' : Type u_6", "α : Type u_7", "inst✝⁸ : LinearOrderedField R", "inst✝⁷ : LinearOrderedField R'", "inst✝⁶ : AddCommGroup E", "inst✝⁵ : AddCommGroup F", "inst✝⁴ : LinearOrderedAddCommGroup α", "inst✝³ : Module R E", "inst✝² : Module R F", "inst✝¹ : Module R α", "inst✝ : OrderedSMul R α", "s✝ : Set E", "i j : ι", "c : R", "t✝ : Finset ι", "w : ι → R", "z : ι → E", "s : Finset ι", "t : Finset ι'", "ws : ι → R", "zs : ι → E", "wt : ι' → R", "zt : ι' → E", "hws : ∑ i ∈ s, ws i = 1", "hwt : ∑ i ∈ t, wt i = 1", "a b : R", "hab : a + b = 1"], "goal": "a • s.centerMass ws zs + b • t.centerMass wt zt = (s.disjSum t).centerMass (Sum.elim (fun i => a * ws i) fun j => b * wt j) (Sum.elim zs zt)"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_4", "CommSemiring R"], "goal": "MvPolynomial.comap (AlgHom.id R (MvPolynomial σ R)) = id"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f✝ g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "R : ℝ", "f : X → ℝ", "hR : 0 ≤ R", "hfint : IntegrableOn f {x | R < f x} μ", "hμ : μ {x | R < f x} ≠ 0", "this : IntegrableOn (fun x => R) {x | R < f x} μ"], "goal": "0 ≤ᶠ[ae (μ.restrict {x | R < f x})] fun a => f a - R"}}
+{"state": {"context": ["α : Sort u_1", "x y : α", "Decidable (x = y)"], "goal": "x = y ∨ x ≠ y"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "CommSemiring R", "φ ψ : MvPolynomial σ R"], "goal": "↑φ = ↑ψ ↔ φ = ψ"}}
+{"state": {"context": ["α β : Type u"], "goal": "⇑(prodEquivPiFinTwo α β).symm = fun f => (f 0, f 1)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : Measure α", "f✝ g : α → F'", "s : Set α", "hμ : NeZero μ", "f : α → F'", "hμ_finite : IsFiniteMeasure μ", "h_meas : StronglyMeasurable (μ[f|⊥])", "c : F'", "h_eq : μ[f|⊥] = fun x => c"], "goal": "(fun x => c) = fun x => (μ Set.univ).toReal⁻¹ • ∫ (x : α), f x ∂μ"}}
+{"state": {"context": ["α : Type u_1", "Monoid α", "DecidableRel fun x x_1 => x ∣ x_1", "a b : α", "h : multiplicity.Finite a b"], "goal": "a ^ (multiplicity a b).get h ∣ b"}}
+{"state": {"context": ["β : Type w", "α : Type w₂", "γ : Type w₃", "C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Unique β", "f : β → C", "s : Cocone (Discrete.functor f)"], "goal": "{ pt := f default, ι := Discrete.natTrans fun x => match x with | { as := j } => eqToHom ⋯ }.ι.app default ≫ (fun s => s.ι.app default) s = s.ι.app default"}}
+{"state": {"context": ["α β : HeytAlg", "e : ↑α ≃o ↑β", "a : ↑α"], "goal": "(HeytAlg.Iso.mk e).hom a = e a"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VAdd α β", "s : Set α", "t : Set β", "b : β"], "goal": "b ∈ s +ᵥ t ↔ ∃ x ∈ s, ∃ y ∈ t, x +ᵥ y = b"}}
+{"state": {"context": ["R : Type u_2", "NonAssocSemiring R", "S : Subsemiring Rᵐᵒᵖ"], "goal": "S.unop.op = S"}}
+{"state": {"context": ["M : Type u_2", "AddCommGroup M", "R : Type u_3", "CommRing R", "Module R M", "Module.Finite R M", "f : Module.End R M"], "goal": "∃ p, p.Monic ∧ (Polynomial.aeval f) p = 0"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "CommSemiring R", "Nontrivial R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "NonUnitalRing A", "StarRing A", "TopologicalSpace A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "NonUnitalContinuousFunctionalCalculus R p", "f : R → R", "a : A", "hf : ¬f 0 = 0"], "goal": "cfcₙ f a = 0"}}
+{"state": {"context": ["ι : Type u_1", "C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasZeroMorphisms C", "c : ComplexShape ι", "K L M K' L' : HomologicalComplex C c", "inst✝ : CategoryWithHomology C", "f : K ⟶ L"], "goal": "quasiIso C c f ↔ QuasiIso f"}}
+{"state": {"context": ["R : Type u_1", "NonAssocSemiring R", "NoZeroDivisors R", "Nontrivial R", "p : ℕ", "NeZero p", "CharP R p"], "goal": "Fact (Nat.Prime p)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n : ℕ", "hpos : 0 < n", "a✝ : Nontrivial R", "prod_monic : (∏ i ∈ n.properDivisors, cyclotomic i R).Monic", "h : (∏ i ∈ n.properDivisors, cyclotomic i R).degree = ⊥"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f : α → β", "TopologicalSpace.PseudoMetrizableSpace β", "MeasurableSpace β", "BorelSpace β", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "Measurable (MeasureTheory.AEStronglyMeasurable.mk f hf)"}}
+{"state": {"context": ["C : Type u_1", "C₁ : Type u_2", "C₂ : Type u_3", "C₃ : Type u_4", "D₁ : Type u_5", "D₂ : Type u_6", "D₃ : Type u_7", "inst✝⁹ : Category.{u_10, u_1} C", "inst✝⁸ : Category.{?u.40310, u_2} C₁", "inst✝⁷ : Category.{?u.40314, u_3} C₂", "inst✝⁶ : Category.{?u.40318, u_4} C₃", "inst✝⁵ : Category.{u_8, u_5} D₁", "inst✝⁴ : Category.{u_9, u_6} D₂", "inst✝³ : Category.{?u.40330, u_7} D₃", "W : MorphismProperty C", "L₁ : C ⥤ D₁", "inst✝² : L₁.IsLocalization W", "L₂ : C ⥤ D₂", "inst✝¹ : L₂.IsLocalization W", "L₃ : C ⥤ D₃", "inst✝ : L₃.IsLocalization W", "X Y Z : C", "G : D₁ ⥤ D₂", "e : L₁ ⋙ G ≅ L₂", "f : L₁.obj X ⟶ L₁.obj Y"], "goal": "(homEquiv W L₁ L₂) f = e.inv.app X ≫ G.map f ≫ e.hom.app Y"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁵ : Category.{u_4, u_1} C", "inst✝⁴ : Category.{u_6, u_2} D", "W : MorphismProperty C", "inst✝³ : W.HasLeftCalculusOfFractions", "X Y : C", "φ : W.LeftFraction X Y", "E : Type u_3", "inst✝² : Category.{u_5, u_3} E", "L₁ : C ⥤ D", "L₂ : C ⥤ E", "inst✝¹ : L₁.IsLocalization W", "inst✝ : L₂.IsLocalization W", "e : L₁ ⋙ (Localization.uniq L₁ L₂ W).functor ≅ L₂ := Localization.compUniqFunctor L₁ L₂ W", "this : IsIso (L₂.map φ.s)"], "goal": "(Localization.uniq L₁ L₂ W).functor.map (φ.map L₁ ⋯) = (Localization.compUniqFunctor L₁ L₂ W).hom.app X ≫ φ.map L₂ ⋯ ≫ (Localization.compUniqFunctor L₁ L₂ W).inv.app Y"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{?u.440575, u_1} C", "n m : ℕ", "F G : ComposableArrows C n", "f g : ComposableArrows C 5", "app₀ : f.obj' 0 ⋯ ≅ g.obj' 0 ⋯", "app₁ : f.obj' 1 ⋯ ≅ g.obj' 1 ⋯", "app₂ : f.obj' 2 ⋯ ≅ g.obj' 2 ⋯", "app₃ : f.obj' 3 ⋯ ≅ g.obj' 3 ⋯", "app₄ : f.obj' 4 ⋯ ≅ g.obj' 4 ⋯", "app₅ : f.obj' 5 ⋯ ≅ g.obj' 5 ⋯", "w₀ : f.map' 0 1 ⋯ ⋯ ≫ app₁.hom = app₀.hom ≫ g.map' 0 1 ⋯ ⋯", "w₁ : f.map' 1 2 ⋯ ⋯ ≫ app₂.hom = app₁.hom ≫ g.map' 1 2 ⋯ ⋯", "w₂ : f.map' 2 3 ⋯ ⋯ ≫ app₃.hom = app₂.hom ≫ g.map' 2 3 ⋯ ⋯", "w₃ : f.map' 3 4 ⋯ ⋯ ≫ app₄.hom = app₃.hom ≫ g.map' 3 4 ⋯ ⋯", "w₄ : f.map' 4 5 ⋯ ⋯ ≫ app₅.hom = app₄.hom ≫ g.map' 4 5 ⋯ ⋯"], "goal": "g.map' 0 1 ⋯ ⋯ ≫ app₁.inv = app₀.inv ≫ f.map' 0 1 ⋯ ⋯"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s✝ t✝ : Set X", "f : X → Y", "y : Y", "hf : Tendsto f (coLindelof X) (𝓝 y)", "hfc : Continuous f", "l : Filter Y", "hne : l.NeBot", "inst✝ : CountableInterFilter l", "hle : l ≤ 𝓟 (insert y (range f))", "s : Set Y", "hsy : s ∈ 𝓝 y", "t : Set Y", "htl : t ∈ l", "hd : Disjoint s t", "K : Set X", "hKc : IsLindelof K", "hKs : Kᶜ ⊆ f ⁻¹' s"], "goal": "∃ x ∈ insert y (range f), ClusterPt x l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ε : Type u_5", "ι : Type u_6", "f : α →. β ⊕ α", "a : α", "b : β", "h : b ∈ f.fix a", "h₁ : Acc (fun x y => Sum.inr x ∈ f y) a", "h₂ : (f a).Dom", "a'✝ : α", "heq✝ : (f a).get h₂ = Sum.inr a'✝", "h₃ : b ∈ WellFounded.fixF (fun a IH => Part.assert (f a).Dom fun hf => match e : (f a).get hf with | Sum.inl b => Part.some b | Sum.inr a' => IH a' ⋯) a'✝ ⋯"], "goal": "Sum.inl b ∈ f a ∨ ∃ a', Sum.inr a' ∈ f a ∧ b ∈ f.fix a'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α →. β", "s t : Set β"], "goal": "f.preimage (s ∩ t) ⊆ f.preimage s ∩ f.preimage t"}}
+{"state": {"context": ["X : Type u_2", "EMetricSpace X", "MeasurableSpace X", "BorelSpace X"], "goal": "(MeasureTheory.Measure.mkMetric fun x => ⊤) = ⊤"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x p₁ p₂ : P", "hx : x ∉ s", "hp₁ : p₁ ∈ s", "hp₂ : p₂ ∈ s", "t : R", "ht : 0 < t"], "goal": "s.SSameSide (t • (x -ᵥ p₁) +ᵥ p₂) x"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "C₃ : Type u_3", "C₄ : Type u_4", "C₁₂ : Type u_5", "CategoryTheory.Category.{u_16, u_1} C₁", "CategoryTheory.Category.{u_15, u_2} C₂", "CategoryTheory.Category.{u_13, u_3} C₃", "CategoryTheory.Category.{u_12, u_4} C₄", "CategoryTheory.Category.{u_14, u_5} C₁₂", "F₁₂ : CategoryTheory.Functor C₁ (CategoryTheory.Functor C₂ C₁₂)", "G : CategoryTheory.Functor C₁₂ (CategoryTheory.Functor C₃ C₄)", "I₁ : Type u_7", "I₂ : Type u_8", "I₃ : Type u_9", "J : Type u_10", "r : I₁ × I₂ × I₃ → J", "ρ₁₂ : CategoryTheory.GradedObject.BifunctorComp₁₂IndexData r", "X₁ : CategoryTheory.GradedObject I₁ C₁", "X₂ : CategoryTheory.GradedObject I₂ C₂", "X₃ : CategoryTheory.GradedObject I₃ C₃", "(((CategoryTheory.GradedObject.mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p", "(((CategoryTheory.GradedObject.mapBifunctor G ρ₁₂.I₁₂ I₃).obj\n            (CategoryTheory.GradedObject.mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj\n        X₃).HasMap\n    ρ₁₂.q", "i₁ : I₁", "i₂ : I₂", "i₃ : I₃", "j : J", "h : r (i₁, i₂, i₃) = j", "i₁₂ : ρ₁₂.I₁₂", "h₁₂ : ρ₁₂.p (i₁, i₂) = i₁₂"], "goal": "CategoryTheory.GradedObject.ιMapBifunctor₁₂BifunctorMapObj F₁₂ G ρ₁₂ X₁ X₂ X₃ i₁ i₂ i₃ j h =\n  CategoryTheory.CategoryStruct.comp\n    ((G.map (CategoryTheory.GradedObject.ιMapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂ i₁ i₂ i₁₂ h₁₂)).app (X₃ i₃))\n    (CategoryTheory.GradedObject.ιMapBifunctorMapObj G ρ₁₂.q\n      (CategoryTheory.GradedObject.mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ i₁₂ i₃ j ⋯)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type v", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "r : ℝ≥0∞", "n : ℕ", "f : E → F", "x : E", "s : Set E", "h : HasFiniteFPowerSeriesOnBall f p x (n + 1) r"], "goal": "HasFiniteFPowerSeriesOnBall (fderiv 𝕜 f) p.derivSeries x n r"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "s : Set α", "TopologicalSpace β", "f : α → β", "hf : ContinuousOn f s", "hf' : Function.Injective f", "h : IsTotallyDisconnected (f '' s)"], "goal": "IsTotallyDisconnected s"}}
+{"state": {"context": ["s : String", "p : Char → Bool"], "goal": "s.all p = s.data.all p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁵ : MeasurableSpace δ", "inst✝⁴ : NormedAddCommGroup β", "inst✝³ : NormedAddCommGroup γ", "𝕜 : Type u_5", "inst✝² : NormedAddCommGroup 𝕜", "inst✝¹ : SMulZeroClass 𝕜 β", "inst✝ : BoundedSMul 𝕜 β", "c : 𝕜", "f : α → β", "hfi : ∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤"], "goal": "∫⁻ (a : α), ↑‖(c • f) a‖₊ ∂μ < ⊤"}}
+{"state": {"context": ["α : Type u_1", "s t u : Set α", "hst : s ⊆ t", "htu : t ⊆ u", "x : ↑s"], "goal": "Set.inclusion htu (Set.inclusion hst x) = Set.inclusion ⋯ x"}}
+{"state": {"context": ["R : Type u_1", "M₁ : Type u_2", "M₂ : Type u_3", "CommRing R", "AddCommGroup M₁", "AddCommGroup M₂", "Module R M₁", "Module R M₂", "Q₁ : QuadraticForm R M₁", "Q₂ : QuadraticForm R M₂", "m₁ : M₁"], "goal": "(CliffordAlgebra.toProd Q₁ Q₂) ((CliffordAlgebra.ι Q₁) m₁ ᵍ⊗ₜ[R] 1) =\n  (CliffordAlgebra.ι (QuadraticMap.prod Q₁ Q₂)) (m₁, 0)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "x y z : α", "ε ε₁ ε₂ : ℝ≥0∞", "s✝ t s : Set α", "hs : ∀ ε > 0, ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε", "x₀ : α", "hx₀ : x₀ ∈ s"], "goal": "∃ t ⊆ s, t.Countable ∧ s ⊆ closure t"}}
+{"state": {"context": ["o : Ordinal.{u_1}"], "goal": "aleph' o < succ (aleph' o)"}}
+{"state": {"context": ["m : Type u → Type v", "α ε : Type u", "Monad m", "x : m α"], "goal": "(ExceptT.lift x).run = Except.ok <$> x"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s t : Set α", "p : α → Prop"], "goal": "(∀ᵐ (x : α) ∂μ.restrict (s ∪ t), p x) ↔ (∀ᵐ (x : α) ∂μ.restrict s, p x) ∧ ∀ᵐ (x : α) ∂μ.restrict t, p x"}}
+{"state": {"context": ["R : Type u_1", "CommRing R"], "goal": "↑ArithmeticFunction.zetaUnit = ↑ArithmeticFunction.zeta"}}
+{"state": {"context": ["β : Type u_2", "γ : Type u_3", "LinearOrder β", "PartialOrder γ", "h : WellFounded fun x x_1 => x < x_1", "f g : β → γ", "hf : StrictMono f", "hg : StrictMono g"], "goal": "Set.range f = Set.range g ↔ f = g"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "S : C", "self : CategoryTheory.PreOneHypercover S", "i₁ i₂ : self.I₀", "j : self.I₁ i₁ i₂"], "goal": "CategoryTheory.CategoryStruct.comp (self.p₁ j) (self.f i₁) = CategoryTheory.CategoryStruct.comp (self.p₂ j) (self.f i₂)"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "k : K", "n d : R", "hd : d ∈ nonZeroDivisors R", "hk : k * (algebraMap R K) ↑(n, ⟨d, hd⟩).2 = (algebraMap R K) (n, ⟨d, hd⟩).1", "hd' : d ≠ 0"], "goal": "Ideal.span {d} ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "L : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "inst✝⁶ : Field K", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝³ : H.IsCartanSubalgebra", "inst✝² : IsTriangularizable K (↥H) L", "inst✝¹ : IsKilling K L", "inst✝ : CharZero K", "α : Weight K (↥H) L", "hα : α.IsNonZero", "e : L", "heα : e ∈ rootSpace H ⇑α", "he₀ : e ≠ 0", "f' : L", "hfα : f' ∈ rootSpace H (-⇑α)", "hf : ((killingForm K L) e) f' ≠ 0", "hef✝ : ⁅e, f'⁆ = ((killingForm K L) e) f' • ↑((cartanEquivDual H).symm (Weight.toLinear K (↥H) L α))", "h : ↥H := ⟨⁅e, f'⁆, ⋯⟩", "hh : α h ≠ 0", "f : L := (2 * (α h)⁻¹) • f'", "hef : ⁅⁅e, f⁆, e⁆ = 2 • e"], "goal": "⁅⁅e, f⁆, f⁆ = -(2 • f)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V"], "goal": "default = G.selfColoring.toPartition"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : InnerProductSpace ℝ E", "inst✝³ : CompleteSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : InnerProductSpace ℝ F", "inst✝ : CompleteSpace F", "K : ProperCone ℝ E", "f : E →L[ℝ] F", "b : F", "h : b ∉ map f K", "C : ConvexCone ℝ F := ↑↑(map f K)", "y : F", "hxy : ∀ x ∈ C, 0 ≤ ⟪x, y⟫_ℝ", "hyb : ⟪y, b⟫_ℝ < 0", "x : E", "hxK : x ∈ K"], "goal": "∃ x_1 ∈ ↑↑K, f x_1 = f x"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "G : SimpleGraph V", "G' : SimpleGraph V'", "u v : V", "p : G.Walk u v"], "goal": "(induce {v_1 | v_1 ∈ p.support} G).Connected"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "idx : Type u_2", "inst✝ : CommRing R", "hp : Fact (Nat.Prime p)", "Φ : MvPolynomial idx ℤ", "φ : ℕ → MvPolynomial (idx × ℕ) ℤ", "h : ∀ (n : ℕ), (bind₁ φ) (W_ ℤ n) = (bind₁ fun i => (rename (Prod.mk i)) (W_ ℤ n)) Φ"], "goal": "∀ (n : ℕ), (bind₁ fun k => (map (Int.castRingHom ℚ)) (φ k)) (W_ ℚ n) = (bind₁ fun i => (rename (Prod.mk i)) (W_ ℚ n)) ((map (Int.castRingHom ℚ)) Φ)"}}
+{"state": {"context": ["α : Type u_1", "MulOneClass α", "Preorder α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "a b c : α", "hbc : b < c", "ha : a < 1"], "goal": "b * a < c"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s s' : Set ℝ", "t₀ a : ℝ", "ha : a ≠ 0", "ε : ℝ", "hε : ε > 0", "h : IsIntegralCurveOn γ v (Metric.ball t₀ ε)"], "goal": "Metric.ball (t₀ / a) (ε / |a|) = {t | t * a ∈ Metric.ball t₀ ε}"}}
+{"state": {"context": ["α : Type u_1", "KleeneAlgebra α", "a b : α", "hb : 1 ≤ b"], "goal": "a * b ≤ b → a∗ ≤ b"}}
+{"state": {"context": ["R : Type u1", "inst✝² : CommRing R", "M : Type u2", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "d d' : Module.Dual R M", "m : CliffordAlgebra Q", "x : M", "hx : (contractLeft d) ((contractLeft d) m) = 0"], "goal": "(contractLeft d) ((contractLeft d) ((ι Q) x * m)) = 0"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "s : AList β"], "goal": "AList.extract a s = (AList.lookup a s, AList.erase a s)"}}
+{"state": {"context": ["α : Type u_1", "DivisionRing α", "a : α", "n : ℤ", "d : ℕ"], "goal": "Mathlib.Meta.NormNum.IsRat a n d → ↑d ≠ 0"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : LinearOrderedAddCommGroup α", "a b x : α", "hx : x ≤ a ∧ x ≤ -b ∨ -a ≤ x ∧ b ≤ x"], "goal": "x ∈ Iic (a, b).1 ∪ Ici (a, b).2"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : Monoid G", "a b x✝ y : G", "n✝ m : ℕ", "x : G", "n : ℕ", "hn : n ≠ 0", "dvd : n ∣ orderOf x"], "goal": "orderOf (x ^ n) = orderOf x / n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "a : α", "b : β", "h : (s ×ˢ t).Nonempty", "st : s.Nonempty ∧ t.Nonempty"], "goal": "s ×ˢ t ⊆ s₁ ×ˢ t₁ ↔ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "β : Type u_3", "m' mΩ : MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "hm' : m' ≤ mΩ", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "Zero β", "One β", "m : MeasurableSpace β", "s : ι → Set Ω", "hs : ProbabilityTheory.iCondIndepSet m' hm' s μ"], "goal": "ProbabilityTheory.iCondIndepFun m' hm' (fun _n => m) (fun n => (s n).indicator fun _ω => 1) μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "u : δ → δ → Prop", "f : r ↪r s", "g : t ↪r u"], "goal": "∀ {a b : α × γ}, Prod.Lex s u ({ toFun := Prod.map ⇑f ⇑g, inj' := ⋯ } a) ({ toFun := Prod.map ⇑f ⇑g, inj' := ⋯ } b) ↔ Prod.Lex r t a b"}}
+{"state": {"context": ["X : Type u", "s : Set X", "TopologicalSpace X"], "goal": "IsClosed s ↔ ∀ (x : X), (∃ᶠ (y : X) in 𝓝 x, y ∈ s) → x ∈ s"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁵ : Group G", "inst✝⁴ : Group G'", "inst✝³ : Group G''", "A : Type u_4", "inst✝² : AddGroup A", "M : Type u_5", "S : Type u_6", "inst✝¹ : DivInvMonoid M", "inst✝ : SetLike S M", "hSM : SubgroupClass S M", "H K : S", "x : M", "hx : x ∈ K", "n : ℕ"], "goal": "x ^ n ∈ K"}}
+{"state": {"context": ["B : Type u₁", "inst✝¹ : Bicategory B", "C : Type u₂", "inst✝ : Bicategory C", "F G H I : OplaxFunctor B C", "η θ : F ⟶ G", "Γ : η ⟶ θ", "ι : G ⟶ H", "a b : B", "f : a ⟶ b"], "goal": "F.map f ◁ Γ.app b ▷ ι.app b ≫ (α_ (F.map f) (θ.app b) (ι.app b)).inv ≫ θ.naturality f ▷ ι.app b ≫ (α_ (θ.app a) (G.map f) (ι.app b)).hom ≫ θ.app a ◁ ι.naturality f ≫ (α_ (θ.app a) (ι.app a) (H.map f)).inv = (α_ (F.map f) (η.app b) (ι.app b)).inv ≫ η.naturality f ▷ ι.app b ≫ (α_ (η.app a) (G.map f) (ι.app b)).hom ≫ Γ.app a ▷ (G.map f ≫ ι.app b) ≫ θ.app a ◁ ι.naturality f ≫ (α_ (θ.app a) (ι.app a) (H.map f)).inv"}}
+{"state": {"context": ["α : Type u_2", "SemilatticeInf α", "a : α"], "goal": "SupPrime (OrderDual.toDual a) ↔ InfPrime a"}}
+{"state": {"context": ["α : Type u_1", "l : Filter α", "u v : α → ℝ"], "goal": "((fun x => |u x|) =o[l] fun x => |v x|) ↔ u =o[l] v"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝⁴ : CommRing R", "inst✝³ : CommRing S", "inst✝² : Algebra R S", "f : R[X]", "T : Type u_1", "inst✝¹ : CommRing T", "inst✝ : Algebra R T", "h : IsAdjoinRoot S f", "e : S ≃ₐ[R] T", "x✝ : R"], "goal": "(algebraMap R T) x✝ = (((↑↑e).comp h.map).comp C) x✝"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : α →ₘ[μ] E"], "goal": "f ∈ MeasureTheory.Lp E p μ ↔ MeasureTheory.eLpNorm (↑f) p μ < ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f : β → ℝ≥0", "hf : Summable f", "g : β → ℝ≥0", "hgf : ∀ (b : β), (fun i => ↑(g i)) b ≤ (fun i => ↑(f i)) b"], "goal": "Summable g"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommRing R", "E : EllipticCurve R", "inst✝ : Invertible 3"], "goal": "(ofJ0 R).j = 0"}}
+{"state": {"context": ["R : Type u", "CommRing R", "M N : ModuleCat R", "f : M ⟶ N"], "goal": "ModuleCat.MonoidalCategory.tensorHom (𝟙 (ModuleCat.of R R)) f ≫ (ModuleCat.MonoidalCategory.leftUnitor N).hom =\n  (ModuleCat.MonoidalCategory.leftUnitor M).hom ≫ f"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "c : ℂ", "R : ℝ", "h0 : 0 < R", "f : ℂ → E", "y : E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c R \\ {c})", "hd : ∀ z ∈ (ball c R \\ {c}) \\ s, DifferentiableAt ℂ f z", "hy : Tendsto f (𝓝[≠] c) (𝓝 y)"], "goal": "‖(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) - (2 * ↑π * I) • y‖ ≤ 0"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ✝ μ : Measure α", "f : α → ℝ≥0∞", "f_meas : Measurable f", "hf : ∀ᵐ (x : α) ∂μ, f x < ⊤", "g i : α → ℝ≥0∞", "i_meas : Measurable i", "hi : i ≤ f * g", "A : (fun x => (f x)⁻¹ * i x) ≤ g"], "goal": "∫⁻ (a : α), i a ∂μ ≤ ⨆ (_ : (fun x => (f x)⁻¹ * i x) ≤ g), ∫⁻ (a : α), (fun x => (f x)⁻¹ * i x) a ∂μ.withDensity f"}}
+{"state": {"context": ["R : Type u_1", "LinearOrderedField R", "a b : ℍ[R]"], "goal": "Quaternion.normSq (a / b) = Quaternion.normSq a / Quaternion.normSq b"}}
+{"state": {"context": ["α : Type u_1", "a b : PosNum"], "goal": "↑a + ↑b + (↑a + ↑b) + 1 = ↑a + ↑a + (↑b + ↑b + 1)"}}
+{"state": {"context": ["Ω : Type u_1", "m' mΩ : MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "hm' : m' ≤ mΩ", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "s₁ s' : Set (Set Ω)", "s₂ : Set (Set Ω)", "h₁ : ProbabilityTheory.CondIndepSets m' hm' s₁ s' μ"], "goal": "ProbabilityTheory.CondIndepSets m' hm' (s₁ ∩ s₂) s' μ"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α", "x : α", "V : Set (α × α)", "hV : V ∈ 𝓤 α"], "goal": "x ∈ UniformSpace.ball x V"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "Module.Finite R M"], "goal": "Module.support R M = PrimeSpectrum.zeroLocus ↑(Module.annihilator R M)"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "xs ys : List α", "h : xs.Sublist ys"], "goal": "(List.sym n xs).Sublist (List.sym n ys)"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "J' : Type u₂", "CategoryTheory.Category.{v₂, u₂} J'", "e : J ≌ J'", "CategoryTheory.Limits.HasColimitsOfShape J C"], "goal": "CategoryTheory.Limits.HasColimitsOfShape J' C"}}
+{"state": {"context": ["x : ℝ≥0"], "goal": "↑↑x ≠ ⊤"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "CategoryTheory.Limits.HasBinaryBiproduct X Y", "b : CategoryTheory.Limits.BinaryBicone X Y", "hb : b.IsBilimit"], "goal": "(hb.isLimit.conePointUniqueUpToIso (CategoryTheory.Limits.BinaryBiproduct.isLimit X Y)).hom =\n  CategoryTheory.Limits.biprod.lift b.fst b.snd"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a : α", "ha : 0 < a"], "goal": "StrictMono fun x => x / a"}}
+{"state": {"context": ["M : Type u_7", "CommMonoid M"], "goal": "rootsOfUnity 1 M = ⊥"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝² : SeminormedGroup E", "inst✝¹ : SeminormedGroup F", "inst✝ : SeminormedGroup G", "s : Set E", "a a₁ a₂ b b₁ b₂ : E", "r r₁ r₂ : ℝ", "f : E → F", "x : E", "y : F"], "goal": "Tendsto f (𝓝 x) (𝓝 y) ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x' : E), ‖x' / x‖ < δ → ‖f x' / y‖ < ε"}}
+{"state": {"context": ["R : Type u_1", "n : ℕ", "hn : n ≠ 0"], "goal": "(n.factorization.prod fun p k => (σ 0) (p ^ k)) = ∏ x ∈ n.primeFactors, (n.factorization x + 1)"}}
+{"state": {"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝⁴ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝³ : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝² : Ring R", "inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀", "inst✝ : LinearOrderedCommMonoidWithZero Γ'₀", "v : Valuation R Γ₀", "x✝ y✝ z x y : R"], "goal": "max (v x) (v y) ≥ v (x + y)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "OmegaCompletePartialOrder α", "OmegaCompletePartialOrder β", "f g : α →𝒄 β", "x y : α", "h₁ : f ≤ g", "h₂ : x ≤ y"], "goal": "f x ≤ g y"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "s : Set α", "f : α → ℝ", "K : ℝ≥0", "h : LipschitzOnWith K f s", "x : α", "hx : x ∈ s", "y : α", "hy : y ∈ s"], "goal": "f x ≤ f y + ↑K * dist x y"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "inst✝ : (i : ι) → MeasurableSpace (α i)", "P : (J : Finset ι) → Measure ((j : { x // x ∈ J }) → α ↑j)", "I J : Finset ι", "hP : IsProjectiveMeasureFamily P", "S : Set ((i : { x // x ∈ I }) → α ↑i)", "T : Set ((i : { x // x ∈ J }) → α ↑i)", "hT : MeasurableSet T", "h_eq : cylinder I S = cylinder J T", "hJI : J ⊆ I"], "goal": "(P I) S = (P J) T"}}
+{"state": {"context": [], "goal": "Subgroup.normalClosure {⟨finRotate 5, ⋯⟩} = ⊤"}}
+{"state": {"context": ["R : Type uR", "S✝ : Type uS", "A : Type uA", "B : Type uB", "inst✝⁴ : CommSemiring R", "inst✝³ : Ring A", "inst✝² : Algebra R A", "inst✝¹ : Ring B", "inst✝ : Algebra R B", "f : A →ₐ[R] B", "S : Subalgebra R A"], "goal": "comap f (map f S) = S ⊔ adjoin R (⇑f ⁻¹' {0})"}}
+{"state": {"context": ["α β : Lat", "e : ↑α ≃o ↑β", "a : ↑α"], "goal": "(Lat.Iso.mk e).hom.toSupHom a = e a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : Mul β", "f g : FreeSemigroup α →ₙ* β", "h : ⇑f ∘ of = ⇑g ∘ of", "x✝ : FreeSemigroup α", "x : α", "y : FreeSemigroup α", "hx : f (of x) = g (of x)", "hy : f y = g y"], "goal": "f (of x * y) = g (of x * y)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_7", "M : Type u_8", "M₂ : Type u_10", "Semiring R", "Semiring S", "AddCommMonoid M", "AddCommMonoid M₂", "module_M : Module R M", "module_S_M₂ : Module S M₂", "σ : R →+* S", "σ' : S →+* R", "re₁ : RingHomInvPair σ σ'", "re₂ : RingHomInvPair σ' σ", "e : M ≃ₛₗ[σ] M₂"], "goal": "e.trans (LinearEquiv.refl S M₂) = e"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v}", "i : ι"], "goal": "f i ≤ Ordinal.sup f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x✝ y z : E", "δ ε : ℝ", "hδ : 0 < δ", "s : Set E", "x : E", "hs : ENNReal.ofReal δ ≤ infEdist x s"], "goal": "infEdist x (thickening δ s) ≤ infEdist x s - ENNReal.ofReal δ"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "p q : ℚ", "K : Type u_5", "inst✝ : LinearOrderedField K"], "goal": "0 < ↑q ↔ 0 < q"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "f_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y"], "goal": "seminormFromBounded' f ≠ 0"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "CompactSpace X", "S : Set X", "hS : IsClosed S", "hne : S.Nonempty"], "goal": "∃ V ⊆ S, V.Nonempty ∧ IsClosed V ∧ ∀ V' ⊆ V, V'.Nonempty → IsClosed V' → V' = V"}}
+{"state": {"context": ["lc : Lean.Omega.LinearCombo", "i : Int", "v : Lean.Omega.Coeffs"], "goal": "(i * lc).eval v = i * lc.eval v"}}
+{"state": {"context": ["α : Type u_1", "a b c d : ℝ≥0∞", "r p q : ℝ≥0", "x : ℝ≥0∞"], "goal": "x.toReal = 0 ↔ x = 0 ∨ x = ⊤"}}
+{"state": {"context": ["f : Ordinal.{u_2} → Ordinal.{u_2}", "c : Ordinal.{u_2}", "hc : ℵ₀ < c.cof", "hf : ∀ i < c, f i < c", "a : Ordinal.{u_2}"], "goal": "a < c → Ordinal.nfp f a < c"}}
+{"state": {"context": ["α : Type u_1", "f : Char → α → α", "l m r : List Char", "a : α"], "goal": "foldrAux f a { data := l ++ m ++ r } { byteIdx := utf8Len l + utf8Len m } { byteIdx := utf8Len l } = List.foldr f a m"}}
+{"state": {"context": ["α : Type u_1", "𝒜 : Finset (Finset α)", "Fintype α", "DecidableEq α"], "goal": "(Finset.Iic (Fintype.card α)).biUnion 𝒜.slice = 𝒜"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕜₃ : Type u_3", "E : Type u_4", "Eₗ : Type u_5", "F : Type u_6", "Fₗ : Type u_7", "G : Type u_8", "Gₗ : Type u_9", "𝓕 : Type u_10", "inst✝¹⁵ : SeminormedAddCommGroup E", "inst✝¹⁴ : SeminormedAddCommGroup Eₗ", "inst✝¹³ : SeminormedAddCommGroup F", "inst✝¹² : SeminormedAddCommGroup Fₗ", "inst✝¹¹ : SeminormedAddCommGroup G", "inst✝¹⁰ : SeminormedAddCommGroup Gₗ", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NontriviallyNormedField 𝕜₂", "inst✝⁷ : NontriviallyNormedField 𝕜₃", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedSpace 𝕜 Eₗ", "inst✝⁴ : NormedSpace 𝕜₂ F", "inst✝³ : NormedSpace 𝕜 Fₗ", "inst✝² : NormedSpace 𝕜₃ G", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "σ₁₃ : 𝕜 →+* 𝕜₃", "inst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "inst✝ : FunLike 𝓕 E F", "f : E →SL[σ₁₂] F", "K : ℝ≥0", "hf : LipschitzWith K ⇑f", "x : E"], "goal": "‖f x‖ ≤ ↑K * ‖x‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x✝ : Fin n → ℕ", "x : ℝ", "hx : 2 / (1 - 2 / rexp 1) ≤ x", "this : 0 < 1 - 2 / rexp 1"], "goal": "2 ≠ 0"}}
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+{"state": {"context": ["R : Type u_1", "AddMonoid R", "S : AddSubmonoid R", "AddOreLocalization.AddOreSet S", "X : Type u_2", "AddAction R X", "p : R", "r : X", "s : ↥S"], "goal": "p -ₒ s +ᵥ (r -ₒ 0) = p +ᵥ r -ₒ s"}}
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+{"state": {"context": ["b x y : ℝ", "hb : 1 < b", "h : 0 < x", "h₁ : 0 < y"], "goal": "logb b x ≤ logb b y ↔ x ≤ y"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "h : Surjective (image f)", "y : β"], "goal": "∃ a, f a = y"}}
+{"state": {"context": ["X : Type u_2", "EMetricSpace X", "MeasurableSpace X", "OpensMeasurableSpace X", "m : Set X → ℝ≥0∞", "hcl : ∀ (s : Set X), m (closure s) = m s", "r : ℝ≥0∞"], "goal": "(MeasureTheory.OuterMeasure.mkMetric'.pre m r).trim = MeasureTheory.OuterMeasure.mkMetric'.pre m r"}}
+{"state": {"context": ["x : ℝ"], "goal": "deriv Real.tan x = 1 / Real.cos x ^ 2"}}
+{"state": {"context": ["R : Type u_1", "r : R", "CommRing R", "P : Polynomial R"], "goal": "IsUnit (Polynomial.C r + Polynomial.X * P) ↔ IsUnit r ∧ IsNilpotent P"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁴ : Semiring R", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "inst✝¹ : AddCommMonoid M'", "inst✝ : Module R M'", "b b₁ : Basis ι R M", "i✝¹ : ι", "c : R", "x : M", "b' : Basis ι' R M'", "i✝ : ι'", "i : ι"], "goal": "(Finsupp.single (Sum.inr i✝) 1) (Sum.inl i) = 0"}}
+{"state": {"context": ["a b m n p✝ p : ℕ", "pp : Prime p"], "goal": "Icc 1 ((factorization 0) p) = Finset.filter (fun i => p ^ i ∣ 0) (Ico 1 0)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "U : OpenAddSubgroup G", "g : G"], "goal": "g ∈ ↑U ↔ g ∈ U"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝³ : UniformSpace α", "inst✝² : UniformSpace β", "inst✝¹ : UniformSpace γ", "e✝ : β → α", "h_e : UniformInducing e✝", "h_dense : DenseRange e✝", "f✝ : β → γ", "h_f : UniformContinuous f✝", "inst✝ : CompleteSpace γ", "p : α → Prop", "e : α → β", "f : α → γ", "b : β", "s : Set α", "hf : UniformContinuous fun x => f ↑x", "he : UniformEmbedding e", "hd : ∀ (x : β), x ∈ closure (range e)", "hb : closure (e '' s) ∈ 𝓝 b", "hs : IsClosed s", "hp : ∀ x ∈ s, p x", "de : DenseEmbedding e", "de' : DenseEmbedding (DenseEmbedding.subtypeEmb p e)", "ue' : UniformEmbedding (DenseEmbedding.subtypeEmb p e)"], "goal": "∃ c, Tendsto f (comap e (𝓝 b)) (𝓝 c)"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "r : ℝ"], "goal": "Metric.ball 0 r = {x | ‖x‖ < r}"}}
+{"state": {"context": ["α : Type u_2", "InvolutiveInv α", "ι : Sort u_5", "f : ι → α"], "goal": "(Set.range f)⁻¹ = Set.range fun i => (f i)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "a : α"], "goal": "max a 0 + max (-a) 0 = |a|"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X"], "goal": "Monotone fun f => ∃ x ∈ s, ClusterPt x f"}}
+{"state": {"context": ["α✝ : Type u_1", "ι : Type u_2", "γ : Type u_3", "α : Type u_4", "β : Type u_5", "tβ : TopologicalSpace β", "inst✝⁷ : T2Space β", "inst✝⁶ : MeasurableSpace β", "inst✝⁵ : MeasurableSpace α", "s : Set γ", "f✝ : γ → β", "inst✝⁴ : TopologicalSpace γ", "inst✝³ : PolishSpace γ", "inst✝² : MeasurableSpace γ", "hγ : OpensMeasurableSpace γ", "inst✝¹ : Countable ι", "l : Filter ι", "inst✝ : l.IsCountablyGenerated", "f : ι → β → γ", "hf : ∀ (i : ι), Measurable (f i)", "hl : l.NeBot"], "goal": "MeasurableSet {x | ∃ c, Tendsto (fun n => f n x) l (𝓝 c)}"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "NontriviallyNormedField 𝕜", "NontriviallyNormedField 𝕜'", "NormedAlgebra 𝕜 𝕜'", "f : 𝕜 → 𝕜'", "x : 𝕜"], "goal": "logDeriv f x = deriv f x / f x"}}
+{"state": {"context": ["α : Type u_2", "DivisionRing α", "CharZero α", "s : Multiset ℚ"], "goal": "↑s.sum = (Multiset.map Rat.cast s).sum"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "H₁ H₂ : G.Subgraph", "h : Disjoint H₁ H₂"], "goal": "Disjoint H₁.edgeSet H₂.edgeSet"}}
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+{"state": {"context": ["ι : Type u'", "R : Type u_2", "M : Type u_4", "v : ι → M", "Ring R", "AddCommGroup M", "Module R M", "hv : LinearIndependent R v", "l : ι →₀ R"], "goal": "↑(hv.totalEquiv l) = (Finsupp.total ι M R v) l"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α", "n : ℕ"], "goal": "Primrec Mathlib.Vector.ofFn"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "s : Set E", "f : E → E", "f' : E → E →L[ℝ] E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hs : MeasurableSet s", "hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x", "ε : ℝ", "εpos : ε > 0", "δ : ℝ≥0 := ⟨ε, ⋯⟩", "δpos : 0 < δ"], "goal": "∃ f, AEMeasurable f (μ.restrict s) ∧ ∀ᵐ (x : E) ∂μ.restrict s, dist (f x) (f' x) ≤ ε"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "H : Type u_3", "TopologicalSpace H", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "AddCommMonoid G", "TopologicalSpace G", "ChartedSpace H G", "SmoothAdd I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "f : ι → M → G", "h : ∀ (i : ι), Smooth I' I (f i)", "hfin : LocallyFinite fun i => Function.support (f i)"], "goal": "Smooth I' I fun x => ∑ᶠ (i : ι), f i x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "M✝ : Matroid α", "F X Y : Set α", "e : α", "M : Matroid α"], "goal": "M.closure univ = M.E"}}
+{"state": {"context": ["y : ℝ", "hy : 0 < y", "x : ℝ≥0∞"], "goal": "(x ^ (1 / y)) ^ y = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "f✝ : β → γ", "g : γ → α", "δ : Type u_5", "mδ : MeasurableSpace δ", "ξ : Kernel γ δ", "inst✝ : IsSFiniteKernel ξ", "η : Kernel β γ", "κ : Kernel α β", "a : α", "f : δ → ℝ≥0∞", "hf : Measurable f"], "goal": "∫⁻ (b : δ), f b ∂(ξ ∘ₖ η ∘ₖ κ) a = ∫⁻ (b : δ), f b ∂(ξ ∘ₖ (η ∘ₖ κ)) a"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "p : ℝ", "a b : ℝ≥0", "hp1 : 1 ≤ p"], "goal": "a ^ p + b ^ p ≤ (a + b) ^ p⁻¹⁻¹"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "α' : Type u_3", "β : Type u_4", "β' : Type u_5", "γ : Type u_6", "γ' : Type u_7", "l : Type u_8", "m : Type u_9", "n : Type u_10", "p : Type u_11", "q : Type u_12", "r : Type u_13", "l' : Type u_14", "m' : Type u_15", "n' : Type u_16", "p' : Type u_17", "inst✝⁴ : Zero α", "inst✝³ : Zero β", "inst✝² : Zero γ", "inst✝¹ : DecidableEq m", "inst✝ : DecidableEq n", "f : α → β → γ", "hf₁ : ∀ (b : β), f 0 b = 0", "hf₂ : ∀ (a : α), f a 0 = 0", "a : m → α", "b : n → β", "i₁ : m", "i₂ : n", "j₁ : m", "j₂ : n"], "goal": "kroneckerMap f (diagonal a) (diagonal b) (i₁, i₂) (j₁, j₂) = diagonal (fun mn => f (a mn.1) (b mn.2)) (i₁, i₂) (j₁, j₂)"}}
+{"state": {"context": ["z : Complex.UnitDisc"], "goal": "(-z).im = -z.im"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "ConditionallyCompleteLinearOrder β", "TopologicalSpace β", "OrderClosedTopology β", "ι : Sort u_4", "Nonempty ι", "f : α → β", "g : ι → α", "Cf : ContinuousAt f (iInf g)", "Af : Antitone f", "bdd : BddBelow (Set.range g) := by bddDefault"], "goal": "f (⨅ i, g i) = ⨆ i, f (g i)"}}
+{"state": {"context": ["a✝ b✝ c d m n✝ k : ℕ", "p q : ℕ → Prop", "a b n : ℕ"], "goal": "a ^ b % n = (a % n) ^ b % n"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝¹⁷ : CommRing R", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : CommRing T", "inst✝¹⁴ : Algebra R S", "inst✝¹³ : Algebra R T", "K : Type u_4", "L : Type u_5", "inst✝¹² : Field K", "inst✝¹¹ : Field L", "inst✝¹⁰ : Algebra K L", "ι κ : Type w", "inst✝⁹ : Fintype ι", "F : Type u_6", "inst✝⁸ : Field F", "inst✝⁷ : Algebra R L", "inst✝⁶ : Algebra L F", "inst✝⁵ : Algebra R F", "inst✝⁴ : IsScalarTower R L F", "inst✝³ : Algebra K F", "inst✝² : IsScalarTower K L F", "E : Type u_7", "inst✝¹ : Field E", "inst✝ : Algebra K E", "pb : PowerBasis K L", "hE : Splits (algebraMap K E) (minpoly K pb.gen)", "hfx : IsSeparable K pb.gen", "this✝ : DecidableEq E := Classical.decEq E", "this : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb"], "goal": "((minpoly K pb.gen).aroots E).Nodup"}}
+{"state": {"context": ["x : SetTheory.PGame", "i : (-x).RightMoves"], "goal": "(-x).moveRight i = -x.moveLeft (SetTheory.PGame.toRightMovesNeg.symm i)"}}
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+{"state": {"context": ["n✝ : ℕ", "b : Bool", "n : ℕ", "h : n ≠ 0", "ih : digits 2 n = List.map (fun b => bif b then 1 else 0) n.bits"], "goal": "digits 2 (bit b n) = List.map (fun b => bif b then 1 else 0) (b :: n.bits)"}}
+{"state": {"context": ["K : Type u", "inst✝³ : Field K", "V : Type v", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "this : DecidableEq ↑(Basis.ofVectorSpaceIndex K V) := Classical.decEq ↑(Basis.ofVectorSpaceIndex K V)", "j : ↑(Basis.ofVectorSpaceIndex K V)"], "goal": "(∑ x : ↑(Basis.ofVectorSpaceIndex K V), if x = j then 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord x else 0) = 1 ⊗ₜ[K] (Basis.ofVectorSpace K V).coord j"}}
+{"state": {"context": ["α : Type u_1", "n : Nat", "f : Fin n → α"], "goal": "ofFn f = ofFnTR f"}}
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+{"state": {"context": ["α : Type u", "a : α", "s t : Set α", "h : s ⊆ t"], "goal": "insert a s ⊆ insert a t"}}
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+{"state": {"context": [], "goal": "Function.Antiperiodic Real.Angle.sign ↑π"}}
+{"state": {"context": ["p q : ℚ≥0", "hn : p.num = q.num", "hd : p.den = q.den"], "goal": "p = q"}}
+{"state": {"context": ["p : ℝ", "hp₀ : 0 < p", "hp₁ : p < 1", "hp₀' : 0 < 1 / p", "hp₁' : 1 < 1 / p", "f : ℝ≥0 ≃o ℝ≥0 := orderIsoRpow (1 / p) hp₀'", "h₁ : StrictConvexOn ℝ≥0 univ ⇑f"], "goal": "StrictConcaveOn ℝ≥0 univ fun x => x ^ p"}}
+{"state": {"context": ["V : Type u_1", "inst✝² : Fintype V", "inst✝¹ : DecidableEq V", "G : SimpleGraph V", "inst✝ : DecidableRel G.Adj", "n : ℕ", "H : SimpleGraph V", "K : Finset V", "left✝ : K ⊆ univ", "hr : 0 < K.card", "hn : K.card ≤ Fintype.card V", "hG : G.IsTuranMaximal K.card", "h : G.CliqueFree K.card"], "goal": "False"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "PseudoEMetricSpace X", "PseudoEMetricSpace Y", "PseudoEMetricSpace Z", "Cg rg : ℝ≥0", "g : Y → Z", "hg : HolderWith Cg rg g", "Cf rf : ℝ≥0", "f : X → Y", "s : Set X", "hf : HolderOnWith Cf rf f s"], "goal": "HolderOnWith (Cg * Cf ^ ↑rg) (rg * rf) (g ∘ f) s"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "f : ℝ → E", "s : ℂ", "a : ℝ", "ha : 0 < a", "this : IntegrableOn (fun x => (fun t => ↑t ^ (s - 1) • f t) (a * x)) (Ioi 0) volume ↔ IntegrableOn (fun t => ↑t ^ (s - 1) • f t) (Ioi (a * 0)) volume"], "goal": "MellinConvergent (fun t => f (a * t)) s ↔ MellinConvergent f s"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "E : ι → Type u_3", "F : Type u_4", "NormedField 𝕜", "(i : ι) → TopologicalSpace (E i)", "(i : ι) → AddCommGroup (E i)", "(i : ι) → Module 𝕜 (E i)", "AddCommGroup F", "Module 𝕜 F", "UniformSpace F", "UniformAddGroup F", "∀ (i : ι), ContinuousSMul 𝕜 (E i)", "ContinuousConstSMul 𝕜 F", "CompleteSpace F", "T2Space F", "h : RestrictGenTopology {s | Bornology.IsVonNBounded 𝕜 s}"], "goal": "CompleteSpace (ContinuousMultilinearMap 𝕜 E F)"}}
+{"state": {"context": ["V : Type u_1", "SeminormedAddCommGroup V", "h : ∃ x, ‖x‖ ≠ 0"], "goal": "‖NormedAddGroupHom.id V‖ = 1"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "l : List α", "a : α"], "goal": "List.filter (fun x => a == x) l = List.replicate (List.count a l) a"}}
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+{"state": {"context": ["a b : ℝ", "h : a < b"], "goal": "Subalgebra.comap (ContinuousMap.compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm.toContinuousMap)\n    (polynomialFunctions unitInterval) =\n  polynomialFunctions (Set.Icc a b)"}}
+{"state": {"context": ["R : Type u₁", "CommSemiring R", "p : ℕ", "hp : Fact (Nat.Prime p)", "CharP R p", "f : Ring.Perfection R p", "m n : ℕ", "hfm : (Perfection.coeff R p m) f ≠ 0", "hmn : m ≤ n"], "goal": "(Perfection.coeff R p n) f ≠ 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "l : List α"], "goal": "∀ (x : α) (y : Array (List α)), ?g₂ x y.toList = (Array.foldl (fun r l => r.push (x :: l)) y y 0).toList"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "M : Type u_4", "N : Type u_5", "inst✝¹ : One M", "inst✝ : One N", "s t : Set α", "f g : α → M", "a : α"], "goal": "(s.mulIndicator f = fun x => 1) ↔ Disjoint (mulSupport f) s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type u_2", "Pow β γ", "f g : α → β", "l : Filter α", "h : f =ᶠ[l] g", "c : γ"], "goal": "(fun x => f x ^ c) =ᶠ[l] fun x => g x ^ c"}}
+{"state": {"context": ["x : Bool"], "goal": "(xor x !x) = true"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "Group G", "hG : IsPGroup p G", "H : Type u_2", "Group H", "ϕ : G ≃* H"], "goal": "IsPGroup p H"}}
+{"state": {"context": ["k : Type u_1", "inst✝ : Field k", "σ : Type u_2", "I : Ideal (MvPolynomial σ k)", "p : MvPolynomial σ k", "hp : p ∈ I.radical", "x : σ → k", "hx : x ∈ zeroLocus I"], "goal": "p ∈ vanishingIdeal {x}"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝¹ : Semiring R", "p✝ q r : R[X]", "inst✝ : AddCommMonoid S", "f : ℕ → R → S", "hf : ∀ (i : ℕ), f i 0 = 0", "n : ℕ", "p : R[X]", "hn : p.degree < ↑n", "hp : ¬p = 0"], "goal": "∑ i : Fin n, f (↑i) (p.coeff ↑i) = p.sum f"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "f : α → β → β", "b : β", "a : α", "l : List α"], "goal": "scanr f b (a :: l) = foldr f b (a :: l) :: scanr f b l"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "SeminormedAddCommGroup β", "f : BoundedContinuousFunction α β", "IsEmpty α"], "goal": "‖f‖ = 0"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "hs : s.Finite", "h : s ⊆ t", "x : α", "hx : t \\ s = {x}"], "goal": "∃ a, t = insert a s"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "q : ℚ"], "goal": "‖↑q‖ = ↑(padicNorm p q)"}}
+{"state": {"context": ["r r₁ r₂ : ℝ≥0", "x y : ℝ", "α : Type u_1", "s : Finset α", "f : α → ℝ", "hf : ∀ a ∈ s, 0 ≤ f a"], "goal": "∑ i ∈ s, f i = ∑ a ∈ s, ↑(f a).toNNReal"}}
+{"state": {"context": ["ι : Type u_1", "I J : Box ι", "x y l₁ u₁ : ι → ℝ", "h₁ : ∀ (i : ι), l₁ i < u₁ i", "l₂ u₂ : ι → ℝ", "h₂ : ∀ (i : ι), l₂ i < u₂ i", "h : (l₂ ≤ l₁ ∧ u₁ ≤ u₂) ∧ l₁ ≤ l₂ ∧ u₂ ≤ u₁"], "goal": "u₁ = u₂"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "f : α → E", "One F", "NormOneClass F", "TopologicalSpace α", "a : α"], "goal": "(f =O[𝓝[≠] a] fun x => 1) ↔ f =O[𝓝 a] fun x => 1"}}
+{"state": {"context": ["μ : ℝ", "v : ℝ≥0", "c : ℝ", "hc : c ≠ 0", "x : ℝ"], "goal": "|c⁻¹| * gaussianPDFReal (c⁻¹ * μ) (⟨c⁻¹ ^ 2, ⋯⟩ * v) x = |c⁻¹| * gaussianPDFReal (c⁻¹ * μ) (⟨(c ^ 2)⁻¹, ⋯⟩ * v) x"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "inst✝⁶ : Category.{v₁, u₁} C", "inst✝⁵ : Category.{v₁, u₂} D", "i : D ⥤ C", "inst✝⁴ : HasFiniteProducts C", "inst✝³ : Reflective i", "inst✝² : CartesianClosed C", "inst✝¹ : HasFiniteProducts D", "inst✝ : ExponentialIdeal i", "A B : C"], "goal": "((reflectorAdjunction i).homEquiv (A ⨯ B) ((reflector i).obj A ⨯ (reflector i).obj B)).symm (prod.lift prod.snd prod.fst ≫ ((exp.adjunction B).homEquiv A (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))).symm ((unitCompPartialBijective A ⋯).symm (((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) (prod.lift prod.snd prod.fst ≫ ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))).symm ((unitCompPartialBijective B ⋯).symm (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) ((PreservesLimitPair.iso i ((reflector i).obj A) ((reflector i).obj B)).inv ≫ i.map (𝟙 ((reflector i).obj A ⨯ (reflector i).obj B)) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))))) ≫ 𝟙 (i.obj ((reflector i).obj A ⨯ (reflector i).obj B))) = prodComparison (reflector i) A B"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "(i : ι) → Preorder (α i)", "x : (i : ι) → α i", "Nonempty ι"], "goal": "(Set.univ.pi fun i => Set.Iio (x i)) ⊆ Set.Iio x"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R"], "goal": "I ∈ Ideal.isPrincipalSubmonoid R ↔ Submodule.IsPrincipal I"}}
+{"state": {"context": ["s : Set ℝ", "a : ℝ"], "goal": "1 ∈ posTangentConeAt s a ↔ a ∈ closure (Set.Ioi a ∩ s)"}}
+{"state": {"context": ["α : Type u_2", "s : Finset (Set α)"], "goal": "s.inf id = ⋂₀ ↑s"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y"], "goal": "Equivalence ContinuousMap.Homotopic"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "R : α → α → Prop", "DecidableRel R"], "goal": "List.destutter R (List.destutter R l) = List.destutter R l"}}
+{"state": {"context": ["α : Type u", "n : ℕ", "s : Stream' α"], "goal": "s.get n.succ = s.tail.get n"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "s t : Set Y", "h : SeparatedNhds s t", "hf : Continuous f"], "goal": "SeparatedNhds (f ⁻¹' s) (f ⁻¹' t)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R", "M : Type u_2", "AddCommGroup M", "Module R M", "r : AdicCompletion I R", "x : M"], "goal": "((AdicCompletion.ofTensorProductBil I M) r) x = r • (AdicCompletion.of I M) x"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{?u.39244, u_1} C", "inst✝³ : Category.{?u.39248, u_2} D", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "inst✝ : HasHomotopyCofiber φ", "K : CochainComplex C ℤ", "d e : ℤ", "γ : Cochain F K d", "he : 1 + d = e"], "goal": "-1 + e = d"}}
+{"state": {"context": ["G : Type u_1", "Group G", "ι : Type u_2", "hfin : Fintype ι", "H : ι → Type u_3", "(i : ι) → Group (H i)", "ϕ : (i : ι) → H i →* G", "hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)", "hind : CompleteLattice.Independent fun i => (ϕ i).range", "hinj : ∀ (i : ι), Function.Injective ⇑(ϕ i)"], "goal": "Function.Injective ⇑(MonoidHom.noncommPiCoprod ϕ hcomm)"}}
+{"state": {"context": ["ι : Type uι", "inst✝⁵ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝¹ : SeminormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p q : FreeAddMonoid (𝕜 × ((i : ι) → E i))"], "goal": "(List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) (FreeAddMonoid.toList p)).sum + (List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) (FreeAddMonoid.toList q)).sum ≤ (List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) p).sum + (List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) q).sum"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "∀ (a b : C), CategoryTheory.Limits.HasCoproductsOfShape (a ⟶ b) D", "c : C", "Y : CategoryTheory.Functor C D", "e : C"], "goal": "((CategoryTheory.evaluationAdjunctionRight D c).counit.app Y).app e = CategoryTheory.Limits.Sigma.desc fun h => Y.map h"}}
+{"state": {"context": ["X : Type u", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : NormalSpace X", "s : Set X", "hs : IsClosed s", "𝕜 : Type v", "inst✝⁴ : RCLike 𝕜", "inst✝³ : TietzeExtension 𝕜", "E : Type w", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "f : ↑s →ᵇ E", "hf : ¬‖f‖ = 0"], "goal": "∃ g, ‖g‖ = ‖f‖ ∧ g.restrict s = f"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "I : Type u_3", "e : I → R", "DecidableEq I"], "goal": "OrthogonalIdempotents e ↔ ∀ (i j : I), e i * e j = if i = j then e i else 0"}}
+{"state": {"context": ["R : Type u_1", "CommMonoid R", "R' : Type u_2", "CommMonoidWithZero R'", "χ : MulChar R R'", "n : ℕ", "hn : n ≠ 0", "a : R"], "goal": "(χ ^ n) a = χ a ^ n"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f✝ g : Perm α", "x y : α", "p : β → Prop", "inst✝ : DecidablePred p", "f : α ≃ Subtype p", "a : α", "ha : g a ≠ a", "ha' : ∀ ⦃y : α⦄, g y ≠ y → g.SameCycle a y", "b : β", "hb : (g.extendDomain f) b ≠ b"], "goal": "(g.extendDomain f).SameCycle (↑(f a)) b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁴ : TopologicalSpace α", "inst✝³ : LinearOrder α", "inst✝² : OrderTopology α", "inst✝¹ : NoMinOrder α", "inst✝ : DenselyOrdered α", "a : α", "s : Set α"], "goal": "(∃ u, toDual a < toDual u ∧ Icc (toDual a) (toDual u) ⊆ ⇑ofDual ⁻¹' s) ↔ ∃ l < a, Icc l a ⊆ s"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ : R", "a b : ℍ[R,c₁,c₂]"], "goal": "(a + b).re = a.re + b.re"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f : CategoryTheory.Arrow (CategoryTheory.Arrow C)"], "goal": "(CategoryTheory.Square.fromArrowArrowFunctor.obj f).X₁ = f.left.left"}}
+{"state": {"context": ["α : Type u_1", "p : Multiset α → Sort u_3", "n : ℕ", "H : (t₁ : Multiset α) → ({t₂ : Multiset α} → Multiset.card t₂ ≤ n → t₁ < t₂ → p t₂) → Multiset.card t₁ ≤ n → p t₁", "s : Multiset α"], "goal": "Multiset.strongDownwardInduction H s = H s fun {t₂} ht _hst => Multiset.strongDownwardInduction H t₂ ht"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "f : α → E", "hf : MeasureTheory.Memℒp f p μ", "hp_ne_zero : p ≠ 0", "hp_ne_top : p ≠ ⊤"], "goal": "MeasureTheory.Memℒp (fun x => ‖f x‖ ^ p.toReal) 1 μ"}}
+{"state": {"context": ["α : Type u_1", "l l' : List α", "a b : α"], "goal": "l.concat a = l'.concat b ↔ l = l' ∧ a = b"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "H : IsOpenImmersion f", "U : Opens ↑↑Y", "hU : ↑U ⊆ Set.range ⇑f.base"], "goal": "(f.c.app (op U) ≫ X.presheaf.map (eqToHom ⋯)) ≫ inv (f.c.app (op ((opensFunctor f).obj ((Opens.map f.base).obj U)))) = Y.presheaf.map (eqToHom ⋯).op"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Ring R", "p q : R[X]", "h : q.degree ≤ p.degree ∧ p ≠ 0", "hq : q.Monic", "hp : p.leadingCoeff ≠ 0", "hq0 : q ≠ 0"], "goal": "q.degree ≤ p.degree"}}
+{"state": {"context": ["M : Type u_8", "M₂ : Type u_10", "AddCommMonoid M", "AddCommMonoid M₂", "e : M ≃+ M₂"], "goal": "⇑e.toNatLinearEquiv = ⇑e"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "k : ℤ_[p]"], "goal": "⟨↑k, ⋯⟩ = k"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a b : α"], "goal": "a ∈ Finset.Icc a b ↔ a ≤ b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "I✝ : Ideal R", "h : ∀ (P : Ideal R), P.IsPrime → P.jacobson = P", "I : Ideal R", "hI : I.IsRadical", "x : R", "hx : ∀ ⦃I_1 : Ideal R⦄, I_1 ∈ {J | I ≤ J ∧ J.IsMaximal} → x ∈ I_1", "P : Ideal R", "hP : I ≤ P ∧ P.IsPrime"], "goal": "∀ ⦃I : Ideal R⦄, I ∈ {J | P ≤ J ∧ J.IsMaximal} → x ∈ I"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "f : PadicSeq p", "hf : ¬f ≈ 0", "v1 v2 : ℕ"], "goal": "stationaryPoint hf ≤ stationaryPoint hf"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "NontriviallyNormedField 𝕜₁", "𝕜₂ : Type u_2", "NontriviallyNormedField 𝕜₂", "σ : 𝕜₁ →+* 𝕜₂", "iσ : RingHomIsometric σ", "B : Type u_3", "F₁ : Type u_4", "NormedAddCommGroup F₁", "NormedSpace 𝕜₁ F₁", "E₁ : B → Type u_5", "(x : B) → AddCommGroup (E₁ x)", "(x : B) → Module 𝕜₁ (E₁ x)", "TopologicalSpace (Bundle.TotalSpace F₁ E₁)", "F₂ : Type u_6", "NormedAddCommGroup F₂", "NormedSpace 𝕜₂ F₂", "E₂ : B → Type u_7", "(x : B) → AddCommGroup (E₂ x)", "(x : B) → Module 𝕜₂ (E₂ x)", "TopologicalSpace (Bundle.TotalSpace F₂ E₂)", "TopologicalSpace B", "(x : B) → TopologicalSpace (E₁ x)", "FiberBundle F₁ E₁", "VectorBundle 𝕜₁ F₁ E₁", "(x : B) → TopologicalSpace (E₂ x)", "FiberBundle F₂ E₂", "VectorBundle 𝕜₂ F₂ E₂", "∀ (x : B), TopologicalAddGroup (E₂ x)", "∀ (x : B), ContinuousSMul 𝕜₂ (E₂ x)", "x₀ : B"], "goal": "(trivializationAt (F₁ →SL[σ] F₂) (Bundle.ContinuousLinearMap σ E₁ E₂) x₀).source =\n  Bundle.TotalSpace.proj ⁻¹' ((trivializationAt F₁ E₁ x₀).baseSet ∩ (trivializationAt F₂ E₂ x₀).baseSet)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : LinearOrderedSemifield α", "l : Filter β", "f : β → α", "r c : α", "n : ℕ", "h : Tendsto (fun x => c * x ^ n) atTop atTop"], "goal": "n ≠ 0"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "M : Mon_ C"], "goal": "M.X ◁ M.mul ≫ M.mul = (α_ M.X M.X M.X).inv ≫ M.mul ▷ M.X ≫ M.mul"}}
+{"state": {"context": ["M : Type u_1", "CancelCommMonoidWithZero M", "UniqueFactorizationMonoid M", "q : Associates M", "n : ℕ", "hn : n ≠ 0", "c : Fin (n + 1) → Associates M", "h₁ : StrictMono c", "h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i", "hq : q ≠ 0"], "goal": "q = c 1 ^ n"}}
+{"state": {"context": ["b : ℂ", "hb : b.re < 0", "c d : ℂ", "hb' : b ≠ 0"], "goal": "Integrable (fun x => cexp (b * ↑x ^ 2 + c * ↑x + d)) volume"}}
+{"state": {"context": ["α : Type u_1", "Subsingleton α", "s : Set α"], "goal": "s.ncard ≤ 1"}}
+{"state": {"context": ["M : Type u_2", "AddCommGroup M", "𝕜 : Type u_7", "Field 𝕜", "Module 𝕜 M", "f : M →ₗ[𝕜] M", "hf : LinearMap.det f ≠ 1"], "goal": "FiniteDimensional 𝕜 M"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p : R[X]", "hp : p.Monic", "hu : ¬IsUnit p"], "goal": "0 < p.degree"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Abelian C", "inst✝¹ : HasProjectiveResolutions C", "inst✝ : EnoughProjectives C", "Z : C", "n : ℕ"], "goal": "(ShortComplex.mk ((ChainComplex.of (fun n => (ChainComplex.mkAux (over Z) (syzygies (Projective.π Z)) (syzygies (d (Projective.π Z))) (d (Projective.π Z)) (d (d (Projective.π Z))) ⋯ (fun S => ⟨syzygies S.f, ⟨d S.f, ⋯⟩⟩) n).X₃) (fun n => (ChainComplex.mkAux (over Z) (syzygies (Projective.π Z)) (syzygies (d (Projective.π Z))) (d (Projective.π Z)) (d (d (Projective.π Z))) ⋯ (fun S => ⟨syzygies S.f, ⟨d S.f, ⋯⟩⟩) n).g) ⋯).d (n + 1 + 1) (n + 1)) ((ChainComplex.of (fun n => (ChainComplex.mkAux (over Z) (syzygies (Projective.π Z)) (syzygies (d (Projective.π Z))) (d (Projective.π Z)) (d (d (Projective.π Z))) ⋯ (fun S => ⟨syzygies S.f, ⟨d S.f, ⋯⟩⟩) n).X₃) (fun n => (ChainComplex.mkAux (over Z) (syzygies (Projective.π Z)) (syzygies (d (Projective.π Z))) (d (Projective.π Z)) (d (d (Projective.π Z))) ⋯ (fun S => ⟨syzygies S.f, ⟨d S.f, ⋯⟩⟩) n).g) ⋯).d (n + 1) n) ⋯).Exact"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝¹ : CancelCommMonoid α", "inst✝ : CancelCommMonoid β", "A : Set α", "B : Set β", "f : α → β", "n : ℕ", "hf : IsMulFreimanIso n A B f", "s t : Multiset α", "hsA : ∀ ⦃x : α⦄, x ∈ s → x ∈ A", "htA : ∀ ⦃x : α⦄, x ∈ t → x ∈ A", "hmn : 0 ≤ n", "hs : card s = 0", "ht : card t = 0"], "goal": "(map f s).prod = (map f t).prod ↔ s.prod = t.prod"}}
+{"state": {"context": ["G : Type u_3", "AddCommGroup G", "a b c : G", "h : c + a = b"], "goal": "a = b - c"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹² : CommRing R", "I : Ideal R", "M✝ : Type u_2", "inst✝¹¹ : AddCommGroup M✝", "inst✝¹⁰ : Module R M✝", "N : Type u_3", "inst✝⁹ : AddCommGroup N", "inst✝⁸ : Module R N", "P : Type u_4", "inst✝⁷ : AddCommGroup P", "inst✝⁶ : Module R P", "T : Type u_5", "inst✝⁵ : AddCommGroup T", "inst✝⁴ : Module (AdicCompletion I R) T", "ι : Type u_6", "inst✝³ : DecidableEq ι", "M : ι → Type u_7", "inst✝² : (i : ι) → AddCommGroup (M i)", "inst✝¹ : (i : ι) → Module R (M i)", "inst✝ : Fintype ι", "f : AdicCauchySequence I (⨁ (j : ι), M j)", "n : ℕ"], "goal": "∑ x : ι, Submodule.Quotient.mk ((lof R ι M x) ((component R ι M x) (↑f n))) = Submodule.Quotient.mk (↑f n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l₁ : List α", "l₂ : List β", "a : α", "b : β"], "goal": "(a, b) ∈ l₁ ×ˢ l₂ ↔ a ∈ l₁ ∧ b ∈ l₂"}}
+{"state": {"context": ["xl xr : Type u_1", "xL : xl → SetTheory.PGame", "xR : xr → SetTheory.PGame", "i : xl"], "goal": "(xL i).LF (SetTheory.PGame.mk xl xr xL xR)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝² : Category.{?u.1617, u_1} C", "inst✝¹ : Category.{?u.1621, u_2} D", "inst✝ : Category.{?u.1625, u_3} E", "F : C ⥤ Karoubi D", "X✝ Y✝ : Karoubi C", "f : X✝ ⟶ Y✝"], "goal": "(F.map f.f).f = ((fun P => { X := (F.obj P.X).X, p := (F.map P.p).f, idem := ⋯ }) X✝).p ≫ (F.map f.f).f ≫ ((fun P => { X := (F.obj P.X).X, p := (F.map P.p).f, idem := ⋯ }) Y✝).p"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "hH : Finite (G ⧸ H)", "val✝ : Fintype (G ⧸ H)"], "goal": "H.index ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "x y z : α", "H : IsRelPrime x (y * z)"], "goal": "IsRelPrime x z"}}
+{"state": {"context": ["b c d : ℝ≥0∞", "r p q a : ℝ≥0", "h0 : ↑a ≠ 0"], "goal": "StrictMono fun x => ↑a * x"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{u₂, u₁} C", "inst✝¹ : PreGaloisCategory C", "F : C ⥤ FintypeCat", "inst✝ : FiberFunctor F", "X : C", "h : IsInitial X → False", "hzero : Nat.card ↑(F.obj X) = 0"], "goal": "Nonempty (IsInitial X)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹⁵ : Category.{u_4, u_1} C", "inst✝¹⁴ : Category.{u_3, u_2} D", "L : C ⥤ D", "inst✝¹³ : HasShift C ℤ", "inst✝¹² : Preadditive C", "inst✝¹¹ : HasZeroObject C", "inst✝¹⁰ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "inst✝⁹ : Pretriangulated C", "inst✝⁸ : HasShift D ℤ", "inst✝⁷ : L.CommShift ℤ", "W : MorphismProperty C", "inst✝⁶ : L.IsLocalization W", "inst✝⁵ : W.IsCompatibleWithTriangulation", "inst✝⁴ : W.HasLeftCalculusOfFractions", "inst✝³ : Preadditive D", "inst✝² : HasZeroObject D", "inst✝¹ : ∀ (n : ℤ), (shiftFunctor D n).Additive", "inst✝ : L.Additive", "T₁ T₂ : Triangle C", "hT₁ : T₁ ∈ distinguishedTriangles", "hT₂ : T₂ ∈ distinguishedTriangles", "a : L.obj T₁.obj₁ ⟶ L.obj T₂.obj₁", "b : L.obj T₁.obj₂ ⟶ L.obj T₂.obj₂", "fac✝ : L.map T₁.mor₁ ≫ b = a ≫ L.map T₂.mor₁", "α : W.LeftFraction T₁.obj₁ T₂.obj₁", "hα : a = α.map L ⋯", "β : W.LeftFraction α.Y' T₂.obj₂", "hβ : T₂.mor₁ ≫ β.s = α.s ≫ β.f", "γ : W.LeftFraction T₁.obj₂ β.Y'", "hγ : b ≫ L.map β.s = γ.map L ⋯", "this✝ : IsIso (L.map β.s)", "this : IsIso (L.map γ.s)", "Z₂ : C", "σ : γ.Y' ⟶ Z₂", "hσ : W σ", "fac : (α.f ≫ β.f ≫ γ.s) ≫ σ = (T₁.mor₁ ≫ γ.f) ≫ σ"], "goal": "∃ φ, φ.hom₁ = a ∧ φ.hom₂ = b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f✝ : α → β", "E✝ I✝ s : Set α", "M : Matroid α", "N✝ : Matroid β", "E B I : Set α", "N : Matroid β", "f : α → β"], "goal": "N.comapOn (f ⁻¹' N.E) f = N.comap f"}}
+{"state": {"context": ["α : Type u", "x : α", "BooleanAlgebra α"], "goal": "x ⊔ xᶜ = ⊤"}}
+{"state": {"context": ["ι : Type u_2", "α✝ : ι → Type ?u.29822", "α : ι → Type u_1", "inst✝ : (i : ι) → MeasurableSpace (α i)"], "goal": "∃ i i_1, ∃ (_ : MeasurableSet i_1), ∅ = cylinder i i_1"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "γ : Sort u_3", "f : α → β → γ", "hf : Function.Injective2 f", "a : α"], "goal": "Function.Injective (f a)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : α → β → Prop", "ca : Computation α", "cb : Computation β"], "goal": "Computation.LiftRel R ca cb.think ↔ Computation.LiftRel R ca cb"}}
+{"state": {"context": ["R S Rₘ Sₘ : Type u", "inst✝¹³ : CommRing R", "inst✝¹² : CommRing S", "inst✝¹¹ : CommRing Rₘ", "inst✝¹⁰ : CommRing Sₘ", "M : Submonoid R", "inst✝⁹ : Algebra R S", "inst✝⁸ : Algebra R Sₘ", "inst✝⁷ : Algebra S Sₘ", "inst✝⁶ : Algebra R Rₘ", "inst✝⁵ : Algebra Rₘ Sₘ", "inst✝⁴ : IsScalarTower R Rₘ Sₘ", "inst✝³ : IsScalarTower R S Sₘ", "inst✝² : IsLocalization M Rₘ", "inst✝¹ : IsLocalization (Submonoid.map (algebraMap R S) M) Sₘ", "inst✝ : FormallySmooth R S", "this : FormallySmooth S Sₘ"], "goal": "FormallySmooth Rₘ Sₘ"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : AddCommGroup E", "inst✝⁶ : Module ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "mE : MeasurableSpace E", "tE : TopologicalSpace E", "inst✝⁴ : TopologicalAddGroup E", "inst✝³ : BorelSpace E", "inst✝² : T2Space E", "inst✝¹ : ContinuousSMul ℝ E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "g : E → ℝ", "h1 : g 0 = 0", "h2 : ∀ (x : E), g (-x) = g x", "h3 : ∀ (x y : E), g (x + y) ≤ g x + g y", "h4 : ∀ {x : E}, g x = 0 → x = 0", "h5 : ∀ (r : ℝ) (x : E), g (r • x) ≤ |r| * g x", "p : ℝ", "hp : 0 < p", "F : Type u_1 := E", "this✝³ : NormedAddCommGroup F := NormedAddCommGroup.mk ⋯", "this✝² : NormedSpace ℝ F := NormedSpace.mk ⋯", "this✝¹ : TopologicalSpace F := UniformSpace.toTopologicalSpace", "this✝ : MeasurableSpace F := borel F", "this : BorelSpace F", "φ : E ≃L[ℝ] F := (LinearEquiv.refl ℝ E).toContinuousLinearEquiv", "ν : Measure F := map (⇑φ) μ"], "goal": "μ {x | g x < 1} = ENNReal.ofReal ((∫ (x : E), Real.exp (-g x ^ p) ∂μ) / Real.Gamma (↑(finrank ℝ E) / p + 1))"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a b c d : α", "h : a ≤ b", "δ : α", "hδ : 0 ≤ δ", "t : ↑(Icc a b)", "n : ℕ", "ht : t ∈ Icc (addNSMul h δ n) (addNSMul h δ (n + 1))"], "goal": "|a + (n + 1) • δ - (a + n • δ)| ≤ δ"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "S : SnakeInput C", "A₀ : C", "x₃ : A₀ ⟶ S.L₀.X₃", "hx₃ : x₃ ≫ S.δ = 0", "A₁ : C", "π₁ : A₁ ⟶ A₀", "hπ₁ : Epi π₁", "p : A₁ ⟶ S.P", "hp : π₁ ≫ x₃ = p ≫ pullback.snd S.L₁.g S.v₀₁.τ₃"], "goal": "∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ x₃ = x₁ ≫ S.L₁'.f"}}
+{"state": {"context": ["G : Type w", "inst✝¹ : TopologicalSpace G", "μ : Content G", "inst✝ : R1Space G", "U : ℕ → Opens G", "h3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), (fun s => ↑(μ.toFun s)) (t.sup K) ≤ ∑ i ∈ t, (fun s => ↑(μ.toFun s)) (K i)", "K : Compacts G", "hK : ↑K ⊆ ↑(⨆ i, U i)", "t : Finset ℕ", "ht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)", "K' : ℕ → Set G", "h1K' : ∀ (i : ℕ), IsCompact (K' i)", "h2K' : ∀ (i : ℕ), K' i ⊆ (SetLike.coe ∘ U) i", "h3K' : ↑K = ⋃ i ∈ t, K' i", "L : ℕ → Compacts G := fun n => { carrier := K' n, isCompact' := ⋯ }"], "goal": "(fun s => ↑(μ.toFun s)) K ≤ ∑' (i : ℕ), μ.innerContent (U i)"}}
+{"state": {"context": ["z : ℂ", "E : Type u_1", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "𝕜 : Type u_2", "inst✝ : RCLike 𝕜", "h : RCLike.im RCLike.I = 1", "x✝ : 𝕜"], "goal": "‖{ toFun := __spread✝⁻⁰.toFun, map_add' := ⋯, map_smul' := ⋯, invFun := __spread✝⁻⁰.invFun, left_inv := ⋯, right_inv := ⋯ } x✝‖ = ‖x✝‖"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedAddCommGroup G", "ι : Type u_5", "fi : Filter ι", "inst✝ : Fact (1 ≤ p)", "f : ι → ↥(Lp E p μ)", "f_lim : α → E", "f_lim_ℒp : Memℒp f_lim p μ"], "goal": "Tendsto f fi (𝓝 (Memℒp.toLp f_lim f_lim_ℒp)) ↔ Tendsto (fun n => eLpNorm (↑↑(f n) - f_lim) p μ) fi (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : CancelCommMonoidWithZero α", "m : Associates α", "p : α", "hp : Prime (Associates.mk p)", "hle : m ≤ Associates.mk p"], "goal": "m = 1 ∨ m = Associates.mk p"}}
+{"state": {"context": ["α : Type u_5", "β : Type u_6", "Monoid α", "Mul β", "MulAction α β", "IsScalarTower α β β", "SMulCommClass α β β", "a : α × β"], "goal": "smulMulHom a = a.1 • a.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : α → ℝ≥0∞", "h : ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0", "φ : α →ₛ ℝ≥0", "hle : ∀ (x : α), ↑(↑φ x) ≤ f x", "b : ℝ≥0∞", "hbφ : b < (SimpleFunc.map ofNNReal φ).lintegral μ + ε", "hb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → (SimpleFunc.map ofNNReal i).lintegral μ ≤ b", "ψ : α →ₛ ℝ≥0", "hψ : ∀ (x : α), ↑(↑ψ x) ≤ f x"], "goal": "(SimpleFunc.map ofNNReal (ψ - φ)).lintegral μ < ε"}}
+{"state": {"context": ["M : Type u_1", "Add M", "a b : Mᵈᵃᵃ"], "goal": "DomAddAct.mk.symm (a + b) = DomAddAct.mk.symm b + DomAddAct.mk.symm a"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "StarRing R", "NonUnitalSemiring A", "StarRing A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "StarModule R A", "S : Set (NonUnitalStarSubalgebra R A)"], "goal": "↑(sInf S) = ⋂ s ∈ S, ↑s"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "g : β → α"], "goal": "Set.BijOn g (Function.fixedPoints (f ∘ g)) (Function.fixedPoints (g ∘ f))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "ι : Type u_5", "l : Filter ι", "inst✝ : l.IsCountablyGenerated", "F : ι → α → ℝ≥0∞", "f bound : α → ℝ≥0∞", "hF_meas : ∀ᶠ (n : ι) in l, Measurable (F n)", "h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a", "h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤", "h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) l (𝓝 (f a))"], "goal": "∀ (x : ℕ → ι), Tendsto x atTop l → Tendsto ((fun n => ∫⁻ (a : α), F n a ∂μ) ∘ x) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "R₂ : Type u_3", "R₃ : Type u_4", "M : Type u_5", "M₁ : Type u_6", "M₂ : Type u_7", "M₃ : Type u_8", "ι : Type u_9", "inst✝¹¹ : Semiring R", "inst✝¹⁰ : Semiring R₂", "inst✝⁹ : Semiring R₃", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : AddCommMonoid M₁", "inst✝⁶ : AddCommMonoid M₂", "inst✝⁵ : AddCommMonoid M₃", "inst✝⁴ : Module R M", "inst✝³ : Module R M₁", "inst✝² : Module R₂ M₂", "inst✝¹ : Module R₃ M₃", "σ₁₂ : R →+* R₂", "σ₂₃ : R₂ →+* R₃", "σ₁₃ : R →+* R₃", "inst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃", "f✝ : M →ₛₗ[σ₁₂] M₂", "g✝ : M₂ →ₛₗ[σ₂₃] M₃", "f g : M →ₗ[R] M", "h : Commute f g", "p : Submodule R M", "hf : MapsTo ⇑f ↑p ↑p", "hg : MapsTo ⇑g ↑p ↑p"], "goal": "f.restrict hf * g.restrict hg = g.restrict hg * f.restrict hf"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "s : Set α"], "goal": "upperBounds (l '' s) = u ⁻¹' upperBounds s"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_3", "E : Type u_5", "AddGroup G", "AddAction G α", "MeasurableSpace α", "NormedAddCommGroup E", "s t : Set α", "μ : MeasureTheory.Measure α", "MeasurableSpace G", "MeasurableVAdd G α", "MeasureTheory.VAddInvariantMeasure G α μ", "Countable G", "hs : MeasureTheory.IsAddFundamentalDomain G s μ", "ht : MeasureTheory.IsAddFundamentalDomain G t μ", "f : α → E", "hf : ∀ (g : G) (x : α), f (g +ᵥ x) = f x"], "goal": "MeasureTheory.IntegrableOn f s μ ↔ MeasureTheory.IntegrableOn f t μ"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : Field K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : NoZeroSMulDivisors R M", "f : End R M", "μ₁ μ₂ : R", "hμ : μ₁ ≠ μ₂", "k l : ℕ", "a✝ : Nontrivial M", "this : IsReduced R"], "goal": "Disjoint ((f.genEigenspace μ₁) k) ((f.genEigenspace μ₂) l)"}}
+{"state": {"context": ["F : Type v", "Field F", "W : WeierstrassCurve.Jacobian F", "P Q : Fin 3 → F", "hPz : P z ≠ 0", "hQz : Q z ≠ 0"], "goal": "W.negAddY P Q =\n  ((P y * Q z ^ 3 - Q y * P z ^ 3) *\n        (W.addX P Q * (P z * Q z) ^ 2 - P x * Q z ^ 2 * WeierstrassCurve.Jacobian.addZ P Q ^ 2) +\n      P y * Q z ^ 3 * WeierstrassCurve.Jacobian.addZ P Q ^ 3) /\n    (P z * Q z) ^ 3"}}
+{"state": {"context": ["R : Type u", "CommRing R", "n : Type v", "DecidableEq n", "Fintype n", "N : Type w", "AddCommGroup N", "Module R N", "b : Basis n R N", "M : Matrix n n R"], "goal": "minpoly R ((Matrix.toLin b b) M) = minpoly R M"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M"], "goal": "QuotientMap ⇑DomAddAct.mk.symm"}}
+{"state": {"context": ["n : ℕ", "i j : ZMod (2 * n)"], "goal": "QuaternionGroup.xa i * QuaternionGroup.xa j = QuaternionGroup.a (↑n + j - i)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : IsDomain R", "inst✝¹ : CommRing S", "inst✝ : Algebra R S", "z : S", "inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0", "p : R[X]", "p_ne_zero : p ≠ 0", "px : (aeval z) p = 0", "a : R := p.leadingCoeff"], "goal": "∃ x y, y ≠ 0 ∧ z * (algebraMap R S) y = ↑x"}}
+{"state": {"context": ["���� : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹³ : NormedAddCommGroup E", "inst✝¹² : NormedAddCommGroup E'", "inst✝¹¹ : NormedAddCommGroup E''", "inst✝¹⁰ : NormedAddCommGroup F", "f f' : G → E", "g✝ g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : NormedSpace 𝕜 E'", "inst✝⁶ : NormedSpace 𝕜 E''", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : NormedSpace 𝕜 F", "n : ℕ∞", "inst✝³ : CompleteSpace F", "inst✝² : MeasurableSpace G", "μ ν : Measure G", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : BorelSpace G", "g : G → E'' →L[𝕜] E'", "hf : LocallyIntegrable f μ", "hcg : HasCompactSupport g", "hg : Continuous g", "x₀ : G", "x : E''"], "goal": "((f ⋆[precompR E'' L, μ] g) x₀) x = (f ⋆[L, μ] fun a => (g a) x) x₀"}}
+{"state": {"context": ["b v : Ordinal.{u_1}", "hb : b ≠ 0", "u : Ordinal.{u_1}", "hv : v ≠ 0", "w : Ordinal.{u_1}"], "goal": "0 < b ^ u * v + w"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : DecidableEq α", "inst✝³ : DecidableEq β", "inst✝² : SemilatticeSup α", "inst✝¹ : OrderBot α", "inst✝ : DecidableRel Disjoint", "s s₁ s₂ t t₁ t₂ u : Finset α", "a b c : α"], "goal": "(∃ x, ∃ a ∈ s, ∃ b ∈ t, Disjoint a b ∧ a ⊔ b = x) → ∃ x, x ∈ s"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "hP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f"], "goal": "∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P.universally f → P.universally (f ∣_ U)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "r : ℝ≥0∞", "hf : HasFPowerSeriesOnBall f p x r"], "goal": "AnalyticAt 𝕜 f x"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Field K", "inst✝² : IsDedekindDomain A", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "J : Ideal A", "hJ : J ≠ ⊤", "ι : Type u_4", "s : Finset ι", "f : ι → K", "j : ι", "hjs : j ∈ s", "hjf : f j ≠ 0", "I : FractionalIdeal A⁰ K := spanFinset A s f"], "goal": "∃ a, (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧ ∃ i ∈ s, a * f i ∉ ↑J"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f✝ g : X → E", "s t : Set X", "μ ν : Measure X", "l l' : Filter X", "f : X → ℝ", "hf : 0 ≤ᶠ[ae (μ.restrict s)] f", "hfi : IntegrableOn f s μ"], "goal": "NullMeasurableSet (f ⁻¹' {0}ᶜ) (μ.restrict s)"}}
+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝² : TopologicalSpace α✝", "α : Type u_5", "inst✝¹ : TopologicalSpace α", "β : Type u_6", "inst✝ : TopologicalSpace β", "s u : Set α", "t v : Set β", "a : α", "b : β", "hu : u ∈ 𝓝[s] a", "hv : v ∈ 𝓝[t] b"], "goal": "u ×ˢ v ∈ 𝓝[s] a ×ˢ 𝓝[t] b"}}
+{"state": {"context": ["α β : CompleteLat", "e : ↑α ≃o ↑β", "a : ↑α"], "goal": "(CompleteLat.Iso.mk e).hom.toFun a = e a"}}
+{"state": {"context": ["M : Type u_1", "MulOneClass M", "Pow M ℕ", "NatPowAssoc M", "x : M", "m n : ℕ"], "goal": "x ^ (m * n) = (x ^ m) ^ n"}}
+{"state": {"context": ["x : ℝ", "h : 0 < x", "h' : x ≤ 1"], "goal": "x - x ^ 3 / 4 < Real.sin x"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X Y : Type u", "Ring X", "Ring Y", "Algebra R X", "Algebra R Y", "e : X ≃ₐ[R] Y"], "goal": "algEquivIsoAlgebraIso.hom e = e.toAlgebraIso"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "m n : ℕ", "h : n < m", "f : ℕ → ℚ := fun i => ↑(n.choose i) * (↑(m.choose i))⁻¹", "hf : f = fun i => ↑(n.choose i) * (↑(m.choose i))⁻¹", "i : ℕ", "h₁✝ : i < n + 1", "h₁ : i ≤ n", "h₂ : i < m", "h₃ : i ≤ m", "hi₄ : ↑i + 1 ≠ 0", "this : ↑(m.choose (i + 1) * (i + 1)) = ↑(m.choose i * (m - i))"], "goal": "f i - f (i + 1) = ↑(n.choose i) * (↑m - ↑n) / ((↑m - ↑i) * ↑(m.choose i))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddCommMonoid α", "TopologicalSpace α", "P : Prop", "Decidable P", "x : β → ¬P → α"], "goal": "(∑' (b : β), if h : P then 0 else x b h) = if h : P then 0 else ∑' (b : β), x b h"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "n✝ : ℕ", "x : α", "inst✝ : StrictOrderedRing α", "hx : x + 1 ≤ 0", "n : ℕ"], "goal": "if Even n then ∑ i ∈ range n, x ^ i ≤ 0 else 1 ≤ ∑ i ∈ range n, x ^ i"}}
+{"state": {"context": ["f₁ f₂ : CircleDeg1Lift", "h₁ : IsUnit f₁", "h₂ : IsUnit f₂", "h : f₁.translationNumber = f₂.translationNumber"], "goal": "∃ F, Function.Semiconj ⇑F ⇑f₁ ⇑f₂"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "V₂ : Type u_5", "P : Type u_10", "P₂ : Type u_11", "NormedField 𝕜", "SeminormedAddCommGroup V", "NormedSpace 𝕜 V", "PseudoMetricSpace P", "NormedAddTorsor V P", "SeminormedAddCommGroup V₂", "NormedSpace 𝕜 V₂", "PseudoMetricSpace P₂", "NormedAddTorsor V₂ P₂", "f : P →ᵃⁱ[𝕜] P₂", "s : Set P"], "goal": "EMetric.diam (⇑f '' s) = EMetric.diam s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Zero γ", "f : α → β"], "goal": "Function.extend f 0 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "P : Cubic R", "Semiring R", "ha : P.a = 1"], "goal": "P.toPoly.Monic"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "D : Type u_3", "E : Type u_4", "F : Type u_5", "G : Type u_6", "V : Type u_7", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : SMulCommClass ℝ 𝕜 F", "f : 𝓢(E, F)", "k : ℕ", "x₀ : E"], "goal": "‖x₀‖ ^ k * ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 k 0) f"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : CommSemiring R", "A : Type u", "inst✝⁷ : TopologicalSpace A", "inst✝⁶ : Semiring A", "inst✝⁵ : Algebra R A", "inst✝⁴ : TopologicalSemiring A", "s : Subalgebra R A", "B : Type u_2", "inst✝³ : TopologicalSpace B", "inst✝² : Ring B", "inst✝¹ : TopologicalRing B", "inst✝ : Algebra R B", "f : B →ₐ[R] A", "f' : B ≃ₜ A", "w : ⇑f = ⇑f'"], "goal": "comap f s.topologicalClosure = (comap f s).topologicalClosure"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M₁ M₂ : Bimod X Y", "f : M₁ ⟶ M₂", "N : Bimod Y Z"], "goal": "((α_ M₁.X Y.X N.X).hom ≫ M₁.X ◁ N.actLeft) ≫ f.hom ▷ N.X = f.hom ▷ Y.X ▷ N.X ≫ (α_ M₂.X Y.X N.X).hom ≫ M₂.X ◁ N.actLeft"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "J₁ J₂ : GrothendieckTopology C", "c : (X : C) → ClosureOperator (Sieve X)", "hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)", "X : C", "S : Sieve X", "hS : (c X) S = ⊤", "R : Sieve X", "hR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ (fun X => {S | (c X) S = ⊤}) Y"], "goal": "∀ (Y : C) (f : Y ⟶ X), S.arrows f → ((c X) R).arrows f"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r : ℚ"], "goal": "↑p ≠ 0"}}
+{"state": {"context": ["n : ℕ∞", "𝕂 : Type u_1", "inst✝⁴ : RCLike 𝕂", "E' : Type u_2", "inst✝³ : NormedAddCommGroup E'", "inst✝² : NormedSpace 𝕂 E'", "F' : Type u_3", "inst✝¹ : NormedAddCommGroup F'", "inst✝ : NormedSpace 𝕂 F'", "f : E' → F'", "hf✝ : ContDiff 𝕂 1 f", "x : E'", "K : ℝ≥0", "t : Set E'", "ht : t ∈ 𝓝 ?m.51584", "hf : LipschitzOnWith K f t"], "goal": "∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : ℕ → α → ℝ≥0∞", "hf : ∀ (n : ℕ), AEMeasurable (f n) μ", "h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x"], "goal": "∫⁻ (a : α), (⨆ i, f i) a ∂μ = ⨆ n, ∫⁻ (a : α), f n a ∂μ"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p✝ q✝ : ℝ≥0", "p q : ℝ", "hq : 0 ≤ q", "h : q ≤ p"], "goal": "ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q"}}
+{"state": {"context": ["α : Type u_5", "Preorder α"], "goal": "IsStronglyCoatomic α ↔ ∀ (a b : α), a < b → ∃ x, a ≤ x ∧ x ⋖ b"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "CategoryTheory.HasExt C", "X : C", "n : ℕ"], "goal": "∀ {X_1 Y : C} (f : X_1 ⟶ Y),\n  (CategoryTheory.Abelian.extFunctorObj X n).map f = AddCommGrp.ofHom ((CategoryTheory.Abelian.Ext.mk₀ f).postcomp X ⋯)"}}
+{"state": {"context": ["V₁ : Type u_3", "V₂ : Type u_4", "V₃ : Type u_5", "SeminormedAddCommGroup V₁", "SeminormedAddCommGroup V₂", "SeminormedAddCommGroup V₃", "f : NormedAddGroupHom V₁ V₂", "g : NormedAddGroupHom V₂ V₃", "h : g.comp f = 0", "v : V₁"], "goal": "↑((NormedAddGroupHom.ker.lift f g h) v) = f v"}}
+{"state": {"context": ["n : ℕ", "X : Type u_1", "Zero X", "f : Fin n → X"], "goal": "SSet.Augmented.StandardSimplex.shiftFun f 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M", "p : Submodule R M", "x : M"], "goal": "p.mkQ x = Submodule.Quotient.mk x"}}
+{"state": {"context": ["α : Type u_1", "NormedOrderedAddGroup α", "s : Set α"], "goal": "↑(lowerClosure (interior s)) ⊆ interior ↑(lowerClosure s)"}}
+{"state": {"context": ["P : Type u_1", "Preorder P", "OrderTop P"], "goal": "Order.Ideal.principal ⊤ = ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "G₀ : Type u_7", "TopologicalSpace β", "GroupWithZero G₀", "MulAction G₀ β", "ContinuousConstSMul G₀ β"], "goal": "∀ {x : MeasurableSpace α} {c : G₀},\n  c ≠ 0 → ((MeasureTheory.StronglyMeasurable fun x => c • f x) ↔ MeasureTheory.StronglyMeasurable f)"}}
+{"state": {"context": ["α : Type u_1", "cut : α → Ordering", "f : α → α", "t : Batteries.RBNode α"], "goal": "Batteries.RBNode.modify cut f t = Batteries.RBNode.alter cut (Option.map f) t"}}
+{"state": {"context": ["C : Type uC", "inst✝³ : Category.{vC, uC} C", "D : Type uD", "inst✝² : Category.{vD, uD} D", "G✝ : C ⥤ D", "J : GrothendieckTopology C", "K : GrothendieckTopology D", "A : Type u_1", "inst✝¹ : Category.{?u.9506, u_1} A", "G : C ⥤ D", "inst✝ : G.IsLocallyFaithful K", "ℱ : SheafOfTypes K", "X Y : C", "i₁ i₂ : X ⟶ Y", "e : G.map i₁ = G.map i₂", "s t : ℱ.val.obj (op (G.obj X))", "h : ∀ ⦃Z : C⦄ (j : Z ⟶ X), j ≫ i₁ = j ≫ i₂ → ℱ.val.map (G.map j).op s = ℱ.val.map (G.map j).op t"], "goal": "∀ ⦃Y_1 : D⦄ ⦃f : Y_1 ⟶ G.obj X⦄, (Sieve.functorPushforward G (Sieve.equalizer i₁ i₂)).arrows f → ℱ.val.map f.op s = ℱ.val.map f.op t"}}
+{"state": {"context": ["S : Type u_5", "LinearOrderedSemifield S", "R : Type u_6", "DivisionSemiring R", "abv : R → S", "IsAbsoluteValue abv", "a : R"], "goal": "abv a⁻¹ = (abv a)⁻¹"}}
+{"state": {"context": ["E : Type u", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℂ E", "F : Type v", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℂ F", "inst✝ : CompleteSpace F", "f : ℂ → F", "z w : ℂ", "r : ℝ := dist w z", "hd : DiffContOnCl ℂ f (ball z r)", "hz : IsMaxOn (norm ∘ f) (closedBall z r) z", "hw : w ∈ closedBall z r", "hw_lt : ‖f w‖ < ‖f z‖", "hr : 0 < r", "hsub : sphere z r ⊆ closedBall z r"], "goal": "‖f w‖ / abs (w - z) < ‖f z‖ / r"}}
+{"state": {"context": ["δ : Type u_1", "δ' : Type u_2", "π : δ → Type u_3", "inst✝³ : (x : δ) → MeasurableSpace (π x)", "μ : (i : δ) → Measure (π i)", "inst✝² : ∀ (i : δ), SigmaFinite (μ i)", "inst✝¹ : DecidableEq δ", "s✝ t : Finset δ", "f✝ g✝ : ((i : δ) → π i) → ℝ≥0∞", "x y : (i : δ) → π i", "i : δ", "inst✝ : Fintype δ", "s : Finset δ", "f g : ((i : δ) → π i) → ℝ≥0∞", "hf : Measurable f", "hg : Measurable g", "hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ", "h : IsEmpty ((i : δ) → π i)"], "goal": "∫⁻ (x : (i : δ) → π i), f x ∂Measure.pi μ ≤ ∫⁻ (x : (i : δ) → π i), g x ∂Measure.pi μ"}}
+{"state": {"context": ["k : Type u_1", "E : Type u_2", "LinearOrderedField k", "OrderedAddCommGroup E", "Module k E", "OrderedSMul k E", "f : k → E", "a b r : k", "hab : a < b", "h₀ : 0 < r", "h₁ : r < 1"], "goal": "(AffineMap.lineMap (f a) (f b)) r ≤ f ((AffineMap.lineMap a b) r) ↔\n  slope f ((AffineMap.lineMap a b) r) b ≤ slope f a ((AffineMap.lineMap a b) r)"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a", "n : ℕ", "b : ℤ√↑(Pell.d a1)", "h1 : 1 ≤ b", "hp : IsPell b", "h : b ≤ pellZd a1 (n + 1)", "a1p : 0 ≤ { re := ↑a, im := 1 }", "am1p : 0 ≤ { re := ↑a, im := -1 }", "a1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1", "ha : { re := ↑a, im := 1 } ≤ b"], "goal": "{ re := ↑a, im := 1 } * { re := ↑a, im := -1 } ≤ b * { re := ↑a, im := -1 }"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "s : Finset α"], "goal": "s.card ≤ Fintype.card α"}}
+{"state": {"context": ["n m : ℕ"], "goal": "¬↑n + 1 + ↑m = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "ι : Type u_4", "v : ι → E", "hv : Function.Injective v", "f : ι ≃ ↑(Set.range v) := Equiv.ofInjective v hv", "h : Orthonormal 𝕜 v"], "goal": "Orthonormal 𝕜 (v ∘ ⇑(Equiv.ofInjective v hv).symm)"}}
+{"state": {"context": ["R : Type u₁", "Ring R", "f : R →+* R", "hf : f = RingHom.id R", "M N : ModuleCat R", "φ : M ⟶ N"], "goal": "φ ≫ (ModuleCat.restrictScalarsId'App f hf N).inv =\n  (ModuleCat.restrictScalarsId'App f hf M).inv ≫ (ModuleCat.restrictScalars f).map φ"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x y : V", "h : o.oangle x y = ↑(-π / 2)"], "goal": "⟪y, x⟫_ℝ = 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "W : CategoryTheory.MorphismProperty C", "X Y : C", "z₁ z₂ : W.LeftFraction X Y", "h : CategoryTheory.MorphismProperty.LeftFractionRel z₁ z₂"], "goal": "CategoryTheory.MorphismProperty.LeftFractionRel z₂ z₁"}}
+{"state": {"context": ["α : Type u", "a : α", "l : List α", "n m : ℕ"], "goal": "((a :: l).rotate' (n + 1)).rotate' m = (a :: l).rotate' (n + 1 + m)"}}
+{"state": {"context": ["n i : Nat", "h : i < (list n).length"], "goal": "(enum n).data[i] = cast ⋯ ⟨i, h⟩"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "δ : Type u_3", "ι : Type u_4", "m0 : MeasurableSpace α", "inst✝ : MeasurableSpace β", "μ✝ ν ν₁ ν₂ : Measure α", "s✝ t : Set α", "μ : Measure α", "h this : SigmaFinite μ", "s : ℕ → Set α := spanningSets μ", "hs_univ : ⋃ i, s i = univ", "hs_meas : ∀ (i : ℕ), MeasurableSet (s i)"], "goal": "μ univ < ⊤"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "CommRing R", "CommMonoidWithZero R'", "χ : MulChar R R'", "n : ℕ", "hn : Odd n", "hχ : χ ^ n = 1"], "goal": "χ (-1) = 1"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁶ : Field F", "inst✝⁵ : Field E", "inst✝⁴ : Algebra F E", "K : Type w", "inst✝³ : Field K", "inst✝² : Algebra F K", "inst✝¹ : Algebra E K", "inst✝ : IsScalarTower F E K", "q : ℕ", "h2✝ : ∀ (x : K), ∃ n, x ^ q ^ n ∈ (algebraMap E K).range", "h1✝ : ∀ (x : E), ∃ n, x ^ q ^ n ∈ (algebraMap F E).range", "h✝ : ExpChar F q", "this : ExpChar E q", "x : K", "n : ℕ", "y : E", "h2 : (algebraMap E K) y = x ^ q ^ n", "m : ℕ", "z : F", "h1 : (algebraMap F E) z = y ^ q ^ m"], "goal": "(algebraMap F K) z = x ^ q ^ (n + m)"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "x : α"], "goal": "(∃ x, x ∈+ l) ↔ ¬l.Nodup"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "hf : Function.Surjective f", "s t : Set β"], "goal": "Disjoint (f ⁻¹' s) (f ⁻¹' t) ↔ Disjoint s t"}}
+{"state": {"context": [], "goal": "∀ {X Y : TwoP} (f : X ⟶ Y) (a : X.toBipointed.X), (TwoP.swap.map f).toFun a = f.toFun a"}}
+{"state": {"context": ["b : Bool"], "goal": "(b || b) = b"}}
+{"state": {"context": ["a x z✝ z : ℂ", "hre : z.re < 0"], "goal": "z.arg ≤ π / 2 ↔ z.im < 0"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "inst✝³ : CommRing R", "inst✝² : LieRing L", "inst✝¹ : LieAlgebra R L", "inst✝ : HasTrivialRadical R L", "h : IsLieAbelian L"], "goal": "Subsingleton L"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "StrictOrderedCommRing R", "AddCommGroup V", "Module R V", "AddTorsor V P", "s : AffineSubspace R P", "x y : P"], "goal": "s.SSameSide x y ↔ s.SSameSide y x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : ProbabilityTheory.Kernel α β", "E : Type u_4", "NormedAddCommGroup E", "ProbabilityTheory.IsSFiniteKernel κ", "f : α → β → E", "hf : MeasureTheory.StronglyMeasurable (Function.uncurry f)"], "goal": "MeasurableSet {x | MeasureTheory.Integrable (f x) (κ x)}"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : Fintype ι", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace ℝ F", "inst✝² : FiniteDimensional ℝ F", "inst✝¹ : MeasurableSpace F", "inst✝ : BorelSpace F", "b : OrthonormalBasis ι ℝ F", "e_3✝ : MeasureSpace.toMeasurableSpace = inst✝¹"], "goal": "volume = map (⇑b.measurableEquiv.symm) (EuclideanSpace.basisFun ι ℝ).toBasis.addHaar"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "motive : (u v : V) → G.Walk u v → Sort u_1", "Hnil : {u : V} → motive u u SimpleGraph.Walk.nil", "Hconcat : {u v w : V} → (p : G.Walk u v) → (h : G.Adj v w) → motive u v p → motive u w (p.concat h)", "u : V"], "goal": "SimpleGraph.Walk.concatRec Hnil Hconcat SimpleGraph.Walk.nil = Hnil"}}
+{"state": {"context": ["i : Nat", "α✝ : Type u_1", "xs : List α✝", "j : Nat", "h₀ : i < xs.length", "h₁ : xs.Nodup", "h₂ : xs.get? i = xs.get? j"], "goal": "i = j"}}
+{"state": {"context": ["𝕜 : Type u_1", "LinearOrderedAddCommGroup 𝕜", "p x y : 𝕜"], "goal": "↑(x + y) = ↑x + ↑y"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I✝ J K L I : Ideal R", "hI : I.IsPrime", "s : Finset R"], "goal": "s.val.prod ∈ I ↔ ∃ p ∈ s, p ∈ I"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "P : (x : CliffordAlgebra Q) → x ∈ evenOdd Q 1 → Prop", "ι : ∀ (v : M), P ((CliffordAlgebra.ι Q) v) ⋯", "add : ∀ (x y : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 1) (hy : y ∈ evenOdd Q 1), P x hx → P y hy → P (x + y) ⋯", "ι_mul_ι_mul : ∀ (m₁ m₂ : M) (x : CliffordAlgebra Q) (hx : x ∈ evenOdd Q 1), P x hx → P ((CliffordAlgebra.ι Q) m₁ * (CliffordAlgebra.ι Q) m₂ * x) ⋯", "x : CliffordAlgebra Q", "hx : x ∈ evenOdd Q 1", "ιv : CliffordAlgebra Q"], "goal": "∀ (h : ιv ∈ LinearMap.range (CliffordAlgebra.ι Q) ^ ZMod.val 1), P ιv ⋯"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_3, u_2} D", "G : C ⥤ D", "A : Type w", "CategoryTheory.Category.{w', w} A", "J : CategoryTheory.GrothendieckTopology C", "K : CategoryTheory.GrothendieckTopology D", "∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F", "CategoryTheory.HasWeakSheafify J A", "CategoryTheory.HasWeakSheafify K A", "G.IsCocontinuous J K", "G.IsContinuous J K", "F : Dᵒᵖ ⥤ A"], "goal": "CategoryTheory.toSheafify J (G.op ⋙ F) ≫ ((G.pushforwardContinuousSheafificationCompatibility A J K).hom.app F).val =\n  CategoryTheory.whiskerLeft G.op (CategoryTheory.toSheafify K F)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f f₁ f₂ : Filter α", "g g₁ g₂ : Filter β", "m : α → β", "m' : β → γ", "s : Set α", "t : Set β", "h : g = ⊥"], "goal": "comap m ⊥ = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι✝ : Type u_3", "ε : ℝ≥0∞", "hε : ε ≠ 0", "ι : Type u_4", "inst✝ : Countable ι", "w δ' : ι → ℝ≥0", "Hpos : ∀ (i : ι), 0 < δ' i", "Hsum : ∑' (i : ι), ↑(δ' i) < ε", "this : ∀ (i : ι), 0 < max 1 (w i)"], "goal": "∑' (i : ι), (fun i => ↑(w i)) i * ↑((fun i => δ' i / max 1 (w i)) i) < ε"}}
+{"state": {"context": ["i₁ i₂ : String.Pos"], "goal": "i₁ = i₂ ↔ i₁.byteIdx = i₂.byteIdx"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "x y : ProjectiveSpectrum 𝒜"], "goal": "x ≤ y ↔ y ∈ closure {x}"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Fintype β", "DecidableEq α", "f : β → Set α", "(w : β) → Fintype ↑(f w)"], "goal": "(⋃ x, f x).toFinset = Finset.univ.biUnion fun x => (f x).toFinset"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "p₁ p₂ p₃ : P", "h : ∠ p₁ p₂ p₃ = π / 2", "h0 : p₁ ≠ p₂ ∨ p₃ = p₂"], "goal": "0 < ∠ p₂ p₃ p₁"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "MulOneClass M", "MulOneClass N", "s : Submonoid M", "t : Submonoid N"], "goal": "s.prod t = ⊥ ↔ s = ⊥ ∧ t = ⊥"}}
+{"state": {"context": ["a : Nat", "b c : Nat", "h : b ≤ c"], "goal": "a + b - c = a - (c - b)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "OrderedCommRing R", "StarRing R", "MetricSpace R", "TopologicalRing R", "ContinuousStar R", "∀ (α : Type u_1) [inst : TopologicalSpace α], StarOrderedRing C(α, R)", "TopologicalSpace A", "Ring A", "StarRing A", "PartialOrder A", "StarOrderedRing A", "Algebra R A", "ContinuousFunctionalCalculus R p", "NonnegSpectrumClass R A", "a : A", "ha : p a", "f g : C(↑(spectrum R a), R)"], "goal": "(cfcHom ha) f ≤ (cfcHom ha) g ↔ f ≤ g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_5", "f : α → β", "SeminormedAddCommGroup β", "NormedSpace ℝ β", "m : MeasurableSpace α", "c : ℝ", "hf : MeasureTheory.StronglyMeasurable f", "hc : 0 ≤ c", "n : ℕ", "x : α"], "goal": "‖↑(hf.approxBounded c n) x‖ ≤ c"}}
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+{"state": {"context": ["n : ℕ", "a b : ZMod n"], "goal": "(a + b).cast = if ↑n ≤ a.cast + b.cast then a.cast + b.cast - ↑n else a.cast + b.cast"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝¹ : DecidableEq α", "inst✝ : DecidableEq β", "f : α → β", "y : α", "s : Multiset α", "ih : y ∈ s → map f (s.erase y) = (map f s).erase (f y)", "h : y ∈ y ::ₘ s"], "goal": "map f ((y ::ₘ s).erase y) = (map f (y ::ₘ s)).erase (f y)"}}
+{"state": {"context": ["R : Type u_1", "F : Type u_3", "AddCommGroup F", "TopologicalSpace F", "TopologicalAddGroup F", "Field R", "Module R F", "ContinuousConstSMul R F", "x : F", "t : R", "ht : t ≠ 0"], "goal": "IsOpenMap ⇑(AffineMap.homothety x t)"}}
+{"state": {"context": ["n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "CategoryTheory.Limits.HasCokernel S.f", "hg : S.g = 0"], "goal": "(CategoryTheory.ShortComplex.LeftHomologyData.ofHasCokernel S hg).π = CategoryTheory.Limits.cokernel.π S.f"}}
+{"state": {"context": ["α : Type u_1", "r : ℕ"], "goal": "Set.Sized r ∅"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝ : Field K", "P : K[X]", "hP : ¬P = 0"], "goal": "↑(Multiplicative.ofAdd (-↑((Associates.mk (Polynomial.idealX K).asIdeal).count (Associates.mk (Ideal.span {P})).factors))) = ↑(Multiplicative.ofAdd (-↑((Associates.mk (idealX K).asIdeal).count (Associates.mk (Ideal.span {↑P})).factors)))"}}
+{"state": {"context": ["α : Type u", "CompleteLattice α", "f : α →o α"], "goal": "Function.IsFixedPt (⇑f) (OrderHom.gfp f)"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁷ : OrderedRing R", "inst✝⁶ : AddCommGroup V", "inst✝⁵ : Module R V", "inst✝⁴ : AddTorsor V P", "inst✝³ : AddCommGroup V'", "inst✝² : Module R V'", "inst✝¹ : AddTorsor V' P'", "inst✝ : NoZeroSMulDivisors R V", "w x y z : P", "h₁ : Wbtw R w x z", "h₂ : Sbtw R x y z"], "goal": "Sbtw R w y z"}}
+{"state": {"context": ["α : Type u_2", "CompleteLattice α", "self : IsAtomistic α", "b : α"], "goal": "∃ s, b = sSup s ∧ ∀ a ∈ s, IsAtom a"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "a : R"], "goal": "(Polynomial.C a).mirror = Polynomial.C a"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u : V", "p : G.Walk u u"], "goal": "p.length = 0 ↔ p = SimpleGraph.Walk.nil"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p✝ : R", "k n p : ℕ", "hp : Nat.Prime p", "hn : p ∣ n", "hn₀ : n ≠ 0", "hq : ∀ (y : ℕ), Nat.Prime y → y ∣ n → y = p"], "goal": "n.primeFactorsList.Subperm (p ^ n.factorization p).primeFactorsList"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Type u_3", "f : α → β", "m : ι → OuterMeasure β", "s : Set α", "hs : s.Nonempty", "t : ℕ → Set β", "ht : s ⊆ ⋃ i, f ⁻¹' t i", "k : ℕ"], "goal": "⨅ i, (m i) (f '' (f ⁻¹' t k)) ≤ ⨅ i, (m i) (t k)"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedBooleanAlgebra α", "a b : α"], "goal": "a ≤ a ∆ b ↔ Disjoint a b"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : ∀ (a b : C), HasCoproductsOfShape (a ⟶ b) D", "c : C", "d : D", "F : C ⥤ D", "f : (evaluationLeftAdjoint D c).obj d ⟶ F", "x : C", "g : c ⟶ x"], "goal": "(Sigma.ι (fun x => d) g ≫ Sigma.desc fun h => (Sigma.ι (fun x => d) (𝟙 c) ≫ f.app c) ≫ F.map h) = Sigma.ι (fun x => d) g ≫ f.app x"}}
+{"state": {"context": ["α : Type u", "s : Set α", "DecidablePred fun x => x ∈ s", "a : α", "H : a ∉ s", "b : PUnit.{u + 1}"], "goal": "(Equiv.Set.insert H).symm (Sum.inr b) = ⟨a, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "l : List α", "n : ℕ", "f : List α → β", "r : List β"], "goal": "sublistsLenAux n l f r = map f (sublistsLen n l) ++ r"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder β", "f : α → β", "l : Filter α", "a : α", "l' : Filter α", "h : IsMaxFilter f l a", "hl : l' ≤ l"], "goal": "IsMaxFilter f l' a"}}
+{"state": {"context": ["X : Type u", "TopologicalSpace X", "t : Set X", "f : Filter X", "ht : IsCompact t", "hf : Disjoint f (Filter.cocompact X)"], "goal": "f ×ˢ 𝓝ˢ t ≤ 𝓝ˢ (Set.univ ×ˢ t)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "hm : 0 < m", "q :\n  ↑↑(AlgebraicGeometry.Spec.locallyRingedSpaceObj\n          (CommRingCat.of (HomogeneousLocalization.Away 𝒜 f))).toPresheafedSpace"], "goal": "AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.carrier.asIdeal f_deg hm q ≠ ⊤"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝²⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝²³ : NormedAddCommGroup E", "inst✝²² : NormedSpace 𝕜 E", "H : Type u_3", "inst✝²¹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝²⁰ : TopologicalSpace M", "inst✝¹⁹ : ChartedSpace H M", "Is : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝¹⁸ : NormedAddCommGroup E'", "inst✝¹⁷ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁶ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝¹⁵ : TopologicalSpace M'", "inst✝¹⁴ : ChartedSpace H' M'", "I's : SmoothManifoldWithCorners I' M'", "F : Type u_8", "inst✝¹³ : NormedAddCommGroup F", "inst✝¹² : NormedSpace 𝕜 F", "G : Type u_9", "inst✝¹¹ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N : Type u_10", "inst✝¹⁰ : TopologicalSpace N", "inst✝⁹ : ChartedSpace G N", "Js : SmoothManifoldWithCorners J N", "F' : Type u_11", "inst✝⁸ : NormedAddCommGroup F'", "inst✝⁷ : NormedSpace 𝕜 F'", "G' : Type u_12", "inst✝⁶ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_13", "inst✝⁵ : TopologicalSpace N'", "inst✝⁴ : ChartedSpace G' N'", "J's : SmoothManifoldWithCorners J' N'", "F₁ : Type u_14", "inst✝³ : NormedAddCommGroup F₁", "inst✝² : NormedSpace 𝕜 F₁", "F₂ : Type u_15", "inst✝¹ : NormedAddCommGroup F₂", "inst✝ : NormedSpace 𝕜 F₂", "f✝ f₁ : M → M'", "s s₁ t : Set M", "x : M", "m n : ℕ∞", "x₀ : N", "f : N → M → M'", "g : N → M", "hf : ContMDiffAt (J.prod I) I' n (uncurry f) (x₀, g x₀)", "hg : ContMDiffAt J I m g x₀", "hmn : m + 1 ≤ n", "h4f : ContinuousAt (fun x => f x (g x)) x₀"], "goal": "ContMDiffAt J 𝓘(𝕜, E →L[𝕜] E') m (inTangentCoordinates I I' g (fun x => f x (g x)) (fun x => mfderiv I I' (f x) (g x)) x₀) x₀"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝² : TopologicalSpace α", "inst✝¹ : Preorder α", "inst✝ : OrderTopology α", "a : α", "ha : ∃ l, l < a"], "goal": "𝓝[≤] a = ⨅ l, ⨅ (_ : l < a), 𝓟 (Ioc l a)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "a b C : ℝ", "f g : ℂ → E", "hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0)", "hB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * Complex.abs z ^ c)", "hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C", "him : ∀ x ≤ 0, ‖f (↑x * I)‖ ≤ C", "z : ℂ", "hz_re : 0 ≤ z.re", "hz_im : 0 ≤ z.im", "H : MapsTo Neg.neg (Ioi 0 ×ℂ Ioi 0) (Iio 0 ×ℂ Iio 0)"], "goal": "‖(f ∘ Neg.neg) z‖ ≤ C"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a : α", "f✝ g : α → β", "inst✝ : CommMonoid β", "s : Finset α", "f : α → β", "this✝ : DecidableEq α", "this : DecidableEq β"], "goal": "∀ (g : (a : α) → a ∈ s → α), (∀ (a : α) (ha : a ∈ s), f a * f (g a ha) = 1) → (∀ (a : α) (ha : a ∈ s), f a ≠ 1 → g a ha ≠ a) → ∀ (g_mem : ∀ (a : α) (ha : a ∈ s), g a ha ∈ s), (∀ (a : α) (ha : a ∈ s), g (g a ha) ⋯ = a) → ∏ x ∈ s, f x = 1"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : Affine R", "x y L : R"], "goal": "C (eval (C L * (X - C x) + C y) (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆))) = (Y - C (C L * (X - C x) + C y)) * (-Y - C (C W.a₁ * X + C W.a₃) - C (C L * (X - C x) + C y)) + (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f g : E → F", "p pf pg : FormalMultilinearSeries 𝕜 E F", "x : E", "r r' : ℝ≥0∞", "hg : f =ᶠ[𝓝 x] g", "r₁ : ℝ≥0∞", "h₁ : HasFPowerSeriesOnBall f p x r₁"], "goal": "HasFPowerSeriesAt g p x"}}
+{"state": {"context": [], "goal": "Rat.pnatDen 0 = 1"}}
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+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K L : HomologicalComplex C c", "φ : K ⟶ L", "i : ι", "K.HasHomology i", "L.HasHomology i"], "goal": "K.homologyπ i ≫ HomologicalComplex.homologyMap φ i = HomologicalComplex.cyclesMap φ i ≫ L.homologyπ i"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "inst✝¹ : Category.{v, u} V", "inst✝ : Preadditive V", "c : ComplexShape ι", "C D E : HomologicalComplex V c", "f✝ g : C ⟶ D", "h k : D ⟶ E", "i✝ : ι", "f : (i j : ι) → C.X i ⟶ D.X j", "i i' : ι", "w : c.Rel i i'"], "goal": "(dNext i) f = C.d i i' ≫ f i' i"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Add α", "Preorder α", "Preorder β", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "f g : β → α", "hf : Monotone f", "hg : StrictMono g"], "goal": "StrictMono fun x => f x + g x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x : Fin n → ℕ", "hN : 2 ≤ N", "hN' : N ≤ 4096"], "goal": "↑N ≤ rexp (4 ^ 2)"}}
+{"state": {"context": ["i : Nat"], "goal": "Nat.testBit 1 i = true ↔ i = 0"}}
+{"state": {"context": ["X : Type u", "inst✝³ : TopologicalSpace X", "inst✝² : SeparableSpace X", "inst✝¹ : NormalSpace X", "s : Set X", "hs : IsClosed s", "inst✝ : DiscreteTopology ↑s", "h : 𝔠 ≤ #↑s", "t : Set X", "htc : t.Countable", "htd : Dense t", "this : Countable ↑t"], "goal": "2 ^ 𝔠 ≤ 𝔠"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {l₁ l₂ : List α}, l₁ <:+ l₂ → l₁ <:+: l₂"}}
+{"state": {"context": ["α : Type u", "AddCommGroup α", "Preorder α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "a b c d : α", "hab : a ≤ b", "hcd : c ≤ d"], "goal": "a - d ≤ b - c"}}
+{"state": {"context": ["x : ℝ"], "goal": "DifferentiableAt ℝ Real.sinh x"}}
+{"state": {"context": ["a t : ℝ", "ht : 0 < t", "this : HasSum (fun n => -I * ↑↑n * cexp (2 * ↑π * I * ↑a * ↑↑n) * ↑(rexp (-π * ↑↑n ^ 2 * t)) + -I * ↑(-↑n) * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) * ↑(rexp (-π * ↑(-↑n) ^ 2 * t))) ↑(sinKernel (↑a) t)"], "goal": "HasSum (fun x => ↑(2 * ↑x * Real.sin (2 * π * a * ↑x) * rexp (-π * ↑x ^ 2 * t))) ↑(sinKernel (↑a) t)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "I J : Ideal R", "h : I ≤ J"], "goal": "Ideal.comap (Ideal.Quotient.mk I) (Ideal.map (Ideal.Quotient.mk I) J) = J"}}
+{"state": {"context": ["X : Type u_1", "x : X"], "goal": "((FundamentalGroupoid.equiv X).symm x).as = x"}}
+{"state": {"context": ["z : ℂ", "x : ℝ"], "goal": "(z / ↑x).im = z.im / x"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α✝", "μ✝ ν : Measure α✝", "α : Type u_5", "mα : MeasurableSpace α", "f : ℕ → α → ℝ≥0∞", "F : α → ℝ≥0∞", "μ : Measure α", "hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ", "hf_tendsto : Tendsto (fun i => ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))", "hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i => f i a", "h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a", "h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤", "h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ⊤"], "goal": "∀ᵐ (a : α) ∂μ, Tendsto (fun i => f i a) atTop (𝓝 (F a))"}}
+{"state": {"context": ["i n : ℕ", "hi : 2 ≤ i", "hn : 1 ≤ n"], "goal": "i + n.factorial < (i + n).factorial"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ l : List α", "a : α", "n : ℕ", "hn : n ≠ 0", "h : l.length ≤ n - 1"], "goal": "l.length + 1 ≤ n"}}
+{"state": {"context": [], "goal": "∫ (x : ℝ) in Set.Ioi 0, rexp (-x) = 1"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2", "hnsol : Polynomial.eval a F ≠ 0"], "goal": "Tendsto (fun n => ‖Polynomial.eval a (Polynomial.derivative F)‖ * T_gen p F a ^ 2 ^ n) atTop (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "k p : ℕ", "f : ℕ → α", "h : Nat.Prime p"], "goal": "∏ x ∈ (p ^ k).properDivisors, f x = ∏ x ∈ Finset.range k, f (p ^ x)"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝³ : Category.{v₂, u₂} D", "F : C ⥤ D", "inst✝² : IsFilteredOrEmpty D", "inst✝¹ : F.Full", "inst✝ : F.Faithful", "h : ∀ (d : D), ∃ c, Nonempty (d ⟶ F.obj c)", "this : IsFilteredOrEmpty C"], "goal": "F.Final"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp : p ≠ 2", "h₁ : p % 8 % 2 = 1", "h₂ : p % 8 < 8"], "goal": "p % 8 ≠ 5 ∧ p % 8 ≠ 7 ↔ p % 8 = 1 ∨ p % 8 = 3"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "Function.Injective Basis.repr"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.cos x ^ 2 = 1 / 2 + Real.cos (2 * x) / 2"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "CategoryTheory.Limits.HasBinaryBiproduct X Y"], "goal": "CategoryTheory.IsPushout 0 0 CategoryTheory.Limits.biprod.inl CategoryTheory.Limits.biprod.inr"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "c : α", "s : Finset α", "hS : s.Nontrivial"], "goal": "(s \\ {c}).Nonempty"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_3", "Fintype ι", "(i : ι) → MeasurableSpace (α i)", "μ : (i : ι) → MeasureTheory.Measure (α i)", "∀ (i : ι), MeasureTheory.SigmaFinite (μ i)", "(i : ι) → PartialOrder (α i)", "∀ (i : ι), MeasureTheory.NoAtoms (μ i)", "f : (i : ι) → α i"], "goal": "(Set.univ.pi fun i => Set.Iio (f i)) =ᵐ[MeasureTheory.Measure.pi μ] Set.Iic f"}}
+{"state": {"context": ["α : Type u_2", "Mul α", "IsLeftCancelMul α", "DecidableEq α", "t : Finset α", "s : Finset α", "hs : s.Nonempty"], "goal": "t.card ≤ (s * t).card"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_5", "CommSemiring R", "Finite ι"], "goal": "(Pi.basisFun R ι).toMatrix = Matrix.transpose"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α", "n : ℤ"], "goal": "(c ∈ Set.Ico a (a + p) ∧ ∃ z, b - z • p = c) → toIcoMod hp a b = c"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u", "β : Type v", "inst✝¹ : PseudoEMetricSpace α", "inst✝ : PseudoEMetricSpace β", "x✝ y : α", "s t : Set α", "Φ : α → β", "x : α", "E : Set α"], "goal": "0 < infEdist x E ↔ x ∉ closure E"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝¹ : DistribLattice α", "inst✝ : OrderBot α", "s✝ : Finset ι", "f✝ : ι → α", "s : Finset ι", "t : Finset ι'", "f : ι × ι' → α"], "goal": "(s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i')) ∧ t.SupIndep fun i' => s.sup fun i => f (i, i')"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VSub α β", "f g : Filter β", "f.NeBot", "g.NeBot"], "goal": "(f -ᵥ g).NeBot"}}
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+{"state": {"context": ["y z : ℝ", "hyz : z < y", "x : ℝ≥0", "hx0 : 0 < ↑x", "hx1 : ↑x < 1"], "goal": "↑x ^ y < ↑x ^ z"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁵ : Group G", "inst✝⁴ : Group G'", "inst✝³ : Group G''", "A : Type u_4", "inst✝² : AddGroup A", "N : Type u_5", "inst✝¹ : Group N", "C : Type u_6", "inst✝ : CommGroup C", "s t : Subgroup C", "y : C", "hy : y ∈ s", "z : C", "hz : z ∈ t"], "goal": "y * z ∈ s ⊔ t"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "Zero R"], "goal": "Function.Injective Unitization.inr"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : Semiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "t : Finset M", "h : (↑t).Finite"], "goal": "(span R ↑t).FG"}}
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+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.Limits.HasPullbacks C", "X Y : CategoryTheory.Dial C"], "goal": "(X.tensorObj Y).src = (X.src ⨯ Y.src)"}}
+{"state": {"context": ["R : Type u_2", "CommSemiring R", "a : R"], "goal": "↑(Polynomial.C a) = (PowerSeries.C R) a"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x : E"], "goal": "⟪x, 0⟫_𝕜 = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_5", "ι : Type u_8", "NontriviallyNormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "p : SeminormFamily 𝕜 E ι", "Nonempty ι", "t : TopologicalSpace E", "hp : WithSeminorms p", "q : Seminorm 𝕜 E", "hq : Continuous ⇑q"], "goal": "∃ s C, C ≠ 0 ∧ q ≤ C • s.sup p"}}
+{"state": {"context": ["ι : Type u_6", "DecidableEq ι", "i : ι", "Y : ι → Type u_7", "(j : ι) → TopologicalSpace (Y j)", "f : (j : ι) → Y j"], "goal": "(Homeomorph.piSplitAt i Y) f = (f i, fun j => f ↑j)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "f : Filter α", "inst✝ : ConditionallyCompleteLinearOrder α", "a : α", "hf : autoParam (IsCobounded (fun x x_1 => x ≤ x_1) f) _auto✝", "h : a < f.limsSup"], "goal": "∃ᶠ (n : α) in f, a < n"}}
+{"state": {"context": ["n : ℕ", "R : Type u_1", "AddGroupWithOne R", "S : Type u_2", "AddGroupWithOne S", "a : ZMod n"], "goal": "a.cast.1 = a.cast"}}
+{"state": {"context": ["b : Bool"], "goal": "(true = b) = (b = true)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝¹ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "inst✝ : IsFiniteKernel ν", "a : α", "s : Set β", "hs : MeasurableSet s", "A : Set γ", "hA : MeasurableSet A"], "goal": "∫ (x : γ) in A, κ.density ν a x s ∂ν a = ((κ a) (A ×ˢ s)).toReal"}}
+{"state": {"context": ["R : Type u", "CommRing R", "IsDedekindDomain R", "K : Type v", "Field K", "Algebra R K", "IsFractionRing R K", "v : IsDedekindDomain.HeightOneSpectrum R", "x : Rˣ"], "goal": "v.valuationOfNeZero ((Units.map ↑(algebraMap R K)) x) = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "f : α → β", "t : Set α", "h : Set.InjOn f s"], "goal": "f '' (s \\ t) = f '' s \\ f '' (s ∩ t)"}}
+{"state": {"context": ["α : Sort u", "op : α → α → α", "self : Std.Commutative op", "a b : α"], "goal": "op a b = op b a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹¹ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝¹⁰ : NormedAddCommGroup D", "inst✝⁹ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "G : Type uG", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedSpace 𝕜 G", "X : Type u_2", "inst✝² : NormedAddCommGroup X", "inst✝¹ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "inst✝ : Subsingleton F"], "goal": "ContDiffAt 𝕜 n (fun x => 0) x"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "s : Set α", "hs : MeasurableSet s"], "goal": "(μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "r : R", "x : CliffordAlgebra Q"], "goal": "star (r • x) = r • star x"}}
+{"state": {"context": ["M : Type u", "G : Type v", "X : Type w", "inst✝⁶ : PseudoEMetricSpace X", "inst✝⁵ : Group G", "inst✝⁴ : MulAction G X", "inst✝³ : IsometricSMul G X", "inst✝² : DivInvMonoid M", "inst✝¹ : PseudoEMetricSpace M", "inst✝ : IsometricSMul Mᵐᵒᵖ M", "a b c : M"], "goal": "edist (a / c) (b / c) = edist a b"}}
+{"state": {"context": ["a b c : ℝ≥0∞"], "goal": "a ≠ ⊤ → (b + a < c + a ↔ b < c)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "f : α → β", "U : ι → Set β", "hU : iUnion U = univ"], "goal": "Bijective f ↔ ∀ (i : ι), Bijective ((U i).restrictPreimage f)"}}
+{"state": {"context": ["R : Type u_4", "NormedRing R", "CompleteSpace R", "x : R", "h : ‖x‖ < 1"], "goal": "‖∑' (n : ℕ), x ^ n‖ ≤ ‖1‖ - 1 + (1 - ‖x‖)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Group α", "inst✝ : DecidableEq α", "x y : Finset α × Finset α", "s t : Finset α", "hs : s.Nonempty", "ht : t.Nonempty", "ih : ∀ (a b : Finset α), a.Nonempty → b.Nonempty → DevosMulRel (a, b) (s, t) → minOrder α ≤ ↑(a * b).card ∨ a.card + b.card ≤ (a * b).card + 1", "hst : s.card ≤ t.card", "a : α", "ha : a ∈ ↑s", "b : α", "hb : b ∈ ↑s", "hab : a ≠ b", "g : α", "hg : g ≠ 1", "hgs : (s ∩ op g • s).Nonempty", "hsg : (s ∩ op g • s).card < s.card", "aux1 : ((mulETransformLeft g (s, t)).1 * (mulETransformLeft g (s, t)).2).card ≤ ((s, t).1 * (s, t).2).card"], "goal": "minOrder α ≤ ↑(s * t).card ∨ s.card + t.card ≤ (s * t).card + 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "TopologicalSpace β", "TopologicalSpace.PseudoMetrizableSpace β", "MeasurableSpace β", "BorelSpace β", "ι : Type u_3", "Countable ι", "Nonempty ι", "μ : MeasureTheory.Measure α", "f : ι → α → β", "L : Filter ι", "L.IsCountablyGenerated", "hf : ∀ (n : ι), AEMeasurable (f n) μ", "h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Filter.Tendsto (fun n => f n x) L (𝓝 l)"], "goal": "∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) L (𝓝 (f_lim x))"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G"], "goal": "AddGroup.FG G ↔ ∃ n S, S.card = n ∧ AddSubgroup.closure ↑S = ⊤"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : NormedSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "s u : Set P", "hu : IsOpen u", "hsu : s ⊆ u", "h : AffineIndependent ℝ Subtype.val", "q : P", "hq : q ∈ s", "ε : ℝ", "ε0 : 0 < ε", "hεu : Metric.closedBall q ε ⊆ u"], "goal": "∃ t, s ⊆ t ∧ t ⊆ u ∧ AffineIndependent ℝ Subtype.val ∧ affineSpan ℝ t = ⊤"}}
+{"state": {"context": ["a : ℝ", "ha : 0 ≤ a", "t : ℝ", "ht : 0 < t"], "goal": "‖HurwitzKernelBounds.F_nat 1 a t‖ ≤\n  rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 + a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))"}}
+{"state": {"context": ["G : Type u_1", "Group G", "self : IsKleinFour G"], "goal": "Monoid.exponent G = 2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝³ : TopologicalSpace α", "inst✝² : LinearOrder α", "inst✝¹ : OrderTopology α", "inst✝ : SecondCountableTopology α", "a✝ : Nontrivial α", "s : Set α := {x | ∃ y, x ⋖ y}", "y : α → α", "hy : ∀ x ∈ s, x ⋖ y x", "Hy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x"], "goal": "{x | ∃ y, x ⋖ y}.Countable"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "f : α → E", "g : α → F", "l : Filter α", "h : f =Θ[l] g"], "goal": "f =O[l] g"}}
+{"state": {"context": ["R : Type u", "Semiring R", "f g : R[X]"], "goal": "Polynomial.derivative (f + g) = Polynomial.derivative f + Polynomial.derivative g"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α", "K : Set α", "S : Set (Set α)", "hK : IsCompact K", "hopen : ∀ s ∈ S, IsOpen s", "hcover : K ⊆ ⋃₀ S"], "goal": "∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, UniformSpace.ball x V ⊆ s"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Submonoid R", "S : Type u_2", "CommRing S", "Algebra R S", "Sₘ : Type u_5", "CommRing Sₘ", "Algebra S Sₘ", "Algebra R Sₘ", "IsScalarTower R S Sₘ", "IsLocalization (Algebra.algebraMapSubmonoid S M) Sₘ", "y : ↥M"], "goal": "IsUnit ((algebraMap R Sₘ) ↑y)"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "S₂ : Type w", "S₃ : Type x", "σ : Type u_1", "a a' a₁ a₂ : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝⁴ : CommSemiring R", "inst✝³ : CommSemiring S₁", "p q : MvPolynomial σ R", "f✝ : R →+* S₁", "inst✝² : CommSemiring S₂", "inst✝¹ : Algebra R S₁", "inst✝ : Algebra R S₂", "f : S₁ →ₐ[R] S₂", "r : R", "h₁ : (algebraMap R (MvPolynomial σ S₁)) r = C ((algebraMap R S₁) r)", "h₂ : (algebraMap R (MvPolynomial σ S₂)) r = C ((algebraMap R S₂) r)"], "goal": "↑↑(map ↑f) ((algebraMap R (MvPolynomial σ S₁)) r) = (algebraMap R (MvPolynomial σ S₂)) r"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝² : PseudoMetricSpace α", "inst✝¹ : Nonempty β", "inst✝ : SemilatticeSup β", "u : ℕ → α", "hu : CauchySeq u", "N : ℕ", "hN : ∀ n ≥ N, dist (u n) (u N) < 1", "R : NNReal := (Finset.range N).sup fun n => nndist (u n) (u N)"], "goal": "∃ R > 0, ∀ (n : ℕ), dist (u n) (u N) < R"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "x : (K →+* ℂ) → ℂ", "h_zero : (commMap K) x = 0", "h_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))"], "goal": "x = 0"}}
+{"state": {"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "hyp : n = 0 ∨ ¬(of v).TerminatedAt (n - 1)"], "goal": "↑(fib (n + 1)) ≤ ((of v).contsAux (n + 1)).b"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "B' : Type u_3", "inst✝⁴ : CommRing A", "inst✝³ : Ring B", "inst✝² : Ring B'", "inst✝¹ : Algebra A B", "inst✝ : Algebra A B'", "x : B", "hx : (minpoly A x).degree = 1", "h : IsIntegral A x", "key : (Polynomial.aeval x) (minpoly A x) = 0"], "goal": "x ∈ (algebraMap A B).range"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "a b : G₀", "h : a * b = 1"], "goal": "a⁻¹ = b"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "CategoryTheory.Category.{v₁, u₂} 𝒳", "CategoryTheory.Category.{v₂, u₁} 𝒮", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ⟶ b", "self : p.IsHomLift f φ"], "goal": "CategoryTheory.IsHomLiftAux p f φ"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "inst✝ : PseudoEMetricSpace α", "x y z : α", "ε ε₁ ε₂ : ℝ≥0∞", "s t : Set α", "h : 0 < ε"], "goal": "x ∈ ball x ε"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Semiring R", "f✝ : R[X]", "S : Type u_2", "inst✝¹ : CommSemiring S", "i : R →+* S", "x : S", "inst✝ : Invertible x", "N : ℕ", "f : R[X]", "hf : f.natDegree ≤ N"], "goal": "eval₂ i (⅟x) (reflect N f) = 0 ↔ eval₂ i (⅟x) (reflect N f) * x ^ N = 0"}}
+{"state": {"context": ["α : Type ua", "UniformSpace α"], "goal": "((𝓤 α).lift' fun s => s ○ s) = 𝓤 α"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "X : C"], "goal": "(β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹ : Category.{u_3, u_1} C", "inst✝ : Category.{u_4, u_2} D", "F F' : C ⥤ D", "G : D ⥤ C", "adj1 : F ⊣ G", "adj2 : F' ⊣ G", "x : D"], "goal": "(whiskerLeft G (adj1.leftAdjointUniq adj2).hom ≫ adj2.counit).app x = adj1.counit.app x"}}
+{"state": {"context": ["n m : ℕ", "hnm : n < m"], "goal": "filter (fun x => decide (x ≤ n)) (Ico n m) = [n]"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "J₁ J₂ : GrothendieckTopology C", "X Y : C", "f : Y ⟶ X", "S : Sieve X"], "goal": "J₁.close (Sieve.pullback f S) ≤ Sieve.pullback f (J₁.close S)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : ConditionallyCompleteLinearOrder α", "inst✝ : Nonempty ι", "f : ι → α", "s : Set α", "x : α", "hne : s.Nonempty", "hbdd : BddAbove s", "hlim : ¬IsSuccLimit (sSup s)", "y : α", "hy : y ⋖ sSup s"], "goal": "sSup s ∈ s"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁸ : Semiring R", "inst✝⁷ : Semiring R₂", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup M₂", "inst✝³ : Module R₂ M₂", "τ₁₂ : R →+* R₂", "inst✝² : RingHomSurjective τ₁₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F τ₁₂ M M₂", "f : F", "p : Submodule R M", "x y : M", "hy : y ∈ ↑p", "e : f y = f x"], "goal": "x ∈ p ⊔ LinearMap.ker f"}}
+{"state": {"context": ["M : Type u_1", "CancelCommMonoidWithZero M", "q : Associates M", "n : ℕ", "hn : n ≠ 0", "c : Fin (n + 1) → Associates M", "h₁ : StrictMono c", "h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i", "hq : q ≠ 0"], "goal": "Irreducible (c 1)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁸ : NontriviallyNormedField 𝕜", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : TopologicalSpace H", "inst✝⁴ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝³ : NormedAddCommGroup E'", "inst✝² : NormedSpace 𝕜 E'", "inst✝¹ : TopologicalSpace H'", "inst✝ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "x : M", "hy : x ∈ f.source"], "goal": "map (↑(f.extend I)) (𝓝 x) = 𝓝[range ↑I] ↑(f.extend I) x"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "z : K"], "goal": "|RCLike.im z| ≤ ‖z‖"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "D : Type u_2", "CategoryTheory.Category.{u_4, u_2} D", "F : C ⥤ D", "X Y : AlgebraicGeometry.PresheafedSpace C", "f : X ⟶ Y"], "goal": "(F.mapPresheaf.map f).c = CategoryTheory.whiskerRight f.c F"}}
+{"state": {"context": ["E : Type u_1", "AddCommGroup E", "Module ℝ E", "TopologicalSpace E", "ContinuousAdd E", "ContinuousSMul ℝ E", "h : 1 < Module.rank ℝ E", "x : E"], "goal": "IsPathConnected {x}ᶜ"}}
+{"state": {"context": ["K : Type u", "inst✝⁶ : CommRing K", "G₀ : Type u_1", "L : Type u_2", "R : Type u_3", "S : Type u_4", "F : Type u_5", "inst✝⁵ : CommGroupWithZero G₀", "inst✝⁴ : Field L", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : FunLike F R[X] S[X]", "inst✝ : MonoidHomClass F R[X] S[X]", "φ : F", "hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰", "n : R[X]", "d : ↥R[X]⁰"], "goal": "(if h : φ ↑d ∈ S[X]⁰ then { toFractionRing := Localization.mk (φ n) ⟨φ ↑d, h⟩ } else 0) = { toFractionRing := Localization.mk (φ n) ⟨φ ↑d, ⋯⟩ }"}}
+{"state": {"context": [], "goal": "φ⁻¹ = -ψ"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : FiniteDimensional ℝ E", "s : Set E", "hs : IsOpen s", "h's : s.Nonempty", "ι : Type u_1 := { f // support f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 }", "T : Set ι", "T_count : T.Countable", "hT : ⋃ f ∈ T, support ↑f = s", "g0 : ℕ → ι", "hg : T = range g0", "g : ℕ → E → ℝ := fun n => ↑(g0 n)", "g_s : ∀ (n : ℕ), support (g n) ⊆ s", "s_g : ∀ x ∈ s, ∃ n, x ∈ support (g n)"], "goal": "∃ f, support f = s ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommRing R", "Ring S", "f : R →+* S", "I : Type u_3", "Fintype I", "e : I → R", "h : ∀ x ∈ RingHom.ker f, IsNilpotent x", "he : ∀ (i : I), IsIdempotentElem (e i)", "he' : CompleteOrthogonalIdempotents (⇑f ∘ e)"], "goal": "CompleteOrthogonalIdempotents e"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "a : ℤ", "ha : ↑a ≠ 0"], "goal": "(quadraticChar (ZMod p)) ↑(a ^ 2) = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕝 : Type u_2", "E : Type u_3", "F : Type u_4", "β : Type u_5", "inst✝⁸ : OrderedRing 𝕜", "inst✝⁷ : TopologicalSpace E", "inst✝⁶ : TopologicalSpace F", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : AddCommGroup F", "inst✝³ : Module 𝕜 E", "inst✝² : Module 𝕜 F", "s t : Set E", "x y : E", "inst✝¹ : Nontrivial 𝕜", "inst✝ : DenselyOrdered 𝕜", "hs : StrictConvex 𝕜 s", "hx : x ∈ s", "hy : y ∈ s", "h : openSegment 𝕜 x y ⊆ frontier s", "a : 𝕜", "ha₀ : 0 < a", "ha₁ : a < 1"], "goal": "x = y"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "s : Set (PrimeSpectrum R)"], "goal": "(PrimeSpectrum.vanishingIdeal s).IsRadical"}}
+{"state": {"context": ["ι : Type uι", "inst✝¹⁰ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁹ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝⁶ : SeminormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "E' : ι → Type u_1", "E'' : ι → Type u_2", "inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E' i)", "inst✝³ : (i : ι) → NormedSpace 𝕜 (E' i)", "inst✝² : (i : ι) → SeminormedAddCommGroup (E'' i)", "inst✝¹ : (i : ι) → NormedSpace 𝕜 (E'' i)", "g : (i : ι) → E' i →L[𝕜] E'' i", "f : (i : ι) → E i →L[𝕜] E' i", "inst✝ : DecidableEq ι", "i : ι", "u : E i →L[𝕜] E' i"], "goal": "update (fun j => ↑(f j)) i ↑u = fun j => ↑(update f i u j)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "L : CategoryTheory.Functor C D", "R : CategoryTheory.Functor D C", "h : L ⊣ R"], "goal": "h.toComonad.ε = h.counit"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : AddCommMonoid E", "inst✝² : AddCommMonoid F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜", "h : B.SeparatingRight", "y : F", "hy : y ∈ B.polar Set.univ", "x : E"], "goal": "(B x) y = 0"}}
+{"state": {"context": ["α : Type u", "ConditionallyCompleteLinearOrder α", "TopologicalSpace α", "OrderTopology α", "DenselyOrdered α", "s : Set α", "h : s.OrdConnected"], "goal": "IsPreconnected s"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "p1 p2 p3 : P"], "goal": "dist p1 p3 * dist p1 p3 = dist p1 p2 * dist p1 p2 + dist p3 p2 * dist p3 p2 - 2 * dist p1 p2 * dist p3 p2 * Real.cos (∠ p1 p2 p3)"}}
+{"state": {"context": ["G : Type u_1", "inst✝ : Group G", "z : G"], "goal": "z ∈ center G ↔ ∀ (g : G), g * z = z * g"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "CommSemiring R", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "Module R M", "Module R N", "Module R P", "f : M →ₗ[R] N →ₗ[R] P", "p : Submodule R M", "q : Submodule R N"], "goal": "Submodule.map₂ f p q = Submodule.span R (Set.image2 (fun m n => (f m) n) ↑p ↑q)"}}
+{"state": {"context": ["E : Type u", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "F : Type v", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℂ F", "f : E → F", "z : E", "r : ℝ", "hd : DiffContOnCl ℂ f (ball z r)", "hz : IsMaxOn (norm ∘ f) (ball z r) z", "w : E", "hw : dist z w ≤ r", "hne : z ≠ w", "e : ℂ → E := ⇑(lineMap z w)"], "goal": "(norm ∘ f) w = Function.const E ‖f z‖ w"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : CommGroupWithZero G���", "a b c d : G₀", "hb : b ≠ 0", "hc : c ≠ 0"], "goal": "a = c / d * b ↔ a / c = b / d"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "s : ι → Set α", "S : Set ι"], "goal": "generateFrom (piiUnionInter (fun k => {s k}) S) = generateFrom {t | ∃ k ∈ S, s k = t}"}}
+{"state": {"context": ["R : Type u", "M : Type v", "ι : Type u_3", "AddCommMonoid R", "AddCommMonoid M", "s : Finset ι", "f : ι → R"], "goal": "TrivSqZeroExt.inl (∑ i ∈ s, f i) = ∑ i ∈ s, TrivSqZeroExt.inl (f i)"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : Ring S", "inst✝⁶ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁵ : CommRing A", "inst✝⁴ : CommRing B", "inst✝³ : IsDomain B", "inst✝² : Algebra A B", "K : Type u_6", "inst✝¹ : Field K", "inst✝ : Algebra K S", "x : S", "h : IsIntegral K x", "g : Fin (minpoly K x).natDegree → K", "i : Fin (minpoly K x).natDegree", "hg : (aeval x) (∑ x_1 : Fin (minpoly K x).natDegree, (monomial ↑x_1) (g x_1)) = 0"], "goal": "g i = 0"}}
+{"state": {"context": ["a b r : ℤ", "hr : 0 < r", "x : ℤ"], "goal": "((a ≤ x ∧ x < b) ∧ ∃ c, x = c * r) ↔ ∃ a_1, (a ≤ a_1 * r ∧ a_1 * r < b) ∧ { toFun := fun x => x * r, inj' := ⋯ } a_1 = x"}}
+{"state": {"context": ["d : ℤ", "h₀ : 0 < d", "hd : ¬IsSquare d", "ξ : ℝ := √↑d", "hξ : Irrational ξ", "M : ℤ", "hM₁ : 2 * |ξ| + 1 < ↑M", "hM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite", "m : ℤ", "hm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite", "hm₀ : m ≠ 0", "this : NeZero m.natAbs", "f : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q => (↑q.num, ↑q.den)", "q₁ : ℚ", "h₁ : q₁.num ^ 2 - d * ↑q₁.den ^ 2 = m", "q₂ : ℚ", "h₂ : q₂.num ^ 2 - d * ↑q₂.den ^ 2 = m", "hne : q₁ ≠ q₂", "hqf : f q₁ = f q₂"], "goal": "∃ x y, x ^ 2 - d * y ^ 2 = 1 ∧ y ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "K : Type u_2", "inst✝⁴ : Field K", "inst✝³ : Algebra R K", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "f : R[X]", "hf : f.Monic", "g : K[X]", "hg : g ∣ map (algebraMap R K) f", "g_ne_0 : g ≠ 0", "g_mul_dvd : g * C g.leadingCoeff⁻¹ ∣ map (algebraMap R K) f", "algeq : ↥(integralClosure R K) ≃ₐ[R] R := ((integralClosure R K).equivOfEq ⊥ ⋯).trans (Algebra.botEquivOfInjective ⋯)", "this : (algebraMap R K).comp (↑algeq).toRingHom = (integralClosure R K).toSubring.subtype", "H : ∃ q, map (algebraMap (↥(integralClosure R K)) K) q = g * C g.leadingCoeff⁻¹"], "goal": "∃ g', map (algebraMap R K) g' = g * C g.leadingCoeff⁻¹"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "p : R[X]", "Semiring S", "n : ℕ", "f : ℕ → R → S[X]"], "goal": "(p.sum f).coeff n = p.sum fun a b => (f a b).coeff n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝² : UniformSpace α", "inst✝¹ : UniformSpace β", "inst✝ : UniformSpace γ", "f : α → β", "hf : UniformInducing f"], "goal": "CompleteSpace α ↔ IsComplete (range f)"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedAddCommGroup F", "NormedSpace ℝ F", "I : BoxIntegral.Box ι", "Fintype ι", "l : BoxIntegral.IntegrationParams", "f : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y : F", "h : BoxIntegral.HasIntegral I l f vol y"], "goal": "BoxIntegral.Integrable I l f vol"}}
+{"state": {"context": ["m : Type u_1", "A : Type u_3", "Fintype m", "CommRing A", "IsDomain A", "DecidableEq m", "M : Matrix m m A", "hM : M.det ≠ 0", "v : m → A", "hv : Matrix.vecMul v M = 0"], "goal": "v = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : MetricSpace α", "β : Type u", "inst✝ : Nonempty β", "p : TauPackage β α", "h : ¬{i | ¬∃ b, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty", "x : Ordinal.{u}", "hxy : p.index x = p.index x", "x_le_y : x ≤ x"], "goal": "x = x"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "J : GrothendieckTopology C", "A : Type u'", "inst✝ : Category.{v', u'} A", "I : Type u_1", "Y : I → C", "hY : J.CoversTop Y", "F : Sheaf J (Type u_2)", "x y : ↑F.val.sections", "h : ∀ (i : I), ↑x (Opposite.op (Y i)) = ↑y (Opposite.op (Y i))", "W : Cᵒᵖ"], "goal": "↑x W = ↑y W"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "s : Set α", "t : Set β", "u : Set γ"], "goal": "⇑(Equiv.prodAssoc α β γ).symm ⁻¹' (s ×ˢ t) ×ˢ u = s ×ˢ t ×ˢ u"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "hp : p ≠ 2"], "goal": "p % 8 ≠ 5 ∧ p % 8 ≠ 7 ↔ p % 8 = 1 ∨ p % 8 = 3"}}
+{"state": {"context": ["M : Type u", "AddMonoid M", "α : Type v", "AddAction M α", "a₁ a₂ : α"], "goal": "a₂ ∈ AddAction.orbit M a₁ ↔ ∃ x, x +ᵥ a₁ = a₂"}}
+{"state": {"context": ["α : Type u_1", "G₀ : Type u_3", "GroupWithZero G₀", "TopologicalSpace G₀", "HasContinuousInv₀ G₀", "x : G₀", "l : Filter α", "f : α → G₀", "hx : x ≠ 0"], "goal": "Filter.Tendsto (fun x => (f x)⁻¹) l (𝓝 x⁻¹) ↔ Filter.Tendsto f l (𝓝 x)"}}
+{"state": {"context": ["V : Type u", "inst✝² : Category.{v, u} V", "inst✝¹ : HasZeroMorphisms V", "ι : Type u_1", "c : ComplexShape ι", "T : Type u_2", "inst✝ : Category.{?u.78, u_2} T", "C : HomologicalComplex (T ⥤ V) c", "t : T", "i j k : ι", "x✝¹ : c.Rel i j", "x✝ : c.Rel j k"], "goal": "(fun i j => (C.d i j).app t) i j ≫ (fun i j => (C.d i j).app t) j k = 0"}}
+{"state": {"context": ["α : Sort u", "b c : Prop", "Decidable b", "Decidable c", "x y u v : α", "h_c : b ↔ c", "h_t : x = u", "h_e : y = v"], "goal": "(if b then x else y) = if c then u else v"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "D : GlueData C", "inst✝ : HasLimits C", "i j : D.J", "U : Opens ↑↑(D.U i)"], "goal": "D.diagramOverOpenπ U i ≫ (D.f i j).c.app (op ((Opens.map (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))).base).obj (⋯.functor.obj U))) ≫ (D.V (i, j)).presheaf.map ((Opens.map (D.f i j).base).op.map ((((eqToIso ⋯).inv ≫ (homOfLE ⋯).op) ≫ (opensFunctor (D.f i i)).op.map (𝟙 ((Opens.map (D.f i i).base).op.obj (op U)) ≫ eqToHom ⋯)) ≫ (eqToHom ⋯).op) ≫ eqToHom ⋯) = limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (i, j)))"}}
+{"state": {"context": ["w z : ℂ"], "goal": "(Complex.orientation.kahler w) z = (starRingEnd ℂ) w * z"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Ring α", "inst✝ : CharZero α", "n✝¹ n✝ : ℤ", "h : decide (n✝¹ = n✝) = false"], "goal": "¬↑n✝¹ = ↑n✝"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝⁸ : CommSemiring R", "inst✝⁷ : CommSemiring S", "inst✝⁶ : Semiring A", "inst✝⁵ : Semiring B", "inst✝⁴ : Algebra R S", "inst✝³ : Algebra R A", "inst✝² : Algebra S A", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R S A", "s✝ t : Set A", "f : A →ₐ[R] B", "s : Set A"], "goal": "(fun x => x) '' s ⊆ (Subalgebra.comap f (adjoin R (⇑f '' s))).carrier"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "H : Type u_3", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "CategoryTheory.Category.{u_6, u_3} H", "L : CategoryTheory.Functor C D", "F : CategoryTheory.Functor C H", "L.HasPointwiseRightKanExtension F", "Y₁ Y₂ : D", "f : Y₁ ⟶ Y₂"], "goal": "(L.pointwiseRightKanExtension F).map f =\n  CategoryTheory.Limits.limit.lift ((CategoryTheory.StructuredArrow.proj Y₂ L).comp F)\n    { pt := CategoryTheory.Limits.limit ((CategoryTheory.StructuredArrow.proj Y₁ L).comp F),\n      π :=\n        {\n          app := fun g =>\n            CategoryTheory.Limits.limit.π ((CategoryTheory.StructuredArrow.proj Y₁ L).comp F)\n              ((CategoryTheory.StructuredArrow.map f).obj g),\n          naturality := ⋯ } }"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "f : X → Y", "f_cont : SeqContinuous f", "K : Set X", "K_cpt : IsSeqCompact K", "ys : ℕ → Y", "xs : ℕ → X", "xs_in_K : ∀ (n : ℕ), xs n ∈ K", "fxs_eq_ys : ∀ (n : ℕ), f (xs n) = ys n", "a : X", "a_in_K : a ∈ K", "phi : ℕ → ℕ", "phi_mono : StrictMono phi", "xs_phi_lim : Tendsto (xs ∘ phi) atTop (𝓝 a)"], "goal": "Tendsto (ys ∘ phi) atTop (𝓝 (f a))"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "ι : Type u_2", "c : ComplexShape ι", "CategoryTheory.Preadditive C", "CategoryTheory.CategoryWithHomology C", "(HomologicalComplex.quasiIso C c).HasLocalization", "c.QFactorsThroughHomotopy C"], "goal": "(HomotopyCategory.quasiIso C c).IsInvertedBy HomologicalComplexUpToQuasiIso.Qh"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "x : L", "m : M", "f : M →ₗ[R] R"], "goal": "⁅x, f⁆ m = -f ⁅x, m⁆"}}
+{"state": {"context": ["n : ℕ", "x✝ : Fin (n + 1)", "i : ℕ", "h : i < n + 1"], "goal": "(i, n - i) ∈ antidiagonal n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : Semiring R", "inst✝ : NoZeroDivisors R", "p✝ q✝ p q : R[X]", "h2 : p ≠ 0 ∧ q ≠ 0"], "goal": "p.natDegree ≤ p.natDegree + q.natDegree"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "f g : α → E", "s t : Set α", "μ ν : Measure α"], "goal": "IntegrableOn f univ μ ↔ Integrable f μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "Zero M", "f : α × β → M"], "goal": "(Function.support fun a b => f (a, b)) = Prod.fst '' Function.support f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℝ E", "s t : Set E", "x : E", "a : ℝ", "hs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s", "hs₂ : Absorbs ℝ s {x}", "hx : x ∉ a • s", "r : ℝ", "hr : r > 0", "h : ∀ (c : ℝ), r ≤ ‖c‖ → {x} ⊆ c • s"], "goal": "a ≤ gauge s x"}}
+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : DecidableEq (K →+* ℂ)", "α : Type u_2", "β : Type u_3", "inst✝¹ : Fintype α", "inst✝ : Fintype β", "a : Matrix α β (𝓞 K)", "ha : a ≠ 0", "hs : ∀ (a_1 : α × (K →+* ℂ)), (fun l => ((NumberField.house.newBasis K).repr (a a_1.1 l.1 * (NumberField.house.newBasis K) l.2)) a_1.2) = 0 a_1"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "h : x ∈ span R (↑p ∪ ↑p')"], "goal": "∀ x ∈ ↑p ∪ ↑p', ∃ y ∈ p, ∃ z ∈ p', y + z = x"}}
+{"state": {"context": ["X : Type u_1", "n : Type u_5", "R : Type u_8", "AddCommMonoid R", "TopologicalSpace R", "f : X → Matrix n n R", "hf : Summable f"], "goal": "Summable fun x => (f x).diag"}}
+{"state": {"context": ["α : Type u", "r : α → α → Prop", "DecidableRel r", "a : α"], "goal": "List.orderedInsert r a [] = [a]"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "M : Type u_4", "N : Type u_5", "inst✝ : MulOneClass M", "s t : Set α", "f✝ g✝ : α → M", "a : α", "f g : α → M", "h : Disjoint (mulSupport f) (mulSupport g)", "x : α", "hx : x ∈ mulSupport g", "this : f x = 1"], "goal": "(f * g) x = g x"}}
+{"state": {"context": [], "goal": "Real.log 2 * 2 ≤ √(Real.log 8)"}}
+{"state": {"context": ["L : FirstOrder.Language", "ι : Type v", "Preorder ι", "G : ι → Type w", "(i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "IsDirected ι fun x x_1 => x ≤ x_1", "DirectedSystem G fun i j h => ⇑(f i j h)", "Nonempty ι", "C : FirstOrder.Language.DirectLimit G f → Prop", "z : FirstOrder.Language.DirectLimit G f", "ih : ∀ (i : ι) (x : G i), C ((FirstOrder.Language.DirectLimit.of L ι G f i) x)"], "goal": "C z"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "FiniteDimensional ℝ E", "MeasurableSpace E", "BorelSpace E", "C : ℝ≥0", "f : E → ℝ", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "v : E"], "goal": "MeasureTheory.LocallyIntegrable (fun x => lineDeriv ℝ f x v) μ"}}
+{"state": {"context": ["α : Type u_2", "CompleteLattice α", "IsCompactlyGenerated α", "a : α", "s : Set α", "h : DirectedOn (fun x x_1 => x ≤ x_1) s"], "goal": "sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "f : α → β", "Preorder α", "Preorder β", "h : StrictAntiOn f s"], "goal": "StrictAnti (f ∘ Subtype.val)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "E' : Type u_3", "F : Type u_4", "G : Type u_5", "G' : Type u_6", "𝕜 : Type u_7", "p : ℝ≥0∞", "inst✝¹³ : RCLike 𝕜", "inst✝¹² : NormedAddCommGroup E", "inst✝¹¹ : InnerProductSpace 𝕜 E", "inst✝¹⁰ : CompleteSpace E", "inst✝⁹ : NormedAddCommGroup E'", "inst✝⁸ : InnerProductSpace 𝕜 E'", "inst✝⁷ : CompleteSpace E'", "inst✝⁶ : NormedSpace ℝ E'", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedAddCommGroup G'", "inst✝¹ : NormedSpace ℝ G'", "inst✝ : CompleteSpace G'", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "hm : m ≤ m0", "hs : MeasurableSet s", "hμs : μ s ≠ ⊤", "f : ↥(Lp ℝ 2 μ)", "hf : ↑↑f =ᶠ[ae (μ.restrict s)] 0"], "goal": "Measurable fun x => ↑‖↑↑f x‖₊"}}
+{"state": {"context": ["C : Type u_1", "H : Type u_3", "D : Type u_5", "CategoryTheory.Category.{u_7, u_1} C", "CategoryTheory.Category.{u_8, u_3} H", "CategoryTheory.Category.{u_9, u_5} D", "F' : CategoryTheory.Functor D H", "L : CategoryTheory.Functor C D", "F : CategoryTheory.Functor C H", "α : F ⟶ L.comp F'", "F'.IsLeftKanExtension α", "G : CategoryTheory.Functor C H", "i : F ≅ G", "G' : CategoryTheory.Functor D H", "β : G ⟶ L.comp G'", "G'.IsLeftKanExtension β"], "goal": "(F'.leftKanExtensionUniqueOfIso α i G' β).inv =\n  G'.descOfIsLeftKanExtension β F' (CategoryTheory.CategoryStruct.comp i.inv α)"}}
+{"state": {"context": ["F : Type u_1", "E : Type u_2", "inst✝³ : Field F", "inst✝² : Field E", "inst✝¹ : Algebra F E", "inst✝ : Algebra.IsAlgebraic F E", "this : FiniteDimensional F E", "α : E", "hprim : F⟮α⟯ = ⊤", "f : F[X] := minpoly F α", "G : Type u_2 := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }", "hfin : Finite G"], "goal": "Finite (IntermediateField F E)"}}
+{"state": {"context": ["R : Type u", "M : Type v", "M₂ : Type w", "Semiring R", "AddCommMonoid M", "AddCommMonoid M₂", "Module R M", "Module R M₂"], "goal": "LinearMap.fst R M₂ M ∘ₗ ↑(LinearEquiv.prodComm R M M₂) = LinearMap.snd R M M₂"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : LinearOrderedRing α", "inst✝² : FloorRing α", "z : ℤ", "a : α", "k : Type u_4", "inst✝¹ : LinearOrderedField k", "inst✝ : FloorRing k", "b : k", "n : ℕ", "hn : 0 < ↑n", "this : ∀ {l : ℤ}, 0 ≤ l → fract (↑l / ↑n) = ↑(l % ↑n) / ↑n", "m₀ : ℕ", "q : ℤ := ⌈↑m₀ / ↑n⌉", "m₁ : ℤ := q * ↑n - ↑m₀", "hm₁ : 0 ≤ m₁"], "goal": "fract (↑(-↑m₀) / ↑n) = ↑(-↑m₀ % ↑n) / ↑n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "CommMonoid α", "TopologicalSpace α", "f : β → γ → α", "b : β", "c : γ", "hfb : ∀ (b' : β), b' ≠ b → f b' c = 1", "hfc : ∀ (b' : β) (c' : γ), c' ≠ c → f b' c' = 1"], "goal": "∏' (b' : β) (c' : γ), f b' c' = f b c"}}
+{"state": {"context": ["R : Type u₁", "Ring R", "f : R →+* R", "hf : f = RingHom.id R", "M : ModuleCat R", "x : ↑M"], "goal": "(ModuleCat.restrictScalarsId'App f hf M).hom x = x"}}
+{"state": {"context": ["n✝ m n : ℕ", "a b : Fin n"], "goal": "↑(a + b) = if n ≤ ↑a + ↑b then ↑a + ↑b - n else ↑a + ↑b"}}
+{"state": {"context": ["R : Type u_1", "Rₛ : Type u_2", "inst✝¹¹ : CommSemiring R", "S : Submonoid R", "inst✝¹⁰ : CommSemiring Rₛ", "inst✝⁹ : Algebra R Rₛ", "hT : IsLocalization S Rₛ", "M : Type u_3", "M' : Type u_4", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : Module R M", "inst✝⁶ : Module Rₛ M", "inst✝⁵ : IsScalarTower R Rₛ M", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : Module R M'", "inst✝² : Module Rₛ M'", "inst✝¹ : IsScalarTower R Rₛ M'", "f : M →ₗ[R] M'", "inst✝ : IsLocalizedModule S f", "ι : Type u_5", "v : ι → M", "hv : ∀ (s : Finset ι) (g : ι → R), ∑ i ∈ s, g i • v i = 0 → ∀ i ∈ s, g i = 0", "t : Finset ι", "g : ι → Rₛ", "hg : ∑ i ∈ t, g i • (⇑f ∘ v) i = 0", "i : ι", "hi : i ∈ t"], "goal": "g i = 0"}}
+{"state": {"context": ["F A : Class"], "goal": "F ′ A ∈ Class.univ"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve.Affine R", "S : Type v", "CommRing S", "f : R →+* S", "x₁ x₂ y₁ L : R"], "goal": "(WeierstrassCurve.map W f).toAffine.negAddY (f x₁) (f x₂) (f y₁) (f L) = f (W.negAddY x₁ x₂ y₁ L)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "self : CategoryTheory.CategoryWithHomology C", "S : CategoryTheory.ShortComplex C"], "goal": "S.HasHomology"}}
+{"state": {"context": ["n : ℕ", "f : Mathlib.Vector ℕ n →. ℕ", "g : Mathlib.Vector ℕ (n + 1) →. ℕ", "hf : Nat.Partrec' f", "hg : Nat.Partrec' g"], "goal": "Nat.Partrec' fun v => (f v).bind fun a => g (a ::ᵥ v)"}}
+{"state": {"context": ["z τ : ℂ"], "goal": "jacobiTheta₂' (z + τ) τ =\n  cexp (-↑π * Complex.I * (τ + 2 * z)) * (jacobiTheta₂' z τ - 2 * ↑π * Complex.I * jacobiTheta₂ z τ)"}}
+{"state": {"context": ["r : ℝ", "n : ℕ", "n.AtLeastTwo"], "goal": "OfNat.ofNat n < r.toNNReal ↔ ↑n < r"}}
+{"state": {"context": ["R : Type u_1", "R₁ : Type u_2", "M : Type u_5", "M₁ : Type u_6", "CommRing R", "CommRing R₁", "AddCommGroup M₁", "Module R₁ M₁", "AddCommGroup M", "Module R M", "I₁ I₂ : R₁ →+* R", "B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] M", "N : Submodule R₁ M₁", "m : M₁"], "goal": "m ∈ N.orthogonalBilin B ↔ ∀ n ∈ N, B.IsOrtho n m"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_3", "inst✝⁷ : TopologicalSpace B", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "EB : Type u_4", "inst✝³ : NormedAddCommGroup EB", "inst✝² : NormedSpace 𝕜 EB", "HB : Type u_5", "inst✝¹ : TopologicalSpace HB", "inst✝ : ChartedSpace HB B", "IB : ModelWithCorners 𝕜 EB HB", "e : PartialHomeomorph (B × F) (B × F)", "s : ↑e.source → Set (B × F)", "hs : ∀ (x : ↑e.source), IsOpen (s x)", "hsp : ∀ (x : ↑e.source), ↑x ∈ s x", "φ : ↑e.source → B → F ≃L[𝕜] F", "u : ↑e.source → Set B", "hu : ∀ (x : ↑e.source), IsOpen (u x)", "hφ : ∀ (x : ↑e.source), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(φ x x_1)) (u x)", "h2φ : ∀ (x : ↑e.source), SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x_1 => ↑(φ x x_1).symm) (u x)", "heφ : ∀ (x : ↑e.source), (e.restr (s x)).EqOnSource (FiberwiseLinear.partialHomeomorph (φ x) ⋯ ⋯ ⋯)", "hesu : ∀ (p : ↑e.source), e.source ∩ s p = u p ×ˢ univ", "hu' : ∀ (p : ↑e.source), (↑p).1 ∈ u p", "heu : ∀ (p : ↑e.source) (q : B × F), q.1 ∈ u p → q ∈ e.source"], "goal": "∃ U, e.source = U ×ˢ univ ∧ ∀ x ∈ U, ∃ φ u, ∃ (hu : IsOpen u) (_ : u ⊆ U) (_ : x ∈ u) (hφ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x)) u) (h2φ : SmoothOn IB 𝓘(𝕜, F →L[𝕜] F) (fun x => ↑(φ x).symm) u), (e.restr (u ×ˢ univ)).EqOnSource (FiberwiseLinear.partialHomeomorph φ hu ⋯ ⋯)"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "S₂ : Type w", "S₃ : Type x", "σ : Type u_1", "a a' a₁ a₂ : R", "e : ℕ", "s : σ →₀ ℕ", "inst✝ : CommSemiring R", "f : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)", "g : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X"], "goal": "∀ (p : MvPolynomial PUnit.{?u.2149 + 1} R), (f.comp g) p = p"}}
+{"state": {"context": ["J : Type v", "CategoryTheory.Category.{w, v} J", "F : J ⥤ CommGrp"], "goal": "CategoryTheory.Limits.HasLimit F ↔ Small.{u, max u v} ↑(F ⋙ CategoryTheory.forget CommGrp).sections"}}
+{"state": {"context": ["q : Turing.PartrecToTM2.Λ'"], "goal": "Turing.PartrecToTM2.tr q.copy =\n  Turing.PartrecToTM2.pop' Turing.PartrecToTM2.K'.rev\n    (Turing.TM2.Stmt.branch Option.isSome\n      (Turing.PartrecToTM2.push' Turing.PartrecToTM2.K'.main\n        (Turing.PartrecToTM2.push' Turing.PartrecToTM2.K'.stack (Turing.TM2.Stmt.goto fun x => q.copy)))\n      (Turing.TM2.Stmt.goto fun x => q))"}}
+{"state": {"context": ["J : Type v", "F : J → TypeMax", "j : J"], "goal": "(↾fun x => ⟨j, x⟩) ≫ (CategoryTheory.Limits.Types.coproductIso F).inv = CategoryTheory.Limits.Sigma.ι F j"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedHeytingAlgebra α", "a b : α"], "goal": "a ⇔ b ⇔ (a ⊔ b) = a ⊓ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "l✝ m : Language α", "a b x : List α", "f : α → β", "l : Language α"], "goal": "(map f) l∗ = ((map f) l)∗"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Group α", "MulAction α β", "s t : Set β", "a : α"], "goal": "a • (s ∩ t) = a • s ∩ a • t"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "c₁ c₂ : R"], "goal": "(CliffordAlgebraQuaternion.quaternionBasis c₁ c₂).j = (CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (0, 1)"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "hS : S ≤ IsUnit.submonoid M", "x : ↥S.leftInv"], "goal": "(S.leftInvEquiv hS).symm (S.fromLeftInv x) = x"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p : ℝ≥0∞", "(i : α) → NormedAddCommGroup (E i)", "DecidableEq α", "hp : 0 < p.toReal", "f : (i : α) → E i", "i : α"], "goal": "‖lp.single p i (f i)‖ = ‖f i‖"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝ : Semiring R", "n : ℕ", "f : R[X]"], "goal": "(∀ i ≥ n, f ∈ LinearMap.ker (lcoeff R i)) ↔ f.support.sup WithBot.some < ↑n"}}
+{"state": {"context": ["M : Type u_1", "Add M", "c d : AddCon M"], "goal": "c ⊔ d = addConGen fun x y => c x y ∨ d x y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "T : Set α → F →L[ℝ] F'", "h_add : FinMeasAdditive μ T", "f : α →ₛ G", "hf : Integrable (↑f) μ", "g : G → F", "hg : g 0 = 0", "T_empty : T ∅ = 0", "hfp : ∀ x ∈ f.range, x ≠ 0 → μ (↑f ⁻¹' {x}) ≠ ⊤", "a : α", "hb : ↑f a ∈ f.range", "h0 : ¬g (↑f a) = 0", "h_left_eq : (T (↑(map g f) ⁻¹' {g (↑f a)})) (g (↑f a)) = (T (↑f ⁻¹' ↑(filter (fun b => g b = g (↑f a)) f.range))) (g (↑f a))", "h_left_eq' : (T (↑f ⁻¹' ↑(filter (fun b => g b = g (↑f a)) f.range))) (g (↑f a)) = (T (⋃ y ∈ filter (fun b => g b = g (↑f a)) f.range, ↑f ⁻¹' {y})) (g (↑f a))"], "goal": "∀ i ∈ filter (fun b => g b = g (↑f a)) f.range, μ (↑f ⁻¹' {i}) ≠ ⊤"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "p : α → Prop", "inst✝¹ : DecidablePred p", "inst✝ : LocallyFiniteOrderBot α", "a : Subtype p", "hp : ∀ ⦃a x : α⦄, x ≤ a → p a → p x"], "goal": "map (Embedding.subtype p) (Iio a) = Iio ↑a"}}
+{"state": {"context": [], "goal": "¬FermatLastTheoremFor 1"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : AddCommMonoid R", "α : Type u_3", "s : SummableFamily Γ R α", "g : Γ", "hg : g ∈ s.hsum.support"], "goal": "g ∈ ⋃ a, (s a).support"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "M : Type u_4", "AddMonoid M", "CategoryTheory.HasShift C M", "X Y : C", "m₀ : M", "hm₀ : m₀ = 0", "f : X ⟶ Y"], "goal": "(CategoryTheory.ShiftedHom.homEquiv m₀ hm₀) f = CategoryTheory.ShiftedHom.mk₀ m₀ hm₀ f"}}
+{"state": {"context": ["α : Type u", "m : OuterMeasure α", "s✝ s₁ s₂ : Set α", "s : ℕ → Set α", "h : ∀ (i : ℕ), m.IsCaratheodory (s i)", "hd : Pairwise (Disjoint on s)", "n : ℕ", "this : ∑ x ∈ Finset.range n, m (s x) = m (⋃ i, ⋃ (_ : i < n), s i)"], "goal": "∑ a ∈ Finset.range n, m (s a) ≤ m (⋃ i, s i)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n ν : ℕ", "h : n < ν"], "goal": "bernsteinPolynomial R n ν = 0"}}
+{"state": {"context": ["x : ℝ"], "goal": "↑x = 1 ↔ x = 1"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "SuccOrder α", "a : α", "NoMaxOrder α"], "goal": "¬Order.IsSuccLimit a ↔ a ∈ Set.range Order.succ"}}
+{"state": {"context": ["α : Type u_1", "t : ℕ → Set α", "n : ℕ", "s : Set α", "hs : s ∈ memPartition t n"], "goal": "MeasurableSet s"}}
+{"state": {"context": ["α : Type u_1", "F' : Type u_4", "𝕜 : Type u_7", "RCLike 𝕜", "NormedAddCommGroup F'", "NormedSpace 𝕜 F'", "NormedSpace ℝ F'", "CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "hm : m ≤ m0", "MeasureTheory.SigmaFinite (μ.trim hm)", "c : 𝕜", "f : α → F'"], "goal": "MeasureTheory.condexpL1 hm μ (c • f) = c • MeasureTheory.condexpL1 hm μ f"}}
+{"state": {"context": ["s : Set ℝ"], "goal": "frontier (Complex.re ⁻¹' s) = Complex.re ⁻¹' frontier s"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁵ : MeasurableSpace X", "inst✝⁴ : NormedAddCommGroup E", "𝕜 : Type u_5", "inst✝³ : NormedField 𝕜", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "p : ℝ≥0∞", "μ : Measure X", "inst✝ : NormedSpace ℝ E", "s : Set X", "this : Fact (1 ≤ 1)", "h_comp : (fun f => ∫ (x : X) in s, ↑↑f x ∂μ) = integral (μ.restrict s) ∘ fun f => ↑↑((LpToLpRestrictCLM X E ℝ μ 1 s) f)"], "goal": "Continuous fun f => ∫ (x : X) in s, ↑↑f x ∂μ"}}
+{"state": {"context": ["k G : Type u", "Field k", "Monoid G", "V : FDRep k G"], "goal": "((CategoryTheory.forget₂ (FDRep k G) (Rep k G)).obj V).ρ = V.ρ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "UniformSpace β", "𝔖 : Set (Set α)", "f : α →ᵤ[𝔖] β"], "goal": "𝓝 f = ⨅ s ∈ 𝔖, ⨅ V ∈ 𝓤 β, 𝓟 {g | ∀ x ∈ s, ((UniformOnFun.toFun 𝔖) f x, (UniformOnFun.toFun 𝔖) g x) ∈ V}"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝¹ : EMetricSpace X", "inst✝ : EMetricSpace Y", "μ : OuterMeasure X", "hm : μ.IsMetric", "t : Set X", "ht : t ∈ {s | IsClosed s}", "s : Set X", "S : ℕ → Set X := fun n => {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEdist x t}", "Ssep : ∀ (n : ℕ), IsMetricSeparated (S n) t", "Ssep' : ∀ (n : ℕ), IsMetricSeparated (S n) (s ∩ t)"], "goal": "μ (s ∩ t) + μ (s \\ t) ≤ μ s"}}
+{"state": {"context": ["n : Nat"], "goal": "(range n).length = n"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "W : MorphismProperty C", "J : Type u_1", "inst✝¹ : W.RespectsIso", "inst✝ : HasProductsOfShape J C", "hW : ∀ (X₁ X₂ : J → C) (f : (j : J) → X₁ j ⟶ X₂ j), (∀ (j : J), W (f j)) → W (Pi.map f)", "X₁ X₂ : Discrete J ⥤ C", "c₁ : Cone X₁", "c₂ : Cone X₂", "hc₁ : IsLimit c₁", "hc₂ : IsLimit c₂", "f : X₁ ⟶ X₂", "hf : W.functorCategory (Discrete J) f", "φ : (j : J) → X₁.obj { as := j } ⟶ X₂.obj { as := j } := fun j => f.app { as := j }", "hf' : W (Pi.map φ)"], "goal": "W (hc₂.lift { pt := c₁.pt, π := c₁.π ≫ f })"}}
+{"state": {"context": ["k n : ℕ"], "goal": "(∀ a ∈ antidiagonal n, Pairwise (fun a_2 b => Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a.1 a_2) (Fin.cons a.1 b)) (antidiagonalTuple k a.2)) ∧ Pairwise (fun a₁ a₂ => ∀ a ∈ antidiagonalTuple k a₁.2, ∀ a_2 ∈ antidiagonalTuple k a₂.2, Pi.Lex (fun x x_1 => x < x_1) (fun x x x_1 => x < x_1) (Fin.cons a₁.1 a) (Fin.cons a₂.1 a_2)) (antidiagonal n)"}}
+{"state": {"context": ["a b c d : ℝ≥0∞", "r p✝ q✝ : ℝ≥0", "p q : ℝ", "h : 0 < q"], "goal": "ENNReal.ofReal p < ENNReal.ofReal q ↔ p < q"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]"], "goal": "Polynomial.X * p.divX + Polynomial.C (p.coeff 0) = p"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type x", "UniformSpace β", "F : ι → α → β", "f : α → β", "s : Set α", "l : Filter ι", "l.IsCountablyGenerated", "h : ∀ (u : ℕ → ι), Filter.Tendsto u Filter.atTop l → TendstoUniformlyOn (fun n => F (u n)) f Filter.atTop s"], "goal": "TendstoUniformlyOn F f l s"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : FiniteDimensional ℝ E", "inst✝² : MeasurableSpace E", "inst✝¹ : BorelSpace E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "r : ℝ", "hnr : ↑(finrank ℝ E) < r", "hr : 0 < r", "h_meas : Measurable fun ω => (1 + ‖ω‖) ^ (-r)", "h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)", "h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))", "f : ℝ → ℝ≥0∞ := fun t => μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))", "mB : ℝ≥0∞ := μ (Metric.ball 0 1)"], "goal": "lintegral (volume.restrict (Ioi 0)) f < ⊤"}}
+{"state": {"context": ["a b : Rat", "g ad bd : Nat", "hg : g = a.den.gcd b.den", "had : ad = a.den / g", "hbd : bd = b.den / g"], "goal": "let den := ad * b.den;\nlet num := a.num * ↑bd - b.num * ↑ad;\nnum.natAbs.gcd g = num.natAbs.gcd den"}}
+{"state": {"context": ["M : Type u_1", "inst✝² : CommMonoid M", "inst✝¹ : TopologicalSpace M", "m m' : M", "G : Type u_2", "inst✝ : CommGroup G", "g g' : G", "f : ℤ → M", "hf : HasProd f m", "this : Injective Int.negSucc", "u : Finset ℤ", "v' : Finset ℕ", "hv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'"], "goal": "∏ x ∈ image Nat.cast v' ∪ image Int.negSucc v', f x = ∏ b ∈ v', f ↑b * f (Int.negSucc b)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.Limits.HasCoequalizers C", "R S T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "(X : C) →\n    CategoryTheory.Limits.PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁}\n      (CategoryTheory.MonoidalCategory.tensorRight X)"], "goal": "Bimod.TensorBimod.X P Q ◁ T.one ≫ Bimod.TensorBimod.actRight P Q = (ρ_ (Bimod.TensorBimod.X P Q)).hom"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "F : LaxMonoidalFunctor C D", "A : Mon_ C"], "goal": "F.obj A.X ◁ (F.ε ≫ F.map A.one) ≫ F.μ A.X A.X ≫ F.map A.mul = (ρ_ (F.obj A.X)).hom"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "TopologicalSpace α", "UniformSpace β"], "goal": "(𝓤 C(α, β)).HasBasis (fun p => IsCompact p.1 ∧ p.2 ∈ 𝓤 β) fun p => {fg | ∀ x ∈ p.1, (fg.1 x, fg.2 x) ∈ p.2}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : β → β → Prop", "r : α → α → Prop", "b : β", "h : ¬s b b", "a : α"], "goal": "(RelEmbedding.prodLexMkRight r h) a = (a, b)"}}
+{"state": {"context": ["z : ℂ"], "goal": "|z.re / Complex.abs z| ≤ 1"}}
+{"state": {"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝² : TopologicalSpace G", "inst✝¹ : Group G", "inst✝ : TopologicalGroup G", "K V : Set G", "hK : IsCompact K", "hV : (interior V).Nonempty", "t : Finset G", "ht : K ⊆ ⋃ x ∈ t, interior ((fun x_1 => x * x_1) ⁻¹' V)"], "goal": "∃ t, K ⊆ ⋃ g ∈ t, (fun x => g * x) ⁻¹' V"}}
+{"state": {"context": ["G : Type u_2", "MeasurableSpace G", "TopologicalSpace G", "BorelSpace G", "μ : MeasureTheory.Measure G", "AddGroup G", "TopologicalAddGroup G", "μ.IsAddLeftInvariant", "μ.Regular", "hμ : μ ≠ 0", "s : Set G", "hs : IsOpen s"], "goal": "μ s ≠ 0 ↔ s.Nonempty"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "x : ↥S.leftInv"], "goal": "↑(S.fromLeftInv x) * ↑x = 1"}}
+{"state": {"context": ["J : Type w", "K : Type u_1", "C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "f : J → C", "inst✝¹ : HasBiproduct f", "p : J → Prop", "inst✝ : HasBiproduct (Subtype.restrict p f)", "j j✝ : Subtype p"], "goal": "(ι f ↑j ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j✝ = ι (Subtype.restrict p f) j ≫ π (Subtype.restrict p f) j✝"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X X' : C", "f : X ⟶ X'"], "goal": "f =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.leftUnitor X).inv\n    (CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.whiskerLeft (𝟙_ C) f)\n      (CategoryTheory.MonoidalCategory.leftUnitor X').hom)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁴ : CommRing R", "inst✝³ : CommRing A", "inst✝² : Field K", "inst✝¹ : IsDedekindDomain R", "v : HeightOneSpectrum R", "inst✝ : Algebra R K", "hK : IsFractionRing R K", "x : K"], "goal": "(∀ (i : MaximalSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯) → ∀ (i : HeightOneSpectrum R), x ∈ Localization.subalgebra.ofField K i.asIdeal.primeCompl ⋯"}}
+{"state": {"context": ["R : Type u", "L₁ : Type v", "L₂ : Type w", "CommRing R", "LieRing L₁", "LieRing L₂", "LieAlgebra R L₁", "LieAlgebra R L₂", "e : L₁ ≃ₗ⁅R⁆ L₂", "x : L₁"], "goal": "e.symm (e x) = x"}}
+{"state": {"context": ["X : Type u_1", "α : Type u_2", "α' : Type u_3", "β : Type u_4", "γ : Type u_5", "δ : Type u_6", "M : Type u_7", "E : Type u_8", "R : Type u_9", "ι : Type u_10", "inst✝³ : TopologicalSpace X", "inst✝² : Zero R", "inst✝¹ : Zero M", "inst✝ : SMulWithZero R M", "s : ι → X → R", "h✝ : LocallyFinite fun i => support (s i)", "f : ι → X → M", "i : ι", "x : X", "h : s i x = 0"], "goal": "(s i • f i) x = 0"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "x y : V", "h : InnerProductGeometry.angle x y = π"], "goal": "⟪x, y⟫_ℝ = -(‖x‖ * ‖y‖)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : Rel α β", "I I' : Set β", "h : I' ∈ R.rightFixedPoints", "h₁ : I ≤ I'"], "goal": "R.leftDual (R.rightDual I) ≤ I'"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "D : GlueData C", "i j k : D.J", "U : Opens ↑↑(pullback (D.f i j) (D.f i k))", "e : ∀ {Z : C} (h : (D.V (k, i)).presheaf.obj (op ((opensFunctor (pullback.fst (D.f k i) (D.f k j))).obj { carrier := ⇑(D.t' k i j).base ⁻¹' ↑U, is_open' := ⋯ })) ⟶ Z), invApp (pullback.snd (D.f i j) (D.f i k)) U ≫ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) ≫ (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ h = (D.t' k i j).c.app (op U) ≫ invApp (pullback.fst (D.f k i) (D.f k j)) { carrier := ⇑(D.t' k i j).base ⁻¹' ↑U, is_open' := ⋯ } ≫ h"], "goal": "invApp (pullback.snd (D.f i j) (D.f i k)) U ≫ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) = invApp (pullback.snd (D.f i j) (D.f i k)) U ≫ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) ≫ (D.V (k, i)).presheaf.map (eqToHom ⋯) ≫ (D.V (k, i)).presheaf.map (eqToHom ⋯)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℝ E", "s t : Set E", "x✝ : E", "a : ℝ", "x : E"], "goal": "sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = 0 x"}}
+{"state": {"context": [], "goal": "Nat.primeFactors 1 = ∅"}}
+{"state": {"context": ["ι : Sort u_1", "f : ι → ℕ"], "goal": "⨅ i, ↑(f i) < ⊤ ↔ Nonempty ι"}}
+{"state": {"context": ["α : Type u_1", "g : GenContFract α", "n : ℕ", "gp : GenContFract.Pair α", "s_nth_eq : g.s.get? n = some gp"], "goal": "g.partNums.get? n = some gp.a"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "inst✝¹⁹ : CommRing A", "inst✝¹⁸ : Field K", "inst✝¹⁷ : CommRing B", "inst✝¹⁶ : Field L", "inst✝¹⁵ : Algebra A K", "inst✝¹⁴ : Algebra B L", "inst✝¹³ : Algebra A B", "inst✝¹² : Algebra K L", "inst✝¹¹ : Algebra A L", "inst✝¹⁰ : IsScalarTower A K L", "inst✝⁹ : IsScalarTower A B L", "inst✝⁸ : IsDomain A", "inst✝⁷ : IsFractionRing A K", "inst✝⁶ : IsIntegralClosure B A L", "inst✝⁵ : IsFractionRing B L", "inst✝⁴ : FiniteDimensional K L", "inst✝³ : Algebra.IsSeparable K L", "inst✝² : IsDomain B", "inst✝¹ : IsIntegrallyClosed A", "inst✝ : NoZeroSMulDivisors A B", "I : Submodule B (FractionRing B)", "x : L"], "goal": "x ∈ comap (↑(FractionRing.algEquiv B L).toLinearEquiv.symm) (traceDual A (FractionRing A) I) ↔ x ∈ (comap (↑(FractionRing.algEquiv B L).toLinearEquiv.symm) I)ᵛ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "f : α → β", "hf : Function.Surjective f", "m : MeasureTheory.OuterMeasure β"], "goal": "(MeasureTheory.OuterMeasure.map f) ((MeasureTheory.OuterMeasure.comap f) m) = m"}}
+{"state": {"context": ["xl xr : Type u", "x : PGame"], "goal": "-x ≈ 0 ↔ x ≈ 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a b c : α"], "goal": "Set.Ioo a b ∩ Set.Ioi c = Set.Ioo (max a c) b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x : Fin n → ℕ"], "goal": "ThreeAPFree (⇑(map (2 * d - 1)) '' ↑(sphere n d k))"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_3", "Fintype ι", "AddCommGroup E", "Module ℝ E", "b : Basis ι ℝ E"], "goal": "parallelepiped ⇑b = {x | ∀ (i : ι), (b.repr x) i ∈ Set.Icc 0 1}"}}
+{"state": {"context": ["n : ℕ", "n_large : 512 ≤ n"], "goal": "n * (2 * n) ^ (2 * n).sqrt * 4 ^ (2 * n / 3) ≤ 4 ^ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : LinearOrderedField α", "inst✝² : Ring β", "abv : β → α", "inst✝¹ : IsAbsoluteValue abv", "f✝ g : ℕ → β", "a : ℕ → α", "inst✝ : Archimedean α", "f : ℕ → β", "n : ℕ", "r : α", "hr0 : 0 ≤ r", "hr1 : r < 1", "h : ∀ (m : ℕ), n ≤ m → abv (f m.succ) ≤ r * abv (f m)"], "goal": "IsCauSeq abv fun m => ∑ n ∈ range m, f n"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e e' : PartialHomeomorph M H", "f✝ f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s t : Set M", "x : M", "Q : (H → H) → Set H → H → Prop", "hG : G.LocalInvariantProp G' P", "f : M → M'"], "goal": "ContinuousWithinAt f s x ∧ P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) (↑(chartAt H x).symm ⁻¹' s) (↑(chartAt H x) x) ↔ ContinuousWithinAt f s x ∧ P (↑(chartAt H' (f x)) ∘ f ∘ ↑(chartAt H x).symm) ((chartAt H x).target ∩ ↑(chartAt H x).symm ⁻¹' (s ∩ f ⁻¹' (chartAt H' (f x)).source)) (↑(chartAt H x) x)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "W : Affine R", "x₁ x₂ y₁ L : R"], "goal": "evalEval (W.addX x₁ x₂ L) (L * (W.addX x₁ x₂ L - x₁) + y₁) (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆)) = 0 ↔ eval (W.addX x₁ x₂ L) (eval (C L * (X - C x₁) + C y₁) (Y ^ 2 + C (C W.a₁ * X + C W.a₃) * Y - C (X ^ 3 + C W.a₂ * X ^ 2 + C W.a₄ * X + C W.a₆))) = 0"}}
+{"state": {"context": ["α : Type u", "op : α → α → α", "ha : Std.Associative op", "l : List α", "a₁ a₂ : α"], "goal": "op (List.foldl op a₁ l) a₂ = op a₁ (List.foldr (fun x x_1 => op x x_1) a₂ l)"}}
+{"state": {"context": ["R' : Type u_6", "CommRing R'", "x y r : R'", "hr : r ∈ R'⁰"], "goal": "x * r ∣ y * r ↔ x ∣ y"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "X : C", "h : CategoryTheory.yoneda.obj X ⋙ CategoryTheory.uliftFunctor.{u₁, v₃} ≅ F.cones", "Y : C", "f : Y ⟶ X"], "goal": "CategoryTheory.Limits.IsLimit.OfNatIso.coneOfHom h f = (CategoryTheory.Limits.IsLimit.OfNatIso.limitCone h).extend f"}}
+{"state": {"context": [], "goal": "(LinearMap.toMatrix Complex.basisOneI Complex.basisOneI) Complex.conjAe.toLinearMap = !![1, 0; 0, -1]"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E"], "goal": "FiniteDimensional.finrank F ↥⊥ = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Set α", "t : Set β", "f : α → β", "p : β → Prop", "hf : Set.SurjOn f s t", "hf' : Set.MapsTo f s t"], "goal": "(∀ y ∈ t, p y) ↔ ∀ x ∈ s, p (f x)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "s : Finset α"], "goal": "s.card + sᶜ.card = Fintype.card α"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "IsLocalization M S", "Q : Type u_4", "CommSemiring Q", "Algebra R Q", "IsLocalization M Q", "x : R", "y : ↥M"], "goal": "(IsLocalization.algEquiv M S Q).symm (IsLocalization.mk' Q x y) = IsLocalization.mk' S x y"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "s : Set R", "a✝ a b : R", "dvd : a ∣ b"], "goal": "torsionBy R M a ≤ torsionBy R M b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : Kernel α (β × ℝ)", "ν : Kernel α β", "f : α × β → StieltjesFunction", "inst✝ : IsFiniteKernel κ", "hf : IsCondKernelCDF f κ ν", "a : α", "s : Set β", "hs : MeasurableSet s", "h_dir : Directed (fun x y => x ⊆ y) fun q => Iic ↑q"], "goal": "∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) (⋃ r, Iic ↑r) ∂ν a = (κ a) (⋃ i, s ×ˢ Iic ↑i)"}}
+{"state": {"context": ["α : Type u", "l : List α"], "goal": "l.cyclicPermutations ≠ []"}}
+{"state": {"context": ["R : Type u_1", "G : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : IsDomain R", "inst✝² : Group G", "inst✝¹ : Fintype G", "inst✝ : DecidableEq G", "f : G →* R", "hf : Injective ⇑f", "n : ℕ", "hn : 0 < n", "g₀ : G", "this : DecidableEq R"], "goal": "(filter (fun g => g ^ n = g₀) univ).card ≤ Multiset.card (nthRoots n (f g₀))"}}
+{"state": {"context": ["M : Type u_1", "TopologicalSpace M"], "goal": "Continuous AddOpposite.op"}}
+{"state": {"context": ["ι : Type u_1", "ι₂ : Type u_2", "ι₃ : Type u_3", "R : Type u_4", "inst✝¹⁰ : CommSemiring R", "R₁ : Type u_5", "R₂ : Type u_6", "s : ι → Type u_7", "inst✝⁹ : (i : ι) → AddCommMonoid (s i)", "inst✝⁸ : (i : ι) → Module R (s i)", "M : Type u_8", "inst✝⁷ : AddCommMonoid M", "inst✝⁶ : Module R M", "E : Type u_9", "inst✝⁵ : AddCommMonoid E", "inst✝⁴ : Module R E", "F : Type u_10", "inst✝³ : AddCommMonoid F", "t : ι → Type u_11", "inst✝² : (i : ι) → AddCommMonoid (t i)", "inst✝¹ : (i : ι) → Module R (t i)", "inst✝ : IsEmpty ι", "x : ⨂[R] (i : ι), s i"], "goal": "∀ (r : R) (f : (i : ι) → s i), (fun r => r • (tprod R) isEmptyElim) ({ toFun := ⇑(lift (constOfIsEmpty R s 1)), map_add' := ⋯, map_smul' := ⋯ }.toFun (r • (tprod R) f)) = r • (tprod R) f"}}
+{"state": {"context": ["X Y : CommSemiRingCat", "f g : X ⟶ Y", "w : ∀ (x : ↑X), f x = g x"], "goal": "f = g"}}
+{"state": {"context": ["α : Type u_1", "E' : Type u_6", "F'' : Type u_10", "SeminormedAddCommGroup E'", "NormedAddCommGroup F''", "g'' : α → F''", "l : Filter α"], "goal": "(fun x => 0) =Θ[l] g'' ↔ g'' =ᶠ[l] 0"}}
+{"state": {"context": ["m n✝ a✝ b✝ c d : ℤ", "a b n : ℕ"], "goal": "↑a % ↑n = ↑b % ↑n ↔ ↑(a % n) = ↑(b % n)"}}
+{"state": {"context": ["c : Cardinal.{u_1}", "h : ℵ₀ ≤ c"], "goal": "c ^< ℵ₀ ≤ c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "MeasurableSpace α", "Inv β", "f : MeasureTheory.SimpleFunc α β", "x : α"], "goal": "↑f⁻¹ x = (↑f x)⁻¹"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_3", "β : Sort u_2", "S : ι → Set α", "f : (i : ι) → ↑(S i) → β", "hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩", "hS : Set.iUnion S = Set.univ", "i : ι", "x : α", "hx : x ∈ S i"], "goal": "Set.liftCover S f hf hS x = f i ⟨x, hx⟩"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "inst✝² : Group M", "inst✝¹ : Group N", "inst✝ : Group P", "c : Con M", "n : ℕ", "w x : M", "h : c w x"], "goal": "c (w ^ Int.ofNat n) (x ^ Int.ofNat n)"}}
+{"state": {"context": ["q : ℚ", "n : ℤ", "hn : n ≠ 0"], "goal": "n /. n = 1"}}
+{"state": {"context": ["x : ℝ", "hx : |x| ≤ 1"], "goal": "|Real.exp x - 1 - x| ≤ x ^ 2"}}
+{"state": {"context": ["a b : Ordinal.{u}", "hab : a * b = b", "ha : 0 < a", "hb : ¬b = 0"], "goal": "nfp (fun x => a * x) 1 ≤ b"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.HasShift C ℤ", "A B : CategoryTheory.Pretriangulated.Triangle C", "h : A = B"], "goal": "(CategoryTheory.eqToHom h).hom₂ = CategoryTheory.eqToHom ⋯"}}
+{"state": {"context": ["R : Type u_3", "CommRing R"], "goal": "Ideal.IsJacobson R → ∀ {I : Ideal R}, I.IsRadical → I.jacobson = I"}}
+{"state": {"context": ["α β : Type u", "β' : Type v", "inst✝ : Infinite α", "leq : lift.{v, u} #α = lift.{u, v} #β'"], "goal": "↑2 ^ lift.{v, u} #α = 2 ^ lift.{v, u} #α"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "D : Type u₄", "CategoryTheory.Category.{v₄, u₄} D", "F : J ⥤ C", "H H' : C ⥤ D", "α : H ≅ H'", "X : CategoryTheory.Limits.Cone F"], "goal": "((CategoryTheory.Functor.functorialityCompPostcompose α).inv.app X).hom = α.inv.app X.pt"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)", "R : Type u₁", "CommSemiring R", "CharP R p", "S : Type u₂", "CommSemiring S", "CharP S p", "φ : R →+* S"], "goal": "PerfectionMap.map p ⋯ ⋯ φ = Perfection.map p φ"}}
+{"state": {"context": ["k : ℕ", "n : ℕ", "hk : k ≠ 0"], "goal": "(n ^ k).primeFactors = n.primeFactors"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝ : LE α", "S : Set (UpperSet α)", "s t : UpperSet α", "a : α", "f : ι → UpperSet α"], "goal": "a ∈ ⋃ i, ↑(f i) ↔ ∃ i, a ∈ f i"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α"], "goal": "Primrec some"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "hGε : ↑(G.edgeDensity U V) ≤ ε ^ 5 / 50"], "goal": "↑(G.edgeDensity U V) ^ 2 - ε ^ 5 / 25 ≤ ((∑ ab ∈ (chunk hP G ε hU).parts.product (chunk hP G ε hV).parts, ↑(G.edgeDensity ab.1 ab.2)) / 16 ^ P.parts.card) ^ 2"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "inst✝⁹ : CommSemiring R", "inst✝⁸ : StarRing R", "inst✝⁷ : MetricSpace R", "inst✝⁶ : TopologicalSemiring R", "inst✝⁵ : ContinuousStar R", "inst✝⁴ : TopologicalSpace A", "inst✝³ : Ring A", "inst✝² : StarRing A", "inst✝¹ : Algebra R A", "inst✝ : ContinuousFunctionalCalculus R p", "f✝ g : R → R", "a✝ : A", "ha✝ : autoParam (p a✝) _auto✝", "hf✝ : autoParam (ContinuousOn f✝ (spectrum R a✝)) _auto✝", "hg : autoParam (ContinuousOn g (spectrum R a✝)) _auto✝", "r : R", "f : R → R", "a : A", "hf : autoParam (ContinuousOn f (spectrum R a)) _auto✝", "ha : autoParam (p a) _auto✝"], "goal": "cfc (fun x => r + f x) a = (algebraMap R A) r + cfc f a"}}
+{"state": {"context": ["ι : Type u", "β : ι → Type v", "(i : ι) → AddGroup (β i)", "b : ℤ", "v : Π₀ (i : ι), β i", "i : ι"], "goal": "(b • v) i = b • v i"}}
+{"state": {"context": ["β₁ : Type u_1", "β₂ : Type u_2", "α : Type u_3", "f : β₁ → β₂", "g₁ : α → β₁ → β₁", "g₂ : α → β₂ → β₂", "l : List α", "init : β₁", "H : ∀ (x : α) (y : β₁), g₂ x (f y) = f (g₁ x y)"], "goal": "List.foldr g₂ (f init) l = f (List.foldr g₁ init l)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedField α", "s : Finset ι", "f : ι → α", "hs : 0 < ↑s.card"], "goal": "(∑ i ∈ s, f i) ^ 2 * ↑s.card ≤ (↑s.card * ∑ i ∈ s, f i ^ 2) * ↑s.card"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "Ring R", "AddCommGroup V", "TopologicalSpace V", "TopologicalSpace R", "Module R V"], "goal": "SeparatingDual R V ↔ ∀ (x : V), x ≠ 0 → ∃ f, f x ≠ 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "E : Type u_2", "inst✝¹ : AddCommGroup E", "inst✝ : Module R E", "p : Submodule R E", "f : E →ₗ[R] E", "h : IsProj p f"], "goal": "f = (p.prodEquivOfIsCompl (ker f) ⋯).conj (id.prodMap 0)"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "V' : Type u_3", "P : Type u_4", "P' : Type u_5", "inst✝⁶ : LinearOrderedField R", "inst✝⁵ : AddCommGroup V", "inst✝⁴ : Module R V", "inst✝³ : AddTorsor V P", "inst✝² : AddCommGroup V'", "inst✝¹ : Module R V'", "inst✝ : AddTorsor V' P'", "s : AffineSubspace R P", "x y : P", "h : s.WOppSide x y"], "goal": "∃ p ∈ s, Wbtw R x p y"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "δ : Type u_4", "m0 : MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "ν : MeasureTheory.Measure β", "f : α → β", "g g' : β → δ", "hf : MeasureTheory.Measure.QuasiMeasurePreserving f μ ν", "h : g =ᵐ[ν] g'"], "goal": "g ∘ f =ᵐ[μ] g' ∘ f"}}
+{"state": {"context": ["R : Type u", "inst✝⁵ : Ring R", "ι : Type v", "inst✝⁴ : Preorder ι", "G : ι → Type w", "inst✝³ : DecidableEq ι", "inst✝² : (i : ι) → AddCommGroup (G i)", "f : (i j : ι) → i ≤ j → G i →+ G j", "P : Type u₁", "inst✝¹ : AddCommGroup P", "g : (i : ι) → G i →+ P", "Hg : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (g i) x", "inst✝ : IsDirected ι fun x x_1 => x ≤ x_1", "h✝ : Nonempty ι", "injective : ∀ (i : ι) (a : G i), (g i) a = 0 → a = 0", "z : DirectLimit G f", "hz : (lift G f P g Hg) z = 0"], "goal": "z = 0"}}
+{"state": {"context": ["ι : Type u_1", "σ : Type u_2", "R : Type u_4", "DecidableEq ι", "Semiring R", "SetLike σ R", "AddSubmonoidClass σ R", "A : ι → σ", "AddRightCancelMonoid ι", "SetLike.GradedMonoid A", "r : ⨁ (i : ι), ↥(A i)", "i : ι", "r' : ↥(A i)", "j : ι"], "goal": "↑((r * (DirectSum.of (fun i => ↥(A i)) i) r') (j + i)) = ↑(r j) * ↑r'"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "NontriviallyNormedField R", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module R (E x)", "NormedAddCommGroup F", "NormedSpace R F", "TopologicalSpace B", "TopologicalSpace (Bundle.TotalSpace F E)", "(x : B) → TopologicalSpace (E x)", "FiberBundle F E", "e : Trivialization F Bundle.TotalSpace.proj", "Trivialization.IsLinear R e", "b : B", "hb : b ∈ e.baseSet"], "goal": "⇑(Trivialization.continuousLinearEquivAt R e b hb).symm = e.symm b"}}
+{"state": {"context": ["B : Type u_1", "TopologicalSpace B", "F₁ : Type u_2", "TopologicalSpace F₁", "E₁ : B → Type u_3", "TopologicalSpace (Bundle.TotalSpace F₁ E₁)", "F₂ : Type u_4", "TopologicalSpace F₂", "E₂ : B → Type u_5", "TopologicalSpace (Bundle.TotalSpace F₂ E₂)", "e₁ : Trivialization F₁ Bundle.TotalSpace.proj", "e₂ : Trivialization F₂ Bundle.TotalSpace.proj", "(x : B) → Zero (E₁ x)", "(x : B) → Zero (E₂ x)"], "goal": "(e₁.prod e₂).baseSet = e₁.baseSet ∩ e₂.baseSet"}}
+{"state": {"context": ["α : Type u_1", "inst✝² : Fintype α", "inst✝¹ : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : ℝ", "A : Finset α", "hA : ∃ A_1, ∃ (hA : A_1 ∈ P.parts), A ∈ (chunk hP G ε hA).parts"], "goal": "A.card = m ∨ A.card = m + 1"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u", "CommRing R", "s : Finset ι", "f : ι → Ideal R", "a b : ι", "hp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime", "I : Ideal R"], "goal": "↑I ⊆ ⋃ i ∈ ↑s, ↑(f i) ↔ ∃ i ∈ s, I ≤ f i"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : Type u_5", "x : ι → A", "CommRing R", "CommRing A", "Algebra R A", "hx : AlgebraicIndependent R x", "b : ↥(Algebra.adjoin R (Set.range x))"], "goal": "hx.aevalEquiv.symm b = Function.surjInv ⋯ b"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "NormedField 𝕜", "AddCommGroup V", "Module 𝕜 V", "e₁ e₂ : ENorm 𝕜 V"], "goal": "e₁ = e₂ ↔ ∀ (x : V), ↑e₁ x = ↑e₂ x"}}
+{"state": {"context": ["α : Type u_2", "Group α", "s t : Set α", "x : α"], "goal": "x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "x : α × β"], "goal": "IsBot x ↔ IsBot x.1 ∧ IsBot x.2"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Ring R", "n : ℕ", "h : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n", "hn' : n > 0", "hn : 1 < n", "p : ℕ", "hp : Nat.Prime p", "hpe : p ∣ (eval 1 (cyclotomic n ℤ)).natAbs", "this✝ : Fact (Nat.Prime p)", "this : ↑n = (∏ x ∈ n.divisors.erase 1 \\ image (fun t => p ^ (t + 1)) (range (padicValNat p n)), eval 1 (cyclotomic x ℤ)) * ∏ x ∈ image (fun t => p ^ (t + 1)) (range (padicValNat p n)), eval 1 (cyclotomic x ℤ)"], "goal": "∀ x ∈ range (padicValNat p n), ∀ y ∈ range (padicValNat p n), p ^ (x + 1) = p ^ (y + 1) → x = y"}}
+{"state": {"context": ["M : Type u_1", "Add M", "x : M"], "goal": "x ∉ ⊥"}}
+{"state": {"context": ["x y : ℝ", "h : x * y < 1", "hy : 0 < y"], "goal": "arctan x + arctan y < π / 2"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "Y : C", "X : CategoryTheory.SimplicialObject C", "n b q : ℕ", "φ : Y ⟶ X _[n + 1]", "v : AlgebraicTopology.DoldKan.HigherFacesVanish q φ", "hnbq : n + 1 = b + q"], "goal": "AlgebraicTopology.DoldKan.HigherFacesVanish q (φ ≫ X.σ ⟨b, ⋯⟩)"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "TopologicalSpace H", "TopologicalSpace M", "ChartedSpace H M", "TopologicalSpace H'", "TopologicalSpace M'", "ChartedSpace H' M'", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "P : (H → H') → Set H → H → Prop", "g : M → M'", "s t : Set M", "x : M", "hG : G.LocalInvariantProp G' P", "ht : t ∈ 𝓝[s] x"], "goal": "ChartedSpace.LiftPropWithinAt P g (s ∩ t) x ↔ ChartedSpace.LiftPropWithinAt P g s x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f : ℝ → ℝ", "x x₀ : ℝ", "n : ℕ", "hx : x₀ < x", "hf : ContDiffOn ℝ (↑n) f (Icc x₀ x)", "hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)", "gcont : ContinuousOn id (Icc x₀ x)", "gdiff : ∀ x_1 ∈ Ioo x₀ x, HasDerivAt id ((fun x => 1) x_1) x_1"], "goal": "∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x') ^ n / ↑n ! * (x - x₀)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "c✝ : E", "f : α → E", "hf : AEStronglyMeasurable f μ", "s : Set α", "p : ℝ≥0∞", "hs : MeasurableSet s", "c : E", "hμsc : c = 0 ∨ μ s ≠ ⊤"], "goal": "Memℒp (s.indicator fun x => c) p μ"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "p p₀ p₁ p₂ p1 p2 p3 : P", "h : dist p3 p1 = dist p3 p2"], "goal": "∠ p3 (midpoint ℝ p1 p2) p2 = π / 2"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ"], "goal": "LSeries.term (f - g) s = LSeries.term f s - LSeries.term g s"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "f✝¹ f✝ : ℂ → E", "hf✝ : Differentiable ℂ f✝", "c z : ℂ", "f : ℂ → ℂ", "hf : Differentiable ℂ f", "n : ℕ"], "goal": "(↑n !)⁻¹ * iteratedDeriv n f c * (z - c) ^ n = (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "X Y Z : C", "f g : X ⟶ Y", "h : Y ⟶ Z", "inst✝¹ : IsIso f", "inst✝ : IsIso h"], "goal": "(f ≫ h) ≫ inv h ≫ inv f = 𝟙 X"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "M' : Type u_4", "X : Type u_5", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "inst✝⁴ : ChartedSpace H M", "inst✝³ : TopologicalSpace H'", "inst✝² : TopologicalSpace M'", "inst✝¹ : ChartedSpace H' M'", "inst✝ : TopologicalSpace X", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e e' : PartialHomeomorph M H", "f f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "g g' : M → M'", "s t : Set M", "x : M", "Q : (H → H) → Set H → H → Prop", "hG : G.LocalInvariantProp G' P", "h : LiftPropWithinAt P g s x", "hs : s ∈ 𝓝 x"], "goal": "LiftPropAt P g x"}}
+{"state": {"context": ["R : Type u", "CommRing R", "r : R"], "goal": "(AdjoinRoot.of (Polynomial.C r * Polynomial.X - 1)) r * AdjoinRoot.root (Polynomial.C r * Polynomial.X - 1) = 1"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : CategoryTheory.Center C"], "goal": "(X ⊗ Y).fst = X.fst ⊗ Y.fst"}}
+{"state": {"context": ["R : Type u_1", "P : Cubic R", "Semiring R", "ha : P.a = 0", "hb : P.b = 0", "hc : P.c = 1"], "goal": "P.toPoly.Monic"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : CompleteSpace E", "H : Type u_2", "inst✝³ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : ChartedSpace H M", "inst✝ : SmoothManifoldWithCorners I M", "γ γ' : ℝ → M", "v : (x : M) → TangentSpace I x", "s✝ s' : Set ℝ", "t₀ : ℝ", "x₀ : M", "hx : I.IsInteriorPoint x₀", "left✝ : ContinuousAt (fun x => { proj := x, snd := v x }) x₀", "hv : ContDiffWithinAt ℝ 1 (↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘ (fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm) (range ↑I) (↑(extChartAt I x₀) x₀)", "f : ℝ → E", "hf1 : f t₀ = ↑(extChartAt I x₀) x₀", "hf2 : ∀ᶠ (y : ℝ) in 𝓝 t₀, HasDerivAt f ((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘ (fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm) (f y)).2 y", "a : ℝ", "ha : a > 0", "hf2' : ∀ y ∈ Metric.ball t₀ a, HasDerivAt f ((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘ (fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm) (f y)).2 y", "hcont : ∀ A ∈ 𝓝 (↑(extChartAt I x₀) x₀), f ⁻¹' A ∈ 𝓝 t₀", "hnhds : ∀ᶠ (x' : ℝ) in 𝓝 t₀, f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 x'", "s : Set ℝ", "hs : s ∈ 𝓝 t₀", "haux : ∀ y ∈ s, HasDerivAt f ((↑(extChartAt I.tangent { proj := x₀, snd := v x₀ }) ∘ (fun x => { proj := x, snd := v x }) ∘ ↑(extChartAt I x₀).symm) (f y)).2 y ∧ f ⁻¹' interior (extChartAt I x₀).target ∈ 𝓝 y", "t : ℝ", "ht : t ∈ s", "xₜ : M := ↑(extChartAt I x₀).symm (f t)", "h : HasDerivAt f ((tangentCoordChange I xₜ x₀ xₜ) (v xₜ)) t", "hf3 : f t ∈ interior (extChartAt I x₀).target", "hf3' : f t ∈ (extChartAt I x₀).target", "hft1 : ↑(extChartAt I x₀).symm (f t) ∈ (extChartAt I x₀).source"], "goal": "HasMFDerivAt 𝓘(ℝ, ℝ) I (↑(extChartAt I x₀).symm ∘ f) t (ContinuousLinearMap.smulRight 1 (v ((↑(extChartAt I x₀).symm ∘ f) t)))"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Group α", "inst✝² : LinearOrder α", "a✝ b✝ : α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c : α", "inst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1"], "goal": "a⁻¹ < b ∧ a < b ↔ b⁻¹ < a ∧ a < b"}}
+{"state": {"context": ["ι : Type u'", "R : Type u_2", "M : Type u_4", "v : ι → M", "Ring R", "AddCommGroup M", "Module R M", "Nontrivial R", "hv : LinearIndependent R v", "x : ι", "f : ι →₀ R", "h : x ∉ f.support"], "goal": "(Finsupp.total ι M R v) f ≠ v x"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "f : R →+* S", "x y : R[X]"], "goal": "x ∣ y → Polynomial.map f x ∣ Polynomial.map f y"}}
+{"state": {"context": ["J K : Type v", "inst✝³ : SmallCategory J", "inst✝² : SmallCategory K", "C : Type u", "inst✝¹ : Category.{v, u} C", "F : J ⥤ K ⥤ C", "G : J × K ⥤ C", "inst✝ : HasColimits C", "j : J", "k : K"], "goal": "colimit.ι ((curry.obj G).obj j) k ≫ colimit.ι (curry.obj G ⋙ colim) j ≫ (colimitIsoColimitCurryCompColim G).inv ≫ (HasColimit.isoOfEquivalence (Prod.braiding K J) (Iso.refl ((Prod.braiding K J).functor ⋙ G))).inv ≫ (colimitIsoColimitCurryCompColim (Prod.swap K J ⋙ G)).hom = colimit.ι ((curry.obj (Prod.swap K J ⋙ G)).obj k) j ≫ colimit.ι (curry.obj (Prod.swap K J ⋙ G) ⋙ colim) k"}}
+{"state": {"context": ["a x z✝ z : ℂ", "hre : z.re < 0", "him : z.im = 0"], "goal": "Tendsto (fun x => Real.arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π))"}}
+{"state": {"context": ["G : Type u", "Group G", "F : Type v", "Field F", "MulSemiringAction G F", "Fintype G", "x : F"], "goal": "FixedPoints.minpoly G F x = minpoly (↥(FixedPoints.subfield G F)) x"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : MeasurableSpace E", "inst✝⁵ : BorelSpace E", "inst✝⁴ : FiniteDimensional ℝ E", "μ : Measure E", "inst✝³ : μ.IsAddHaarMeasure", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "s✝ s : Set E", "x : E", "h : Tendsto (fun r => μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 1)", "t : Set E", "ht : MeasurableSet t", "h't : μ t ≠ 0"], "goal": "∀ᶠ (r : ℝ) in 𝓝[>] 0, (s ∩ ({x} + r • t)).Nonempty"}}
+{"state": {"context": ["M : Type u_3", "Monoid M", "s : Submonoid M"], "goal": "↑s * ↑s = ↑s"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "self : CategoryTheory.OplaxFunctor B C", "a b : B", "f : a ⟶ b"], "goal": "self.map₂ (CategoryTheory.Bicategory.leftUnitor f).hom =\n  CategoryTheory.CategoryStruct.comp (self.mapComp (CategoryTheory.CategoryStruct.id a) f)\n    (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (self.mapId a) (self.map f))\n      (CategoryTheory.Bicategory.leftUnitor (self.map f)).hom)"}}
+{"state": {"context": ["μ : YoungDiagram"], "goal": "YoungDiagram.ofRowLens μ.rowLens ⋯ = μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "a : Ordinal.{u_4}"], "goal": "0 - a = 0"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝¹ : PartialOrder Γ", "inst✝ : Zero R", "a b : Γ", "r : R", "x : HahnSeries Γ R", "g : Γ", "h : x.orderTop ≠ ⊤", "hx : x ≠ 0", "hg : ⋯.min ⋯ = g"], "goal": "x.coeff (⋯.min ⋯) ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : Rel α β"], "goal": "r.comp ⊥ = ⊥"}}
+{"state": {"context": ["a b : Int"], "goal": "a.mod b = a - b * a.div b"}}
+{"state": {"context": ["K : Type u_1", "F : Type u_2", "R : Type u_3", "inst✝³ : DivisionRing K", "Γ₀ : Type u_4", "Γ'₀ : Type u_5", "Γ''₀ : Type u_6", "inst✝² : LinearOrderedCommMonoidWithZero Γ''₀", "inst✝¹ : Ring R", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "x✝ y✝ z x y : R", "g : Γ₀", "hx : v x ≤ g", "hy : v y ≤ g"], "goal": "v (x + -y) ≤ g"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁸ : TopologicalSpace X", "inst✝⁷ : Zero X", "A : Type u_2", "inst✝⁶ : TopologicalSpace A", "inst✝⁵ : NonUnitalRing A", "inst✝⁴ : StarRing A", "inst✝³ : Module ℝ A", "inst✝² : IsScalarTower ℝ A A", "inst✝¹ : SMulCommClass ℝ A A", "inst✝ : TopologicalRing A", "φ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A", "r : ℝ", "f : C(X, ℝ)₀"], "goal": "φ (r • f).toNNReal - φ (-(r • f)).toNNReal = r • (φ f.toNNReal - φ (-f).toNNReal)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "a b c : R", "x : R", "e : ℕ", "pq_pf : a + b = c"], "goal": "x ^ e * a + x ^ e * b = x ^ e * c"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s s₁ s₂ t t₁ t₂ u : Set α"], "goal": "¬s ⊆ t ↔ ∃ a ∈ s, a ∉ t"}}
+{"state": {"context": ["R : Type u", "S✝ : Type v", "A : Type w", "B : Type u₁", "M : Type v₁", "inst✝¹² : CommSemiring R", "inst✝¹¹ : Semiring S✝", "inst✝¹⁰ : AddCommMonoid A", "inst✝⁹ : Algebra R S✝", "inst✝⁸ : Module S✝ A", "inst✝⁷ : Module R A", "inst✝⁶ : IsScalarTower R S✝ A", "S : Type u_1", "T : Type u_2", "inst✝⁵ : CommSemiring S", "inst✝⁴ : Semiring T", "inst✝³ : Algebra R S", "inst✝² : Algebra R T", "inst✝¹ : Algebra S T", "inst✝ : IsScalarTower R S T", "x : S", "a : Set S", "hx : x ∈ span R a"], "goal": "∃ y ∈ span R a, (↑R (Algebra.linearMap S T)) y = (algebraMap S T) x"}}
+{"state": {"context": ["E : Type u_1", "β : Type u_2", "inst✝⁷ : AddCommGroup E", "inst✝⁶ : TopologicalSpace E", "inst✝⁵ : Module ℝ E", "inst✝⁴ : TopologicalAddGroup E", "inst✝³ : ContinuousSMul ℝ E", "inst✝² : OrderedAddCommGroup β", "inst✝¹ : Module ℝ β", "inst✝ : OrderedSMul ℝ β", "s : Set E", "f : E → β", "a : E", "a_in_s : a ∈ s", "h_localmin : IsLocalMinOn f s a", "h_conv : ConvexOn ℝ s f", "x : E", "x_in_s : x ∈ s", "g : ℝ →ᵃ[ℝ] E := AffineMap.lineMap a x", "hg0 : g 0 = a"], "goal": "x ∈ {x | (fun x => f a ≤ f x) x}"}}
+{"state": {"context": ["ι : Type u_1", "I J : BoxIntegral.Box ι", "π₁ π₂ : BoxIntegral.TaggedPrepartition I"], "goal": "J ∈ π₁.infPrepartition π₂.toPrepartition ↔ J ∈ π₂.infPrepartition π₁.toPrepartition"}}
+{"state": {"context": ["α β : Type u", "ι : Type u_1", "f : ι → Type u"], "goal": "(prod fun i => #(f i)) = 0 ↔ ∃ i, (fun i => #(f i)) i = 0"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_15", "NormedDivisionRing 𝕜", "c : 𝕜", "hc : c ≠ 0", "f : α → 𝕜", "l : Filter α"], "goal": "Asymptotics.IsBigOWith ‖c‖⁻¹ l f fun x => c * f x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C"], "goal": "∀ {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y), CategoryTheory.ShortComplex.π₂.map f = f.τ₂"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a b : α", "ha : 0 < a", "hb : 0 < b"], "goal": "1 / a ≤ b ↔ 1 / b ≤ a"}}
+{"state": {"context": ["r : ℝ", "f : ℕ → ℝ"], "goal": "(ofSeq f).IsSt r ↔ Germ.Tendsto (ofSeq f) (𝓝 r)"}}
+{"state": {"context": ["X Y : AlgebraicGeometry.Scheme", "f : X ⟶ Y", "self : AlgebraicGeometry.Surjective f"], "goal": "Function.Surjective ⇑f.val.base"}}
+{"state": {"context": ["ι : Sort u_4", "f : ι → ℝ≥0∞", "a : ℝ≥0∞"], "goal": "iSup f * a = ⨆ i, f i * a"}}
+{"state": {"context": ["F : Type u", "Field F"], "goal": "2 ≠ 0 ∨ 3 ≠ 0"}}
+{"state": {"context": ["x : ℝ", "hx : x ≠ 0"], "goal": "HasDerivAt (fun x => √x * log x) ((2 + log x) / (2 * √x)) x"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "e : S₁ ≅ S₂", "h₁ : S₁.LeftHomologyData", "h₂ : S₂.LeftHomologyData"], "goal": "(CategoryTheory.ShortComplex.leftHomologyMapIso' e h₁ h₂).hom = CategoryTheory.ShortComplex.leftHomologyMap' e.hom h₁ h₂"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X"], "goal": "IsIrreducible s ↔ ∀ (t : Finset (Set X)), (∀ z ∈ t, IsClosed z) → s ⊆ ⋃₀ ↑t → ∃ z ∈ t, s ⊆ z"}}
+{"state": {"context": [], "goal": "Real.Gamma 1 = 1"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℝ≥0∞"], "goal": "HasSum f (⨆ s, ∑ a ∈ s, f a)"}}
+{"state": {"context": ["X : Type u_1", "inst✝⁴ : MeasurableSpace X", "inst✝³ : TopologicalSpace X", "inst✝² : OpensMeasurableSpace X", "inst✝¹ : T0Space X", "inst✝ : CompletelyRegularSpace X", "μ : ↑(range diracProba)", "F : Filter ↑(range diracProba)", "key : Tendsto (diracProba ∘ ⇑diracProbaEquiv.symm) F (𝓝 (diracProba (diracProbaEquiv.symm μ))) ↔ Tendsto (⇑diracProbaEquiv.symm) F (𝓝 (diracProbaEquiv.symm μ))"], "goal": "diracProba (diracProbaEquiv.symm μ) = ↑μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "R : Type u_4", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : R", "h : ‖x‖ < 1"], "goal": "(1 - x) * ∑' (i : ℕ), x ^ i = 1"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "f g : ι → ℝ≥0", "p q : ℝ", "hpq : p.IsConjExponent q", "hf : ∑ i ∈ s, f i ^ p ≤ 1", "hg : ∑ i ∈ s, g i ^ q ≤ 1", "hp_ne_zero : p.toNNReal ≠ 0"], "goal": "∑ i ∈ s, f i * g i ≤ 1"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.Limits.HasZeroMorphisms D", "F : C ⥤ D", "F.PreservesZeroMorphisms", "X Y : C", "CategoryTheory.Limits.HasBinaryBiproduct X Y", "CategoryTheory.Limits.PreservesBinaryBiproduct X Y F", "W : C", "f : W ⟶ X", "g : W ⟶ Y"], "goal": "CategoryTheory.Limits.biprod.lift (F.map f) (F.map g) ≫ (F.mapBiprod X Y).inv =\n  F.map (CategoryTheory.Limits.biprod.lift f g)"}}
+{"state": {"context": ["M : Type u_2", "α : Type u_3", "SMul M α", "MeasurableSpace M", "MeasurableSpace α", "self : MeasurableSMul M α", "x : α"], "goal": "Measurable fun x_1 => x_1 • x"}}
+{"state": {"context": ["G : Type u_1", "inst✝ : Group G", "H K : Subgroup G", "S T : Set G"], "goal": "(∀ (g : G), ∃! s, ↑↑s = ↑g) ↔ ∀ (q : Quotient (QuotientGroup.leftRel H)), ∃! s, Quotient.mk'' ↑s = q"}}
+{"state": {"context": ["R : Type u", "CommRing R", "C : WeierstrassCurve.VariableChange R"], "goal": "C.comp WeierstrassCurve.VariableChange.id = C"}}
+{"state": {"context": ["m n : ℕ", "cop : m.Coprime n", "x : ℕ", "ha : n * x ≠ 0", "y : ℕ", "hb : m * y ≠ 0", "h : n * x * m + m * y * n = m * n"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "ι : Type u_7", "f : ι → α", "h : DenseRange f", "x : α"], "goal": "(𝓝[Set.range f] x).NeBot"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Ring R", "p✝ q✝ p q : R[X]", "this✝ : DecidableEq R := Classical.decEq R", "hp0 : ¬p = 0", "hq : q.Monic", "h : q.degree ≤ p.degree", "this : Nontrivial R"], "goal": "(p /ₘ q).degree ≤ p.degree"}}
+{"state": {"context": ["α : Type u_1", "Unique α", "s : Finset α"], "goal": "s.Nonempty ↔ s = {default}"}}
+{"state": {"context": ["B : Type u_3", "S : B", "M : Type u_4", "AddCommMonoid M", "SetLike B M", "AddSubmonoidClass B M", "ι : Type u_5", "t : Finset ι", "f : ι → M", "h : ∀ c ∈ t, f c ∈ S"], "goal": "∑ c ∈ t, f c ∈ S"}}
+{"state": {"context": ["A : Type u_2", "CommRing A", "IsDomain A", "h : IsDedekindDomainInv A"], "goal": "IsDedekindDomain A"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "MeasurableSpace E", "m : MeasurableSpace Ω", "ℙ : MeasureTheory.Measure Ω", "μ : MeasureTheory.Measure E", "MeasureTheory.IsFiniteMeasure ℙ", "X : Ω → E"], "goal": "(fun x => ENNReal.ofReal (MeasureTheory.pdf X ℙ μ x).toReal) =ᵐ[μ] MeasureTheory.pdf X ℙ μ"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "υ : Type u_3", "R : Type u_4", "CommSemiring R", "f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R", "g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R"], "goal": "MvPolynomial.comap (g.comp f) = MvPolynomial.comap f ∘ MvPolynomial.comap g"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕝 : Type u_3", "𝕝₂ : Type u_4", "E : Type u_5", "F : Type u_6", "G : Type u_7", "ι : Type u_8", "ι' : Type u_9", "inst✝² : NormedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "p : SeminormFamily 𝕜 E ι", "U : Set E"], "goal": "U ∈ p.basisSets ↔ ∃ i r, 0 < r ∧ U = (i.sup p).ball 0 r"}}
+{"state": {"context": ["n : ℕ"], "goal": "(2 * n + 1).div2 = n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "a a₁ a₂ b b₁ b₂ c d : α", "h : b₁ < b₂", "x : α"], "goal": "x ∈ Ioc a₁ b₁ ∩ Ioo a₂ b₂ ↔ x ∈ Ioc (max a₁ a₂) b₁"}}
+{"state": {"context": ["α : Type u", "x y z : α", "GeneralizedBooleanAlgebra α", "hz : z ≤ y", "hx : x ≤ y"], "goal": "Disjoint z (y \\ x) ↔ z ≤ x"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Field K", "inst✝² : IsDedekindDomain A", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "p : K[X]", "hp : p ≠ 0"], "goal": "factors (span {p}) = Multiset.map (fun q => span {q}) (factors p)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : MeasurableSpace E", "inst✝ : OpensMeasurableSpace E", "R : ℝ≥0", "p : ℝ≥0∞", "f : ℕ → α → E", "hfmeas : ∀ (n : ℕ), Measurable (f n)", "hp0 : ¬p.toReal = 0", "hp : p ≠ 0", "hp' : p ≠ ⊤", "hbdd : ∀ (n : ℕ), (∫⁻ (x : α), ↑‖f n x‖₊ ^ p.toReal ∂μ) ^ p.toReal⁻¹ ≤ ↑R"], "goal": "liminf (fun n => ∫⁻ (a : α), ↑‖f n a‖₊ ^ p.toReal ∂μ) atTop ≤ ↑R ^ p.toReal"}}
+{"state": {"context": ["R : Type u", "inst✝¹¹ : CommSemiring R", "S : Submonoid R", "Rₚ : Type v", "inst✝¹⁰ : CommSemiring Rₚ", "inst✝⁹ : Algebra R Rₚ", "inst✝⁸ : IsLocalization S Rₚ", "M : Type w", "inst✝⁷ : AddCommMonoid M", "inst✝⁶ : Module R M", "Mₚ : Type t", "inst✝⁵ : AddCommMonoid Mₚ", "inst✝⁴ : Module R Mₚ", "inst✝³ : Module Rₚ Mₚ", "inst✝² : IsScalarTower R Rₚ Mₚ", "f : M →ₗ[R] Mₚ", "inst✝¹ : IsLocalizedModule S f", "inst✝ : Finite R M", "T : Finset M", "hT : Submodule.span R ↑T = ⊤", "x : Mₚ", "y : M", "m : ↥S", "hyx : IsLocalizedModule.mk' f y m = x", "hy : y ∈ Submodule.span R ↑T", "this : f y ∈ Submodule.map f (Submodule.span R ↑T)"], "goal": "x ∈ Submodule.span Rₚ ↑(Finset.image (⇑f) T)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "n : ℕ"], "goal": "(algebraMap R[X] R[T;T⁻¹]) (Polynomial.X ^ n) = LaurentPolynomial.T ↑n"}}
+{"state": {"context": ["x : SetTheory.PGame"], "goal": "x ≈ x"}}
+{"state": {"context": ["R : Type u1", "CommRing R", "M : Type u2", "AddCommGroup M", "Module R M", "N : Type u4", "AddCommGroup N", "Module R N", "f : M →ₗ[R] N", "g : N →ₗ[R] M"], "goal": "Function.LeftInverse ⇑(ExteriorAlgebra.map g) ⇑(ExteriorAlgebra.map f) ↔ Function.LeftInverse ⇑g ⇑f"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "R : Type u_1", "inst✝ : NonAssocSemiring R", "f : (k : ℕ) → R →+* ZMod (p ^ k)", "f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1", "r : R", "n : ℕ"], "goal": "(lift f_compat) r - ↑((f n) r).val ∈ Ideal.span {↑p ^ n}"}}
+{"state": {"context": ["R✝ : Type u", "F : Type u_1", "R : Type u_2", "S : Type u_3", "inst✝⁷ : NonUnitalSemiring R", "inst✝⁶ : PartialOrder R", "inst✝⁵ : StarRing R", "inst✝⁴ : StarOrderedRing R", "inst✝³ : NonUnitalSemiring S", "inst✝² : PartialOrder S", "inst✝¹ : StarRing S", "inst✝ : StarOrderedRing S", "f : R →⋆ₙ+* S", "x p : R", "hp : p ∈ AddSubmonoid.closure (range fun s => star s * s)"], "goal": "∃ p_1 ∈ AddSubmonoid.closure (range fun s => star s * s), f (x + p) = f x + p_1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : TopCat", "F : TopCat.Presheaf C X", "ι : Type w", "U : ι → TopologicalSpace.Opens ↑X", "Y : TopologicalSpace.Opens ↑X", "hY : Y = iSup U"], "goal": "(F.whiskerIsoMapGenerateCocone U hY).hom.hom = F.map (CategoryTheory.eqToHom ⋯)"}}
+{"state": {"context": ["n : ℕ", "i : ZMod (2 * n)"], "goal": "QuaternionGroup.xa i ^ 4 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ : Set ℕ", "inst✝² : Infinite ↑s✝", "inst✝¹ : DecidablePred fun x => x ∈ s✝", "s : Set ℕ", "inst✝ : DecidablePred fun x => x ∈ s", "x : ↑s"], "goal": "(List.filter (fun x => decide (x ∈ s)) (List.range ↑x)).length = (Finset.filter (fun x => x ∈ s) (Finset.range ↑x)).card"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "s : Set E"], "goal": "(convexHull 𝕜) s = ⋃ t, ⋃ (_ : ↑t ⊆ s), ⋃ (_ : AffineIndependent 𝕜 Subtype.val), (convexHull 𝕜) ↑t"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "T2Space X", "f : X → Y", "g : Y → X", "h : Function.LeftInverse f g", "hf : Continuous f", "hg : Continuous g"], "goal": "IsClosed (Set.range g)"}}
+{"state": {"context": ["V : Type u_1", "R : Type u_2", "inst✝³ : Fintype V", "inst✝² : DecidableEq V", "G : SimpleGraph V", "inst✝¹ : DecidableRel G.Adj", "inst✝ : Ring R", "i : V"], "goal": "(lapMatrix R G *ᵥ fun x => 1) i = 0 i"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ Cᵒᵖ", "c : CategoryTheory.Limits.Cone F.leftOp", "j : J"], "goal": "(CategoryTheory.Limits.coconeOfConeLeftOp c).ι.app j = (c.π.app (Opposite.op j)).op"}}
+{"state": {"context": ["α : Type u_2", "Add α", "IsRightCancelAdd α", "DecidableEq α", "s : Finset α", "a : α"], "goal": "(s + {a}).card = s.card"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "DecidableEq n", "Fintype n", "DecidableEq m", "Fintype m", "R : Type v", "CommRing R", "A : Matrix m m R", "C : Matrix n m R", "D : Matrix n n R"], "goal": "(Matrix.fromBlocks A 0 C D).det = A.det * D.det"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "a : ℍ[R]"], "goal": "a.im.imI = a.imI"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "Monoid R", "NonUnitalNonAssocSemiring A", "DistribMulAction R A", "Star A"], "goal": "⇑1 = id"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : NonAssocRing M", "n✝ : ℤ", "x✝¹ x✝ : M", "n : ℕ"], "goal": "x✝¹ * ↑↑n * x✝ = x✝¹ * (↑↑n * x✝)"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "r : R", "n : ℕ"], "goal": "(Polynomial.opRingEquiv R) (MulOpposite.op (Polynomial.C r * Polynomial.X ^ n)) =\n  Polynomial.C (MulOpposite.op r) * Polynomial.X ^ n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "CategoryTheory.HasShift C ℤ", "T : CategoryTheory.Pretriangulated.Triangle C"], "goal": "T.rotate.mor₃ = -(CategoryTheory.shiftFunctor C 1).map T.mor₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝ : MeasurableSpace β", "g : α → β", "hg : MeasurableEmbedding g", "f : β → ℝ≥0∞", "f₀ : α →ₛ ℝ≥0∞", "hf₀ : ↑f₀ ≤ fun a => f (g a)"], "goal": "f₀.lintegral μ ≤ ⨆ g_1, ⨆ (_ : ↑g_1 ≤ fun a => f a), g_1.lintegral (Measure.map g μ)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "l : List α", "x : α", "h : x ∉ l"], "goal": "l.formPerm x = x"}}
+{"state": {"context": ["a : Int", "H : a ≤ 0"], "goal": "↑a.natAbs = -a"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "hfc : StrictConcaveOn ℝ S f", "hfd : ∀ x ∈ S, DifferentiableAt ℝ f x"], "goal": "StrictAntiOn (deriv f) S"}}
+{"state": {"context": ["G : Type w", "inst✝² : TopologicalSpace G", "μ : Content G", "inst✝¹ : R1Space G", "S : MeasurableSpace G", "inst✝ : BorelSpace G", "U : Set G", "hU : U ∈ {s | IsOpen s}", "U' : Opens G", "this✝¹ : Nonempty { L // ↑L ⊆ ↑U' ∩ U }", "L : Compacts G", "L' : Compacts G := { carrier := closure ↑L, isCompact' := ⋯ }", "hL : ↑L ⊆ ↑U' ∧ ↑L ⊆ U", "hL'U : ↑L' ⊆ U", "hL'U' : ↑L' ⊆ ↑U'", "this✝ : ↑U' \\ U ⊆ ↑U' \\ ↑L'", "this : Nonempty { M // ↑M ⊆ ↑U' \\ closure ↑L }"], "goal": "↑(μ.toFun L') + ⨆ x, ↑(μ.toFun ↑x) ≤ μ.outerMeasure ↑U'"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p"], "goal": "⇑(Multiset.countPAddMonoidHom p) = Multiset.countP p"}}
+{"state": {"context": ["F : Type u_1", "Field F", "E : Type u_2", "Field E", "Algebra F E"], "goal": "FiniteDimensional.finrank (↥⊤) E = 1"}}
+{"state": {"context": ["V : Type u_1", "V' : Type u_2", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : NormedAddCommGroup V'", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : InnerProductSpace ℝ V'", "inst✝¹ : Fact (finrank ℝ V = 2)", "inst✝ : Fact (finrank ℝ V' = 2)", "o : Orientation ℝ V (Fin 2)", "x : V"], "goal": "↑(↑(‖x‖ ^ 2)).arg = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "MeasurableSpace β", "f : α → β", "μ : MeasureTheory.Measure α", "h : Measurable f"], "goal": "AEMeasurable f μ"}}
+{"state": {"context": ["R : Type u_3", "OrderedSemiring R", "x y : R", "n : ℕ", "hx : 0 ≤ x", "hy : 0 ≤ y", "hn : n ≠ 0"], "goal": "x ^ n + y ^ n ≤ (x + y) ^ n"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I J : Ideal R", "N : Submodule R M", "hN : N.FG", "hIN : N ≤ I • N", "hIjac : I ≤ J.jacobson"], "goal": "N ≤ J • N"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "u₁ u₂ : Mˣ"], "goal": "Commute ↑u₁ ↑u₂ → Commute u₁ u₂"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "I B X : Set α", "hB : M.Base B", "hBX : M.Basis (B ∩ X) X", "e : α", "he : (e ∈ M.E ∧ e ∉ X) ∧ e ∈ B", "B' : Set α", "hB' : M.Base B'", "hem : B' ⊆ B ∪ X ∧ e ∉ B'", "f : α", "hfb : f ∈ B' \\ B", "hBf : M.Base (insert f (B \\ {e}))", "hi : M.Indep (insert f (B ∩ X))"], "goal": "False"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type x", "k : Type y", "A : Type z", "a b : R", "m n : ℕ", "inst✝ : CommRing R", "f : R[X]", "x y : R"], "goal": "eval₂ (RingHom.id R) (x + y) f = f.sum fun e a => a * (x ^ e + ↑e * x ^ (e - 1) * y + ↑(Polynomial.polyBinomAux1 x y e a) * y ^ 2)"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : CommSemiring R", "S : Type v", "inst✝ : CommSemiring S", "f g : R[X]", "h : (f * g).Separable", "this : IsCoprime f (derivative f * g + f * derivative g)"], "goal": "IsCoprime f g"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "𝒜 : Finset (Finset α)", "r : ℕ", "hr : r ≠ 0", "hr' : r ≤ Fintype.card α", "h𝒜 : 𝒜.card * r ≤ (∂ 𝒜).card * (Fintype.card α - r + 1)"], "goal": "↑𝒜.card / ↑((Fintype.card α).choose r) ≤ ↑(∂ 𝒜).card / ↑((Fintype.card α).choose (r - 1))"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H"], "goal": "↑I.symm = ↑I.symm"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Ring R", "AddCommGroup M", "Module R M", "p : Submodule R M", "S : Type u_3", "SMul S R", "SMul S M", "IsScalarTower S R M", "r : S", "x : M"], "goal": "Submodule.Quotient.mk (r • x) = r • Submodule.Quotient.mk x"}}
+{"state": {"context": ["K : Type u_2", "Field K", "w : NumberField.InfinitePlace K"], "goal": "¬w.IsReal ↔ w.IsComplex"}}
+{"state": {"context": ["Γ : Type u_1", "Γ' : Type u_2", "Inhabited Γ", "Inhabited Γ'", "f : Turing.PointedMap Γ Γ'", "l : List Γ"], "goal": "Turing.Tape.map f (Turing.Tape.mk₁ l) = Turing.Tape.mk₁ (List.map f.f l)"}}
+{"state": {"context": ["C : CategoryTheory.Cat"], "goal": "CategoryTheory.nerveFunctor.obj C = CategoryTheory.nerve ↑C"}}
+{"state": {"context": [], "goal": "∀ {α α_1 : Type u_1} {f : α → α_1}, f <$> none = none"}}
+{"state": {"context": ["p : Prop"], "goal": "{p}ᶜ = {¬p}"}}
+{"state": {"context": ["X : Type u_1", "MeasurableSpace X", "TopologicalSpace X", "OpensMeasurableSpace X", "T0Space X", "CompletelyRegularSpace X"], "goal": "Continuous ⇑MeasureTheory.diracProbaEquiv.symm"}}
+{"state": {"context": ["R : Type u_2", "NonUnitalRing R", "p : RingSeminorm R"], "goal": "⇑p.toAddGroupSeminorm = ⇑p"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_4", "AddCommMonoid α", "StarAddMonoid α", "A : Matrix n n α"], "goal": "(A + A.conjTranspose).IsHermitian"}}
+{"state": {"context": ["C : Type u", "inst✝⁹ : Category.{v, u} C", "X Y : C", "D : Type u₂", "inst✝⁸ : Category.{w, u₂} D", "E : Type u₃", "inst✝⁷ : Category.{w', u₃} E", "F : C ⥤ D", "G : D ⥤ E", "A A' B B' : C", "inst✝⁶ : HasBinaryProduct A B", "inst✝⁵ : HasBinaryProduct A' B'", "inst✝⁴ : HasBinaryProduct (F.obj A) (F.obj B)", "inst✝³ : HasBinaryProduct (F.obj A') (F.obj B')", "inst✝² : HasBinaryProduct (G.obj (F.obj A)) (G.obj (F.obj B))", "inst✝¹ : HasBinaryProduct ((F ⋙ G).obj A) ((F ⋙ G).obj B)", "inst✝ : IsIso (prodComparison F A B)"], "goal": "inv (prodComparison F A B) ≫ F.map prod.fst = prod.fst"}}
+{"state": {"context": ["x : ℝ", "hx : 0 ≤ x"], "goal": "x + 1 ≤ Real.exp x"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ : R"], "goal": "Quaternion.imI 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝⁷ : MonoidWithZero M₀", "inst✝⁶ : GroupWithZero G₀", "inst✝⁵ : Nontrivial M₀", "inst✝⁴ : MonoidWithZero M₀'", "inst✝³ : FunLike F G₀ M₀", "inst✝² : MonoidWithZeroHomClass F G₀ M₀", "inst✝¹ : FunLike F' G₀ M₀'", "f✝ : F", "a : G₀", "inst✝ : MonoidWithZeroHomClass F' G₀ M₀'", "f g : F'", "h : f a = g a", "ha : a ≠ 0"], "goal": "f a⁻¹ = g a⁻¹"}}
+{"state": {"context": ["α : Type u_2", "DivisionRing α", "CharZero α", "s : List ℚ"], "goal": "↑s.prod = (List.map Rat.cast s).prod"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_3", "LinearOrder ι", "LocallyFiniteOrderBot ι", "IsWellOrder ι fun x x_1 => x < x_1", "f : ι → E", "c : ι"], "goal": "Submodule.span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic c) = Submodule.span 𝕜 (f '' Set.Iic c)"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E", "inst✝⁵ : Algebra F E", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "inst✝² : Algebra E K", "inst✝¹ : IsScalarTower F E K", "inst✝ : Algebra.IsAlgebraic F E"], "goal": "Cardinal.lift.{w, v} (sepDegree F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)"}}
+{"state": {"context": ["R : Type u", "Semiring R", "n : ℕ", "a : R", "ha : a ≠ 0"], "goal": "(Polynomial.C a * Polynomial.X ^ n).natDegree = n"}}
+{"state": {"context": ["n : ℕ"], "goal": "Ordinal.toNatOrdinal ↑n = ↑n"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "self : Cᴹᵒᵖ"], "goal": "(CategoryTheory.MonoidalOpposite.unmopBraidedFunctor C).obj self = self.unmop"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "𝒜₁ 𝒜₂ : Finset (Finset α)", "r : ℕ", "h₁ : Finset.Colex.IsInitSeg 𝒜₁ r", "h₂ : Finset.Colex.IsInitSeg 𝒜₂ r"], "goal": "𝒜₁ ⊆ 𝒜₂ ∨ 𝒜₂ ⊆ 𝒜₁"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "CommRing R", "x : WittVector p R"], "goal": "x.verschiebungFun.coeff 0 = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : CategoryTheory.Comma (𝟭 (CategoryTheory.SimplicialObject C)) (CategoryTheory.SimplicialObject.const C)"], "goal": "(𝟙 X).right = 𝟙 X.right"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommSemiring R", "Semiring S", "Algebra R S", "M N : Submodule R S", "hc : ∀ (m : ↥M) (n : ↥N), Commute ↑m ↑n"], "goal": "N.mulMap M = M.mulMap N ∘ₗ ↑(TensorProduct.comm R ↥N ↥M)"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "v : ℝ → E → E", "s : ℝ → Set E", "K : ℝ≥0", "f g f' g' : ℝ → E", "a b t₀ εf εg δ : ℝ", "hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)", "hf : ContinuousOn f (Icc a b)", "hf' : ∀ t ∈ Ioc a b, HasDerivWithinAt f (v t (f t)) (Iic t) t", "hfs : ∀ t ∈ Ioc a b, f t ∈ s t", "hg : ContinuousOn g (Icc a b)", "hg' : ∀ t ∈ Ioc a b, HasDerivWithinAt g (v t (g t)) (Iic t) t", "hgs : ∀ t ∈ Ioc a b, g t ∈ s t", "hb : f b = g b", "hv' : ∀ (t : ℝ), LipschitzOnWith K (Neg.neg ∘ v (-t)) (s (-t))", "hmt1 : MapsTo Neg.neg (Icc (-b) (-a)) (Icc a b)"], "goal": "EqOn f g (Icc a b)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "OrderedSemiring 𝕜", "AddCommMonoid E", "SMul 𝕜 E"], "goal": "↑⊥ = ∅"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommRing R", "n✝ : ℕ", "inst✝² : IsDomain R", "n : ℕ", "K : Type u_2", "inst✝¹ : Field K", "μ : K", "inst✝ : NeZero ↑n", "hnpos : 0 < n", "hμ : X - C μ ∣ cyclotomic n K", "hμn : orderOf μ ∣ n", "hnμ : orderOf μ ≠ n", "ho : 0 < orderOf μ", "i : ℕ", "hiμ : X - C μ ∣ cyclotomic i K", "hio : i ∣ orderOf μ", "key : i < n", "key' : i ∣ n", "hni : {i, n} ⊆ n.divisors"], "goal": "False"}}
+{"state": {"context": ["n p : ℕ", "pp : Prime p", "k a : ℕ"], "goal": "a ∈ (p ^ k).divisors ↔ a ∈ map { toFun := fun x => p ^ x, inj' := ⋯ } (range (k + 1))"}}
+{"state": {"context": ["ι : Type u_1", "σ : Type u_2", "R : Type u_4", "DecidableEq ι", "Semiring R", "SetLike σ R", "AddSubmonoidClass σ R", "A : ι → σ", "CanonicallyOrderedAddCommMonoid ι", "SetLike.GradedMonoid A", "i : ι", "r : ↥(A i)", "r' : ⨁ (i : ι), ↥(A i)", "n : ι", "h : ¬i ≤ n"], "goal": "↑(((DirectSum.of (fun i => ↥(A i)) i) r * r') n) = 0"}}
+{"state": {"context": ["F : Type w", "R : Type u", "S : Type v", "T : Type u_1", "inst✝⁴ : NonUnitalNonAssocRing R", "inst✝³ : NonUnitalNonAssocRing S", "inst✝² : NonUnitalNonAssocRing T", "inst✝¹ : FunLike F R S", "inst✝ : NonUnitalRingHomClass F R S", "g : S →ₙ+* T", "f : R →ₙ+* S", "s : Set R", "p : ↥(closure s) → Prop", "a : ↥(closure s)", "mem : ∀ (x : R) (hx : x ∈ s), p ⟨x, ⋯⟩", "zero : p 0", "add : ∀ (x y : ↥(closure s)), p x → p y → p (x + y)", "neg : ∀ (x : ↥(closure s)), p x → p (-x)", "mul : ∀ (x y : ↥(closure s)), p x → p y → p (x * y)", "b : R", "hb : b ∈ closure s"], "goal": "∃ (x : b ∈ closure s), p ⟨b, x⟩"}}
+{"state": {"context": ["ι : Type u_1", "s : Finset ι", "(j : ι) → Decidable (j ∈ s)", "π : ι → Type u_3", "t : Set ι", "t' : (i : ι) → Set (π i)", "f g : (i : ι) → π i", "hf : f ∈ t.pi t'", "hg : g ∈ t.pi t'"], "goal": "s.piecewise f g ∈ t.pi t'"}}
+{"state": {"context": ["v i : Nat"], "goal": "(BitVec.allOnes v).getLsb i = decide (i < v)"}}
+{"state": {"context": ["G : Type u_1", "Monoid G", "x : G", "h : IsOfFinOrder x"], "goal": "0 < orderOf x"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "CompactSpace Y", "CompactSpace X", "W : TopologicalSpace.Clopens (X × Y)"], "goal": "∃ I, W = I.sup fun i => i.1 ×ˢ i.2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "R : Type u_3", "n : Type u_4", "m : Type u_5", "inst✝¹ : Zero α", "inst✝ : DecidableEq n", "A : Matrix n n α", "h : A.IsDiag", "i j : n", "hij : i ≠ j"], "goal": "diagonal A.diag i j = A i j"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : Z.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "i : 𝒰.J"], "goal": "(AlgebraicGeometry.Scheme.Pullback.openCoverOfBase 𝒰 f g).map i =\n  CategoryTheory.Limits.pullback.map (CategoryTheory.Limits.pullback.snd f (��.map i))\n    (CategoryTheory.Limits.pullback.snd g (𝒰.map i)) f g (CategoryTheory.Limits.pullback.fst f (𝒰.map i))\n    (CategoryTheory.Limits.pullback.fst g (𝒰.map i)) (𝒰.map i) ⋯ ⋯"}}
+{"state": {"context": ["E : Type u_4", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "K : Submodule ℝ E"], "goal": "K.orthogonalBilin bilinFormOfRealInner = Kᗮ"}}
+{"state": {"context": ["A : Type u₁", "CategoryTheory.Category.{v₁, u₁} A", "B : Type u₂", "CategoryTheory.Category.{v₂, u₂} B", "T : Type u₃", "CategoryTheory.Category.{v₃, u₃} T", "L : CategoryTheory.Functor A T", "R : CategoryTheory.Functor B T", "X : CategoryTheory.Comma L R"], "goal": "(CategoryTheory.CategoryStruct.id X).right = CategoryTheory.CategoryStruct.id X.right"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "κ : ι → Sort u_5", "LE α", "f : (i : ι) → κ i → LowerSet α"], "goal": "↑(⨅ i, ⨅ j, f i j) = ⋂ i, ⋂ j, ↑(f i j)"}}
+{"state": {"context": ["α : Type u", "AddMonoidWithOne α", "n : ℕ", "n.AtLeastTwo"], "goal": "⊤ ≠ OfNat.ofNat n"}}
+{"state": {"context": ["α✝ : Type u", "L L₁ L₂ L₃ L₄ : List (α✝ × Bool)", "β✝ : Type v", "f : α✝ → β✝", "x✝ y : FreeGroup α✝", "α : Type ?u.60273", "β : Type ?u.60274", "e : α ≃ β", "x : FreeGroup β"], "goal": "(map ⇑e) ((map ⇑e.symm) x) = x"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "E : Type u_2", "inst✝⁴ : NormedAddCommGroup E", "𝕜 : Type u_3", "inst✝³ : RCLike 𝕜", "inst✝² : NormedSpace 𝕜 E", "G : Type u_4", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f : ι → E → G", "g : E → G", "f' : ι → E → E →L[𝕜] G", "g' : E → E →L[𝕜] G", "x : E", "hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)", "hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2", "hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))", "A : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E", "hl : l ≠ ⊥", "this : l.NeBot", "hfg' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (f' n.1.1 n.2 - f' n.1.2 n.2) < ε", "ε : ℝ", "hε : ε > 0"], "goal": "∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) ((↑‖n.2 - x‖)⁻¹ • (f n.1.1 n.2 - f n.1.1 x) - (↑‖n.2 - x‖)⁻¹ • (f n.1.2 n.2 - f n.1.2 x)) < ε"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "f : Arrow C", "inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x => f.left) fun x => f.hom", "S : SplitEpi f.hom", "n : ℕ"], "goal": "(s f S n ≫ WidePullback.base fun x => f.hom) = WidePullback.base fun x => f.hom"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "m : Multiset α"], "goal": "m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype fun x => x ∈ m.toEnumFinset) = m.coeEmbedding"}}
+{"state": {"context": ["ι : Type u_1", "β : ι → Type u_2", "x : (i : ι) → β i", "i : ι"], "goal": "toLex x i = x i"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π₁ π₂ : BoxIntegral.Prepartition I", "h : Disjoint π₁.iUnion π₂.iUnion"], "goal": "(π₁.disjUnion π₂ h).iUnion = π₁.iUnion ∪ π₂.iUnion"}}
+{"state": {"context": ["α : Type u_3", "Preorder α", "self : SuccOrder α", "a : α"], "goal": "a ≤ SuccOrder.succ a"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : CancelCommMonoidWithZero α", "inst✝² : UniqueFactorizationMonoid α", "inst✝¹ : DecidableEq (Associates α)", "inst✝ : (p : Associates α) → Decidable (Irreducible p)", "m p : Associates α", "h0 : m ≠ 0", "hp : Irreducible p", "a✝ : Nontrivial α"], "goal": "0 < p.count m.factors → p ≤ m"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f g : α → ℝ≥0∞", "hf : Measurable f", "hg : Measurable g", "s : Set α"], "goal": "∫⁻ (x : α) in s, max (f x) (g x) ∂μ = ∫⁻ (x : α) in s ∩ {x | f x ≤ g x}, g x ∂μ + ∫⁻ (x : α) in s ∩ {x | g x < f x}, f x ∂μ"}}
+{"state": {"context": ["K : Type u", "Field K", "A : ValuationSubring K"], "goal": "0 ∈ A"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Monoid α", "inst✝² : Monoid β", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "a b : α", "h : 0 < multiplicity a b"], "goal": "a ^ 1 ∣ b"}}
+{"state": {"context": ["a : Nat", "b : Nat", "h : 0 < a"], "goal": "0 < a / a.gcd b"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "𝕜 : Type u_3", "inst✝¹⁰ : _root_.RCLike 𝕜", "E : Type u_4", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : InnerProductSpace 𝕜 E", "E' : Type u_5", "inst✝⁷ : NormedAddCommGroup E'", "inst✝⁶ : InnerProductSpace 𝕜 E'", "F : Type u_6", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : InnerProductSpace ℝ F", "F' : Type u_7", "inst✝³ : NormedAddCommGroup F'", "inst✝² : InnerProductSpace ℝ F'", "inst✝¹ : Fintype ι", "v : Set E", "A : ι → Submodule 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "b : OrthonormalBasis ι ℝ ℝ", "this✝ : Unique ι", "this : b default = 1 ∨ b default = -1"], "goal": "(⇑b = fun x => 1) ∨ ⇑b = fun x => -1"}}
+{"state": {"context": ["R : Type u_1", "EuclideanDomain R", "GCDMonoid R", "p q : R", "hp : p ≠ 0"], "goal": "p / gcd p q ≠ 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "P Q R : C", "f : P ⟶ Q", "g : Q ⟶ R", "CategoryTheory.StrongMono (CategoryTheory.CategoryStruct.comp f g)"], "goal": "CategoryTheory.StrongMono f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f : α → β", "Add β", "ContinuousAdd β", "hf : MeasureTheory.AEStronglyMeasurable f μ", "c : β"], "goal": "MeasureTheory.AEStronglyMeasurable (fun x => c + f x) μ"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁹ : CommRing R", "inst✝⁸ : IsDomain R", "inst✝⁷ : CommRing S", "L : Type u_3", "inst✝⁶ : Field L", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra S L", "inst✝³ : Algebra R L", "inst✝² : IsScalarTower R S L", "inst✝¹ : IsIntegralClosure S R L", "inst✝ : Algebra.IsAlgebraic R L", "inj : Function.Injective ⇑(algebraMap R L)", "a b : S", "hb : b ≠ 0", "c : ↥(integralClosure R L)", "d : R", "d_ne : d ≠ 0", "hx : (algebraMap S L) a / (algebraMap S L) b * (algebraMap R L) d = ↑c"], "goal": "(algebraMap S L) (d • a) = (algebraMap S L) (b * mk' S ↑c ⋯)"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "CommSemiring R", "CommSemiring S", "Semiring A", "Algebra R S", "Algebra R A", "Algebra S A", "IsScalarTower R S A", "s : Set A"], "goal": "Algebra.adjoin S ↑(Algebra.adjoin R s) = Algebra.adjoin S s"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E", "h : x ∉ tsupport f"], "goal": "fderiv 𝕜 f x = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁶ : MeasurableSpace δ", "inst✝⁵ : NormedAddCommGroup β", "inst✝⁴ : NormedAddCommGroup γ", "𝕜 : Type u_5", "inst✝³ : NontriviallyNormedField 𝕜", "inst✝² : CompleteSpace 𝕜", "E : Type u_6", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f : α → 𝕜", "c : E", "hc : c ≠ 0", "a✝ : AEStronglyMeasurable f μ"], "goal": "∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤ ∧ ↑‖c‖₊ < ⊤ ∨ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ = 0 ∨ ↑‖c‖₊ = 0 ↔ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤"}}
+{"state": {"context": ["ι : Type u", "X : Type v", "TopologicalSpace X", "s : Set X", "f : BumpCovering ι X s", "i : ι", "x : X"], "goal": "∃ t, ⇑(f.toPartitionOfUnity i) =ᶠ[𝓝 x] ⇑(f i) * ⇑(∏ j ∈ Finset.filter (fun j => WellOrderingRel j i) t, (1 - f j))"}}
+{"state": {"context": ["P : Type u_1", "inst✝¹ : Preorder P", "p : P", "ι : Type u_2", "inst✝ : Encodable ι", "𝒟 : ι → Cofinal P", "i : ι"], "goal": "(match Encodable.decode (Encodable.encode i) with | none => sequenceOfCofinals p 𝒟 (Encodable.encode i) | some i_1 => (𝒟 i_1).above (sequenceOfCofinals p 𝒟 (Encodable.encode i))) ∈ 𝒟 i"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹⁴ : CommRing R", "inst✝¹³ : CommRing S", "inst✝¹² : Algebra R S", "P : Generators R S", "R' : Type u'", "S' : Type v'", "inst✝¹¹ : CommRing R'", "inst✝¹⁰ : CommRing S'", "inst✝⁹ : Algebra R' S'", "P' : Generators R' S'", "inst✝⁸ : Algebra R R'", "inst✝⁷ : Algebra S S'", "inst✝⁶ : Algebra R S'", "inst✝⁵ : IsScalarTower R R' S'", "inst✝⁴ : IsScalarTower R S S'", "T : Type uT", "inst✝³ : CommRing T", "inst✝² : Algebra R T", "inst✝¹ : Algebra S T", "inst✝ : IsScalarTower R S T", "f g : P.Hom P'", "x : ↥P.ker"], "goal": "↑(P'.ker.cotangentEquivIdeal (mk ⟨f.toAlgHom ↑x, ⋯⟩ - mk ⟨g.toAlgHom ↑x, ⋯⟩).val) = ↑(P'.ker.cotangentEquivIdeal (P'.ker.toCotangent ((f.subToKer g) ↑x)))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝³ : MeasurableSpace δ", "inst✝² : NormedAddCommGroup β", "inst✝¹ : NormedAddCommGroup γ", "E : Type u_5", "inst✝ : NormedAddCommGroup E", "f : α → E", "h : HasFiniteIntegral f μ", "s : Set α"], "goal": "HasFiniteIntegral f (μ.restrict s)"}}
+{"state": {"context": ["C₁ : Type u₁", "C₂ : Type u₂", "C₃ : Type u₃", "C₄ : Type u₄", "CategoryTheory.Category.{v₁, u₁} C₁", "CategoryTheory.Category.{v₂, u₂} C₂", "CategoryTheory.Category.{v₃, u₃} C₃", "CategoryTheory.Category.{v₄, u₄} C₄", "T : CategoryTheory.Functor C₁ C₂", "L : CategoryTheory.Functor C₁ C₃", "R : CategoryTheory.Functor C₂ C₄", "B : CategoryTheory.Functor C₃ C₄", "w : CategoryTheory.TwoSquare T L R B", "X₂ : C₂", "X₃ : C₃", "g : R.obj X₂ ⟶ B.obj X₃", "f : w.StructuredArrowRightwards g"], "goal": "(CategoryTheory.TwoSquare.EquivalenceJ.functor w g).obj f =\n  CategoryTheory.CostructuredArrow.mk (CategoryTheory.StructuredArrow.homMk f.right.hom ⋯)"}}
+{"state": {"context": ["n : ℕ"], "goal": "0 < n.succ"}}
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+{"state": {"context": ["R : Type u", "Ring R", "p q : R[X]", "h : q.natDegree < p.natDegree"], "goal": "(p - q).natDegree = p.natDegree"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s✝ : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a : α", "f : β × γ → ℝ≥0∞", "hf : Measurable f", "s : Set β", "t : Set γ", "hs : MeasurableSet s", "ht : MeasurableSet t"], "goal": "∫⁻ (z : β × γ) in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫⁻ (x : β) in s, ∫⁻ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a"}}
+{"state": {"context": ["X✝ : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝² : MeasurableSpace X✝", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "f✝ g : X✝ → E", "s✝ t : Set X✝", "μ✝ ν : Measure X✝", "l l' : Filter X✝", "X : Type u_5", "m m0 : MeasurableSpace X", "μ : Measure X", "hm : m ≤ m0", "f : X → E", "hf_meas : StronglyMeasurable f", "s : Set X", "hs : MeasurableSet s"], "goal": "∫ (x : X) in s, f x ∂μ = ∫ (x : X) in s, f x ∂μ.trim hm"}}
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+{"state": {"context": ["α : Type u", "Preorder α", "s t : Set α", "h : s ⊆ t"], "goal": "BddAbove t → BddAbove s"}}
+{"state": {"context": ["ι : Type u_1", "N : Type u_4", "TopologicalSpace N", "Monoid N", "ContinuousMul N", "T2Space N", "f : ι → Nˣ", "r₁ r₂ : N", "l : Filter ι", "l.NeBot", "h₁ : Filter.Tendsto (fun x => ↑(f x)) l (𝓝 r₁)", "h₂ : Filter.Tendsto (fun x => ↑(f x)⁻¹) l (𝓝 r₂)"], "goal": "↑(h₁.units h₂) = r₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "c : α", "AddCommGroup α", "Neg β", "h : Function.Antiperiodic f c", "a : α"], "goal": "Function.Antiperiodic (fun x => f (x - a)) c"}}
+{"state": {"context": ["x y : ℝ", "h : 1 < x * y", "hx : 0 < x"], "goal": "Real.arctan x + Real.arctan y = Real.arctan ((x + y) / (1 - x * y)) + Real.pi"}}
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+{"state": {"context": ["R : Type u_1", "M : Type u_2", "N : Type u_4", "ι : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "TopologicalSpace M", "AddCommMonoid N", "Module R N", "TopologicalSpace N", "f : M [⋀^ι]→L[R] N", "v : ι → M", "hv : ¬Function.Injective v"], "goal": "f v = 0"}}
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+{"state": {"context": ["u v m n : ℤ"], "goal": "¬n % 2 = 1 ↔ n % 2 = 0"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "CommRing K", "Field L", "Algebra K L", "NoZeroSMulDivisors K L", "Algebra.IsAlgebraic K L", "ϕ : L ≃ₐ[K] L"], "goal": "(Algebra.IsAlgebraic.algEquivEquivAlgHom K L) ϕ = ↑ϕ"}}
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+{"state": {"context": ["x y : ℝ", "h0 : x < 0", "h1 : -1 < x", "h0' : 0 < -x", "h1' : -x < 1"], "goal": "log (-x) < 0"}}
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+{"state": {"context": ["w : Nat", "x y : BitVec w", "pred : ∀ (i : Fin w), x.getLsb ↑i = y.getLsb ↑i"], "goal": "x = y"}}
+{"state": {"context": ["α : Type u_3", "m : MeasurableSpace α", "E : Type u_4", "MeasurableSpace E", "MeasurableSingletonClass E", "Countable E", "f g : α → E", "hf : Measurable f", "hg : Measurable g"], "goal": "MeasurableSet {x | f x = g x}"}}
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+{"state": {"context": ["α : Type u_1", "LinearOrderedCommGroupWithZero α", "a b c : α", "h : c ≠ 0", "hab : a * c ≤ b * c"], "goal": "a ≤ b"}}
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+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "R : Type u_4", "SMul R Y", "r : R", "f : LocallyConstant X Y", "x : X"], "goal": "(r • f) x = r • f x"}}
+{"state": {"context": ["L : Language", "M : Type w", "N : Type u_1", "P : Type u_2", "inst✝² : L.Structure M", "inst✝¹ : L.Structure N", "inst✝ : L.Structure P", "α : Type u'", "β : Type v'"], "goal": "lift.{max (max u u') v, max (max u u') v} #((k : ℕ) × L.Term (α ⊕ Fin k) ⊕ (n : ℕ) × L.Relations n ⊕ ℕ) ≤ max ℵ₀ (lift.{max (max u u') v, max (max u u') v} (lift.{max u v, u'} #α + lift.{u', max u v} L.card)) ∧ ℵ₀ ≤ max ℵ₀ (lift.{max (max u u') v, max (max u u') v} (lift.{max u v, u'} #α + lift.{u', max u v} L.card))"}}
+{"state": {"context": ["α : Type u", "s t : Set α"], "goal": "Disjoint (t \\ s) s"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "s : Set A", "z : A"], "goal": "z ∈ StarSubalgebra.centralizer R s ↔ ∀ g ∈ s, g * z = z * g ∧ star g * z = z * star g"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "p : α → Prop", "DecidablePred p", "LocallyFiniteOrder α", "a b : Subtype p", "hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x"], "goal": "Finset.map (Function.Embedding.subtype p) (Finset.Ioo a b) = Finset.Ioo ↑a ↑b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "x y : R", "inst✝ : Semiring R", "h_comm : Commute x y", "m n k : ℕ", "hx : x ^ m = 0", "hy : y ^ n = 0", "h : m + n ≤ k + 1"], "goal": "∑ m ∈ Finset.antidiagonal k, k.choose m.1 • (x ^ m.1 * y ^ m.2) = 0"}}
+{"state": {"context": ["n m : SimplexCategory", "f : n ⟶ m"], "goal": "CategoryTheory.Epi f ↔ Function.Surjective ⇑(SimplexCategory.Hom.toOrderHom f)"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "Ring k", "AddCommGroup V1", "Module k V1", "AffineSpace V1 P1", "p₁ p₂ p₃ p₄ : P1", "c : k"], "goal": "(AffineMap.lineMap p₁ p₂) c -ᵥ (AffineMap.lineMap p₃ p₄) c = (AffineMap.lineMap (p₁ -ᵥ p₃) (p₂ -ᵥ p₄)) c"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "LinearOrder α", "ClosedIciTopology α", "a : α"], "goal": "IsOpen (Set.Iio a)"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝¹¹ : CommSemiring ι", "inst✝¹⁰ : Module ι (Additive ℤˣ)", "inst✝⁹ : DecidableEq ι", "𝒜 : ι → Type u_5", "ℬ : ι → Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)", "inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)", "inst✝⁵ : (i : ι) → Module R (𝒜 i)", "inst✝⁴ : (i : ι) → Module R (ℬ i)", "inst✝³ : DirectSum.GRing 𝒜", "inst✝² : DirectSum.GRing ℬ", "inst✝¹ : DirectSum.GAlgebra R 𝒜", "inst✝ : DirectSum.GAlgebra R ℬ", "a : ⨁ (i : ι), 𝒜 i", "b : ℬ 0"], "goal": "↑(gradedComm R 𝒜 ℬ) ∘ₗ (mk R (⨁ (i : ι), 𝒜 i) (⨁ (i : ι), ℬ i)).flip ((lof R ι ℬ 0) b) = (mk R (⨁ (i : ι), ℬ i) (⨁ (i : ι), 𝒜 i)) ((lof R ι ℬ 0) b)"}}
+{"state": {"context": ["R : Type u", "A : Type u_1", "inst✝¹ : CommRing R", "inst✝ : CommRing A", "p : R[X]", "x : R", "P : Ideal R", "hP : x ∈ P", "n✝ : ℕ", "i : n✝ < (p.scaleRoots x).natDegree"], "goal": "(p.scaleRoots x).coeff n✝ ∈ P"}}
+{"state": {"context": ["α : Type u_1", "Lattice α", "CommGroup α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α"], "goal": "mabs a / a = a⁻ᵐ ^ 2"}}
+{"state": {"context": ["𝓕 : Type u_1", "α✝ : Type u_2", "E✝ : Type u_3", "F : Type u_4", "inst✝³ : SeminormedCommGroup E✝", "inst✝² : SeminormedCommGroup F", "a a₁ a₂ b b₁ b₂ : E✝", "r r₁ r₂ : ℝ", "α : Type u_5", "E : Type u_6", "inst✝¹ : SeminormedCommGroup E", "inst✝ : PseudoEMetricSpace α", "K Kf Kg : ℝ≥0", "f g : α → E", "s : Set α", "x✝ : α", "hf : AntilipschitzWith Kf f", "hg : LipschitzWith Kg g", "hK : Kg < Kf⁻¹", "this : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace ⋯", "x y : α"], "goal": "dist x y ≤ ↑(Kf⁻¹ - Kg)⁻¹ * dist (f x * g x) (f y * g y)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁵ : RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : InnerProductSpace 𝕜 E", "ι : Type u_3", "inst✝² : LinearOrder ι", "inst✝¹ : LocallyFiniteOrderBot ι", "inst✝ : IsWellOrder ι fun x x_1 => x < x_1", "i j : ι", "hij : i < j", "b : Basis ι 𝕜 E", "this : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt 𝕜 ⇑b '' Set.Iio j)"], "goal": "(b.repr (gramSchmidt 𝕜 (⇑b) i)) j = 0"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "QuasiSober α", "IrreducibleSpace α"], "goal": "IsGenericPoint (genericPoint α) ⊤"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a : α", "s : Finset α", "h : a ∈ s"], "goal": "s ∩ {a} = {a}"}}
+{"state": {"context": ["α : Type u", "x : FreeRing α"], "goal": "(FreeAbelianGroup.lift ⇑(FreeMonoid.lift FreeCommRing.of)) x = (FreeAbelianGroup.lift ((fun α_1 => FreeAbelianGroup.of α_1) ∘ fun l => ↑l)) x"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹² : CommRing R", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R M", "I : Ideal R", "S : Type u_3", "inst✝⁹ : CommSemiring S", "inst✝⁸ : Algebra S R", "inst✝⁷ : Module S M", "inst✝⁶ : IsScalarTower S R M", "R' : Type u_4", "M' : Type u_5", "inst✝⁵ : CommRing R'", "inst✝⁴ : AddCommGroup M'", "inst✝³ : Module R' M'", "inst✝² : Algebra R R'", "inst✝¹ : Module R M'", "inst✝ : IsScalarTower R R' M'", "p : R[X]", "q : PolynomialModule R M", "r : R"], "goal": "∀ (a : ℕ) (b : M), (eval r) (p • (single R a) b) = Polynomial.eval r p • (eval r) ((single R a) b)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R", "S : Type u_2", "CommSemiring S", "Algebra R S", "IsLocalization M S", "a₁ b₁ : R", "a₂ b₂ : ↥M", "H : ↑a₂ * b₁ = ↑b₂ * a₁"], "goal": "IsLocalization.mk' S a₁ a₂ = IsLocalization.mk' S b₁ b₂"}}
+{"state": {"context": ["x : SetTheory.PGame", "i : x.RightMoves"], "goal": "(-x).moveLeft (SetTheory.PGame.toLeftMovesNeg i) = -x.moveRight i"}}
+{"state": {"context": ["R : Type u", "Semiring R", "S T : Ideal R", "x : R"], "goal": "x ∈ T → x ∈ S ⊔ T"}}
+{"state": {"context": ["α : Type u_1", "as bs cs : List α", "h : as ++ bs = cs ++ bs"], "goal": "as = cs"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "k : ℕ", "a : R", "inst✝ : Invertible 2"], "goal": "dickson 1 1 1 = 2 * (Chebyshev.T R ↑1).comp (C ⅟2 * X)"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "x : K"], "goal": "‖(starRingEnd K) x + x‖ ^ 2 = RCLike.re ((starRingEnd K) x + x) ^ 2"}}
+{"state": {"context": ["m : ℤ", "hm₀ : m ≠ 0", "hm₁ : m ≠ 1", "x : ℝ", "hx : 0 < x"], "goal": "0 < (∏ i ∈ Finset.range 2, (↑m - ↑i)) * x ^ (m - ↑2)"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "s : Set α", "f : α →₀ M"], "goal": "(Finsupp.lsubtypeDomain s) f = Finsupp.subtypeDomain (fun x => x ∈ s) f"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "M : Type u_2", "inst✝³ : AddCommMonoid M", "inst✝² : Module R M", "A : Type u_3", "inst✝¹ : Semiring A", "inst✝ : Algebra R A", "f : M →ₗ[R] A", "g : TensorAlgebra R M →ₐ[R] A"], "goal": "g.toLinearMap ∘ₗ ι R = f ↔ (lift R).symm g = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "N : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "f : α → β", "v : α →₀ M", "g : M → N", "h0 : g 0 = 0", "hadd : ∀ (x y : M), g (x + y) = g x + g y"], "goal": "Finsupp.mapDomain f (Finsupp.mapRange g h0 v) = Finsupp.mapRange g h0 (Finsupp.mapDomain f v)"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜₂ : Type u_2", "𝕝 : Type u_3", "𝕝₂ : Type u_4", "E : Type u_5", "F : Type u_6", "G : Type u_7", "ι : Type u_8", "ι' : Type u_9", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : Module 𝕜 E", "inst✝² : Nonempty ι", "inst✝¹ : TopologicalSpace E", "p : SeminormFamily 𝕜 E ι", "inst✝ : T1Space E", "hp : WithSeminorms p", "x : E", "hx : x ≠ 0", "this : ∀ ⦃x y : E⦄, x ⤳ y → x = y"], "goal": "∃ i, (p i) x ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NormedField 𝕜", "SeminormedAddCommGroup E", "NormedSpace 𝕜 E", "s : Set E", "h : Bornology.IsBounded s"], "goal": "Bornology.IsVonNBounded 𝕜 s"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "K : Type u_3", "M : Type u_4", "M₂ : Type u_5", "V : Type u_6", "S✝ : Type u_7", "inst✝⁷ : Semiring R", "inst✝⁶ : AddCommMonoid M", "inst✝⁵ : Module R M", "x : M", "p p' : Submodule R M", "inst✝⁴ : Semiring R₂", "σ₁₂ : R →+* R₂", "inst✝³ : AddCommMonoid M₂", "inst✝² : Module R₂ M₂", "F : Type u_8", "inst✝¹ : FunLike F M M₂", "inst✝ : SemilinearMapClass F σ₁₂ M M₂", "s t : Set M", "S : Finset M"], "goal": "CompleteLattice.IsCompactElement (⨆ x ∈ ↑S, span R {x})"}}
+{"state": {"context": ["n : ℕ", "l : List ℕ", "h₁ : l.prod = n", "h₂ : ∀ p ∈ l, Prime p"], "goal": "l ~ n.primeFactorsList"}}
+{"state": {"context": ["k : Type u_1", "Field k", "K : Type u_2", "Field K", "Algebra k K", "σ : K ≃ₐ[k] K", "w : NumberField.InfinitePlace K", "x : K"], "goal": "(σ • w) x = w (σ.symm x)"}}
+{"state": {"context": ["a✝ b✝ c✝ d✝ : Ordinal.{u}", "a b c a' : Ordinal.{u_1}", "ha : a' < a", "d : Ordinal.{u_1}", "hd : d < b ♯ c"], "goal": "a' ⨳ (b ♯ c) ♯ a ⨳ d < a ⨳ b ♯ a ⨳ c ♯ a' ⨳ d"}}
+{"state": {"context": ["M : Type u_1", "S T : Set M", "inst✝ : MulOneClass M", "x✝ : M"], "goal": "1 * x✝ = x✝ * 1"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : NonemptyInterval α", "a : α"], "goal": "↑s = Interval.pure a ↔ s = NonemptyInterval.pure a"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "CommMonoid M", "f : α → M", "h : ∀ (x : α), f x = 1"], "goal": "∏ᶠ (i : α), f i = 1"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X", "H : IsPreirreducible s", "f : X → Y", "hf : ContinuousOn f s", "u v : Set Y", "hu : IsOpen u", "hv : IsOpen v", "x : X", "hx : x ∈ s", "hxu : x ∈ f ⁻¹' u", "y : X", "hy : y ∈ s", "hyv : y ∈ f ⁻¹' v", "u' : Set X", "hu' : IsOpen u'", "u'_eq : f ⁻¹' u ∩ s = u' ∩ s", "v' : Set X", "hv' : IsOpen v'", "v'_eq : f ⁻¹' v ∩ s = v' ∩ s", "this : (s ∩ u').Nonempty → (s ∩ v').Nonempty → (s ∩ (u' ∩ v')).Nonempty"], "goal": "(f '' s ∩ (u ∩ v)).Nonempty"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "this : ∀ (r : ℝ≥0) {n : ℕ}, 0 < n → (↑r ≤ 1 / ↑(‖p n‖₊ ^ (1 / ↑n)) ↔ ‖p n‖₊ * r ^ n ≤ 1)", "r : ℝ≥0", "hr : ↑r < liminf (fun n => 1 / ↑(‖p n‖₊ ^ (1 / ↑n))) atTop"], "goal": "↑r ≤ p.radius"}}
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+{"state": {"context": ["J : Type w", "C : Type u", "inst✝ : Category.{v, u} C", "X B : C", "F : Discrete PEmpty.{1} ⥤ Over B", "s : Cone F", "m : s.pt ⟶ { pt := mk (𝟙 B), π := { app := fun p => p.as.elim, naturality := ⋯ } }.pt", "x✝ : ∀ (j : Discrete PEmpty.{1}), m ≫ { pt := mk (𝟙 B), π := { app := fun p => p.as.elim, naturality := ⋯ } }.π.app j = s.π.app j"], "goal": "m.left = s.pt.hom"}}
+{"state": {"context": ["c : Cardinal.{u_2}", "H : c.IsRegular"], "goal": "0 < c.ord"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "One α", "Preorder α", "a : α", "Nonempty β"], "goal": "1 ≤ Function.const β a ↔ 1 ≤ a"}}
+{"state": {"context": ["n : ℕ", "A : Type u_2", "AddGroup A", "f : { f // f ↑n = 0 }"], "goal": "Function.Injective ⇑((ZMod.lift n) f) ↔ ∀ (m : ℤ), ↑f m = 0 → ↑m = 0"}}
+{"state": {"context": ["k G : Type u", "inst✝¹ : CommRing k", "inst✝ : Group G", "A : Rep k G"], "goal": "(HomologicalComplex.cyclesIsoSc' (inhomogeneousCochains A) 1 2 3 isoTwoCocycles.proof_2 isoTwoCocycles.proof_3).hom ≫ cyclesMap { τ₁ := ↑(oneCochainsLequiv A), τ₂ := ↑(twoCochainsLequiv A), τ₃ := ↑(threeCochainsLequiv A), comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫ (shortComplexH2 A).iCycles = iCocycles A 2 ≫ ↑(twoCochainsLequiv A)"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Fintype α", "inst✝² : DecidableEq α", "P : Finpartition univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "inst✝¹ : DecidableRel G.Adj", "ε : ℝ", "U : Finset α", "hU✝ : U ∈ P.parts", "V : Finset α", "𝒜 : Finset (Finset α)", "s : Finset α", "inst✝ : Nonempty α", "hPα : P.parts.card * 16 ^ P.parts.card ≤ Fintype.card α", "hPε : 100 ≤ 4 ^ P.parts.card * ε ^ 5", "hε₁ : ε ≤ 1", "hU : U ∈ P.parts", "hV : V ∈ P.parts", "A B : Finset (Finset α)", "hA : A ⊆ (chunk hP G ε hU).parts", "hB : B ⊆ (chunk hP G ε hV).parts", "this : (1 + ε ^ 5 / 49) * ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) ≤ ↑(G.edgeDensity (A.biUnion id) (B.biUnion id)) + ε ^ 5 / 49", "x y : Finset α", "hx : x ∈ A", "hy : y ∈ B"], "goal": "B.Nonempty"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "E : Type u_3", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f g : α → E", "hf : Integrable f μ", "hg : Integrable g μ", "i : Set α", "hi : MeasurableSet i"], "goal": "∫ (x : α) in i, (f + g) x ∂μ = ∫ (x : α) in i, f x ∂μ + ∫ (x : α) in i, g x ∂μ"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : LocalRing R", "inst✝² : CommSemiring S", "inst✝¹ : Nontrivial S", "f : R →+* S", "inst✝ : IsLocalRingHom f", "hf : Function.Surjective ⇑f", "a b : R", "hab : IsUnit (f (a + b))"], "goal": "IsUnit (f a) ∨ IsUnit (f b)"}}
+{"state": {"context": ["α : Type u_1", "CommMonoid α", "u : (Associates α)ˣ"], "goal": "u = 1"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "x : R", "hu : IsUnit x", "y z : R"], "goal": "IsCoprime y (z * x) ↔ IsCoprime y z"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "n : ℕ", "hn : 1 < n"], "goal": "(Polynomial.cyclotomic n R).coeff 0 = 1"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "β : Type u_3", "TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s : Set α"], "goal": "omegaLimit f ϕ s = ⋂ u ∈ f, closure (Set.image2 ϕ u s)"}}
+{"state": {"context": ["J : Type u₁", "inst✝² : Category.{v₁, u₁} J", "K : Type u₂", "inst✝¹ : Category.{v₂, u₂} K", "C : Type u₃", "inst✝ : Category.{v₃, u₃} C", "F : J ⥤ C", "t : Cone F", "X : C", "h : yoneda.obj X ⋙ uliftFunctor.{u₁, v₃} ≅ F.cones", "s : Cone F"], "goal": "(limitCone h).extend (homOfCone h s) = coneOfHom h (homOfCone h s)"}}
+{"state": {"context": ["ι : Sort u_1", "M : Type u_2", "A : Type u_3", "B : Type u_4", "inst✝ : Mul M", "S : ι → Subsemigroup M", "C : M → Prop", "x₁ : M", "hx₁ : x₁ ∈ closure (⋃ i, ↑(S i))", "mem : ∀ (i : ι), ∀ x₂ ∈ S i, C x₂", "mul : ∀ (x y : M), C x → C y → C (x * y)"], "goal": "C x₁"}}
+{"state": {"context": ["R : Type u", "A : Type v", "inst✝⁴ : CommRing R", "inst✝³ : CommRing A", "inst✝² : IsDomain A", "inst✝¹ : Algebra R A", "inst✝ : NoZeroSMulDivisors R A", "g : ↑{x | IsAlgebraic R x} → R[X]", "hg₁ : ∀ (x : ↑{x | IsAlgebraic R x}), g x ≠ 0", "hg₂ : ∀ (x : ↑{x | IsAlgebraic R x}), (aeval ↑x) (g x) = 0", "x : ↑{x | IsAlgebraic R x}"], "goal": "↑x ∈ (g x).rootSet A"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_5", "NormedAddCommGroup G", "NormedSpace ℝ G", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α"], "goal": "∫ (x : α), 0 ∂μ = 0"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Preadditive C", "X : SimplicialObject C", "Y : C", "q a m : ℕ", "φ : Y ⟶ X _[m + 1 + 1]", "v : HigherFacesVanish q φ", "j : Fin (m + 1 + 1)", "hj₁ : m + 1 + 1 ≤ ↑j + (q + 1)", "hqn : ¬m + 1 < q", "ha : q + a = m + 1", "hj₂ : ¬a = ↑j", "haj : a < ↑j"], "goal": "φ ≫ X.δ j.succ = φ ≫ X.δ ⟨a + 1, ⋯⟩ ≫ X.σ ⟨a, ⋯⟩ ≫ X.δ j.succ"}}
+{"state": {"context": ["n : Nat"], "goal": "0 < n → n - 1 + 1 = n"}}
+{"state": {"context": ["V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Limits.HasZeroMorphisms V", "C : CochainComplex V ℕ", "X : V", "f : X ⟶ C.X 0", "w : f ≫ C.d 0 1 = 0", "i j : ℕ"], "goal": "(C.augment f w).d (i + 1) (j + 1) = C.d i j"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "A X : C", "CategoryTheory.Closed A"], "goal": "CategoryTheory.MonoidalClosed.curry\n    (CategoryTheory.CategoryStruct.id\n      (CategoryTheory.MonoidalCategory.tensorObj A ((CategoryTheory.Functor.id C).obj X))) =\n  (CategoryTheory.ihom.coev A).app X"}}
+{"state": {"context": ["α : Type u_1", "Nontrivial α", "x : α"], "goal": "∃ y, y ≠ x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ l : List α", "a : α", "n : ℕ", "hn : n ≠ 0", "h : n < (a :: l).length"], "goal": "n - 1 < l.length"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "p₁ p₂ : P"], "goal": "p₁ ∈ affineSpan k {p₂} ↔ p₁ = p₂"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "SuccOrder α", "a b : α", "hb : Order.IsSuccLimit b", "ha : a < b"], "goal": "Order.succ a < b"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "γ : Type u_3", "LinearOrder γ", "f : α → γ"], "goal": "LowerSemicontinuous f ↔ ∀ (y : γ), IsClosed (f ⁻¹' Set.Iic y)"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "f : Equiv.Perm α"], "goal": "f.support.card ≠ 1"}}
+{"state": {"context": ["p : ℝ", "hp : 1 ≤ p"], "goal": "ConvexOn ℝ (Ici 0) fun x => x ^ p"}}
+{"state": {"context": ["i : Nat"], "goal": "List.iota (i + 1) = (i + 1) :: List.iota i"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x => f.left) fun x => f.hom", "X : Augmented C", "F : Arrow C", "G : Augmented.toArrow.obj X ⟶ F", "x : SimplexCategoryᵒᵖ", "i : Fin ((Opposite.unop x).len + 1)"], "goal": "(X.left.map ((SimplexCategory.mk 0).const (Opposite.unop x) i).op ≫ G.left) ≫ F.hom = X.hom.app x ≫ G.right"}}
+{"state": {"context": ["Γ : Type u_1", "R : Type u_2", "inst✝² : PartialOrder Γ", "inst✝¹ : Zero R", "a b : Γ", "r : R", "inst✝ : Zero Γ", "x : HahnSeries Γ R", "hx : x ≠ 0"], "goal": "x.coeff (⋯.min ⋯) ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι✝ : Sort u_4", "ι'✝ : Sort u_5", "l��� l' : Filter α", "p✝ : ι✝ → Prop", "s✝ : ι✝ → Set α", "t✝ : Set α", "i : ι✝", "p' : ι'✝ → Prop", "s' : ι'✝ → Set α", "i' : ι'✝", "ι : Type u_6", "ι' : ι → Type u_7", "dom : Set ι", "hdom : dom.Nonempty", "l : ι → Filter α", "s : (i : ι) → ι' i → Set α", "p : (i : ι) → ι' i → Prop", "hl : ∀ i ∈ dom, (l i).HasBasis (p i) (s i)", "h : DirectedOn (l ⁻¹'o GE.ge) dom", "t : Set α"], "goal": "t ∈ ⨅ i ∈ dom, l i ↔ ∃ i, (i.fst ∈ dom ∧ p i.fst i.snd) ∧ s i.fst i.snd ⊆ t"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{u₂, u₁} C", "inst✝³ : PreGaloisCategory C", "F : C ⥤ FintypeCat", "inst✝² : FiberFunctor F", "X A : C", "inst✝¹ : Nonempty ↑(F.obj X)", "inst✝ : IsConnected A", "f : X ⟶ A"], "goal": "Function.Surjective (F.map f)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "f : R[X]", "g : S[X]", "h : f.support ⊆ g.support"], "goal": "f.degree ≤ g.degree"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "SemilatticeSup α", "OrderBot α", "s : Finset β", "t : Finset γ", "f : β → γ → α"], "goal": "(s.sup fun b => t.sup (f b)) = t.sup fun c => s.sup fun b => f b c"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹⁰ : CommRing R", "inst✝⁹ : LieRing L", "inst✝⁸ : LieAlgebra R L", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "inst✝⁵ : LieRingModule L M", "inst✝⁴ : LieModule R L M", "k✝ : ℕ", "N : LieSubmodule R L M", "M₂ : Type w₁", "inst✝³ : AddCommGroup M₂", "inst✝² : Module R M₂", "inst✝¹ : LieRingModule L M₂", "inst✝ : LieModule R L M₂", "k : ℕ", "hM : lowerCentralSeries R L M k = ⊥", "x : L", "m : M"], "goal": "((toEnd R L M) x ^ k) m = 0 m"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E : Type u_3", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "Dist E", "l : Filter ι", "f f' : ι → α → E", "g g' : α → E", "h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i", "h_right : g =ᵐ[μ] g'", "h_tendsto : MeasureTheory.TendstoInMeasure μ f l g"], "goal": "MeasureTheory.TendstoInMeasure μ f' l g'"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a b : α"], "goal": "Set.Iic b ⊆ Set.Iic a ∪ Set.Ioc a b"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : NormalSpace X", "u : ι → Set X", "s : Set X", "v : PartialRefinement u s", "hs : IsClosed s", "i : ι", "hi : i ∉ v.carrier", "I : s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ v.toFun i", "C : IsClosed (s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ)"], "goal": "∃ v', v < v'"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝⁸ : CommSemiring R", "inst✝⁷ : CommSemiring S", "inst✝⁶ : Semiring A", "inst✝⁵ : Semiring B", "inst✝⁴ : Algebra R S", "inst✝³ : Algebra R A", "inst✝² : Algebra S A", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R S A", "s t : Set A", "x : A", "hx : x ∈ adjoin R s", "p : (x : A) → x ∈ adjoin R s → Prop", "mem : ∀ (x : A) (h : x ∈ s), p x ⋯", "algebraMap : ∀ (r : R), p ((_root_.algebraMap R A) r) ⋯", "add : ∀ (x : A) (hx : x ∈ adjoin R s) (y : A) (hy : y ∈ adjoin R s), p x hx → p y hy → p (x + y) ⋯", "mul : ∀ (x : A) (hx : x ∈ adjoin R s) (y : A) (hy : y ∈ adjoin R s), p x hx → p y hy → p (x * y) ⋯"], "goal": "p x hx"}}
+{"state": {"context": ["C₁ : Type u_1", "C₂ : Type u_2", "D₁ : Type u_3", "D₂ : Type u_4", "CategoryTheory.Category.{u_8, u_1} C₁", "CategoryTheory.Category.{u_7, u_2} C₂", "CategoryTheory.Category.{u_6, u_3} D₁", "CategoryTheory.Category.{u_5, u_4} D₂", "G : CategoryTheory.Functor C₁ C₂", "F : CategoryTheory.Functor C₂ C₁", "adj : G ⊣ F", "L₁ : CategoryTheory.Functor C₁ D₁", "L₂ : CategoryTheory.Functor C₂ D₂", "W₂ : CategoryTheory.MorphismProperty C₂", "L₂.IsLocalization W₂", "G' : CategoryTheory.Functor D₁ D₂", "F' : CategoryTheory.Functor D₂ D₁", "CategoryTheory.CatCommSq G L₁ L₂ G'", "CategoryTheory.CatCommSq F L₂ L₁ F'", "X₂ : C₂"], "goal": "(CategoryTheory.Adjunction.Localization.η adj L₁ L₂ W₂ G' F').app (L₂.obj X₂) =\n  CategoryTheory.CategoryStruct.comp (G'.map ((CategoryTheory.CatCommSq.iso F L₂ L₁ F').inv.app X₂))\n    (CategoryTheory.CategoryStruct.comp ((CategoryTheory.CatCommSq.iso G L₁ L₂ G').inv.app (F.obj X₂))\n      (L₂.map (adj.counit.app X₂)))"}}
+{"state": {"context": ["α : Type u_1"], "goal": "(match (motive := Bool → Num → Num) true with | true => Num.bit1 | false => Num.bit0) 0 = 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁵ : MetricSpace α", "inst✝⁴ : SecondCountableTopology α", "inst✝³ : MeasurableSpace α", "inst✝² : BorelSpace α", "μ : Measure α", "inst✝¹ : IsLocallyFiniteMeasure μ", "inst✝ : IsUnifLocDoublingMeasure μ", "p : ℕ → Prop", "s : ℕ → Set α", "M : ℝ", "hM : 0 < M", "r : ℕ → ℝ", "h₁ : blimsup (fun i => cthickening (r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p", "h₂ : blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => cthickening (r i) (s i)) atTop p", "hr : Tendsto (fun i => M * r i) atTop (𝓝 0)", "hr' : ∀ᶠ (i : ℕ) in atTop, p i → 0 < M * r i"], "goal": "blimsup (fun i => thickening (M * r i) (s i)) atTop p =ᶠ[ae μ] blimsup (fun i => thickening (r i) (s i)) atTop p"}}
+{"state": {"context": ["α : Type u_1", "s : Set α"], "goal": "SetSemiring.down (Set.up s) = s"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a : α", "l : List α"], "goal": "l ⊆ List.insert a l"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "x : E", "y : NormedSpace.Dual 𝕜 F"], "goal": "Continuous fun a => y (a x)"}}
+{"state": {"context": ["α : Type u_1", "Encodable α", "n : ℕ", "a : α"], "goal": "Encodable.decode₂ α n = some a ↔ Encodable.encode a = n"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "K L : CochainComplex C ℤ", "n : ℤ", "γ : CochainComplex.HomComplex.Cochain K L n", "a n' : ℤ", "hn' : n + a = n'"], "goal": "(γ.leftShift a n' hn').rightShift a n hn' = (a * n' + a * (a - 1) / 2).negOnePow • γ.shift a"}}
+{"state": {"context": ["β : Type u_2", "α : Type u_4", "Finite (α ⊕ β)"], "goal": "Finite β"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : Semiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "s : Set M", "hs : s.Finite"], "goal": "span R ↑hs.toFinset = span R s"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝¹ : NormedAddCommGroup E", "f : ℝ → E", "a b : ℝ", "μ : Measure ℝ", "inst✝ : NoAtoms μ", "hab : a ≤ b"], "goal": "IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ"}}
+{"state": {"context": ["ι✝ : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝⁴ : EMetricSpace X", "inst✝³ : EMetricSpace Y", "inst✝² : MeasurableSpace X", "inst✝¹ : BorelSpace X", "β : Type u_4", "ι : β → Type u_5", "inst✝ : ∀ (n : β), Countable (ι n)", "s : Set X", "l : Filter β", "r : β → ℝ≥0∞", "hr : Tendsto r l (𝓝 0)", "t : (n : β) → ι n → Set X", "ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), diam (t n i) ≤ r n", "hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i", "m : ℝ≥0∞ → ℝ≥0∞", "this : (n : β) → Encodable (ι n)", "ε : ℝ≥0∞", "hε : 0 < ε", "c : ℝ≥0∞", "hc : liminf (fun n => ∑' (i : ι n), m (diam (t n i))) l < c", "n : β", "hn : ∑' (i : ι n), m (diam (t n i)) < c", "hrn : r n < ε", "htn : ∀ (i : ι n), diam (t n i) ≤ r n", "hstn : s ⊆ ⋃ i, t n i", "u : ℕ → Set X := fun j => ⋃ b ∈ decode₂ (ι n) j, t n b"], "goal": "⨅ t, ⨅ (_ : s ⊆ iUnion t), ⨅ (_ : ∀ (n : ℕ), diam (t n) ≤ ε), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (diam (t n)) ≤ c"}}
+{"state": {"context": ["u₁ u₂ : ℤˣ"], "goal": "u₁ / u₂ = u₁ * u₂"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝ : Fintype ι", "s t : Set (ι → ℝ)", "a₁ a₂ b₁ b₂ x y : ι → ℝ", "δ : ℝ"], "goal": "MonotoneOn (dist x) (Ici x)"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : Abelian C", "P : C", "hP : PreservesFiniteColimits (preadditiveCoyonedaObj (op P))"], "goal": "(preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Preadditive C", "U : CategoryTheory.Comonad C", "U.Additive", "F G : U.Coalgebra", "α β : F ⟶ G"], "goal": "(α + β).f = α.f + β.f"}}
+{"state": {"context": ["a b : ℝ", "n✝ m n : ℕ", "hc : Continuous fun u => u ^ m * (1 - u ^ 2) ^ n", "x✝ : ℝ"], "goal": "cos x✝ * cos x✝ = 1 - sin x✝ * sin x✝"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "m✝ m : Multiset α", "x : m.ToType"], "goal": "m.coeEmbedding x ∈ m.toEnumFinset"}}
+{"state": {"context": ["m n : ℤ", "hne : m ≠ n"], "goal": "1 ≤ |↑m - ↑n|"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : CommGroup G", "a✝ b✝ c✝ d a b c : G"], "goal": "a * b / (a / c) = b * c"}}
+{"state": {"context": ["n m : ℕ", "hm : m ≠ 0"], "goal": "n ∣ m → n ≤ m"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedField α", "a b c d : α", "n : ℤ"], "goal": "a / b ≤ 0 ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b"}}
+{"state": {"context": ["α : Type u", "a : α"], "goal": "[a].reverse = [a]"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s t : Set α", "inst✝ : Preorder α"], "goal": "s.chainHeight = ⨆ i ∈ s, (s ∩ Ici i).chainHeight"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝³ : Category.{v₂, u₂} D", "J : Type v₁", "inst✝² : SmallCategory J", "inst✝¹ : FinCategory J", "K✝ : J ⥤ C", "F : C ⥤ D", "inst✝ : RepresentablyFlat F", "c✝ : Cone K✝", "hc✝ : IsLimit c✝", "s✝ : Cone (K✝ ⋙ F)", "K : J ⥤ C", "c : Cone K", "hc : IsLimit c", "s : Cone (K ⋙ F)", "f₁ f₂ : s.pt ⟶ F.obj c.pt", "h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j", "h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j", "α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }", "α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X => eqToHom ⋯, naturality := ⋯ }", "c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) := (Cones.postcompose (whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)", "c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) := (Cones.postcompose (whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₂)", "c₀ : Cone (biconeMk J c₁ c₂) := IsCofiltered.cone (biconeMk J c₁ c₂)", "g₁ : c₀.pt ⟶ c₁.pt := c₀.π.app Bicone.left", "g₂ : c₀.pt ⟶ c₂.pt := c₀.π.app Bicone.right", "this✝¹ : ∀ (j : J), g₁.right ≫ c.π.app j = g₂.right ≫ c.π.app j", "this✝ : c.extend g₁.right = c.extend g₂.right", "this : g₁.right = g₂.right"], "goal": "f₁ = f₂"}}
+{"state": {"context": ["E : Type u_4", "NormedAddCommGroup E", "NormedSpace ℝ E", "FiniteDimensional ℝ E", "s : Set E", "h : (interior s).Nonempty"], "goal": "dimH s = ↑(FiniteDimensional.finrank ℝ E)"}}
+{"state": {"context": ["m n✝ n a p : ℕ", "inst✝ : Fact (Nat.Prime p)", "k : ℕ", "h : ↑k ≠ 0"], "goal": "k.Coprime p"}}
+{"state": {"context": ["G : Type u_3", "Group G", "a b : G"], "goal": "a * b * a⁻¹ = 1 ↔ b = 1"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "Disjoint s ∅"}}
+{"state": {"context": ["α : Type u", "l₁ l₂ : List α", "a : α"], "goal": "List.takeD l₁.length (l₁ ++ l₂) a = l₁"}}
+{"state": {"context": ["α : Type u_2", "PartialOrder α", "a b : α", "H : ∀ (c : α), a ≤ c ↔ b ≤ c"], "goal": "a = b"}}
+{"state": {"context": ["a b : Prop", "h : a = True"], "goal": "(a ↔ b) = b"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "s : Multiset α"], "goal": "Multiset.card s = Multiset.countP p s + Multiset.countP (fun x => ¬p x) s"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {p q : List α}, p ++ q = [] ↔ p = [] ∧ q = []"}}
+{"state": {"context": [], "goal": "RingHom.RespectsIso @RingHom.FinitePresentation"}}
+{"state": {"context": ["p : ℕ", "w✝ : 1 < p", "p' : ℕ := p - 2", "z : p = p' + 2", "w : 1 < p' + 2"], "goal": "¬Nat.Prime (mersenne p) → ¬LucasLehmerTest p"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E", "𝔸 : Type u_6", "NormedRing 𝔸", "NormedAlgebra 𝕜 𝔸", "a : E → 𝔸", "ha : DifferentiableAt 𝕜 a x", "b : 𝔸"], "goal": "DifferentiableAt 𝕜 (fun y => b * a y) x"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasInitial C", "h : ∀ (A : C) (f : A ⟶ ⊥_ C), IsIso f", "I A : C", "f : A ⟶ I", "hI : IsInitial I", "this : IsIso (f ≫ hI.to (⊥_ C))"], "goal": "f ≫ hI.to (⊥_ C) ≫ inv (f ≫ hI.to (⊥_ C)) = 𝟙 A"}}
+{"state": {"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "r : ℝ", "z : K"], "goal": "im (r • z) = r * im z"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_4", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup G", "ι : Type u_5", "Nonempty ι", "LinearOrder ι", "f : ι → α → G", "f_lim : α → G", "h_lim : ∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun n => f n x) Filter.atTop (𝓝 (f_lim x))"], "goal": "MeasureTheory.eLpNorm f_lim ⊤ μ = essSup (fun x => Filter.liminf (fun m => ↑‖f m x‖₊) Filter.atTop) μ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "C' : Type u₃", "CategoryTheory.Category.{v₃, u₃} C'", "D' : Type u₄", "CategoryTheory.Category.{v₄, u₄} D'", "iC : CategoryTheory.Functor C C'", "iD : CategoryTheory.Functor D D'", "L' : CategoryTheory.Functor C' D'", "R' : CategoryTheory.Functor D' C'", "adj : L' ⊣ R'", "hiC : iC.FullyFaithful", "hiD : iD.FullyFaithful", "L : CategoryTheory.Functor C D", "R : CategoryTheory.Functor D C", "comm1 : iC.comp L' ≅ L.comp iD", "comm2 : iD.comp R' ≅ R.comp iC", "X : C"], "goal": "iC.map ((adj.restrictFullyFaithful hiC hiD comm1 comm2).unit.app X) =\n  CategoryTheory.CategoryStruct.comp (adj.unit.app (iC.obj X))\n    (CategoryTheory.CategoryStruct.comp (R'.map (comm1.hom.app X)) (comm2.hom.app (L.obj X)))"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "s : Set α"], "goal": "s.IsPWO ↔ ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ g, Monotone (f ∘ ⇑g)"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : CommMonoid M", "S : Submonoid M", "hS : S ≤ IsUnit.submonoid M", "x : ↥S"], "goal": "↑((S.leftInvEquiv hS).symm x) * ↑x = 1"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : LinearOrderedField α", "β : Type u_2", "inst✝² : Field β", "abv : β → α", "inst✝¹ : IsAbsoluteValue abv", "inst✝ : IsComplete β abv", "f✝ : CauSeq β abv", "hf✝ : ¬f✝.LimZero", "hl : f✝.lim ≠ 0", "g f : CauSeq β abv", "hf : ¬f.LimZero", "h₂ : g - f * f.inv hf * g = 1 * g - f * f.inv hf * g", "h₃ : f * f.inv hf * g = f * f.inv hf * g", "h₄ : g - f * f.inv hf * g = (1 - f * f.inv hf) * g", "h₅ : g - f * f.inv hf * g = g * (1 - f * f.inv hf)", "h₆ : g - f * f.inv hf * g = g * (1 - f.inv hf * f)"], "goal": "(g * (1 - f.inv hf * f)).LimZero"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "inst✝³ : Fintype n", "inst✝² : Fintype o", "inst✝¹ : CommRing R", "inst✝ : Nontrivial R"], "goal": "rank 0 = 0"}}
+{"state": {"context": ["G : Type u", "ι : Type u_1", "inst✝¹ : Fintype ι", "M : ι → Type u_2", "inst✝ : (i : ι) → Monoid (M i)"], "goal": "(univ.lcm fun x => exponent (M x)) ∣ exponent ((i : ι) → M i)"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "F : J ⥤ C", "self : CategoryTheory.Limits.HasColimit F"], "goal": "Nonempty (CategoryTheory.Limits.ColimitCocone F)"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : E → ℂ", "f' : E →L[ℂ] ℂ", "x : E", "s : Set E", "c : ℂ", "hf : HasFDerivWithinAt f f' s x", "h0 : c ≠ 0 ∨ f x ≠ 0"], "goal": "HasFDerivWithinAt (fun x => c ^ f x) ((c ^ f x * Complex.log c) • f') s x"}}
+{"state": {"context": ["α : Sort u", "β : Sort v"], "goal": "Function.Injective fun e => ⇑e"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "A : Type u_3", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Algebra R A", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "p : Submodule R M", "a : A", "m : M", "hm : m ∈ p"], "goal": "a • 1 ⊗ₜ[R] m ∈ baseChange A p"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "E₁ : ι → Type wE₁", "G : Type wG", "Fintype ι", "NontriviallyNormedField 𝕜", "(i : ι) → SeminormedAddCommGroup (E₁ i)", "(i : ι) → NormedSpace 𝕜 (E₁ i)", "SeminormedAddCommGroup G", "NormedSpace 𝕜 G", "f : ContinuousMultilinearMap 𝕜 E₁ G", "k : ℕ", "x : (i : ι) → E₁ i"], "goal": "‖f.iteratedFDeriv k x‖ ≤ ↑((Fintype.card ι).descFactorial k) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k)"}}
+{"state": {"context": ["α : Type u", "s : Stream'.WSeq α", "n : ℕ"], "goal": "s.tail.get? n = s.get? (n + 1)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing A", "inst✝³ : Field K", "inst✝² : IsDedekindDomain A", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "p : K[X]", "hp : p ≠ 0", "this : ∀ q ∈ Multiset.map (fun q => span {q}) (factors p), Prime q"], "goal": "factors (span {p}) = Multiset.map (fun q => span {q}) (factors p)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Div β", "a b : β"], "goal": "Function.const α a / Function.const α b = Function.const α (a / b)"}}
+{"state": {"context": ["n : ℕ", "α : TypeVec.{u_1} (n + 1)", "p : α.Arrow (TypeVec.repeat (n + 1) Prop)"], "goal": "TypeVec.dropFun (TypeVec.ofSubtype p) = TypeVec.ofSubtype (TypeVec.dropFun p)"}}
+{"state": {"context": ["m n : ℕ", "hn : n ≠ 0"], "goal": "m % n = m ↔ m < n"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "NormalSpace X", "s t : Set X", "hs : IsClosed s", "ht : IsOpen t", "hst : s ⊆ t"], "goal": "∃ u, IsOpen u ∧ s ⊆ u ∧ closure u ⊆ t"}}
+{"state": {"context": ["α : Type u_1", "DistribLattice α", "OrderBot α", "a b c : α", "hb : Disjoint a b", "hc : Disjoint a c"], "goal": "Disjoint a (b ⊔ c)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "OrderedCommSemiring R", "Nontrivial R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀", "TopologicalSpace A", "NonUnitalRing A", "StarRing A", "PartialOrder A", "StarOrderedRing A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "NonUnitalContinuousFunctionalCalculus R p", "f g : R → R", "a : A", "h : ∀ x ∈ σₙ R a, f x ≤ g x", "hf : ContinuousOn f (σₙ R a) := by cfc_cont_tac", "hg : ContinuousOn g (σₙ R a) := by cfc_cont_tac", "hf0 : f 0 = 0 := by cfc_zero_tac", "hg0 : g 0 = 0 := by cfc_zero_tac"], "goal": "cfcₙ f a ≤ cfcₙ g a"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "T1Space X", "DecidableEq X", "TopologicalSpace Y", "f : X → Y", "s : Set X", "x : X", "y : Y"], "goal": "ContinuousOn (Function.update f x y) s ↔ ContinuousOn f (s \\ {x}) ∧ (x ∈ s → Filter.Tendsto f (𝓝[s \\ {x}] x) (𝓝 y))"}}
+{"state": {"context": ["I : Type u", "f : I → Type v", "(i : I) → Neg (f i)", "DecidableEq I", "f₁ : (i : I) → f i", "i : I", "x₁ : f i"], "goal": "Function.update (-f₁) i (-x₁) = -Function.update f₁ i x₁"}}
+{"state": {"context": ["p : ℕ", "α : Type u_4", "β : Type u_5", "f : α → β", "hf : Function.Injective f"], "goal": "Function.Injective (WittVector.mapFun f)"}}
+{"state": {"context": ["k : ℕ", "K : Type u", "Field K", "CharZero K", "ζ : K", "IsCyclotomicExtension {2 ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(2 ^ (k + 1))"], "goal": "(Algebra.norm ℤ) (hζ.toInteger ^ 2 ^ k - 1) = (-2) ^ 2 ^ k"}}
+{"state": {"context": ["f : ℝ → ℝ", "a : ℝ", "h : IsLocalMax f a"], "goal": "deriv f a = 0"}}
+{"state": {"context": ["α : Type u", "a b c : α", "LE α", "h₁ : a = b", "h₂ : b ≤ c"], "goal": "a ≤ c"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "V2 : Type u_4", "P2 : Type u_5", "Ring k", "AddCommGroup V1", "Module k V1", "AffineSpace V1 P1", "AddCommGroup V2", "Module k V2", "AffineSpace V2 P2", "f : P1 →ᵃ[k] P2"], "goal": "f.comp (AffineMap.id k P1) = f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : PseudoEMetricSpace α", "s : Set α"], "goal": "diam (closure s) = diam s"}}
+{"state": {"context": ["a : Nat", "b c : Nat", "h : b ≤ c"], "goal": "(a + b = c) = (a = c - b)"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "CommSemiring R", "CommRing A", "Algebra R A", "𝒜 : ℕ → Submodule R A", "GradedAlgebra 𝒜", "t t' : Set (ProjectiveSpectrum 𝒜)"], "goal": "ProjectiveSpectrum.vanishingIdeal t ⊔ ProjectiveSpectrum.vanishingIdeal t' ≤ ProjectiveSpectrum.vanishingIdeal (t ∩ t')"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "inst✝² : Mul G", "inst✝¹ : Mul H", "A B : Finset G", "a0 b0 : G", "inst✝ : DecidableEq H", "f : G →ₙ* H", "hf : ∀ ⦃a b c d : G⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d", "h : UniqueMul (Finset.image (⇑f) A) (Finset.image (⇑f) B) (f a0) (f b0)", "a b : G", "ha : a ∈ A", "hb : b ∈ B", "ab : a * b = a0 * b0"], "goal": "f a * f b = f a0 * f b0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝ : ConditionallyCompleteLattice α", "f : Filter β", "u v : β → α", "p : β → Prop", "h : ∀ᶠ (a : β) in f, p a → u a = v a"], "goal": "blimsup u f p = blimsup v f p"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_4, u_1} C", "inst✝³ : Category.{u_5, u_2} D", "L✝ : C ⥤ D", "W✝ : MorphismProperty C", "E : Type u_3", "inst✝² : Category.{u_6, u_3} E", "inst✝¹ : L✝.IsLocalization W✝", "L : C ⥤ D", "W : MorphismProperty C", "inst✝ : L.IsLocalization W", "F₁ F₂ : D ⥤ E", "τ τ' : F₁ ⟶ F₂", "h : ∀ (X : C), τ.app (L.obj X) = τ'.app (L.obj X)"], "goal": "τ = τ'"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m m0 : MeasurableSpace α", "μ : Measure α", "s t : Set α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "p : ℝ≥0∞", "f g : α → E", "hf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn f s μ", "hg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn g s μ", "hfg_eq : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, f x ∂μ = ∫ (x : α) in s, g x ∂μ", "hf : AEFinStronglyMeasurable f μ", "hg : AEFinStronglyMeasurable g μ", "hfg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∫ (x : α) in s, (f - g) x ∂μ = 0", "hfg_int : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → IntegrableOn (f - g) s μ"], "goal": "f - g =ᶠ[ae μ] 0"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_3", "RCLike 𝕜", "E : Type u_4", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "Fintype ι", "v : Basis ι 𝕜 E", "hv : Orthonormal 𝕜 ⇑v"], "goal": "⇑(v.toOrthonormalBasis hv) = ⇑v"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "𝒜 : Finset (Finset α)", "h𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜", "h𝒜₀ : ∅ ∉ 𝒜"], "goal": "∑ s ∈ 𝒜, (↑((Fintype.card α).choose s.card))⁻¹ ≤ AhlswedeZhang.infSum 𝒜"}}
+{"state": {"context": ["α : Type u_1", "a b c d : ℝ≥0∞", "r p q : ℝ≥0"], "goal": "⋂ n, Ioi ↑n = {⊤}"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "a : ℍ[R]"], "goal": "a + star a = 2 * ↑a.re"}}
+{"state": {"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : Fintype α", "s✝ t✝ : Finset α", "a b : α", "s t : Finset α", "h : _root_.Disjoint s t"], "goal": "(s.disjUnion t h).dens = s.dens + t.dens"}}
+{"state": {"context": ["P : MorphismProperty Scheme", "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop", "inst✝² : HasRingHomProperty P Q", "X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "inst✝¹ : IsAffine X", "inst✝ : IsAffine Y"], "goal": "(∀ (i : (Scheme.openCoverOfIsIso (𝟙 X)).J), Q (Scheme.Hom.app ((Scheme.openCoverOfIsIso (𝟙 X)).map i ≫ f) ⊤)) ↔ Q (Scheme.Hom.app f ⊤)"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "E : Type w", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f✝ f₀ f₁ g : 𝕜 → F", "f' f₀' f₁' g' : F", "x : 𝕜", "s t : Set 𝕜", "L L₁ L₂ : Filter 𝕜", "f : 𝕜 → F", "x₀ : 𝕜", "C : ℝ", "hC₀ : 0 ≤ C", "hlip : ∀ᶠ (x : 𝕜) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"], "goal": "‖deriv f x₀‖ ≤ C"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r r' : α → α → Prop", "inst✝¹ : LinearOrder β", "inst✝ : PartialOrder γ", "h : WellFounded fun x x_1 => x < x_1", "f g : β → γ", "hf : StrictMono f", "hg : StrictMono g", "hfg : range f = range g", "a : β"], "goal": "∀ (x : β), (∀ y < x, f y = g y) → f x = g x"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "x : R", "hu : IsUnit x", "y z : R"], "goal": "IsCoprime (y * x) z ↔ IsCoprime y z"}}
+{"state": {"context": ["α : Type u_2", "s : Set α", "Preorder α", "SupSet α", "Inhabited ↑s"], "goal": "sSup ∅ = default"}}
+{"state": {"context": ["l m₁ : List Char", "c : Char", "m₂ r : List Char"], "goal": "{ str := { data := l ++ (m₁ ++ c :: m₂) ++ r }, startPos := { byteIdx := utf8Len l }, stopPos := { byteIdx := utf8Len l + utf8Len (m₁ ++ c :: m₂) } }.prev { byteIdx := utf8Len m₁ + c.utf8Size } = { byteIdx := utf8Len m₁ }"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "s : Multiset ι", "f : ι → Ideal R", "P : Ideal R", "hp : P.IsPrime"], "goal": "(Multiset.map f s).prod ≤ P ↔ ∃ i ∈ s, f i ≤ P"}}
+{"state": {"context": ["R : Type uR", "inst✝⁴ : Semiring R", "S : Type uS", "inst✝³ : CommSemiring S", "T : Type uT", "A : Type uA", "inst✝² : Semiring A", "inst✝¹ : Algebra S A", "r✝ : R → R → Prop", "inst✝ : Semiring T", "r : R → R → Prop", "F : RingQuot r →+* T"], "goal": "(fun f => preLift ⋯) ((fun F => ⟨F.comp (mkRingHom r), ⋯⟩) F) = F"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u", "i : Fin (n + 1)"], "goal": "⇑(Equiv.piFinSuccAbove α i) = fun f => (f i, i.removeNth f)"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "a a' : R", "s : σ →₀ ℕ", "CommSemiring R"], "goal": "MvPolynomial.C a * (MvPolynomial.monomial s) a' = (MvPolynomial.monomial s) (a * a')"}}
+{"state": {"context": ["m n : ℕ", "hn : 1 < n", "h : n ! = m !", "hnm : m < n"], "goal": "n = m"}}
+{"state": {"context": ["α β : Type u", "Infinite α", "Zero β", "Nontrivial β"], "goal": "#(α →₀ β) = max #α #β"}}
+{"state": {"context": ["n : ℕ∞", "𝕂 : Type u_1", "inst✝⁸ : RCLike 𝕂", "E' : Type u_2", "inst✝⁷ : NormedAddCommGroup E'", "inst✝⁶ : NormedSpace 𝕂 E'", "F' : Type u_3", "inst✝⁵ : NormedAddCommGroup F'", "inst✝⁴ : NormedSpace 𝕂 F'", "E : Type u_4", "F : Type u_5", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : E → F", "s : Set E", "x : E", "hf : ContDiffWithinAt ℝ 1 f s x", "hs : Convex ℝ s"], "goal": "∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t"}}
+{"state": {"context": ["α : Type u", "L₁ L₂ : List (α × Bool)", "x : α", "b : Bool"], "goal": "FreeGroup.Red.Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂)"}}
+{"state": {"context": ["n m : Nat"], "goal": "n * (m + 1) = n * m + n"}}
+{"state": {"context": ["R : Type u₁", "CommSemiring R", "p : ℕ", "hp : Fact (Nat.Prime p)", "CharP R p"], "goal": "(frobenius (Ring.Perfection R p) p).comp (Perfection.pthRoot R p) = RingHom.id (Ring.Perfection R p)"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x : V", "h : x ≠ 0", "r : ℝ", "hr : ¬r = 0"], "goal": "o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.rotation ↑(π / 2)) x - x) = ↑(Real.arctan r⁻¹)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y X' Y' : C", "f g : X ⟶ Y", "f' g' : X' ⟶ Y'", "p : X ⟶ X'", "q : Y ⟶ Y'", "wf : CategoryTheory.CategoryStruct.comp f q = CategoryTheory.CategoryStruct.comp p f'", "wg : CategoryTheory.CategoryStruct.comp g q = CategoryTheory.CategoryStruct.comp p g'"], "goal": "(CategoryTheory.Limits.parallelPairHom f g f' g' p q wf wg).app CategoryTheory.Limits.WalkingParallelPair.zero = p"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.ConcreteCategory C", "J : Type w", "CategoryTheory.Category.{r, w} J", "F : J ⥤ C", "CategoryTheory.Limits.PreservesColimit F (CategoryTheory.forget C)", "CategoryTheory.IsFiltered J", "CategoryTheory.Limits.HasColimit F", "i j : J", "x : (CategoryTheory.forget C).obj (F.obj i)", "y : (CategoryTheory.forget C).obj (F.obj j)"], "goal": "(CategoryTheory.Limits.colimit.ι F i) x = (CategoryTheory.Limits.colimit.ι F j) y ↔ ∃ k f g, (F.map f) x = (F.map g) y"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝³ : DivisionRing K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "f : End K V", "m : ℕ", "hm : finrank K V ≤ m", "k : ℕ", "h_k_le : k ≤ finrank K V", "hk : LinearMap.ker (f ^ k) = LinearMap.ker (f ^ k.succ)"], "goal": "LinearMap.ker (f ^ m) = LinearMap.ker (f ^ finrank K V)"}}
+{"state": {"context": ["α : Type u", "x : WithOne α"], "goal": "x ≠ 1 ↔ ∃ a, ↑a = x"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{u₂, u₁} C", "inst✝¹ : PreGaloisCategory C", "F : C ⥤ FintypeCat", "inst✝ : FiberFunctor F", "X : C", "h : IsInitial X → False", "he : IsEmpty ↑(F.obj X)"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "s : Finset (ι × ℝ)", "p : ι × ℝ"], "goal": "BoxIntegral.Prepartition.splitMany I (insert p s) =\n  BoxIntegral.Prepartition.splitMany I s ⊓ BoxIntegral.Prepartition.split I p.1 p.2"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "X✝ Y✝ Z : C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "inst✝¹ : HasPullbacks C", "X Y : C", "g : Y ⟶ X", "inst✝ : Mono g", "f₂ : Subobject Y"], "goal": "∀ (f₁ : Subobject Y), (map g).obj (f₁ ⊓ f₂) = (map g).obj f₁ ⊓ (map g).obj f₂"}}
+{"state": {"context": [], "goal": "List.iota 0 = List.nil"}}
+{"state": {"context": ["R : Type u_1", "AddMonoidWithOne R", "p : ℕ", "CharP R p", "p.AtLeastTwo"], "goal": "OfNat.ofNat p = 0"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y : Mon_ C", "M N : Bimod X Y", "f : M ⟶ N"], "goal": "X.X ◁ f.hom ≫ colimit.ι (parallelPair (X.mul ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)) WalkingParallelPair.one = X.X ◁ f.hom ≫ (((λ_ (X.X ⊗ N.X)).inv ≫ 𝟙_ C ◁ N.actLeft) ≫ X.one ▷ N.X) ≫ coequalizer.π (X.mul ▷ N.X) ((α_ X.X X.X N.X).hom ≫ X.X ◁ N.actLeft)"}}
+{"state": {"context": ["Ω : Type u_1", "m : MeasurableSpace Ω", "p : ℕ", "μ : MeasureTheory.Measure Ω", "hp : p ≠ 0"], "goal": "ProbabilityTheory.centralMoment 0 p μ = 0"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "NormedAddCommGroup E", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : MeasureTheory.SimpleFunc α E", "g : E → ℝ≥0∞", "hf : MeasureTheory.Integrable (↑f) μ", "hg0 : g 0 = 0", "ht : ∀ (b : E), g b ≠ ⊤"], "goal": "MeasureTheory.SimpleFunc.integral μ (MeasureTheory.SimpleFunc.map (ENNReal.toReal ∘ g) f) =\n  (∫⁻ (a : α), g (↑f a) ∂μ).toReal"}}
+{"state": {"context": ["𝕜 : Type u_1", "ι : Type u_2", "κ : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝³ : LinearOrderedField 𝕜", "r : α → β → Prop", "inst✝² : (a : α) → DecidablePred (r a)", "s✝ s₁ s₂ : Finset α", "t✝ t₁ t₂ : Finset β", "a✝ : α", "b : β", "δ : 𝕜", "inst✝¹ : DecidableEq α", "inst✝ : DecidableEq β", "s : Finset α", "t : Finset ι", "f : ι → Finset β", "a : α × β"], "goal": "a ∈ interedges r s (t.biUnion f) ↔ a ∈ t.biUnion fun b => interedges r s (f b)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "J : CategoryTheory.GrothendieckTopology C", "F F' : CategoryTheory.Sheaf J (Type w)", "f : F ⟶ F'"], "goal": "(CategoryTheory.GrothendieckTopology.imageSheafι f).val =\n  (CategoryTheory.GrothendieckTopology.Subpresheaf.sheafify J\n      (CategoryTheory.GrothendieckTopology.imagePresheaf f.val)).ι"}}
+{"state": {"context": ["J : Type u_1", "CategoryTheory.Category.{u_2, u_1} J", "K : CategoryTheory.Functor J (Type u)", "c : CategoryTheory.Limits.Cocone K", "hc : CategoryTheory.Limits.IsColimit c", "lc : CategoryTheory.Limits.Cocone (K.comp CategoryTheory.uliftFunctor.{v, u})", "f : c.pt → lc.pt"], "goal": "f = CategoryTheory.Limits.Types.descFun hc lc ↔ ∀ (j : J), f ∘ c.ι.app j = lc.ι.app j ∘ ULift.up"}}
+{"state": {"context": ["α : Type u_1", "E' : Type u_6", "F' : Type u_7", "SeminormedAddCommGroup E'", "SeminormedAddCommGroup F'", "f' : α → E'", "g' : α → F'", "l : Filter α"], "goal": "f' =O[l] fun x => (f' x, g' x)"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "UniqueFactorizationMonoid α", "DecidableEq (Associates α)", "m p : Associates α", "h₁ : m ≠ 0", "h₂ : Irreducible p", "k : ℕ"], "goal": "p ^ k ≤ m ↔ k ≤ Associates.bcount ⟨p, h₂⟩ m.factors"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "s : Set X"], "goal": "𝓝ˢ s = 𝓟 s ↔ IsOpen s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ✝ : Measure α", "inst✝ : NormedAddCommGroup β", "μ : Measure α", "p : ℝ≥0∞", "f✝ : ℕ → α → β", "g✝ : α → β", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "f : ℕ → α → β", "g : α → β", "hf : ∀ (n : ℕ), StronglyMeasurable (f n)", "hg : StronglyMeasurable g", "hg' : Memℒp g p μ", "hui : UnifIntegrable f p μ", "hut : UnifTight f p μ", "hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))", "ε : ℝ≥0∞", "hε : ε > 0", "hfinε : ε ≠ ⊤", "hμ : ¬μ = 0"], "goal": "∃ N, ∀ n ≥ N, eLpNorm (f n - g) p μ ≤ ε"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Finset α", "t : Finset β", "a : α"], "goal": "Sum.inl a ∈ s.disjSum t ↔ a ∈ s"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "F : CategoryTheory.Pseudofunctor B C", "a b c d : B", "f : a ⟶ b", "g : b ⟶ c", "h : c ⟶ d"], "goal": "CategoryTheory.CategoryStruct.comp (F.mapComp f (CategoryTheory.CategoryStruct.comp g h)).hom\n    (CategoryTheory.Bicategory.whiskerLeft (F.map f) (F.mapComp g h).hom) =\n  CategoryTheory.CategoryStruct.comp (F.map₂ (CategoryTheory.Bicategory.associator f g h).inv)\n    (CategoryTheory.CategoryStruct.comp (F.mapComp (CategoryTheory.CategoryStruct.comp f g) h).hom\n      (CategoryTheory.CategoryStruct.comp (CategoryTheory.Bicategory.whiskerRight (F.mapComp f g).hom (F.map h))\n        (CategoryTheory.Bicategory.associator (F.map f) (F.map g) (F.map h)).hom))"}}
+{"state": {"context": ["α : Type u_1", "β✝ : Type u_2", "inst✝¹ : LinearOrder α", "x y : α", "β : Type u_3", "inst✝ : LinearOrder β", "x' y' : β", "h : cmp x y = cmp x' y'"], "goal": "cmp y x = cmp y' x'"}}
+{"state": {"context": ["x : ℂ", "n : ℕ", "hx : Complex.abs x / ↑n.succ ≤ 1 / 2"], "goal": "Complex.abs (Complex.exp x - ∑ m ∈ Finset.range n, x ^ m / ↑m.factorial) ≤ Complex.abs x ^ n / ↑n.factorial * 2"}}
+{"state": {"context": ["α : Type u_2", "PartialOrder α", "LocallyFiniteOrder α", "a b : α", "DecidableEq α", "h : a ≤ b"], "goal": "Finset.Icc a b \\ Finset.Ioc a b = {a}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "NontriviallyNormedField 𝕜", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f g : E → F", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "n : ℕ", "hf : HasFiniteFPowerSeriesAt f p x n", "hg : f =ᶠ[𝓝 x] g"], "goal": "HasFiniteFPowerSeriesAt g p x n"}}
+{"state": {"context": ["J : Type v₁", "CategoryTheory.SmallCategory J", "C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "F : CategoryTheory.Functor J C", "c₁ c₂ : CategoryTheory.Limits.Cone F"], "goal": "∀ {X Y : CategoryTheory.Bicone J} (f : X ⟶ Y),\n  (CategoryTheory.biconeMk J c₁ c₂).map f =\n    CategoryTheory.BiconeHom.casesOn (motive := fun a a_1 x =>\n      X = a →\n        Y = a_1 →\n          HEq f x →\n            ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n              (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n      f\n      (fun h =>\n        Eq.ndrec (motive := fun {X} =>\n          (f : X ⟶ Y) →\n            Y = CategoryTheory.Bicone.left →\n              HEq f CategoryTheory.BiconeHom.left_id →\n                ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n                  (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n          (fun f h =>\n            Eq.ndrec (motive := fun {Y} =>\n              (f : CategoryTheory.Bicone.left ⟶ Y) →\n                HEq f CategoryTheory.BiconeHom.left_id →\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.left ⟶\n                    (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n              (fun f h =>\n                CategoryTheory.CategoryStruct.id\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.left))\n              ⋯ f)\n          ⋯ f)\n      (fun h =>\n        Eq.ndrec (motive := fun {X} =>\n          (f : X ⟶ Y) →\n            Y = CategoryTheory.Bicone.right →\n              HEq f CategoryTheory.BiconeHom.right_id →\n                ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n                  (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n          (fun f h =>\n            Eq.ndrec (motive := fun {Y} =>\n              (f : CategoryTheory.Bicone.right ⟶ Y) →\n                HEq f CategoryTheory.BiconeHom.right_id →\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.right ⟶\n                    (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n              (fun f h =>\n                CategoryTheory.CategoryStruct.id\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.right))\n              ⋯ f)\n          ⋯ f)\n      (fun j h =>\n        Eq.ndrec (motive := fun {X} =>\n          (f : X ⟶ Y) →\n            Y = CategoryTheory.Bicone.diagram j →\n              HEq f (CategoryTheory.BiconeHom.left j) →\n                ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n                  (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n          (fun f h =>\n            Eq.ndrec (motive := fun {Y} =>\n              (f : CategoryTheory.Bicone.left ⟶ Y) →\n                HEq f (CategoryTheory.BiconeHom.left j) →\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.left ⟶\n                    (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n              (fun f h => c₁.π.app j) ⋯ f)\n          ⋯ f)\n      (fun j h =>\n        Eq.ndrec (motive := fun {X} =>\n          (f : X ⟶ Y) →\n            Y = CategoryTheory.Bicone.diagram j →\n              HEq f (CategoryTheory.BiconeHom.right j) →\n                ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n                  (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n          (fun f h =>\n            Eq.ndrec (motive := fun {Y} =>\n              (f : CategoryTheory.Bicone.right ⟶ Y) →\n                HEq f (CategoryTheory.BiconeHom.right j) →\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) CategoryTheory.Bicone.right ⟶\n                    (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n              (fun f h => c₂.π.app j) ⋯ f)\n          ⋯ f)\n      (fun {j k} f_1 h =>\n        Eq.ndrec (motive := fun {X} =>\n          (f : X ⟶ Y) →\n            Y = CategoryTheory.Bicone.diagram k →\n              HEq f (CategoryTheory.BiconeHom.diagram f_1) →\n                ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) X ⟶\n                  (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n          (fun f h =>\n            Eq.ndrec (motive := fun {Y} =>\n              (f : CategoryTheory.Bicone.diagram j ⟶ Y) →\n                HEq f (CategoryTheory.BiconeHom.diagram f_1) →\n                  ((fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j)\n                      (CategoryTheory.Bicone.diagram j) ⟶\n                    (fun X => CategoryTheory.Bicone.casesOn X c₁.pt c₂.pt fun j => F.obj j) Y))\n              (fun f h => F.map f_1) ⋯ f)\n          ⋯ f)\n      ⋯ ⋯ ⋯"}}
+{"state": {"context": ["R : Type u", "L : Type v", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "L₂ : Type w", "inst✝¹ : LieRing L₂", "inst✝ : LieAlgebra R L₂", "f : L →ₗ⁅R⁆ L₂", "K K' : LieSubalgebra R L", "K₂ : LieSubalgebra R L₂", "s t : Set L", "h : s ⊆ t"], "goal": "s ⊆ ↑(lieSpan R L t)"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_2", "F' : Type u_3", "𝕜 : Type u_4", "p : ℝ≥0∞", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace 𝕜 F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : CompleteSpace F'", "m m0 : MeasurableSpace α", "μ : Measure α", "f✝ g : α → F'", "s : Set α", "ι : Type u_5", "f : ι → α → F'", "hf : ∀ i ∈ ∅, Integrable (f i) μ"], "goal": "μ[∑ i ∈ ∅, f i|m] =ᶠ[ae μ] ∑ i ∈ ∅, μ[f i|m]"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "Fact (FiniteDimensional.finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E", "h : ⟪x, y⟫_ℝ = 0"], "goal": "|(o.areaForm x) y| = ‖x‖ * ‖y‖"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "BEq α", "a : α", "l : Batteries.AssocList α β"], "goal": "Batteries.AssocList.contains a l = l.toList.any fun x => x.fst == a"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "x : 𝕜"], "goal": "HasDerivAt (fun x => -x) (-1) x"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_4", "R : Type u_13", "Norm F", "SeminormedRing R", "c : ℝ", "g : α → F", "l : Filter α", "f : α → R", "h : Asymptotics.IsBigOWith c l f g", "c' : R"], "goal": "Asymptotics.IsBigOWith (‖c'‖ * c) l (fun x => c' * f x) g"}}
+{"state": {"context": ["n m : ℕ", "h : n < m"], "goal": "[n] ++ Ico n.succ m = n :: Ico (n + 1) m"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_6", "m : ι → MeasurableSpace α"], "goal": "MeasurableSpace.generateFrom (⋃ n, {t | MeasurableSet t}) = ⨆ n, m n"}}
+{"state": {"context": ["α : Type u_1", "M : IndepMatroid α"], "goal": "∀ (x : Set α), M.matroid.Base x = (x ∈ maximals (fun x x_1 => x ⊆ x_1) {I | M.Indep I})"}}
+{"state": {"context": ["X : Scheme", "inst✝ : IsIntegral X", "U V : X.Opens", "i : U ⟶ V", "H : Nonempty ↑↑(↑U).toPresheafedSpace", "x : ↑Γ(X, V)", "hx : X.basicOpen ((X.presheaf.map i.op) x) = ⊥"], "goal": "X.basicOpen x = ⊥"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "inst✝ : FiniteDimensional ℝ V", "s : Set V", "h : affineSpan ℝ s = ⊤"], "goal": "(interior ((convexHull ℝ) s)).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "Zero M", "p : α → Prop", "DecidablePred p", "f : α →₀ M", "a : α"], "goal": "(Finsupp.filter p f) a = if p a then f a else 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : MeasurableSpace α", "μ ν : Measure α", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "inst✝ : TopologicalSpace δ", "f : α →ₘ[μ] β", "g : { f // AEStronglyMeasurable f μ } := Quotient.out' f", "this : g = ⟨↑g, ⋯⟩"], "goal": "↑f =ᶠ[ae μ] ↑g"}}
+{"state": {"context": ["α : Type u_2", "OrderedCommMonoid α", "s : Multiset α", "n : α", "h : ∀ x ∈ s, x ≤ n"], "goal": "s.prod ≤ n ^ Multiset.card s"}}
+{"state": {"context": ["K : Type u_1", "Field K", "x : RatFunc K"], "goal": "Valued.v x = (Polynomial.idealX K).valuation x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "ι' : Type u_4", "inst✝ : CompleteLattice α", "s : Set α", "hs : SetIndependent s", "t : ι → α", "ht✝ ht : Independent t", "i : ι", "a✝ : i ∈ {i | t i ≠ ⊥}", "j : ι", "hj : t j ≠ ⊥", "h : t i = t j", "contra : i ≠ j"], "goal": "False"}}
+{"state": {"context": ["a c b d : Nat", "hac : a < c", "hbd : b ≤ d", "hd : 0 < d"], "goal": "a * b < c * d"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a b c d : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "hab : IntervalIntegrable f μ a b", "hbc : IntervalIntegrable f μ b c", "hac : IntervalIntegrable f μ a c"], "goal": "MeasurableSet (Ioc a c)"}}
+{"state": {"context": ["R : Type u_2", "L : Type u_3", "M : Type u_4", "CommRing R", "LieRing L", "LieAlgebra R L", "LieAlgebra.IsNilpotent R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "LieModule.IsNilpotent R L M", "x : L"], "goal": "LieModule.posFittingCompOf R M x = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝² : MeasurableSpace α", "inst✝¹ : TopologicalSpace α", "μ✝ : Measure α", "inst✝ : OpensMeasurableSpace α", "μ : Measure α", "s : ℕ → Set α", "h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular", "h' : ∀ (n : ℕ), IsOpen (s n)", "h'' : univ ⊆ ⋃ n, s n", "A : Set α", "hA : MeasurableSet A", "r : ℝ≥0∞", "hr : r > μ A", "HA : μ A < ⊤"], "goal": "∃ U ⊇ A, IsOpen U ∧ μ U < r"}}
+{"state": {"context": ["k : ℕ"], "goal": "derivative (bernoulli k) = ↑k * bernoulli (k - 1)"}}
+{"state": {"context": ["J : Type u'", "inst✝¹ : Category.{v', u'} J", "C : Type u", "inst✝ : Category.{v, u} C", "F : J ⥤ PresheafedSpace C", "j₁ j₂ j₃ : J", "f : j₁ ⟶ j₂", "g : j₂ ⟶ j₃", "U : (Opens ↑↑(F.obj j₃))ᵒᵖ"], "goal": "(F.map (f ≫ g)).c.app U = (F.map g).c.app U ≫ ((pushforward C (F.map g).base).map (F.map f).c).app U ≫ (pushforwardEq ⋯ (F.obj j₁).presheaf).hom.app U"}}
+{"state": {"context": ["X Y : TopCat", "f g : C(↑X, ↑Y)", "H : f.Homotopy g", "x₀ x₁ : ↑X", "p : fromTop x₀ ⟶ fromTop x₁", "x : ↑X"], "goal": "⟦H.evalAt x⟧ = eqToHom ⋯ ≫ (π.map H.uliftMap).map ((prodToProdTop (TopCat.of (ULift.{u, 0} ↑I)) X).map (⟦unitInterval.upath01⟧, 𝟙 (fromTop x))) ≫ eqToHom ⋯"}}
+{"state": {"context": ["m n : Nat"], "goal": "Int.negOfNat n * Int.negSucc m = Int.ofNat (n * m.succ)"}}
+{"state": {"context": ["x y z : ℝ", "n : ℕ", "hx : 0 = x ∨ 0 < x", "hxy : x < y", "hz : 0 < z"], "goal": "x ^ z < y ^ z"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "∀ (a b : C), CategoryTheory.Limits.HasProductsOfShape (a ⟶ b) D", "c : C"], "goal": "∀ {X Y : D} (f : X ⟶ Y) (t : C),\n  ((CategoryTheory.evaluationRightAdjoint D c).map f).app t =\n    CategoryTheory.Limits.Pi.lift fun g =>\n      CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Pi.π (fun x => X) g) f"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "t✝ : Finset α", "m n✝ : ℕ", "m_pos : m > 0", "s : Finset α", "ih : ∀ t ⊂ s, ∀ {a b : ℕ} {P : Finpartition t}, a * m + b * (m + 1) = t.card → ∃ Q, (∀ x ∈ Q.parts, x.card = m ∨ x.card = m + 1) ∧ (∀ x ∈ P.parts, (x \\ (filter (fun y => y ⊆ x) Q.parts).biUnion id).card ≤ m) ∧ (filter (fun i => i.card = m + 1) Q.parts).card = b", "a b : ℕ", "P : Finpartition s", "hs : a * m + b * (m + 1) = s.card", "hab : 0 < a ∨ 0 < b", "n : ℕ := if 0 < a then m else m + 1", "hn : n = if 0 < a then m else m + 1", "hn₀ : 0 < n", "hn₁ : n ≤ m + 1", "hn₂ : n ≤ a * m + b * (m + 1)", "hn₃ : (if 0 < a then a - 1 else a) * m + (if 0 < a then b else b - 1) * (m + 1) = s.card - n", "u : Finset α", "hu₁ : u ∈ P.parts", "hu₂ : m + 1 ≤ u.card", "t : Finset α", "htu : t ⊆ u", "htn : t.card = n", "ht : t.Nonempty"], "goal": "∃ Q, (∀ x ∈ Q.parts, x.card = m ∨ x.card = m + 1) ∧ (∀ x ∈ P.parts, (x \\ (filter (fun y => y ⊆ x) Q.parts).biUnion id).card ≤ m) ∧ (filter (fun i => i.card = m + 1) Q.parts).card = b"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝ : CommSemiring R", "φ ψ : R[X]"], "goal": "↑φ = 1 ↔ φ = 1"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : Nontrivial E", "c : ℝ", "hc : 0 ≤ c", "x : E", "hx : ‖x‖ ≠ 0"], "goal": "‖c • ‖x‖⁻¹ • x‖ = c"}}
+{"state": {"context": ["α : Type u_1", "LT α", "a : WithTop α"], "goal": "¬⊤ < a"}}
+{"state": {"context": ["R : Type w", "inst✝² : Ring R", "J : Type u", "inst✝¹ : Category.{v, u} J", "F : J ⥤ ModuleCat R", "inst✝ : HasColimit (F ⋙ forget₂ (ModuleCat R) AddCommGrp)", "r s : R", "j : J"], "goal": "(F.obj j).smul r ≫ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j + (F.obj j).smul s ≫ colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j = colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j ≫ colimMap { app := fun j => (F.obj j).smul r, naturality := ⋯ } + colimit.ι (F ⋙ forget₂ (ModuleCat R) AddCommGrp) j ≫ colimMap { app := fun j => (F.obj j).smul s, naturality := ⋯ }"}}
+{"state": {"context": ["X : Type u", "ι : Type u_1", "TopologicalSpace X", "s : Finset ι", "f : ι → Set X", "hf : ∀ i ∈ s, IsLindelof (f i)"], "goal": "IsLindelof (⋃ i ∈ s, f i)"}}
+{"state": {"context": ["α : Type u_1", "SubtractionMonoid α", "a : α"], "goal": "-a ≠ 0 ↔ a ≠ 0"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "S : Submonoid M"], "goal": "S.leftInv.leftInv ≤ S"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "f : G →g G'", "f' : G' →g G''", "u✝ v✝ u' v' : V", "p✝ : G.Walk u✝ v✝", "u v : V", "p : G.Walk u v", "H : SimpleGraph V", "hp : ∀ e ∈ p.edges, e ∈ H.edgeSet"], "goal": "(p.transfer H hp).reverse = p.reverse.transfer H ⋯"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "n : ℤ"], "goal": "Polynomial.Chebyshev.T R (n - 1) = 2 * Polynomial.X * Polynomial.Chebyshev.T R n - Polynomial.Chebyshev.T R (n + 1)"}}
+{"state": {"context": ["I : Type u", "AddMonoid I", "C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.MonoidalCategory C", "X₁ X₂ X₃ : CategoryTheory.GradedObject I C", "X₂.HasTensor X₃", "X₁.HasTensor (CategoryTheory.GradedObject.Monoidal.tensorObj X₂ X₃)", "Y₁ Y₂ Y₃ : CategoryTheory.GradedObject I C", "Y₂.HasTensor Y₃", "Y₁.HasTensor (CategoryTheory.GradedObject.Monoidal.tensorObj Y₂ Y₃)", "f₁ : X₁ ⟶ Y₁", "f₂ : X₂ ⟶ Y₂", "f₃ : X₃ ⟶ Y₃", "i₁ i₂ i₃ j : I", "h : i₁ + i₂ + i₃ = j"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.GradedObject.Monoidal.ιTensorObj₃ X₁ X₂ X₃ i₁ i₂ i₃ j h)\n    (CategoryTheory.GradedObject.Monoidal.tensorHom f₁ (CategoryTheory.GradedObject.Monoidal.tensorHom f₂ f₃) j) =\n  CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.MonoidalCategory.tensorHom (f₁ i₁) (CategoryTheory.MonoidalCategory.tensorHom (f₂ i₂) (f₃ i₃)))\n    (CategoryTheory.GradedObject.Monoidal.ιTensorObj₃ Y₁ Y₂ Y₃ i₁ i₂ i₃ j h)"}}
+{"state": {"context": ["α : Type u_1", "A : Type u_3", "AddGroup A", "GroupWithZero α", "DistribMulAction α A", "a : α", "ha : a ≠ 0", "S T : AddSubgroup A"], "goal": "a • S ≤ a • T ↔ S ≤ T"}}
+{"state": {"context": ["𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "p : FormalMultilinearSeries 𝕜 E F", "x : E"], "goal": "HasFPowerSeriesWithinAt f p Set.univ x ↔ HasFPowerSeriesAt f p x"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : LeftCancelMonoid G", "x✝ y a : G", "m n : ℕ", "x : G", "h : ¬IsOfFinOrder x", "s : Set G := (fun n => x ^ n) '' {n | 0 < n}", "hs : s ⊆ {y | ¬IsOfFinOrder y}"], "goal": "{y | ¬IsOfFinOrder y}.Infinite"}}
+{"state": {"context": ["b : ℝ", "hb : 0 < b", "s : ℝ", "hs : -1 < s"], "goal": "AEStronglyMeasurable (fun x => (-x) ^ s * rexp (-b * x ^ 2)) (volume.restrict (Ioi 0))"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "hf : HasFDerivAt f f' x", "v : E", "α : Type u_6", "c : α → 𝕜", "l : Filter α", "hc : Tendsto (fun n => ‖c n‖) l atTop", "U : Set E", "hU : U ∈ 𝓝 v", "y : α", "hy : c y ≠ 0"], "goal": "(fun n => c n • (c n)⁻¹ • v) y ∈ U"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β"], "goal": "Surjective (image f) ↔ Surjective f"}}
+{"state": {"context": ["M : Type u_2", "AddCommGroup M", "K : Type u_3", "Field K", "Module K M", "f g : Module.End K M", "FiniteDimensional K M", "PerfectField K", "comm : Commute f g", "hf : f.IsSemisimple", "hg : g.IsSemisimple"], "goal": "(f * g).IsSemisimple"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s s₁ t✝ u : Set E", "f✝ f₁ : E → F", "g✝ : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n✝ : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "f : E → F → G", "g : E → F", "t : Set F", "n : ℕ∞", "hf : ContDiffWithinAt 𝕜 n (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀)", "hg : ContDiffWithinAt 𝕜 m g s x₀", "ht : UniqueDiffOn 𝕜 t", "hmn : m + 1 ≤ n", "hx₀ : x₀ ∈ s", "hst : s ⊆ g ⁻¹' t"], "goal": "∀ᶠ (x : E) in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : PartialOrder α", "inst✝¹ : LocallyFiniteOrder α", "a b c : α", "inst✝ : DecidableEq α", "h : a ≤ b"], "goal": "Icc a b \\ Ico a b = {b}"}}
+{"state": {"context": ["F : CategoryTheory.Limits.WalkingParallelPair ⥤ TopCat", "U : Set ↑(CategoryTheory.Limits.colimit F)"], "goal": "IsOpen U ↔ IsOpen (⇑(CategoryTheory.Limits.colimit.ι F CategoryTheory.Limits.WalkingParallelPair.one) ⁻¹' U)"}}
+{"state": {"context": ["a b : Ordinal.{u_1}"], "goal": "¬a < b → ¬a.toPGame ⧏ b.toPGame"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝¹ : TopologicalSpace α", "s✝ t u✝ v✝ : Set α", "inst✝ : TopologicalSpace β", "s : Set α", "H : IsPreconnected s", "f : α → β", "hf : ContinuousOn f s", "u v : Set β", "hu : IsOpen u", "hv : IsOpen v", "x : α", "xs : x ∈ s", "xu : f x ∈ u", "y : α", "ys : y ∈ s", "yv : f y ∈ v", "u' : Set α", "hu' : IsOpen u'", "u'_eq : f ⁻¹' u ∩ s = u' ∩ s", "v' : Set α", "hv' : IsOpen v'", "v'_eq : f ⁻¹' v ∩ s = v' ∩ s", "huv : s ⊆ u' ∪ v'", "z : α", "hz : z ∈ s ∩ (u' ∩ v')"], "goal": "(f '' s ∩ (u ∩ v)).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "κ η : ProbabilityTheory.Kernel α γ"], "goal": "κ.rnDeriv η = fun a x => ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) / ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_5", "mγ : MeasurableSpace γ", "κ : ProbabilityTheory.Kernel (α × β) γ", "a : β × α", "g : γ → ℝ≥0∞"], "goal": "∫⁻ (c : γ), g c ∂κ.swapLeft a = ∫⁻ (c : γ), g c ∂κ a.swap"}}
+{"state": {"context": ["ι : Type v", "Preorder ι", "G : ι → Type w", "DecidableEq ι", "(i : ι) → AddCommGroup (G i)", "f : (i j : ι) → i ≤ j → G i →+ G j", "IsDirected ι fun x x_1 => x ≤ x_1"], "goal": "AddCommGroup.DirectLimit.map (fun i => AddMonoidHom.id (G i)) ⋯ = AddMonoidHom.id (AddCommGroup.DirectLimit G f)"}}
+{"state": {"context": ["R₁ : Type u_3", "CommRing R₁", "K : Type u_4", "Field K", "Algebra R₁ K", "frac : IsFractionRing R₁ K", "IsDomain R₁", "IsNoetherianRing R₁", "I : FractionalIdeal R₁⁰ K"], "goal": "IsNoetherian R₁ ↥↑I"}}
+{"state": {"context": ["μ : ℝ", "v : ℝ≥0", "h : v ≠ 0"], "goal": "∫⁻ (x : ℝ), ENNReal.ofReal (ProbabilityTheory.gaussianPDFReal μ v x) = 1"}}
+{"state": {"context": ["T : Type u₁", "CategoryTheory.Category.{v₁, u₁} T", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor D T", "X : T"], "goal": "∀ (X_1 : CategoryTheory.Comma (F.comp (CategoryTheory.Functor.id T)) (CategoryTheory.Functor.fromPUnit X)),\n  ((CategoryTheory.CostructuredArrow.toOver F X).obj X_1).left = F.obj X_1.left"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s✝ t✝ u v : Finset α", "a b : α", "s t : Finset α", "x : α", "h : x ∉ t"], "goal": "insert x ↑s \\ ↑t = insert x (↑s \\ ↑t)"}}
+{"state": {"context": ["u : Lean.Level", "α : Type u_1", "inst✝¹ : DivisionRing α", "inst✝ : CharZero α", "n d : ℕ", "inv✝ : Invertible ↑d"], "goal": "IsRat (↑(-Int.ofNat n.succ) * ⅟↑d)⁻¹ (-Int.ofNat d) n.succ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "P : CategoryTheory.Functor Cᵒᵖ (Type w)"], "goal": "(CategoryTheory.sheafOfTypesBotEquiv.inverse.obj P).val = P"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : α → E"], "goal": "⨍ (x : α), f x ∂0 = 0"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u_1", "x : (i : Fin (n + 1)) → α i", "s : (i : Fin (n + 1)) → Finset (α i)"], "goal": "x ∈ piFinset s ↔ init x ∈ piFinset (init s) ∧ x (last n) ∈ s (last n)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "CommMonoidWithZero α", "NoZeroDivisors α", "Nontrivial α", "DecidableEq α", "LT α", "s : Finset ι", "f : ι → WithBot α", "h : ∀ i ∈ s, f i ≠ ⊥"], "goal": "⊥ < ∏ i ∈ s, f i"}}
+{"state": {"context": ["R : Type u_1", "inst✝³ : CommRing R", "S : Submonoid R", "P : Type u_2", "inst✝² : CommRing P", "inst✝¹ : Algebra R P", "loc : IsLocalization S P", "inst✝ : NoZeroSMulDivisors R P", "hS : 0 ∉ S", "I : FractionalIdeal S P", "hI : I.num = ⊥"], "goal": "↑I ≤ ⊥"}}
+{"state": {"context": ["a b : Rat"], "goal": "{ num := a.num.div ↑(a.num.natAbs.gcd b.den) * b.num.div ↑(b.num.natAbs.gcd a.den), den := a.den / b.num.natAbs.gcd a.den * (b.den / a.num.natAbs.gcd b.den), den_nz := ⋯, reduced := ⋯ } = normalize (a.num * b.num) (a.den * b.den) ⋯"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "s : Set α", "inst✝ : CompleteLattice α", "a : α", "S : Set ↑(Iic a)", "f : ι → ↑(Iic a)", "p : ι → Prop"], "goal": "↑(⨅ i, f i) = a ⊓ ⨅ i, ↑(f i)"}}
+{"state": {"context": ["s : Set EReal"], "goal": "s ∈ 𝓝 ⊤ ↔ ∃ y, Set.Ioi ↑y ⊆ s"}}
+{"state": {"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₃ : Type u_3", "J : Type u_4", "Zero I₂", "r : I₁ × I₂ × I₃ → J", "π : I₁ × I₃ → J", "self : CategoryTheory.GradedObject.TriangleIndexData r π", "i₁ : I₁"], "goal": "self.p₁₂ (i₁, 0) = i₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "a b : α", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "f : α → β"], "goal": "((StrictMono fun a => map f ↑a) ∧ ∀ (x : α), map f ⊥ < map f ↑x) ↔ StrictMono f"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommSemiring R", "M : Type u_2", "inst✝⁸ : AddCommMonoid M", "inst✝⁷ : Module R M", "N : Type u_3", "inst✝⁶ : AddCommMonoid N", "inst✝⁵ : Module R N", "ι : Type u_4", "inst✝⁴ : DecidableEq ι", "S : Type u_5", "inst✝³ : CommSemiring S", "inst✝² : Algebra R S", "inst✝¹ : Module S M", "inst✝ : IsScalarTower R S M", "p : ι →₀ R", "n : N", "i : ι"], "goal": "((finsuppScalarLeft R N ι) (p ⊗ₜ[R] n)) i = (p.sum fun i m => Finsupp.single i (m • n)) i"}}
+{"state": {"context": ["x z : EReal", "h : x < z"], "goal": "∃ y, x < ↑y ∧ ↑y < z"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁶ : TopologicalSpace α", "inst✝⁵ : TopologicalSpace β", "inst✝⁴ : LinearOrder α", "inst✝³ : LinearOrder β", "inst✝² : OrderTopology α", "inst✝¹ : OrderTopology β", "t : Set α", "x : α", "inst✝ : (𝓝 x).IsCountablyGenerated", "htx : IsLUB t x", "not_mem : x ∉ t", "ht : t.Nonempty", "v : ℕ → α", "hvx : Tendsto v atTop (𝓝[≤] x)", "hvt : ∀ (n : ℕ), v n ∈ t"], "goal": "∃ u, StrictMono u ∧ (∀ (n : ℕ), u n < x) ∧ Tendsto u atTop (𝓝 x) ∧ ∀ (n : ℕ), u n ∈ t"}}
+{"state": {"context": ["F : Type u → Type w", "G : Type v → Type u", "Applicative F", "Applicative G", "LawfulApplicative F", "LawfulApplicative G", "α β γ : Type v", "x : Functor.Comp F G α", "f : Functor.Comp F G (α → β)", "g : Functor.Comp F G (β → γ)"], "goal": "(Seq.seq g fun x_1 => Seq.seq f fun x_2 => x) = Seq.seq (Seq.seq (Function.comp <$> g) fun x => f) fun x_1 => x"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "CommRing R", "AddCommGroup M", "Module R M", "Invertible 2", "Q : QuadraticMap R M R", "hB : Q.Anisotropic"], "goal": "LinearMap.SeparatingLeft (QuadraticMap.associated' Q)"}}
+{"state": {"context": ["x y : ℂ"], "goal": "SameRay ℝ x y ↔ x = 0 ∨ y = 0 ∨ x.arg = y.arg"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "f : α → ℝ", "a : α", "hf : ContinuousAt f a", "h₀ : f a ≠ 0"], "goal": "ContinuousAt (fun x => Real.log (f x)) a"}}
+{"state": {"context": ["M : Type u_2", "N : Type u_3", "MulOneClass M", "MulOneClass N", "f : M →* N"], "goal": "∀ (a : Mᵐᵒᵖ), (MonoidHom.op f) a = (MulOpposite.op ∘ ⇑f ∘ MulOpposite.unop) a"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x✝ y✝ z : X", "ι : Type u_3", "F : Set X", "f : X → Y", "hf : Inducing f", "hF : F.Nonempty ∧ ∀ a ∈ F, ∀ a_2 ∈ F, JoinedIn (f '' F) (f a) (f a_2)", "x : X", "hx : x ∈ F", "y : X", "hy : y ∈ F"], "goal": "JoinedIn F x y"}}
+{"state": {"context": ["K : Type u_3", "Field K", "a b c : K", "hc : c ≠ 0"], "goal": "b - a / c = (b * c - a) / c"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X₁₁ X₁₂ X₁₃ X₂₁ X₂₂ X₂₃ : C", "h₁₁ : X₁₁ ⟶ X₁₂", "h₁₂ : X₁₂ ⟶ X₁₃", "h₂₁ : X₂₁ ⟶ X₂₂", "h₂₂ : X₂₂ ⟶ X₂₃", "v₁₁ : X₁₁ ⟶ X₂₁", "v₁₂ : X₁₂ ⟶ X₂₂", "v₁₃ : X₁₃ ⟶ X₂₃", "s : CategoryTheory.IsPushout h₁₁ v₁₁ v₁₂ h₂₁", "e : h₁₂ ≫ v₁₃ = v₁₂ ≫ h₂₂"], "goal": "CategoryTheory.IsPushout (h₁₁ ≫ h₁₂) v₁₁ v₁₃ (h₂₁ ≫ h₂₂) ↔ CategoryTheory.IsPushout h₁₂ v₁₂ v₁₃ h₂₂"}}
+{"state": {"context": ["R : Type u", "CommRing R", "P Q : Fin 3 → R", "hPz : P z = 0", "hQz : Q z ≠ 0"], "goal": "¬P ≈ Q"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "Complex.normSq ((o.kahler x) y) = (‖x‖ * ‖y‖) ^ 2"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j : ι", "hi : c.prev j = i", "h : K.d i j = 0", "K.HasHomology j"], "goal": "(K.isoHomologyπ i j hi h).inv ≫ K.homologyπ j = 𝟙 (K.homology j)"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b", "z : 𝓞 K", "hz : S'.c = λ * z", "x : 𝓞 K", "hx : S'.a ^ 3 - 1 = λ ^ 4 * x", "y : 𝓞 K", "hy : S'.b ^ 3 - 1 = λ ^ 4 * y"], "goal": "λ ^ 4 ∣ S'.a ^ 3 - 1 ∧ λ ^ 4 ∣ S'.b ^ 3 + 1 ∨ λ ^ 4 ∣ S'.a ^ 3 + 1 ∧ λ ^ 4 ∣ S'.b ^ 3 - 1"}}
+{"state": {"context": ["M : Type u_3", "AddCommGroup M", "S : AddSubgroup M"], "goal": "↑(AddSubgroup.toIntSubmodule S) = ↑S"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "β : Type u_2", "g : α → β", "y : α", "q : Fin n → α"], "goal": "g ∘ Fin.cons y q = Fin.cons (g y) (g ∘ q)"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "s✝ : Set E", "s_conv✝ : Convex ℝ s✝", "f✝ : E → F", "f'✝ : E → E →L[ℝ] F", "f''✝ : E →L[ℝ] E →L[ℝ] F", "hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x", "x✝ : E", "xs✝ : x✝ ∈ s✝", "hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝", "s : Set E", "s_conv : Convex ℝ s", "f : E → F", "f' : E → E →L[ℝ] F", "f'' : E →L[ℝ] E →L[ℝ] F", "hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x", "x : E", "xs : x ∈ s", "hx : HasFDerivWithinAt f' f'' (interior s) x", "v w y : E", "hy : y ∈ interior s", "z : E", "hz : z = (1 / 4) • (y - x)", "A : ∀ (m : E), Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 y)"], "goal": "(f'' v) w = (f'' w) v"}}
+{"state": {"context": ["α : Type u", "p : α → Bool", "a : α", "l : List α", "h : a ∈ List.filter p l"], "goal": "a ∈ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "o : Ordinal.{u_4}", "h : ¬∃ a, o = succ a"], "goal": "o.pred ≤ o"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝¹¹ : NontriviallyNormedField 𝕜", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "inst✝⁸ : TopologicalSpace H", "inst✝⁷ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁶ : NormedAddCommGroup E'", "inst✝⁵ : NormedSpace 𝕜 E'", "inst✝⁴ : TopologicalSpace H'", "inst✝³ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝² : ChartedSpace H M", "inst✝¹ : ChartedSpace H' M'", "inst✝ : I.Boundaryless", "x : M"], "goal": "(extChartAt I x).target ∈ 𝓝 (↑(extChartAt I x) x)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "K K' : LieSubalgebra R L", "h : K ≤ K'"], "goal": "LieSubalgebra.ofLe h = LieSubalgebra.comap K'.incl K"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s : Set E", "f : E → F", "n : ℕ∞", "m : ℕ", "h : ContDiffOn 𝕜 n f s", "hmn : ↑m ≤ n", "hs : UniqueDiffOn 𝕜 s"], "goal": "ContinuousOn (iteratedFDerivWithin 𝕜 m f s) s"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β i)", "inst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)", "i : ι", "f : (i : ι) → β i", "s : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }"], "goal": "i ∈ ↑s ∧ f i ≠ 0 ↔ f i ≠ 0"}}
+{"state": {"context": ["a b : ℕ", "hb : Odd b", "ha₀ : a ≠ 0"], "goal": "J(↑a | b) = J(↑a | b % (4 * a))"}}
+{"state": {"context": ["b : ℂ", "V : Type u_1", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : FiniteDimensional ℝ V", "inst✝¹ : MeasurableSpace V", "inst✝ : BorelSpace V", "hb : 0 < b.re", "c : ℂ", "w : V", "e : EuclideanSpace ℝ (Fin (FiniteDimensional.finrank ℝ V)) ≃ₗᵢ[ℝ] V := (stdOrthonormalBasis ℝ V).repr.symm"], "goal": "∫ (v : V), cexp (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ) = (↑π / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * cexp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))"}}
+{"state": {"context": ["α : Sort u_4", "β : Sort u_5", "Finite β", "f : α → β", "H : Function.Injective f"], "goal": "Finite α"}}
+{"state": {"context": ["R S : Type u", "M : Type v", "M'✝ : Type v'", "M₁ : Type v", "ι : Type w", "ι' : Type w'", "η : Type u₁'", "φ : η → Type u_1", "inst✝⁷ : Ring R", "inst✝⁶ : CommRing S", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : AddCommGroup M'✝", "inst✝³ : AddCommGroup M₁", "inst✝² : Module R M", "inst✝¹ : Module R M'✝", "inst✝ : Module R M₁", "M' : Submodule R M", "ι₁ : Type u_2", "ι₂ : Type u_3", "f : ι₁ → ↥M'", "hf : LinearIndependent R f", "g : ι₂ → M", "hg : LinearIndependent R (Submodule.Quotient.mk ∘ g)", "x : M", "h₁ : x ∈ span R (range fun x => ↑(f x))", "h₂ : x ∈ span R (range g)"], "goal": "x = 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a✝ b c d : R", "n✝ m : ℕ", "inst✝¹ : Ring R", "inst✝ : Nontrivial R", "n : ℕ", "hn : 0 < n", "a : R"], "goal": "X ^ n - C a ≠ 0"}}
+{"state": {"context": ["α✝ : Type u_1", "ι : Type u_2", "inst✝⁶ : TopologicalSpace α✝", "α : Type u_3", "β : Type u_4", "t : TopologicalSpace α", "inst✝⁵ : PolishSpace α", "inst✝⁴ : MeasurableSpace α", "inst✝³ : BorelSpace α", "tβ : TopologicalSpace β", "inst✝² : MeasurableSpace β", "inst✝¹ : OpensMeasurableSpace β", "f : α → β", "inst✝ : SecondCountableTopology ↑(range f)", "hf : Measurable f", "b : Set (Set ↑(range f))", "b_count : b.Countable", "hb : IsTopologicalBasis b", "this✝ : Countable ↑b", "T : ↑b → TopologicalSpace α", "Tt : ∀ (s : ↑b), T s ≤ t", "Tpolish : ∀ (s : ↑b), PolishSpace α", "h✝ : ∀ (s : ↑b), IsClosed (rangeFactorization f ⁻¹' ↑s)", "Topen : ∀ (s : ↑b), IsOpen (rangeFactorization f ⁻¹' ↑s)", "t' : TopologicalSpace α", "t'T : ∀ (i : ↑b), t' ≤ T i", "t't : t' ≤ t", "t'_polish : PolishSpace α", "this : Continuous (rangeFactorization f)"], "goal": "Continuous f"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "U : Set α"], "goal": "ConnectedComponents.mk ⁻¹' (ConnectedComponents.mk '' U) = ⋃ x ∈ U, connectedComponent x"}}
+{"state": {"context": ["α : Type u_1"], "goal": "FreeMonoid.ofList [] = 1"}}
+{"state": {"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "X Y Z : Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g", "i j k : 𝒰.J"], "goal": "t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.snd (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map k) ≫ pullback.snd (𝒰.map i ≫ f) g = pullback.snd (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.fst (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map k) ≫ pullback.snd (𝒰.map i ≫ f) g"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G"], "goal": "H.Saturated ↔ ∀ (n : ℕ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "α : Type u_3"], "goal": "card α = 0 ↔ IsEmpty α"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_2", "Monoid M", "Semiring R", "MulSemiringAction M R", "a : M", "s : Set R"], "goal": "a • Subsemiring.closure s = Subsemiring.closure (a • s)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f g : CategoryTheory.Arrow C", "CategoryTheory.Limits.HasImage f.hom", "CategoryTheory.Limits.HasImage g.hom", "sq : f ⟶ g", "m : CategoryTheory.Limits.ImageMap sq"], "goal": "CategoryTheory.Limits.factorThruImage f.hom ≫ m.map = sq.left ≫ CategoryTheory.Limits.factorThruImage g.hom"}}
+{"state": {"context": [], "goal": "Nat.Primrec' fun v => v.head - v.tail.head"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ φ' : S₁ ⟶ S₂", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData"], "goal": "CategoryTheory.ShortComplex.homologyMap' (φ + φ') h₁ h₂ =\n  CategoryTheory.ShortComplex.homologyMap' φ h₁ h₂ + CategoryTheory.ShortComplex.homologyMap' φ' h₁ h₂"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝⁴ : CommRing R", "inst✝³ : IsDedekindDomain R", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "v : HeightOneSpectrum R", "r : ↥R⁰", "x : FiniteAdeleRing R K", "hx : ∃ x_1 ∈ ↑(Submodule.span (R_hat R K) {↑↑r}), ∃ y ∈ ↑(Submodule.span (R_hat R K) {↑↑r}), x_1 * y = x"], "goal": "x ∈ ↑(Submodule.span (R_hat R K) {↑↑r})"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "CommMonoid β", "s : Finset α", "p : α → Prop", "_hp : DecidablePred p", "f g : α → γ", "h : γ → β"], "goal": "∏ x ∈ s, h (if p x then f x else g x) =\n  (∏ x ∈ Finset.filter p s, h (f x)) * ∏ x ∈ Finset.filter (fun x => ¬p x) s, h (g x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "𝕜 : Type u_4", "inst✝ : NontriviallyNormedField 𝕜", "m : ℤ", "x : 𝕜"], "goal": "ContinuousAt (fun x => x ^ m) x → x ≠ 0 ∨ 0 ≤ m"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s✝ s₁ s₂ t t₁ t₂ u s : Set α"], "goal": "¬s.Nonempty ↔ s = ∅"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι✝ : Type u_3", "ε : ℝ≥0∞", "hε : ε ≠ 0", "ι : Type u_4", "inst✝ : Countable ι", "x : ℝ≥0", "right✝ : ↑x < ε", "h0r : 0 < ↑x", "hrε : ↑x < ε"], "goal": "∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∑' (i : ι), ↑(ε' i) < ε"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "X : AlgebraicGeometry.PresheafedSpace C"], "goal": "X.toRestrictTop.base = (TopologicalSpace.Opens.inclusionTopIso ↑X).inv"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "s : Set α", "p : ℝ≥0∞", "hs : MeasurableSet s", "c : E", "hμsc : c = 0 ∨ μ s ≠ ⊤"], "goal": "MeasureTheory.Memℒp (s.indicator fun x => c) p μ"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "α : Type u_3", "R : Type u_4", "S : Type u_5", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "f : σ → τ", "d : σ →₀ ℕ", "r : R"], "goal": "(rename f) ((monomial d) r) = (monomial (Finsupp.mapDomain f d)) r"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "x : E", "n : ℕ∞", "hf : ContDiffAt ℝ n f x"], "goal": "ContDiffAt ℝ n (fun x => Real.exp (f x)) x"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "C : β → Type u_7", "(b : β) → SemilatticeInf (C b)", "s : Finset α", "H : s.Nonempty", "f : α → (b : β) → C b", "b : β"], "goal": "s.inf' H f b = s.inf' H fun a => f a b"}}
+{"state": {"context": ["F : Type u_3", "Field F", "Finite F", "hF : ringChar F ≠ 2"], "goal": "∃ a, ¬IsSquare a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "inst✝ : NormedAddCommGroup E", "p : ℝ≥0∞", "μ : Measure α", "f : α → E", "hp_ne_zero : p ≠ 0", "hp_ne_top : p ≠ ⊤", "hf : AEStronglyMeasurable f μ", "ε : ℝ≥0∞", "x : α"], "goal": "ε ≤ ↑‖f x‖₊ ↔ ε ^ p.toReal ≤ ↑‖f x‖₊ ^ p.toReal"}}
+{"state": {"context": ["θ : ℂ", "n : ℤ"], "goal": "Polynomial.eval (Complex.cos θ) (Polynomial.Chebyshev.U ℂ n) * Complex.sin θ = Complex.sin ((↑n + 1) * θ)"}}
+{"state": {"context": ["α : Type u", "Monoid α", "self : αˣ"], "goal": "self.inv * ↑self = 1"}}
+{"state": {"context": ["n : ℕ"], "goal": "Monoid.exponent (DihedralGroup n) = lcm n 2"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "D : Type u'", "inst✝⁵ : Category.{v, u'} D", "inst✝⁴ : HasFiniteProducts C", "inst✝³ : HasFiniteProducts D", "F : C ⥤ D", "L : D ⥤ C", "inst✝² : CartesianClosed C", "inst✝¹ : CartesianClosed D", "inst✝ : PreservesLimitsOfShape (Discrete WalkingPair) F", "h : L ⊣ F", "A : C", "conjeq : (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) ((mateEquiv h h) (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L (prod.functor.map (h.counit.app A)))) = (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)) (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L (prod.functor.map (h.counit.app A)))", "B : C"], "goal": "(h.unit.app (F.obj A ⨯ F.obj B) ≫ F.map (prodComparison L (F.obj A) (F.obj B) ≫ prod.map (h.counit.app A) (h.counit.app B))) ≫ prodComparison F A B = 𝟙 (F.obj A ⨯ F.obj B)"}}
+{"state": {"context": ["x y : UInt16"], "goal": "(x * y).toNat = x.toNat * y.toNat % UInt16.size"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "T2Space Y", "s : Set X", "hs : IsCompact (closure s)", "f : X → Y", "hf : ContinuousOn f (closure s)"], "goal": "f '' closure s = closure (f '' s)"}}
+{"state": {"context": ["α : Type u_1", "SemilatticeInf α", "OrderBot α", "a b c : α", "h : Disjoint (a ⊓ b) c", "hle : b ≤ c"], "goal": "Disjoint a b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : Affine R", "f : R →+* S", "x₁ x₂ y₁ L : R"], "goal": "span {(mk W) (C (X - C (W.addX x₁ x₂ L))), (mk W) (Y - C (C (-(L * (W.addX x₁ x₂ L - x₁) + y₁) - W.a₁ * W.addX x₁ x₂ L - W.a₃)) + C (X - C (W.addX x₁ x₂ L)) * C (C (W.a₁ + L)))} = span {(mk W) (C (X - C (W.addX x₁ x₂ L))), (mk W) (C (C W.a₁ * X + C W.a₃) + C (C L * (X - C x₁) + C y₁) + Y)}"}}
+{"state": {"context": ["m : Type u_3", "n : Type u_4", "α : Type u_5", "Fintype m", "Fintype n", "SeminormedAddCommGroup α", "A : Matrix m n α"], "goal": "‖A‖ = ↑(Finset.univ.sup fun i => ∑ j : n, ‖A i j‖₊)"}}
+{"state": {"context": ["μ : YoungDiagram", "self : SemistandardYoungTableau μ", "i j1 j2 : ℕ"], "goal": "j1 < j2 → (i, j2) ∈ μ → self.entry i j1 ≤ self.entry i j2"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "g₁ g₂ : β → α", "fa : Filter α", "fb : Filter β", "hleft : ∀ᶠ (x : α) in fa, g₁ (f x) = x", "hright : ∀ᶠ (y : β) in fb, f (g₂ y) = y", "htendsto : Filter.Tendsto g₂ fb fa"], "goal": "g₁ =ᶠ[fb] g₂"}}
+{"state": {"context": ["α : Type u_1"], "goal": "⇑FreeAddMonoid.toList ∘ ⇑FreeAddMonoid.ofList = id"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : LinearOrderedField α", "a b c d : α", "n : ℤ", "hb : b < 0"], "goal": "1 < a / b ↔ 0 < b ∧ b < a ∨ b < 0 ∧ a < b"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalNonAssocRing R", "I J : TwoSidedIdeal R"], "goal": "I < J ↔ ↑I ⊂ ↑J"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "s t : Set α"], "goal": "(s ∪ t).IsPWO ↔ s.IsPWO ∧ t.IsPWO"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "𝒜 ℬ : Finset (Finset α)", "s✝ : Finset α", "h𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜", "h𝒜₀ : ∅ ∉ 𝒜", "s : Finset α", "hs : s ∈ 𝒜"], "goal": "↑s.card ≠ 0"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Fintype α", "Mul β"], "goal": "commProb (α → β) = commProb β ^ Fintype.card α"}}
+{"state": {"context": ["z w : Complex.UnitDisc"], "goal": "(z * w).conj = z.conj * w.conj"}}
+{"state": {"context": ["G : Type u₂", "k : Type u_3", "H : Type u_4", "F : Type u_5", "Semiring k", "Monoid G", "Monoid H", "FunLike F G H", "MonoidHomClass F G H", "f : F"], "goal": "∀ (a : G →₀ k), (MonoidAlgebra.mapDomainRingHom k f) a = (↑(Finsupp.mapDomain.addMonoidHom ⇑f)).toFun a"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s s₁ s₂ : Finset α", "a : α", "f g : α → β", "inst✝¹ : CommMonoid β", "R : Setoid α", "inst✝ : DecidableRel Setoid.r", "h : ∀ x ∈ s, ∏ a ∈ filter (fun y => y ≈ x) s, f a = 1", "x : α", "x_in_s : x ∈ s", "xbar_in_s : Quotient.mk'' x ∈ image Quotient.mk'' s"], "goal": "∏ y ∈ filter (fun x_1 => ⟦x_1⟧ = Quotient.mk'' x) s, f y = 1"}}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field K", "inst✝² inst✝¹ : NumberField K", "inst✝ : IsCyclotomicExtension {n} ℚ K", "ζ : K", "l : ℕ", "hζ : IsPrimitiveRoot ζ l", "hl✝ : l ≠ 0", "hl : NeZero l", "hroot : IsPrimitiveRoot (zeta n ℚ K) ↑n", "r : ℕ", "hr : GCDMonoid.lcm l ↑n = ↑n * r", "ineq : φ ↑n * φ r ≤ φ (GCDMonoid.lcm l ↑n)", "key : φ r ≤ 1"], "goal": "l ∣ 2 * ↑n"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t✝ : Set X", "l : Filter X", "inst✝ : CountableInterFilter l", "hs : IsLindelof s", "H : ∀ x ∈ s, Disjoint (𝓝 x) l", "U : X → Set X", "hUl : ∀ x ∈ s, (U x)ᶜ ∈ l", "hxU : ∀ x ∈ s, x ∈ U x", "hUo : ∀ x ∈ s, IsOpen (U x)", "t : Set X", "htc : t.Countable", "hts : ∀ x ∈ t, x ∈ s", "hst : s ⊆ ⋃ x ∈ t, U x"], "goal": "Disjoint (𝓝ˢ s) l"}}
+{"state": {"context": ["K : Type v", "V : Type w", "Field K", "AddCommGroup V", "Module K V", "FiniteDimensional K V", "f : Module.End K V", "μ : K", "h : (minpoly K f).IsRoot μ"], "goal": "f.HasEigenvalue μ"}}
+{"state": {"context": ["X : LocallyRingedSpace", "r : ↑(Γ.obj (op X))"], "goal": "(toOpen (↑(Γ.obj (op X))) (basicOpen 1) ≫ (structureSheaf ↑(Γ.obj (op X))).val.map (eqToHom ⋯).op) ≫ X.toΓSpec.val.c.app (op ⊤) = 𝟙 (Γ.obj (op X))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "∀ (n : ℕ) (f : CategoryTheory.Arrow C), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom", "f : CategoryTheory.Arrow C"], "goal": "CategoryTheory.CosimplicialObject.augmentedCechConerve.obj f = f.augmentedCechConerve"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "DecidableRel fun x x_1 => x ≤ x_1", "x : α", "t : Ordnode α", "h : t.Valid"], "goal": "(Ordnode.erase x t).Valid"}}
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+{"state": {"context": ["α : Type u_1", "f : α → α"], "goal": "⋃ n, Function.ptsOfPeriod f ↑n = Function.periodicPts f"}}
+{"state": {"context": ["G : Type u", "inst✝² : Group G", "inst✝¹ : Finite G", "N : Subgroup G", "inst✝ : N.Normal", "h1 : (Nat.card ↥N).Coprime N.index", "h2 : ∀ (G' : Type u) [inst : Group G'] [inst_1 : Finite G'], Nat.card G' < Nat.card G → ∀ {N' : Subgroup G'} [inst_2 : N'.Normal], (Nat.card ↥N').Coprime N'.index → ∃ H', N'.IsComplement' H'", "h3 : ∀ (H : Subgroup G), ¬N.IsComplement' H", "P : Sylow (Nat.card ↥N).minFac ↥N"], "goal": "↑P ≠ ⊥"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "inst✝⁷ : CommRing R", "inst✝⁶ : IsDomain R", "inst✝⁵ : IsPrincipalIdealRing R", "M✝ : Type u_3", "inst✝⁴ : AddCommGroup M✝", "inst✝³ : Module R M✝", "b : ι → M✝", "inst✝² : Finite ι", "O : Type u_4", "inst✝¹ : AddCommGroup O", "inst✝ : Module R O", "M N : Submodule R O", "b'M : Basis ι R ↥M", "N_bot : N ≠ ⊥", "N_le_M : N ≤ M", "this : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ.submoduleImage N < ψ.submoduleImage N", "ϕ : ↥M →ₗ[R] R := this.choose", "ϕ_max : ∀ (ψ : ↥M →ₗ[R] R), ¬this.choose.submoduleImage N < ψ.submoduleImage N", "a : R := generator (ϕ.submoduleImage N)", "a_mem : a ∈ ϕ.submoduleImage N", "a_zero : ¬a = 0", "y : O", "yN : y ∈ N", "ϕy_eq : ϕ ⟨y, ⋯⟩ = a", "_ϕy_ne_zero : ϕ ⟨y, ⋯⟩ ≠ 0"], "goal": "∃ y ∈ M, ∃ a, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N, N' ≤ M' ∧ (∀ (c : R), ∀ z ∈ M', c • y + z = 0 → c = 0) ∧ (∀ (c : R), ∀ z ∈ N', c • a • y + z = 0 → c = 0) ∧ ∀ (n' : ℕ) (bN' : Basis (Fin n') R ↥N'), ∃ bN, ∀ (m' : ℕ) (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R ↥M'), ∃ (hnm : n' + 1 ≤ m' + 1), ∃ bM, ∀ (as : Fin n' → R), (∀ (i : Fin n'), ↑(bN' i) = as i • ↑(bM' (Fin.castLE hn'm' i))) → ∃ as', ∀ (i : Fin (n' + 1)), ↑(bN i) = as' i • ↑(bM (Fin.castLE hnm i))"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_4", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup G", "s : Set α", "c : G", "hs : MeasurableSet s", "hμs : μ s ≠ 0", "hp : p ≠ 0"], "goal": "MeasureTheory.eLpNorm (s.indicator fun x => c) p μ = ↑‖c‖₊ * μ s ^ (1 / p.toReal)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "Finite ι", "DecidableEq α", "t : ι → Finset α"], "goal": "(∀ (s : Finset ι), s.card ≤ (s.biUnion t).card) ↔ ∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝⁹ : CommRing R✝", "I✝ : Ideal R✝", "M✝ : Type u_2", "inst✝⁸ : AddCommGroup M✝", "inst✝⁷ : Module R✝ M✝", "N : Type u_3", "inst✝⁶ : AddCommGroup N", "inst✝⁵ : Module R✝ N", "R : Type u", "inst✝⁴ : CommRing R", "I : Ideal R", "M : Type u", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : IsNoetherianRing R", "inst✝ : Module.Finite R M", "n : ℕ", "f : (Fin n → R) →ₗ[R] M", "hf : Function.Surjective ⇑f"], "goal": "Function.Bijective ⇑(ofTensorProduct I M)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹³ : NormedAddCommGroup E", "inst✝¹² : NormedSpace ℝ E", "inst✝¹¹ : FiniteDimensional ℝ E", "F : Type u_2", "inst✝¹⁰ : NormedAddCommGroup F", "inst✝⁹ : NormedSpace ��� F", "inst✝⁸ : CompleteSpace F", "H : Type u_3", "inst✝⁷ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type u_4", "inst✝⁶ : TopologicalSpace M", "inst✝⁵ : ChartedSpace H M", "inst✝⁴ : SmoothManifoldWithCorners I M", "inst✝³ : MeasurableSpace M", "inst✝² : BorelSpace M", "inst✝¹ : SigmaCompactSpace M", "inst✝ : T2Space M", "f f' : M → F", "μ : Measure M", "U✝ : Set M", "hU : IsOpen U✝", "hSig : IsSigmaCompact U✝", "hf : LocallyIntegrableOn f U✝ μ", "h : ∀ (g : M → ℝ), Smooth I 𝓘(ℝ, ℝ) g → HasCompactSupport g → tsupport g ⊆ U✝ → ∫ (x : M), g x • f x ∂μ = 0", "meas_U : MeasurableSet U✝", "U : Opens M := { carrier := U✝, is_open' := hU }"], "goal": "∀ᵐ (x : ↥U) ∂Measure.comap Subtype.val μ, f ↑x = 0"}}
+{"state": {"context": ["α : Type u_2", "PartialOrder α", "x y : α"], "goal": "x ≤ y → y = x ↔ ¬x < y"}}
+{"state": {"context": ["R : Type u", "Semiring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "AddCommMonoid M", "Module R M", "A : ι → Submodule R M", "h : DirectSum.IsInternal A", "α : ι → Type u_2", "v : (i : ι) → Basis (α i) R ↥(A i)", "x : M", "i : ι", "a : α i", "hx : x ∈ A i"], "goal": "((h.collectedBasis v).repr x) ⟨i, a⟩ = ((v i).repr ⟨x, hx⟩) a"}}
+{"state": {"context": ["R : Type r", "S : Type s", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "W : WeierstrassCurve R", "f : R →+* S"], "goal": "2 * X ^ 6 + C (f W.b₂) * X ^ 5 + 5 * C (f W.b₄) * X ^ 4 + 10 * C (f W.b₆) * X ^ 3 + 10 * C (f W.b₈) * X ^ 2 + C (f W.b₂ * f W.b₈ - f W.b₄ * f W.b₆) * X + C (f W.b₄ * f W.b₈ - f W.b₆ ^ 2) = Polynomial.map f (2 * X ^ 6 + C W.b₂ * X ^ 5 + 5 * C W.b₄ * X ^ 4 + 10 * C W.b₆ * X ^ 3 + 10 * C W.b₈ * X ^ 2 + C (W.b₂ * W.b₈ - W.b₄ * W.b₆) * X + C (W.b₄ * W.b₈ - W.b₆ ^ 2))"}}
+{"state": {"context": ["a b : Ordinal.{v}"], "goal": "Ordinal.lift.{u, v} a < Ordinal.lift.{u, v} b ↔ a < b"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace ℝ E", "ι : Type u_2", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι", "ne : Nonempty ι", "e f : OrthonormalBasis ι ℝ E", "x : Orientation ℝ E ι"], "goal": "Orthonormal ℝ ⇑(e.toBasis.adjustToOrientation x)"}}
+{"state": {"context": ["α : Type u_1", "d : DynkinSystem α", "a : Set α", "h : d.Has aᶜ"], "goal": "d.Has a"}}
+{"state": {"context": ["α : Type u_1", "CompleteLattice α", "s : True → α"], "goal": "iSup s = s trivial"}}
+{"state": {"context": ["R : Type uR", "A₁ : Type uA₁", "A₂ : Type uA₂", "CommSemiring R", "Semiring A₁", "Semiring A₂", "Algebra R A₁", "Algebra R A₂", "f : A₁ ≃+* A₂", "hf : ∀ (x : R), f ((algebraMap R A₁) x) = (algebraMap R A₂) x"], "goal": "↑(AlgEquiv.ofRingEquiv hf) = { toFun := ⇑f, invFun := ⇑f.symm, left_inv := ⋯, right_inv := ⋯ }"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "α : Type u_12", "DecidableEq l", "DecidableEq m", "Zero α", "v : l ⊕ m → α"], "goal": "(Matrix.diagonal v).toBlocks₁₁ = Matrix.diagonal fun i => v (Sum.inl i)"}}
+{"state": {"context": ["C : FloatCfg", "this : ↑prec ≤ ↑(2 * emax)"], "goal": "emin + ↑prec - 1 ≤ ↑emax"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "p : ι → Prop", "DecidablePred p", "f g : ι → Set α"], "goal": "(⋂ i, if p i then f i else g i) = (⋂ i, ⋂ (_ : p i), f i) ∩ ⋂ i, ⋂ (_ : ¬p i), g i"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "S : CategoryTheory.ShortComplex C", "S.HasHomology", "h : S.RightHomologyData"], "goal": "S.Exact ↔ CategoryTheory.Mono h.g'"}}
+{"state": {"context": ["ι : Type u_1", "α : ι → Type u_2", "(i : ι) → MeasurableSpace (α i)", "P : (J : Finset ι) → MeasureTheory.Measure ((j : { x // x ∈ J }) → α ↑j)", "I J : Finset ι", "hP : MeasureTheory.IsProjectiveMeasureFamily P", "hJI : J ⊆ I"], "goal": "(P I) Set.univ = (P J) Set.univ"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "p : G.Walk u v"], "goal": "u ∈ p.support"}}
+{"state": {"context": ["n m : ℕ", "hm : NeZero m", "h : n ∣ m", "x : ℕ", "hx : x.Coprime n", "ps : Finset ℕ := Finset.filter (fun p => ¬p ∣ x) m.primeFactors"], "goal": "∃ k, (x + k * n).Coprime m"}}
+{"state": {"context": ["𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "CompleteSpace F", "f g : E → F", "s : Set E", "x : E", "hf : AnalyticWithinAt 𝕜 f s x", "hs : f =ᶠ[𝓝[s] x] g", "hx : f x = g x"], "goal": "AnalyticWithinAt 𝕜 g s x"}}
+{"state": {"context": ["a x✝ z x y : ℂ"], "goal": "↑x.arg = ↑y.arg ↔ x.arg = y.arg"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "H : Subgroup (Perm α)", "d : DecidablePred fun x => x ∈ H", "τ : Perm α", "h0 : Nat.Prime (Fintype.card α)", "h1 : Fintype.card α ∣ Fintype.card ↥H", "h2 : τ ∈ H", "h3 : τ.IsSwap", "this : Fact (Nat.Prime (Fintype.card α))", "σ : ↥H", "hσ : orderOf σ = Fintype.card α", "hσ1 : orderOf ↑σ = Fintype.card α", "hσ2 : (↑σ).IsCycle", "hσ3 : (↑σ).support = ⊤"], "goal": "H = ⊤"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_2", "β : Type u_3", "f : α → β", "m : ι → MeasureTheory.OuterMeasure β"], "goal": "(MeasureTheory.OuterMeasure.comap f) (⨅ i, m i) = ⨅ i, (MeasureTheory.OuterMeasure.comap f) (m i)"}}
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+{"state": {"context": ["m n : ℕ"], "goal": "↑(m + n) = ↑m + ↑n"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "s : Set P"], "goal": "⊥ < affineSpan k s ↔ s.Nonempty"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommSemiring R", "σ : Type u_2", "s : Multiset σ", "r : σ → R", "k : ℕ", "h : k ≤ card s"], "goal": "k ≤ card s"}}
+{"state": {"context": ["K : Type u_1", "S : Type u_2", "inst✝¹⁰ : Field K", "inst✝⁹ : CommRing S", "inst✝⁸ : Algebra K S", "R : Type u_3", "inst✝⁷ : CommRing R", "inst✝⁶ : Algebra R S", "inst✝⁵ : Algebra R K", "inst✝⁴ : IsScalarTower R K S", "A : Type u_4", "inst✝³ : CommRing A", "inst✝² : Algebra R A", "inst✝¹ : Algebra S A", "inst✝ : IsScalarTower R S A", "B✝ : PowerBasis S A", "hB✝ : IsIntegral R B✝.gen", "B B' : PowerBasis K S", "P : R[X]", "h : (aeval B.gen) P = B'.gen", "hB : IsIntegral R B.gen", "hmin : minpoly K B.gen = Polynomial.map (algebraMap R K) (minpoly R B.gen)", "i : Fin B.dim", "j : Fin B'.dim"], "goal": "IsIntegral R ((B.basis.repr ((fun i => B'.gen ^ ↑i) j)) i)"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrderedField α", "a b c : CauSeq α abs", "εa : α", "εa0 : εa > 0", "ia : ℕ", "ha : ∀ j ≥ ia, εa ≤ ↑(c - a) j", "εb : α", "εb0 : εb > 0", "ib : ℕ", "hb : ∀ j ≥ ib, εb ≤ ↑(c - b) j"], "goal": "a ⊔ b < c"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "inst✝¹ : Semiring α", "inst✝ : Semiring β", "a b : α", "m n✝ : ℕ", "ha : Even a", "n : ℕ", "x✝ : n + 1 ≠ 0"], "goal": "Even (a ^ n * a)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "n✝ n : ℕ", "hpos : 0 < n"], "goal": "Irreducible (cyclotomic n ℚ)"}}
+{"state": {"context": ["X : TopCat", "p₀ : ↑X", "inst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)", "C : Type v", "inst✝¹ : Category.{u, v} C", "A : C", "inst✝ : HasTerminal C", "y : ↑X", "h : p₀ ⤳ y", "U V : (OpenNhds y)ᵒᵖ", "inc : U ⟶ V"], "goal": "eqToHom ⋯ ≫ (fun U => eqToHom ⋯) V = (fun U => eqToHom ⋯) U ≫ ((Functor.const (OpenNhds y)ᵒᵖ).obj A).map inc"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : α → ℝ≥0∞", "hf : Measurable f"], "goal": "Measurable fun x => ↑(f x)"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "e : E ≃ₛₗᵢ[σ₁₂] E₂"], "goal": "AntilipschitzWith 1 ⇑e"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "μ : MeasureTheory.Measure α", "MeasureTheory.SFinite μ", "f : α × β → ℝ≥0∞", "hf : Measurable f"], "goal": "Measurable fun y => ∫⁻ (x : α), f (x, y) ∂μ"}}
+{"state": {"context": ["ι : Type uι", "inst✝⁵ : Fintype ι", "𝕜 : Type u𝕜", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : ι → Type uE", "inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)", "inst✝² : (i : ι) → NormedSpace 𝕜 (E i)", "F : Type uF", "inst✝¹ : SeminormedAddCommGroup F", "inst✝ : NormedSpace ��� F", "p q : FreeAddMonoid (𝕜 × ((i : ι) → E i))"], "goal": "(List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) (FreeAddMonoid.toList p) ++ List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) (FreeAddMonoid.toList q)).sum ≤ (List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) p).sum + (List.map (fun p => ‖p.1‖ * ∏ i : ι, ‖p.2 i‖) q).sum"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "Semiring k", "Mul G", "A : Type u₃", "NonUnitalNonAssocSemiring A", "Module k A", "IsScalarTower k A A", "SMulCommClass k A A", "f : G →ₙ* A", "a : MonoidAlgebra k G"], "goal": "((MonoidAlgebra.liftMagma k) f) a = Finsupp.sum a fun m t => t • f m"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "CommMonoid M", "f : α → M", "s : Set α", "p : M → Prop", "hp₀ : p 1", "hp₁ : ∀ (x y : M), p x → p y → p (x * y)", "hp₂ : ∀ x ∈ s, p (f x)"], "goal": "p (∏ᶠ (i : α) (_ : i ∈ s), f i)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F G : CategoryTheory.Functor C D", "α : F.op ≅ G.op"], "goal": "(CategoryTheory.NatIso.removeOp α).hom = CategoryTheory.NatTrans.removeOp α.hom"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "G'' : Type u_3", "inst✝⁵ : Group G", "inst✝⁴ : Group G'", "inst✝³ : Group G''", "A : Type u_4", "inst✝² : AddGroup A", "H K✝ : Subgroup G", "k : Set G", "N : Type u_5", "inst✝¹ : Group N", "P : Type u_6", "inst✝ : Group P", "f : G ≃* N", "K : Subgroup G", "x : N"], "goal": "f.symm x ∈ ↑K ↔ f.symm x ∈ K"}}
+{"state": {"context": ["a b : ℕ+"], "goal": "(Ico a b).card = (Ico ↑a ↑b).card"}}
+{"state": {"context": ["α✝ : Type u_1", "r : α✝ → α✝ → Prop", "c : Cardinal.{u}", "h : ℵ₀ ≤ c", "α : Type u", "αe : ⟦α⟧ = succ c"], "goal": "c < (succ c).ord.cof"}}
+{"state": {"context": ["cf : Code"], "goal": "encode cf < encode cf.rfind'"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "S T : NonUnitalSubalgebra R A", "h : S ≤ T", "x : A", "hx : x ∈ S"], "goal": "(NonUnitalSubalgebra.inclusion h) ⟨x, hx⟩ = ⟨x, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "Finite α", "s t : Set α", "h : #↑s = #↑t"], "goal": "#↑sᶜ = #↑tᶜ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "h :\n  ∀ (J : Type w) {𝒥 : CategoryTheory.SmallCategory J},\n    CategoryTheory.FinCategory J → CategoryTheory.Limits.HasColimitsOfShape J C"], "goal": "CategoryTheory.Limits.HasFiniteColimits C"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "κ : ι → Sort u_5", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "s : Set α", "p : α → Prop", "a✝ : α", "e : α ↪o β", "he : IsUpperSet (range ⇑e)", "a : α"], "goal": "⇑e '' Ioi a = Ioi (e a)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : InvolutiveInv α", "s t : Set α", "a : α", "ι : Sort u_5", "f : ι → α"], "goal": "Inv.inv '' range f = range fun i => (f i)⁻¹"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "D : Type u_1", "inst✝³ : Category.{u_2, u_1} D", "F : C ⥤ D", "G : D ⥤ C", "adj : F ⊣ G", "inst✝² : G.Full", "inst✝¹ : G.Faithful", "I : D", "hI : Injective (G.obj I)", "X Y : D", "f : X ⟶ I", "g : X ⟶ Y", "inst✝ : Mono g", "this : PreservesLimitsOfSize.{0, 0, u_2, v₁, u_1, u₁} G", "w : G.obj Y ⟶ G.obj I", "h : G.map g ≫ w = G.map f"], "goal": "g ≫ inv (adj.counit.app Y) ≫ F.map w ≫ adj.counit.app I = f"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I : Box ι", "i : ι", "x : ℝ", "y✝ y : ι → ℝ"], "goal": "(y ∈ univ.pi fun i_1 => Ioc (update I.lower i (max x (I.lower i)) i_1) (I.upper i_1)) ↔ y ∈ ↑I ∩ {y | x < y i}"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommSemiring R", "f g : ArithmeticFunction R", "hf : f.IsMultiplicative", "hg : g.IsMultiplicative", "m n : ℕ", "cop : m.Coprime n"], "goal": "∀ a ∈ m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal, f a.1.1 * g a.1.2 * (f a.2.1 * g a.2.2) = f (match a with | ((i, j), k, l) => (i * k, j * l)).1 * g (match a with | ((i, j), k, l) => (i * k, j * l)).2"}}
+{"state": {"context": ["α : Type u_1", "a : α", "x : Option (Option α)", "h : a ∈ x.join"], "goal": "some a ∈ x"}}
+{"state": {"context": ["a b c : Prop", "h : a ↔ b"], "goal": "a → c ↔ b → c"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝¹³ : CommRing R", "inst✝¹² : LieRing L", "inst✝¹¹ : LieAlgebra R L", "inst✝¹⁰ : AddCommGroup M", "inst✝⁹ : Module R M", "inst✝⁸ : LieRingModule L M", "inst✝⁷ : LieModule R L M", "k✝ : ℕ", "N : LieSubmodule R L M", "L₂ : Type u_1", "M₂ : Type u_2", "inst✝⁶ : LieRing L₂", "inst✝⁵ : LieAlgebra R L₂", "inst✝⁴ : AddCommGroup M₂", "inst✝³ : Module R M₂", "inst✝² : LieRingModule L₂ M₂", "inst✝¹ : LieModule R L₂ M₂", "f : L →ₗ⁅R⁆ L₂", "g : M →ₗ[R] M₂", "hf : Surjective ⇑f", "hg : Surjective ⇑g", "hfg : ∀ (x : L) (m : M), ⁅f x, g m⁆ = g ⁅x, m⁆", "inst✝ : LieModule.IsNilpotent R L M", "k : ℕ", "hk : lowerCentralSeries R L M k = ⊥"], "goal": "LieModule.IsNilpotent R L₂ M₂"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "e : Sym2 V", "inst✝¹ : Fintype V", "inst✝ : DecidableEq V", "v w : V", "h : v ≠ w"], "goal": "{v, w}.toFinset ⊆ Set.univ.toFinset"}}
+{"state": {"context": ["α : Type u_3", "Preorder α", "a : α"], "goal": "grade α a = a"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α"], "goal": "toAntisymmetrization (fun x x_1 => x ≤ x_1) a ≤ toAntisymmetrization (fun x x_1 => x ≤ x_1) b ↔ a ≤ b"}}
+{"state": {"context": ["x : ℝ*", "h : x ≠ 0"], "goal": "x.Infinitesimal ↔ x⁻¹⁻¹.Infinitesimal"}}
+{"state": {"context": ["ι : Type u_1", "M : ι → Type u_2", "inst✝¹ : (i : ι) → Monoid (M i)", "N : Type u_3", "inst✝ : Monoid N", "f g : CoprodI M →* N", "h : ∀ (i : ι), f.comp of = g.comp of", "x✝ : (i : ι) × M i", "i : ι", "x : M i"], "goal": "(g.comp of) x = (g.comp (conGen (Rel M)).mk') (FreeMonoid.of ⟨i, x⟩)"}}
+{"state": {"context": ["R : Type u_3", "CommRing R", "c₁ c₂ : R", "a b : ℍ[R,c₁,c₂]"], "goal": "(a * b).imI = a.re * b.imI + a.imI * b.re - c₂ * a.imJ * b.imK + c₂ * a.imK * b.imJ"}}
+{"state": {"context": ["α : Type u_2", "CompleteLattice α", "a : α"], "goal": "(FrameHom.id α) a = a"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "x : G₀", "h : x ≠ 0"], "goal": "Function.Injective fun y => x * y"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "α : Type u_2", "f : OnePoint X → α", "l : Filter α"], "goal": "Filter.Tendsto f (𝓝 OnePoint.infty) l ↔\n  Filter.Tendsto f (pure OnePoint.infty) l ∧ Filter.Tendsto (f ∘ OnePoint.some) (Filter.coclosedCompact X) l"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p✝ : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedAddCommGroup G", "f : α → F", "p : ℝ≥0", "hp : p ≠ 0"], "goal": "eLpNorm f (↑p) μ ^ ↑p = ∫⁻ (x : α), ↑‖f x‖₊ ^ ↑p ∂μ"}}
+{"state": {"context": ["ι : Type u_4", "M : Type u_5", "N : Type u_7", "αs : ι → Type u_13", "Zero M", "l : (i : ι) × αs i →₀ M", "AddCommMonoid N", "f : (i : ι) × αs i → M → N"], "goal": "l.sum f = ∑ i ∈ l.splitSupport, (l.split i).sum fun a b => f ⟨i, a⟩ b"}}
+{"state": {"context": ["z : ℍ", "x : Fin 2 → ℤ", "hx : x ≠ 0", "H1 : max (x 0).natAbs (x 1).natAbs = (x 1).natAbs"], "goal": "1 ≤ (↑(x 1) / ‖x‖) ^ 2"}}
+{"state": {"context": ["α : Type u_1", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup F", "f : α → F"], "goal": "MeasureTheory.eLpNorm (-f) p μ = MeasureTheory.eLpNorm f p μ"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "J : GrothendieckTopology C", "D : Type w", "inst✝⁵ : Category.{max v u, w} D", "inst✝⁴ : ConcreteCategory D", "inst✝³ : PreservesLimits (forget D)", "inst✝² : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D", "inst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)", "inst✝ : (X : C) → PreservesColimitsOfShape (J.Cover X)ᵒᵖ (forget D)", "P : Cᵒᵖ ⥤ D", "hsep : ∀ (X : C) (S : J.Cover X) (x y : (forget D).obj (P.obj (op X))), (∀ (I : S.Arrow), (P.map I.f.op) x = (P.map I.f.op) y) → x = y", "X : C", "S : J.Cover X", "s : Meq (J.plusObj P) S", "inj : ∀ (X : C), Function.Injective ⇑((J.toPlus P).app (op X))", "T : (I : S.Arrow) → J.Cover I.Y", "t : (I : S.Arrow) → Meq P (T I)", "ht : ∀ (I : S.Arrow), ↑s I = mk (t I)", "B : J.Cover X := S.bind T"], "goal": "∃ t, Meq.mk S t = s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f : α → β", "s : Set α", "x : α", "h : ContinuousOn f s", "hx : s ∈ 𝓝 x"], "goal": "ContinuousAt f x"}}
+{"state": {"context": ["k : Type u_1", "P₁ : Type u_2", "P₂ : Type u_3", "V₁ : Type u_6", "V₂ : Type u_7", "Ring k", "AddCommGroup V₁", "Module k V₁", "AddTorsor V₁ P₁", "AddCommGroup V₂", "Module k V₂", "AddTorsor V₂ P₂", "TopologicalSpace P₁", "TopologicalSpace P₂", "f : P₁ ≃ᵃL[k] P₂", "s : Set P₂"], "goal": "⇑f.symm '' s = ⇑f ⁻¹' s"}}
+{"state": {"context": ["R : Type u_3", "Γ₀ : Type u_4", "Ring R", "LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "x y : R", "h : v y < v x"], "goal": "v (x - y) = v x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Ω : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "inst✝⁴ : MeasurableSpace Ω", "inst✝³ : StandardBorelSpace Ω", "inst✝² : Nonempty Ω", "inst✝¹ : CountableOrCountablyGenerated α β", "κ : Kernel α (β × Ω)", "inst✝ : IsFiniteKernel κ", "f : β × Ω → ℝ≥0∞", "hf : Measurable f", "a : α", "t : Set Ω", "ht : MeasurableSet t"], "goal": "∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in Set.univ ×ˢ t, f x ∂κ a"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_3", "E : Type u_4", "MeasurableSpace M", "MeasurableSpace α", "NormedAddCommGroup E", "μ : MeasureTheory.Measure α", "p : ℝ≥0∞", "VAdd M α", "MeasureTheory.VAddInvariantMeasure M α μ", "MeasurableVAdd M α", "c : Mᵈᵃᵃ", "f g : ↥(MeasureTheory.Lp E p μ)"], "goal": "c +ᵥ (f - g) = c +ᵥ f - (c +ᵥ g)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "f : α → β"], "goal": "Monotone (⇑OrderDual.toDual ∘ f ∘ ⇑OrderDual.ofDual) ↔ Monotone f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "UniformSpace α", "UniformSpace β", "f : α → β", "h : UniformContinuous f", "a : α"], "goal": "SeparationQuotient.map f (SeparationQuotient.mk a) = SeparationQuotient.mk (f a)"}}
+{"state": {"context": ["α : Type u", "a : α", "s₁ : Stream' α", "s₂ : Stream' α"], "goal": "a ∈ s₁ → a ∈ s₁ ⋈ s₂"}}
+{"state": {"context": ["E : Type u_2", "Norm E", "x : E"], "goal": "‖{ down := x }‖ = ‖x‖"}}
+{"state": {"context": ["α : Type u", "Countable α"], "goal": "Set.univ.Countable"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "X' : Type u_4", "Y : Type u_5", "Z : Type u_6", "α : Type u_7", "α' : Type u_8", "β : Type u_9", "β' : Type u_10", "γ : Type u_11", "𝓕 : Type u_12", "tX : TopologicalSpace X", "tY : TopologicalSpace Y", "tZ : TopologicalSpace Z", "uα : UniformSpace α", "uβ : UniformSpace β", "uγ : UniformSpace γ", "t : κ → TopologicalSpace X'", "F : ι → X' → α", "x₀ : X'", "k : κ", "hk : EquicontinuousAt F x₀"], "goal": "EquicontinuousAt F x₀"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_5"], "goal": "∀ {x : MeasurableSpace α} [inst : MeasurableSpace E] [inst_1 : AddGroup E] [inst_2 : MeasurableAdd E] {g : α → E},\n  Measurable g → ∀ (f : MeasureTheory.SimpleFunc α E), Measurable (↑f + g)"}}
+{"state": {"context": ["G : Type u_2", "MeasurableSpace G", "AddGroup G", "MeasurableAdd G", "μ : MeasureTheory.Measure G", "μ.IsAddRightInvariant", "x : G"], "goal": "Filter.map (fun h => h + x) (MeasureTheory.ae μ) = MeasureTheory.ae μ"}}
+{"state": {"context": ["β : Type u_1", "G : Type u_2", "M : Type u_3", "inst✝² : Monoid M", "inst✝¹ : Preorder M", "inst✝ : CovariantClass M M (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "x : M", "hx : x ≤ 1", "n : ℕ"], "goal": "x ^ (n + 1) ≤ 1"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "r : ℝ", "hr : r ≠ 0", "x y : E", "s t : Set E"], "goal": "μ ({x} + r • s) / μ ({y} + r • t) = μ s / μ t"}}
+{"state": {"context": ["α : Type u_1", "x : α → ℕ"], "goal": "0 x = 0"}}
+{"state": {"context": ["w : Nat", "x : BitVec w"], "goal": "x.msb = true ↔ 2 * x.toNat ≥ 2 ^ w"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "Q : QuadraticForm R M", "i' : ZMod 2", "x' : ↥(evenOdd Q i')"], "goal": "∀ (m : M), (GradedAlgebra.ι Q) m * (GradedAlgebra.ι Q) m = (algebraMap R (⨁ (i : ZMod 2), ↥(evenOdd Q i))) (Q m)"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "∀ (i : 𝒰.J), CategoryTheory.Limits.HasPullback (𝒰.map i ≫ f) g"], "goal": "AlgebraicGeometry.Scheme.Pullback.p1 𝒰 f g ≫ f = AlgebraicGeometry.Scheme.Pullback.p2 𝒰 f g ≫ g"}}
+{"state": {"context": ["α : Type u", "Order.Frame α", "s : Set α", "a : α"], "goal": "a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "DiscreteTopology α", "f : C(α, β)", "a : α"], "goal": "(ContinuousMap.equivFnOfDiscrete α β) f a = f a"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "LinearOrder α", "OrderTopology α", "NoMaxOrder α", "DenselyOrdered α", "a : α", "s : Set α"], "goal": "s ∈ 𝓝[>] a ↔ ∃ u ∈ Set.Ioi a, Set.Ioc a u ⊆ s"}}
+{"state": {"context": ["M : Type u_2", "AddZeroClass M", "S₁ : AddSubmonoid M", "S₂ : AddSubmonoid Mᵃᵒᵖ"], "goal": "S₁.op ≤ S₂ ↔ S₁ ≤ S₂.unop"}}
+{"state": {"context": ["R : Type u_1", "p m n : ℕ", "inst✝² : CommSemiring R", "inst✝¹ : ExpChar R p", "inst✝ : PerfectRing R p", "x : R"], "goal": "(iterateFrobeniusEquiv R p 1) x = x ^ p"}}
+{"state": {"context": ["q : ℍ"], "goal": "‖NormedSpace.exp ℝ q‖ = ‖NormedSpace.exp ℝ q.re‖"}}
+{"state": {"context": ["α : Type u_1", "f : α → ENNReal", "s : Finset α", "h : ∑ a ∈ s, f a = 1", "h' : ∀ a ∉ s, f a = 0", "a : α"], "goal": "(PMF.ofFinset f s h h') a = f a"}}
+{"state": {"context": ["M : Type u_6", "CommGroup M", "f : ℕ → M", "n : ℕ"], "goal": "∏ i ∈ Finset.range n, f (i + 1) / f i = f n / f 0"}}
+{"state": {"context": ["K : SSet", "Δ : SimplexCategoryᵒᵖ", "x : (K ⊗ 𝟙_ SSet).obj Δ"], "goal": "(ρ_ K).hom.app Δ x = x.1"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s' : Set α", "x : α", "p : Filter ι", "p' : Filter α", "g : ι → α", "inst✝ : TopologicalSpace α", "hx : x ∈ s", "L : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[s] x, ∃ F, ContinuousWithinAt F s x ∧ ∀ y ∈ t, (f y, F y) ∈ u", "u₀ : Set (β × β)", "hu₀ : u₀ ∈ 𝓤 β", "u₁ : Set (β × β)", "h₁ : u₁ ∈ 𝓤 β", "u₁₀ : u₁ ○ u₁ ⊆ u₀", "u₂ : Set (β × β)", "h₂ : u₂ ∈ 𝓤 β", "hsymm : ∀ {a b : β}, (a, b) ∈ u₂ → (b, a) ∈ u₂", "u₂₁ : u₂ ○ u₂ ⊆ u₁"], "goal": "u₀ ∈ map (fun x_1 => (f x_1, f x)) (𝓝[s] x)"}}
+{"state": {"context": ["n : ℕ", "m : ℤ", "hnpos : 0 < n", "N : ℤ", "P : 1 ≠ 0", "C : N.natAbs.Coprime 1", "hxr : m * ↑1 ^ n = N ^ n", "hv : ¬∃ y, ↑{ num := N, den := 1, den_nz := P, reduced := C } = ↑y", "c1 : ↑↑1 ≠ 0", "c2 : ↑↑1 ^ n ≠ 0", "hdivn : 1 ∣ N.natAbs"], "goal": "False"}}
+{"state": {"context": ["E : Type u", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℂ E", "F : Type v", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℂ F", "inst✝ : CompleteSpace F", "f : ℂ → F", "z w : ℂ", "r : ℝ := dist w z", "hd : DiffContOnCl ℂ f (ball z r)", "hz : IsMaxOn (norm ∘ f) (closedBall z r) z", "hw : w ∈ closedBall z r", "hw_lt : ‖f w‖ < ‖f z‖", "hr : 0 < r", "hsub : sphere z r ⊆ closedBall z r"], "goal": "‖(w - z)⁻¹ • f w‖ < ‖f z‖ / r"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Ring R", "Ring S", "f : R →+* S", "hf : Function.Injective ⇑f", "s : Subring R"], "goal": "Subring.comap f (Subring.map f s) = s"}}
+{"state": {"context": ["α : Type u_1", "self : Matroid α", "I : Set α"], "goal": "self.Indep I ↔ ∃ B, self.Base B ∧ I ⊆ B"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "S : CategoryTheory.Subgroupoid C"], "goal": "S.IsThin ↔ ∀ (c : ↑S.objs), Subsingleton ↑(S.arrows ↑c ↑c)"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "X Y Z : C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : HasPullbacks C", "f : X ⟶ Y", "g : Y ⟶ Z", "t : MonoOver Z"], "goal": "(pullback (f ≫ g)).obj (Quotient.mk'' t) = (pullback f).obj ((pullback g).obj (Quotient.mk'' t))"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "CategoryTheory.Category.{v, u} C", "AddCommMonoid A", "CategoryTheory.HasShift C A", "X Y : C", "f : X ⟶ Y", "i j : A"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.shiftComm X i j).hom\n    ((CategoryTheory.shiftFunctor C i).map ((CategoryTheory.shiftFunctor C j).map f)) =\n  CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor C j).map ((CategoryTheory.shiftFunctor C i).map f))\n    (CategoryTheory.shiftComm Y i j).hom"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s : Finset α", "x y : α", "a : α × α", "ha₁ : a.1 ∈ s", "ha₂ : a.2 ∈ s", "ha : a.1 ≠ a.2", "h : Rel α a (x, y)", "hx : x ∈ s", "hy : y ∈ s", "hxy : x ≠ y", "hxy' : y ≠ x", "this : filter (fun z => Sym2.mk z = s(x, y)) s.offDiag = {(x, y), (y, x)}"], "goal": "x = y → ¬y = x"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "E : Type u₃", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Category.{v₃, u₃} E", "F : C × D ⥤ E", "W : C", "X Y Z : D", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "F.map (𝟙 W, f ≫ g) = F.map (𝟙 W, f) ≫ F.map (𝟙 W, g)"}}
+{"state": {"context": ["R : Type u_2", "V : Type u_3", "W : Type u_4", "P : Type u_6", "NormedAddCommGroup V", "MetricSpace P", "NormedAddTorsor V P", "NormedAddCommGroup W", "NormedField R", "NormedSpace R V", "NormedSpace R W", "t : R", "f : P →ᴬ[R] W"], "goal": "(t • f).contLinear = t • f.contLinear"}}
+{"state": {"context": ["R : Type u_1", "Add R", "PartialOrder R", "a : R", "ha : AddLECancellable a"], "goal": "IsAddLeftRegular a"}}
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+{"state": {"context": ["a b c : Int", "h : a < b + c"], "goal": "-b + a < c"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "s : Set P", "p : P", "hp : p ∈ s"], "goal": "vectorSpan k s = Submodule.span k ((fun x => x -ᵥ p) '' s)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "I : FractionalIdeal S P", "n : ℕ"], "goal": "↑(I ^ n) = ↑I ^ n"}}
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+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "F : Sym2 α → Sym2 β → γ", "a₁ a₂ : α", "b₁ b₂ : β"], "goal": "↑(Sym2.lift₂.symm F) a₁ a₂ b₁ b₂ = F s(a₁, a₂) s(b₁, b₂)"}}
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+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F : Type u_4", "Norm E", "Norm F", "f : α → E", "g : α → F", "l : Filter α"], "goal": "f =O[l] g → ∃ c, Asymptotics.IsBigOWith c l f g"}}
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+{"state": {"context": ["α✝ : Type u_1", "β✝ : Type u_2", "γ : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝¹ : Fintype α", "inst✝ : Fintype β", "f : α ≃ β"], "goal": "card α = card β"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁹ : CommRing R", "inst✝⁸ : AddCommGroup M", "inst✝⁷ : Module R M", "inst✝⁶ : AddCommGroup N", "inst✝⁵ : Module R N", "inst✝⁴ : CommSemiring S", "inst✝³ : Algebra S R", "inst✝² : Module S N", "inst✝¹ : IsScalarTower S R N", "inst✝ : Invertible 2", "B₁ : BilinMap R M R", "Q✝ : QuadraticMap R M N", "Q : QuadraticForm R M", "hB₁ : associated' Q ≠ 0"], "goal": "¬∀ (x : M), Q x = 0"}}
+{"state": {"context": ["𝕜 : Type u", "ι : Type v", "ι' : Type v'", "n : ℕ", "E : ι → Type wE", "E₁ : ι → Type wE₁", "E' : ι' → Type wE'", "Ei : Fin n.succ → Type wEi", "G : Type wG", "G' : Type wG'", "inst✝¹⁴ : Fintype ι", "inst✝¹³ : Fintype ι'", "inst✝¹² : NontriviallyNormedField 𝕜", "inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)", "inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)", "inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)", "inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)", "inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)", "inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)", "inst✝⁵ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)", "inst✝⁴ : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f : ContinuousMultilinearMap 𝕜 Ei G", "m : (i : Fin n.succ) → Ei i"], "goal": "f.curryRight.uncurryRight m = f m"}}
+{"state": {"context": ["R : Type u_1", "CommMonoid R", "R' : Type u_2", "CommMonoidWithZero R'", "x : R"], "goal": "(MulChar.trivial R R') x = if IsUnit x then 1 else 0"}}
+{"state": {"context": ["α : Type u", "CommSemiring α", "x y : α", "n : ℕ"], "goal": "(∑ i ∈ Finset.range n, (x + y) ^ i * y ^ (n - 1 - i)) * x + y ^ n = (x + y) ^ n"}}
+{"state": {"context": ["m : Type u_2", "o : Type u_4", "α : Type u_12", "DecidableEq o", "DecidableEq m", "Fintype o", "Fintype m", "Semiring α", "M : o → Matrix m m α", "n : ℕ"], "goal": "Matrix.blockDiagonal (M ^ n) = Matrix.blockDiagonal M ^ n"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDedekindDomain R", "S : Type u_2", "CommRing S", "Algebra R S", "Module.Free R S", "Module.Finite R S", "p : Ideal R", "hp0 : p ≠ ⊥", "p.IsPrime", "Sₚ : Type u_3", "CommRing Sₚ", "Algebra S Sₚ", "IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ", "Algebra R Sₚ", "IsScalarTower R S Sₚ", "IsDedekindDomain Sₚ", "IsDomain S", "P : Ideal Sₚ", "hP : P.IsPrime", "hP0 : P ≠ ⊥"], "goal": "P ∈ UniqueFactorizationMonoid.normalizedFactors (Ideal.map (algebraMap R Sₚ) p)"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "α : Type u_11"], "goal": "⋃ x, {x} = univ"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "v : VitaliFamily μ", "E : Type u_2", "NormedAddCommGroup E", "SecondCountableTopology α", "BorelSpace α", "MeasureTheory.IsLocallyFiniteMeasure μ", "NormedSpace ℝ E", "CompleteSpace E", "f : α → E", "hf : MeasureTheory.LocallyIntegrable f μ"], "goal": "∀ᵐ (x : α) ∂μ, Filter.Tendsto (fun a => ⨍ (y : α) in a, f y ∂μ) (v.filterAt x) (𝓝 (f x))"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "TopologicalSpace W", "f : Y × Z → W", "g : X → Y", "h : X → Z", "x : X", "hf : ContinuousAt f (g x, h x)", "hg : ContinuousAt g x", "hh : ContinuousAt h x"], "goal": "ContinuousAt (fun x => f (g x, h x)) x"}}
+{"state": {"context": ["n : ℕ", "hn : n > 0"], "goal": "↑(harmonic (n + 1)) - log ↑(n + 1) < ↑(harmonic n) - log ↑n"}}
+{"state": {"context": ["R : Type u_1", "ι : Type u_2", "inst✝³ : Semiring R", "φ : ι → Type u_3", "inst✝² : (i : ι) → AddCommMonoid (φ i)", "inst✝¹ : (i : ι) → Module R (φ i)", "inst✝ : DecidableEq ι", "i j : ι"], "goal": "proj i ∘ₗ stdBasis R φ j = diag j i"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "s : Set M", "Fintype ↑s", "card_lt : s.toFinset.card < FiniteDimensional.finrank R M"], "goal": "Submodule.span R s < ⊤"}}
+{"state": {"context": ["a b n : ℕ"], "goal": "n.floorRoot a ≠ 0 ↔ n ≠ 0 ∧ a ≠ 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_4", "E : B → Type u_6", "NontriviallyNormedField 𝕜", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "TopologicalSpace (Bundle.TotalSpace F E)", "(x : B) → TopologicalSpace (E x)", "EB : Type u_7", "NormedAddCommGroup EB", "NormedSpace 𝕜 EB", "HB : Type u_8", "TopologicalSpace HB", "IB : ModelWithCorners 𝕜 EB HB", "n : ℕ∞", "TopologicalSpace B", "ChartedSpace HB B", "FiberBundle F E", "s : Set (Bundle.TotalSpace F E)"], "goal": "ContMDiffOn (IB.prod 𝓘(𝕜, F)) IB n Bundle.TotalSpace.proj s"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "E : Type u_2", "AddCommGroup E", "Module R E", "F : Type u_3", "AddCommGroup F", "Module R F", "f g : E →ₗ.[R] F", "h : f.graph = g.graph"], "goal": "f = g"}}
+{"state": {"context": ["H : Type u_1", "H' : Type u_3", "M' : Type u_4", "TopologicalSpace H", "TopologicalSpace H'", "TopologicalSpace M'", "ChartedSpace H' M'", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "f f' : PartialHomeomorph M' H'", "P : (H → H') → Set H → H → Prop", "hG : G.LocalInvariantProp G' P", "g : H → M'", "x : H", "s : Set H", "hf : f ∈ StructureGroupoid.maximalAtlas M' G'", "xf : g x ∈ f.source", "hf' : f' ∈ StructureGroupoid.maximalAtlas M' G'", "xf' : g x ∈ f'.source", "hgs : ContinuousWithinAt g s x"], "goal": "P (↑f ∘ g) s x ↔ P (↑f' ∘ g) s x"}}
+{"state": {"context": ["α : Type u_4", "self : BiheytingAlgebra α", "a b c : α"], "goal": "a \\ b ≤ c ↔ a ≤ b ⊔ c"}}
+{"state": {"context": ["x✝ y x : ℝ", "hx : 0 ≤ x", "n j : ℕ", "hj : j ≥ n"], "goal": "↑(const _root_.abs (∑ i ∈ range n, x ^ i / ↑i !)) j ≤ ↑⟨fun n => (↑(exp' ↑x) n).re, ⋯⟩ j"}}
+{"state": {"context": ["R : Type u", "a b c d : R", "Semiring R"], "goal": "(Polynomial.C a * Polynomial.X ^ 3 + Polynomial.C b * Polynomial.X ^ 2 + Polynomial.C c * Polynomial.X +\n      Polynomial.C d).natDegree ≤\n  3"}}
+{"state": {"context": ["𝕜 : Type u", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "E : Type v", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "F : Type w", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace 𝕜 F", "F' : Type x", "inst✝⁵ : AddCommGroup F'", "inst✝⁴ : Module 𝕜 F'", "inst✝³ : TopologicalSpace F'", "inst✝² : TopologicalAddGroup F'", "inst✝¹ : ContinuousSMul 𝕜 F'", "inst✝ : CompleteSpace 𝕜", "c : 𝕜", "hc : 1 < ‖c‖", "R : ℝ", "hR : ‖c‖ < R", "h : ¬FiniteDimensional 𝕜 E", "this : IsSymm E fun x y => 1 ≤ ‖x - y‖", "s : Finset E"], "goal": "∃ y, ‖y‖ ≤ R ∧ ∀ x ∈ s, 1 ≤ ‖x - y‖"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "inst✝⁵ : HasColimits C", "X Y Z : TopCat", "inst✝⁴ : ConcreteCategory C", "inst✝³ : PreservesFilteredColimits (forget C)", "inst✝² : HasLimits C", "inst✝¹ : PreservesLimits (forget C)", "inst✝ : (forget C).ReflectsIsomorphisms", "F G : Sheaf C X", "f : F ⟶ G", "U : Opens ↑X", "hinj : ∀ (x : ↥U), Function.Injective ⇑((stalkFunctor C ↑x).map f.val)", "hsurj : ∀ (t : (forget C).obj (G.val.obj (op U))) (x : ↥U), ∃ V, ∃ (_ : ↑x ∈ V), ∃ iVU s, (f.val.app (op V)) s = (G.val.map iVU.op) t", "t : (forget C).obj (G.val.obj (op U))", "V : ↥U → Opens ↑X", "mV : ∀ (x : ↥U), ↑x ∈ V x", "iVU : (x : ↥U) → V x ⟶ U", "sf : (x : ↥U) → (forget C).obj (F.val.obj (op (V x)))", "heq : ∀ (x : ↥U), (f.val.app (op (V x))) (sf x) = (G.val.map (iVU x).op) t", "V_cover : U ≤ iSup V", "x y : ↥U"], "goal": "∀ (x_1 : ↥(V x ⊓ V y)), (F.presheaf.germ x_1) ((F.val.map ((V x).infLELeft (V y)).op) (sf x)) = (F.presheaf.germ x_1) ((F.val.map ((V x).infLERight (V y)).op) (sf y))"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "π : ι → Set (Set α)", "S T : Set ι", "hST : S ⊆ T"], "goal": "piiUnionInter π S ⊆ piiUnionInter π T"}}
+{"state": {"context": ["G : Type u_1", "Group G", "N : Type u_5", "Group N", "f : G →* N", "H : Subgroup N"], "goal": "Subgroup.map f (Subgroup.comap f H) = f.range ⊓ H"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β"], "goal": "Set.graphOn f ∅ = ∅"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Semiring R", "AddCommMonoid M", "Module R M", "f g : ArithmeticFunction R", "h : ArithmeticFunction M"], "goal": "(f * g) • h = f • g • h"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "A : Type u_3", "S : Type u_4", "inst✝¹ : Group G", "inst✝ : AddGroup A", "s : Set G", "p : (x : G) → x ∈ closure s → Prop", "one : p 1 ⋯", "x : G", "mul_left : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ Submonoid.closure (s ∪ s⁻¹)), p y ⋯ → p (x * y) ⋯", "mul_left_inv : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ Submonoid.closure (s ∪ s⁻¹)), p y ⋯ → p (x⁻¹ * y) ⋯"], "goal": "∀ (h : x ∈ Submonoid.closure (s ∪ s⁻¹)), p x ⋯"}}
+{"state": {"context": ["a✝ b✝ c d✝ m n k : ℕ", "p q : ℕ → Prop", "a b d : ℕ", "hdb : d ∣ b", "h : a < b"], "goal": "a / d < b / d"}}
+{"state": {"context": ["G : Type v", "Group G", "PseudoEMetricSpace G", "IsometricSMul Gᵐᵒᵖ G", "c : G"], "goal": "(IsometryEquiv.divRight c).symm = IsometryEquiv.mulRight c"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "s t : Set α", "inst✝¹ : Preorder α", "inst✝ : NoMaxOrder α", "a : α", "ha : ∀ b ∈ s, (fun x x_1 => x ≤ x_1) b a"], "goal": "Bounded (fun x x_1 => x < x_1) s"}}
+{"state": {"context": [], "goal": "↑1 = 1"}}
+{"state": {"context": ["C : Type u_1", "ι : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Abelian C", "c : ComplexShape ι", "S : CategoryTheory.ShortComplex (HomologicalComplex C c)", "hS : S.ShortExact", "i j : ι", "hij : c.Rel i j", "A : C", "x₃ : A ⟶ S.X₃.X i", "hx₃ : x₃ ≫ S.X₃.d i j = 0", "x₂ : A ⟶ S.X₂.X i", "hx₂ : x₂ ≫ S.g.f i = x₃", "x₁ : A ⟶ S.X₁.X j", "hx₁ : x₁ ≫ S.f.f j = x₂ ≫ S.X₂.d i j", "k : ι", "hk : c.next j = k"], "goal": "S.X₃.liftCycles x₃ j ⋯ hx₃ ≫ S.X₃.homologyπ i ≫ hS.δ i j hij = S.X₁.liftCycles x₁ k hk ⋯ ≫ S.X₁.homologyπ j"}}
+{"state": {"context": ["n : ℕ"], "goal": "Fintype.card (Fin n) = n"}}
+{"state": {"context": ["R : Type uR", "M : Type uM", "CommRing R", "AddCommGroup M", "Module R M", "Q : QuadraticForm R M", "A : Type uA", "Ring A", "Algebra R A", "f : CliffordAlgebra.EvenHom Q A"], "goal": "∀ (a : ↥(CliffordAlgebra.even Q)),\n  (CliffordAlgebra.even.lift.aux f) a = (((CliffordAlgebra.foldr Q (CliffordAlgebra.even.lift.fFold f) ⋯) (1, 0)) ↑a).1"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "inst✝⁴ : MetricSpace α", "inst✝³ : MeasurableSpace α", "inst✝² : OpensMeasurableSpace α", "inst✝¹ : SecondCountableTopology α", "μ : Measure α", "inst✝ : IsLocallyFiniteMeasure μ", "s : Set α", "t : Set ι", "C : ℝ≥0", "r : ι → ℝ", "c : ι → α", "B : ι → Set α", "hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)", "μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)", "ht : ∀ a ∈ t, (interior (B a)).Nonempty", "h't : ∀ a ∈ t, IsClosed (B a)", "hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x", "R : α → ℝ", "hR0 : ∀ (x : α), 0 < R x", "hR1 : ∀ (x : α), R x ≤ 1", "hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ⊤", "t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}", "u : Set ι", "ut' : u ⊆ t'", "u_disj : u.PairwiseDisjoint B", "hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b"], "goal": "∃ u ⊆ t, u.Countable ∧ u.PairwiseDisjoint B ∧ μ (s \\ ⋃ a ∈ u, B a) = 0"}}
+{"state": {"context": ["α : Type u", "inst✝² : UniformSpace α", "β : Type v", "γ : Type w", "inst✝¹ : UniformSpace β", "inst✝ : UniformSpace γ", "f : Set (α × α) → Set (CauchyFilter α × CauchyFilter α) := fun s => {p | s ∈ ↑p.2 ×ˢ ↑p.1}", "h₁ : map Prod.swap ((𝓤 α).lift' gen) = (𝓤 α).lift' f", "h₂ : (𝓤 α).lift' f ≤ (𝓤 α).lift' gen"], "goal": "map Prod.swap ((𝓤 α).lift' gen) ≤ (𝓤 α).lift' gen"}}
+{"state": {"context": ["α β : Sort u_4", "p : α → Prop", "q : β → Prop", "a : { x // p x }", "b : { y // q y }", "h : α = β", "h' : HEq p q"], "goal": "HEq a b ↔ HEq ↑a ↑b"}}
+{"state": {"context": ["a : ℝ"], "goal": "¬MeasureTheory.IntegrableOn (fun x => x⁻¹) (Set.Ici a) MeasureTheory.volume"}}
+{"state": {"context": ["α : Type u_1", "a : α", "s t : Multiset α"], "goal": "a ::ₘ s ⊆ t ↔ a ∈ t ∧ s ⊆ t"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommSemiring R", "CommRing S", "Algebra R S", "S' : Subalgebra R S", "s : Set S", "l : ↑s →₀ S", "hs : (Finsupp.total (↑s) S S Subtype.val) l = 1", "hs' : s ⊆ ↑S'", "hl : ∀ (i : ↑s), l i ∈ S'", "x : S", "H : ∀ (r : ↑s), ∃ n, ↑r ^ n • x ∈ S'"], "goal": "x ∈ S'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β"], "goal": "(Setoid.ker f).classes ⊆ Set.range fun y => {x | f x = y}"}}
+{"state": {"context": ["α : Type u_1", "E B I : Set α", "hIE : I ⊆ E"], "goal": "(Matroid.uniqueBaseOn I E).Base B ↔ B = I"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s : Finset α", "t : Finset β"], "goal": "_root_.Disjoint (map Embedding.inl s) (map Embedding.inr t)"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "inst✝⁴ : RCLike 𝕜", "E : Type u_3", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "cplt : CompleteSpace E", "G : ι → Type u_4", "inst✝¹ : (i : ι) → NormedAddCommGroup (G i)", "inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)", "V : (i : ι) → G i →ₗᵢ[𝕜] E", "hV : OrthogonalFamily 𝕜 G V", "i : ι", "x : G i"], "goal": "hV.linearIsometry (lp.single 2 i x) = (V i) x"}}
+{"state": {"context": ["x : ℂ"], "goal": "-π < x.arg"}}
+{"state": {"context": ["b o : Ordinal.{u}", "hb : 1 < b", "x : Ordinal.{u} × Ordinal.{u}"], "goal": "x ∈ Ordinal.CNF b o → x.2 < b"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f : ℕ → Ω → ℝ", "N n m : ℕ", "ω : Ω", "ℱ : Filtration ℕ m0", "inst✝ : IsFiniteMeasure μ", "hf : Submartingale f ℱ μ", "hfN : ∀ (ω : Ω), a ≤ f N ω", "hfzero : 0 ≤ f 0", "hab : a < b"], "goal": "(b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (∑ k ∈ Finset.range N, upcrossingStrat a b f N k * (f (k + 1) - f k)) x ∂μ"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H K L : AddSubgroup G", "hHK : H ≤ K", "hKL : K ≤ L"], "goal": "H.relindex K * K.relindex L = H.relindex L"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "G : K ⥤ C", "e : K ≌ J", "α : e.functor ⋙ F ≅ G"], "goal": "(CategoryTheory.Limits.Cones.equivalenceOfReindexing e α).inverse =\n  CategoryTheory.Limits.Cones.postcompose α.inv ⋙\n    CategoryTheory.Limits.Cones.whiskering e.inverse ⋙ CategoryTheory.Limits.Cones.postcompose (e.invFunIdAssoc F).hom"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "I J : Ideal R"], "goal": "PrimeSpectrum.zeroLocus ↑(I ⊔ J) = PrimeSpectrum.zeroLocus ↑I ∩ PrimeSpectrum.zeroLocus ↑J"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedAddCommGroup β", "NormedAddCommGroup γ", "K K' : ℝ≥0", "f : α → β", "g : β → γ", "hg : LipschitzWith K g", "hg' : AntilipschitzWith K' g", "g0 : g 0 = 0"], "goal": "MeasureTheory.Integrable (g ∘ f) μ ↔ MeasureTheory.Integrable f μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "s t : Finset α", "a b : α", "n : ℕ", "h : a ∉ s", "m : { x // x ∈ (insert a s).sym n }"], "goal": "(Sym.filterNe a ↑m).snd ∈ s.sym (n - ↑(Sym.filterNe a ↑m).fst)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace ℝ E", "n : ℕ", "_i : Fact (finrank ℝ E = n + 1)", "o : Orientation ℝ E (Fin (n + 1))", "e : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o"], "goal": "(-o).volumeForm = -o.volumeForm"}}
+{"state": {"context": ["Ω : Type u_1", "E : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : CompleteSpace E", "f✝ : ℕ → Ω → E", "ℱ : Filtration ℕ m0", "n✝ : ℕ", "inst✝ : SigmaFiniteFiltration μ ℱ", "f g : ℕ → Ω → E", "hf : Martingale f ℱ μ", "hg : Adapted ℱ fun n => g (n + 1)", "hg0 : g 0 = 0", "hgint : ∀ (n : ℕ), Integrable (g n) μ", "n : ℕ", "h : ℕ → Ω → E := f - martingalePart (f + g) ℱ μ", "hhdef : h = f - martingalePart (f + g) ℱ μ", "hh : h = predictablePart (f + g) ℱ μ - g", "hhpred : Adapted ℱ fun n => h (n + 1)"], "goal": "martingalePart (f + g) ℱ μ n =ᶠ[ae μ] f n"}}
+{"state": {"context": ["G : Type u_1", "Group G", "IsCoatomic (Subgroup G)", "K : Subgroup G", "h : K ⊔ frattini G = ⊤"], "goal": "K = ⊤"}}
+{"state": {"context": ["R : Type u_1", "P : Cubic R", "Semiring R", "ha : P.a = 0"], "goal": "P.toPoly.degree ≤ 2"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "a b : α", "f : α ↪o β", "hab : a ⩿ b", "h : (Set.range ⇑f).OrdConnected"], "goal": "f a ⩿ f b"}}
+{"state": {"context": ["G : Type u", "inst✝ : AddCommGroup G", "hG : AddGroup.FG G", "n : ℕ", "ι : Type", "fι : Fintype ι", "p : ι → ℤ", "hp : ∀ (i : ι), Irreducible (p i)", "e : ι → ℕ", "f : G ≃ₗ[ℤ] (Fin n →₀ ℤ) × ⨁ (i : ι), ℤ ⧸ Submodule.span ℤ {p i ^ e i}"], "goal": "∃ n ι x p, ∃ (_ : ∀ (i : ι), Nat.Prime (p i)), ∃ e, Nonempty (G ≃+ (Fin n →₀ ℤ) × ⨁ (i : ι), ZMod (p i ^ e i))"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_4", "AddZeroClass M", "s t : Set α", "a : α", "h : a ∉ s ∩ t", "f : α → M"], "goal": "(s ∪ t).indicator f a = s.indicator f a + t.indicator f a"}}
+{"state": {"context": ["n b a : Nat"], "goal": "a < n → b % n < n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "E : Type u_3", "Norm E", "F : Type u_4", "Norm F", "e : α ≃ₜ β", "b : β", "f : β → E", "g : β → F"], "goal": "f =O[𝓝 b] g ↔ (f ∘ ⇑e) =O[𝓝 (e.symm b)] (g ∘ ⇑e)"}}
+{"state": {"context": ["a b c d m n✝ k : ℕ", "p q : ℕ → Prop", "n : ℕ"], "goal": "(1 + n).pred = n"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "m : Ordnode α", "y : α", "r : Ordnode α"], "goal": "(l.node3L x m y r).dual = r.dual.node3R y m.dual x l.dual"}}
+{"state": {"context": ["α : Type u_1"], "goal": "∀ (a : α), Function.Embedding.coeWithTop a = ↑a"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : Fintype Ω", "s : Set Ω"], "goal": "MeasurableSet ↑Finset.univ"}}
+{"state": {"context": ["m : Type u", "n : Type u'", "α : Type v", "Fintype n", "DecidableEq n", "CommRing α", "A : Matrix n n α", "Invertible A"], "goal": "Function.Injective fun x => A * x"}}
+{"state": {"context": ["α : Type u", "ι : Type x", "PseudoEMetricSpace α", "f : ι → Function.End α", "K : ι → ℝ≥0", "h : ∀ (i : ι), LipschitzWith (K i) (f i)", "l : List ι"], "goal": "LipschitzWith (List.map K l).prod (List.map f l).prod"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type v", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "CompleteSpace 𝕜", "r : ℝ", "rpos : 0 < r", "h : IsCompact (Metric.closedBall 0 r)"], "goal": "FiniteDimensional 𝕜 E"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "RCLike 𝕂", "NormedRing 𝔸", "NormedAlgebra 𝕂 𝔸"], "goal": "0 < (NormedSpace.expSeries 𝕂 𝔸).radius"}}
+{"state": {"context": ["𝕜 : Type u_1", "X : Type u_2", "inst✝² : RCLike 𝕜", "inst✝¹ : TopologicalSpace X", "inst✝ : CompactSpace X", "A : StarSubalgebra 𝕜 C(X, 𝕜)", "hA : A.SeparatesPoints", "I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM", "key : LinearMap.range I ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSubmodule A.toSubalgebra)).topologicalClosure", "f : C(X, 𝕜)", "f_re : C(X, ℝ) := { toFun := ⇑re, continuous_toFun := ⋯ }.comp f", "f_im : C(X, ℝ) := { toFun := ⇑im, continuous_toFun := ⋯ }.comp f"], "goal": "f ∈ A.topologicalClosure"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "TopologicalSpace α", "OrderTopology α", "a : α", "(𝓝[<] a).NeBot"], "goal": "(𝓝ˢ (Set.Ici a)).HasBasis (fun x => x < a) Set.Ici"}}
+{"state": {"context": ["α : Type u_1", "p : α → Bool", "l : List α", "n : ℕ", "x : α", "hx : x ∈ takeWhile p l.reverse"], "goal": "p x = true"}}
+{"state": {"context": ["M : Type u_6", "N : Type u_7", "Mul M", "Mul N", "f : M ≃* N", "x y : M"], "goal": "f (x * y) = f x * f y"}}
+{"state": {"context": ["b : String"], "goal": "Levenshtein.stringLengthCost.insert b = b.length"}}
+{"state": {"context": ["J : Type u₁", "CategoryTheory.Category.{v₁, u₁} J", "K : Type u₂", "CategoryTheory.Category.{v₂, u₂} K", "C : Type u₃", "CategoryTheory.Category.{v₃, u₃} C", "F : J ⥤ C", "s : CategoryTheory.Limits.Cone F", "G : K ⥤ C", "t : CategoryTheory.Limits.Cone G", "P : CategoryTheory.Limits.IsLimit s", "Q : CategoryTheory.Limits.IsLimit t", "e : J ≌ K", "w : e.functor ⋙ G ≅ F"], "goal": "(P.conePointsIsoOfEquivalence Q e w).inv =\n  P.lift ((CategoryTheory.Limits.Cones.equivalenceOfReindexing e w).functor.obj t)"}}
+{"state": {"context": ["α : Type u", "Semiring α", "PartialOrder α", "a b : α", "PosMulStrictMono α", "MulPosMono α", "h1 : 0 ≤ a", "h2 : a < b"], "goal": "a * a < b * b"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "U : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ", "a b : (x : ↥(unop U)) → at ↑x", "ha : (isLocallyFraction 𝒜).pred a", "hb : (isLocallyFraction 𝒜).pred b", "x : ↥(unop U)", "Va : Opens ↑(ProjectiveSpectrum.top 𝒜)", "ma : ↑x ∈ Va", "ia : Va ⟶ unop U", "ja : ℕ", "ra : A", "ra_mem : ra ∈ 𝒜 ja", "sa : A", "sa_mem : sa ∈ 𝒜 ja", "hwa : ∀ (x : ↥Va), ↑⟨sa, sa_mem⟩ ∉ (↑x).asHomogeneousIdeal", "wa : ∀ (x : ↥Va), (fun x => a ((fun x => ⟨↑x, ⋯⟩) x)) x = HomogeneousLocalization.mk { deg := ja, num := ⟨ra, ra_mem⟩, den := ⟨sa, sa_mem⟩, den_mem := ⋯ }", "Vb : Opens ↑(ProjectiveSpectrum.top 𝒜)", "mb : ↑x ∈ Vb", "ib : Vb ⟶ unop U", "jb : ℕ", "rb : A", "rb_mem : rb ∈ 𝒜 jb", "sb : A", "sb_mem : sb ∈ 𝒜 jb", "hwb : ∀ (x : ↥Vb), ↑⟨sb, sb_mem⟩ ∉ (↑x).asHomogeneousIdeal", "wb : ∀ (x : ↥Vb), (fun x => b ((fun x => ⟨↑x, ⋯⟩) x)) x = HomogeneousLocalization.mk { deg := jb, num := ⟨rb, rb_mem⟩, den := ⟨sb, sb_mem⟩, den_mem := ⋯ }"], "goal": "∃ V, ∃ (_ : ↑x ∈ V), ∃ i, (isFractionPrelocal 𝒜).pred fun x => (a + b) ((fun x => ⟨↑x, ⋯⟩) x)"}}
+{"state": {"context": ["Ω : Type u_1", "mΩ : MeasurableSpace Ω", "μ : Measure Ω", "f g : Ω → ℝ≥0∞", "X✝ Y✝ : Ω → ℝ", "β : Type u_2", "inst✝² : MeasurableSpace β", "X Y : Ω → β", "inst✝¹ : NormedDivisionRing β", "inst✝ : BorelSpace β", "hXY : IndepFun X Y μ", "h'XY : Integrable (X * Y) μ", "hX : AEStronglyMeasurable X μ", "hY : AEStronglyMeasurable Y μ", "h'X : ¬X =ᶠ[ae μ] 0", "I : ∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ ≠ 0"], "goal": "HasFiniteIntegral Y μ"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : OrderedAddCommGroup α", "a b c : α"], "goal": "(fun x => x - a) ⁻¹' Ico b c = Ico (b + a) (c + a)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "c : ComplexShape ι", "c' : ComplexShape ι'", "C : Type u_3", "CategoryTheory.Category.{u_4, u_3} C", "CategoryTheory.Limits.HasZeroMorphisms C", "K : HomologicalComplex C c'", "e : c.Embedding c'", "e.IsTruncGE", "∀ (i' : ι'), K.HasHomology i'", "i j : ι", "hij : c.Rel i j", "i' j' : ι'", "hi' : e.f i = i'", "hj' : e.f j = j'", "hi : e.BoundaryGE i"], "goal": "(K.truncGE' e).d i j = (K.truncGE'XIsoOpcycles e hi' hi).hom ≫ K.fromOpcycles i' j' ≫ (K.truncGE'XIso e hj' ⋯).inv"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "t : Set α", "p : α → Prop", "ht : ∀ᵐ (x : α) ∂μ.restrict t, p x", "htc : ∀ᵐ (x : α) ∂μ.restrict tᶜ, p x"], "goal": "∀ᵐ (x : α) ∂μ, p x"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "B A₁ A₂ : C", "f : A₁ ⟶ B", "g : A₂ ⟶ B", "CategoryTheory.Mono f", "CategoryTheory.Mono g", "h : CategoryTheory.Subobject.mk f ≤ CategoryTheory.Subobject.mk g"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.Subobject.ofMkLEMk f g h) g = f"}}
+{"state": {"context": ["w : Nat", "α : Type u_1", "f : Fin w → α → α × Bool", "state : Nat → α", "value : BitVec w", "a : α", "init : state 0 = a", "step : ∀ (i : Fin w), f i (state ↑i) = (state (↑i + 1), value.getLsb ↑i)"], "goal": "BitVec.iunfoldr f a = (state w, BitVec.truncate w value)"}}
+{"state": {"context": ["X : AlgebraicGeometry.LocallyRingedSpace", "r : ↑(AlgebraicGeometry.LocallyRingedSpace.Γ.obj (Opposite.op X))"], "goal": "X.toΓSpecFun ⁻¹' (PrimeSpectrum.basicOpen r).carrier = (X.toRingedSpace.basicOpen r).carrier"}}
+{"state": {"context": ["n : ℕ", "a : Composition n", "b : Composition a.length", "i : Fin b.length"], "goal": "(a.sigmaCompositionAux b ⟨↑i, ⋯⟩).length = b.blocksFun i"}}
+{"state": {"context": ["δ' : Type u_5", "π : δ' → Type u_6", "(x : δ') → MeasurableSpace (π x)", "DecidableEq δ'", "l : List δ'", "hnd : l.Nodup", "h : ∀ (i : δ'), i ∈ l"], "goal": "⇑(MeasurableEquiv.piMeasurableEquivTProd hnd h).symm = List.TProd.elim' h"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "s : Finset { x // p x }", "a : α", "h : ¬p a"], "goal": "a ∉ Finset.map (Function.Embedding.subtype fun x => p x) s"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : DecidableEq β", "inst✝¹ : Group α", "inst✝ : MulAction α β", "s t : Finset β", "a : α", "b : β"], "goal": "↑s ⊆ a • ↑t ↔ a⁻¹ • ↑s ⊆ ↑t"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g f' g' : X ⟶ Y", "α : CategoryTheory.Limits.parallelPair f g ⟶ CategoryTheory.Limits.parallelPair f' g'", "c : CategoryTheory.Limits.Fork f g"], "goal": "CategoryTheory.Limits.Fork.ι ((CategoryTheory.Limits.Cones.postcompose α).obj c) =\n  CategoryTheory.CategoryStruct.comp c.ι (α.app CategoryTheory.Limits.WalkingParallelPair.zero)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s t : Finset α", "x : α × α"], "goal": "s.diag.card = s.card"}}
+{"state": {"context": ["α : Type u_2", "HeytingAlgebra α", "a b : α"], "goal": "(a ⊔ b)ᶜ = aᶜ ⊓ bᶜ"}}
+{"state": {"context": ["B : Type u₁", "inst✝⁴ : Category.{v₁, u₁} B", "C : Type u₂", "inst✝³ : Category.{v₂, u₂} C", "D : Type u₃", "inst✝² : Category.{v₃, u₃} D", "E : Type u₄", "inst✝¹ : Category.{v₄, u₄} E", "H : Type u₅", "inst✝ : Category.{v₅, u₅} H", "F₁ F₂ : C × D ⥤ E", "h : curry.obj F₁ = curry.obj F₂"], "goal": "F₁ = F₂"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "s✝ s₁ s₂ : Finset α", "a✝ : α", "f✝ g : α → β", "inst✝² : CommMonoid β", "inst✝¹ : DecidableEq α", "M : Type u_6", "inst✝ : CommMonoid M", "f : α → M", "a : α", "s : List α", "IH : (List.map f s).prod = ∏ m ∈ s.toFinset, f m ^ count m s", "has : a ∉ s.toFinset"], "goal": "f a * ∏ m ∈ s.toFinset, f m ^ count m s = ∏ x ∈ insert a s.toFinset, f x ^ count x (a :: s)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝³ : TopologicalSpace α", "s t u v : Set α", "inst✝² : TopologicalSpace β", "inst✝¹ : TotallyDisconnectedSpace β", "f✝ : α → β", "S : Set α", "hS : IsPreconnected S", "T : Set β", "inst✝ : DiscreteTopology ↑T", "f : α → β", "hc : ContinuousOn f S", "hTm : MapsTo f S T", "x y : α", "hx : x ∈ S", "hy : y ∈ S", "F : ↑S → ↑T := MapsTo.restrict f S T hTm"], "goal": "f x = f y"}}
+{"state": {"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "h : 1 < finrank ℚ K", "x : 𝓞 K", "h_nz : x ≠ 0", "h_bd : ↑|(Algebra.norm ℚ) ↑x| ≤ (4 / π) ^ NrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|", "h_nm : 1 ≤ ↑|(Algebra.norm ℚ) ↑x|"], "goal": "4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤ ↑|discr K|"}}
+{"state": {"context": ["a : Prop", "h : (¬a) = True"], "goal": "a = False"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝¹ : Field K", "inst✝ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "hcycl : IsCyclotomicExtension {p ^ k} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ k)", "x : K", "h : IsIntegral ℤ x"], "goal": "x ∈ adjoin ℤ {ζ}"}}
+{"state": {"context": [], "goal": "π ^ 4 / 90 = -(8 * π ^ 4 * ↑(bernoulli 4)) / ↑4!"}}
+{"state": {"context": ["D : GlueData", "α : Type u", "inst✝ : TopologicalSpace α", "J : Type u", "U : J → Opens α", "s : Set ↑(ofOpenSubsets U).glued", "hs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑((ofOpenSubsets U).ι i) ⁻¹' s)"], "goal": "∀ x ∈ ⇑(fromOpenSubsetsGlue U) '' s, ∃ t ⊆ ⇑(fromOpenSubsetsGlue U) '' s, IsOpen t ∧ x ∈ t"}}
+{"state": {"context": ["α : Type u_2", "LinearOrder α", "s : Finset α"], "goal": "s.max = ⊥ ↔ s = ∅"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Monoid R", "S : Submonoid R", "inst✝¹ : OreSet S", "X : Type u_2", "inst✝ : MulAction R X", "r₁ : R", "s₁ : ↥S", "r₂ : R", "s₂ : ↥S", "r₃ : X", "s₃ : ↥S", "ra : R", "sa : ↥S", "ha : ↑sa * r₁ = ra * ↑s₂", "rb : R", "sb : ↥S", "hb : ↑sb * r₂ = rb * ↑s₃", "rc : R", "sc : ↥S", "hc : ↑sc * ra = rc * ↑sb"], "goal": "(r₁ /ₒ s₁) • (rb • r₃ /ₒ (sb * s₂)) = rc • rb • r₃ /ₒ (sc * sa * s₁)"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y : ℝ", "hfc : StrictConcaveOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hfd : DifferentiableWithinAt ℝ f S y"], "goal": "derivWithin f S y < slope f x y"}}
+{"state": {"context": ["m n k : ℕ"], "goal": "↑m >>> (↑n + ↑k) = ↑m >>> ↑n >>> ↑k"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α", "h : a ≤ b"], "goal": "Set.Iio a ⊆ Set.Iic b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : MeasurableSpace α", "inst✝² : MeasurableSpace β", "inst✝¹ : MeasurableSpace γ", "inst✝ : MeasurableSpace δ", "μa : Measure α", "μb : Measure β", "μc : Measure γ", "μd : Measure δ", "f : α → β", "hf : MeasurePreserving f μa μb", "s : Set β", "hs : MeasurableSet s"], "goal": "map f (μa.restrict (f ⁻¹' s)) = μb.restrict s"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "β : Type u_3", "AddGroup G", "AddAction G β", "VAdd α β", "VAddCommClass G α β", "a : α", "b : β"], "goal": "AddAction.stabilizer G b ≤ AddAction.stabilizer G (a +ᵥ b)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b : R", "m n✝ : ℕ", "ι : Type y", "inst✝ : Semiring R", "p q r : R[X]", "hp : p.Monic", "n : ℕ"], "goal": "(p ^ n).nextCoeff = n • p.nextCoeff"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "α : Type u'", "m n : ℕ", "h : m = n", "h' : m ≤ n", "φ : L.BoundedFormula α m", "v : α → M", "xs : Fin n → M"], "goal": "(FirstOrder.Language.BoundedFormula.castLE h' φ).Realize v xs ↔ φ.Realize v (xs ∘ Fin.cast h)"}}
+{"state": {"context": ["α : Type u", "b : Set (Set α)", "hc : b.Countable"], "goal": "SecondCountableTopology α"}}
+{"state": {"context": ["q : ℚ"], "goal": "sqrt (q * q) = |q|"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "G : Type u_4", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "x : E", "L : Filter E", "f₂ : E → F × G", "f₂' : E →L[𝕜] F × G", "h : HasFDerivAtFilter f₂ f₂' x L"], "goal": "HasFDerivAtFilter (fun x => (f₂ x).1) ((ContinuousLinearMap.fst 𝕜 F G).comp f₂') x L"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Algebra R A", "𝒜 : ℕ → Submodule R A", "inst✝ : GradedAlgebra 𝒜", "f : A", "m : ℕ", "f_deg : f ∈ 𝒜 m", "q : ↑↑(Spec A⁰_ f).toPresheafedSpace", "a b : A", "ha : a ∈ carrier f_deg q", "hb : b ∈ carrier f_deg q", "i : ℕ", "g : ℕ → A⁰_ f := fun j => (m + m).choose j • if h2 : m + m < j then 0 else if h1 : j ≤ m then HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } else HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } * HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }", "j : ℕ", "hj : j ∈ range (m + m + 1)"], "goal": "Localization.mk ((m + m).choose j • ((proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m + m - j))) ⟨f ^ (i + i), ⋯⟩ = HomogeneousLocalization.val (g j)"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝¹¹ : CommRing R✝", "S : Submonoid R✝", "P : Type u_2", "inst✝¹⁰ : CommRing P", "inst✝⁹ : Algebra R✝ P", "loc : IsLocalization S P", "R₁ : Type u_3", "inst✝⁸ : CommRing R₁", "K✝ : Type u_4", "inst✝⁷ : Field K✝", "inst✝⁶ : Algebra R₁ K✝", "inst✝⁵ : IsFractionRing R₁ K✝", "inst✝⁴ : IsDomain R₁", "R : Type u_6", "inst✝³ : CommRing R", "K : Type u_5", "inst✝² : Field K", "inst✝¹ : Algebra R K", "inst✝ : IsFractionRing R K", "I : FractionalIdeal R⁰ K", "hI : I ≠ 0", "a : R", "J : Ideal R", "haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J", "h : Ideal.span {a} = 0"], "goal": "False"}}
+{"state": {"context": ["𝕜 : Type u_1", "LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : StrictConvexOn 𝕜 s f", "x y z : 𝕜", "hx : x ∈ s", "hz : z ∈ s", "hxy : x < y", "hyz : y < z"], "goal": "(z - x) * f y < (z - y) * f x + (y - x) * f z"}}
+{"state": {"context": ["x : ℝ"], "goal": "-deriv arcsin x = -(1 / √(1 - x ^ 2))"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "x : α", "y : ℤ"], "goal": "round (x + ↑y) = round x + y"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝¹ : CommMonoid α", "inst✝ : CommMonoid β", "s t : Multiset α", "a : α", "m✝ : Multiset ι", "f g : ι → α", "m : Multiset α"], "goal": "prod 0 = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "N : Matroid β", "E B : Set α", "h : Set.BijOn f E N.E"], "goal": "(N.comapOn E f).Base B ↔ N.Base (f '' B) ∧ B ⊆ E"}}
+{"state": {"context": ["R : Type u", "Field R", "p q : R[X]"], "goal": "q * p.div q + p.mod q = p"}}
+{"state": {"context": ["K : Type u_1", "R : Type u_2", "inst✝³ : Ring K", "inst✝² : NoZeroDivisors K", "inst✝¹ : Nontrivial K", "G : Subgroup Kˣ", "inst✝ : Fintype ↥G", "n : ℕ := Fintype.card ↥G", "nzero : ↑(Fintype.card ↥G) = 0", "p : ℕ", "char_p : CharP K p", "hd : p ∣ n"], "goal": "False"}}
+{"state": {"context": ["ιa : Type u_1", "ιb : Type u_2", "inst✝¹⁰ : Fintype ιa", "inst✝⁹ : Fintype ιb", "R' : Type u_3", "Mᵢ : Type u_4", "N₁ : Type u_5", "N₂ : Type u_6", "inst✝⁸ : CommSemiring R'", "inst✝⁷ : AddCommGroup N₁", "inst✝⁶ : Module R' N₁", "inst✝⁵ : AddCommGroup N₂", "inst✝⁴ : Module R' N₂", "inst✝³ : AddCommMonoid Mᵢ", "inst✝² : Module R' Mᵢ", "inst✝¹ : DecidableEq ιa", "inst✝ : DecidableEq ιb", "a : MultilinearMap R' (fun x => Mᵢ) N₁", "b : MultilinearMap R' (fun x => Mᵢ) N₂"], "goal": "domCoprod' ((∑ σ : Perm ιa, Perm.sign σ • domDomCongr σ a) ⊗ₜ[R'] ∑ σ : Perm ιb, Perm.sign σ • domDomCongr σ b) = ∑ x : Perm ιa, ∑ x_1 : Perm ιb, Perm.sign x • Perm.sign x_1 • domCoprod' (domDomCongr x a ⊗ₜ[R'] domDomCongr x_1 b)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4"], "goal": "(fun x => !x) ∘ isNone = isSome"}}
+{"state": {"context": [], "goal": "Embedding ENNReal.toEReal"}}
+{"state": {"context": ["α : Type u", "a : α", "s t : Stream'.Seq α"], "goal": "(Stream'.Seq.cons a s).append t = Stream'.Seq.cons a (s.append t)"}}
+{"state": {"context": ["α : Type u", "DivisionMonoid α", "a b c : α", "h : IsUnit c"], "goal": "a = b / c ↔ a * c = b"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "p : MvPolynomial σ R"], "goal": "p.support = ∅ ↔ p = 0"}}
+{"state": {"context": ["z : ℂ", "hz : 1 + -z ∈ slitPlane"], "goal": "log (1 + -z) = -z * ∫ (t : ℝ) in 0 ..1, (1 - t • z)⁻¹"}}
+{"state": {"context": ["X Y Z : SemiNormedGrp", "f : X ⟶ Y", "g : Y ⟶ Z", "w : f ≫ g = 0"], "goal": "‖SemiNormedGrp.explicitCokernelDesc w‖ ≤ ‖g‖"}}
+{"state": {"context": ["p : ℕ+", "k : ℕ", "K : Type u", "inst✝² : Field K", "inst✝¹ : CharZero K", "ζ : K", "hp : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hodd : p ≠ 2", "this : NumberField K := numberField {p ^ (k + 1)} ℚ K"], "goal": "(Algebra.norm ℤ) (hζ.toInteger - 1) = ↑↑p"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "γ : Type u_3", "α : Type u_4", "inst✝ : Infinite α", "this : DecidableEq α := Classical.decEq α", "m n : ℕ", "h : Infinite.natEmbeddingAux α m = Infinite.natEmbeddingAux α n", "hmlen : m ≤ n", "hmn✝ : ¬m = n", "hmn : m < n"], "goal": "False"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "α : X ≅ Y", "f : CategoryTheory.Aut X", "n : ℤ"], "goal": "α.conjAut (f ^ n) = α.conjAut f ^ n"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "F : CategoryTheory.Functor C D", "h : Function.Surjective F.obj"], "goal": "F.EssSurj"}}
+{"state": {"context": ["α : Type u_3", "a : Multiset α", "l : List (Multiset α)"], "goal": "l.sum.Disjoint a ↔ ∀ b ∈ l, b.Disjoint a"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "i : n", "A : ↥(Subgroup.center (Matrix.SpecialLinearGroup n R))"], "goal": "(Matrix.SpecialLinearGroup.center_equiv_rootsOfUnity' i) A = rootsOfUnity.mkOfPowEq (↑↑A i i) ⋯"}}
+{"state": {"context": [], "goal": "Real.sin Real.pi = 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : Preorder α", "l : Ordnode α", "x : α", "o₁ : WithBot α", "o₂ : WithTop α", "hl : Valid' o₁ l ↑x", "rs : ℕ", "rl : Ordnode α", "rx : α", "rr : Ordnode α", "hr : Valid' (↑x) (Ordnode.node rs rl rx rr) o₂", "H1 : ¬l.size + (Ordnode.node rs rl rx rr).size ≤ 1", "H2 : delta * l.size ≤ rl.size + rr.size", "H3 : 2 * (rl.size + rr.size) ≤ 9 * l.size + 3 ∨ rl.size + rr.size ≤ 2", "H3_0 : l.size = 0 → rl.size + rr.size ≤ 2", "H3p : l.size > 0 → 2 * (rl.size + rr.size) ≤ 9 * l.size + 3", "ablem : ∀ {a b : ℕ}, 1 ≤ a → a + b ≤ 2 → b ≤ 1", "hlp : l.size > 0 → ¬rl.size + rr.size ≤ 1"], "goal": "Valid' o₁ (l.rotateL x (Ordnode.node rs rl rx rr)) o₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "n d k N : ℕ", "x : Fin n → ℕ", "hN : 2 ≤ N", "hN' : N ≤ 4096"], "goal": "↑N ≤ rexp 16"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝⁵ : CompleteLattice β", "γ : Type u_3", "mγ : MeasurableSpace γ", "f : α → γ", "g : γ → β", "inst✝⁴ : MeasurableSpace β", "inst✝³ : TopologicalSpace β", "inst✝² : SecondCountableTopology β", "inst✝¹ : OrderClosedTopology β", "inst✝ : OpensMeasurableSpace β", "hg : Measurable g", "hf : AEMeasurable f μ"], "goal": "essSup g (Measure.map f μ) = essSup (g ∘ f) μ"}}
+{"state": {"context": ["R : Type u", "S : Type u₁", "Monoid R", "Monoid S", "φ : R →* S", "A : Type v", "B : Type w", "NonUnitalNonAssocSemiring A", "DistribMulAction R A", "NonUnitalNonAssocSemiring B", "DistribMulAction S B", "f : A →ₛₙₐ[φ] B"], "goal": "f 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ", "f : Fin n → α"], "goal": "(Mathlib.Vector.mOfFn fun i => Part.some (f i)) = Part.some (Mathlib.Vector.ofFn f)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y"], "goal": "Continuous ⇑h"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "p p₀ p₁ p₂ p1 p2 p3 p4 : P", "h : InnerProductGeometry.angle (p2 -ᵥ p3) (p4 -ᵥ p3) = π"], "goal": "∠ p2 p3 p1 + ∠ p4 p3 p1 = π"}}
+{"state": {"context": ["ι : Type u_11", "α : Type u_12", "s : Set ι", "hs : s.Nonempty", "f : ι → Set α", "t : Set α"], "goal": "⋂ i ∈ s, f i ∩ t = (⋂ i ∈ s, f i) ∩ t"}}
+{"state": {"context": ["α : Type u_1", "cmp : α → α → Ordering", "t : Batteries.RBSet α cmp", "cut : α → Ordering", "f : Option α → Option α", "self : t.AlterWF cut f"], "goal": "Batteries.RBNode.WF cmp (Batteries.RBNode.alter cut f t.val)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "κ : ι → Type u_5", "s s₁ s₂ : Finset ι", "i : ι", "a : α", "f g : ι → α", "inst✝ : CommRing α", "n k : ℕ"], "goal": "∏ i ∈ range k, (↑n - ↑i) = ↑(∏ i ∈ range k, (n - i))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "M : Type u_5", "M' : Type u_6", "N : Type u_7", "P : Type u_8", "G : Type u_9", "H : Type u_10", "R : Type u_11", "S : Type u_12", "inst✝ : Zero M", "a : α", "m : M", "f : α →₀ M"], "goal": "(∃ a_1, f a_1 ≠ 0 ∧ { toFun := fun a => (a, f a), inj' := ⋯ } a_1 = (a, m)) ↔ f a = m ∧ m ≠ 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝¹ : CancelCommMonoidWithZero α", "inst✝ : NormalizedGCDMonoid α", "s s₁ : Finset β", "f✝ f g : β → α", "hfg : ∀ a ∈ s₁, f a = g a"], "goal": "s₁.gcd f = s₁.gcd g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "f : ℕ → α → ℝ≥0∞", "F : α → ℝ≥0∞", "hf : ∀ (n : ℕ), AEMeasurable (f n) μ", "h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n => f n x", "h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (F x))", "this : Monotone fun n => ∫⁻ (x : α), f n x ∂μ"], "goal": "F =ᶠ[ae μ] fun a => ⨆ n, f n a"}}
+{"state": {"context": ["i : Fin 2", "hi : i ≠ 0"], "goal": "i = 1"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α", "F : Set α", "x : α", "h : x ∉ F"], "goal": "connectedComponentIn F x = ∅"}}
+{"state": {"context": ["G : Type u", "inst✝¹ : Groupoid G", "inst✝ : IsFreeGroupoid G", "a b : G", "p : a ⟶ b", "X : Type u := FreeGroup (WeaklyConnectedComponent (Generators G))", "f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)", "F : G ⥤ CategoryTheory.SingleObj X := differenceFunctor f", "a✝ b✝ : Generators G", "e✝ : a✝ ⟶ b✝"], "goal": "WeaklyConnectedComponent.mk b✝ = WeaklyConnectedComponent.mk (let_fun this := a✝; this)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t : Set X"], "goal": "(∀ (f : Ultrafilter X), ↑f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x ↑f) ↔ ∀ (f : Ultrafilter X), ↑f ≤ 𝓟 s → ∃ x ∈ s, ↑f ≤ 𝓝 x"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "NormedAddCommGroup E", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "NormedSpace ℝ E", "CompleteSpace E", "f : ↥(MeasureTheory.Lp E 1 μ)"], "goal": "‖MeasureTheory.L1.integral f‖₊ ≤ ‖f‖₊"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "E : Type u_5", "NormedAddCommGroup E", "𝕜 : Type u_6", "NontriviallyNormedField 𝕜", "NormedSpace 𝕜 E", "H : Type u_7", "NormedAddCommGroup H", "NormedSpace 𝕜 H", "φ : α → H", "L : H ≃L[𝕜] E"], "goal": "MeasureTheory.Integrable (fun a => L (φ a)) μ ↔ MeasureTheory.Integrable φ μ"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "β : Type u", "g : F (α ::: β) → β", "a : (P F).A", "f' : (P F).drop.B a ⟹ α", "f : (P F).last.B a → (P F).W α"], "goal": "g (abs ⟨a, splitFun f' fun i => (P F).wRec (fun a f' _f rec => g (abs ⟨a, splitFun f' rec⟩)) (f i)⟩) = g (abs ⟨a, splitFun f' (((P F).wRec fun a f' _f rec => g (abs ⟨a, splitFun f' rec⟩)) ∘ f)⟩)"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "s : Set α"], "goal": "TotallyBounded s ↔ ∀ ε > 0, ∃ t, t.Finite ∧ s ⊆ ⋃ y ∈ t, EMetric.ball y ε"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "f : 𝕜 →ᵃ[𝕜] E"], "goal": "Differentiable 𝕜 ⇑f"}}
+{"state": {"context": ["J : Type w", "inst✝³ : SmallCategory J", "C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasLimitsOfShape J C", "inst✝ : MonoidalCategory C", "X : J ⥤ C", "j : J"], "goal": "(λ_ (limit X)).hom ≫ limit.π X j = (limit.lift ((Functor.const J).obj (𝟙_ C)) { pt := 𝟙_ C, π := { app := fun j => 𝟙 (𝟙_ C), naturality := ⋯ } } ⊗ 𝟙 (limit X)) ≫ ((limit.π (𝟙_ (J ⥤ C)) j ⊗ 𝟙 (limit X)) ≫ (𝟙 (𝟙_ C) ⊗ limit.π X j)) ≫ (λ_ (X.obj j)).hom"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_9", "Semiring R", "AddCommMonoid M", "Module R M", "σ : R →+* R", "RingHomId σ"], "goal": "⇑LinearMap.id' = id"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁶ : LinearOrder α", "inst✝⁵ : ConditionallyCompleteLinearOrder β", "inst✝⁴ : TopologicalSpace β", "inst✝³ : OrderTopology β", "f : α → β", "hf : Monotone f", "x y : α", "inst✝² : TopologicalSpace α", "inst✝¹ : OrderTopology α", "inst✝ : SecondCountableTopology β"], "goal": "{x | ¬ContinuousWithinAt f (Ioi x) x}.Countable"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A", "Star A", "S T : NonUnitalStarSubalgebra R A", "h : ∀ (x : A), x ∈ S ↔ x ∈ T"], "goal": "S = T"}}
+{"state": {"context": ["x✝ y✝ x y : ℝ", "h : 0 < x"], "goal": "x / √x = √x"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : LeftCancelMonoid G", "x✝ y a : G", "m n✝ : ℕ", "x : G", "hx : IsOfFinOrder x", "n : ℕ", "hn : ∃ m, x ^ m = x ^ n"], "goal": "(finEquivPowers x hx).symm ⟨x ^ n, hn⟩ = ⟨n % orderOf x, ⋯⟩"}}
+{"state": {"context": ["C : Type u", "A : Type u_1", "inst✝² : Category.{v, u} C", "inst✝¹ : AddMonoid A", "inst✝ : HasShift C A", "a : A", "X : C"], "goal": "(shiftFunctorAdd C 0 a).hom.app X = eqToHom ⋯ ≫ (shiftFunctor C a).map ((shiftFunctorZero C A).inv.app X)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α", "n : ℤ"], "goal": "¬a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b"}}
+{"state": {"context": ["n : ℕ"], "goal": "Multiset.card (Multiset.range n) = n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "ConditionallyCompleteLinearOrder β", "f : α → β", "TopologicalSpace β", "FirstCountableTopology β", "OrderTopology β", "hf : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) (MeasureTheory.ae μ) f := by isBoundedDefault"], "goal": "μ {y | essSup f μ < f y} = 0"}}
+{"state": {"context": ["n : ℕ"], "goal": "↑(bif (↑n).bodd then 1 else 0) + 2 * (↑n).div2 = ↑n"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_3", "M : Type u_5", "M₂ : Type u_7", "Semiring R", "Semiring R₂", "AddCommMonoid M", "AddCommMonoid M₂", "module_M : Module R M", "module_M₂ : Module R₂ M₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "re₁₂ : RingHomInvPair σ₁₂ σ₂₁", "re₂₁ : RingHomInvPair σ₂₁ σ₁₂", "e : M ≃ₛₗ[σ₁₂] M₂", "p : Submodule R M"], "goal": "Submodule.map (↑e) p = Submodule.comap (↑e.symm) p"}}
+{"state": {"context": ["α : Type u", "β : Type v", "l : α → β", "u : β → α", "Preorder α", "PartialOrder β", "gi : GaloisInsertion l u"], "goal": "Function.Injective u"}}
+{"state": {"context": ["α : Type u_2", "Zero α"], "goal": "Filter.NeBot 0"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "M : Comon_ C", "X✝ Y✝ : (Mon_ Cᵒᵖ)ᵒᵖ", "f : X✝ ⟶ Y✝"], "goal": "f.unop.hom.unop ≫ ((fun A => Mon_OpOpToComon_obj' C (unop A)) Y✝).counit = ((fun A => Mon_OpOpToComon_obj' C (unop A)) X✝).counit"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "AddCommMonoid A", "Module R A", "Coalgebra R A", "a : A"], "goal": "(TensorProduct.assoc R A A A) ((LinearMap.rTensor A Coalgebra.comul) (Coalgebra.comul a)) =\n  (LinearMap.lTensor A Coalgebra.comul) (Coalgebra.comul a)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "R : α → β → Prop", "ca ca' : Computation α", "cb cb' : Computation β", "ha : ca.Equiv ca'", "hb : cb.Equiv cb'"], "goal": "Computation.LiftRel R ca cb ↔ Computation.LiftRel R ca' cb'"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "f : α → β", "s t : Multiset α", "H : s ⊆ t"], "goal": "Multiset.map f s ⊆ Multiset.map f t"}}
+{"state": {"context": ["x y : ℂ", "s1 : cos x = cos ((x + y) / 2) * cos ((x - y) / 2) - sin ((x + y) / 2) * sin ((x - y) / 2)", "s2 : cos ((x + y) / 2 - (x - y) / 2) = cos ((x + y) / 2) * cos ((x - y) / 2) + sin ((x + y) / 2) * sin ((x - y) / 2)"], "goal": "cos x - cos y = -2 * sin ((x + y) / 2) * sin ((x - y) / 2)"}}
+{"state": {"context": ["b o : Ordinal.{u_1}", "hbo : o < b"], "goal": "log b o = 0"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝² : Semiring α", "inst✝¹ : AddCommMonoid β", "inst✝ : Module α β", "a b : α", "s : Set β", "x : β", "hx : x ∈ s"], "goal": "(fun x => (a + b) • x) x ∈ a • s + b • s"}}
+{"state": {"context": ["M : Type u_1", "Mul M", "c : Con M", "C : c.Quotient → Prop", "q : c.Quotient", "H : ∀ (x : M), C ↑x"], "goal": "C q"}}
+{"state": {"context": ["a : ℕ", "a1 : 1 < a"], "goal": "0 < Pell.d a1"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_3", "NormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "A : Set E", "TopologicalSpace E", "ContinuousSMul 𝕜 E", "hA : Balanced 𝕜 A"], "goal": "Balanced 𝕜 (0 ∪ interior A)"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_4", "AddGroup α", "A : Matrix n n α"], "goal": "(-A).IsDiag ↔ A.IsDiag"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "f g : M₁ →SL[σ₁₂] M₂"], "goal": "↑f = ↑g ↔ f = g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : MeasurableSpace α", "M : Type u_3", "inst✝² : AddCommMonoid M", "inst✝¹ : TopologicalSpace M", "inst✝ : OrderedAddCommMonoid M", "s : SignedMeasure α", "i j : Set α"], "goal": "∃ x, ∀ x_1 ∈ s.measureOfNegatives, x ≤ x_1"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁴ : MeasurableSpace X", "ι : Type u_5", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "inst✝¹ : CompleteSpace E", "μ : Measure X", "l : Filter X", "inst✝ : l.IsMeasurablyGenerated", "f : X → E", "b : E", "h : Tendsto f (l ⊓ ae μ) (𝓝 b)", "hfm : StronglyMeasurableAtFilter f l μ", "hμ : μ.FiniteAtFilter l", "s : ι → Set X", "li : Filter ι", "hs : Tendsto s li l.smallSets", "m : optParam (ι → ℝ) fun i => (μ (s i)).toReal", "hsμ : autoParam ((fun i => (μ (s i)).toReal) =ᶠ[li] m) _auto✝"], "goal": "(fun s => ∫ (x : X) in s, f x ∂μ - (μ s).toReal • b) =o[l.smallSets] fun s => (μ s).toReal"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "FunLike F G H", "b : H", "AddZeroClass G", "One G", "Add H", "AddConstMapClass F G H 1 b", "f : F"], "goal": "f 1 = f 0 + b"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "f : α → β", "hf : Function.Surjective f", "p : β → Prop"], "goal": "(∀ (y : β), p y) ↔ ∀ (x : α), p (f x)"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "f : α → β", "hf : Function.Bijective f", "x : β"], "goal": "f ((Equiv.ofBijective f hf).symm x) = x"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "CommSemiring R", "IsEmpty σ", "p : MvPolynomial σ R"], "goal": "p = MvPolynomial.C (MvPolynomial.coeff 0 p)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : Semiring R", "f : R[X]", "r : R", "n : ℕ"], "goal": "(C r * X ^ n).eraseLead = 0"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "Base : Set α → Prop", "exch✝ : ExchangeProperty Base", "B B' B₁ B₂ : Set α", "exch : ExchangeProperty Base", "hB₁ : Base B₁", "hB₂ : Base B₂", "hinf : (B₂ \\ B₁).encard = ⊤"], "goal": "(B₁ \\ B₂).encard �� (B₂ \\ B₁).encard"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Submonoid R", "S : Type u_2", "CommRing S", "Algebra R S", "IsLocalization M S", "N : Type u_5", "AddCommGroup N", "Module R N", "Module S N", "IsScalarTower R S N", "x : N", "a : Set N"], "goal": "x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z, x = IsLocalization.mk' S 1 z • y"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e e' : PartialHomeomorph X Y", "h : e ≈ e'"], "goal": "Set.EqOn (↑e) (↑e') e.source"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "F : Type u_3", "TopologicalSpace B", "NontriviallyNormedField 𝕜", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "φ φ' : B → F ≃L[𝕜] F", "U U' : Set B", "hU : IsOpen U", "hφ : ContinuousOn (fun x => ↑(φ x)) U", "h2φ : ContinuousOn (fun x => ↑(φ x).symm) U", "hU' : IsOpen U'", "hφ' : ContinuousOn (fun x => ↑(φ' x)) U'", "h2φ' : ContinuousOn (fun x => ↑(φ' x).symm) U'", "b : B", "v : F"], "goal": "↑(FiberwiseLinear.partialHomeomorph φ hU hφ h2φ ≫ₕ FiberwiseLinear.partialHomeomorph φ' hU' hφ' h2φ') (b, v) =\n  (b, (φ' b) ((φ b) v))"}}
+{"state": {"context": [], "goal": "∀ {x : Sort u_1} {cmp : x → x → Ordering} [inst : Batteries.TransCmp cmp] {x_1 y z : x},\n  cmp x_1 y = Ordering.lt → cmp y z ≠ Ordering.gt → cmp x_1 z = Ordering.lt"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹³ : NormedAddCommGroup E", "inst✝¹² : NormedSpace 𝕜 E", "E' : Type u_3", "inst✝¹¹ : NormedAddCommGroup E'", "inst✝¹⁰ : NormedSpace 𝕜 E'", "H : Type u_4", "inst✝⁹ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "H' : Type u_5", "inst✝⁸ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_6", "inst✝⁷ : TopologicalSpace M", "inst✝⁶ : ChartedSpace H M", "inst✝⁵ : SmoothManifoldWithCorners I M", "M' : Type u_7", "inst✝⁴ : TopologicalSpace M'", "inst✝³ : ChartedSpace H' M'", "inst✝² : SmoothManifoldWithCorners I' M'", "F : Type u_8", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : TangentBundle I H"], "goal": "↑(chartAt (ModelProd H E) p) = ⇑(TotalSpace.toProd H E)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝ : PseudoEMetricSpace α", "f g : α → ℝ", "Kf Kg : ℝ≥0", "hf : LipschitzWith Kf f", "a : ℝ"], "goal": "LipschitzWith Kf fun x => min (f x) a"}}
+{"state": {"context": ["α✝ α : Type u", "inst✝² : Monoid α", "a b : α", "inst✝¹ : Invertible a", "inst✝ : Invertible b", "h : a = b"], "goal": "⅟a = ⅟b"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a b : α", "ha : 0 < a", "hb : 0 < b"], "goal": "a < 1 / b ↔ b < 1 / a"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α", "n : ℤ"], "goal": "(∃ z, b = a + z • p) → toIocMod hp a b = a + p"}}
+{"state": {"context": ["ι : Type u", "α : Type v", "inst✝¹ : DecidableEq α", "t : ι → Finset α", "inst✝ : Fintype ι", "n : ℕ", "hn : Fintype.card ι = n.succ", "ht : ∀ (s : Finset ι), s.card ≤ (s.biUnion t).card", "ih : ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n → (∀ (s' : Finset ι'), s'.card ≤ (s'.biUnion t').card) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x", "s : Finset ι", "hs : s.Nonempty", "hns : s ≠ univ", "hus : s.card = (s.biUnion t).card", "this : DecidableEq ι", "card_ι'_le : Fintype.card { x // x ∈ s } ≤ n", "t' : { x // x ∈ s } → Finset α := fun x' => t ↑x'", "f' : { x // x ∈ s } → α", "hf' : Function.Injective f'", "hsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x"], "goal": "∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝¹ : OmegaCompletePartialOrder α", "inst✝ : CompleteLattice β", "f g : α →o β", "hf : Continuous f", "hg : Continuous g"], "goal": "∀ f_1 ∈ {f, g}, Continuous f_1"}}
+{"state": {"context": ["p q : ℝ", "h : p.IsConjExponent q"], "goal": "p⁻¹ - 1 = -q⁻¹"}}
+{"state": {"context": ["G : Type w", "AddGroup G", "TopologicalSpace G", "TopologicalAddGroup G", "a : G"], "goal": "IsClosedMap fun x => a - x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "CategoryTheory.Limits.HasZeroMorphisms C", "f : X ⟶ Y", "CategoryTheory.Limits.HasKernel f", "Z : C", "h : Y ⟶ Z", "CategoryTheory.Limits.HasKernel (f ≫ h)"], "goal": "CategoryTheory.Limits.kernelSubobject f ≤ CategoryTheory.Limits.kernelSubobject (f ≫ h)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "f g : R"], "goal": "((toBasicOpen R f).comp (algebraMap R (Localization.Away f))) g = (toOpen R (PrimeSpectrum.basicOpen f)) g"}}
+{"state": {"context": ["α : Type u", "inst✝¹ : Monoid α", "a b c : α", "inst✝ : Invertible c"], "goal": "a * ⅟c = b ↔ a = b * c"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "a b : α", "hab : a < b"], "goal": "Set.Ioo a b ∪ {a} = Set.Ico a b"}}
+{"state": {"context": ["S : Type u_6", "LinearOrderedRing S", "a : S"], "goal": "AbsoluteValue.abs a = |a|"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "x : α"], "goal": "⋃ n, Metric.closedBall x ↑n = Set.univ"}}
+{"state": {"context": ["α : Type u_1", "γ : Type u_2", "mα : MeasurableSpace α", "mγ : MeasurableSpace γ", "κ η : Kernel α γ", "hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ", "inst✝ : IsFiniteKernel η", "hκη : κ ≤ η", "a : α", "x : γ", "hx_le_one : (∂κ a/∂η a) x ≤ 1 x"], "goal": "κ.rnDerivAux η a x ≤ 1 x"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "ι : Type u_5", "ι' : Type u_6", "CommRing R", "AddCommGroup L", "Module R L", "AddCommGroup M", "Module R M", "φ : L →ₗ[R] Module.End R M", "Fintype ι", "Fintype ι'", "DecidableEq ι", "DecidableEq ι'", "Module.Free R M", "Module.Finite R M", "b : Basis ι R L", "b' : Basis ι' R L", "k : ℕ", "H : (φ.polyCharpoly b).coeff k = 0"], "goal": "(φ.polyCharpoly b').coeff k = 0"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "UniformSpace α", "f : Filter β", "u : β → α", "a : α"], "goal": "Filter.Tendsto u f (𝓝 a) ↔ Filter.Tendsto (fun x => (u x, a)) f (𝓤 α)"}}
+{"state": {"context": ["M₀ : Type u_2", "CommMonoidWithZero M₀", "a : M₀", "ha : ⟦a⟧ ∈ (Associates M₀)⁰"], "goal": "associatesNonZeroDivisorsEquiv ⟨⟦a⟧, ha⟩ = ⟦⟨a, ⋯⟩⟧"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x✝ y f' : ℝ", "hfc : ConvexOn ℝ S f", "hfd : ∀ x ∈ S, DifferentiableAt ℝ f x", "x : ℝ", "hx hy : x ∈ S", "hxy : x ≤ x"], "goal": "deriv f x ≤ deriv f x"}}
+{"state": {"context": ["B : Type u₁", "CategoryTheory.Bicategory B", "C : Type u₂", "CategoryTheory.Bicategory C", "F G : CategoryTheory.OplaxFunctor B C", "η : F ⟶ G", "a : B"], "goal": "(CategoryTheory.Bicategory.rightUnitor η).hom.app a = (CategoryTheory.Bicategory.rightUnitor (η.app a)).hom"}}
+{"state": {"context": ["R' : Type u_5", "M' : Type u_6", "CommSemiring R'", "AddCommMonoid M'", "Module R' M'", "s : Set R'", "N : Submodule R' M'"], "goal": "↑(Ideal.span s) • N = s • N"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "ι : Type u_4", "DecidableEq ι", "V : ι → Submodule 𝕜 E", "hV : OrthogonalFamily 𝕜 (fun i => ↥(V i)) fun i => (V i).subtypeₗᵢ", "x : DirectSum ι fun i => ↥(V i)", "i : ι", "CompleteSpace ↥(V i)"], "goal": "(orthogonalProjection (V i)) ((DirectSum.coeAddMonoidHom V) x) = x i"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "hf : Measurable f", "hf_ne_zero : ∀ᵐ (x : α) ∂μ, f x ≠ 0", "hf_ne_top : ∀ᵐ (x : α) ∂μ, f x ≠ ⊤"], "goal": "((μ.withDensity f).withDensity fun x => (f x)⁻¹) = μ"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f g : CategoryTheory.Arrow C", "∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout f.left (fun x => f.right) fun x => f.hom", "∀ (n : ℕ), CategoryTheory.Limits.HasWidePushout g.left (fun x => g.right) fun x => g.hom", "F : f ⟶ g"], "goal": "(CategoryTheory.Arrow.mapAugmentedCechConerve F).left = F.left"}}
+{"state": {"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop", "hb : b ≠ 0", "ha : 2 ≤ a", "h : a ^ n ∣ b"], "goal": "a ^ n < a ^ b"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{?u.147899, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "ψ : S₃ ⟶ S₁"], "goal": "(ψ ≫ φ₁).τ₃ = ψ.τ₃ ≫ h.h₃ + (ψ.τ₃ ≫ h.h₂) ≫ S₂.g + (ψ ≫ φ₂).τ₃"}}
+{"state": {"context": ["E : Type u_5", "SeminormedAddCommGroup E", "R' : Type u_12", "Ring R'", "Module R' E", "p q : Submodule R' E", "h : p = q"], "goal": "(LinearIsometryEquiv.ofEq p q h).symm = LinearIsometryEquiv.ofEq q p ⋯"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ≥0∞", "hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤", "ε : ℝ≥0∞", "hε : ε ≠ 0"], "goal": "∃ s, MeasurableSet s ∧ μ s < ⊤ ∧ ∫⁻ (a : α) in sᶜ, f a ∂μ < ε"}}
+{"state": {"context": ["a t : ℝ", "ht : 0 < t"], "goal": "HasSum (fun n => rexp (-π * (↑n + a) ^ 2 * t)) (evenKernel (↑a) t)"}}
+{"state": {"context": ["R : Type u", "a : R", "Semiring R", "ha : a ≠ 0"], "goal": "(Polynomial.C a * Polynomial.X).degree = 1"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝² : UniformSpace α", "inst✝¹ : UniformSpace β", "inst✝ : UniformSpace γ", "f₁ : α → β", "f₂ : α → γ", "h₁ : UniformContinuous f₁", "h₂ : UniformContinuous f₂"], "goal": "UniformContinuous fun a => (f₁ a, f₂ a)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "F G H : C ⥤ D", "X : C"], "goal": "(α_ F G H).hom.app X = (α_ (F.obj X) (G.obj X) (H.obj X)).hom"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝² : MeasurableSpace Ω", "ι : Type u_2", "L : Filter ι", "μ : Measure Ω", "μs : ι → Measure Ω", "inst✝¹ : IsProbabilityMeasure μ", "inst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)", "E : Set Ω", "E_mble : MeasurableSet E", "h : limsup (fun i => (μs i) E) L ≤ μ E", "hne : L.NeBot", "meas_Ec : μ Eᶜ = 1 - μ E", "meas_i_Ec : ∀ (i : ι), (μs i) Eᶜ = 1 - (μs i) E", "obs : liminf (fun i => 1 - (μs i) E) L = liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L"], "goal": "1 - μ E ≤ liminf ((fun x => 1 - x) ∘ fun i => (μs i) E) L"}}
+{"state": {"context": ["α : Type u", "f : Filter α"], "goal": "Filter.comap id f = f"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "E₁ : Type u_9", "E₂ : Type u_10", "UniformSpace E₁", "UniformSpace E₂", "AddCommGroup E₁", "AddCommGroup E₂", "Module R₁ E₁", "Module R₂ E₂", "UniformAddGroup E₁", "UniformAddGroup E₂", "e : E₁ ≃ₛₗ[σ₁₂] E₂", "h₁ : Continuous ⇑e", "h₂ : Continuous ⇑e.symm"], "goal": "UniformEmbedding ⇑e"}}
+{"state": {"context": ["α : Sort u_1", "β : Sort u_2", "p : α → β → Prop"], "goal": "(∃ x, p x.fst x.snd) ↔ ∃ a b, p a b"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "Zero α", "s : Finset ι", "f : ι →₀ α", "t : ι → Finset α"], "goal": "f ∈ s.finsupp t ↔ f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i"}}
+{"state": {"context": ["k : Type u₁", "G : Type u₂", "H : Type u_1", "R : Type u_2", "inst✝⁵ : Semiring k", "α : Type u_3", "β : Type u_4", "α₂ : Type u_5", "inst✝⁴ : Semiring β", "inst✝³ : Mul α", "inst✝² : Mul α₂", "F : Type u_6", "inst✝¹ : FunLike F α α₂", "inst✝ : MulHomClass F α α₂", "f : F", "x y : MonoidAlgebra β α"], "goal": "(sum x fun a b => sum y fun a_1 b_1 => Finsupp.single (f a * f a_1) (b * b_1)) = sum (mapDomain (⇑f) x) fun a₁ b₁ => sum (mapDomain (⇑f) y) fun a₂ b₂ => single (a₁ * a₂) (b₁ * b₂)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "LinearOrder α", "TopologicalSpace β", "a : α", "f : α → β"], "goal": "ContinuousAt f a ↔ ContinuousWithinAt f (Set.Iio a) a ∧ ContinuousWithinAt f (Set.Ioi a) a"}}
+{"state": {"context": ["R : Type u_6", "A : Type u_7", "M : Type u_8", "Semiring R", "Semiring A", "Module R A", "AddCommMonoid M", "Module R M", "Module A M", "IsScalarTower R A M", "Module.Finite R A", "Module.Finite A M"], "goal": "Module.Finite R M"}}
+{"state": {"context": ["α : Type u", "Semiring α", "LinearOrder α", "ExistsAddOfLE α", "MulPosStrictMono α", "CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "ContravariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x < x_1", "a b c : α", "h : c < 0"], "goal": "a * c < b * c ↔ b < a"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "m : M"], "goal": "(TensorAlgebra.GradedAlgebra.ι R M) m =\n  (DirectSum.of (fun i => ↥(LinearMap.range (TensorAlgebra.ι R) ^ i)) 1) ⟨(TensorAlgebra.ι R) m, ⋯⟩"}}
+{"state": {"context": ["α : Type u_2", "M : Type u_4", "CanonicallyOrderedCommMonoid M", "a : α", "s : Set α", "f g : α → M", "hfg : a ∈ s → f a ≤ g a"], "goal": "s.mulIndicator f a ≤ g a"}}
+{"state": {"context": ["V : Type u_1", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "Fact (FiniteDimensional.finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x : V"], "goal": "(o.rotation ↑π) x = -x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Sort w", "α : Type u_1", "β : Type u_2", "x : X", "s s₁ s₂ t : Set X", "p p₁ p₂ : X → Prop", "inst✝ : TopologicalSpace X", "h : IsClosed s", "A : frontier s = s \\ interior s", "B : interior (frontier s) ⊆ interior s", "C : interior (frontier s) ⊆ frontier s", "this : interior (frontier s) ⊆ interior s ∩ (s \\ interior s)"], "goal": "interior (frontier s) = ∅"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "OrderBot α", "LocallyFiniteOrder α"], "goal": "Finset.Iio = Finset.Ico ⊥"}}
+{"state": {"context": ["n : ℕ"], "goal": "(Tree.treesOfNumNodesEq n).card = catalan n"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : Ring S", "f : R[X]", "inst✝ : Algebra R S", "h : IsAdjoinRootMonic S f", "x : S", "i : Fin f.natDegree"], "goal": "(Finsupp.comapDomain Fin.val (h.modByMonicHom x).toFinsupp ⋯) i = (h.modByMonicHom x).coeff ��i"}}
+{"state": {"context": ["I : Type u", "LinearOrder I", "C : Set (I → Bool)", "l : Profinite.NobelingProof.Products I", "J K : I → Prop", "h : ∀ a ∈ ↑l, J a", "(j : I) → Decidable (J j)", "(j : I) → Decidable (K j)", "hJK : ∀ (i : I), J i → K i"], "goal": "⇑(Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.π C J) l) ∘\n    Profinite.NobelingProof.ProjRestricts C hJK =\n  ⇑(Profinite.NobelingProof.Products.eval (Profinite.NobelingProof.π C K) l)"}}
+{"state": {"context": ["𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "NormedAddCommGroup E", "NormedSpace ℝ E", "μ : MeasureTheory.Measure ℝ", "RCLike 𝕜", "NormedSpace 𝕜 E", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "a b : ℝ", "φ : ℝ → F →L[𝕜] E", "hφ : IntervalIntegrable φ μ a b", "v : F"], "goal": "(∫ (x : ℝ) in a..b, φ x ∂μ) v = ∫ (x : ℝ) in a..b, (φ x) v ∂μ"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁵ : Semiring R", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "inst✝² : AddCommMonoid M'", "inst✝¹ : Module R M'", "b✝ b₁ : Basis ι R M", "i : ι", "c : R", "x✝ : M", "inst✝ : NoZeroDivisors R", "b : Basis ι R M", "N : Submodule R M", "x : M", "hx : x ∈ N", "x_ne : x ≠ 0"], "goal": "LinearIndependent R (Subtype.val ∘ fun x_1 => ⟨x, hx⟩)"}}
+{"state": {"context": ["σ₁ : Type u_1", "σ₂ : Type u_2", "f₁ : σ₁ → Option σ₁", "f₂ : σ₂ → Option σ₂", "tr : σ₁ → σ₂ → Prop", "H : Respects f₁ f₂ tr", "a₁ : σ₁", "a₂ : σ₂", "aa : tr a₁ a₂", "b₁ : σ₁", "ab✝ : Reaches f₁ a₁ b₁", "ab : TransGen (fun a b => b ∈ f₁ a) a₁ b₁"], "goal": "∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂"}}
+{"state": {"context": ["M : Type u_1", "CommMonoid M", "S : Submonoid M", "N : Type u_2", "CommMonoid N", "P : Type u_3", "CommMonoid P", "f : S.LocalizationMap N", "g : S.LocalizationMap P", "x y : M"], "goal": "f.toMap x = f.toMap y ↔ g.toMap x = g.toMap y"}}
+{"state": {"context": ["β : Type u_2", "AddCommMonoid β", "f : Fin 5 → β"], "goal": "∑ i : Fin 5, f i = f 0 + f 1 + f 2 + f 3 + f 4"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_4", "InvolutiveStar α", "A : Matrix n n α"], "goal": "A.conjTranspose.IsHermitian ↔ A.IsHermitian"}}
+{"state": {"context": [], "goal": "Fact (0 < 1)"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℂ E", "f : E → ℂ", "x : E", "s : Set E", "hf : DifferentiableWithinAt ℂ f s x", "hxs : UniqueDiffWithinAt ℂ s x"], "goal": "fderivWithin ℂ (fun x => Complex.sin (f x)) s x = Complex.cos (f x) • fderivWithin ℂ f s x"}}
+{"state": {"context": ["E : Type u_3", "SeminormedAddGroup E", "s : Set E", "hs : Bornology.IsBounded s"], "goal": "∃ R > 0, ∀ x ∈ s, ‖x‖ ≤ R"}}
+{"state": {"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I J✝ : Box ι", "i : ι", "x : ℝ", "inst✝ : Finite ι", "π : Prepartition I", "t : Finset (ι × ℝ)", "ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J", "J : Box ι", "hJ : J ∈ (splitMany I t).filter fun J => ↑J ⊆ π.iUnion"], "goal": "∃ I' ∈ π, J ≤ I'"}}
+{"state": {"context": ["u : XgcdType", "a b : ℕ+"], "goal": "(↑a - 1 + 1 + 0, 0 + (↑b - 1 + 1)) = (↑a, ↑b)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "uniformSpace : UniformSpace α", "ι : Sort u_1", "s : ι → Set α", "hs : ∀ (i : ι), IsComplete (s i)", "U : Set (α × α)", "hU : U ∈ 𝓤 α", "hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j", "S : Set α := ⋃ i, s i", "l : Filter α", "hl : Cauchy l", "hls : S ∈ l", "hl_ne : l.NeBot", "hl' : ∀ s ∈ 𝓤 α, ∃ t ∈ l, t ×ˢ t ⊆ s"], "goal": "∃ x ∈ S, l ≤ 𝓝 x"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝³ : DivisionRing k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "p₁ p₂ p₃ p₄ p₅ : P", "h₁ : p₁ ∈ affineSpan k {p₄, p₅}", "h₂ : p₂ ∈ affineSpan k {p₄, p₅}", "h₃ : p₃ ∈ affineSpan k {p₄, p₅}"], "goal": "{p₁, p₂, p₃, p₄} ⊆ {p₁, p₂, p₃, p₄, p₅}"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "_i : Fact (1 ≤ p)", "hp_ne_top : p ≠ ⊤", "P : (α → E) → Prop", "h_ind : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ⊤ → P (s.indicator fun x => c)", "h_add :\n  ∀ ⦃f g : α → E⦄,\n    Disjoint (Function.support f) (Function.support g) →\n      MeasureTheory.Memℒp f p μ → MeasureTheory.Memℒp g p μ → P f → P g → P (f + g)", "h_closed : IsClosed {f | P ↑↑f}", "h_ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MeasureTheory.Memℒp f p μ → P f → P g", "f : α → E"], "goal": "MeasureTheory.Memℒp f p μ → P f"}}
+{"state": {"context": ["R✝ S : Type u", "inst✝⁶ : CommRing R✝", "inst✝⁵ : CommRing S", "M : Submonoid R✝", "N : Submonoid S", "R' S' : Type u", "inst✝⁴ : CommRing R'", "inst✝³ : CommRing S'", "f : R✝ →+* S", "inst✝² : Algebra R✝ R'", "inst✝¹ : Algebra S S'", "R : Type u_1", "inst✝ : CommRing R", "h : ∀ (J : Ideal R) (x : J.IsMaximal), (fun R hR => IsReduced R) (Localization.AtPrime J) OreLocalization.instCommRing", "x : R", "hx : IsNilpotent x"], "goal": "x = 0"}}
+{"state": {"context": ["K : Type u_1", "Field K", "s : SimpContFract K", "n : ℕ", "not_terminatedAt_n : ¬(↑s).TerminatedAt n"], "goal": "(↑s).nums n * (↑s).dens (n + 1) - (↑s).dens n * (↑s).nums (n + 1) = (-1) ^ (n + 1)"}}
+{"state": {"context": ["e : ONote", "n : ℕ+", "a o : ONote", "h₁ : (e.oadd n a).NF", "h₂ : o.NF", "this✝¹ : a.NF", "h' : (a.add o).repr = a.repr + o.repr", "e' : ONote", "n' : ℕ+", "a' : ONote", "h : a.add o = e'.oadd n' a'", "nf : (e'.oadd n' a').NF", "this✝ : e.NF", "this : e'.NF", "ee : (e.cmp e').Compares e e'"], "goal": "(match e.cmp e' with | Ordering.lt => e'.oadd n' a' | Ordering.eq => e.oadd (n + n') a' | Ordering.gt => e.oadd n (e'.oadd n' a')).repr = ω ^ e.repr * ↑↑n + (ω ^ e'.repr * ↑↑n' + a'.repr)"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "S₂ : Type w", "S₃ : Type x", "σ : Type u_1", "a a' a₁ a₂ : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S₁", "p q✝ : MvPolynomial σ R", "f q : R →+* S₁", "r : R", "hr : ∀ (r' : R), q r' = 0 ↔ r ∣ r'", "φ : MvPolynomial σ R"], "goal": "C r ∣ φ ↔ (map q) φ = 0"}}
+{"state": {"context": ["R : Type u", "L : Type v", "L' : Type w₂", "inst✝⁴ : CommRing R", "inst✝³ : LieRing L", "inst✝² : LieAlgebra R L", "inst✝¹ : LieRing L'", "inst✝ : LieAlgebra R L'", "f : L →ₗ⁅R⁆ L'", "I : LieIdeal R L", "J : LieIdeal R L'", "I₁ I₂ : LieIdeal R L", "h : Function.Surjective ⇑f", "y₁ : ↥I₁", "h₁ : f ↑y₁ ∈ map f I₁", "y₂ : ↥I₂", "h₂ : f ↑y₂ ∈ map f I₂"], "goal": "⁅↑⟨f ↑y₁, h₁⟩, ↑⟨f ↑y₂, h₂⟩⁆ ∈ ⇑↑f '' {m | ∃ x n, ⁅↑x, ↑n⁆ = m}"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "s : Finset α", "f : α → β", "DivisionCommMonoid β"], "goal": "∏ x ∈ s, (f x)⁻¹ = (∏ x ∈ s, f x)⁻¹"}}
+{"state": {"context": ["M : Type u_12", "N : Type u_13", "F : Type u_14", "Mul M", "Mul N", "FunLike F M N", "MulHomClass F M N", "f : F", "hf : Function.Bijective ⇑f", "a : M"], "goal": "(MulEquiv.ofBijective f hf) a = f a"}}
+{"state": {"context": ["ι : Type u_1", "σ : Type u_2", "R : Type u_3", "A : Type u_4", "inst✝⁵ : Semiring A", "inst✝⁴ : DecidableEq ι", "inst✝³ : AddMonoid ι", "inst✝² : SetLike σ A", "inst✝¹ : AddSubmonoidClass σ A", "𝒜 : ι → σ", "inst✝ : GradedRing 𝒜", "κ : Sort u_5", "f : κ → Ideal A", "s : κ → Set ↥(homogeneousSubmonoid 𝒜)", "hs : ∀ (i : κ), f i = span (Subtype.val '' s i)"], "goal": "(fun i => f i) = fun i => span (Subtype.val '' s i)"}}
+{"state": {"context": ["α : Type u_1", "AddCommMonoid α", "x x' : α", "h : x = x'"], "goal": "x = Mathlib.Tactic.Abel.term 1 x' 0"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "s : Set α", "ε : ℝ≥0"], "goal": "EMetric.diam (Metric.cthickening (↑ε) s) ≤ EMetric.diam s + 2 * ↑ε"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝⁵ : Ring R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "ι✝ : Type w", "ι' : Type w'", "inst✝² : StrongRankCondition R", "ι : Type u_1", "inst✝¹ : Fintype ι", "v : ι → M", "i : LinearIndependent R v", "w : Set M", "inst✝ : Fintype ↑w", "s : range v ≤ ↑(span R w)"], "goal": "Injective ⇑?f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "r : ℝ≥0∞", "hr : 1 ≤ r"], "goal": "∑' (n : ℕ), r ^ n = (1 - r)⁻¹"}}
+{"state": {"context": ["n : ℕ", "c : Composition n", "h : 0 < n"], "goal": "0 < c.length"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝ : LinearOrderedField 𝕜", "x✝ y✝ z x y : 𝕜", "h : y ≤ x"], "goal": "[x-[𝕜]y] = Icc (min x y) (max x y)"}}
+{"state": {"context": ["x y : Bool"], "goal": "x ≠ y → x = !y"}}
+{"state": {"context": ["X : Type u_2", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "ι : Type u_6", "l : Filter ι", "F : ι → C(X, Y)", "f : C(X, Y)"], "goal": "Filter.Tendsto F l (𝓝 f) ↔\n  ∀ (K : Set X),\n    IsCompact K → Filter.Tendsto (fun i => ContinuousMap.restrict K (F i)) l (𝓝 (ContinuousMap.restrict K f))"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace ℝ E", "φ ψ : ℝ → ℝ", "s : Set E", "a b m : ℝ", "x y : E", "f : E → ℝ", "ha : 0 ≤ a", "hb : 0 ≤ b", "hab : a + b = 1"], "goal": "a * (f x + m / 2 * ‖x‖ ^ 2) + b * (f y + m / 2 * ‖y‖ ^ 2) - m / 2 * ‖a • x + b • y‖ ^ 2 = a * f x + b * f y + m / 2 * a * b * ‖x - y‖ ^ 2"}}
+{"state": {"context": ["I : Type u", "LinearOrder I", "IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "o : Ordinal.{u}", "hC : IsClosed C", "hsC : Profinite.NobelingProof.contained C (Order.succ o)", "ho : o < Ordinal.type fun x x_1 => x < x_1"], "goal": "(CategoryTheory.ShortComplex.mk (ModuleCat.ofHom (Profinite.NobelingProof.πs C o))\n    (ModuleCat.ofHom (Profinite.NobelingProof.Linear_CC' C hsC ho)) ⋯).Exact"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "r✝ : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "r : α → α → Prop", "inst✝ : IsWellOrder α r", "h0 : 0 < type r", "a : α"], "goal": "¬r a (enum r 0 h0)"}}
+{"state": {"context": ["f : Type u₀ → Type u₁", "self : EquivFunctor f", "α : Type u₀"], "goal": "EquivFunctor.map (Equiv.refl α) = id"}}
+{"state": {"context": ["f : ℕ → ℕ"], "goal": "Primrec f ↔ Nat.Primrec f"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "IsDedekindDomain R", "v : IsDedekindDomain.HeightOneSpectrum R", "r : R"], "goal": "v.intValuationDef r < 1 ↔ v.asIdeal ∣ Ideal.span {r}"}}
+{"state": {"context": ["G : Type u_1", "inst✝³ : Group G", "ι : Type u_2", "hdec : DecidableEq ι", "hfin : Fintype ι", "H : ι → Type u_3", "inst✝² : (i : ι) → Group (H i)", "ϕ : (i : ι) → H i →* G", "f g : (i : ι) → H i", "hcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)", "inst✝¹ : Finite ι", "inst✝ : (i : ι) → Fintype (H i)", "hcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))", "val✝ : Fintype ι", "this : DecidableEq ι := Classical.decEq ι", "i : ι"], "goal": "(fun i => (ϕ i).range) i ⊓ ⨆ j, ⨆ (_ : j ≠ i), (fun i => (ϕ i).range) j ≤ ⊥"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G"], "goal": "H.addSubgroupOf ⊥ = ⊤"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "f : PowerSeries R", "n : ℕ"], "goal": "(HahnSeries.ofPowerSeries ℤ R) (f ^ n) = (HahnSeries.ofPowerSeries ℤ R) f ^ n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝³ : CommGroup α", "inst✝² : UniformSpace α", "inst✝¹ : UniformGroup α", "f g : β → α", "a a₁ a₂ : α", "inst✝ : CompleteSpace α"], "goal": "Multipliable f ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e"}}
+{"state": {"context": ["α : Type u_4"], "goal": "Nat.card α = if h : Finite α then Fintype.card α else 0"}}
+{"state": {"context": ["m : Type u_2", "α : Type v", "NonAssocSemiring α", "Fintype m", "DecidableEq m", "x : ℕ", "x.AtLeastTwo", "v : m → α"], "goal": "v ᵥ* OfNat.ofNat x = MulOpposite.op (OfNat.ofNat x) • v"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "E✝ : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedAddCommGroup E✝", "p : ℝ≥0∞", "f✝ : ι → α → E✝", "g✝ : α → E✝", "E : Type u_4", "inst✝ : NormedAddCommGroup E", "f : ι → α → E", "g : α → E", "l : Filter ι", "δ : ℝ", "hδ : 0 < δ", "hfg : ∀ ε > 0, ∀ᶠ (x : ι) in l, essSup (fun x_1 => ↑‖(f x - g) x_1‖₊) μ ≤ ε", "ε : ℝ≥0∞", "hε : ε > 0"], "goal": "∀ᶠ (x : ι) in l, μ {x_1 | δ ≤ dist (f x x_1) (g x_1)} ≤ ε"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "UniformSpace α", "AddGroup α", "UniformAddGroup α", "UniformSpace β", "AddGroup β", "UniformAddGroup β", "f : α →+ β", "hf : Continuous ⇑f", "a : α"], "goal": "(f.completion hf) (↑α a) = ↑β (f a)"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m : MeasurableSpace Ω", "X : Ω → ℝ", "p : ℕ", "μ : Measure Ω", "t : ℝ", "inst✝ : IsProbabilityMeasure μ"], "goal": "cgf X μ 0 = 0"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝³⁷ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³⁶ : NormedAddCommGroup E", "inst✝³⁵ : NormedSpace 𝕜 E", "H : Type u_3", "inst✝³⁴ : TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "inst✝³³ : TopologicalSpace M", "inst✝³² : ChartedSpace H M", "inst✝³¹ : SmoothManifoldWithCorners I M", "E' : Type u_5", "inst✝³⁰ : NormedAddCommGroup E'", "inst✝²⁹ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝²⁸ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝²⁷ : TopologicalSpace M'", "inst✝²⁶ : ChartedSpace H' M'", "inst✝²⁵ : SmoothManifoldWithCorners I' M'", "E'' : Type u_8", "inst✝²⁴ : NormedAddCommGroup E''", "inst✝²³ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝²² : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝²¹ : TopologicalSpace M''", "inst✝²⁰ : ChartedSpace H'' M''", "F : Type u_11", "inst✝¹⁹ : NormedAddCommGroup F", "inst✝¹⁸ : NormedSpace 𝕜 F", "G : Type u_12", "inst✝¹⁷ : TopologicalSpace G", "J : ModelWithCorners 𝕜 F G", "N : Type u_13", "inst✝¹⁶ : TopologicalSpace N", "inst✝¹⁵ : ChartedSpace G N", "inst✝¹⁴ : SmoothManifoldWithCorners J N", "F' : Type u_14", "inst✝¹³ : NormedAddCommGroup F'", "inst✝¹² : NormedSpace 𝕜 F'", "G' : Type u_15", "inst✝¹¹ : TopologicalSpace G'", "J' : ModelWithCorners 𝕜 F' G'", "N' : Type u_16", "inst✝¹⁰ : TopologicalSpace N'", "inst✝⁹ : ChartedSpace G' N'", "inst✝⁸ : SmoothManifoldWithCorners J' N'", "F₁ : Type u_17", "inst✝⁷ : NormedAddCommGroup F₁", "inst✝⁶ : NormedSpace 𝕜 F₁", "F₂ : Type u_18", "inst✝⁵ : NormedAddCommGroup F₂", "inst✝⁴ : NormedSpace 𝕜 F₂", "F₃ : Type u_19", "inst✝³ : NormedAddCommGroup F₃", "inst✝² : NormedSpace 𝕜 F₃", "F₄ : Type u_20", "inst✝¹ : NormedAddCommGroup F₄", "inst✝ : NormedSpace 𝕜 F₄", "e : PartialHomeomorph M H", "e' : PartialHomeomorph M' H'", "f f₁ : M → M'", "s s₁ t : Set M", "x : M", "m n : ℕ∞"], "goal": "𝓝[↑(extChartAt I x).symm ⁻¹' insert x s ∩ range ↑I] ↑(extChartAt I x) x = 𝓝[insert (↑(extChartAt I x) x) (↑(extChartAt I x).symm ⁻¹' s ∩ range ↑I)] ↑(extChartAt I x) x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y : C", "f g : X ⟶ Y", "t : CategoryTheory.Limits.Fork f g", "ht : CategoryTheory.Limits.IsLimit t", "Z : C", "h : { h // CategoryTheory.CategoryStruct.comp h f = CategoryTheory.CategoryStruct.comp h g }"], "goal": "(CategoryTheory.Limits.Fork.IsLimit.homIso ht Z).symm h = ↑(CategoryTheory.Limits.Fork.IsLimit.lift' ht ↑h ⋯)"}}
+{"state": {"context": ["a b : Int"], "goal": "a + b = b + a"}}
+{"state": {"context": ["M : Type u", "X : Type w", "PseudoEMetricSpace X", "SMul M X", "IsometricSMul M X", "c : M", "s : Set X"], "goal": "EMetric.diam (c • s) = EMetric.diam s"}}
+{"state": {"context": ["R : Type u", "K : Type v", "L : Type z", "p : R", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : Algebra K L", "inst✝⁶ : Algebra R L", "inst✝⁵ : Algebra R K", "inst✝⁴ : IsScalarTower R K L", "inst✝³ : Algebra.IsSeparable K L", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "B : PowerBasis K L", "hp : _root_.Prime p", "hBint : IsIntegral R B.gen", "z : L", "Q : R[X]", "hQ : (aeval B.gen) Q = p • z", "hzint : IsIntegral R z", "hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})", "this : Module.Finite K L := finite B", "P : R[X] := minpoly R B.gen", "n : ℕ", "hn : B.dim = n.succ", "finrank_K_L : FiniteDimensional.finrank K L = B.dim", "deg_K_P : (minpoly K B.gen).natDegree = B.dim", "deg_R_P : P.natDegree = B.dim"], "goal": "p ∣ Q.coeff 0"}}
+{"state": {"context": ["R : Type u", "Semiring R", "x : R"], "goal": "IsUnit (Polynomial.C x) ↔ IsUnit x"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α"], "goal": "μ s = (MeasureTheory.inducedOuterMeasure (fun s x => μ s) ⋯ ⋯) s"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "x✝ y : R", "inst✝⁴ : MonoidWithZero R", "inst✝³ : MonoidWithZero S", "F : Type u_3", "inst✝² : FunLike F R S", "inst✝¹ : MonoidWithZeroHomClass F R S", "f : F", "hf : Injective ⇑f", "inst✝ : IsReduced S", "x : R", "hx : IsNilpotent x"], "goal": "f x = f 0"}}
+{"state": {"context": ["b c : Int", "a : Int"], "goal": "a + b ≤ a + c ↔ b ≤ c"}}
+{"state": {"context": ["C : ℝ → Prop", "x : ℝ", "h : ∀ (y : CauSeq ℚ abs), C (Real.mk y)"], "goal": "C x"}}
+{"state": {"context": ["K R : Type v", "V M : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : AddCommGroup M", "inst✝³ : Module R M", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V", "f : End K V", "μ : K", "n : ℕ"], "goal": "LinearMap.range (f * (f - (algebraMap K (End K V)) μ) ^ n) = LinearMap.range ((f - (algebraMap K (End K V)) μ) ^ n * f)"}}
+{"state": {"context": ["n a : ℕ", "ha : n / 2 < a", "ha' : a < n"], "goal": "(↑a).valMinAbs = ↑a - ↑n"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "σ : Type v", "M✝ : Type w", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : AddCommGroup M✝", "inst✝ : Module R M✝", "inst : IsNoetherianRing R", "I : Ideal R[X]", "M : Submodule R R := ⋯.min (Set.range I.leadingCoeffNth) ⋯", "hm : M ∈ Set.range I.leadingCoeffNth", "N : ℕ", "HN : I.leadingCoeffNth N = M", "s : Finset R[X]", "hs : Submodule.span R ↑s = I.degreeLE ↑N", "hm2 : ∀ (k : ℕ), I.leadingCoeffNth k ≤ M", "hs2 : ∀ {x : R[X]}, x ∈ I.degreeLE ↑N → x ∈ Ideal.span ↑s", "this : Submodule.span R[X] ↑s = Ideal.span ↑s", "p : R[X]", "hp : p ∈ I", "k : ℕ", "hn : p.natDegree = k"], "goal": "p ∈ Ideal.span ↑s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "a a₁ a₂ b b₁ b₂ c d x✝ : α"], "goal": "x✝ ∈ Ioo a b ∩ Iio c ↔ x✝ ∈ Ioo a (min b c)"}}
+{"state": {"context": ["ι : Type v", "β : ι → Type w", "(i : ι) → AddCommGroup (β i)", "g₁ g₂ : DirectSum ι fun i => β i", "i : ι"], "goal": "(g₁ - g₂) i = g₁ i - g₂ i"}}
+{"state": {"context": ["X : TwoP", "a : ((𝟭 TwoP).obj X).toBipointed.X"], "goal": "(TwoP.swapEquiv.counitIso.inv.app X).toFun a = a"}}
+{"state": {"context": ["α : Type u_1", "a : α", "p : α → Bool", "l : List α"], "goal": "a ∈ List.eraseP p l → a ∈ l"}}
+{"state": {"context": [], "goal": "arctan 0 = 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NormedRing R", "inst✝ : CompleteSpace R", "x : Rˣ", "a✝ : Nontrivial R"], "goal": "∃ ε > 0, ∀ ⦃y : R⦄, dist y 0 < ε → inverse (↑x + y) = inverse (1 + ↑x⁻¹ * y) * ↑x⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "I J : BoxIntegral.Box ι", "h : J ≤ I"], "goal": "(BoxIntegral.Prepartition.single I J h).IsPartition ↔ J = I"}}
+{"state": {"context": ["α : Sort u", "r : α → α → Prop", "x y : α", "h₁ : Acc r x", "h₂ : r y x"], "goal": "Acc r y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m m0 : MeasurableSpace α", "p : ENNReal", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "hm : m ≤ m0", "f : α → E", "hf : MeasureTheory.Memℒp f p (μ.trim hm)"], "goal": "MeasureTheory.Memℒp f p μ"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : Ring R", "E : Type u_2", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module R E", "F : Type u_3", "inst✝³ : AddCommGroup F", "inst✝² : Module R F", "G : Type u_4", "inst✝¹ : AddCommGroup G", "inst✝ : Module R G", "c : Set (E →ₗ.[R] F)", "hc : DirectedOn (fun x x_1 => x ≤ x_1) c"], "goal": "∃ f, { domain := sSup (domain '' c), toFun := f } ∈ upperBounds c"}}
+{"state": {"context": ["C : Type u", "n : CategoryTheory.FreeMonoidalCategory.NormalMonoidalObject C"], "goal": "(𝟙_ (CategoryTheory.FreeMonoidalCategory C)).normalizeObj n = n"}}
+{"state": {"context": ["a b : ℝ", "n : ℕ"], "goal": "∀ x ∈ [[a, b]], DifferentiableAt ℝ sin x"}}
+{"state": {"context": ["F : Type u", "K : Type v", "L : Type w", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Field F", "f : K[X]", "hf : f.natDegree ≠ 0", "g : K[X]", "hg : f = f.factor * g"], "goal": "(map (AdjoinRoot.of f.factor) f).IsRoot (AdjoinRoot.root f.factor)"}}
+{"state": {"context": ["α : Type u_2", "LE α", "OrderTop α"], "goal": "TopHom.dual.symm (BotHom.id αᵒᵈ) = TopHom.id α"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l l' : List α", "hl : l.Nodup", "hl' : l'.Nodup", "hn : 2 ≤ l.length", "hn' : 2 ≤ l'.length"], "goal": "(∀ (x : α), l.formPerm x = x ∨ l'.formPerm x = x) ↔ ∀ ⦃a : α⦄, a ∈ l → a ∈ l' → False"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝¹ : TopologicalSpace α", "s✝ t u v : Set α", "inst✝ : PreconnectedSpace α", "s : ι → Set α", "h_disj : Pairwise (Disjoint on s)", "h_clopen : ∀ (i : ι), IsClopen (s i)", "h_nonempty : ∀ (i : ι), s i ≠ ∅", "i j : ι", "h_ne : s i ∩ s j = ∅"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s✝ s₁ s₂ t✝ t₁ t₂ u : Set α", "x : α", "s t : Set α"], "goal": "s \\ {x} ⊆ t ↔ s ⊆ insert x t"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "Semiring R", "StarMul R", "TrivialStar R", "AddCommGroup A", "Module R A", "StarAddMonoid A", "StarModule R A", "Invertible 2", "x : A", "hx : IsSelfAdjoint x"], "goal": "↑((selfAdjointPart R) x) = x"}}
+{"state": {"context": ["R : Type u_3", "S : Type u_5", "T : Type u_7", "NonUnitalNonAssocSemiring R", "NonUnitalNonAssocSemiring S", "NonUnitalNonAssocSemiring T", "f : R →ₙ+* S", "g : R →ₙ+* T"], "goal": "(NonUnitalRingHom.snd S T).comp (f.prod g) = g"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "X Y : C", "f : X ⟶ Y", "A B : Cᵒᵖ", "g : A ⟶ B"], "goal": "(image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalNonAssocRing R", "x : R", "S : NonUnitalSubring R"], "goal": "x ∈ nonUnitalSubalgebraOfNonUnitalSubring S ↔ x ∈ S"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "l : Filter ι", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "OpensMeasurableSpace α", "a b : ι → α", "A B : α", "MeasureTheory.NoAtoms μ", "ha : Filter.Tendsto a l (𝓝 A)", "hb : Filter.Tendsto b l (𝓝 B)"], "goal": "MeasureTheory.AECover (μ.restrict (Set.Ico A B)) l fun i => Set.Ico (a i) (b i)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommSemiring R", "inst✝¹ : TopologicalSpace R", "inst✝ : TopologicalSemiring R", "X : Set R", "x✝ : R", "a✝ : ↑X"], "goal": "((↑↑{ toFun := fun p => p.toContinuousMapOn X, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }).toFun ((algebraMap R R[X]) x✝)) a✝ = ((algebraMap R C(↑X, R)) x✝) a✝"}}
+{"state": {"context": ["C : Type u_1", "inst✝ : Category.{?u.17202, u_1} C", "F G : ℕ ⥤ C", "app : (n : ℕ) → F.obj n ⟶ G.obj n", "naturality : ∀ (n : ℕ), F.map (homOfLE ⋯) ≫ app (n + 1) = app n ≫ G.map (homOfLE ⋯)", "k : ℕ"], "goal": "∀ ⦃i j : ℕ⦄ (hk : j = i + k), F.map (homOfLE ⋯) ≫ app j = app i ≫ G.map (homOfLE ⋯)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "s✝ s' : Set α", "x : α", "p✝ : Filter ι", "g : ι → α", "inst✝ : UniformSpace β", "𝔖 : Set (Set α)", "h : 𝔖.Nonempty", "h' : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖", "p : ι → Prop", "s : ι → Set (β × β)", "hb : (𝓤 β).HasBasis p s"], "goal": "(⨅ i ∈ 𝔖, 𝓤 (α →ᵤ[𝔖] β)).HasBasis (fun Si => Si.1 ∈ 𝔖 ∧ p Si.2) fun Si => UniformOnFun.gen 𝔖 Si.1 (s Si.2)"}}
+{"state": {"context": ["V : Type u_1", "G : SimpleGraph V", "v : V", "Fintype ↑(G.neighborSet v)", "w : V"], "goal": "w ∈ G.neighborFinset v ↔ G.Adj v w"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "inst✝⁵ : CommRing A", "inst✝⁴ : IsDomain A", "inst✝³ : UniqueFactorizationMonoid A", "inst✝² : Field K", "inst✝¹ : Algebra A K", "inst✝ : IsFractionRing A K", "p : A[X]", "r : K", "hr : (aeval r) p = 0", "j : ℕ", "hj : j ≠ 0"], "goal": "num A r ∣ (p.scaleRoots ↑(den A r)).coeff j * num A r ^ j"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "F' : Type u_7", "Norm E", "SeminormedAddCommGroup F'", "c : ℝ", "f : α → E", "g' : α → F'", "l : Filter α"], "goal": "(Asymptotics.IsBigOWith c l f fun x => ‖g' x‖) ↔ Asymptotics.IsBigOWith c l f g'"}}
+{"state": {"context": ["k : ℕ", "hk : k ≠ 0", "this : 2 ^ (2 * k - 1) = 2 ^ (2 * k) / 2"], "goal": "(-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)! = (-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! * ↑(bernoulli (2 * k))"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "n : ℕ", "s : Simplex k P n", "i : Fin (n + 1)", "i✝ : Fin (0 + 1)"], "goal": "s.points (({i}.orderEmbOfFin ⋯) i✝) = s.points i"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "f : α → ℚ → ℝ", "hf : ProbabilityTheory.IsMeasurableRatCDF f", "a : α"], "goal": "(hf.stieltjesFunction a).measure Set.univ = 1"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_2", "N : Type u_3", "P : Type u_4", "inst✝² : DecidableEq α", "inst✝¹ : DecidableEq N", "inst✝ : AddGroup N", "f f₁ f₂ g g₁ g₂ : α →₀ N"], "goal": "(f + g).neLocus f = g.support"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "CoalgebraStruct R A"], "goal": "↑(BialgEquiv.refl R A) = CoalgEquiv.refl R A"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace ℝ E", "inst✝³ : CompleteSpace E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "inst✝ : CompleteSpace F", "μ ν : Measure α", "s t N : Set α", "f : α → ℝ≥0∞", "hμ : μ s ≠ 0", "hs : NullMeasurableSet s μ", "hint : ∫⁻ (a : α) in s, f a ∂μ ≠ ⊤", "hμ₁ : μ s = ⊤"], "goal": "0 < μ {x | x ∈ s ∧ ⨍⁻ (a : α) in s, f a ∂μ ≤ f x}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "f : α → γ", "g : β → γ", "μ : autoParam (MeasureTheory.Measure α) _auto✝", "ν : autoParam (MeasureTheory.Measure β) _auto✝¹", "self : ProbabilityTheory.IdentDistrib f g μ ν"], "goal": "AEMeasurable g ν"}}
+{"state": {"context": ["α : Type u_1", "I : Set α", "M : Matroid α", "M.Finitary"], "goal": "M.Indep I ↔ ∀ J ⊆ I, J.Finite → M.Indep J"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "S : AffineSpace V P", "ι : Type u_4", "v : V", "x : k", "s : Set ι", "p : ι → P", "b : P"], "goal": "(∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = x ∧ v = (fs.weightedVSubOfPoint p b) w) ↔\n  ∃ fs w, ∑ i ∈ fs, w i = x ∧ v = (fs.weightedVSubOfPoint (fun i => p ↑i) b) w"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "J : Type u₂", "inst✝ : Category.{v₂, u₂} J", "X : Type v₂", "α : Type u_1", "Z : α → C", "f f' : Fan Z", "c : Cofan fun x => op (Z x)", "hf : IsLimit f", "hf' : IsLimit f'", "hc : IsColimit c", "j : α"], "goal": "((opProductIsoCoproduct' hf hc).hom ≫ (opProductIsoCoproduct' hf' hc).inv).unop ≫ f.proj j = (hf.conePointUniqueUpToIso hf').op.inv.unop ≫ f.π.app { as := j }"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "D : Type u��", "CategoryTheory.Category.{v₂, u₂} D", "G : C ⥤ D", "f : X ⟶ Z", "g : Y ⟶ Z", "CategoryTheory.Limits.HasPullback f g", "CategoryTheory.Limits.HasPullback (G.map f) (G.map g)", "W : C", "h : W ⟶ X", "k : W ⟶ Y", "w : h ≫ f = k ≫ g"], "goal": "G.map (CategoryTheory.Limits.pullback.lift h k w) ≫ CategoryTheory.Limits.pullbackComparison G f g =\n  CategoryTheory.Limits.pullback.lift (G.map h) (G.map k) ⋯"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝⁵ : CommSemiring R", "inst✝⁴ : Semiring A", "inst✝³ : Algebra R A", "inst✝² : Semiring B", "inst✝¹ : Algebra R B", "S : Subalgebra R A", "ι : Type u_4", "inst✝ : Nonempty ι", "K : ι → Subalgebra R A", "dir : Directed (fun x x_1 => x ≤ x_1) K", "f : (i : ι) → ↥(K i) →ₐ[R] B", "hf : ∀ (i j : ι) (h : K i ≤ K j), f i = (f j).comp (inclusion h)", "T : Subalgebra R A", "hT : T = iSup K", "i : ι", "x : ↥(K i)", "hx : ↑x ∈ T"], "goal": "Set.iUnionLift (fun i => ↑(K i)) (fun i x => (f i) x) ⋯ ↑T ⋯ ⟨↑x, hx⟩ = (f i) x"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "α : Type u_4", "TopologicalSpace X", "UniformSpace α", "F : ι → X → α", "TopologicalSpace ι", "T2Space α", "𝔖 : Set (Set X)", "𝔖_compact : ∀ K ∈ 𝔖, IsCompact K", "F_clemb : ClosedEmbedding (⇑(UniformOnFun.ofFun 𝔖) ∘ F)", "s : Set ι", "s_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn (F ∘ Subtype.val) K", "s_pointwiseCompact : ∀ K ∈ 𝔖, ∀ x ∈ K, ∃ Q, IsCompact Q ∧ ∀ i ∈ s, F i x ∈ Q"], "goal": "IsCompact (closure s)"}}
+{"state": {"context": ["X : Grp"], "goal": "Grp.uliftFunctor.obj X = Grp.of (ULift.{v, u} ↑X)"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : CancelCommMonoidWithZero M", "q : Associates M", "n : ℕ", "c : Fin (n + 1) → Associates M", "h₁ : StrictMono c", "h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i", "i : Fin (n + 1)", "hr : 1 = c i"], "goal": "IsUnit (c 0)"}}
+{"state": {"context": ["R : Type u", "inst✝² : Ring R", "J : Type v", "inst✝¹ : SmallCategory J", "inst✝ : IsFiltered J", "F : J ⥤ ModuleCatMax R", "r s : R", "x : ↑(M F)"], "goal": "∀ (a : (j : J) × ((((F ⋙ forget₂ (ModuleCat R) AddCommGrp) ⋙ forget₂ AddCommGrp AddGrp) ⋙ forget₂ AddGrp AddMonCat) ⋙ forget AddMonCat).obj j), (r + s) • Quot.mk (Types.Quot.Rel ((((F ⋙ forget₂ (ModuleCat R) AddCommGrp) ⋙ forget₂ AddCommGrp AddGrp) ⋙ forget₂ AddGrp AddMonCat) ⋙ forget AddMonCat)) a = r • Quot.mk (Types.Quot.Rel ((((F ⋙ forget₂ (ModuleCat R) AddCommGrp) ⋙ forget₂ AddCommGrp AddGrp) ⋙ forget₂ AddGrp AddMonCat) ⋙ forget AddMonCat)) a + s • Quot.mk (Types.Quot.Rel ((((F ⋙ forget₂ (ModuleCat R) AddCommGrp) ⋙ forget₂ AddCommGrp AddGrp) ⋙ forget₂ AddGrp AddMonCat) ⋙ forget AddMonCat)) a"}}
+{"state": {"context": ["α : Type u", "CommGroup α", "LE α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b c : α"], "goal": "a⁻¹ ≤ b / c ↔ c ≤ a * b"}}
+{"state": {"context": ["x y : ℝ", "hx : ¬x = 0", "hy : ¬y = 0"], "goal": "-(x * y * log (x * y)) = y * -(x * log x) + x * -(y * log y)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W' : WeierstrassCurve.Jacobian R", "P : Fin 3 → R", "u : R"], "goal": "W'.dblU (u • P) = u ^ 4 * W'.dblU P"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "F G : CategoryTheory.ComposableArrows C 1", "left : F.left = G.left", "right : F.right = G.right", "w :\n  F.hom =\n    CategoryTheory.CategoryStruct.comp (CategoryTheory.eqToHom left)\n      (CategoryTheory.CategoryStruct.comp G.hom (CategoryTheory.eqToHom ⋯))"], "goal": "F = G"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_5", "CompleteLattice α", "p : ι → Prop", "f : Subtype p → α"], "goal": "iSup f = ⨆ i, ⨆ (h : p i), f ⟨i, h⟩"}}
+{"state": {"context": ["R : Type u", "L : Type v", "inst✝⁹ : CommRing R", "inst✝⁸ : LieRing L", "inst✝⁷ : LieAlgebra R L", "M : Type u_1", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : LieRingModule L M", "inst✝³ : LieModule R L M", "M' : Type u_2", "inst✝² : AddCommGroup M'", "inst✝¹ : Module R M'", "inst✝ : LieRingModule L M'", "e : M ≃ₗ⁅R,L⁆ M'"], "goal": "e.range = ⊤"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "s t✝ u : Finset α", "P : Finpartition s", "a : α", "ha : a ∈ s", "t : Finset α", "ht : t ∈ P.parts", "ht' : a ∈ t"], "goal": "∃! t, t ∈ P.parts ∧ a ∈ t"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "f : ℂ → E", "c : ℂ", "R : ℝ", "hR : 0 ≤ R", "hf : ContinuousOn f (Metric.sphere c R)"], "goal": "CircleIntegrable f c R"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "α : Type u_2", "inst✝ : MulAction G α", "s : Set α", "hs : s.Finite", "g : G", "this✝ : Fintype ↑s", "this : Fintype ↑(g • s)"], "goal": "g ∈ stabilizer G s ↔ g • s ⊆ s"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Multiset α"], "goal": "Multiset.toDFinsupp s = Multiset.toDFinsupp t ↔ s = t"}}
+{"state": {"context": ["X Y : Scheme", "U : X.Opens", "hU : IsAffineOpen U", "f : ↑Γ(X, U)"], "goal": "(Spec Γ(X, U)).basicOpen ((Scheme.Hom.app hU.fromSpec U) f) = PrimeSpectrum.basicOpen f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "DecidableEq β", "x : β", "e : α ≃ { y // y ≠ x }", "b : { y // y ≠ x }"], "goal": "(↑((Equiv.optionSubtype x).symm e)).symm ↑b = some (e.symm b)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve.Projective R", "P : Fin 3 → R"], "goal": "(MvPolynomial.eval P) W.polynomialX = W.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W.a₂ * P 0 * P 2 + W.a₄ * P 2 ^ 2)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "A' : Type u_1", "B : Type u_2", "a b : R", "n : ℕ", "inst✝¹⁰ : CommSemiring R", "inst✝⁹ : Semiring A", "inst✝⁸ : CommSemiring A'", "inst✝⁷ : Semiring B", "inst✝⁶ : Algebra R A", "inst✝⁵ : Algebra R A'", "inst✝⁴ : Algebra R B", "p q : R[X]", "x : A", "inst✝³ : CommSemiring S", "inst✝² : Algebra S R", "inst✝¹ : Algebra S A'", "inst✝ : Algebra S B", "g : R →ₐ[S] A'", "y x✝ : A'"], "goal": "(aevalTower (Algebra.ofId S A') x✝) X = (aeval x✝) X"}}
+{"state": {"context": ["α : Type v", "M : Type u", "AddMonoid M", "AddAction M α", "m : M", "a : α"], "goal": "AddAction.period m a ∣ addOrderOf m"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "fa : α → α", "fb : β → β", "f : α → β", "g : β → γ", "s t : Set α", "h : Semiconj f fa fb", "ha : InjOn fa s", "hf : InjOn f (fa '' s)", "x : α", "hx : x ∈ s", "y : α", "hy : y ∈ s", "H : f (fa x) = f (fa y)"], "goal": "f x = f y"}}
+{"state": {"context": ["ι : Type u_1", "M : ι → Type u_2", "inst✝³ : (i : ι) → Monoid (M i)", "N : Type u_3", "inst✝² : Monoid N", "inst✝¹ : (i : ι) → DecidableEq (M i)", "inst✝ : DecidableEq ι", "C : Word M → Prop", "h_empty : C empty", "h_smul : ∀ (i : ι) (m : M i) (w : Word M), C w → C (of m • w)", "w : Word M"], "goal": "C w"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommSemiring R", "x y z : R", "H : IsCoprime (y * z) x"], "goal": "IsCoprime y x"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Groupoid C", "unop : C"], "goal": "(CategoryTheory.Groupoid.invFunctor C).obj unop = Opposite.op unop"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α"], "goal": "M.map id ⋯ = M"}}
+{"state": {"context": ["ι : Type u_1", "σ : Type u_2", "A : Type u_4", "Semiring A", "DecidableEq ι", "AddMonoid ι", "SetLike σ A", "AddSubmonoidClass σ A", "𝒜 : ι → σ", "GradedRing 𝒜"], "goal": "Ideal.IsHomogeneous 𝒜 ⊥"}}
+{"state": {"context": ["L : FirstOrder.Language", "T : L.Theory", "α : Type w", "p : T.CompleteType α"], "goal": "FirstOrder.Language.Theory.IsMaximal ↑p"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_4", "β : Type u_5", "SMul M α", "SMul M β", "a : M", "c : β"], "goal": "a • Sum.inr c = Sum.inr (a • c)"}}
+{"state": {"context": ["K : Type u_7", "L : Type u_8", "Field K", "LieRing L", "LieAlgebra K L", "Module.Finite K L", "x : L"], "goal": "LieAlgebra.rank K L ≤ FiniteDimensional.finrank K ↥(LieSubalgebra.engel K x)"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "InnerProductSpace ℝ E", "v : E"], "goal": "ContinuousOn (stereoToFun v) {x | ((innerSL ℝ) v) x ≠ 1}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁴ : PseudoMetricSpace α", "inst✝³ : ProperSpace α", "x : α", "r✝ : ℝ", "s : Set α", "inst✝² : TopologicalSpace β", "inst✝¹ : ConditionallyCompleteLinearOrder β", "inst✝ : OrderTopology β", "f : α → β", "a z : α", "r : ℝ", "hf : ContinuousOn f (closedBall a r)", "hz : z ∈ closedBall a r", "hf1 : ∀ z' ∈ closedBall a r \\ ball a r, f z < f z'"], "goal": "∃ z ∈ ball a r, IsLocalMin f z"}}
+{"state": {"context": ["R : Type x", "MulOneClass R", "HasDistribNeg R", "a : R"], "goal": "Commute a (-1)"}}
+{"state": {"context": ["x : ℝ"], "goal": "Real.sin x ∈ Set.Icc (-1) 1"}}
+{"state": {"context": [], "goal": ""}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "n : ℕ", "s : Simplex ℝ P (n + 2)"], "goal": "(affineCombination ℝ univ s.pointsWithCircumcenter) ((↑(n + 2 + 1) / ↑(n + 2 - 1)) • (centroidWeightsWithCircumcenter univ - circumcenterWeightsWithCircumcenter (n + 2)) + circumcenterWeightsWithCircumcenter (n + 2)) = (affineCombination ℝ univ s.pointsWithCircumcenter) (mongePointWeightsWithCircumcenter n)"}}
+{"state": {"context": ["K : Type u_2", "L : Type u_3", "LieRing L", "Field K", "LieAlgebra K L", "FiniteDimensional K L", "H : LieSubalgebra K L", "H.IsCartanSubalgebra", "LieModule.IsTriangularizable K (↥H) L", "LieAlgebra.IsKilling K L", "CharZero K", "α : LieModule.Weight K (↥H) L"], "goal": "(LieModule.traceForm K (↥H) L).orthogonal (Submodule.span K {LieAlgebra.IsKilling.coroot α}) = LieModule.Weight.ker"}}
+{"state": {"context": ["α : Type u_1", "PseudoMetricSpace α", "MeasurableSpace α", "OpensMeasurableSpace α", "x : α", "ε : ℝ"], "goal": "MeasurableSet (Metric.closedBall x ε)"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "AddCommGroup M", "Module R M", "LieRingModule L M"], "goal": "Function.Injective LieSubmodule.toSubmodule"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "α ε β : Type u_1", "Monad m", "x : m α", "f : α → ExceptCpsT ε m β"], "goal": "(ExceptCpsT.lift x >>= f).run = do\n  let a ← x\n  (f a).run"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "H : Type u_2", "TopologicalSpace H", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "TopologicalSpace G", "ChartedSpace H G", "Group G", "LieGroup I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "f : M → G", "x₀ : M", "hf : SmoothAt I' I f x₀"], "goal": "SmoothAt I' I (fun x => (f x)⁻¹) x₀"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Abelian C", "X : C", "h : ∀ {Z : C} (f : X ⟶ Z) [inst : CategoryTheory.Epi f], CategoryTheory.IsIso f ↔ f ≠ 0"], "goal": "CategoryTheory.Simple X"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommSemiring R", "inst✝³ : TopologicalSpace R", "inst✝² : TopologicalSemiring R", "inst✝¹ : StarRing R", "inst✝ : ContinuousStar R", "s : Set R"], "goal": "(polynomialFunctions s).starClosure = adjoin R {(toContinuousMapOnAlgHom s) X}"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b : α"], "goal": "b - toIcoDiv hp a b • p = toIcoMod hp a b"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "𝕜 : Type u_2", "inst✝³ : RCLike 𝕜", "G : Type u_3", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "f : ι → 𝕜 → G", "g : 𝕜 → G", "f' : ι → 𝕜 → G", "g' : 𝕜 → G", "x✝ : 𝕜", "inst✝ : l.NeBot", "hf' : TendstoUniformly f' g' l", "hf : ∀ᶠ (n : ι) in l, ∀ (x : 𝕜), HasDerivAt (f n) (f' n x) x", "hfg : ∀ (x : 𝕜), Tendsto (fun n => f n x) l (𝓝 (g x))", "x : 𝕜"], "goal": "HasDerivAt g (g' x) x"}}
+{"state": {"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "this : HasSum (fun n => -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑n ^ s / 2 + -(-I * ↑(↑n).sign) * cexp (-(2 * ↑π * I * ↑a * ↑n)) / ↑n ^ s / 2) (sinZeta (↑a) s)", "n : ℕ", "h : n ≠ 0"], "goal": "(cexp (-(2 * ↑π * ↑a * ↑n) * I) - cexp (2 * ↑π * ↑a * ↑n * I)) * I / 2 / ↑n ^ s = -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑n ^ s / 2 + -(-I * ↑(↑n).sign) * cexp (-(2 * ↑π * I * ↑a * ↑n)) / ↑n ^ s / 2"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "v w : V"], "goal": "(∀ (x : V), G.Reachable x v ↔ G.Reachable x w) → G.Reachable v w"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "E : Type u_3", "F : Type u_4", "A : Type u_5", "inst✝² : NormedAddCommGroup E", "inst✝¹ : CompleteSpace E", "inst✝ : NormedSpace ℝ E", "a b : ℝ", "f g : ℝ → E", "μ : Measure ℝ", "hf : IntervalIntegrable f μ a b", "hg : IntervalIntegrable g μ a b"], "goal": "∫ (x : ℝ) in a..b, f x + g x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ + ∫ (x : ℝ) in a..b, g x ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "f : α × β → ℚ → ℝ", "κ : ProbabilityTheory.Kernel α (β × ℝ)", "ν : ProbabilityTheory.Kernel α β", "self : ProbabilityTheory.IsRatCondKernelCDFAux f κ ν", "a : α", "A : Set β", "_hA : MeasurableSet A", "q : ℚ"], "goal": "∫ (c : β) in A, f (a, c) q ∂ν a = ((κ a) (A ×ˢ Set.Iic ↑q)).toReal"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "inst✝³ : CommSemiring R", "inst✝² : Ring A", "inst✝¹ : Algebra R A", "inst✝ : NoZeroDivisors A", "x : A", "hx : x ≠ 0", "this : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }"], "goal": "Function.Injective ⇑(mulLeft R x)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "t : Finset (α × β × γ)", "b : β", "c : γ"], "goal": "(SimpleGraph.TripartiteFromTriangles.graph t).Adj (Sum3.in₂ c) (Sum3.in₁ b) ↔ ∃ a, (a, b, c) ∈ t"}}
+{"state": {"context": ["a x✝ y✝ : ℂ", "ha : a ≠ 0", "x y : ℂ"], "goal": "angle (x * a) (y * a) = angle x y"}}
+{"state": {"context": ["n : ℕ"], "goal": "List.Nat.antidiagonalTuple 2 n = List.map (fun i => ![i.1, i.2]) (List.Nat.antidiagonal n)"}}
+{"state": {"context": ["b : ℂ", "hb : 0 < b.re", "c T : ℝ", "hT : 0 ≤ T", "vert_norm_bound : ∀ {T : ℝ}, 0 ≤ T → ∀ {c y : ℝ}, |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))"], "goal": "ℝ"}}
+{"state": {"context": ["α : Type u_1", "ms : MeasurableSpace α", "m : MeasureTheory.OuterMeasure α", "h : ms ≤ m.caratheodory"], "goal": "(m.toMeasure h).toOuterMeasure = m.trim"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β✝ : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β✝ i)", "s : Finset ι", "x : (i : ↑↑s) → β✝ ↑i", "i✝ : ι", "f✝ : Π₀ (i : ι), β✝ i", "b✝ : β✝ i✝", "j✝ : ι", "β : ι → Type u_1", "inst✝ : (i : ι) → AddZeroClass (β i)", "f : Π₀ (i : ι), β i", "i : ι", "b : β i", "j : ι", "h : i ≠ j"], "goal": "(update i f b) j = (erase i f + single i b) j"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l f : Filter α", "g : Filter β", "s : Set (β → α × β)", "hs : s ∈ map Prod.mk f", "t : Set β", "ht : t ∈ g"], "goal": "s.seq t ∈ (map (fun b a => (a, b)) g).seq f"}}
+{"state": {"context": ["m n : ℕ", "h : m ≤ n", "this : n - m + m = n"], "goal": "(n - m + m - m).gcd (n - m + m) = (n - m).gcd m"}}
+{"state": {"context": ["G : Type u_1", "inst✝¹ : Group G", "φ : G →* G", "inst✝ : Finite G", "hφ : FixedPointFree ⇑φ", "h2 : Function.Involutive ⇑φ", "g h : G"], "goal": "Commute g h"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "T : Type u_3", "inst✝⁹ : CommRing R", "inst✝⁸ : Ring S", "inst✝⁷ : Algebra R S", "A : Type u_4", "B : Type u_5", "inst✝⁶ : CommRing A", "inst✝⁵ : CommRing B", "inst✝⁴ : IsDomain B", "inst✝³ : Algebra A B", "K : Type u_6", "inst✝² : Field K", "S' : Type u_7", "inst✝¹ : Semiring S'", "inst✝ : Algebra R S'", "pb : PowerBasis R S", "f✝ g : S →ₐ[R] S'", "h : f✝ pb.gen = g pb.gen", "f : R[X]"], "goal": "f✝ ((aeval pb.gen) f) = g ((aeval pb.gen) f)"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "l : List (Sigma β)", "DecidableEq α", "a : α"], "goal": "l.NodupKeys → (List.kerase a l).NodupKeys"}}
+{"state": {"context": ["X : Type u", "x : X", "TopologicalSpace X", "ιX : Sort u_3", "ιF : Sort u_4", "pX : ιX → Prop", "sX : ιX → Set X", "pF : ιF → Prop", "sF : ιF → Set X", "F : Filter X", "hX : (𝓝 x).HasBasis pX sX", "hF : F.HasBasis pF sF"], "goal": "ClusterPt x F ↔ ∀ ⦃i : ιX⦄, pX i → ∀ ⦃j : ιF⦄, pF j → (sX i ∩ sF j).Nonempty"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "a b c✝ d : ℝ≥0∞", "r p q : ℝ≥0", "x y z ε ε₁ ε₂ : ℝ≥0∞", "s✝ : Set ℝ≥0∞", "R : Type u_4", "inst✝¹ : SMul R ℝ≥0∞", "inst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞", "s : Set ℝ≥0∞", "c : R"], "goal": "c • sSup s = ⨆ i ∈ s, c • i"}}
+{"state": {"context": ["σ : Type u_1", "inst✝² : Fintype σ", "inst✝¹ : DecidableEq σ", "R : Type u_2", "inst✝ : CommRing R", "k : ℕ", "a : ℕ × ℕ", "ha : a ∈ antidiagonal k"], "goal": "∑ y : σ, ∑ x ∈ powersetCard a.1 univ, ((-1) ^ x.card * ∏ a ∈ x, X a) * X y ^ (k - x.card) = ∑ i : σ, (∑ i ∈ powersetCard a.1 univ, (-1) ^ a.1 * ∏ a ∈ i, X a) * X i ^ a.2"}}
+{"state": {"context": ["θ ψ : ℝ"], "goal": "Real.cos θ = Real.cos ψ ↔ ↑θ = ↑ψ ∨ ↑θ = -↑ψ"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y"], "goal": "Continuous Sum.inl"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "L : Type u_2", "inst✝¹ : Field L", "inst✝ : NumberField L", "f : K ≃ₐ[ℚ] L", "f₀ : 𝓞 K ≃ₗ[ℤ] 𝓞 L := (AlgEquiv.restrictScalars ℤ f).mapIntegralClosure.toLinearEquiv", "x✝ : Module.Free.ChooseBasisIndex ℤ (𝓞 K)"], "goal": "f ((algebraMap (𝓞 K) K) ((RingOfIntegers.basis K) x✝)) = (algebraMap (𝓞 L) L) (f₀ ((RingOfIntegers.basis K) x✝))"}}
+{"state": {"context": ["X : Type u_7", "Y : Type u_8", "TopologicalSpace X", "TopologicalSpace Y", "self : PartialHomeomorph X Y"], "goal": "IsOpen self.source"}}
+{"state": {"context": ["n : ℕ", "j : Fin n", "i : Fin (n + 1)", "hi : i ≠ 0"], "goal": "i.pred hi < j ↔ i < j.succ"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "T : D", "Y : C", "S : CategoryTheory.Functor C D", "f : S.obj Y ⟶ T"], "goal": "(CategoryTheory.CostructuredArrow.mk f).hom = f"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "n : ℕ"], "goal": "(truncate n).comp ((fromPadicInt p).comp (toPadicInt p)) = (truncate n).comp (RingHom.id (𝕎 (ZMod p)))"}}
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+{"state": {"context": [], "goal": "Cardinal.lift.{max v u, u} = Cardinal.lift.{v, u}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "f : α × β → ℚ → ℝ", "κ : ProbabilityTheory.Kernel α (β × ℝ)", "ν : ProbabilityTheory.Kernel α β", "self : ProbabilityTheory.IsRatCondKernelCDF f κ ν"], "goal": "Measurable f"}}
+{"state": {"context": ["x : ℝ", "n : ℤ"], "goal": "Real.tan (x + ↑n * Real.pi) = Real.tan x"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝³ : Ring k", "inst✝² : AddCommGroup V", "inst✝¹ : Module k V", "inst✝ : AffineSpace V P", "ι : Type u_4", "s : Set V", "p : P", "h : Submodule.span k (range Subtype.val) = ⊤"], "goal": "(affineSpan k ({p} ∪ (fun v => v +ᵥ p) '' s)).direction = ⊤.direction"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : PseudoMetricSpace α", "f : ℕ → α", "a : α", "h : Summable fun n => dist (f n) (f n.succ)", "ha : Tendsto f atTop (𝓝 a)"], "goal": "dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "M : Type u_2", "AddCommGroup M", "Module R M", "N : Type u_3", "AddCommGroup N", "Module R N", "Module.Free R M", "Module.Finite R M", "Module.Free R N", "Module.Finite R N"], "goal": "(LinearMap.llcomp R M N M).compr₂ (LinearMap.trace R M) = (LinearMap.llcomp R N M N).flip.compr₂ (LinearMap.trace R N)"}}
+{"state": {"context": ["f : ℝ → ℝ", "f' x : ℝ", "s : Set ℝ", "hf : HasDerivWithinAt f f' s x"], "goal": "HasDerivWithinAt (fun x => Real.cos (f x)) (-Real.sin (f x) * f') s x"}}
+{"state": {"context": ["β : Type u_2", "PseudoMetricSpace β", "x y : β"], "goal": "nndist { down := x } { down := y } = nndist x y"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "M : Type u_3", "inst✝⁸ : CommRing R", "inst✝⁷ : IsDomain R", "inst✝⁶ : Field K", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "inst✝³ : Module.Finite R M", "inst✝² : Module.Free R M", "inst✝¹ : Module K M", "inst✝ : Module.Finite K M", "φ : Module.End R M", "h : IsNilpotent φ"], "goal": "IsNilpotent ((LinearMap.toMatrix (chooseBasis R M) (chooseBasis R M)) φ)"}}
+{"state": {"context": ["α✝ : Sort u", "β✝ : Sort v", "γ : Sort w", "α : Sort u_1", "β : Type u_2", "f : α → β", "hf : Injective f", "a : α"], "goal": "(ofInjective f hf) ((ofInjective f hf).symm ⟨f a, ⋯⟩) = (ofInjective f hf) a"}}
+{"state": {"context": ["V : Type u_1", "Fintype V", "G : SimpleGraph V", "DecidableRel G.Adj", "r : ℕ", "hr : 0 < r", "hn : r ≤ Fintype.card V", "hG : G.IsTuranMaximal r"], "goal": "¬G.CliqueFree r"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "f g✝ : α → β", "c c₁ c₂ x✝ : α", "inst✝¹ : Add α", "inst✝ : Add γ", "h : Periodic f c", "g : AddHom γ α", "g_inv : α → γ", "hg : RightInverse g_inv ⇑g", "x : γ"], "goal": "(f ∘ ⇑g) (x + g_inv c) = (f ∘ ⇑g) x"}}
+{"state": {"context": ["G : Type u_1", "AddLeftCancelMonoid G", "Finite G", "n : ℕ", "x : G"], "goal": "addOrderOf (n • x) = addOrderOf x / (addOrderOf x).gcd n"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type w", "L.Structure M", "α : Type u'", "β : Type v'", "γ : Type u_3", "Finite γ", "f : α → β ⊕ γ", "φ : L.Formula α", "v : β → M"], "goal": "(FirstOrder.Language.Formula.iExs f φ).Realize v ↔ ∃ i, φ.Realize fun a => Sum.elim v i (f a)"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "Semiring R", "AddCommMonoid M", "Module R M"], "goal": "⊥.FG"}}
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+{"state": {"context": ["U : Type u_1", "Quiver U", "Quiver.HasInvolutiveReverse U", "u : U", "e : Quiver.Star u"], "goal": "((Quiver.starEquivCostar u) e).fst = e.fst"}}
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+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "P : Finpartition Finset.univ", "hP : P.IsEquipartition", "G : SimpleGraph α", "DecidableRel G.Adj", "ε : ℝ"], "goal": "(SzemerediRegularity.increment hP G ε).IsEquipartition"}}
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+{"state": {"context": ["a : SetTheory.PGame"], "goal": "⟦-a⟧ = -⟦a⟧"}}
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+{"state": {"context": ["R : Type u", "CommRing R", "IsPrincipalIdealRing R", "IsDomain R", "r : R"], "goal": "Associated (Submodule.IsPrincipal.generator (Ideal.span {r})) r"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type u_1", "N : Type u_2", "L.Structure M", "L.Structure N", "f : M ↪ₑ[L] N"], "goal": "M ≅[L] N"}}
+{"state": {"context": ["α : Type u_1", "p q : α → Prop", "h : ∀ {x : α}, q x → p x"], "goal": "∀ (a : Subtype q), ↑↑((Equiv.subtypeSubtypeEquivSubtype h).symm a) = ↑a"}}
+{"state": {"context": ["α : Type u_1", "UniformSpace α", "Group α", "UniformGroup α"], "goal": "𝓤 α = Filter.comap (fun x => x.1 / x.2) (𝓝 1)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "n : ℕ", "hn : 0 < 1 + ε ↑n"], "goal": "0 ≤ 1 + ε ↑n"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "A : ι → Type u_2", "(i : ι) → SMul α (A i)", "a : α", "x : GradedMonoid A"], "goal": "(a • x).fst = x.fst"}}
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+{"state": {"context": ["X : Type u_1"], "goal": "Finsupp.toFreeAbelianGroup.comp FreeAbelianGroup.toFinsupp = AddMonoidHom.id (FreeAbelianGroup X)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t✝ s : Set α", "t : Set β", "f : α → β", "hf : MapsTo f s t", "f_inj : InjOn f s"], "goal": "(f '' s).encard ≤ t.encard"}}
+{"state": {"context": ["R : Type u", "M M₁ : Type v", "M' : Type v'", "ι : Type w", "inst✝⁸ : Ring R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup M'", "inst✝⁵ : AddCommGroup M₁", "inst✝⁴ : Module R M", "inst✝³ : Module R M'", "inst✝² : Module R M₁", "inst✝¹ : NoZeroSMulDivisors R M", "inst✝ : StrongRankCondition R", "this : Nontrivial R", "h : ∀ (w : M), ∃ c, c • 0 = w"], "goal": "finrank R M ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "a : α", "Subsingleton α"], "goal": "Finset.univ = {a}"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝ : ConditionallyCompleteLinearOrder R", "f : S → Tropical (WithTop R)"], "goal": "untrop (∑ i ∈ ∅, f i) = sInf (untrop ∘ f '' ↑∅)"}}
+{"state": {"context": ["α : Type u_1", "f : Filter α", "f.NeBot", "m : α → NNReal", "x : ℝ"], "goal": "Filter.Tendsto (fun a => ↑(m a)) f (𝓝 x) ↔ ∃ (hx : 0 ≤ x), Filter.Tendsto m f (𝓝 ⟨x, hx⟩)"}}
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+{"state": {"context": ["R : Type u_2", "A : Type u_3", "B : Type u_4", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "Semiring B", "StarRing B", "Algebra R B", "StarModule R B", "f : A →⋆ₐ[R] B", "hf : Function.Injective ⇑f"], "goal": "Function.Injective (StarSubalgebra.map f)"}}
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+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "p₁ p₂ p₃ : P", "h : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫_ℝ = 0", "h0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0"], "goal": "dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "a : α", "s : Finset α", "h : a ∉ s"], "goal": "(insert a s).toList.Perm (a :: s.toList)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "M : Type u_5", "AddCommMonoid M", "f : α → M", "s : Set β", "g : β → α", "hg : Set.InjOn g s"], "goal": "∑ᶠ (i : α) (_ : i ∈ g '' s), f i = ∑ᶠ (j : β) (_ : j ∈ s), f (g j)"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "SeminormedRing R", "SeminormedAddCommGroup M", "x : tsze R M"], "goal": "‖x‖₊ = ‖x.fst‖₊ + ‖x.snd‖₊"}}
+{"state": {"context": ["p : ℝ", "hp₀ : 0 < p", "hp₁ : p < 1", "hp₀' : 0 < 1 / p", "hp₁' : 1 < 1 / p", "f : ℝ≥0 ≃o ℝ≥0 := orderIsoRpow (1 / p) hp₀'", "h₁ : StrictConvexOn ℝ≥0 univ ⇑f", "h₂ : ∀ (x : ℝ≥0), f.symm x = x ^ p", "x : ℝ≥0", "mx : x ∈ univ", "y : ℝ≥0", "my : y ∈ univ", "hxy : x ≠ y", "a b : ℝ≥0", "ha : 0 < a", "hb : 0 < b", "hab : a + b = 1"], "goal": "a • f.symm x + b • f.symm y < f.symm (a • x + b • y)"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X : Type v", "Ring X", "Bialgebra R X"], "goal": "↑(BialgebraCat.of R X).toBundled = X"}}
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+{"state": {"context": ["R : Type u", "CommRing R", "f : R[X]"], "goal": "(AdjoinRoot.isAdjoinRoot f).map = AdjoinRoot.mk f"}}
+{"state": {"context": ["m : Type u_2", "α : Type v", "Fintype m", "NonUnitalNonAssocSemiring α", "v : m → α"], "goal": "(fun x => 0) ⬝ᵥ v = 0"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "¬(x ≠ 0 ∧ y ≠ 0) → ¬(o.kahler x) y ≠ 0"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "inst✝⁸ : CommSemiring R", "inst✝⁷ : CommSemiring S", "inst✝⁶ : Semiring A", "inst✝⁵ : Semiring B", "inst✝⁴ : Algebra R S", "inst✝³ : Algebra R A", "inst✝² : Algebra S A", "inst✝¹ : Algebra R B", "inst✝ : IsScalarTower R S A", "s t : Set A"], "goal": "Subalgebra R A → Set A"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure μ", "h : NeZero μ", "c : ℝ≥0∞"], "goal": "⨍⁻ (_x : α), c ∂μ = c"}}
+{"state": {"context": ["α : Type u_1", "l m : Language α"], "goal": "(l * m).reverse = m.reverse * l.reverse"}}
+{"state": {"context": ["F : Type u", "E✝ : Type v", "inst✝⁷ : Field F", "inst✝⁶ : Field E✝", "inst✝⁵ : Algebra F E✝", "K : Type w", "inst✝⁴ : Field K", "inst✝³ : Algebra F K", "E : Type v", "inst✝² : Field E", "inst✝¹ : Algebra F E", "inst✝ : Algebra.IsSeparable F E", "hfd : FiniteDimensional F E", "L : IntermediateField F E", "x : E", "h : finSepDegree F ↥L = finrank F ↥L", "M : IntermediateField (↥L) E := (↥L)⟮x⟯", "heq : finSepDegree F ↥L * finSepDegree ↥L ↥M = finrank F ↥L * finrank ↥L ↥M", "this : Algebra.IsAlgebraic ↥L ↥M"], "goal": "finSepDegree F ↥(restrictScalars F M) = finrank F ↥(restrictScalars F M)"}}
+{"state": {"context": ["n : ℕ"], "goal": "sqrt ↑n = ↑n.sqrt"}}
+{"state": {"context": ["R : Type u", "CommRing R", "X Y : Type u", "Ring X", "Ring Y", "Algebra R X", "Algebra R Y", "i : AlgebraCat.of R X ≅ AlgebraCat.of R Y"], "goal": "algEquivIsoAlgebraIso.inv i = i.toAlgEquiv"}}
+{"state": {"context": ["M : Type u_1", "m₁ m₂ : CancelMonoid M", "h_mul : HMul.hMul = HMul.hMul"], "goal": "m₁ = m₂"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "NontriviallyNormedField 𝕂", "NormedCommRing 𝔸", "NormedAlgebra 𝕂 𝔸", "CompleteSpace 𝔸", "CharZero 𝕂", "x : 𝔸", "hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius"], "goal": "HasFDerivAt (NormedSpace.exp 𝕂) (NormedSpace.exp 𝕂 x • 1) x"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝¹ : NormedAddCommGroup G'", "inst✝ : NormedSpace 𝕜 G'", "f f₀ f₁ g : E → F", "f' f₀' f₁' g' e : E →L[𝕜] F", "x : E", "s t : Set E", "L L₁ L₂ : Filter E", "hL : L ≤ 𝓝 x", "h : HasFDerivAtFilter f f' x L", "this✝ : Tendsto (fun x' => f x' - f x) L (𝓝 0)", "this : Tendsto (fun x_1 => f x_1 - f x + f x) L (𝓝 (0 + f x))"], "goal": "Tendsto f L (𝓝 (f x))"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "α : Type v", "NonUnitalNonAssocRing α", "Fintype n", "v : n → α", "A : Matrix m n α"], "goal": "A *ᵥ -v = -(A *ᵥ v)"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "Semiring R", "TopologicalSpace F", "TopologicalSpace B", "TopologicalSpace (Bundle.TotalSpace F E)", "AddCommMonoid F", "Module R F", "(x : B) → AddCommMonoid (E x)", "(x : B) → Module R (E x)", "e : Trivialization F Bundle.TotalSpace.proj", "Trivialization.IsLinear R e", "b : B", "hb : b ∈ e.baseSet"], "goal": "Trivialization.linearMapAt R e b = ↑(Trivialization.linearEquivAt R e b hb)"}}
+{"state": {"context": ["M : Type u_1", "inst✝ : CommMonoid M", "a : Stream' M", "s : Finset ℕ", "hs : s.Nonempty"], "goal": "∏ i ∈ s, a.get i ∈ FP a"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a b : α", "DenselyOrdered α"], "goal": "Set.Iio a ⊆ Set.Iic b ↔ a ≤ b"}}
+{"state": {"context": ["S : Set ℝ", "f : ℝ → ℝ", "x y : ℝ", "hfc : StrictConcaveOn ℝ S f", "hx : x ∈ S", "hy : y ∈ S", "hxy : x < y", "hfd : DifferentiableWithinAt ℝ f (Set.Iio y) y"], "goal": "derivWithin f (Set.Iio y) y < slope f x y"}}
+{"state": {"context": ["G : Type u_1", "GroupWithZero G", "a : G", "ha : a ≠ 0"], "goal": "⇑(Equiv.mulRight₀ a ha) = fun x => x * a"}}
+{"state": {"context": ["R : Type u_1", "Mul R", "Add R", "self : RightDistribClass R", "a b c : R"], "goal": "(a + b) * c = a * c + b * c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "a b✝ c d : ℝ≥0∞", "r p q : ℝ≥0", "x✝¹ y✝ z ε ε₁ ε₂ : ℝ≥0∞", "s : Set ℝ≥0∞", "b : ℝ≥0", "x✝ : ⊤ ≠ ⊤ ∨ ↑b ≠ ⊤", "x : ℝ≥0", "y : ℝ≥0∞ × ℝ≥0∞", "hy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)"], "goal": "y.2 + ↑x < y.1"}}
+{"state": {"context": ["α : Type u", "Semiring α", "LinearOrder α", "a b : α", "IsRightCancelAdd α", "NoZeroDivisors α", "ZeroLEOneClass α", "ExistsAddOfLE α", "PosMulMono α", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x < x_1"], "goal": "a * a + b * b = 0 ↔ a = 0 ∧ b = 0"}}
+{"state": {"context": ["Ω : Type u_1", "ι : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "a b : ℝ", "f✝ : ℕ → Ω → ℝ", "N✝ n m : ℕ", "ω✝ : Ω", "f : ℕ → Ω → ℝ", "N : ℕ", "ω : Ω", "hab : a < b", "h : ∀ (n : ℕ), upperCrossingTime a b f N n ω ≠ N", "this : StrictMono fun n => upperCrossingTime a b f N n ω", "k : ℕ", "hk : N < (fun n => upperCrossingTime a b f N n ω) k"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "E : Type u_2", "PseudoEMetricSpace E", "s : Set α"], "goal": "LowerSemicontinuous fun f => eVariationOn f s"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s : Set E", "ι : Type u_6", "u : Finset ι", "A : ι → E → F", "h : ∀ i ∈ u, DifferentiableOn 𝕜 (A i) s"], "goal": "DifferentiableOn 𝕜 (fun y => ∑ i ∈ u, A i y) s"}}
+{"state": {"context": ["K : Type u_2", "L : Type u_3", "LieRing L", "Field K", "LieAlgebra K L", "FiniteDimensional K L", "H : LieSubalgebra K L", "H.IsCartanSubalgebra", "LieModule.IsTriangularizable K (↥H) L", "LieAlgebra.IsKilling K L", "PerfectField K", "α : LieModule.Weight K (↥H) L"], "goal": "↑(LieAlgebra.corootSpace ⇑α) =\n  Submodule.span K {(LieAlgebra.IsKilling.cartanEquivDual H).symm (LieModule.Weight.toLinear K (↥H) L α)}"}}
+{"state": {"context": ["M : Type u_1", "AddZeroClass M", "S : AddSubmonoid M"], "goal": "Nontrivial ↥S ↔ ∃ x ∈ S, x ≠ 0"}}
+{"state": {"context": ["β : Type u_2", "ι : Sort u_4", "f : ι → Set β"], "goal": "⋃₀ Set.range f = ⋃ x, f x"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "C✝ : ℝ", "f g✝ : ℂ → E", "z : ℂ", "C : ℝ", "hC₀ : 0 < C", "a b : ℝ", "hza : a - b < z.im", "hle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C", "hzb : z.im < a + b", "hle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C", "hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + b))", "hab : a - b < a + b", "hb : 0 < b", "hπb : 0 < π / 2 / b", "c : ℝ", "hc : c < π / 2 / b", "B : ℝ", "hO : f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo (a - b) (a + b))] fun z => expR (B * expR (c * |z.re|))", "d : ℝ", "hd : d < π / 2 / b", "hcd : c < d", "hd₀ : 0 < d", "hb' : d * b < π / 2", "aff : ℂ → ℂ := fun w => ↑d * (w - ↑a * I)", "g : ℝ → ℂ → ℂ := fun ε w => cexp (↑ε * (cexp (aff w) + cexp (-aff w)))", "ε : ℝ", "ε₀ : ε < 0"], "goal": "‖g ε z • f z‖ ≤ C"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "r : Rel α β", "l₁ : Filter α", "l₂ : Filter β"], "goal": "(∀ s ∈ l₂, r.core s ∈ l₁) ↔ l₁ ≤ rcomap r l₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "p : ι → Prop", "s : ι → Set α", "𝓕 : Filter ι", "a : α"], "goal": "liminf s cofinite = {x | {n | x ∉ s n}.Finite}"}}
+{"state": {"context": ["μ : ℝ", "v : ℝ≥0", "c : ℝ", "hv : ¬v = 0", "hc : ¬c = 0", "e : ℝ ≃ᵐ ℝ := (Homeomorph.mulLeft₀ c hc).symm.toMeasurableEquiv", "he' : ∀ (x : ℝ), HasDerivAt (⇑e) ((fun x => c⁻¹) x) x", "s' : Set ℝ", "hs' : MeasurableSet s'", "x : ℝ"], "goal": "|c⁻¹| * |c| = 1"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts", "this : Continuous eval"], "goal": "chaar K₀ ∈ eval ⁻¹' {1}"}}
+{"state": {"context": ["α : Type u_1", "a : α", "Preorder α", "x : ↑(Set.Iic a)"], "goal": "x = ⊤ ↔ ↑x = a"}}
+{"state": {"context": ["α : Type u", "J : Type w", "inst✝³ : SmallCategory J", "inst✝² : FinCategory J", "inst✝¹ : SemilatticeInf α", "inst✝ : OrderTop α", "x y : α"], "goal": "limit (pair x y) = Finset.univ.inf (pair x y).obj"}}
+{"state": {"context": ["x✝ y x : ℝ", "h1 : 0 < x", "h2 : x ≤ 1", "this : log (1 / x) < 1 / x"], "goal": "|log x * x| < 1"}}
+{"state": {"context": ["x y : UInt8"], "goal": "(x * y).toNat = x.toNat * y.toNat % UInt8.size"}}
+{"state": {"context": ["w : List ℕ", "hw : List.Sorted (fun x x_1 => x ≥ x_1) w", "hpos : ∀ x ∈ w, 0 < x"], "goal": "¬False ∧ ∀ n < w.length, ∃ (h : n < w.length), 0 < w.get ⟨n, ⋯⟩"}}
+{"state": {"context": ["α : Type u", "EMetricSpace α", "CompleteSpace α"], "goal": "IsClosed (Set.range TopologicalSpace.NonemptyCompacts.toCloseds)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "UniformSpace α", "UniformSpace β", "f : α → β", "self : UniformEmbedding f"], "goal": "Function.Injective f"}}
+{"state": {"context": ["R : CommRingCat", "f : ↑R", "x : ↑↑(Spec R).toPresheafedSpace"], "goal": "IsUnit (((Spec R).presheaf.germ ⟨x, _root_.trivial⟩) ((Scheme.ΓSpecIso R).inv f)) ↔ x ∈ PrimeSpectrum.basicOpen f"}}
+{"state": {"context": ["n p : ℕ"], "goal": "p ∈ n.primeFactorsList ↔ Nat.Prime p ∧ p ∣ n ∧ n ≠ 0"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r✝ r : ℚ", "h : ‖↑r‖ ≤ 1", "norm_denom_lt : ‖↑r.den‖ < 1"], "goal": "False"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u", "f : (i : Fin (n + 1)) → α i"], "goal": "Fin.removeNth 0 f = Fin.tail f"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝ : UniformSpace α", "x : α", "V : Set (α × α)", "V_in : V ∈ 𝓤 α", "w : Set (α × α)", "w_in : w ∈ 𝓤 α", "w_symm : SymmetricRel w", "w_sub : w ○ w ⊆ V", "this : ball x w ×ˢ ball x w ∈ 𝓝 (x, x)", "u v : α", "u_in : (u, v).1 ∈ ball x w", "v_in : (u, v).2 ∈ ball x w"], "goal": "(u, v) ∈ V"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁷ : TopologicalSpace α", "inst✝⁶ : TopologicalSpace β", "inst✝⁵ : LinearOrder α", "inst✝⁴ : LinearOrder β", "inst✝³ : OrderTopology α", "inst✝² : OrderTopology β", "inst✝¹ : DenselyOrdered α", "inst✝ : FirstCountableTopology α", "x y : α", "h : x < y", "u : ℕ → α", "hu_anti : StrictAnti u", "hu_mem : ∀ (n : ℕ), u n ∈ Ioo x y", "hux : Tendsto u atTop (𝓝 x)", "v : ℕ → α", "hv_mono : StrictMono v", "hv_mem : ∀ (n : ℕ), v n ∈ Ioo (u 0) y", "hvy : Tendsto v atTop (𝓝 y)"], "goal": "∃ u v, StrictAnti u ∧ StrictMono v ∧ (∀ (k : ℕ), u k ∈ Ioo x y) ∧ (∀ (l : ℕ), v l ∈ Ioo x y) ∧ (∀ (k l : ℕ), u k < v l) ∧ Tendsto u atTop (𝓝 x) ∧ Tendsto v atTop (𝓝 y)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "p : G.Walk v u", "h : G.Adj u v"], "goal": "(SimpleGraph.Walk.cons h p).IsCycle ↔ p.IsPath ∧ s(u, v) ∉ p.edges"}}
+{"state": {"context": ["x y a : ℝ", "ha : 0 < a", "hb : 0 < 1 - a", "hab : a + (1 - a) = 1", "hx : 0 < x", "hy : 0 < y"], "goal": "0 < Gamma y"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α"], "goal": "[TopologicalSpace.NoetherianSpace α, WellFounded fun s t => s < t, ∀ (s : Set α), IsCompact s,\n    ∀ (s : TopologicalSpace.Opens α), IsCompact ↑s].TFAE"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : E → ℝ", "s : Set E", "hc : DifferentiableOn ℝ f s"], "goal": "DifferentiableOn ℝ (fun x => Real.cos (f x)) s"}}
+{"state": {"context": ["R : Type u_3", "M : Type u_4", "N : Type u_5", "CommSemiring R", "AddCommMonoid M", "Module R M", "AddCommMonoid N", "Module R N", "Q : QuadraticMap R M N", "Q' : M → N", "h : Q' = ⇑Q"], "goal": "Q.copy Q' h = Q"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "l : Filter α", "h : ∀ (x : α), {x}ᶜ ∈ l", "s : Set α", "hs : sᶜ.Finite"], "goal": "s ∈ l"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "Field K", "AddCommGroup V", "Module K V", "A : Projectivization.Subspace K V", "x : Projectivization K V"], "goal": "x ∈ A.carrier ↔ x ∈ A"}}
+{"state": {"context": ["α : Type u", "Group α", "LE α", "CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α"], "goal": "(OrderIso.mulRight a).toEquiv = Equiv.mulRight a"}}
+{"state": {"context": ["a b : Nat"], "goal": "(¬a < b) = (b ≤ a)"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommRing R", "S : Finset R", "hS : Ideal.span ↑S = ⊤", "hN✝ : ∀ (s : { x // x ∈ S }), IsNoetherianRing (Localization.Away ↑s)", "I : ℕ →o Submodule R R", "floc : (s : R) → R →+* Localization.Away s := fun s => algebraMap R (Localization.Away s)", "suitableN : R → Set ℕ := fun s => {n | ∀ (m : ℕ), n ≤ m → Ideal.map (floc s) (I n) = Ideal.map (floc s) (I m)}", "minN : R → ℕ := fun s => sInf (suitableN s)", "hSuit : ∀ (s : { x // x ∈ S }), minN ↑s ∈ suitableN ↑s", "N : ℕ := S.sup minN", "hN : ∀ (s : { x // x ∈ S }), minN ↑s ≤ N", "n : ℕ", "hn : N ≤ n"], "goal": "⨅ x, Ideal.comap (algebraMap R (Localization.Away ↑x)) (Ideal.map (algebraMap R (Localization.Away ↑x)) (I N)) = ⨅ x, Ideal.comap (algebraMap R (Localization.Away ↑x)) (Ideal.map (algebraMap R (Localization.Away ↑x)) (I n))"}}
+{"state": {"context": ["α : Type u_1", "Group α", "LinearOrder α", "a b : α", "h : mabs a = b"], "goal": "a = b ∨ a = b⁻¹"}}
+{"state": {"context": ["m : Type u_1 → Type u_2", "α : Type u_3", "β : Type u_1", "inst✝ : Monad m", "as : Array α", "b : β", "f : α → β → m (ForInStep β)"], "goal": "forIn (List.drop 0 as.data) b f = forIn as.data b f"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "x : Y", "h : x ∈ e.target"], "goal": "↑e.symm x ∈ e.source"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Monoid α", "Monoid β", "f : α →* β", "hf : Function.Surjective ⇑f"], "goal": "Function.Surjective (ConjClasses.map f)"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "J : Type u₂", "inst✝² : Category.{v₂, u₂} J", "X✝ : Type v₂", "X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝¹ : HasPullback f g", "inst✝ : HasPushout f.op g.op"], "goal": "(pushout.inl f.op g.op ≫ (pullbackIsoUnopPushout f g).hom.op).unop = (pullback.fst f g).op.unop"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "f : ℝ → E", "a b : ℝ", "g : ℝ → E", "fdiff : DifferentiableOn ℝ f (Set.Icc a b)", "gdiff : DifferentiableOn ℝ g (Set.Icc a b)", "hderiv : Set.EqOn (derivWithin f (Set.Icc a b)) (derivWithin g (Set.Icc a b)) (Set.Ico a b)", "hi : f a = g a", "y : ℝ"], "goal": "y ∈ Set.Icc a b → f y = g y"}}
+{"state": {"context": ["𝕜₁ : Type u_1", "𝕜₂ : Type u_2", "inst✝¹⁰ : NontriviallyNormedField 𝕜₁", "inst✝⁹ : NormedField 𝕜₂", "σ₁₂ : 𝕜₁ →+* 𝕜₂", "M₁ : Type u_3", "M₂ : Type u_4", "inst✝⁸ : SeminormedAddCommGroup M₁", "inst✝⁷ : AddCommGroup M₂", "inst✝⁶ : NormedSpace 𝕜₁ M₁", "inst✝⁵ : Module 𝕜₂ M₂", "inst✝⁴ : UniformSpace M₂", "inst✝³ : UniformAddGroup M₂", "inst✝² : ContinuousConstSMul 𝕜₂ M₂", "inst✝¹ : T2Space M₂", "inst✝ : CompleteSpace M₂", "u : M₁ →SL[σ₁₂] M₂", "hu : u ∈ closure {f | IsCompactOperator ⇑f}"], "goal": "u ∈ {f | IsCompactOperator ⇑f}"}}
+{"state": {"context": ["G : Type u", "Group G", "F : Type v", "Field F", "MulSemiringAction G F", "Fintype G", "x : F"], "goal": "(FixedPoints.minpoly G F x).Monic"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s t m : Multiset α"], "goal": "m.dedup.toFinset = m.toFinset"}}
+{"state": {"context": ["α : Type u_1", "P : Set α → Prop", "m : (s : Set α) → P s → ℝ≥0∞", "P0 : P ∅", "m0 : m ∅ P0 = 0", "PU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)", "msU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯", "m_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂", "f : α ≃ α", "Pm : ∀ (s : Set α), P (⇑f ⁻¹' s) ↔ P s", "mm : ∀ (s : Set α) (hs : P s), m (⇑f ⁻¹' s) ⋯ = m s hs", "A : Set α"], "goal": "(MeasureTheory.inducedOuterMeasure m P0 m0) (⇑f ⁻¹' A) = (MeasureTheory.inducedOuterMeasure m P0 m0) A"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "B : C", "α : Type u_2", "X : α → C", "π : (a : α) → X a ⟶ B", "self : CategoryTheory.EffectiveEpiFamilyStruct X π", "W : C", "e : (a : α) → X a ⟶ W", "h :\n  ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),\n    CategoryTheory.CategoryStruct.comp g₁ (π a₁) = CategoryTheory.CategoryStruct.comp g₂ (π a₂) →\n      CategoryTheory.CategoryStruct.comp g₁ (e a₁) = CategoryTheory.CategoryStruct.comp g₂ (e a₂)", "m : B ⟶ W"], "goal": "(∀ (a : α), CategoryTheory.CategoryStruct.comp (π a) m = e a) → m = self.desc e ⋯"}}
+{"state": {"context": ["σ K : Type u", "Fintype K", "Field K", "Finite σ", "p : MvPolynomial σ K", "h : ∀ (v : σ → K), (MvPolynomial.eval v) p = 0", "hp : p ∈ MvPolynomial.restrictDegree σ K (Fintype.card K - 1)"], "goal": "p = 0"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "n : ℕ"], "goal": "(p * Polynomial.X).coeff (n + 1) = p.coeff n"}}
+{"state": {"context": ["f : ℝ → ℝ", "f' x p : ℝ", "hf : HasDerivAt f f' x", "hx : f x ≠ 0 ∨ 1 ≤ p"], "goal": "HasDerivAt (fun y => f y ^ p) (f' * p * f x ^ (p - 1)) x"}}
+{"state": {"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "inst✝⁸ : Fintype n", "inst✝⁷ : Fintype o", "inst✝⁶ : CommRing R", "inst✝⁵ : Finite m", "inst✝⁴ : DecidableEq n", "M₁ : Type u_6", "M₂ : Type u_7", "inst✝³ : AddCommGroup M₁", "inst✝² : AddCommGroup M₂", "inst✝¹ : Module R M₁", "inst✝ : Module R M₂", "A : Matrix m n R", "v₁ : Basis m R M₁", "v₂ : Basis n R M₂", "val✝ : Fintype m", "e₁ : (m → R) ≃ₗ[R] M₁ := (Pi.basisFun R m).equiv v₁ (Equiv.refl m)", "e₂ : (n → R) ≃ₗ[R] M₂ := (Pi.basisFun R n).equiv v₂ (Equiv.refl n)", "range_e₂ : LinearMap.range e₂ = ⊤", "i : n", "aux₁ : ((toLin (Pi.basisFun R n) (Pi.basisFun R m)) A) ((Pi.basisFun R n) i) = ∑ j : m, A j i • (Pi.basisFun R m) j", "aux₂ : ∀ (e : n ≃ n), ((Pi.basisFun R n).equiv v₂ e) ((Pi.basisFun R n) i) = v₂ (e i)"], "goal": "((↑e₁ ∘ₗ A.mulVecLin) ∘ₗ LinearMap.single i) 1 = (((toLin v₂ v₁) A ∘ₗ ↑e₂) ∘ₗ LinearMap.single i) 1"}}
+{"state": {"context": ["M : Type u_2", "AddZeroClass M", "H : AddSubmonoid M", "b : { b // (fun x => x ∈ H.op) b }"], "goal": "↑(H.equivOp.symm b) = AddOpposite.unop ↑b"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "m : Type u_3", "n : Type u_4", "inst✝³ : DecidableEq m", "inst✝² : DecidableEq n", "inst✝¹ : Fintype m", "inst✝ : Fintype n", "M₁₁ : Matrix m m R", "M₁₂ : Matrix m n R", "M₂₁ : Matrix n m R", "M₂₂ M✝ : Matrix n n R", "i j : n", "M : Matrix n n R", "h : matPolyEquiv (M.charpoly • 1) = matPolyEquiv M.charmatrix.adjugate * (X - C M)"], "goal": "(aeval M) M.charpoly = 0"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemiring α", "FloorSemiring α", "Sub α", "OrderedSub α", "ExistsAddOfLE α", "a : α", "n : ℕ"], "goal": "⌊a - ↑n⌋₊ = ⌊a⌋₊ - n"}}
+{"state": {"context": ["J : Type w", "C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroMorphisms C", "D : Type uD", "inst✝² : Category.{uD', uD} D", "inst✝¹ : HasZeroMorphisms D", "P Q : C", "F : C ⥤ D", "inst✝ : F.PreservesZeroMorphisms", "X✝ Y✝ : BinaryBicone P Q", "f : X✝ ⟶ Y✝"], "goal": "((fun A => { pt := F.obj A.pt, fst := F.map A.fst, snd := F.map A.snd, inl := F.map A.inl, inr := F.map A.inr, inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }) X✝).inl ≫ F.map f.hom = ((fun A => { pt := F.obj A.pt, fst := F.map A.fst, snd := F.map A.snd, inl := F.map A.inl, inr := F.map A.inr, inl_fst := ⋯, inl_snd := ⋯, inr_fst := ⋯, inr_snd := ⋯ }) Y✝).inl"}}
+{"state": {"context": ["k : Type u_1", "V₁ : Type u_2", "P₁ : Type u_3", "Ring k", "AddCommGroup V₁", "Module k V₁", "AffineSpace V₁ P₁", "s : AffineSubspace k P₁"], "goal": "AffineSubspace.map (AffineMap.id k P₁) s = s"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : SemilatticeSup α", "a : α"], "goal": "atTop = comap Subtype.val atTop"}}
+{"state": {"context": ["E : ℕ → Type u_1", "x : (n : ℕ) → E n", "m n : ℕ", "h : m ≤ n"], "goal": "PiNat.cylinder x n ⊆ PiNat.cylinder x m"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B : Set α", "h : M✶.RkPos", "hB : M.Base B"], "goal": "B ⊂ M.E"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁹ : RCLike 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : InnerProductSpace 𝕜 E", "inst✝⁴ : InnerProductSpace 𝕜 F", "inst✝³ : InnerProductSpace 𝕜 G", "inst✝² : FiniteDimensional 𝕜 E", "inst✝¹ : FiniteDimensional 𝕜 F", "inst✝ : FiniteDimensional 𝕜 G", "ι₁ : Type u_5", "ι₂ : Type u_6", "b₁ : Basis ι₁ 𝕜 E", "b₂ : Basis ι₂ 𝕜 F", "A : E →ₗ[𝕜] F", "B : F →ₗ[𝕜] E"], "goal": "A = adjoint B ↔ ∀ (i₁ : ι₁) (i₂ : ι₂), ⟪A (b₁ i₁), b₂ i₂⟫_𝕜 = ⟪b₁ i₁, B (b₂ i₂)⟫_𝕜"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "J : FractionalIdeal S P"], "goal": "J ≤ 1 ↔ ∃ I, ↑I = J"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝³ : CommRing R", "inst✝² : AddCommGroup M", "inst✝¹ : Module R M", "inst✝ : IsArtinian R M", "r : R", "x : M", "n : ℕ", "hn : ∀ (m : ℕ), n ≤ m → LinearMap.range (r ^ n • LinearMap.id) = LinearMap.range (r ^ m • LinearMap.id)"], "goal": "∃ n y, r ^ n.succ • y = r ^ n • x"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u", "B : Type u_3", "inst✝¹⁸ : CommRing A", "inst✝¹⁷ : Field K", "inst✝¹⁶ : CommRing B", "inst✝¹⁵ : Field L", "inst✝¹⁴ : Algebra A K", "inst✝¹³ : Algebra B L", "inst✝¹² : Algebra A B", "inst✝¹¹ : Algebra K L", "inst✝¹⁰ : Algebra A L", "inst✝⁹ : IsScalarTower A K L", "inst✝⁸ : IsScalarTower A B L", "inst✝⁷ : IsDomain A", "inst✝⁶ : IsFractionRing A K", "inst✝⁵ : IsIntegralClosure B A L", "inst✝⁴ : IsFractionRing B L", "inst✝³ : FiniteDimensional K L", "inst✝² : Algebra.IsSeparable K L", "inst✝¹ : IsIntegrallyClosed A", "inst✝ : IsDedekindDomain B", "I J : FractionalIdeal B⁰ L", "hI : I ≠ 0", "hJ : J ≠ 0", "x : L"], "goal": "x ∈ dual A K I ↔ ∀ a ∈ I, ((traceForm K L) x) a ∈ (algebraMap A K).range"}}
+{"state": {"context": ["R : Type u", "Semiring R", "p : R[X]", "h : p.degree = 1"], "goal": "p = Polynomial.C p.leadingCoeff * Polynomial.X + Polynomial.C (p.coeff 0)"}}
+{"state": {"context": ["α : Sort u_1", "p : α → Prop"], "goal": "Function.Injective fun a => ↑a"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t✝ : Set X", "l : Filter X", "hs : IsCompact s", "H : ∀ x ∈ s, Disjoint (𝓝 x) l", "U : X → Set X", "hUl : ∀ x ∈ s, (U x)ᶜ ∈ l", "hxU : ∀ x ∈ s, x ∈ U x", "hUo : ∀ x ∈ s, IsOpen (U x)", "t : Finset X", "hts : ∀ x ∈ t, x ∈ s", "hst : s ⊆ ⋃ x ∈ t, U x"], "goal": "(⋃ x ∈ t, U x)ᶜ ∈ l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "a b c : α"], "goal": "(∀ ⦃c : α⦄, a < c → ¬c < b) ↔ Ioo a b = ∅"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s : Set E", "f : E → F", "x : E", "n : ℕ∞", "m : ℕ", "h : ContDiffWithinAt 𝕜 n f s x", "hmn : ↑m < n", "hs : UniqueDiffOn 𝕜 (insert x s)"], "goal": "DifferentiableWithinAt 𝕜 (iteratedFDerivWithin 𝕜 m f s) s x"}}
+{"state": {"context": ["α : Type u_1", "m : Set α → ℝ≥0∞", "s t : Set α", "h : ∀ (u : Set α), (s ∩ u).Nonempty → (t ∩ u).Nonempty → m u = ⊤"], "goal": "(MeasureTheory.OuterMeasure.boundedBy m) (s ∪ t) =\n  (MeasureTheory.OuterMeasure.boundedBy m) s + (MeasureTheory.OuterMeasure.boundedBy m) t"}}
+{"state": {"context": ["G : Type u", "AddGroup G", "N : AddSubgroup G", "nN : N.Normal", "x : G"], "goal": "(QuotientAddGroup.mk' N) x = ↑x"}}
+{"state": {"context": ["F : Type u_1", "G : Type u_2", "H : Type u_3", "inst✝³ : FunLike F G H", "a : G", "b : H", "inst✝² : AddGroupWithOne G", "inst✝¹ : AddGroup H", "inst✝ : AddConstMapClass F G H 1 b", "f : F", "x : G", "n : ℕ"], "goal": "f (x - ↑n) = f x - n • b"}}
+{"state": {"context": ["f : StieltjesFunction", "hf0 : Tendsto (↑f) atBot (𝓝 0)", "hf1 : Tendsto (↑f) atTop (𝓝 1)"], "goal": "cdf f.measure = f"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "s t : Set ι", "h : s ≠ t"], "goal": "Disjoint ↑(I.splitCenterBox s) ↑(I.splitCenterBox t)"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "inst✝⁹ : ConditionallyCompleteLinearOrder ι", "inst✝⁸ : IsWellOrder ι fun x x_1 => x < x_1", "inst✝⁷ : Countable ι", "inst✝⁶ : TopologicalSpace ι", "inst✝⁵ : OrderTopology ι", "inst✝⁴ : FirstCountableTopology ι", "inst✝³ : TopologicalSpace β", "inst✝² : PseudoMetrizableSpace β", "inst✝¹ : MeasurableSpace β", "inst✝ : BorelSpace β", "f : Filtration ι m", "u : ι → Ω → β", "τ : Ω → ι", "hτ : IsStoppingTime f τ", "N : ι", "hτbdd : ∀ (x : Ω), τ x ≤ N", "s : Set β", "hs : MeasurableSet s", "hf : Adapted f u", "n : ι"], "goal": "MeasurableSet {ω | (fun x => hitting u s (τ x) N x) ω ≤ n}"}}
+{"state": {"context": ["R : Type u", "inst✝ : Ring R", "X✝ : Type u"], "goal": "{ obj := fun X => of R (X →₀ R), map := fun {X Y} f => Finsupp.lmapDomain R R f }.map (𝟙 X✝) = 𝟙 ({ obj := fun X => of R (X →₀ R), map := fun {X Y} f => Finsupp.lmapDomain R R f }.obj X✝)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "φ ψ : ℝ → ℝ", "s : Set E", "a b m : ℝ", "x y : E", "f g : E → ℝ", "hf : UniformConvexOn s φ f"], "goal": "UniformConcaveOn s φ (-f)"}}
+{"state": {"context": ["α : Type u_1", "n : Type u_3", "A B : Matrix n n α", "Sub α", "hA : A.IsSymm", "hB : B.IsSymm"], "goal": "(A - B).IsSymm"}}
+{"state": {"context": ["n : ℕ"], "goal": "1 + (↑n + 1)⁻¹ - 1 = (↑n + 1)⁻¹"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "α : Type u_3", "inst✝² : Group G", "inst✝¹ : Group H", "inst✝ : MulAction G α", "a✝ : G", "s : Subgroup Gᵐᵒᵖ", "a : G", "h : ∀ (b : Gᵐᵒᵖ), a • b ∈ ↑s ↔ b ∈ ↑s"], "goal": "a ∈ s.unop"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "OrderedCommSemiring R", "Nontrivial R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "∀ (α : Type u_1) [inst : Zero α] [inst_1 : TopologicalSpace α], StarOrderedRing C(α, R)₀", "TopologicalSpace A", "NonUnitalRing A", "StarRing A", "PartialOrder A", "StarOrderedRing A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "NonUnitalContinuousFunctionalCalculus R p", "a : A", "ha : p a", "f g : C(↑(σₙ R a), R)₀", "hfg : f ≤ g"], "goal": "(cfcₙHom ha) f ≤ (cfcₙHom ha) g"}}
+{"state": {"context": ["R : Type u", "CommRing R", "U : (TopologicalSpace.Opens (PrimeSpectrum R))ᵒᵖ", "r : R"], "goal": "(algebraMap R ↑((AlgebraicGeometry.Spec.structureSheaf R).val.obj U)) r =\n  (AlgebraicGeometry.StructureSheaf.toOpen R (Opposite.unop U)) r"}}
+{"state": {"context": ["L : FirstOrder.Language", "κ : Cardinal.{w}", "T : L.Theory", "h : κ.Categorical T", "h1 : ℵ₀ ≤ κ", "h2 : Cardinal.lift.{w, max u v} L.card ≤ Cardinal.lift.{max u v, w} κ", "hS : T.IsSatisfiable", "hT : ∀ (M : T.ModelType), Infinite ↑M"], "goal": "T.IsComplete"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{u₂, u₁} C", "inst✝³ : GaloisCategory C", "F✝ : C ⥤ FintypeCat", "inst✝² : FiberFunctor F✝", "A B : C", "inst✝¹ : IsConnected A", "inst✝ : IsGalois B", "f : A ⟶ B", "σ τ : Aut A", "F : C ⥤ FintypeCat := GaloisCategory.getFiberFunctor C", "a : ↑(F.obj A)"], "goal": "autMap f (σ * τ) = autMap f σ * autMap f τ"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Semiring R", "Semiring S", "f : R →+* S", "hf : Function.Injective ⇑f", "p : R[X]"], "goal": "(Polynomial.map f p).nextCoeff = f p.nextCoeff"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f : α → β", "x : α", "y : β", "s t : Set β", "h : ContinuousWithinAt f (f ⁻¹' s) x", "ht : t ∈ 𝓝[s] y", "hxy : y = f x"], "goal": "f ⁻¹' t ∈ 𝓝[f ⁻¹' s] x"}}
+{"state": {"context": ["α : Type u", "β : Type v", "X : Type u_1", "ι : Type u_2", "inst✝¹ : PseudoMetricSpace α", "γ : Type w", "inst✝ : MetricSpace γ", "x y : γ"], "goal": "nndist x y = 0 ↔ x = y"}}
+{"state": {"context": ["P : ℕ → Prop", "DecidablePred P", "n : ℕ", "h : P n"], "goal": "PartENat.find P ≤ ↑n"}}
+{"state": {"context": ["R : Type uR", "inst✝³ : Semiring R", "S : Type uS", "inst✝² : CommSemiring S", "T : Type uT", "A : Type uA", "inst✝¹ : Semiring A", "inst✝ : Algebra S A", "r✝ r : R → R → Prop"], "goal": "∀ (x y : R), { toFun := fun x => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯ }.toFun (x * y) = { toFun := fun x => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯ }.toFun x * { toFun := fun x => { toQuot := Quot.mk (Rel r) x }, map_one' := ⋯ }.toFun y"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "self : LocallyFiniteOrder α", "a b x : α"], "goal": "x ∈ LocallyFiniteOrder.finsetIoc a b ↔ a < x ∧ x ≤ b"}}
+{"state": {"context": ["X Y : Type u"], "goal": "(CategoryTheory.Limits.Types.binaryProductIso X Y).hom ≫ Prod.fst = CategoryTheory.Limits.prod.fst"}}
+{"state": {"context": ["R : Type u", "self : EuclideanDomain R", "a : R", "b : R"], "goal": "b ≠ 0 → ¬EuclideanDomain.r (a * b) a"}}
+{"state": {"context": ["α : Sort u"], "goal": "Function.Surjective Opposite.op"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "X : C"], "goal": "(CategoryTheory.MonoidalOpposite.mopMopEquivalence C).counitIso.inv.app X = 𝟙 X"}}
+{"state": {"context": ["ι : Type u_1", "M₀ : Type u_4", "MulZeroOneClass M₀", "s t : Set ι", "Nontrivial M₀", "h : s.indicator 1 = t.indicator 1"], "goal": "s = t"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "β : Type u_4", "Preorder α", "Preorder β", "f : ι → α", "g : ι → β", "h : Antivary f g", "s : Set ι"], "goal": "AntivaryOn f g s"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "R : Type u_1", "inst✝ : NonAssocSemiring R", "f : (k : ℕ) → R →+* ZMod (p ^ k)", "f_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1", "r : R", "ε : ℝ", "hε : 0 < ε", "ε' : ℚ", "hε' : ↑ε' < ε", "hε'0 : 0 < ε'"], "goal": "∃ N, ∀ n ≥ N, ‖limNthHom f_compat r - ↑(nthHom f r n)‖ < ε"}}
+{"state": {"context": ["α : Type u_1", "R : α → α → Prop", "inst✝ : DecidableRel R", "l : List α"], "goal": "pwFilter R l = l ↔ Pairwise R l"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : ι → α → β", "inst✝ : IsFiniteMeasure μ", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ", "h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε", "g : ι → α → β := fun i => Exists.choose ⋯", "hgmeas : ∀ (i : ι), StronglyMeasurable (g i)"], "goal": "UniformIntegrable f p μ"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "inst✝⁵ : CancelCommMonoidWithZero R", "inst✝⁴ : UniqueFactorizationMonoid R", "inst✝³ : CancelCommMonoidWithZero α", "inst✝² : UniqueFactorizationMonoid α", "β : Type u_3", "inst✝¹ : CancelCommMonoidWithZero β", "inst✝ : DecidableEq α", "s : Finset α", "i : α → ℕ", "p : α", "hps : p ∉ s", "is_prime : ∀ q ∈ insert p s, Prime q", "is_coprime : ∀ q ∈ insert p s, ∀ q' ∈ insert p s, q ∣ q' → q = q'", "hp : Prime p", "d : α", "hdprod : d ∣ ∏ p' ∈ s, p' ^ i p'", "hd : Prime d", "hdp : d ∣ p", "q : α", "q_mem : q ∈ s.val", "q_mem' : q ^ i q ∈ Multiset.map (fun p' => p' ^ i p') s.val", "hdq : d ∣ q ^ i q"], "goal": "p ∈ s"}}
+{"state": {"context": ["z x✝ y✝ : ℝ", "n : ℕ", "hn : n ≠ 0", "x y : ℝ", "h : x < y", "hy : 0 < y", "hn' : 0 < ↑n", "q : ℚ", "hxq : max 0 x ^ (↑n)⁻¹ < ↑q", "hqy : ↑q < y ^ (↑n)⁻¹", "this : 0 ≤ max 0 x ^ (↑n)⁻¹"], "goal": "∃ q, 0 < q ∧ x < ↑q ^ n ∧ ↑q ^ n < y"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "inst✝ : Ring α", "a b : α", "n : ℕ", "ha : Even a", "hb : Odd b"], "goal": "Odd (a + -b)"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E"], "goal": "fderiv 𝕜 id x = ContinuousLinearMap.id 𝕜 E"}}
+{"state": {"context": ["x : ℝ≥0", "h : x ≠ 0", "y : ��"], "goal": "↑x ^ y = ↑(x ^ y)"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "NormedAddCommGroup V", "InnerProductSpace ℝ V", "MetricSpace P", "NormedAddTorsor V P", "hd2 : Fact (FiniteDimensional.finrank ℝ V = 2)", "Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P"], "goal": "-(∡ p₁ p₂ p₃).sign = (∡ p₃ p₂ p₁).sign"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace α", "μ : Measure α", "inst✝¹ : NormedAddCommGroup β", "p : ℝ≥0∞", "f : ι → α → β", "inst✝ : IsFiniteMeasure μ", "hp : 1 ≤ p", "hp' : p ≠ ⊤", "hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ", "h : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal.ofReal ε", "g : ι → α → β := fun i => Exists.choose ⋯"], "goal": "UniformIntegrable f p μ"}}
+{"state": {"context": ["α : Type u_1", "TopologicalSpace α", "LE α"], "goal": "↑⊥ = ∅"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ"], "goal": "CochainComplex.HomComplex.δ 0 1 (CochainComplex.mappingCone.snd φ) =\n  -(↑(CochainComplex.mappingCone.fst φ)).comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯"}}
+{"state": {"context": ["X : TwoP"], "goal": "∀ (a : ((TwoP.swap ⋙ TwoP.swap).obj X).toBipointed.X),\n  (TwoP.swapEquiv.unitIso.inv.app X).toFun a =\n    (TwoP.swap.map (TwoP.swap.map { toFun := id, map_fst := ⋯, map_snd := ⋯ }) ≫\n          TwoP.swap.map { toFun := id, map_fst := ⋯, map_snd := ⋯ } ≫ { toFun := id, map_fst := ⋯, map_snd := ⋯ }).toFun\n      ((𝟙 (TwoP.swap.obj (TwoP.swap.obj X))).toFun a)"}}
+{"state": {"context": ["K : Type u_1", "DecidableEq K", "Γ : K → Type u_2", "Λ : Type u_3", "σ : Type u_4", "M : Λ → Turing.TM2.Stmt Γ Λ σ"], "goal": "Turing.Respects (Turing.TM2.step M) (Turing.TM1.step (Turing.TM2to1.tr M)) Turing.TM2to1.TrCfg"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Fintype α", "inst✝ : Fintype β", "emb : Fintype (α ↪ β)"], "goal": "‖α ↪ β‖ = ‖β‖.descFactorial ‖α‖"}}
+{"state": {"context": ["C : Type u", "inst✝² : Category.{v, u} C", "P✝ : Cᵒᵖ ⥤ Type (max v u)", "X✝¹ : C", "R✝ : Presieve X✝¹", "S✝ : Sieve X✝¹", "inst✝¹ : R✝.hasPullbacks", "P : Cᵒᵖ ⥤ Type w", "X✝ : C", "R : Presieve X✝", "S : Sieve X✝", "B : C", "I : Type", "X : I → C", "π : (i : I) → X i ⟶ B", "inst✝ : (Presieve.ofArrows X π).hasPullbacks", "z₁ z₂ : SecondObj P X π", "h : ∀ (ij : I × I), Pi.π (fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) ij z₁ = Pi.π (fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) ij z₂"], "goal": "∀ (j : Discrete (I × I)), limit.π (Discrete.functor fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) j z₁ = limit.π (Discrete.functor fun ij => P.obj (op (Limits.pullback (π ij.1) (π ij.2)))) j z₂"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "ι : Type u_4", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : Module 𝕜 E", "inst✝² : AddCommGroup F", "inst✝¹ : Module 𝕜 F", "inst✝ : Nonempty ι", "B✝ B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜", "p : SeminormFamily 𝕜 E F := B.toSeminormFamily", "h : (𝓝 0).HasBasis (fun x => 0 < x) (Metric.ball 0)", "h' : (Filter.pi fun x => 𝓝 0).HasBasis (fun If => If.1.Finite ∧ ∀ i ∈ If.1, 0 < If.2 i) fun If => If.1.pi fun i => Metric.ball 0 (If.2 i)", "h'' : (Filter.comap (fun x y => (B x) y) (Filter.pi fun x => 𝓝 0)).HasBasis (fun If => If.1.Finite ∧ ∀ i ∈ If.1, 0 < If.2 i) fun i => (fun x y => (B x) y) ⁻¹' i.1.pi fun i_1 => Metric.ball 0 (i.2 i_1)"], "goal": "∀ (i' : Set E), B.toSeminormFamily.basisSets i' → ∃ i, (i.1.Finite ∧ ∀ i_1 ∈ i.1, 0 < i.2 i_1) ∧ ((fun x y => (B x) y) ⁻¹' i.1.pi fun i_1 => Metric.ball 0 (i.2 i_1)) ⊆ _root_.id i'"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "f : ℂ → E", "s : Set ℂ", "c : ℂ", "hs : s ∈ 𝓝 c", "hd : DifferentiableOn ℂ f (s \\ {c})", "hc : ContinuousAt f c", "x : ℂ", "hx : x ∈ s", "hne : x ≠ c"], "goal": "DifferentiableWithinAt ℂ f s x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "hab : b ≤ a", "hcd : c ≤ d", "h₁ : b ≤ d", "h₂ : c ≤ a"], "goal": "Ico a b ∪ Ico c d = Ico (min a c) (max b d)"}}
+{"state": {"context": ["n : Type u_2", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : Matrix n n R", "i j : n", "hij : i ≠ j"], "goal": "(A.updateRow i (A i + A j)).det = A.det"}}
+{"state": {"context": ["P : Type u_1", "L : Type u_2", "inst✝³ : Membership P L", "inst✝² : ProjectivePlane P L", "inst✝¹ : Finite P", "inst✝ : Finite L", "p₁ p₂ p₃ : P", "l₁ l₂ l₃ : L", "h₂₁ : p₂ ∉ l₁", "h₂₂ : p₂ ∈ l₂", "h₂₃ : p₂ ∈ l₃", "h₃₁ : p₃ ∉ l₁", "h₃₂ : p₃ ∈ l₂", "h₃₃ : p₃ ∉ l₃", "val✝ : Fintype { p // p ∈ l₂ }", "h : mkPoint ⋯ ∈ l₁ ∧ mkPoint ⋯ ∈ l₂"], "goal": "∃ a b c, ¬↑a = ↑b ∧ ¬↑a = ↑c ∧ ¬↑b = ↑c"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "s✝ t✝ u : Multiset α", "a b : α", "s t : Multiset α", "l₁ l₂ : List α"], "goal": "↑(List.foldl List.erase l₁ l₂) = foldl erase ⋯ ↑l₁ ↑l₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p : PMF α", "a : α", "h : p a = 1", "a' : α", "ha' : a' ∈ p.support", "ha : a' ∉ {a}"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "X : Type u_3", "α : Type u_7", "tX : TopologicalSpace X", "uα : UniformSpace α", "p : κ → Prop", "s : κ → Set (α × α)", "F : ι → X → α", "x₀ : X", "hα : (uniformity α).HasBasis p s"], "goal": "EquicontinuousAt F x₀ ↔ ∀ (k : κ), p k → ∀ᶠ (x : X) in nhds x₀, ∀ (i : ι), (F i x₀, F i x) ∈ s k"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "Preorder β", "f : α → β", "s : Set α", "a : α", "TopologicalSpace β", "hf : IsLocalMaxOn f s a", "(𝓝[>] f a).NeBot"], "goal": "¬𝓝 (f a) ≤ Filter.map f (𝓝[s] a)"}}
+{"state": {"context": [], "goal": "↑(-π / 2) ≠ 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Preorder α", "Preorder β", "LocallyFiniteOrder α", "LocallyFiniteOrder β", "DecidableRel fun x x_1 => x ≤ x_1", "p q : α × β"], "goal": "Finset.Icc p q = Finset.Icc p.1 q.1 ×ˢ Finset.Icc p.2 q.2"}}
+{"state": {"context": ["𝓕 : Type u_1", "𝕜 : Type u_2", "α : Type u_3", "ι : Type u_4", "κ : Type u_5", "E : Type u_6", "F : Type u_7", "G : Type u_8", "inst✝¹ : NormedGroup E", "inst✝ : NormedGroup F", "a b : E"], "goal": "0 < ‖a / b‖ ↔ a ≠ b"}}
+{"state": {"context": ["X : Scheme", "inst✝ : IsIntegral X", "U : X.Opens", "x : ↥U", "y : ↑(X.presheaf.obj (op U))", "hy : (X.presheaf.germ x) y = (X.presheaf.germ x) 0", "W : Opens ↑↑X.toPresheafedSpace", "hW : ↑x ∈ W", "iU : W ⟶ U", "e : (X.presheaf.map iU.op) y = (X.presheaf.map iU.op) 0"], "goal": "y = 0"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α", "h : AntisymmRel (fun x x_1 => x ≤ x_1) a b"], "goal": "a ⩿ b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝¹⁴ : CommRing R", "inst✝¹³ : CommRing S", "inst✝¹² : Algebra R S", "P : Generators R S", "R' : Type u'", "S' : Type v'", "inst✝¹¹ : CommRing R'", "inst✝¹⁰ : CommRing S'", "inst✝⁹ : Algebra R' S'", "P' : Generators R' S'", "inst✝⁸ : Algebra R R'", "inst✝⁷ : Algebra S S'", "inst✝⁶ : Algebra R S'", "inst✝⁵ : IsScalarTower R R' S'", "inst✝⁴ : IsScalarTower R S S'", "T : Type uT", "inst✝³ : CommRing T", "inst✝² : Algebra R T", "inst✝¹ : Algebra S T", "inst✝ : IsScalarTower R S T", "f g : P.Hom P'", "x : P.Cotangent"], "goal": "((map f - map g) x).val = ((f.sub g ∘ₗ P.cotangentComplex) x).val"}}
+{"state": {"context": ["R : Type u", "Semiring R", "n : ℕ"], "goal": "Polynomial.erase n 0 = 0"}}
+{"state": {"context": ["R✝ : Type u", "S : Type v", "inst✝⁶ : CommSemiring R✝", "inst✝⁵ : CommSemiring S", "R : Type u", "inst✝⁴ : CommRing R", "inst✝³ : IsNoetherianRing R", "A : Type u", "inst✝² : CommRing A", "inst✝¹ : IsDomain A", "inst✝ : IsNoetherianRing A", "I M : Ideal R", "hgt : ∀ J > M, (fun I => ∃ Z, (Multiset.map asIdeal Z).prod ≤ I) J", "h_prM : ¬M.IsPrime", "htop : ¬M = ⊤", "lt_add : ∀ z ∉ M, M < M + span R {z}", "x : R", "hx : x ∉ M", "y : R", "hy : y ∉ M", "hxy : x * y ∈ M"], "goal": "(fun I => ∃ Z, (Multiset.map asIdeal Z).prod ≤ I) M"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs : ∀ (n : ℕ), s ≠ -↑n", "hs' : a ≠ 0 ∨ s ≠ 1"], "goal": "HurwitzZeta.hurwitzZeta a (1 - s) =\n  (2 * ↑π) ^ (-s) * Complex.Gamma s *\n    (cexp (-↑π * Complex.I * s / 2) * HurwitzZeta.expZeta a s +\n      cexp (↑π * Complex.I * s / 2) * HurwitzZeta.expZeta (-a) s)"}}
+{"state": {"context": ["B : Type u₁", "inst✝² : Bicategory B", "C : Type u₂", "inst✝¹ : Bicategory C", "D : Type u₃", "inst✝ : Bicategory D", "F : Pseudofunctor B C", "a✝ b✝ : B", "f : a✝ ⟶ b✝"], "goal": "(ρ_ (F.map f)).hom = (F.map f ◁ (fun a => (F.mapId a).inv) b✝ ≫ (fun {a b c} f g => (F.mapComp f g).inv) f (𝟙 b✝)) ≫ (F.map₂Iso (ρ_ f)).hom"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "D : Type u_2", "CategoryTheory.Category.{u_4, u_2} D", "F : C ⥤ D", "X Y : AlgebraicGeometry.PresheafedSpace C", "f : X ⟶ Y"], "goal": "(F.mapPresheaf.map f).base = f.base"}}
+{"state": {"context": ["α : Type u", "PseudoMetricSpace α", "β : Type u_3", "l : Filter β", "l.NeBot", "x : α", "r : β → ℝ", "hr : Filter.Tendsto r l Filter.atTop", "hc : ∀ᶠ (i : β) in l, IsCompact (Metric.closedBall x (r i))"], "goal": "ProperSpace α"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝⁶ : MeasurableSpace Ω", "R : Type u_2", "inst✝⁵ : SMul R ℝ≥0", "inst✝⁴ : SMul R ℝ≥0∞", "inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞", "inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞", "inst✝¹ : TopologicalSpace Ω", "inst✝ : OpensMeasurableSpace Ω", "μ : FiniteMeasure Ω", "f g : Ω →ᵇ ℝ≥0", "le_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)", "ω : Ω"], "goal": "(fun a => ↑(f a)) ω ≤ (fun a => ↑(g a) + edist f g) ω"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "inst✝ : CommRing R"], "goal": "(aeval (Function.uncurry fun i => (![] i).coeff)) (wittOne p 0) = 1"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β₁ : Type u_3", "β₂ : Type u_4", "γ₁ : Type u_5", "γ₂ : Type u_6", "f₁ : α₁ → β₁ → Finset γ₁", "f₂ : α₂ → β₂ → Finset γ₂", "g₁ : α₁ → β₂ → Finset γ₁", "g₂ : α₁ → β₂ → Finset γ₂", "a : α₁ ⊕ α₂", "b : β₁ ⊕ β₂", "c₂ : γ₂"], "goal": "Sum.inr c₂ ∈ Finset.sumLexLift f₁ f₂ g₁ g₂ a b ↔\n  (∃ a₁ b₂, a = Sum.inl a₁ ∧ b = Sum.inr b₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨ ∃ a₂ b₂, a = Sum.inr a₂ ∧ b = Sum.inr b₂ ∧ c₂ ∈ f₂ a₂ b₂"}}
+{"state": {"context": ["m n : ℕ", "α : Fin (n + 1) → Type u", "x : α (last n)", "q✝ : (i : Fin (n + 1)) → α i", "p : (i : Fin n) → α i.castSucc", "i✝ : Fin n", "y : α i✝.castSucc", "z : α (last n)", "β : Type u_1", "a : β", "q : Fin n → β", "b : β", "i : Fin (n.succ + 1)", "h : ¬i = 0", "j : Fin (n + 1) := i.pred h", "ji : j = i.pred h", "this : i = j.succ"], "goal": "cons a (snoc q b) i = snoc (cons a q) b i"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : CommMonoid R", "inst✝ : DecidableRel Dvd.dvd", "r a : R"], "goal": "a * a ∣ r → IsUnit a ↔ multiplicity a r ≤ 1 ∨ IsUnit a"}}
+{"state": {"context": ["m : Type u_2", "α : Type v", "AddCommMonoid α", "CommSemigroup α", "Fintype m", "M : (Matrix m m α)ᵐᵒᵖ"], "goal": "(Matrix.transposeRingEquiv m α).symm M = (MulOpposite.unop M)ᵀ"}}
+{"state": {"context": ["R : Type u_1", "S : Type ?u.122514", "inst✝¹⁰ : CommRing R", "inst✝⁹ : CommRing S", "inst✝⁸ : Algebra R S", "inst✝⁷ : LocalRing R", "M : Type u_2", "N : Type ?u.122979", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R M", "inst✝³ : Module R N", "P : Type ?u.124119", "inst✝² : AddCommGroup P", "inst✝¹ : Module R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "hg : Surjective ⇑g", "h : Exact ⇑f ⇑g", "inst✝ : Module.Finite R M"], "goal": "Subsingleton (k ⊗[R] M) ↔ Subsingleton M"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "PseudoEMetricSpace X", "PseudoEMetricSpace Y", "C : ℝ≥0", "f : X → Y"], "goal": "HolderWith C 1 f ↔ LipschitzWith C f"}}
+{"state": {"context": ["b : ℂ", "hb : 0 < b.re"], "goal": "(𝓕 fun x => cexp (-↑π * b * ↑x ^ 2)) = fun t => 1 / b ^ (1 / 2) * cexp (-↑π / b * ↑t ^ 2)"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "f : ι → α → β", "μ : MeasureTheory.Measure α", "hf : ∀ (i : ι), AEMeasurable (f i) μ", "p : α → (ι → β) → Prop", "i : ι"], "goal": "Measurable (aeSeq hf p i)"}}
+{"state": {"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "K₁ K₂ : Compacts G", "h : ↑K₁ ⊆ ↑K₂", "eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁", "this : Continuous eval"], "goal": "clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "D : Type u_1", "CategoryTheory.Category.{u_2, u_1} D", "Z : D", "F : CategoryTheory.Functor C D", "hZ : CategoryTheory.Limits.IsInitial Z", "G : CategoryTheory.Functor (CategoryTheory.WithInitial C) D", "h : CategoryTheory.WithInitial.incl.comp G ≅ F", "hG : G.obj CategoryTheory.WithInitial.star ≅ Z", "X : CategoryTheory.WithInitial C"], "goal": "(CategoryTheory.WithInitial.liftToInitialUnique F hZ G h hG).inv.app X =\n  (match X with\n    | CategoryTheory.WithInitial.of x => h.app x\n    | CategoryTheory.WithInitial.star => hG).inv"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "f g : X → Y", "h : ∀ (x : X), f x ~ᵢ g x"], "goal": "Continuous f ↔ Continuous g"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "P : C → Prop"], "goal": "CategoryTheory.Localization.LeftBousfield.W (CategoryTheory.isoClosure P) =\n  CategoryTheory.Localization.LeftBousfield.W P"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝¹ : OrderedAddCommGroup α", "inst✝ : OrderedAddCommMonoid β", "i : FunLike F α β", "f : F", "iamhc : AddMonoidHomClass F α β", "h : Monotone ⇑f", "a : α"], "goal": "0 ≤ a → f 0 ≤ f a"}}
+{"state": {"context": ["U : Type u_1", "Quiver U", "V : Type u_2", "Quiver V", "φ : U ⥤q V", "u : U"], "goal": "∀ (a : Quiver.Star u), (φ.star u a).fst = φ.obj a.fst"}}
+{"state": {"context": ["ι : Type u", "E : Type v", "F : Type w", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "I J : Box ι", "π : TaggedPrepartition I", "inst✝ : Fintype ι", "l : IntegrationParams", "f g : (ι → ℝ) → E", "vol : ι →ᵇᵃ[⊤] E →L[ℝ] F", "y y' : F", "c c₁ c₂ : ℝ≥0", "ε✝ ε₁ ε₂ : ℝ", "π₁ π₂ : TaggedPrepartition I", "h : Integrable I l f vol", "ε : ℝ", "ε0 : 0 < ε / 2"], "goal": "∃ ia, (∀ (c : ℝ≥0), l.RCond (ia c)) ∧ ∀ x ∈ {π | ∃ c, l.MemBaseSet I c (ia c) π} ×ˢ {π | ∃ c, l.MemBaseSet I c (ia c) π} ∩ {π | π.1.iUnion = π.2.iUnion}, (integralSum f vol x.1, integralSum f vol x.2) ∈ {p | dist p.1 p.2 ≤ ε}"}}
+{"state": {"context": ["J : Type v'", "inst✝⁹ : Category.{u', v'} J", "C : Type u", "inst✝⁸ : Category.{v, u} C", "K : Type u_1", "inst✝⁷ : Category.{?u.207926, u_1} K", "D : Type u_2", "inst✝⁶ : Category.{u_3, u_2} D", "inst✝⁵ : HasColimitsOfShape J C", "Gl : C ⥤ D", "Gr : D ⥤ C", "adj : Gl ⊣ Gr", "inst✝⁴ : Gr.Full", "inst✝³ : Gr.Faithful", "F : J ⥤ D", "c : Cocone (F ⋙ Gr)", "H : IsVanKampenColimit c", "inst✝² : ∀ (X : D) (f : X ⟶ Gl.obj c.pt), HasPullback (Gr.map f) (adj.unit.app c.pt)", "inst✝¹ : (X : D) → (f : X ⟶ Gl.obj c.pt) → PreservesLimit (cospan (Gr.map f) (adj.unit.app c.pt)) Gl", "inst✝ : (X : C) → (i : J) → (f : X ⟶ c.pt) → PreservesLimit (cospan f (c.ι.app i)) Gl", "this✝³ : PreservesLimitsOfSize.{0, 0, u_3, v, u_2, u} Gr", "this✝² : PreservesColimitsOfSize.{u', v', v, u_3, u, u_2} Gl", "F' : J ⥤ D", "c' : Cocone F'", "α : F' ⟶ (F ⋙ Gr) ⋙ Gl", "f : c'.pt ⟶ (Gl.mapCocone c).pt", "h : α ≫ (Gl.mapCocone c).ι = c'.ι ≫ (Functor.const J).map f", "hα : NatTrans.Equifibered α", "hc' : IsColimit c'", "j : J", "α' : F' ⟶ F := α ≫ (F.associator Gr Gl).hom ≫ whiskerLeft F adj.counit ≫ F.rightUnitor.hom", "hα' : NatTrans.Equifibered α'", "hα'' : ∀ (j : J), Gl.map (Gr.map (α'.app j)) = adj.counit.app (F'.toPrefunctor.1 j) ≫ α.app j", "β : F' ⋙ Gr ⋙ Gl ≅ F' := isoWhiskerLeft F' (asIso adj.counit) ≪≫ F'.rightUnitor", "hl : IsColimit ((Cocones.precompose β.hom).obj c') := (IsColimit.precomposeHomEquiv β c').symm hc'", "hr : IsColimit (Gl.mapCocone (colimit.cocone (F' ⋙ Gr))) := isColimitOfPreserves Gl (colimit.isColimit (F' ⋙ Gr))", "this✝¹ : α.app j = β.inv.app j ≫ Gl.map (Gr.map (α'.app j))", "this✝ : f = (hl.coconePointUniqueUpToIso hr).hom ≫ Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })", "this : IsPullback (Gl.map ((colimit.cocone (F' ⋙ Gr)).ι.app j)) (Gl.map ((whiskerRight α' Gr).app j)) (Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })) (Gl.map (c.ι.app j))"], "goal": "IsPullback (c'.ι.app j) (β.inv.app j ≫ Gl.map (Gr.map (α'.app j))) ((hl.coconePointUniqueUpToIso hr).hom ≫ Gl.map (colimit.desc (F' ⋙ Gr) { pt := c.pt, ι := whiskerRight α' Gr ≫ c.ι })) ((Gl.mapCocone c).ι.app j)"}}
+{"state": {"context": ["α : Type u_2", "β : α → Type u_1", "(a : α) → Unique (β a)", "x : α"], "goal": "(Equiv.sigmaUnique α β).symm x = ⟨x, default⟩"}}
+{"state": {"context": ["n : ℕ", "α : Type u_1", "inst✝¹ : PartialOrder α", "inst✝ : DecidableRel LE.le", "m : ℕ", "f : Fin m → α", "a : α", "h_sorted : Monotone f", "j k✝ : Fin m", "h : ¬f k✝ ≤ a", "i : Fin m", "hij : ¬i < k✝", "hia : f i ≤ a"], "goal": "f k✝ ≤ a"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedCommGroupWithZero α", "a : α", "ha : a ≠ 0"], "goal": "1 < a⁻¹ ↔ a < 1"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "G₀ : Type u_3", "M₀ : Type u_4", "R : Type u_5", "inst✝ : MulZeroClass M₀", "f g : ι → M₀", "x : ι", "hfg : x ∈ support fun x => f x * g x", "hf : f x = 0"], "goal": "(fun x => f x * g x) x = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "δ' : Type u_5", "ι : Sort uι", "s t u : Set α", "inst✝ : MeasurableSpace α", "f : α → Bool", "h : MeasurableSet (f ⁻¹' {true})"], "goal": "∀ (x : Bool), MeasurableSet (f ⁻¹' {x})"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝³ : TopologicalSpace α", "s✝ t u v : Set α", "inst✝² : LinearOrder β", "inst✝¹ : SuccOrder β", "inst✝ : IsSuccArchimedean β", "s : β → Set α", "H : ∀ (n : β), IsPreconnected (s n)", "K : ∀ (n : β), (s n ∩ s (succ n)).Nonempty", "i✝ j i : β", "x✝ : i ∈ Ico j i✝"], "goal": "(s i ∩ s (succ i)).Nonempty"}}
+{"state": {"context": ["ι✝ : Type u_1", "α : Type u_2", "M : Matroid α", "F X Y : Set α", "e : α", "ι : Sort u_3", "I J B : Set α", "f x y : α", "hI : M.Indep I", "this : I ⊆ M.E"], "goal": "¬M.Indep (insert x I) ∧ x ∈ M.E ∨ x ∈ I ↔ x ∈ M.E ∧ (¬M.Indep (insert x I) ∨ x ∈ I)"}}
+{"state": {"context": ["S : Type uS", "CommSemiring S", "A : Type uA", "Semiring A", "Algebra S A", "s : A → A → Prop"], "goal": "Function.Surjective ⇑(RingQuot.mkAlgHom S s)"}}
+{"state": {"context": ["α : Type u_1", "l : Ordnode α", "x : α", "r : Ordnode α", "hl : l.Sized", "hr : r.Sized"], "goal": "(l.rotateR x r).dual.Sized"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "κ : ProbabilityTheory.Kernel α β", "μ : MeasureTheory.Measure α", "a : α"], "goal": "(κ ∘ₖ ProbabilityTheory.Kernel.const α μ) a = μ.bind ⇑κ"}}
+{"state": {"context": ["α : Type u_2", "f : Equiv.Perm α", "a : α"], "goal": "f.IsCycleOn {a} ↔ f a = a"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "Field 𝕂", "Ring 𝔸", "Algebra 𝕂 𝔸", "TopologicalSpace 𝔸", "TopologicalRing 𝔸", "T2Space 𝔸", "x : 𝔸ᵐᵒᵖ"], "goal": "NormedSpace.exp 𝕂 (MulOpposite.unop x) = MulOpposite.unop (NormedSpace.exp 𝕂 x)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "X : Mon_ C"], "goal": "(ρ_ X).hom.hom = (ρ_ X.X).hom"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "NontriviallyNormedField 𝕂", "NormedRing 𝔸", "NormedAlgebra 𝕂 𝔸", "CompleteSpace 𝔸", "x : 𝔸", "hx : x ∈ EMetric.ball 0 (NormedSpace.expSeries 𝕂 𝔸).radius"], "goal": "Summable fun n => (↑n.factorial)⁻¹ • x ^ n"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t u : Set Ω", "hs : s.Finite", "ht : t.Finite", "hst : Disjoint s t", "hs' : s.Nonempty", "ht' : t.Nonempty"], "goal": "Measure.count s * (Measure.count s)⁻¹ * Measure.count (s ∩ u) * (Measure.count (s ∪ t))⁻¹ + Measure.count t * (Measure.count t)⁻¹ * Measure.count (t ∩ u) * (Measure.count (s ∪ t))⁻¹ = (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)"}}
+{"state": {"context": [], "goal": "CategoryTheory.Limits.WalkingPair.swap CategoryTheory.Limits.WalkingPair.right = CategoryTheory.Limits.WalkingPair.left"}}
+{"state": {"context": ["m : Type u", "n : Type u'", "DecidableEq m", "R : Type u_2", "Semiring R", "Finite m", "Fintype n", "A : Matrix m n R"], "goal": "Function.Surjective A.mulVec ↔ ∃ B, A * B = 1"}}
+{"state": {"context": ["R : Type u_1", "C : Type u_2", "Semiring R", "CategoryTheory.Category.{u_3, u_2} C", "CategoryTheory.Preadditive C", "CategoryTheory.Linear R C", "S₁ S₂ : CategoryTheory.ShortComplex C", "a : R", "φ : S₁ ⟶ S₂"], "goal": "(a • φ).τ₃ = a • φ.τ₃"}}
+{"state": {"context": ["α : Type u_3", "TopologicalSpace α"], "goal": "Monotone exterior"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "A B A' B' : C", "f : A ⟶ B", "g : A' ⟶ B'", "e : CategoryTheory.Arrow.mk f ≅ CategoryTheory.Arrow.mk g", "h : CategoryTheory.StrongMono f"], "goal": "CategoryTheory.StrongMono g"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F✝ : Type u_3", "inst✝¹ : SeminormedAddCommGroup F✝", "inst✝ : NormedSpace ℝ F✝", "c : 𝕜", "hc : 1 < ‖c‖", "R : ℝ", "hR : ‖c‖ < R", "F : Subspace 𝕜 E", "hFc : IsClosed ↑F", "hF : ∃ x, x ∉ F"], "goal": "∃ x₀, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖"}}
+{"state": {"context": ["A B : AddGrp", "f : A ⟶ B"], "goal": "CategoryTheory.Epi f ↔ AddMonoidHom.range f = ⊤"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "NonAssocSemiring α", "Preorder α", "NonAssocSemiring β", "Preorder β"], "goal": "Function.Injective OrderRingIso.toOrderRingHom"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : CompleteSpace F", "T T' T'' : Set α → E →L[ℝ] F", "C C' C'' : ℝ", "f g : α → E", "hT : DominatedFinMeasAdditive μ T C", "hf : Integrable f μ", "hg : Integrable g μ"], "goal": "setToFun μ T hT (f + g) = setToFun μ T hT f + setToFun μ T hT g"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "f g : Perm α", "x y z : α", "h : (x ≠ y ∧ x ≠ z ∧ ¬False) ∧ y ≠ z ∧ ¬False"], "goal": "{x, y, z} ≤ (swap x y * swap y z).support"}}
+{"state": {"context": ["α : Type u_1", "f g : α → ℝ", "s : Set α"], "goal": "regionBetween f g s ⊆ s ×ˢ univ"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "Preorder α", "Preorder β", "AddZeroClass α", "AddZeroClass β", "f : α →+ β", "h : Monotone (↑f).toFun"], "goal": "⇑{ toAddMonoidHom := f, monotone' := h } = ⇑f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Finite α", "f : α → β", "e : α ≃ β"], "goal": "Function.Injective f ↔ Function.Surjective f"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "c : ComplexShape ι", "c' : ComplexShape ι'", "C : Type u_3", "CategoryTheory.Category.{u_4, u_3} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "K : HomologicalComplex C c", "e : c.Embedding c'", "i' : ι'", "hi' : e.r i' = none"], "goal": "CategoryTheory.Limits.IsZero ((K.extend e).X i')"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "h : MeasureTheory.Memℒp 0 p μ"], "goal": "MeasureTheory.Memℒp.toLp 0 h = 0"}}
+{"state": {"context": ["α : Type u_1", "CommMonoidWithZero α", "WfDvdMonoid α", "a : α", "ha : ¬IsUnit a", "ha0 : a ≠ 0"], "goal": "∃ i, Irreducible i ∧ i ∣ a"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "ι : Type u_2", "E : ι → Type u_3", "(i : ι) → NormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "Fintype ι", "f : ContinuousMultilinearMap 𝕜 E F", "n : ℕ"], "goal": "iteratedFDeriv 𝕜 n ⇑f = f.iteratedFDeriv n"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a b₁ b₂ : α", "h : b₁ ≤ b₂"], "goal": "Finset.Ioc a b₁ ⊆ Finset.Ioc a b₂"}}
+{"state": {"context": ["α : Type u", "β : Type v", "M : Type w", "inst✝¹ : DecidableEq β", "s : Finset α", "t : Finset β", "f : α → β", "w : α → M", "b : M", "n : ℕ", "inst✝ : LinearOrderedCommSemiring M", "ht : t.Nonempty", "hb : ↑s.card ≤ t.card • b"], "goal": "∃ y ∈ t, ↑(filter (fun x => f x = y) s).card ≤ b"}}
+{"state": {"context": ["α : Type u_1", "n : ℕ"], "goal": "(↑n).toZNumNeg = -↑n"}}
+{"state": {"context": ["A : Type u_1", "inst✝¹ : NonUnitalNonAssocCommRing A", "inst✝ : IsCommJordan A", "a b c : A"], "goal": "⁅L a, L (b * b)⁆ + ⁅L a, L (c * c)⁆ + ⁅L a, 2 • L (a * b)⁆ + ⁅L a, 2 • L (c * a)⁆ + ⁅L a, 2 • L (b * c)⁆ + (⁅L b, L (a * a)⁆ + ⁅L b, L (c * c)⁆ + ⁅L b, 2 • L (a * b)⁆ + ⁅L b, 2 • L (c * a)⁆ + ⁅L b, 2 • L (b * c)⁆) + (⁅L c, L (a * a)⁆ + ⁅L c, L (b * b)⁆ + ⁅L c, 2 • L (a * b)⁆ + ⁅L c, 2 • L (c * a)⁆ + ⁅L c, 2 • L (b * c)⁆) = ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (a * b)⁆) + (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) + (⁅L b, L (c * c)⁆ + ⁅L c, L (b * b)⁆ + 2 • (⁅L b, L (b * c)⁆ + ⁅L c, L (b * c)⁆)) + (2 • ⁅L a, L (b * c)⁆ + 2 • ⁅L b, L (c * a)⁆ + 2 • ⁅L c, L (a * b)⁆)"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p : ℝ≥0∞", "(i : α) → NormedAddCommGroup (E i)", "ι : Type u_3", "l : Filter ι", "l.NeBot", "_i : Fact (1 ≤ p)", "F : ι → ↥(lp E p)", "hF : Bornology.IsBounded (Set.range F)", "f : (a : α) → E a", "hf : Filter.Tendsto (id fun i => ↑(F i)) l (nhds f)"], "goal": "Memℓp f p"}}
+{"state": {"context": ["a b m n✝ p n d : ℕ", "hn : n ≠ 0", "hd : d ≠ 0"], "goal": "(∀ (p k : ℕ), Prime p → p ^ k ∣ d → p ^ k ∣ n) → ∀ (p : ℕ), Prime p → d.factorization p ≤ n.factorization p"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "hpi : Prime ↑p", "hp2 : p = 2", "x✝ : Irreducible ↑p", "not_unit✝ : ¬IsUnit ↑p", "h : ∀ (a b : ℤ[i]), ↑p = a * b → IsUnit a ∨ IsUnit b", "this : { re := 1, im := 1 }.norm.natAbs = 1 ∨ { re := 1, im := -1 }.norm.natAbs = 1"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_2", "M : Type u_3", "Zero R", "Zero M", "SMulWithZero R M", "s : Set α", "r : α → R", "f : α → M", "a : α"], "goal": "s.indicator (fun a => r a • f a) a = s.indicator r a • f a"}}
+{"state": {"context": ["α : Type u", "x : α", "a b : Set α"], "goal": "x ∈ a ∪ b ↔ x ∈ a ∨ x ∈ b"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : EuclideanDomain R", "inst✝ : GCDMonoid R", "p✝ q✝ p q : R", "hp : p ≠ 0", "r : R", "hr : p = GCDMonoid.gcd p q * r", "pq0 : GCDMonoid.gcd p q ≠ 0", "r0 : r ≠ 0"], "goal": "r ≠ 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.CategoryStruct.{v, u} C", "a b : C", "f : a ⟶ b"], "goal": "f.toLoc.as = f"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Type u_4", "N : Type u_5", "P : Type u_6", "Q : Type u_7", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "AddCommMonoid Q", "Module R M", "Module R N", "Module R P", "Module R Q", "m : M", "n : N", "p : P", "q : Q"], "goal": "(TensorProduct.tensorTensorTensorAssoc R M N P Q) ((m ⊗ₜ[R] n) ⊗ₜ[R] p ⊗ₜ[R] q) = (m ⊗ₜ[R] n ⊗ₜ[R] p) ⊗ₜ[R] q"}}
+{"state": {"context": ["R : Type u_1", "c₁ c₂ : R", "a : Fin 4 → R"], "goal": "(QuaternionAlgebra.equivTuple c₁ c₂).symm a = { re := a 0, imI := a 1, imJ := a 2, imK := a 3 }"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹² : MeasurableSpace X", "G : Type u_5", "𝕜 : Type u_6", "inst✝¹¹ : TopologicalSpace X", "inst✝¹⁰ : TopologicalSpace Y", "inst✝⁹ : MeasurableSpace Y", "inst✝⁸ : OpensMeasurableSpace Y", "μ : Measure Y", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace 𝕜 F", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace 𝕜 G", "inst✝ : NormedSpace 𝕜 E", "L : F →L[𝕜] G →L[𝕜] E", "f : X → Y → G", "s : Set X", "k : Set Y", "g : Y → F", "hk : IsCompact k", "hf : ContinuousOn (uncurry f) (s ×ˢ univ)", "hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0", "hg : IntegrableOn g k μ", "A : ∀ p ∈ s, Continuous (f p)", "q : X", "hq : q ∈ s", "ε : ℝ", "εpos : ε > 0", "δ : ℝ", "δpos : 0 < δ", "hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε", "v : Set X", "v_mem : v ∈ 𝓝[s] q", "hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ", "I : ∀ p ∈ s, IntegrableOn (fun y => (L (g y)) (f p y)) k μ", "p : X", "hp : p ∈ v", "h'p : p ∈ s"], "goal": "‖∫ (y : Y), (L (g y)) (f p y) ∂μ - ∫ (y : Y), (L (g y)) (f q y) ∂μ‖ < ε"}}
+{"state": {"context": ["ι : Type u", "α : Type v", "inst✝¹ : DecidableEq α", "t : ι → Finset α", "inst✝ : Fintype ι", "n : ℕ", "hn : Fintype.card ι = n.succ", "ht : ∀ (s : Finset ι), s.card ≤ (s.biUnion t).card", "ih : ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n → (∀ (s' : Finset ι'), s'.card ≤ (s'.biUnion t').card) → ∃ f, Function.Injective f ∧ ∀ (x : ι'), f x ∈ t' x", "s : Finset ι", "hs : s.Nonempty", "hns : s ≠ univ", "hus : s.card = (s.biUnion t).card", "this : DecidableEq ι", "card_ι'_le : Fintype.card { x // x ∈ s } ≤ n", "t' : { x // x ∈ s } → Finset α := fun x' => t ↑x'", "f' : { x // x ∈ s } → α", "hf' : Function.Injective f'", "hsf' : ∀ (x : { x // x ∈ s }), f' x ∈ t' x", "ι'' : Set ι := (↑s)ᶜ", "t'' : ↑ι'' → Finset α := fun a'' => t ↑a'' \\ s.biUnion t", "card_ι''_le : Fintype.card ↑ι'' ≤ n", "f'' : ↑ι'' → α", "hf'' : Function.Injective f''", "hsf'' : ∀ (x : ↑ι''), f'' x ∈ t'' x", "f'_mem_biUnion : ∀ (x' : ι) (hx' : x' ∈ s), f' ⟨x', hx'⟩ ∈ s.biUnion t", "f''_not_mem_biUnion : ∀ (x'' : ι) (hx'' : x'' ∉ s), f'' ⟨x'', hx''⟩ ∉ s.biUnion t"], "goal": "∃ f, Function.Injective f ∧ ∀ (x : ι), f x ∈ t x"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "Add M", "Add N", "ι : Sort u_5", "f : AddHom M N", "s : ι → AddSubsemigroup M"], "goal": "AddSubsemigroup.map f (iSup s) = ⨆ i, AddSubsemigroup.map f (s i)"}}
+{"state": {"context": ["α : Type u_1", "SeminormedAddCommGroup α", "r C : ℝ", "f : ℕ → α", "hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n", "n : ℕ"], "goal": "dist (∑ i ∈ Finset.range n, f i) (∑ i ∈ Finset.range (n + 1), f i) ≤ C * r ^ n"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B I J X Y : Set α", "e : α", "hI : M.Indep I", "hIX : I ⊆ X"], "goal": "(∀ (J : Set α), M.Indep J → I ⊆ J → J ⊆ X → I = J) ∧ X ⊆ M.E ↔ ∀ e ∈ X \\ I, M.Dep (insert e I)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "κ η : Kernel α α", "μ : Measure α", "inst✝ : IsSFiniteKernel κ", "hκ : κ.Invariant μ", "hη : η.Invariant μ", "h✝ : IsEmpty α"], "goal": "(κ ∘ₖ η).Invariant μ"}}
+{"state": {"context": ["α : Type u", "β : α → Type v", "DecidableEq α", "a : α", "b : β a", "xs : List (Sigma β)"], "goal": "(⟨a, b⟩ :: xs).toFinmap = Finmap.insert a b xs.toFinmap"}}
+{"state": {"context": ["ι : Type u_2", "i j : ι"], "goal": "(ComplexShape.refl ι).Rel i j = (i = j)"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ : UniformSpace K", "inst✝ : CompletableTopField K", "x : hat K", "h : x ≠ 0", "y : hat K", "y_ne : y ≠ 0", "this : (fun x => ↑K x⁻¹) = (fun y => ↑K y) ∘ fun x => x⁻¹"], "goal": "Cauchy (map (fun x => ↑K x⁻¹) (Filter.comap (↑K) (𝓝 y)))"}}
+{"state": {"context": ["R : Type u", "CommRing R", "P Q : Fin 3 → R", "hz : P z = Q z", "mem : Q z ∈ nonZeroDivisors R"], "goal": "P ≈ Q ↔ P = Q"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e e' : PartialHomeomorph X Y", "hs : Disjoint e.source e'.source", "ht : Disjoint e.target e'.target"], "goal": "e.IsImage e'.source e'.target"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "p a b c : α"], "goal": "a ≡ b [PMOD p] → b ≡ c [PMOD p] → a ≡ c [PMOD p]"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "g : G"], "goal": "g ∈ IsAddSubgroup.trivial G ↔ g = 0"}}
+{"state": {"context": ["m₁ : List Char", "c : Char", "m₂ : List Char", "x✝ : Substring", "h : x✝.Valid", "e : x✝.toString.data = m₁ ++ c :: m₂"], "goal": "x✝.next { byteIdx := utf8Len m₁ } = { byteIdx := utf8Len m₁ + c.utf8Size }"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : Semiring R", "l : Type u_2", "m : Type u_3", "n : Type u_4", "inst✝¹ : Fintype m", "inst✝ : DecidableEq m", "M : Matrix m n R", "i : m", "j : n"], "goal": "((LinearMap.stdBasis R (fun x => R) i) 1 ᵥ* M) j = M i j"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasFiniteProducts C", "CategoryTheory.Limits.HasPullbacks C", "X Y : CategoryTheory.Dial C"], "goal": "(X.braiding Y).hom.F =\n  CategoryTheory.Limits.prod.lift\n    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.snd)\n    (CategoryTheory.CategoryStruct.comp CategoryTheory.Limits.prod.snd CategoryTheory.Limits.prod.fst)"}}
+{"state": {"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : DecidableEq (K →+* ℂ)", "α : Type u_2", "β : Type u_3", "inst✝¹ : Fintype α", "inst✝ : Fintype β", "a : Matrix α β (𝓞 K)", "ha : a ≠ 0", "p q : ℕ", "cardα : Fintype.card α = p", "cardβ : Fintype.card β = q", "h0p : 0 < p", "hpq : p < q", "x : β × (K →+* ℂ) → ℤ", "hxl : x ≠ 0", "hmulvec0 : NumberField.house.asiegel K a *ᵥ x = 0", "hxbound : ‖x‖ ≤ (↑q * ↑(finrank ℚ K) * ‖NumberField.house.asiegel K a‖) ^ (↑p / (↑q - ↑p))", "k : α", "lin_0 : ∀ (u : K →+* ℂ), ∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r)) = 0", "this✝ : 0 = ∑ u : K →+* ℂ, (∑ r : K →+* ℂ, ∑ l : β, ↑(NumberField.house.a' K a k l r u) * ↑(x (l, r))) * (NumberField.house.newBasis K) u", "this : 0 = ∑ y : β, ∑ x_1 : K →+* ℂ, ↑(x (y, x_1)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k y x_1 u) * (NumberField.house.newBasis K) u", "l✝ : β", "l : K →+* ℂ"], "goal": "↑(x (l✝, l)) * ∑ u : K →+* ℂ, ↑(NumberField.house.a' K a k l✝ l u) * (NumberField.house.newBasis K) u = (fun j => a k j) l✝ * (↑(x (l✝, l)) * (NumberField.house.newBasis K) l)"}}
+{"state": {"context": ["a b c : Ordinal.{u}", "h : c.IsLimit"], "goal": "a < b + c ↔ ∃ c' < c, a < b + c'"}}
+{"state": {"context": ["α : Type u_3", "Preorder α", "NoMaxOrder α", "a : α"], "goal": "∀ᶠ (x : α) in Filter.atTop, x ≠ a"}}
+{"state": {"context": ["S : Type u_1", "T : Type u_2", "R : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "A : Type u_7", "inst✝⁶ : CommRing R", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "K : Type v", "inst✝³ : Field K", "hK : Invertible 2", "d : ℕ", "ih : ∀ {V : Type u} [inst : AddCommGroup V] [inst_1 : Module K V] [inst_2 : FiniteDimensional K V] {B : BilinForm K V}, IsSymm B → finrank K V = d → ∃ v, IsOrthoᵢ B ⇑v", "V : Type u", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : FiniteDimensional K V", "B : BilinForm K V", "hB₂ : IsSymm B", "this : Nontrivial V", "hB₁ : B ≠ 0", "x : V", "hd : finrank K ↥((Submodule.span K {x}).orthogonalBilin B) + 1 = d + 1", "hx : ¬IsOrtho B x x", "B' : ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] ↥((Submodule.span K {x}).orthogonalBilin B) →ₗ[K] K := domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)", "v' : Basis (Fin d) K ↥((Submodule.span K {x}).orthogonalBilin B)", "hv₁ : (domRestrict₁₂ B ((Submodule.span K {x}).orthogonalBilin B) ((Submodule.span K {x}).orthogonalBilin B)).IsOrthoᵢ ⇑v'", "b : Basis (Fin (d + 1)) K V := Basis.mkFinCons x v' ⋯ ⋯", "j i : Fin (d + 1)"], "goal": "j ≠ i → (IsOrtho B on Fin.cons x (Subtype.val ∘ ⇑v')) j i"}}
+{"state": {"context": ["n m d : ℕ", "hm : n ∣ m", "hd : m ∣ d"], "goal": "(unitsMap hm).comp (unitsMap hd) = unitsMap ⋯"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "f : ℂ → E", "z : ℂ", "h : Complex.HadamardThreeLines.sSupNormIm f 0 = 0 ∨ Complex.HadamardThreeLines.sSupNormIm f 1 = 0"], "goal": "Complex.HadamardThreeLines.interpStrip f z = 0"}}
+{"state": {"context": ["a b : Prop"], "goal": "¬(a → b) ↔ a ∧ ¬b"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "CommMonoid α", "TopologicalSpace α", "e : γ ≃ β", "f : β → α"], "goal": "∏' (c : γ), f (e c) = ∏' (b : β), f b"}}
+{"state": {"context": ["C : Type u_1", "inst✝³ : Category.{u_3, u_1} C", "inst✝² : Preadditive C", "ι : Type u_2", "c : ComplexShape ι", "F G K : HomologicalComplex C c", "φ : F ⟶ G", "inst✝¹ : HasHomotopyCofiber φ", "inst✝ : DecidableRel c.Rel", "i : ι", "hi : ¬c.Rel i (c.next i)", "A : C", "f g : A ⟶ X φ i", "h : f ≫ sndX φ i = g ≫ sndX φ i"], "goal": "f = g"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁶ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝¹⁵ : NormedAddCommGroup D", "inst✝¹⁴ : NormedSpace 𝕜 D", "E : Type uE", "inst✝¹³ : NormedAddCommGroup E", "inst✝¹² : NormedSpace 𝕜 E", "F : Type uF", "inst✝¹¹ : NormedAddCommGroup F", "inst✝¹⁰ : NormedSpace 𝕜 F", "G : Type uG", "inst✝⁹ : NormedAddCommGroup G", "inst✝⁸ : NormedSpace 𝕜 G", "X : Type u_2", "inst✝⁷ : NormedAddCommGroup X", "inst✝⁶ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f✝ f₁ : E → F", "g✝ : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n✝ : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "R : Type u_3", "inst✝⁵ : NormedRing R", "inst✝⁴ : NormedAlgebra 𝕜 R", "𝕜' : Type u_4", "inst✝³ : NormedField 𝕜'", "inst✝² : NormedAlgebra 𝕜 𝕜'", "inst✝¹ : CompleteSpace 𝕜'", "inst✝ : CompleteSpace 𝕜", "f g : E → 𝕜", "n : ℕ∞", "hf : ContDiff 𝕜 n f", "hg : ContDiff 𝕜 n g", "h0 : ∀ (x : E), g x ≠ 0"], "goal": "ContDiff 𝕜 n fun x => f x / g x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "x y : α", "inst✝³ : DecidableRel f.SameCycle", "inst✝² : DecidableRel g.SameCycle", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "h : 2 ≤ (f.cycleOf x).support.card"], "goal": "f x ≠ x"}}
+{"state": {"context": ["X✝ Y Z : Scheme", "𝒰✝ : X✝.OpenCover", "f : X✝ ⟶ Z", "g : Y ⟶ Z", "inst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f) g", "X : Scheme", "𝒰 : X.OpenCover", "x : ↑↑X.toPresheafedSpace"], "goal": "∃ i, x ∈ Set.range ⇑(𝒰.map i).val.base"}}
+{"state": {"context": ["K : Type u_1", "inst✝³ : Field K", "inst✝² : NumberField K", "inst✝¹ : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝ : DecidableRel fun a b => a ∣ b"], "goal": "λ ^ (3 * S.multiplicity - 2) * FermatLastTheoremForThreeGen.Solution.x S * λ * FermatLastTheoremForThreeGen.Solution.y S * λ * FermatLastTheoremForThreeGen.Solution.z S = ↑S.u * λ ^ (3 * S.multiplicity) * FermatLastTheoremForThreeGen.Solution.w S ^ 3"}}
+{"state": {"context": ["G : Type u_1", "X : Type u_2", "SMul G X"], "goal": "MulAction.IsBlock G ⊥"}}
+{"state": {"context": ["K : Type u", "inst✝³ : Field K", "n : ℕ", "hζ : (primitiveRoots n K).Nonempty", "hn : 0 < n", "a : K", "H : Irreducible (X ^ n - C a)", "L : Type u_1", "inst✝² : Field L", "inst✝¹ : Algebra K L", "inst✝ : IsSplittingField K L (X ^ n - C a)", "α : L", "hα : α ^ n = (algebraMap K L) a"], "goal": "⊤.toSubalgebra = Algebra.adjoin K {α}"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "σ : Type u_4", "inst✝⁶ : DecidableEq ι", "inst✝⁵ : Semiring R", "inst✝⁴ : AddCommMonoid M", "inst✝³ : Module R M", "ℳ : ι → Submodule R M", "inst✝² : Decomposition ℳ", "N : Type u_5", "inst✝¹ : AddCommMonoid N", "inst✝ : Module R N", "f g : M →ₗ[R] N", "h : ∀ (i : ι), f ∘ₗ (ℳ i).subtype = g ∘ₗ (ℳ i).subtype", "i : ι"], "goal": "(f ∘ₗ ↑(decomposeLinearEquiv ℳ).symm) ∘ₗ lof R ι (fun i => ↥(ℳ i)) i = (g ∘ₗ ↑(decomposeLinearEquiv ℳ).symm) ∘ₗ lof R ι (fun i => ↥(ℳ i)) i"}}
+{"state": {"context": ["p : ℕ", "hp : Nat.Prime p", "x : ℕ × ↑p.smoothNumbers"], "goal": "↑((Nat.equivProdNatSmoothNumbers hp) x) = p ^ x.1 * ↑x.2"}}
+{"state": {"context": ["C : Type u", "inst✝⁴ : Category.{v, u} C", "inst✝³ : HasZeroObject C", "inst✝² : HasShift C ℤ", "inst✝¹ : Preadditive C", "inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive", "hC : Pretriangulated C", "Z X : C", "h : Z ⟶ (shiftFunctor C 1).obj X", "Y' : C", "f' : (shiftFunctor C 1).obj X ⟶ Y'", "g' : Y' ⟶ (shiftFunctor C 1).obj Z", "mem : Triangle.mk h f' g' ∈ distinguishedTriangles"], "goal": "∃ Y f g, Triangle.mk f g h ∈ distinguishedTriangles"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹⁰ : LinearOrderedCommRing 𝕜", "inst✝⁹ : LinearOrderedCommRing E", "inst✝⁸ : LinearOrderedAddCommGroup F", "inst✝⁷ : Module 𝕜 E", "inst✝⁶ : Module 𝕜 F", "inst✝⁵ : Module E F", "inst✝⁴ : IsScalarTower 𝕜 E F", "inst✝³ : SMulCommClass 𝕜 E F", "inst✝² : OrderedSMul 𝕜 E", "inst✝¹ : OrderedSMul 𝕜 F", "inst✝ : OrderedSMul E F", "s : Set 𝕜", "f : 𝕜 → E", "g : 𝕜 → F", "hf : ConcaveOn 𝕜 s f", "hg : ConcaveOn 𝕜 s g", "hf₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ f x", "hg₀ : ∀ ⦃x : 𝕜⦄, x ∈ s → 0 ≤ g x", "hfg : AntivaryOn f g s", "x : 𝕜", "hx : x ∈ s", "y : 𝕜", "hy : y ∈ s", "a b : 𝕜", "ha : 0 ≤ a", "hb : 0 ≤ b", "hab : a + b = 1"], "goal": "(a * a) • f x • g x + (b * b) • f y • g y + ((a * b) • f x • g y + (a * b) • f y • g x) = (a • f x) • a • g x + (b • f y) • a • g x + ((a • f x) • b • g y + (b • f y) • b • g y)"}}
+{"state": {"context": ["α β : Type u"], "goal": "Cardinal.mk α ^ Cardinal.mk β = Cardinal.mk (β → α)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Monoid α", "inst✝² : Monoid β", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "a b : α", "m : ℕ", "h : Finite a b", "hm : (multiplicity a b).get h < m"], "goal": "multiplicity a b < ↑m"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝ : Ring R"], "goal": "restriction 0 = 0"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M₁ : Type u_3", "M₂ : Type u_4", "N₁ : Type u_5", "N₂ : Type u_6", "P : Type u_7", "Mᵢ : ι → Type u_8", "Nᵢ : ι → Type u_9", "inst✝¹² : CommSemiring R", "inst✝¹¹ : AddCommMonoid M₁", "inst✝¹⁰ : AddCommMonoid M₂", "inst✝⁹ : AddCommMonoid N₁", "inst✝⁸ : AddCommMonoid N₂", "inst✝⁷ : AddCommMonoid P", "inst✝⁶ : Module R M₁", "inst✝⁵ : Module R M₂", "inst✝⁴ : Module R N₁", "inst✝³ : Module R N₂", "inst✝² : Module R P", "inst✝¹ : Preorder P", "inst✝ : CovariantClass P P (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "Q₁ : QuadraticMap R M₁ P", "Q₂ : QuadraticMap R M₂ P"], "goal": "(∀ (x : M₁ × M₂), 0 ≤ (Q₁.prod Q₂) x) ↔ (∀ (x : M₁), 0 ≤ Q₁ x) ∧ ∀ (x : M₂), 0 ≤ Q₂ x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "κ : Type u_7", "inst✝¹ : TopologicalSpace X", "T : (i : ι) → TopologicalSpace (π i)", "f : X → (i : ι) → π i", "inst✝ : DecidableEq ι", "s : Set ((a : ι) → π a)", "x y : (a : ι) → π a", "I : Finset ι", "t : (i : ι) → Set (π i)", "htx : ∀ (i : ι), t i ∈ 𝓝 (x i)", "hts : (↑I).pi t ⊆ s", "i : ι", "hi : i ∈ ↑I"], "goal": "I.piecewise x y i ∈ t i"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : OrderedRing 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : AddCommGroup F", "inst✝¹ : Module 𝕜 E", "inst✝ : Module 𝕜 F", "x y : E", "s t : Set E", "hs : StarConvex 𝕜 x s"], "goal": "StarConvex 𝕜 (-x) (-s)"}}
+{"state": {"context": ["n k : ℕ", "hk : k ≤ n"], "goal": "n.choose k = n.choose k * (k ! * (n - k)!) / (k ! * (n - k)!)"}}
+{"state": {"context": ["n m : Nat", "h : n.succ ≤ m.succ"], "goal": "Fin.castLE h 0 = 0"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "u v : V", "e : G.Adj u v"], "goal": "SimpleGraph.singletonFinsubgraph v ≤ SimpleGraph.finsubgraphOfAdj e"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "n : ℕ", "inst✝ : CompleteSpace E", "I : Box (Fin (n + 1))", "i : Fin (n + 1)", "f : (Fin (n + 1) → ℝ) → E", "f' : (Fin (n + 1) → ℝ) →L[ℝ] E", "hfc : ContinuousOn f (Box.Icc I)", "x : Fin (n + 1) → ℝ", "hxI : x ∈ Box.Icc I", "a : E", "ε : ℝ", "h0 : 0 < ε", "hε : ∀ y ∈ Box.Icc I, ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖", "c : ℝ≥0", "hc : I.distortion ≤ c", "e : ℝ → (Fin n → ℝ) → Fin (n + 1) → ℝ := i.insertNth", "Hl : I.lower i ∈ Set.Icc (I.lower i) (I.upper i)"], "goal": "‖(∏ j : Fin (n + 1), (I.upper j - I.lower j)) • f' (Pi.single i 1) - (integral (I.face i) ⊥ (f ∘ e (I.upper i)) BoxAdditiveMap.volume - integral (I.face i) ⊥ (f ∘ e (I.lower i)) BoxAdditiveMap.volume)‖ ≤ 2 * ε * ↑c * ∏ j : Fin (n + 1), (I.upper j - I.lower j)"}}
+{"state": {"context": ["σ : Type u_1", "τ : Type u_2", "R : Type u_3", "S : Type u_4", "inst✝² inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "φ ψ : MvPolynomial σ R", "m n : ℕ", "hφ : φ.IsHomogeneous n", "d : σ →₀ ℕ", "hd : coeff d φ ≠ 0"], "goal": "∑ a ∈ d.support, d a ≤ d.sum fun i c => c"}}
+{"state": {"context": ["ι : Type u_1", "inst✝ : Fintype ι", "this : StieltjesFunction.id.measure.IsAddLeftInvariant"], "goal": "volume = StieltjesFunction.id.measure"}}
+{"state": {"context": ["F : Type v'", "R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝¹⁶ : CommSemiring R", "inst✝¹⁵ : StarRing R", "inst✝¹⁴ : NonUnitalSemiring A", "inst✝¹³ : StarRing A", "inst✝¹² : Module R A", "inst✝¹¹ : IsScalarTower R A A", "inst✝¹⁰ : SMulCommClass R A A", "inst✝⁹ : StarModule R A", "inst✝⁸ : NonUnitalSemiring B", "inst✝⁷ : StarRing B", "inst✝⁶ : Module R B", "inst✝⁵ : IsScalarTower R B B", "inst✝⁴ : SMulCommClass R B B", "inst✝³ : StarModule R B", "inst✝² : FunLike F A B", "inst✝¹ : NonUnitalAlgHomClass F R A B", "inst✝ : NonUnitalStarAlgHomClass F R A B", "S : NonUnitalStarSubalgebra R A", "s : Set A", "z : A"], "goal": "(∀ x ∈ s, x * z = z * x) ��� ((∀ x ∈ star s, x * z = z * x) ↔ ∀ x ∈ s, star x * z = z * star x)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ : S₁ ⟶ S₂", "S₁.HasHomology", "S₂.HasHomology"], "goal": "CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ) S₂.homologyι =\n  CategoryTheory.CategoryStruct.comp S₁.homologyι (CategoryTheory.ShortComplex.opcyclesMap φ)"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "_i : Valued R Γ₀", "s : Set R", "x : R"], "goal": "s ∈ 𝓝 x ↔ ∃ γ, {y | v (y - x) < ↑γ} ⊆ s"}}
+{"state": {"context": ["ι✝ : Type u_1", "α : Type u_2", "M : Matroid α", "F X Y : Set α", "e : α", "ι : Sort u_3", "I J B : Set α", "f x y : α", "hI : M.Basis I X", "hY : I ⊆ Y", "h : M.closure X = M.closure Y", "hYE : autoParam (Y ⊆ M.E) _auto✝"], "goal": "M.Basis I Y"}}
+{"state": {"context": ["α : Type u_1", "LE α", "s : Set α", "hs : IsLowerSet s"], "goal": "↑{ carrier := s, lower' := hs } = s"}}
+{"state": {"context": ["n : Type u", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R", "A : Matrix.SpecialLinearGroup n R"], "goal": "↑(Matrix.SpecialLinearGroup.toLin' A) = Matrix.toLin' ↑A"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : NonAssocRing R", "p : ℕ", "inst✝ : CharP R p", "k : ℕ"], "goal": "↑(k % p + p * (k / p)) = ↑(k % p)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimits C", "X : TopCat", "CategoryTheory.ConcreteCategory C", "CategoryTheory.Limits.PreservesFilteredColimits (CategoryTheory.forget C)", "CategoryTheory.Limits.HasLimits C", "CategoryTheory.Limits.PreservesLimits (CategoryTheory.forget C)", "(CategoryTheory.forget C).ReflectsIsomorphisms", "F G : TopCat.Sheaf C X", "f : F ⟶ G", "∀ (x : ↑X), CategoryTheory.IsIso ((TopCat.Presheaf.stalkFunctor C x).map f.val)"], "goal": "CategoryTheory.IsIso f"}}
+{"state": {"context": ["R₁ : Type u_1", "R₂ : Type u_2", "Semiring R₁", "Semiring R₂", "σ₁₂ : R₁ →+* R₂", "σ₂₁ : R₂ →+* R₁", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "M₁ : Type u_4", "TopologicalSpace M₁", "AddCommMonoid M₁", "M₂ : Type u_6", "TopologicalSpace M₂", "AddCommMonoid M₂", "Module R₁ M₁", "Module R₂ M₂", "e : M₁ ≃SL[σ₁₂] M₂", "x : M₂", "y : M₁"], "goal": "y = e.symm x ↔ e y = x"}}
+{"state": {"context": ["x x₁ x₂ x₃ x' y y₁ y₂ y₃ y' : PGame", "h₁ : x₁.Numeric", "h₂ : x₂.Numeric", "h₁₂ : IH24 x₁ x₂ y", "h₂₁ : IH24 x₂ x₁ y", "he : x₁ ≈ x₂"], "goal": "x₁ * y ≤ x₂ * y"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "TopologicalSpace Y", "f : X → Y", "s : Set X", "t : Set Y", "h1 : Set.MapsTo f s t", "x : ↑s", "h2 : ContinuousAt f ↑x"], "goal": "ContinuousAt (Set.MapsTo.restrict f s t h1) x"}}
+{"state": {"context": ["α : Type u", "l : List α", "k : Nat", "hk : k < l.length", "l' : List α"], "goal": "(l ++ l').eraseIdx k = l.eraseIdx k ++ l'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "c : α", "AddGroup α", "InvolutiveNeg β", "h : Function.Antiperiodic f c", "x : α"], "goal": "f (x - c) = -f x"}}
+{"state": {"context": ["α : Type u_1", "Monoid α", "a b : α", "u : αˣ"], "goal": "a ∣ b * ↑u ↔ a ∣ b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "P : Type u_3", "N : Type w", "inst✝⁷ : Ring R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : Module R M", "inst✝⁴ : AddCommGroup N", "inst✝³ : Module R N", "inst✝² : AddCommGroup P", "inst✝¹ : Module R P", "inst✝ : IsNoetherian R M", "f : M →ₗ[R] M"], "goal": "∀ᶠ (n : ℕ) in atTop, ⨆ m, ker (f ^ m) = ker (f ^ n)"}}
+{"state": {"context": [], "goal": "∀ (a : Bool × ℕ), Equiv.boolProdNatEquivNat a = Function.uncurry Nat.bit a"}}
+{"state": {"context": ["A : Type u_1", "K : Type u_2", "L : Type u_3", "B : Type u_4", "inst✝¹³ : CommRing A", "inst✝¹² : CommRing B", "inst✝¹¹ : Algebra A B", "inst✝¹⁰ : Field K", "inst✝⁹ : Field L", "inst✝⁸ : Algebra A K", "inst✝⁷ : IsFractionRing A K", "inst✝⁶ : Algebra B L", "inst✝⁵ : Algebra K L", "inst✝⁴ : Algebra A L", "inst✝³ : IsScalarTower A B L", "inst✝² : IsScalarTower A K L", "inst✝¹ : IsIntegralClosure B A L", "inst✝ : FiniteDimensional K L", "σ : B →ₐ[A] B", "x : B", "this : IsDomain A"], "goal": "((galRestrictHom A K L B).symm σ) ((algebraMap B L) x) = (algebraMap B L) (σ x)"}}
+{"state": {"context": ["R : Type u_1", "CommMonoid R", "R' : Type u_2", "CommMonoidWithZero R'", "χ : MulChar R R'", "a : Rˣ"], "goal": "↑((MulChar.equivToUnitHom χ) a) = χ ↑a"}}
+{"state": {"context": ["F : Type u", "E : Type v", "Field F", "Field E", "Algebra F E", "f : F[X]"], "goal": "(Polynomial.map (algebraMap F E) f).natSepDegree = f.natSepDegree"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VSub α β", "b c : β"], "goal": "{b} -ᵥ {c} = {b -ᵥ c}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "p✝ : FormalMultilinearSeries 𝕜 E F", "r✝ : ℝ≥0", "p : FormalMultilinearSeries 𝕜 E F", "f : F →L[𝕜] G", "r : ℝ≥0", "hr : ↑r < p.radius", "n : ℕ"], "goal": "‖‖f.compFormalMultilinearSeries p n‖‖ ≤ ‖f‖ * ‖‖p n‖‖"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "f₁ f₂ : X ⟶ Z", "g₁ g₂ : Y ⟶ Z", "h₁ : f₁ = f₂", "h₂ : g₁ = g₂", "CategoryTheory.Limits.HasPullback f₁ g₁", "CategoryTheory.Limits.HasPullback f₂ g₂"], "goal": "(CategoryTheory.Limits.pullback.congrHom h₁ h₂).inv =\n  CategoryTheory.Limits.pullback.map f₂ g₂ f₁ g₁ (𝟙 X) (𝟙 Y) (𝟙 Z) ⋯ ⋯"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "I : Ideal R", "hI : I ≠ 0", "hf : (mulSupport fun v => v.maxPowDividing I).Finite", "h_ne_zero : v.maxPowDividing I ≠ 0"], "goal": "¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "F G : C ⥤ D", "α : F ≅ G", "W X X' : D", "Y : C", "f : W ⟶ X", "g : X ⟶ G.obj Y", "f' : W ⟶ X'", "g' : X' ⟶ G.obj Y"], "goal": "f ≫ g ≫ α.inv.app Y = f' ≫ g' ≫ α.inv.app Y ↔ f ≫ g = f' ≫ g'"}}
+{"state": {"context": ["l : List ℕ", "n : ℕ"], "goal": "List.Sorted (fun x x_1 => x < x_1) (Denumerable.raise' l n)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Preadditive C", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "HomologicalComplex.HasHomotopyCofiber φ", "n : ℤ"], "goal": "(CochainComplex.mappingCone φ).d (n - 1) n ≫ (CochainComplex.mappingCone.snd φ).v n n ⋯ =\n  (↑(CochainComplex.mappingCone.fst φ)).v (n - 1) n ⋯ ≫ φ.f n +\n    (CochainComplex.mappingCone.snd φ).v (n - 1) (n - 1) ⋯ ≫ G.d (n - 1) n"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "A : Type u_5", "x : ι → A", "CommRing R", "CommRing A", "Algebra R A", "hx : AlgebraicIndependent R x"], "goal": "RingHom.ker ↑hx.repr = ⊥"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a✝ b : R", "n : ℕ", "inst✝ : CommSemiring R", "a p P : R[X]", "h : ∀ (r : R), r • P = 0 → r = 0", "Q : R[X]", "hQ : P * Q = 0", "l : ℕ", "IH : ∀ m > l, P.coeff m • Q = 0", "hl : (P.coeff l • Q).natDegree = Q.natDegree", "m : ℕ := Q.natDegree", "i j : ℕ", "hij : i + j = l + m", "H : i = l → ¬j = m", "hi : i < l"], "goal": "P.coeff i * Q.coeff j = 0"}}
+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝ : AddMonoid A", "x y : A"], "goal": "y ∈ closure {x} ↔ ∃ n, n • x = y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι✝ : Type u_5", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s s₁ s₂ t : Set α", "ι : Type u_8", "inst✝ : MeasurableSpace α", "μ : Measure α", "As : ι → Set α", "As_mble : ∀ (i : ι), NullMeasurableSet (As i) μ", "As_disj : Pairwise (AEDisjoint μ on As)"], "goal": "∑' (i : ι), μ (As i) ≤ μ (⋃ i, As i)"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "c : R", "f : RingSeminorm R", "hf1 : f 1 ≤ 1", "hc : f c ≠ 0", "hpm : IsPowMul ⇑f", "hf : f 0 = 0"], "goal": "seminormFromConst_seq c f 0 = 0"}}
+{"state": {"context": ["α : Type u_2", "n : ℕ", "xs : Mathlib.Vector α n", "σ₁ : Type", "β : Type u_1", "σ₂ : Type", "f₁ : α → σ₁ → σ₁ × β", "f₂ : α → σ₂ → σ₂ × β", "s₁ : σ₁", "s₂ : σ₂", "R : σ₁ → σ₂ → Prop", "h₀ : R s₁ s₂", "hR : ∀ {s : σ₁} {q : σ₂} (a : α), R s q → R (f₁ a s).1 (f₂ a q).1 ∧ (f₁ a s).2 = (f₂ a q).2"], "goal": "R (Mathlib.Vector.mapAccumr f₁ xs s₁).1 (Mathlib.Vector.mapAccumr f₂ xs s₂).1 ∧\n  (Mathlib.Vector.mapAccumr f₁ xs s₁).2 = (Mathlib.Vector.mapAccumr f₂ xs s₂).2"}}
+{"state": {"context": ["s : (k : ℕ) × (l : ℕ) × { s // s.card = l }"], "goal": "(FormalMultilinearSeries.changeOriginIndexEquiv s).fst = s.fst + s.snd.fst"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "S S' S'' : D", "Y Y' Y'' : C", "T T' : C ⥤ D", "f f' : StructuredArrow S T", "g : f.right ⟶ f'.right", "w : autoParam (f.hom ≫ T.map g = f'.hom) _auto✝"], "goal": "𝟙 S ≫ f'.hom = f.hom ≫ T.map g"}}
+{"state": {"context": ["R : Type u", "S : Type v", "ι : Type w", "a b : R", "m n : ℕ", "inst✝¹ : Semiring R", "p✝ q r : R[X]", "inst✝ : Semiring S", "p : R[X]", "hp : p ≠ 0", "f : R →+* S", "z : S", "hz : eval₂ f z p = 0", "inj : ∀ (x : R), f x = 0 → x = 0", "hlt : 0 ≥ p.natDegree", "A : p = C (p.coeff 0)"], "goal": "False"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁸ : StrictOrderedCommRing R", "M : Type u_2", "N : Type u_3", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup N", "inst✝⁵ : Module R M", "inst✝⁴ : Module R N", "ι : Type u_4", "ι' : Type u_5", "inst✝³ : Fintype ι", "inst✝² : DecidableEq ι", "inst✝¹ : Fintype ι'", "inst✝ : DecidableEq ι'", "e : Basis ι R M", "w : ι → Rˣ"], "goal": "SameRay R (e.unitsSMul w).det ((↑(∏ i : ι, w i)⁻¹ * ∏ i : ι, ↑(w i)) • (e.unitsSMul w).det)"}}
+{"state": {"context": ["R : Type u", "S₁ : Type v", "S₂ : Type w", "S₃ : Type x", "σ : Type u_1", "a a' a₁ a₂ : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S₁", "p✝ q : MvPolynomial σ R", "f : R →+* S₁", "p : MvPolynomial σ R", "x : σ →₀ ℕ"], "goal": "coeff x p = 0 → f (coeff x p) = 0"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasBinaryCoproducts C", "self : CategoryTheory.HasPullbacksOfInclusions C", "X Y Z : C", "f : Z ⟶ X ⨿ Y"], "goal": "CategoryTheory.Limits.HasPullback CategoryTheory.Limits.coprod.inl f"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "FiniteDimensional 𝕜 E", "T : E →ₗ[𝕜] E", "x : E"], "goal": "0 ≤ RCLike.re ⟪x, (LinearMap.adjoint T * T) x⟫_𝕜"}}
+{"state": {"context": ["G : Type u", "self : DivInvMonoid G", "a b : G"], "goal": "a / b = a * b⁻¹"}}
+{"state": {"context": ["α : Type u_1", "M₀ : Type u_2", "G₀ : Type u_3", "M₀' : Type u_4", "G₀' : Type u_5", "F : Type u_6", "F' : Type u_7", "inst✝¹ : MonoidWithZero M₀", "inst✝ : GroupWithZero G₀", "a✝¹ b c d : G₀", "m n : ℕ", "a : G₀", "a✝ : ℕ"], "goal": "a ≠ 0 → a ^ a✝ ≠ 0"}}
+{"state": {"context": ["J : Type u₁", "inst✝² : Category.{v₁, u₁} J", "K : Type u₂", "inst✝¹ : Category.{v₂, u₂} K", "C : Type u", "inst✝ : Category.{v, u} C", "F✝ : J ⥤ C", "L : (J ⥤ C) ⥤ C", "adj : const J ⊣ L", "F : J ⥤ C", "s : Cone F", "j : J", "eq : ((L ⋙ const J).map s.π ≫ adj.counit.app F).app j = (adj.counit.app ((const J).obj s.pt) ≫ (𝟭 (J ⥤ C)).map s.π).app j"], "goal": "(fun s => (adj.homEquiv s.pt F) s.π) s ≫ (coneOfAdj adj F).π.app j = s.π.app j"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "s : Set E", "f : E → F", "x : E", "n : ℕ∞", "h : ContDiffWithinAt 𝕜 n f s x", "t : Set E", "hst : 𝓝[s] x = 𝓝[t] x"], "goal": "ContDiffWithinAt 𝕜 n f t x"}}
+{"state": {"context": ["A : Type u_1", "inst✝³ : Category.{?u.8867, u_1} A", "inst✝² : ConcreteCategory A", "inst✝¹ : ReflectsFiniteLimits (CategoryTheory.forget A)", "F : LightProfiniteᵒᵖ ⥤ A", "inst✝ : PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)", "hF : EqualizerCondition (F ⋙ CategoryTheory.forget A)"], "goal": "Nonempty (PreservesFiniteProducts (F ⋙ CategoryTheory.forget A)) ∧ EqualizerCondition (F ⋙ CategoryTheory.forget A)"}}
+{"state": {"context": ["R : Type u", "Ring R", "s t : Set R", "h : s ⊆ t"], "goal": "Subring.centralizer t ≤ Subring.centralizer s"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁸ : Category.{u_3, u_1} C", "inst✝⁷ : Category.{u_4, u_2} D", "inst✝⁶ : HasZeroMorphisms C", "inst✝⁵ : HasZeroMorphisms D", "S : ShortComplex C", "h₁✝ : S.LeftHomologyData", "h₂✝ : S.RightHomologyData", "F✝ : C ⥤ D", "inst✝⁴ : F✝.PreservesZeroMorphisms", "hl : S.LeftHomologyData", "hr : S.RightHomologyData", "S₁ S₂ : ShortComplex C", "φ : S₁ ⟶ S₂", "hl₁ : S₁.LeftHomologyData", "hr₁ : S₁.RightHomologyData", "hl₂ : S₂.LeftHomologyData", "hr₂ : S₂.RightHomologyData", "h₁ : S₁.HomologyData", "h₂ : S₂.HomologyData", "F : C ⥤ D", "inst✝³ : F.PreservesZeroMorphisms", "inst✝² : S.HasHomology", "inst✝¹ : F.PreservesLeftHomologyOf S", "inst✝ : F.PreservesRightHomologyOf S", "γ : HomologyMapData (𝟙 (S.map F)) (S.map F).homologyData (S.homologyData.map F)", "eq : leftHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.left (S.homologyData.left.map F) ≫ F.map S.homologyData.iso.hom = (S.map F).homologyData.iso.hom ≫ rightHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.right (S.homologyData.right.map F)"], "goal": "(S.map F).homologyData.iso.hom ≫ rightHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.right (S.homologyData.right.map F) ≫ F.map S.homologyData.iso.inv = leftHomologyMap' (𝟙 (S.map F)) (S.map F).homologyData.left (S.homologyData.left.map F)"}}
+{"state": {"context": ["R : Type u_1", "R' : Type u_2", "𝕜 : Type u_3", "𝕜₂ : Type u_4", "𝕜₃ : Type u_5", "𝕝 : Type u_6", "E : Type u_7", "E₂ : Type u_8", "E₃ : Type u_9", "F : Type u_10", "G : Type u_11", "ι : Type u_12", "inst✝⁵ : NormedField 𝕜", "inst✝⁴ : AddCommGroup E", "inst✝³ : NormedSpace ℝ 𝕜", "inst✝² : Module 𝕜 E", "inst✝¹ : Module ℝ E", "inst✝ : IsScalarTower ℝ 𝕜 E", "p : Seminorm 𝕜 E", "x : E", "r : ℝ", "y : E"], "goal": "p (y + -x) < r ↔ y ∈ {x_1 | (⇑p ∘ fun z => z + -x) x_1 < r}"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "e : PartialEquiv α β", "s : Set α"], "goal": "(e.restr s).target = e.target ∩ ↑e.symm ⁻¹' s"}}
+{"state": {"context": ["α : Type u_1", "CancelCommMonoidWithZero α", "NormalizationMonoid α", "UniqueFactorizationMonoid α", "x y : α", "hx : x ≠ 0", "hy : y ≠ 0"], "goal": "DvdNotUnit x y ↔ UniqueFactorizationMonoid.normalizedFactors x < UniqueFactorizationMonoid.normalizedFactors y"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "VAdd α β", "f : Filter α", "g h : Filter β"], "goal": "h ≤ f +ᵥ g ↔ ∀ ⦃s : Set α⦄, s ∈ f → ∀ ⦃t : Set β⦄, t ∈ g → s +ᵥ t ∈ h"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "H : Type u_3", "TopologicalSpace H", "I : ModelWithCorners 𝕜 E H", "M : Type u_4", "TopologicalSpace M", "ChartedSpace H M", "E' : Type u_5", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_6", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "TopologicalSpace M'", "ChartedSpace H' M'", "f f₁ : M → M'", "s : Set M", "x : M", "n : ℕ∞", "h₁ : ∀ y ∈ s, f₁ y = f y", "hx : f₁ x = f x"], "goal": "ContMDiffWithinAt I I' n f₁ s x ↔ ContMDiffWithinAt I I' n f s x"}}
+{"state": {"context": ["J : Type u_1", "C : Type u_2", "CategoryTheory.Category.{u_3, u_1} J", "CategoryTheory.Category.{u_4, u_2} C", "CategoryTheory.Limits.HasZeroMorphisms C", "F : CategoryTheory.Functor J (CategoryTheory.ShortComplex C)"], "goal": "((CategoryTheory.ShortComplex.FunctorEquivalence.inverse J C).obj F).X₁ = F.comp CategoryTheory.ShortComplex.π₁"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "γ : Type u_2", "δ : Type u_3", "ε : Type u_4", "l₁ : List α", "h : l₁.length ≤ [].length"], "goal": "(l₁.zip []).unzip.1 = l₁"}}
+{"state": {"context": ["ι : Type u_1", "π : ι → Type u_4", "(i : ι) → HImp (π i)", "a b : (i : ι) → π i", "i : ι"], "goal": "(a ⇨ b) i = a i ⇨ b i"}}
+{"state": {"context": ["X : Type u_1", "inst✝³ : TopologicalSpace X", "inst✝² : MeasurableSpace X", "inst✝¹ : OpensMeasurableSpace X", "μ : Measure X", "inst✝ : IsFiniteMeasure μ", "f : X →ᵇ ℝ", "c : ℝ"], "goal": "∫ (x : X), (f + const X c) x ∂μ = ∫ (x : X), f x ∂μ + (μ Set.univ).toReal • c"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "AddCommMonoid M", "Module R M", "B : LinearMap.BilinForm R M", "H : B.IsRefl", "x y : M"], "goal": "(B x) y = 0 → (B y) x = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝¹ : PartialOrder α", "inst✝ : LocallyFiniteOrder α", "a b c : α", "h : a < b"], "goal": "Ico a b = cons a (Ioo a b) ⋯"}}
+{"state": {"context": ["H : Type u_1", "M : Type u_2", "H' : Type u_3", "TopologicalSpace H", "TopologicalSpace M", "ChartedSpace H M", "TopologicalSpace H'", "G : StructureGroupoid H", "G' : StructureGroupoid H'", "e e' : PartialHomeomorph M H", "P : (H → H') → Set H → H → Prop", "s : Set M", "x : M", "hG : G.LocalInvariantProp G' P", "g : M → H'", "he : e ∈ StructureGroupoid.maximalAtlas M G", "xe : x ∈ e.source", "he' : e' ∈ StructureGroupoid.maximalAtlas M G", "xe' : x ∈ e'.source"], "goal": "P (g ∘ ↑e.symm) (↑e.symm ⁻¹' s) (↑e x) ↔ P (g ∘ ↑e'.symm) (↑e'.symm ⁻¹' s) (↑e' x)"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "Ring k", "AddCommGroup V", "Module k V", "AffineSpace V P", "n : ℕ", "s : Affine.Simplex k P n"], "goal": "s.reindex (Equiv.refl (Fin (n + 1))) = s"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "β : Type u_3", "δ : Type u_4", "γ : Type u_5", "ι : Type u_6", "inst✝ : MeasureSpace α", "s✝ t s : Set α", "hs : MeasurableSet s"], "goal": "Measure.map Subtype.val volume = volume.restrict s"}}
+{"state": {"context": ["n : ℕ", "F : TypeVec.{u} n → Type u_1", "q : MvQPF F", "h : IsUniform", "α : TypeVec.{u} n", "a : (P F).A", "f : (P F).B a ⟹ α", "i✝ : Fin2 n", "u : α i✝"], "goal": "(∀ (a_1 : (P F).A) (f_1 : (P F).B a_1 ⟹ α), abs ⟨a_1, f_1⟩ = abs ⟨a, f⟩ → u ∈ f_1 i✝ '' univ) ↔ u ∈ f i✝ '' univ"}}
+{"state": {"context": ["τ : Type u_1", "α : Type u_2", "�� : Type u_3", "ι : Type u_4", "inst✝ : TopologicalSpace β", "f : Filter τ", "ϕ : τ → α → β", "s s₁ s₂ : Set α", "c : Set β", "hc₁ : IsCompact c", "n : Set β", "hn₁ : IsOpen n", "hn₂ : ω f ϕ s ⊆ n", "v : Set τ", "hv₁ : v ∈ f", "hv₂ : closure (image2 ϕ v s) ⊆ c", "k : Set β := closure (image2 ϕ v s)", "hk : IsCompact (k \\ n)", "j : Set τ → Set β := fun u => (closure (image2 ϕ (u ∩ v) s))ᶜ"], "goal": "∃ u ∈ f, closure (image2 ϕ u s) ⊆ n"}}
+{"state": {"context": ["α : Type u_1", "BooleanAlgebra α", "a b : α"], "goal": "toBoolRing (a ∆ b) = toBoolRing a + toBoolRing b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝¹ : CommSemiring R", "σ : Type u_3", "inst✝ : CanonicallyLinearOrderedAddCommMonoid M", "w : σ → M", "φ : MvPolynomial σ R", "hw : NonTorsionWeight w", "m : σ →₀ ℕ", "x : σ"], "goal": "m x ≠ 0 → m x • w x = 0 ↔ m x = 0"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "d df dg : ℕ", "a b : R", "f g : R[X]", "h_mul_left : f.natDegree ≤ df", "h_mul_right : g.natDegree ≤ dg"], "goal": "f.coeff df = a → g.coeff dg = b → df + dg ≤ d → (f * g).coeff d = if d = df + dg then a * b else 0"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "NormedAddCommGroup F", "g : E → F", "c : ℝ≥0", "Fact (1 ≤ p)", "hg : LipschitzWith c g", "g0 : g 0 = 0"], "goal": "Continuous (hg.compLp g0)"}}
+{"state": {"context": ["i : ℤ", "j : ℤ", "hj : j ≠ 0"], "goal": "0 < i.gcd j"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "LE α", "LE β", "e : α ≃o β"], "goal": "e.symm.trans e = OrderIso.refl β"}}
+{"state": {"context": ["n m l : ℕ", "hln : l ≤ n"], "goal": "filter (fun x => decide (l ≤ x)) (Ico n m) = Ico (max n l) m"}}
+{"state": {"context": ["α : Type u_1", "CompleteLattice α", "a : α", "s : Set α"], "goal": "sInf (insert a s) = a ⊓ sInf s"}}
+{"state": {"context": ["R : Type u_1", "m₁ : Type u_3", "m₂ : Type u_4", "n₁ : Type u_6", "n₂ : Type u_7", "B₁₁ : Matrix m₁ n₁ R", "B₁₂ : Matrix m₁ n₂ R", "B₂₁ : Matrix m₂ n₁ R", "B₂₂ : Matrix m₂ n₂ R"], "goal": "(B₁₁.fromColumns B₁₂).fromRows (B₂₁.fromColumns B₂₂) = Matrix.fromBlocks B₁₁ B₁₂ B₂₁ B₂₂"}}
+{"state": {"context": ["H : Type u_1", "TopologicalSpace H", "G : StructureGroupoid H", "ClosedUnderRestriction G"], "goal": "G.LocalInvariantProp G G.IsLocalStructomorphWithinAt"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝ : UniformSpace α", "t : Set (α × α)", "h : t ∈ 𝓤 α", "w : Set (α × α)", "w_in : w ∈ 𝓤 α", "w_symm : SymmetricRel w", "r : w ○ w ○ w ⊆ t"], "goal": "Set (α × α)"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : AddCommGroup E", "inst✝ : Module ℝ E", "s t : Set E", "c : ℝ", "hc : 0 ≤ c", "x : E"], "goal": "(c * gauge s x / (c * gauge t x) * c) • x = (c * (gauge s x / gauge t x)) • x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "p : α → Bool", "a b : α", "as : List α", "h✝ : filter p (a :: b :: as) ≠ []", "h' : p ((b :: as).getLast ⋯) = true", "h : (if p a = true then a :: filter p (b :: as) else filter p (b :: as)) ≠ []"], "goal": "(if p a = true then a :: filter p (b :: as) else filter p (b :: as)).getLast ⋯ = (b :: as).getLast ⋯"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "W X Y Z : C", "f : X ⟶ Y", "g : X ⟶ Z", "g' : Z ⟶ W", "CategoryTheory.Limits.HasPushout f g", "CategoryTheory.Limits.HasPushout (CategoryTheory.Limits.pushout.inr f g) g'", "CategoryTheory.Limits.HasPushout f (g ≫ g')"], "goal": "CategoryTheory.Limits.pushout.inl f g ≫\n    CategoryTheory.Limits.pushout.inl (CategoryTheory.Limits.pushout.inr f g) g' ≫\n      (CategoryTheory.Limits.pushoutLeftPushoutInrIso f g g').hom =\n  CategoryTheory.Limits.pushout.inl f (g ≫ g')"}}
+{"state": {"context": ["x y : Cardinal.{u_1}"], "goal": "Cardinal.toNat (x * y) = Cardinal.toNat x * Cardinal.toNat y"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "β : Type u_5", "MeasurableSpace β", "μb : MeasureTheory.Measure β", "f : α → β", "g : ↥(MeasureTheory.Lp E p μb)", "hf : MeasureTheory.MeasurePreserving f μ μb"], "goal": "↑((MeasureTheory.Lp.compMeasurePreserving f hf) g) = (↑g).compMeasurePreserving f hf"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B B₁ B₂ : Set α", "inst✝ : M.RkPos", "hB : M.Base B"], "goal": "B.Nonempty"}}
+{"state": {"context": ["𝕜 : Type u_1", "𝕜' : Type u_2", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "inst✝¹³ : NontriviallyNormedField 𝕜'", "σ : 𝕜 →+* 𝕜'", "σ' : 𝕜' →+* 𝕜", "inst✝¹² : RingHomInvPair σ σ'", "inst✝¹¹ : RingHomInvPair σ' σ", "inst✝¹⁰ : RingHomIsometric σ", "inst✝⁹ : RingHomIsometric σ'", "E : Type u_3", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F✝ : Type u_4", "inst✝⁶ : NormedAddCommGroup F✝", "inst✝⁵ : NormedSpace 𝕜' F✝", "f : E →SL[σ] F✝", "inst✝⁴ : CompleteSpace F✝", "inst✝³ : CompleteSpace E", "F : Type u_5", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace 𝕜 F", "inst✝ : CompleteSpace F", "g : E →ₗ[𝕜] F", "hg : IsClosed ↑g.graph"], "goal": "Continuous ⇑g"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "H : Type u_3", "TopologicalSpace H", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "AddCommMonoid G", "TopologicalSpace G", "ChartedSpace H G", "SmoothAdd I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "s : Set M", "x : M", "t : Finset ι", "f : ι → M → G", "h : ∀ i ∈ t, SmoothWithinAt I' I (f i) s x"], "goal": "SmoothWithinAt I' I (fun x => ∑ i ∈ t, f i x) s x"}}
+{"state": {"context": ["ι : Type u_1", "I✝ J✝ : Box ι", "p : Box ι → Prop", "I : Box ι", "H_nhds : ∀ z ∈ Box.Icc I, ∃ U ∈ 𝓝[Box.Icc I] z, ∀ J ≤ I, ∀ (m : ℕ), z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J", "hpI : ¬p I", "s : Box ι → Set ι", "hs : ∀ J ≤ I, ¬p J → ¬p (J.splitCenterBox (s J))", "J : ℕ → Box ι := fun m => (fun J => J.splitCenterBox (s J))^[m] I"], "goal": "False"}}
+{"state": {"context": ["R : Type u₁", "L : Type u₂", "CommRing R", "LieRing L", "LieAlgebra R L", "A : Type u₃", "Ring A", "Algebra R A", "f : L →ₗ⁅R⁆ A", "x : L"], "goal": "((UniversalEnvelopingAlgebra.lift R) f) ((UniversalEnvelopingAlgebra.mkAlgHom R L) (ιₜ x)) = f x"}}
+{"state": {"context": ["θ ψ : Real.Angle", "h : 2 • θ = 2 • ψ"], "goal": "|θ.sin| = |ψ.sin|"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "ι : Type u_4", "inst✝⁴ : DivisionRing k", "inst✝³ : AddCommGroup V", "inst✝² : Module k V", "inst✝¹ : AffineSpace V P", "inst✝ : Fintype ι", "p : ι → P", "hp : AffineIndependent k p", "h✝ : IsEmpty ι"], "goal": "Fintype.card ι ≤ finrank k ↥(vectorSpan k (Set.range p)) + 1"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "F : C ⥤ D", "inst✝ : F.IsEquivalence", "X Y : C", "f : X ⟶ Y"], "goal": "F.inv.map (F.map f) = (F.asEquivalence.functor ⋙ F.asEquivalence.inverse).map f"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "w z : ι → ℝ", "x : ℝ", "hw' : ∑ i ∈ s, w i = 1", "hx : ∀ i ∈ s, w i ≠ 0 → z i = x"], "goal": "∑ i ∈ s, w i * x = x"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "AddGroup G", "AddAction G α", "g : G", "a : α"], "goal": "AddAction.stabilizer G (g +ᵥ a) = AddSubgroup.map (AddEquiv.toAddMonoidHom (AddAut.conj g)) (AddAction.stabilizer G a)"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁶ : Category.{v₁, u₁} C", "W : MorphismProperty C", "D : Type u₂", "inst✝⁵ : Category.{v₂, u₂} D", "D' : Type u₃", "inst✝⁴ : Category.{v₃, u₃} D'", "L : C ⥤ D", "inst✝³ : L.IsLocalization W", "X Y Z : C", "inst✝² : HasSmallLocalizedHom W X Y", "inst✝¹ : HasSmallLocalizedHom W Y Z", "inst✝ : HasSmallLocalizedHom W X Z", "this✝ : Small.{w, max u₁ v₁} (W.Q.obj X ⟶ W.Q.obj Y) := small_of_hasSmallLocalizedHom W W.Q X Y", "this : Small.{w, max u₁ v₁} (W.Q.obj Y ⟶ W.Q.obj Z) := small_of_hasSmallLocalizedHom W W.Q Y Z", "α : W.Q.obj X ⟶ W.Q.obj Y", "β : W.Q.obj Y ⟶ W.Q.obj Z"], "goal": "(homEquiv W W.Q L) ((homEquiv W W.Q W.Q) ((homEquiv W W.Q W.Q) α ≫ (homEquiv W W.Q W.Q) β)) = (homEquiv W W.Q L) α ≫ (homEquiv W W.Q L) β"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "A : Type uA", "B : Type uB", "M : Type uM", "N : Type uN", "inst✝¹² : CommSemiring R", "inst✝¹¹ : CommSemiring S", "inst✝¹⁰ : Semiring A", "inst✝⁹ : Semiring B", "inst✝⁸ : Algebra R S", "inst✝⁷ : Algebra R A", "inst✝⁶ : Algebra R B", "inst✝⁵ : AddCommMonoid M", "inst✝⁴ : Module R M", "inst✝³ : AddCommMonoid N", "inst✝² : Module R N", "inst✝¹ : Algebra S A", "inst✝ : IsScalarTower R S A", "hA : FiniteType R A"], "goal": "FiniteType S A"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "K : Set α", "hK : IsCompact K"], "goal": "(𝓝ˢ K).HasBasis (fun δ => 0 < δ) fun δ => Metric.thickening δ K"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝⁶ : MeasurableSpace α", "inst✝⁵ : MeasurableSpace α'", "inst✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace β'", "inst✝² : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝¹ : NormedAddCommGroup E", "inst✝ : SFinite ν", "s : Set (α × β)", "h : (μ.prod ν) s = 0"], "goal": "(fun x => ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0"}}
+{"state": {"context": ["E : Type u_3", "NormedAddCommGroup E", "NormedSpace ℝ E", "a b : ℝ", "f g : ℝ → E", "μ : MeasureTheory.Measure ℝ", "h : ∀ᵐ (x : ℝ) ∂μ, x ∈ Ι a b → f x = g x"], "goal": "∫ (x : ℝ) in a..b, f x ∂μ = ∫ (x : ℝ) in a..b, g x ∂μ"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s t : Finset α"], "goal": "Finset.Ico s t = Finset.filter (fun x => s ⊆ x) t.ssubsets"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "S : Type u_2", "hp : Fact (Nat.Prime p)", "inst✝¹ : CommRing R", "inst✝ : CommRing S", "r : R", "n : ℕ"], "goal": "0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "X✝ Y✝ : PartialFun", "f : X✝ ⟶ Y✝", "a : (𝟭 PartialFun).obj X✝", "b : (partialFunToPointed ⋙ pointedToPartialFun).obj Y✝"], "goal": "(b ∈ (f a).bind fun x => Part.some ⟨some x, ⋯⟩) ↔ b ∈ (PFun.toSubtype (fun x => ¬x = none) (Option.elim' none fun a => (f a).toOption) ∘ Subtype.val) ⟨some a, ⋯⟩"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a x : α", "h : x ∈ Set.Iio a"], "goal": "x ∈ Set.Iic a"}}
+{"state": {"context": ["S : Type u_1", "R : Type u_3", "M : Type u_4", "N : Type u_5", "CommRing R", "AddCommGroup M", "AddCommGroup N", "Module R M", "Module R N", "Q : QuadraticMap R M N", "CommSemiring S", "Algebra S R", "Module S M", "IsScalarTower S R M", "Module S N", "IsScalarTower S R N", "a : S", "x y : M"], "goal": "QuadraticMap.polar (⇑Q) (a • x) y = a • QuadraticMap.polar (⇑Q) x y"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝¹ : Category.{?u.25278, u_1} C", "inst✝ : Category.{?u.25282, u_2} D", "F G : C ⥤ Cat", "α : F ⟶ G", "X : Grothendieck F"], "goal": "(eqToHom ⋯).app X.fiber ≫ eqToHom ⋯ = eqToHom ⋯"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedAddCommGroup E", "TopologicalSpace α", "BorelSpace α", "SecondCountableTopologyEither α E", "MeasureTheory.IsFiniteMeasure μ", "f : α →ᵇ E"], "goal": "‖⟨ContinuousMap.toAEEqFun μ f.toContinuousMap, ⋯⟩‖₊ ≤ MeasureTheory.measureUnivNNReal μ ^ p.toReal⁻¹ * ‖f‖₊"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α"], "goal": "⊤ + μ = ⊤"}}
+{"state": {"context": ["R : Type u", "ι : Type v", "M₂ : Type w₂", "M₃ : Type w₃", "Semiring R", "AddCommMonoid M₂", "AddCommMonoid M₃", "Module R M₂", "Module R M₃", "TopologicalSpace M₂", "TopologicalSpace M₃", "Subsingleton ι", "i : ι", "f : M₂ →L[R] M₃"], "goal": "((ContinuousMultilinearMap.ofSubsingleton R M₂ M₃ i) f).toMultilinearMap = (MultilinearMap.ofSubsingleton R M₂ M₃ i) ↑f"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "Semiring R", "φ : MvPowerSeries σ R", "s : σ"], "goal": "Commute φ (MvPowerSeries.X s)"}}
+{"state": {"context": ["a : Rat"], "goal": "(-a).den = a.den"}}
+{"state": {"context": ["μ : YoungDiagram", "x : ℕ", "hx : ∃ a ∈ List.range (μ.colLen 0), μ.rowLen a = x"], "goal": "0 < x"}}
+{"state": {"context": ["α : Type u_1", "x y : List α", "h : x <+: y", "hx : x ≠ []"], "goal": "y ≠ []"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "σ : Type u_3", "AddCommMonoid M", "w : σ → M", "m : M"], "goal": "MvPolynomial.weightedHomogeneousSubmodule R w m = Finsupp.supported R R {d | (Finsupp.weight w) d = m}"}}
+{"state": {"context": [], "goal": "@product = @productTR"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "RCLike 𝕜", "NormedAddCommGroup E", "NormedAddCommGroup F", "InnerProductSpace 𝕜 E", "InnerProductSpace 𝕜 F", "CompleteSpace E", "CompleteSpace F", "A : E →L[𝕜] F", "x : E"], "goal": "‖A x‖ = √(RCLike.re ⟪((ContinuousLinearMap.adjoint A).comp A) x, x⟫_𝕜)"}}
+{"state": {"context": ["G : Type u_1", "P : Type u_2", "AddGroup G", "T : AddTorsor G P", "p q : P"], "goal": "p -ᵥ q ≠ 0 ↔ p ≠ q"}}
+{"state": {"context": ["α : Type u_1", "G : Type u_2", "Group G", "MulAction G α", "g h : G", "comm : Commute g h"], "goal": "(MulAction.fixedBy α g)ᶜ ∈ MulAction.fixedBy (Set α) h"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasTerminal C", "CategoryTheory.Limits.HasBinaryProducts C"], "goal": "comonEquiv.unitIso = CategoryTheory.NatIso.ofComponents (fun A => iso_cartesianComon_ A) ⋯"}}
+{"state": {"context": ["K : Type u_1", "RCLike K", "z : K"], "goal": "RCLike.normSq z = 0 ↔ z = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "t₁ t₂ : TopologicalSpace α", "g : β → α"], "goal": "TopologicalSpace.induced g (t₁ ⊓ t₂) = TopologicalSpace.induced g t₁ ⊓ TopologicalSpace.induced g t₂"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "L' : LieSubalgebra R L"], "goal": "0 ∈ L'"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "m : MeasurableSpace α", "μ ν : Measure α", "inst✝² : MeasurableSpace δ", "inst✝¹ : NormedAddCommGroup β", "inst✝ : NormedAddCommGroup γ", "f : α → ℝ≥0∞", "hfm : AEMeasurable f μ", "hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤"], "goal": "Memℒp (fun x => (f x).toReal) 1 μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β → β", "inst✝¹ : BEq α", "inst✝ : Hashable α", "m : Imp α β", "k : α", "h : m.size = m.buckets.size"], "goal": "m.size = ((m.buckets.update (mkIdx ⋯ (hash k).toUSize).val AssocList.nil ⋯).update (mkIdx ⋯ (hash k).toUSize).val (AssocList.modify k f m.buckets.val[(mkIdx ⋯ (hash k).toUSize).val.toNat]) ⋯).size"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "MeasurableSpace α", "MeasurableSpace β", "f : ι → α → β", "μ : MeasureTheory.Measure α", "p : α → (ι → β) → Prop", "Countable ι", "hf : ∀ (i : ι), AEMeasurable (f i) μ", "hp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x"], "goal": "μ (aeSeqSet hf p)ᶜ = 0"}}
+{"state": {"context": ["M : Type u_5", "N : Type u_6", "AddZeroClass M", "AddZeroClass N"], "goal": "⇑AddEquiv.prodComm = Prod.swap"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "l✝ : List α", "x✝ : α", "l l' : List α", "h : l ~r l'", "hn : l.Nodup", "x : α", "hx : x ∈ l"], "goal": "l.next x hx = l'.next x ⋯"}}
+{"state": {"context": ["α β : Type u", "f : α → β", "x : FreeAddGroup α"], "goal": "f <$> (-x) = -f <$> x"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "One M", "One N", "f : α → M × N"], "goal": "Function.mulSupport f = (Function.mulSupport fun x => (f x).1) ∪ Function.mulSupport fun x => (f x).2"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "(i : ι) → Group (G i)", "Group H", "φ : (i : ι) → H →* G i", "d : Monoid.PushoutI.NormalWord.Transversal φ", "DecidableEq ι", "(i : ι) → DecidableEq (G i)", "w : Monoid.CoprodI.Word G", "i : ι", "h : H", "hw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → g ∈ d.set i", "hφw : ∀ (j : ι) (g : G j), ⟨j, g⟩ ∈ (Monoid.CoprodI.of ((φ i) h) • w).toList → g ∈ d.set j"], "goal": "h = 1"}}
+{"state": {"context": ["G : Type u_1", "Group G", "S : Set G", "hS : Subgroup.closure S = ⊤"], "goal": "Subgroup.center G = ⨅ g, Subgroup.centralizer ↑(Subgroup.zpowers ↑g)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁹ : NormedAddCommGroup D", "inst✝⁸ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "G : Type uG", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "X : Type u_2", "inst✝¹ : NormedAddCommGroup X", "inst✝ : NormedSpace 𝕜 X", "s✝ s₁ t u : Set E", "f✝ f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "ι : Type u_3", "f : ι → E → F", "s : Finset ι", "h : ∀ i ∈ s, ContDiff 𝕜 n fun x => f i x"], "goal": "ContDiff 𝕜 n fun x => ∑ i ∈ s, f i x"}}
+{"state": {"context": ["p : ℝ", "hp1 : 0 < p", "hp2 : p < 1", "hs : -1 ≤ -1", "hs' : -1 ≠ 0"], "goal": "(1 + -1) ^ p < 1 + p * -1"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α", "a : α", "n : ℕ", "h : a ∉ s", "m : { x // x ∈ (insert a s).sym n }"], "goal": "↑((Finset.symInsertEquiv h) m).snd = (Sym.filterNe a ↑m).snd"}}
+{"state": {"context": ["α : Type u", "as : List α", "a : α"], "goal": "as.concat a = as ++ a :: List.nil"}}
+{"state": {"context": ["A : Type u_4", "CommRing A", "IsDomain A", "UniqueFactorizationMonoid A", "K : Type u_5", "Field K", "Algebra A K", "IsFractionRing A K", "x : K"], "goal": "IsRelPrime (IsFractionRing.num A x) ↑(IsFractionRing.den A x)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁸ : TopologicalSpace α", "inst✝⁷ : NormalSpace α", "inst✝⁶ : R1Space α", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : BorelSpace α", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "μ : Measure α", "p : ℝ≥0∞", "inst✝² : NormedSpace ℝ E", "inst✝¹ : WeaklyLocallyCompactSpace α", "inst✝ : μ.Regular", "f : α → E", "hf : Integrable f μ", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ∂μ ≤ ε ∧ Continuous g ∧ Integrable g μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "n✝ m✝ l n m : ℕ"], "goal": "(fun x => x = m) = Eq m"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "f g : Perm (Fin n)"], "goal": "(∏ x ∈ finPairsLT n, if (f * g) x.fst ≤ (f * g) x.snd then -1 else 1) = (∏ x ∈ finPairsLT n, if f x.fst ≤ f x.snd then -1 else 1) * ∏ x ∈ finPairsLT n, if g⁻¹ x.fst ≤ g⁻¹ x.snd then -1 else 1"}}
+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "ha : Polynomial.eval a F = 0", "z' : ℤ_[p]", "hz' : Polynomial.eval z' F = 0", "hnormz' : ‖z' - a‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖", "h : ℤ_[p] := z' - a", "q : ℤ_[p]", "hq : Polynomial.eval (a + h) F = Polynomial.eval a F + Polynomial.eval a (Polynomial.derivative F) * h + q * h ^ 2", "this✝ : (Polynomial.eval a (Polynomial.derivative F) + q * h) * h = 0", "hne : ¬h = 0", "this : Polynomial.eval a (Polynomial.derivative F) + q * h = 0"], "goal": "Polynomial.eval a (Polynomial.derivative F) = -q * h"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C"], "goal": "CategoryTheory.Square.arrowArrowEquivalence'.inverse = CategoryTheory.Square.fromArrowArrowFunctor'"}}
+{"state": {"context": ["a b d : Int", "hda : d ∣ a", "hdb : d ∣ b"], "goal": "a.div d = b.div d ↔ a = b"}}
+{"state": {"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "U : Opens ↑↑Z.toPresheafedSpace"], "goal": "((pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ (pullbackRightPullbackFstIso g (Scheme.Opens.ι U) f).inv) ≫ pullback.snd f (pullback.fst g (Scheme.Opens.ι U)) = ((pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.snd f (g ⁻¹ᵁ U).ι) ≫ (pullbackRestrictIsoRestrict g U).inv"}}
+{"state": {"context": ["R : Type u", "Ring R", "p : R[X]"], "goal": "(-p).degree = p.degree"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℚ → ℝ", "a : α", "self : ProbabilityTheory.IsRatStieltjesPoint f a"], "goal": "Monotone (f a)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "NonarchimedeanAddGroup G", "K : Type u_3", "AddGroup K", "TopologicalSpace K", "NonarchimedeanAddGroup K", "U : Set (G × K)", "hU : U ∈ 𝓝 0"], "goal": "∃ V W, ↑V ×ˢ ↑W ⊆ U"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_4", "𝕜 : Type u_6", "MeasurableSpace α", "NormedAddCommGroup E", "p : ℝ≥0∞", "μ : MeasureTheory.Measure α", "NormedRing 𝕜", "Module 𝕜 E", "BoundedSMul 𝕜 E", "Fact (1 ≤ p)"], "goal": "BoundedSMul 𝕜 ↥(MeasureTheory.Lp.simpleFunc E p μ)"}}
+{"state": {"context": ["E : Type uE", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace ℝ E", "inst✝⁶ : FiniteDimensional ℝ E", "H : Type uH", "inst✝⁵ : TopologicalSpace H", "I : ModelWithCorners ℝ E H", "M : Type uM", "inst✝⁴ : TopologicalSpace M", "inst✝³ : ChartedSpace H M", "inst✝² : SmoothManifoldWithCorners I M", "c : M", "f : SmoothBumpFunction I c", "x✝ : M", "inst✝¹ : T2Space M", "inst✝ : SmoothManifoldWithCorners I M", "x : M", "hx : x ∈ tsupport ↑f"], "goal": "ContMDiffAt I 𝓘(ℝ, ℝ) ⊤ (↑f) x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E"], "goal": "DifferentiableAt 𝕜 (fun x => x) x"}}
+{"state": {"context": ["x : ℝ", "hx✝ : x ≠ 0", "hx : 0 < x"], "goal": "x + 1 < rexp x"}}
+{"state": {"context": ["X : Scheme", "inst✝ : IsIntegral X", "U : X.Opens", "x : ↥U", "y : ↑(X.presheaf.obj (op U))", "hy : (X.presheaf.germ x) y = 0"], "goal": "y = 0"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "s : MeasureTheory.SignedMeasure α", "i : Set α", "n : ℕ", "hn :\n  ¬MeasureTheory.VectorMeasure.restrict s\n        (i \\ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k) ≤\n      MeasureTheory.VectorMeasure.restrict 0\n        (i \\ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)"], "goal": "1 /\n    (↑(MeasureTheory.SignedMeasure.findExistsOneDivLT s\n          (i \\ ⋃ k, ⋃ (_ : k ≤ n), MeasureTheory.SignedMeasure.restrictNonposSeq s i k)) +\n      1) <\n  ↑s (MeasureTheory.SignedMeasure.restrictNonposSeq s i n.succ)"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝¹ : LinearOrderedField 𝕜", "G : SimpleGraph α", "inst✝ : DecidableRel G.Adj", "ε : 𝕜", "s t : Finset α", "a b : α", "h : ¬G.IsUniform ε s t"], "goal": "(⋯.choose, ⋯.choose).2 ⊆ t"}}
+{"state": {"context": ["R : Type u_1", "L : Type u_3", "M : Type u_4", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "Module.Free R M", "Module.Finite R M", "LieAlgebra.IsNilpotent R L", "IsDomain R", "IsPrincipalIdealRing R", "h : LinearMap.ker (LieModule.traceForm R L M) = ⊥"], "goal": "IsLieAbelian L"}}
+{"state": {"context": ["p : ℕ", "q : ℚ", "hq : ¬q = 0"], "goal": "0 ≤ ↑p"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝¹ : MeasureSpace Ω", "inst✝ : IsProbabilityMeasure ℙ", "X : Ω → ℝ", "hint : Integrable X ℙ", "hnonneg : 0 ≤ X", "K : ℕ", "A : Tendsto (fun N => ∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N}) atTop (𝓝 (∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioi ↑j}))", "N : ℕ", "hN : N ∈ Set.Ici K"], "goal": "∑ j ∈ range K, ℙ {ω | X ω ∈ Set.Ioc ↑j ↑N} ≤ ENNReal.ofReal ((∫ (a : Ω), X a) + 1)"}}
+{"state": {"context": ["α : Type u_1", "Mul α", "LE α", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a : α", "b c : α"], "goal": "a * b ≤ a * c ↔ b ≤ c"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : LT α", "a✝ b : α", "a : WithTop α"], "goal": "¬⊤ < a"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝ : CommRing R", "H : ∀ (P : Ideal R), P.IsPrime → IsPrincipal P", "J : Ideal R", "hJ : J ∈ nonPrincipals R", "I : Ideal R", "Ibad : I ∈ nonPrincipals R", "Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_3", "γ : Type u_5", "MeasurableSpace α", "MeasurableSpace β", "MeasurableSpace γ", "X : α → β", "Y : α → γ", "μ : MeasureTheory.Measure α", "hY : AEMeasurable Y μ"], "goal": "(MeasureTheory.Measure.map (fun a => (X a, Y a)) μ).fst = MeasureTheory.Measure.map X μ"}}
+{"state": {"context": ["V : Type u_1", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℝ V", "hd2 : Fact (finrank ℝ V = 2)", "o : Orientation ℝ V (Fin 2)", "x : V", "h : x ≠ 0", "r : ℝ", "hr : ¬r = 0", "hx : x = r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))"], "goal": "o.oangle (-(r • (o.rotation ↑(π / 2)) x) + r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))) (r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))) = ↑(Real.arctan r)"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "inst✝³ : Field A", "inst✝² : Ring B", "inst✝¹ : Algebra A B", "x : B", "inst✝ : IsReduced B", "n : ℕ", "p : A[X]", "dvd : (Polynomial.aeval x) (p ^ n) = 0"], "goal": "(Polynomial.aeval x) p = 0"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : PseudoMetricSpace α", "x y z : ℝ", "h : y ∈ uIcc x z"], "goal": "dist y z ≤ dist x z"}}
+{"state": {"context": ["R : Type u", "CommRing R", "W : WeierstrassCurve.Affine R", "x y : R"], "goal": "W.Equation x (W.negY x y) ↔ W.Equation x y"}}
+{"state": {"context": ["E : Type u_1", "F : Type u_2", "𝓕 : Type u_3", "SeminormedAddGroup E", "SeminormedAddCommGroup F", "FunLike 𝓕 E F", "ZeroAtInftyContinuousMapClass 𝓕 E F", "f : 𝓕", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ r, ∀ (x : E), r < ‖x‖ → ‖f x‖ < ε"}}
+{"state": {"context": ["K : Type u_3", "L : Type u_4", "Field K", "Field L", "Algebra K L", "x : L", "p : K[X]", "aeval_eq : (Polynomial.aeval x) p = 0", "coeff_zero_ne : p.coeff 0 ≠ 0"], "goal": "x⁻¹ = -((Polynomial.aeval x) p.divX / (algebraMap K L) (p.coeff 0))"}}
+{"state": {"context": ["a : Cardinal.{u_1}"], "goal": "a ^ 0 = 1"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹⁰ : Category.{u_6, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝⁹ : Category.{u_5, u_2} A", "inst✝⁸ : HasWeakSheafify J A", "inst✝⁷ : (constantSheaf J A).Faithful", "inst✝⁶ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "D : Type u_3", "inst✝⁵ : Category.{u_4, u_3} D", "K : GrothendieckTopology D", "inst✝⁴ : HasWeakSheafify K A", "G : C ⥤ D", "inst✝³ : ∀ (X : Dᵒᵖ), HasLimitsOfShape (StructuredArrow X G.op) A", "inst✝² : IsDenseSubsite J K G", "ht' : IsTerminal (G.obj t)", "inst✝¹ : (constantSheaf J A).Faithful", "inst✝ : (constantSheaf J A).Full", "F : Sheaf K A", "e : Sheaf J A ≌ Sheaf K A := sheafEquiv G J K A", "this✝ : (constantSheaf K A).Faithful", "this : (constantSheaf K A).Full"], "goal": "e.inverse.obj F ∈ (constantSheaf J A).essImage ↔ F ∈ (constantSheaf K A).essImage"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "o : Type u_4", "m' : Type u_6", "n' : Type u_7", "o' : Type u_8", "R : Type u_11", "A : Type u_12", "Semiring R", "Semiring A", "Module R A", "Fintype n", "Fintype n'", "eₘ : m ≃ m'", "eₙ : n ≃ n'", "eₒ : o ≃ o'", "M : Matrix m n A", "N : Matrix n o A"], "goal": "(Matrix.reindexLinearEquiv R A eₘ eₙ) M * (Matrix.reindexLinearEquiv R A eₙ eₒ) N =\n  (Matrix.reindexLinearEquiv R A eₘ eₒ) (M * N)"}}
+{"state": {"context": ["α : Type u_1", "AddCommGroup α", "l : ℤ", "r : α", "tl : ℤ", "tr t : α", "prl : l = tl", "prr : r = tr", "prt : Mathlib.Tactic.Abel.smulg tl tr = t"], "goal": "Mathlib.Tactic.Abel.smulg l r = t"}}
+{"state": {"context": ["α β : FinBddDistLat", "e : ↑α.toBddDistLat.toDistLat ≃o ↑β.toBddDistLat.toDistLat", "a : ↑β.toBddDistLat.toDistLat"], "goal": "(FinBddDistLat.Iso.mk e).inv.toSupHom a = e.symm a"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "A : Type v", "CommSemiring A", "Algebra R A", "I : Submodule R A"], "goal": "1 ≤ 1 / I ↔ I ≤ 1"}}
+{"state": {"context": ["δ : Type u_1", "δ' : Type u_2", "π : δ → Type u_3", "inst✝³ : (x : δ) → MeasurableSpace (π x)", "μ : (i : δ) → Measure (π i)", "inst✝² : ∀ (i : δ), SigmaFinite (μ i)", "inst✝¹ : DecidableEq δ", "s✝ t : Finset δ", "f✝ g✝ : ((i : δ) → π i) → ℝ≥0∞", "x y : (i : δ) → π i", "i : δ", "inst✝ : Fintype δ", "s : Finset δ", "f g : ((i : δ) → π i) → ℝ≥0∞", "hf : Measurable f", "hg : Measurable g", "hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ"], "goal": "∫⁻ (x : (i : δ) → π i), f x ∂Measure.pi μ ≤ ∫⁻ (x : (i : δ) → π i), g x ∂Measure.pi μ"}}
+{"state": {"context": ["n✝ n : ℕ"], "goal": "∏ x ∈ Icc 1 (n + 1), x ! = sf n + 1"}}
+{"state": {"context": ["X Y : Type u", "f : X ⟶ Y"], "goal": "CategoryTheory.Mono f ↔ Function.Injective f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a✝ a₁ a₂ : α", "b b₁✝ b₂✝ : β", "inst✝³ : LinearOrderedField α", "inst✝² : OrderedAddCommGroup β", "inst✝¹ : Module α β", "a : α", "b₁ b₂ : β", "inst✝ : PosSMulMono α β", "h : a < 0"], "goal": "a⁻¹ • b₁ ≤ b₂ ↔ a • b₂ ≤ b₁"}}
+{"state": {"context": ["E : Type u_1", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : FiniteDimensional ℝ E", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "F : Type u_2", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℝ F", "C D : ℝ≥0", "f g : E → ℝ", "s : Set E", "μ : Measure E", "inst✝ : μ.IsAddHaarMeasure", "hf : LipschitzWith C f", "hg : Integrable g μ", "v : E"], "goal": "∀ᶠ (n : ℝ) in 𝓝[>] 0, ∀ᵐ (a : E) ∂μ, ‖n⁻¹ • (f (a + n • v) - f a) * g a‖ ≤ ↑C * ‖v‖ * ‖g a‖"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_5, u_1} C", "CategoryTheory.Preadditive C", "I₁ : Type u_2", "I₂ : Type u_3", "I₁₂ : Type u_4", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "K : HomologicalComplex₂ C c₁ c₂", "c₁₂ : ComplexShape I₁₂", "DecidableEq I₁₂", "TotalComplexShape c₁ c₂ c₁₂", "K.HasTotal c₁₂", "i₁ i₁' : I₁", "h : c₁.Rel i₁ i₁'", "i₂ : I₂", "i₁₂ : I₁₂"], "goal": "K.d₁ c₁₂ i₁ i₂ i₁₂ =\n  c₁.ε₁ c₂ c₁₂ (i₁, i₂) • (K.d i₁ i₁').f i₂ ≫ K.toGradedObject.ιMapObjOrZero (c₁.π c₂ c₁₂) (i₁', i₂) i₁₂"}}
+{"state": {"context": ["c : Char", "l r : List Char", "it : Iterator", "h : ValidFor (c :: l) r it"], "goal": "ValidFor l (c :: r) it.prev"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedField α", "β : Type u_2", "DivisionRing β", "abv : β → α", "IsAbsoluteValue abv"], "goal": "¬0 ≈ 1"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x y z : X", "ι : Type u_3", "F : Set X", "n : ℕ", "s : Set X", "h : IsPathConnected s", "p : Fin (n + 1) → X", "hp : ∀ (i : Fin (n + 1)), p i ∈ s", "p' : ℕ → X := fun k => if h : k < n + 1 then p ⟨k, h⟩ else p ⟨0, ⋯⟩", "γ : Path (p' 0) (p' n)", "hγ : (∀ i ≤ n, p' i ∈ range ⇑γ) ∧ range ⇑γ ⊆ s", "hpp' : ∀ k < n + 1, p ↑k = p' k", "i : ℕ", "hi : i < n + 1"], "goal": "i = i % n.succ"}}
+{"state": {"context": ["V : Type u", "G₁ G₂ : SimpleGraph V"], "goal": "G₁.edgeSet = G₂.edgeSet ↔ G₁ = G₂"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "SemilatticeInf α", "OrderTop α", "s : Finset β", "f g : β → α", "p : β → Prop", "DecidablePred p"], "goal": "(s.inf fun i => if p i then f i else g i) = (Finset.filter p s).inf f ⊓ (Finset.filter (fun i => ¬p i) s).inf g"}}
+{"state": {"context": ["Ω : Type u_1", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "ℱ : Filtration ℕ m0", "f : ℕ → Ω → ℝ", "ω✝ : Ω", "r : ℝ", "R : ℝ≥0", "inst✝ : IsFiniteMeasure μ", "hf : Submartingale f ℱ μ", "hf0 : f 0 = 0", "hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R", "ht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n => stoppedValue f (leastGE f (↑i) n) ω) atTop (𝓝 c)", "ω : Ω", "hω : ∀ (i : ℕ), ∃ c, Tendsto (fun n => stoppedValue f (leastGE f (↑i) n) ω) atTop (𝓝 c)", "hωb : BddAbove (Set.range fun n => f n ω)"], "goal": "∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c)"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "M : Type u_3", "A : Type u_4", "inst✝² : DecidableEq ι", "inst✝¹ : AddMonoid ι", "inst✝ : Semiring M", "f g : M[ι]", "to_hom : M[ι] →+ ⨁ (x : ι), M := { toFun := toDirectSum, map_zero' := ⋯, map_add' := ⋯ }"], "goal": "(f * g).toDirectSum = f.toDirectSum * g.toDirectSum"}}
+{"state": {"context": ["s : Set ℕ∞"], "goal": "sSup s = 0 ↔ ∀ a ∈ s, a = 0"}}
+{"state": {"context": ["𝕜 : Type u𝕜", "G : Type uG", "E : Type uE", "E' : Type uE'", "E'' : Type uE''", "F : Type uF", "F' : Type uF'", "F'' : Type uF''", "P : Type uP", "inst✝¹⁸ : NormedAddCommGroup E", "inst✝¹⁷ : NormedAddCommGroup E'", "inst✝¹⁶ : NormedAddCommGroup E''", "inst✝¹⁵ : NormedAddCommGroup F", "f f' : G → E", "g✝ g' : G → E'", "x x' : G", "y y' : E", "inst✝¹⁴ : NontriviallyNormedField 𝕜", "inst✝¹³ : NormedSpace 𝕜 E", "inst✝¹² : NormedSpace 𝕜 E'", "inst✝¹¹ : NormedSpace 𝕜 E''", "inst✝¹⁰ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝⁹ : MeasurableSpace G", "μ ν : Measure G", "inst✝⁸ : NormedSpace ℝ F", "inst✝⁷ : SeminormedAddCommGroup G", "inst✝⁶ : BorelSpace G", "inst✝⁵ : SecondCountableTopology G", "inst✝⁴ : μ.IsAddLeftInvariant", "inst✝³ : SFinite μ", "inst✝² : NormedSpace ℝ E", "inst✝¹ : NormedSpace ℝ E'", "inst✝ : CompleteSpace E'", "ι : Type u_1", "g : ι → G → E'", "l : Filter ι", "x₀ : G", "z₀ : E'", "φ : ι → G → ℝ", "k : ι → G", "hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x", "hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1", "hmg : ∀ᶠ (i : ι) in l, AEStronglyMeasurable (g i) μ", "hcg : Tendsto (uncurry g) (l ×ˢ 𝓝 x₀) (𝓝 z₀)", "hk : Tendsto k l (𝓝 x₀)", "hφ : ∀ t ∈ 𝓝 0, ∀ᶠ (x : ι) in l, support (φ x) ⊆ t"], "goal": "Tendsto (fun i => (φ i ⋆[lsmul ℝ ℝ, μ] g i) (k i)) l (𝓝 z₀)"}}
+{"state": {"context": ["R : Type u_4", "k : ℕ", "CommRing R", "ζ : Rˣ", "h : IsPrimitiveRoot ζ k", "i : ℕ"], "goal": "h.zmodEquivZPowers.symm (Additive.ofMul ⟨ζ ^ i, ⋯⟩) = ↑i"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "idx : Type u_2", "CommRing R", "hp : Fact (Nat.Prime p)", "Φ : MvPolynomial idx ℤ", "n : ℕ"], "goal": "(MvPolynomial.aeval fun i => (MvPolynomial.map (Int.castRingHom R)) (wittStructureInt p Φ i)) (W_ ℤ n) =\n  (MvPolynomial.aeval fun i => (MvPolynomial.rename (Prod.mk i)) (W_ R n)) Φ"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "AddCommMonoid α", "AddCommMonoid β", "s : Multiset α", "F : Type u_6", "FunLike F α β", "AddMonoidHomClass F α β", "f : F"], "goal": "(Multiset.map (⇑f) s).sum = f s.sum"}}
+{"state": {"context": ["ι : Type uι", "Fintype ι", "𝕜 : Type u𝕜", "NontriviallyNormedField 𝕜", "E : ι → Type uE", "(i : ι) → SeminormedAddCommGroup (E i)", "(i : ι) → NormedSpace 𝕜 (E i)", "E' : ι → Type u_1", "(i : ι) → SeminormedAddCommGroup (E' i)", "(i : ι) → NormedSpace 𝕜 (E' i)", "f : (i : ι) → E i →L[𝕜] E' i"], "goal": "∀ (a : ⨂[𝕜] (i : ι), E i),\n  ((PiTensorProduct.mapLMultilinear 𝕜 E E') f) a =\n    (PiTensorProduct.liftAux ((PiTensorProduct.tprodL 𝕜).compContinuousLinearMap f).toMultilinearMap) a"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "G : Type u_2", "inst✝ : Group G", "n : ℕ"], "goal": "vectorsProdEqOne G 1 = {1 ::ᵥ Vector.nil}"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "F F' : CategoryTheory.Functor C D", "h : F ≅ F'", "Y : D", "hY : Y ∈ F.essImage"], "goal": "Y ∈ F'.essImage"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "p : Set γ", "f f₁ f₂ f₃ : α → β", "g g₁ g₂ : β → γ", "f' f₁' f₂' : β → α", "g' : γ → β", "a : α", "b : β"], "goal": "f '' s = t → SurjOn f s t ∧ MapsTo f s t"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "A : Type u_3", "AddCommMonoid ι", "DecidableEq ι", "CommRing R", "CommRing A", "Algebra R A", "𝒜 : ι → Submodule R A", "GradedAlgebra 𝒜", "x : Submonoid A", "y : HomogeneousLocalization 𝒜 x", "n : ℕ"], "goal": "(y ^ n).val = y.val ^ n"}}
+{"state": {"context": ["n : Nat"], "goal": "¬n ∈ List.range n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "s✝ t✝ : Finset ℕ", "n : ℕ", "hn : 2 ≤ n", "s t : Finset ℕ", "a : ℕ", "hat : a ∈ t", "has : a ∉ s", "ha : ∀ b ∈ s, b ∉ t → b < a"], "goal": "(fun s => ∑ k ∈ s.ofColex, n ^ k) { ofColex := s } < (fun s => ∑ k ∈ s.ofColex, n ^ k) { ofColex := t }"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "F : Type u_2", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "v : E", "f : E → F", "f' : F", "x₀ : E", "hf : HasLineDerivAt 𝕜 f f' x₀ v", "C : ℝ", "hC₀ : 0 ≤ C", "hlip : ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖"], "goal": "‖f'‖ ≤ C * ‖v‖"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S₁ S₂ : CategoryTheory.ShortComplex C", "e : S₁ ≅ S₂", "S₁.HasLeftHomology", "S₂.HasLeftHomology"], "goal": "(CategoryTheory.ShortComplex.cyclesMapIso e).hom = CategoryTheory.ShortComplex.cyclesMap e.hom"}}
+{"state": {"context": ["R : Type u", "CommRing R", "x y : R"], "goal": "IsCoprime x (-y) ↔ IsCoprime x y"}}
+{"state": {"context": ["V : Type u_1", "α : Type u_2", "DecidableEq α", "DecidableEq V", "A : Matrix V V α", "Zero α", "One α", "i : V"], "goal": "A.compl i i = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "f g : α → β", "s : Set α", "h' : Set.EqOn g f s"], "goal": "ContinuousOn g s ↔ ContinuousOn f s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder β", "f : α → β", "l : List α", "a m : α"], "goal": "a ∈ l → m ∈ List.argmin f l → f m ≤ f a"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "Field K", "CharZero K", "LieRing L", "LieAlgebra K L", "LieAlgebra.IsKilling K L", "FiniteDimensional K L", "H : LieSubalgebra K L", "H.IsCartanSubalgebra", "LieModule.IsTriangularizable K (↥H) L", "α β : LieModule.Weight K (↥H) L", "hα : α.IsNonZero", "β' : LieModule.Weight K (↥H) L", "n : ℤ", "hβ' : ⇑β' = n • ⇑α + ⇑β"], "goal": "↑(LieModule.chainTopCoeff (⇑α) β') = ↑(LieModule.chainTopCoeff (⇑α) β) - n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Type u_1", "π : ι → Type u_2", "inst✝³ : TopologicalSpace α", "s t u v : Set α", "inst✝² : TopologicalSpace β", "inst✝¹ : TotallyDisconnectedSpace β", "f✝ : α → β", "T : Set β", "inst✝ : DiscreteTopology ↑T", "f : α → β", "hne : T.Nonempty", "hS : IsPreconnected ∅", "hc : ContinuousOn f ∅", "hTm : MapsTo f ∅ T"], "goal": "∃ y ∈ T, EqOn f (const α y) ∅"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_3", "β : Sort u_2", "S : ι → Set α", "f : (i : ι) → ↑(S i) → β", "hf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩", "hS : Set.iUnion S = Set.univ", "i : ι", "x : ↑(S i)"], "goal": "Set.liftCover S f hf hS ↑x = f i x"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "e : E", "x : 𝕜"], "goal": "MeromorphicAt (fun x => e) x"}}
+{"state": {"context": ["L : FirstOrder.Language", "L' : FirstOrder.Language", "self : L ≃ᴸ L'"], "goal": "self.toLHom.comp self.invLHom = FirstOrder.Language.LHom.id L'"}}
+{"state": {"context": ["R : Type u", "inst✝⁸ : CommRing R", "I : Ideal R", "M : Type u", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : Module R M", "N : Type u", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R N", "P : Type u", "inst✝³ : AddCommGroup P", "inst✝² : Module R P", "inst✝¹ : IsNoetherianRing R", "inst✝ : Module.Finite R N", "f : M →ₗ[R] N", "hf : Function.Injective ⇑f", "k : ℕ", "hk : ∀ n ≥ k, I ^ n • ⊤ ⊓ range f = I ^ (n - k) • (I ^ k • ⊤ ⊓ range f)", "x : AdicCompletion I M", "a : AdicCauchySequence I M", "hx : (map I f) ((mk I M) a) = 0"], "goal": "(mk I M) a = 0"}}
+{"state": {"context": ["α β : BddLat", "e : ↑α.toLat ≃o ↑β.toLat", "a : ↑β.toLat"], "goal": "(BddLat.Iso.mk e).inv.toSupHom a = e.symm a"}}
+{"state": {"context": ["n : ℕ", "n1 : 1 ≤ n"], "goal": "(Nat.unpair n).1 < n"}}
+{"state": {"context": ["α : Type u", "β : Type v", "δ : Type x", "TopologicalSpace α", "Preorder β", "f : α → β", "TopologicalSpace δ", "s : Set δ", "g : δ → α", "b : δ", "hf : IsLocalMin f (g b)", "hg : ContinuousOn g s", "hb : b ∈ s"], "goal": "IsLocalMinOn (f ∘ g) s b"}}
+{"state": {"context": [], "goal": "∀ {X Y : (CategoryTheory.Limits.WidePullbackShape CategoryTheory.Limits.WalkingPair)ᵒᵖ} (f : X ⟶ Y),\n  CategoryTheory.Limits.walkingCospanOpEquiv.functor.map f =\n    (CategoryTheory.Limits.widePullbackShapeOpMap CategoryTheory.Limits.WalkingPair (Opposite.unop Y) (Opposite.unop X)\n        f.unop).unop"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : LowerSet α", "a : α"], "goal": "(s.erase a).erase a = s.erase a"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "inst✝³ : NormedAddCommGroup F'", "inst✝² : NormedSpace ℝ F'", "inst✝¹ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : CompleteSpace F", "T T' T'' : Set α → E →L[ℝ] F", "C C' C'' : ℝ", "f✝ g : α → E", "hT : DominatedFinMeasAdditive (⊤ • μ) T C", "f : α → E"], "goal": "setToFun (⊤ • μ) T hT f = 0"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "E : Type u_6", "inst✝¹⁰ : MeasurableSpace α", "inst✝⁹ : MeasurableSpace α'", "inst✝⁸ : MeasurableSpace β", "inst✝⁷ : MeasurableSpace β'", "inst✝⁶ : MeasurableSpace γ", "μ μ' : Measure α", "ν ν' : Measure β", "τ : Measure γ", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : SFinite ν", "inst✝³ : NormedSpace ℝ E", "inst✝² : SFinite μ", "E' : Type u_7", "inst✝¹ : NormedAddCommGroup E'", "inst✝ : NormedSpace ℝ E'", "hE : CompleteSpace E"], "goal": "IsClosed {f | ∫ (z : α × β), ↑↑f z ∂μ.prod ν = ∫ (x : α), ∫ (y : β), ↑↑f (x, y) ∂ν ∂μ}"}}
+{"state": {"context": ["n : ℕ", "a b : Fin n"], "goal": "a ≤ b → a < b ∨ a = b"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁴ : MeasurableSpace X", "inst✝³ : NormedAddCommGroup E", "𝕜 : Type u_5", "inst✝² : NormedField 𝕜", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : ℝ≥0∞", "μ : Measure X", "c : 𝕜", "f : ↥(Lp F p μ)", "s : Set X"], "goal": "↑↑(Memℒp.toLp ↑↑(c • f) ⋯) =ᶠ[ae (μ.restrict s)] ↑↑(c • Memℒp.toLp ↑↑f ⋯)"}}
+{"state": {"context": ["R : Type u_1", "NonUnitalNonAssocRing R", "I : TwoSidedIdeal R", "x y : R", "hx : x ∈ I"], "goal": "x * y ∈ I"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "g : β → α", "hf : Injective f", "hg : Injective g", "hβ : Nonempty β", "F : Set α →o Set α := { toFun := fun s => (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }", "s : Set α := OrderHom.lfp F", "hs : (g '' (f '' s)ᶜ)ᶜ = s", "hns : g '' (f '' s)ᶜ = sᶜ", "g' : α → β := invFun g", "g'g : LeftInverse g' g", "hg'ns : g' '' sᶜ = (f '' s)ᶜ"], "goal": "∃ h, Bijective h"}}
+{"state": {"context": ["K : Type u_3", "DivisionRing K", "a b : K", "hab : Commute a b", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "a⁻¹ - b⁻¹ = (b - a) / (a * b)"}}
+{"state": {"context": ["n : Nat", "i : Fin n"], "goal": "i.succ.castSucc.pred ⋯ = i.castSucc"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "NonUnitalNonAssocSemiring A", "Module R A"], "goal": "Function.Injective NonUnitalSubalgebra.toSubmodule"}}
+{"state": {"context": ["a b : ℤ"], "goal": "Fintype.card ↑(Set.Ioc a b) = (b - a).toNat"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "W X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "f' : W ⟶ X", "CategoryTheory.Limits.HasPullback f g", "CategoryTheory.Limits.HasPullback f' (CategoryTheory.Limits.pullback.fst f g)", "CategoryTheory.Limits.HasPullback (f' ≫ f) g"], "goal": "(CategoryTheory.Limits.pullbackRightPullbackFstIso f g f').hom ≫ CategoryTheory.Limits.pullback.snd (f' ≫ f) g =\n  CategoryTheory.Limits.pullback.snd f' (CategoryTheory.Limits.pullback.fst f g) ≫\n    CategoryTheory.Limits.pullback.snd f g"}}
+{"state": {"context": ["α : Type u", "Preorder α", "k j : α", "hkj : k ≤ j"], "goal": "Monotone fun x =>\n  match x with\n  | ⟨i, hi⟩ => ⟨i, ⋯⟩"}}
+{"state": {"context": ["a : ℝ≥0∞", "h : a ≠ ⊤"], "goal": "ENNReal.ofReal a.toReal = a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace β", "f g : α → β", "AddCommGroup β", "TopologicalAddGroup β", "hf : MeasureTheory.AEStronglyMeasurable f μ"], "goal": "MeasureTheory.AEStronglyMeasurable (f + g) μ ↔ MeasureTheory.AEStronglyMeasurable g μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "DecidableEq β", "s : Finset α", "H : Function.Injective f"], "goal": "(Finset.image f s).card = s.card"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r : ℚ", "x✝ : ℤ_[p]", "n : ℕ", "x : ℤ_[p]"], "goal": "x ∈ RingHom.ker (toZModPow n) ↔ x ∈ Ideal.span {↑p ^ n}"}}
+{"state": {"context": ["y : ℝ≥0∞"], "goal": "(⊤ * y).log = ⊤.log + y.log"}}
+{"state": {"context": ["ι : Type u_1", "V : Type u", "CategoryTheory.Category.{v, u} V", "CategoryTheory.Preadditive V", "c : ComplexShape ι", "C D : HomologicalComplex V c", "W : Type u_2", "CategoryTheory.Category.{u_3, u_2} W", "CategoryTheory.Preadditive W", "G : V ⥤ W", "G.Additive", "hom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)"], "goal": "(G.mapHomologicalComplex c).map (Homotopy.nullHomotopicMap' hom) =\n  Homotopy.nullHomotopicMap' fun i j hij => G.map (hom i j hij)"}}
+{"state": {"context": ["C : Type v₁", "inst✝² : Category.{v₁, v₁} C", "D : Type u₂", "inst✝¹ : Category.{v₁, u₂} D", "F : C ⥤ D", "inst✝ : IsFilteredOrEmpty C", "hF : F.Final"], "goal": "∀ {d : D} {c : C} (s s' : d ⟶ F.obj c), ∃ c' t, s ≫ F.map t = s' ≫ F.map t"}}
+{"state": {"context": ["J : Type u₁", "inst✝⁶ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝⁵ : Category.{v₂, u₂} K", "C : Type u", "inst✝⁴ : Category.{v, u} C", "F✝ : J ⥤ C", "D : Type u'", "inst✝³ : Category.{v', u'} D", "F : J ⥤ C", "inst✝² : HasLimit F", "G : C ⥤ D", "inst✝¹ : HasLimit (F ⋙ G)", "E : Type u''", "inst✝ : Category.{v'', u''} E", "H : D ⥤ E", "h : HasLimit ((F ⋙ G) ⋙ H)", "this : HasLimit (F ⋙ G ⋙ H)", "j✝ : J"], "goal": "(H.map (post F G) ≫ post (F ⋙ G) H) ≫ π ((F ⋙ G) ⋙ H) j✝ = post F (G ⋙ H) ≫ π ((F ⋙ G) ⋙ H) j���"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommSemiring R", "inst✝¹ : AddCommMonoid M", "inst✝ : Module R M", "s : Set R", "a : R", "x : M", "hx : ∀ (a : ↑s), ↑a • x = 0"], "goal": "s ⊆ ↑(Ideal.torsionOf R M x)"}}
+{"state": {"context": ["S : CategoryTheory.SheafOfTypes CategoryTheory.typesGrothendieckTopology"], "goal": "(CategoryTheory.equivYoneda' S).inv.val = (CategoryTheory.equivYoneda S.val ⋯).inv"}}
+{"state": {"context": ["n : ℕ+", "K : Type u_1", "inst✝⁴ : Field K", "L : Type u_2", "μ : L", "inst✝³ : CommRing L", "inst✝² : IsDomain L", "hμ✝ : IsPrimitiveRoot μ ↑n", "inst✝¹ : Algebra K L", "inst✝ : IsCyclotomicExtension {n} K L", "h : Irreducible (cyclotomic (↑n) K)", "hζ : IsPrimitiveRoot (zeta n K L) ↑n := zeta_spec n K L", "hμ : ∀ (t : (ZMod ↑n)ˣ), IsPrimitiveRoot (zeta n K L ^ (↑t).val) ↑n := fun t => IsPrimitiveRoot.pow_of_coprime hζ (↑t).val (ZMod.val_coe_unit_coprime t)", "x : (ZMod ↑n)ˣ"], "goal": "(IsPrimitiveRoot.autToPow K ⋯) ((IsPrimitiveRoot.powerBasis K hζ).equivOfMinpoly (IsPrimitiveRoot.powerBasis K ⋯) ⋯) = x"}}
+{"state": {"context": ["x : ℂ", "n : ℕ", "n.AtLeastTwo", "hlt : -π < OfNat.ofNat n * x.arg", "hle : OfNat.ofNat n * x.arg ≤ π", "y : ℂ"], "goal": "x ^ (OfNat.ofNat n * y) = (x ^ OfNat.ofNat n) ^ y"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝³ : Fintype α", "inst✝² : Fintype β", "A : Matrix α β ℤ", "v✝ : β → ℤ", "hn : m < n", "hm : 0 < m", "inst✝¹ : DecidableEq α", "inst✝ : DecidableEq β", "v : β → ℤ", "hv : 0 ≤ v ∧ v ≤ B'", "mulVec_def : A *ᵥ v = fun i => ∑ j : β, A i j * v j", "i : α", "j : β", "a✝ : j ∈ univ", "hsign : 0 ≤ A i j"], "goal": "-(↑B * (A i j)⁻) ≤ v j * A i j"}}
+{"state": {"context": ["α : Sort u", "r : α → α → Prop", "q : Quot r"], "goal": "Exists fun a => Quot.mk r a = q"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : MulZeroClass α", "s✝ t s : Set α"], "goal": "0 * s ⊆ 0"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n : ℕ"], "goal": "Ioo k n ⊆ Ioc k (max (k + 1) n)"}}
+{"state": {"context": ["M : Type u_1", "Zero M", "l : List M", "DecidablePred fun x => l.getD x 0 ≠ 0"], "goal": "⇑l.toFinsupp = fun x => l.getD x 0"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a : α", "h₀ : 0 < a"], "goal": "a⁻¹ < 1 ↔ 1 < a"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "N : ℕ", "τ : ℝ", "a : Besicovitch.SatelliteConfig E N τ", "lastc : a.c (Fin.last N) = 0", "lastr : a.r (Fin.last N) = 1", "hτ : 1 ≤ τ", "δ : ℝ", "hδ1 : τ ≤ 1 + δ / 4", "hδ2 : δ ≤ 1"], "goal": "∃ c', (∀ (n : Fin N.succ), ‖c' n‖ ≤ 2) ∧ Pairwise fun i j => 1 - δ ≤ ‖c' i - c' j‖"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℝ E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : NormedSpace ℝ F", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "inst✝ : NormedAddCommGroup G", "m✝ : MeasurableSpace α", "μ : Measure α", "m : MeasurableSpace α", "T : Set α → F →L[ℝ] F'", "f : α →ₛ F", "x : F", "x✝ : x ∈ f.range", "hx0 : x = 0"], "goal": "(T (↑f ⁻¹' {0})) 0 = 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort x", "f : α → β", "x : ι → Filter α", "y : ι → Filter β", "h : ∀ (i : ι), Filter.Tendsto f (x i) (y i)"], "goal": "Filter.Tendsto f (iSup x) (iSup y)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "a b c : Ordinal.{u_4}", "h : c ≠ 0"], "goal": "a < b / c ↔ c * succ a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "E : α → Type u_2", "p q : ℝ≥0∞", "inst✝⁴ : (i : α) → NormedAddCommGroup (E i)", "𝕜 : Type u_3", "inst✝³ : NormedRing 𝕜", "inst✝² : (i : α) → Module 𝕜 (E i)", "inst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)", "inst✝ : DecidableEq α", "hp : 0 < p.toReal", "f : ↥(lp E p)", "s : Finset α", "F : α → ℝ := fun i => ‖↑f i‖ ^ p.toReal - ‖↑(f - ∑ i ∈ s, lp.single p i (↑f i)) i‖ ^ p.toReal"], "goal": "HasSum (fun b => ‖↑f b‖ ^ p.toReal - ‖↑(f - ∑ i ∈ s, lp.single p i (↑f i)) b‖ ^ p.toReal) (∑ i ∈ s, ‖↑f i‖ ^ p.toReal)"}}
+{"state": {"context": ["R : Type u", "self : EuclideanDomain R", "a b : R"], "goal": "b * EuclideanDomain.quotient a b + EuclideanDomain.remainder a b = a"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α"], "goal": "1 < s.card ↔ ∃ a ∈ s, ∃ b ∈ s, a ≠ b"}}
+{"state": {"context": ["α : Type u", "TopologicalSpace α"], "goal": "Continuous ConnectedComponents.mk"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "AddZeroClass M"], "goal": "∀ (a : α →₀ M) (a_1 : α), Finsupp.coeFnAddHom a a_1 = a a_1"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "τ : Type u_3", "S : Type u_4", "inst✝⁵ : CommSemiring R", "inst✝⁴ : CommSemiring S", "inst✝³ : Fintype σ", "inst✝² : Fintype τ", "n : ℕ", "inst✝¹ : DecidableEq σ", "inst✝ : Nontrivial R"], "goal": "((powersetCard n univ).biUnion fun t => {∑ i ∈ t, Finsupp.single i 1}) = image (fun t => ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)"}}
+{"state": {"context": ["X✝ Y : LightCondSet", "f : X✝ ⟶ Y", "X : LightCondSet", "x✝² x✝¹ : LightProfiniteᵒᵖ", "x✝ : x✝² ⟶ x✝¹", "a✝ : X.val.obj x✝²"], "goal": "(X.val.map x✝ ≫ (fun S x => { toFun := fun s => X.val.map (LightProfinite.const (Opposite.unop S) s).op x, continuous_toFun := ⋯ }) x✝¹) a✝ = ((fun S x => { toFun := fun s => X.val.map (LightProfinite.const (Opposite.unop S) s).op x, continuous_toFun := ⋯ }) x✝² ≫ X.toTopCat.toLightCondSet.val.map x✝) a✝"}}
+{"state": {"context": ["D : CategoryTheory.GlueData (Type v)", "x : D.glued"], "goal": "∃ i y, D.ι i y = x"}}
+{"state": {"context": ["K : Type u_1", "v : K", "n : ℕ", "LinearOrderedField K", "FloorRing K", "ifp_succ_n : GenContFract.IntFractPair K", "succ_nth_stream_eq : GenContFract.IntFractPair.stream v (n + 1) = some ifp_succ_n"], "goal": "1 ≤ ifp_succ_n.b"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁵ : Category.{v₁, u₁} C", "inst✝⁴ : MonoidalCategory C", "inst✝³ : BraidedCategory C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "inst✝¹ : MonoidalCategory D", "inst✝ : BraidedCategory D", "F : LaxBraidedFunctor C D", "A : CommMon_ C", "this : ∀ (X Y : C), F.μ X Y ≫ F.map (β_ X Y).hom = (β_ (F.obj X) (F.obj Y)).hom ≫ F.μ Y X"], "goal": "(F.μ A.X A.X ≫ F.map (β_ A.X A.X).hom) ≫ F.map A.mul = F.μ A.X A.X ≫ F.map A.mul"}}
+{"state": {"context": ["x y : ℝ*"], "goal": "¬x.InfinitePos → y.InfiniteNeg → (x + y).InfiniteNeg"}}
+{"state": {"context": ["R : Type u_1", "α : Type u_2", "M : Type u_3", "DivisionSemiring R", "AddCommMonoid M", "Module R M", "CharZero R", "f : α → α", "x : α", "h : Function.IsFixedPt f x", "g : α → M", "n : ℕ", "hn : n ≠ 0"], "goal": "birkhoffAverage R f g n x = g x"}}
+{"state": {"context": ["S : Type u", "inst✝ : State S", "s t✝ : S", "m : t✝ ∈ r s", "h : turnBound s = 0", "t : turnBound t✝ < turnBound s"], "goal": "False"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f g : E → F", "hg : Differentiable 𝕜 g"], "goal": "(Differentiable 𝕜 fun y => f y + g y) ↔ Differentiable 𝕜 f"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "Fintype α", "f : Equiv.Perm α", "x : α", "h : f (f x) ≠ x"], "goal": "(Equiv.swap x (f x) * f).support = f.support \\ {x}"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_3", "G : Type u_5", "F' : Type u_7", "Norm E", "Norm G", "SeminormedAddCommGroup F'", "l : Filter α", "f : α → E", "g : α → F'", "k : α → G", "h₁ : f =O[l] g", "h₂ : g =Θ[l] k"], "goal": "f =O[l] k"}}
+{"state": {"context": ["R : Type u", "S : Type v", "inst✝² : CommRing R", "inst✝¹ : Ring S", "f : R[X]", "inst✝ : Algebra R S", "h : IsAdjoinRootMonic S f", "z : S", "i : ℕ", "hi : f.natDegree ≤ i"], "goal": "{ toFun := Polynomial.coeff, map_add' := ⋯ } (h.modByMonicHom z) i = 0"}}
+{"state": {"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "self : TotalComplexShape c₁ c₂ c₁₂", "i₁ i₁' : I₁", "h : c₁.Rel i₁ i₁'", "i₂ : I₂"], "goal": "c₁₂.Rel (TotalComplexShape.π c₁ c₂ c₁₂ (i₁, i₂)) (TotalComplexShape.π c₁ c₂ c₁₂ (i₁', i₂))"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α✝ : Type u_3", "β : Type u_4", "γ : Type u_5", "f : α✝ → β → β", "op : α✝ → α✝ → α✝", "inst✝³ : Monoid α✝", "inst✝² : Monoid β", "inst✝¹ : FunLike F α✝ β", "α : Type u_6", "inst✝ : CommMonoid α", "s : Multiset α"], "goal": "s.noncommProd ⋯ = s.prod"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "Field K", "AddCommGroup V", "Module K V", "T : Projectivization.Subspace K V", "v w : V", "hv : v ≠ 0", "hw : w ≠ 0", "hvw : v + w ≠ 0"], "goal": "Projectivization.mk K v hv ∈ T → Projectivization.mk K w hw ∈ T → Projectivization.mk K (v + w) hvw ∈ T"}}
+{"state": {"context": ["X : Type u_1", "E : Type u_3", "MeasurableSpace X", "TopologicalSpace X", "NormedAddCommGroup E", "f : X → E", "μ : MeasureTheory.Measure X", "OpensMeasurableSpace X", "a b : X", "MeasureTheory.IsFiniteMeasureOnCompacts μ", "Preorder X", "CompactIccSpace X", "T2Space X", "hf : Continuous f"], "goal": "MeasureTheory.IntegrableOn f (Set.Icc a b) μ"}}
+{"state": {"context": ["P : ℕ∞ → Prop", "a : ℕ∞", "h0 : P 0", "hsuc : ∀ (n : ℕ), P ↑n → P ↑n.succ", "htop : (∀ (n : ℕ), P ↑n) → P ⊤"], "goal": "P a"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Category.{u_4, u_2} D", "CategoryTheory.Preadditive C", "W : CategoryTheory.MorphismProperty C", "X Y : C", "φ : W.LeftFraction₂ X Y", "F : CategoryTheory.Functor C D", "hF : W.IsInvertedBy F", "CategoryTheory.Preadditive D", "F.Additive"], "goal": "φ.add.map F hF = φ.fst.map F hF + φ.snd.map F hF"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : E", "𝔸 : Type u_6", "NormedRing 𝔸", "NormedAlgebra 𝕜 𝔸", "a b : E → 𝔸", "a' b' : E →L[𝕜] 𝔸", "ha : HasFDerivAt a a' x", "hb : HasFDerivAt b b' x"], "goal": "HasFDerivAt (fun y => a y * b y) (a x • b' + a'.smulRight (b x)) x"}}
+{"state": {"context": ["C : Type u", "inst✝ : Groupoid C", "S : Subgroupoid C", "c : C", "hc : c ∈ S.objs", "d : C", "hd : d ∈ S.objs", "hcd : S.hom.obj ⟨c, hc⟩ = S.hom.obj ⟨d, hd⟩"], "goal": "⟨c, hc⟩ = ⟨d, hd⟩"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "o : Ordinal.{u}"], "goal": "∃ ι f, lsub f = o ∧ #ι = sInf {a | ∃ ι f, lsub f = o ∧ #ι = a}"}}
+{"state": {"context": ["R : Type u_1", "R₂ : Type u_2", "E : Type u_5", "E₂ : Type u_6", "Semiring R", "Semiring R₂", "σ₁₂ : R →+* R₂", "σ₂₁ : R₂ →+* R", "RingHomInvPair σ₁₂ σ₂₁", "RingHomInvPair σ₂₁ σ₁₂", "SeminormedAddCommGroup E", "SeminormedAddCommGroup E₂", "Module R E", "Module R₂ E₂", "e : E ≃ₛₗᵢ[σ₁₂] E₂"], "goal": "⇑e.toIsometryEquiv = ⇑e"}}
+{"state": {"context": ["α : Type u_1", "G : SimpleGraph α", "c : G.Coloring Bool", "u v : α", "p : G.Walk u v"], "goal": "Even p.length ↔ (c u = true ↔ c v = true)"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "A : Type u_3", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Category.{u_5, u_2} D", "CategoryTheory.Category.{u_7, u_3} A", "J : CategoryTheory.GrothendieckTopology C", "K : CategoryTheory.GrothendieckTopology D", "H : C ⥤ D", "F G : Dᵒᵖ ⥤ A", "f : F ⟶ G", "CategoryTheory.ConcreteCategory A", "H.IsCocontinuous J K", "CategoryTheory.Presheaf.IsLocallyInjective K f"], "goal": "CategoryTheory.Presheaf.IsLocallyInjective J (CategoryTheory.whiskerLeft H.op f)"}}
+{"state": {"context": ["α : Type u_1", "UniformSpace α", "pkg : AbstractCompletion α", "p : pkg.space → Prop", "a : pkg.space", "hp : IsClosed {a | p a}", "ih : ∀ (a : α), p (pkg.coe a)"], "goal": "p a"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "K : AddSubgroup G"], "goal": "AddSubgroup.map (AddMonoidHom.id G) K = K"}}
+{"state": {"context": ["R : Type u", "S : Type u_1", "inst✝ : Ring R", "a✝ : Nontrivial R", "hR : IsField R[X]"], "goal": "False"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "f : ι → α", "p : α → Prop"], "goal": "(∃ a ∈ Set.range f, p a) ↔ ∃ i, p (f i)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "SMul α β", "Preorder α", "LinearOrder β", "Zero α"], "goal": "PosSMulStrictMono α β ↔ PosSMulReflectLE α β"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝ : Group G", "x y : G", "i : ℤ"], "goal": "↑(orderOf x) ∣ i ↔ x ^ i = 1"}}
+{"state": {"context": ["f g : ℕ → ℂ", "s : ℂ", "hf : LSeriesSummable f s", "hg : LSeriesSummable g s"], "goal": "L (f ⍟ g) s = L f s * L g s"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "I : LieIdeal R L", "k l : ℕ", "h : l ≤ k"], "goal": "LieAlgebra.derivedSeriesOfIdeal R L k I ≤ LieAlgebra.derivedSeriesOfIdeal R L l I"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_5", "N : Type u_7", "P : Type u_8", "Zero M", "Zero N", "Zero P", "f : ZeroHom N P", "f₂ : ZeroHom M N"], "goal": "Finsupp.mapRange.zeroHom (f.comp f₂) = (Finsupp.mapRange.zeroHom f).comp (Finsupp.mapRange.zeroHom f₂)"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H : AddSubgroup G"], "goal": "H.subtype.ker = ⊥"}}
+{"state": {"context": ["b : ℕ", "b_ge_two : 2 ≤ b", "p : ℕ", "p_prime : Prime p", "p_gt_two : 2 < p", "not_dvd : ¬p ∣ b * (b ^ 2 - 1)", "A : ℕ := (b ^ p - 1) / (b - 1)", "B : ℕ := (b ^ p + 1) / (b + 1)", "hi_A : 1 < A", "hi_B : 1 < B", "hi_AB : 1 < A * B", "hi_b : 0 < b", "hi_p : 1 ≤ p", "hi_bsquared : 0 < b ^ 2 - 1", "hi_bpowtwop : 1 ≤ b ^ (2 * p)", "hi_bpowpsubone : 1 ≤ b ^ (p - 1)", "p_odd : Odd p", "AB_not_prime : ¬Prime (A * B)", "AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1)", "hd : b ^ 2 - 1 ∣ b ^ (2 * p) - 1", "ha₁ : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b)", "ha₂ : 2 ∣ b ^ p + b", "ha₃ : p ∣ b ^ (p - 1) - 1", "ha₄ : b ^ 2 - 1 ∣ b ^ (p - 1) - 1", "ha₅ : 2 * p * (b ^ 2 - 1) ∣ (b ^ 2 - 1) * (A * B - 1)", "ha₇ : A * B ∣ b ^ (2 * p) - 1", "q : ℕ", "hq : A * B - 1 = 2 * p * q", "ha₈ : b ^ (2 * p) - 1 ∣ b ^ (A * B - 1) - 1"], "goal": "(A * B).ProbablePrime b"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι : Sort w", "κ : ι → Sort w'", "minAx : MinimalAxioms α", "f : (i : ι) → κ i → α", "x✝ : CompleteLattice α := minAx.toCompleteLattice", "g : ((i : ι) → κ i) → ι", "a : ι", "ha : ∀ (b : κ a), ∃ f, ∃ (h : a = g f), h ▸ b = f (g f)"], "goal": "⨅ i, f (g i) (i (g i)) ≤ ⨆ i, ⨅ j, f i j"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "LinearOrder α", "LinearOrder β", "e : α ↪o β", "x y : α"], "goal": "⇑e ⁻¹' Ι (e x) (e y) = Ι x y"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "P1 : Type u_3", "V2 : Type u_4", "P2 : Type u_5", "V3 : Type u_6", "P3 : Type u_7", "V4 : Type u_8", "P4 : Type u_9", "inst✝¹² : Ring k", "inst✝¹¹ : AddCommGroup V1", "inst✝¹⁰ : Module k V1", "inst✝⁹ : AffineSpace V1 P1", "inst✝⁸ : AddCommGroup V2", "inst✝⁷ : Module k V2", "inst✝⁶ : AffineSpace V2 P2", "inst✝⁵ : AddCommGroup V3", "inst✝⁴ : Module k V3", "inst✝³ : AffineSpace V3 P3", "inst✝² : AddCommGroup V4", "inst✝¹ : Module k V4", "inst✝ : AffineSpace V4 P4", "p₀ p₁ : P1", "c : k"], "goal": "(lineMap 1 0) (1 - c) = c"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : DecidableEq α", "δ : α → Sort u_2", "m : Multiset α", "a : α", "b : δ a", "f : (a' : α) → a' ∈ m → δ a'", "δ' : α → Sort u_3", "φ : ⦃a' : α⦄ → δ a' → δ' a'", "a' : α", "ha' : a' ∈ a ::ₘ m"], "goal": "(fun h => ⋯ ▸ φ b) = fun h => φ (⋯ ▸ b)"}}
+{"state": {"context": ["x✝ x : ℝ", "h1 : 0 < x", "h2 : x < π / 2", "U : Set ℝ := Ico 0 (π / 2)", "intU : interior U = Ioo 0 (π / 2)"], "goal": "x < tan x"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "p : FormalMultilinearSeries 𝕜 E F", "i : E ≃L[𝕜] F", "h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i", "h0 : p 0 = 0"], "goal": "(p.leftInv i).comp (p.comp (p.rightInv i)) = ((p.leftInv i).comp p).comp (p.rightInv i)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "f : α → β", "g : β → γ"], "goal": "FreeAbelianGroup.map (g ∘ f) = (FreeAbelianGroup.map g).comp (FreeAbelianGroup.map f)"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "𝕜 : Type u_2", "RCLike 𝕜", "E : Type u_3", "NormedAddCommGroup E", "NormedSpace ℝ E", "NormedSpace 𝕜 E", "H : Type u_4", "NormedAddCommGroup H", "NormedSpace 𝕜 H", "F : H → α → E", "x₀ : H", "bound : α → ℝ", "ε : ℝ", "F' : α → H →L[𝕜] E", "ε_pos : 0 < ε", "hF_meas : ∀ᶠ (x : H) in 𝓝 x₀, MeasureTheory.AEStronglyMeasurable (F x) μ", "hF_int : MeasureTheory.Integrable (F x₀) μ", "hF'_meas : MeasureTheory.AEStronglyMeasurable F' μ", "h_lip : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x => F x a) (Metric.ball x₀ ε)", "bound_integrable : MeasureTheory.Integrable bound μ", "h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀"], "goal": "MeasureTheory.Integrable F' μ ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀"}}
+{"state": {"context": ["R : Type u", "Semiring R", "x : R"], "goal": "Polynomial.eval x 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁴ : TopologicalSpace α", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : (i : ι) → TopologicalSpace (π i)", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "inst✝ : TopologicalSpace δ", "f : α → β", "s : Set α", "t : Set β", "hf : ContinuousOn f s", "hs : IsClosed s", "ht : IsClosed t"], "goal": "IsClosed (s ∩ f ⁻¹' t)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "𝕜 : Type u_4", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace ℝ F", "p : ℝ≥0∞", "G : Type u_5", "F' : Type u_6", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedAddCommGroup F'", "inst✝¹ : NormedSpace ℝ F'", "m : MeasurableSpace α", "μ : Measure α", "inst✝ : DecidablePred fun x => x ≠ 0", "f : α →ₛ F", "s : Finset F", "hs : filter (fun x => x ≠ 0) f.range ⊆ s", "x : F", "hx✝ : x ∉ f.range ∨ x = 0", "hx : x ∉ f.range"], "goal": "(μ (↑f ⁻¹' {x})).toReal • x = 0"}}
+{"state": {"context": ["α : Type u_1", "LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a b c : α"], "goal": "toIocDiv hp a (b - c) = toIocDiv hp (a + c) b"}}
+{"state": {"context": ["β : Type u_2", "ι : Sort u_4", "s : ι → Set β", "i : ι"], "goal": "s i ⊆ ⋃ i, s i"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "a : α"], "goal": "{a} ⊆ s ↔ a ∈ s"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_3", "β : Type u_4", "LinearOrder α", "LinearOrder β", "f : ι → α", "g : ι → β", "s : Set ι"], "goal": "MonovaryOn f g s ↔ MonovaryOn g f s"}}
+{"state": {"context": ["ι : Type u_1", "R₂ : Type u_4", "M : Type u_6", "CommRing R₂", "AddCommGroup M", "Module R₂ M", "e : Basis ι R₂ M", "w : ι → R₂ˣ", "v : M", "i : ι"], "goal": "((e.unitsSMul w).repr v) i = (w i)⁻¹ • (e.repr v) i"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "NontriviallyNormedField 𝕜", "NonUnitalNormedRing A", "NormedSpace 𝕜 A", "SMulCommClass 𝕜 A A", "IsScalarTower 𝕜 A A", "StarRing A", "CstarRing A", "a : 𝓜(𝕜, A)"], "goal": "‖a.toProd.2‖₊ = ‖a‖₊"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommSemiring R", "Semiring A", "Algebra R A", "x : A"], "goal": "x ∈ ⊤"}}
+{"state": {"context": ["R S : Type u", "inst✝⁹ : CommRing R", "inst✝⁸ : CommRing S", "M✝ : Submonoid R", "N : Submonoid S", "R' S' : Type u", "inst✝⁷ : CommRing R'", "inst✝⁶ : CommRing S'", "f : R →+* S", "inst✝⁵ : Algebra R R'", "inst✝⁴ : Algebra S S'", "inst✝³ : Algebra R S", "inst✝² : Algebra R S'", "inst✝¹ : IsScalarTower R S S'", "M : Submonoid S", "inst✝ : IsLocalization M S'", "x : S", "s : Finset S'", "A : Subalgebra R S", "hA₁ : ↑(finsetIntegerMultiple M s) ⊆ ↑A", "hA₂ : M ≤ A.toSubmonoid", "hx : (algebraMap S S') x ∈ Algebra.adjoin R ↑s", "g : S →ₐ[R] S' := IsScalarTower.toAlgHom R S S'", "y : ↥M := commonDenomOfFinset M s", "hx₁ : ↑y • ↑s = ⇑g '' ↑(finsetIntegerMultiple M s)"], "goal": "∃ m, m • x ∈ A"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "Zero α"], "goal": "NonemptyInterval.toProd 0 = 0"}}
+{"state": {"context": ["α : Type u_1", "R : Type u_3", "SMul R ℝ≥0∞", "IsScalarTower R ℝ≥0∞ ℝ≥0∞", "c : R", "m : MeasureTheory.OuterMeasure α", "s : Set α"], "goal": "(c • m) s = c • m s"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "LinearOrderedField R", "AddCommGroup V", "Module R V", "AddTorsor V P", "s : AffineSubspace R P", "x y z : P", "hxy : s.WOppSide x y", "hyz : s.WSameSide y z", "hy : y ∉ s"], "goal": "s.WOppSide x z"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : _root_.RCLike 𝕜", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : InnerProductSpace ℝ F", "K : Submodule 𝕜 E", "inst✝ : HasOrthogonalProjection K", "u v : E"], "goal": "(reflection (Submodule.span 𝕜 {u})) v = 2 • (⟪u, v⟫_𝕜 / ↑‖u‖ ^ 2) • u - v"}}
+{"state": {"context": ["K : Type u", "V : Type v", "inst✝⁴ : Ring K", "inst✝³ : StrongRankCondition K", "inst✝² : AddCommGroup V", "inst✝¹ : Module K V", "inst✝ : Module.Free K V", "this : Nontrivial K"], "goal": "Module.rank K V = 1 ↔ ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : V), ∃ r, r • v₀ = v"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a : ℝ", "f g : ℂ → E", "l : Filter ℂ", "u : ℂ → ℝ", "this : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖", "cf : ℝ", "hcf : cf < a", "Bf : ℝ", "hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))", "cg : ℝ", "hcg : cg < a", "Bg : ℝ", "hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))"], "goal": "cg ≤ max cf cg"}}
+{"state": {"context": ["J : Type u", "inst✝ : Category.{v, u} J", "F : J ⥤ AddCommGrp", "j j' : J", "f : j ⟶ j'", "x : ↑(F.obj j)"], "goal": "(coconeMorphism F j') ((F.map f) x) = (coconeMorphism F j) x"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "LinearOrderedSemifield α", "l : Filter β", "f : β → α", "r : α", "hr : 0 < r", "hf : Filter.Tendsto f l Filter.atTop"], "goal": "Filter.Tendsto (fun x => r * f x) l Filter.atTop"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : CommRing K", "μ : K", "inst✝¹ : IsDomain K", "inst✝ : CharZero K", "p : ℕ", "h : IsPrimitiveRoot μ 0", "hdiv : ¬p ∣ 0"], "goal": "minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "ι : Type u_4", "v : ι → E", "hv : Orthonormal 𝕜 v", "l₁ l₂ : ι →₀ 𝕜"], "goal": "⟪(Finsupp.total ι E 𝕜 v) l₁, (Finsupp.total ι E 𝕜 v) l₂⟫_𝕜 = l₂.sum fun i y => (starRingEnd 𝕜) (l₁ i) * y"}}
+{"state": {"context": ["G : Type u_1", "H : Type u_2", "A : Type u_3", "a✝ a₁ a₂ b c : G", "inst✝ : Group G", "s✝ s : Set G", "a : G", "h : a ∈ closure s", "x : G", "x✝¹ : InClosure s x", "x✝ : ∃ l, (∀ x ∈ l, x ∈ s ∨ x⁻¹ ∈ s) ∧ l.prod = x", "L : List G", "HL1 : ∀ x ∈ L, x ∈ s ∨ x⁻¹ ∈ s", "HL2 : L.prod = x", "hd : G", "tl : List G", "ih : (List.map Inv.inv tl.reverse).prod = tl.prod⁻¹"], "goal": "(List.map Inv.inv (hd :: tl).reverse).prod = (hd :: tl).prod⁻¹"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X X' : C", "f : X ⟶ X'", "Y Z : C"], "goal": "CategoryTheory.MonoidalCategory.whiskerRight (CategoryTheory.MonoidalCategory.whiskerRight f Y) Z =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Y Z).hom\n    (CategoryTheory.CategoryStruct.comp\n      (CategoryTheory.MonoidalCategory.whiskerRight f (CategoryTheory.MonoidalCategory.tensorObj Y Z))\n      (CategoryTheory.MonoidalCategory.associator X' Y Z).inv)"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹² : CommRing R", "M : Type u_2", "inst✝¹¹ : AddCommGroup M", "inst✝¹⁰ : Module R M", "N : Type u_3", "P : Type u_4", "inst✝⁹ : AddCommGroup N", "inst✝⁸ : Module R N", "inst✝⁷ : AddCommGroup P", "inst✝⁶ : Module R P", "ι : Type u_5", "inst✝⁵ : Module.Free R M", "inst✝⁴ : Module.Finite R M", "inst✝³ : Module.Free R N", "inst✝² : Module.Finite R N", "inst✝¹ : Module.Free R P", "inst✝ : Module.Finite R P", "f : M →ₗ[R] M", "g : N →ₗ[R] N", "h : (trace R (M × N) ∘ₗ prodMapLinear R M N M N R) (f, g) = (id.coprod id ∘ₗ (trace R M).prodMap (trace R N)) (f, g)"], "goal": "(trace R (M × N)) (f.prodMap g) = (trace R M) f + (trace R N) g"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedCoheytingAlgebra α", "a : α"], "goal": "a \\ a = ⊥"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_4, u_1} C", "inst✝³ : Category.{u_3, u_2} D", "inst✝² : Preadditive C", "L : C ⥤ D", "W : MorphismProperty C", "inst✝¹ : L.IsLocalization W", "inst✝ : W.HasLeftCalculusOfFractions", "X Y Z : C", "X' Y' Z' : D", "eX : L.obj X ≅ X'", "eY : L.obj Y ≅ Y'", "eZ : L.obj Z ≅ Z'", "X'' Y'' : C", "eX' : L.obj X'' ≅ X'", "eY' : L.obj Y'' ≅ Y'", "f₁ f₂ : X' ⟶ Y'", "h₁ : 𝟙 X' ≫ add W eX eY f₁ f₂ = add W eX' eY (𝟙 X' ≫ f₁) (𝟙 X' ≫ f₂)"], "goal": "add W eX eY f₁ f₂ = add W eX' eY' f₁ f₂"}}
+{"state": {"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : MonoidalCategory C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "inst✝ : MonoidalCategory D", "F : OplaxMonoidalFunctor C D", "X✝ Y✝ : Comon_ C", "f : X✝ ⟶ Y✝"], "goal": "F.map f.hom ≫ F.map Y✝.comul ≫ F.δ Y✝.X Y✝.X = (F.map X✝.comul ≫ F.δ X✝.X X✝.X) ≫ (F.map f.hom ⊗ F.map f.hom)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Abelian C"], "goal": "∀ {X Y : CategoryTheory.ShortComplex.SnakeInput C} (f : X ⟶ Y),\n  (CategoryTheory.ShortComplex.SnakeInput.functorL₂'.map f).τ₂ = f.f₃.τ₁"}}
+{"state": {"context": ["L : FirstOrder.Language", "M : Type u_1", "L.Structure M", "N : L.Substructure M"], "goal": "N.CG ↔ ∃ S, S.Countable ∧ (FirstOrder.Language.Substructure.closure L).toFun S = N"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace β", "g : α → β", "γ : Type u_4"], "goal": "∀ {x : MeasurableSpace γ} {x_1 : MeasurableSpace α} {f : γ → α} {μ : MeasureTheory.Measure γ}\n  {ν : MeasureTheory.Measure α},\n  MeasureTheory.AEStronglyMeasurable g ν →\n    MeasureTheory.MeasurePreserving f μ ν → MeasureTheory.AEStronglyMeasurable (g ∘ f) μ"}}
+{"state": {"context": ["E : Type u_1", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℝ E", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : E → F", "f' : E →L[ℝ] F", "x : E", "L : Filter E"], "goal": "Tendsto (fun x' => ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) ↔ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : CommGroup α", "inst✝ : DecidableEq α", "A✝ B✝ C✝ A B C : Finset α", "hB : B.Nonempty", "hB' : B ∈ B.powerset.erase ∅", "U : Finset α", "hU : U.Nonempty ∧ U ⊆ B", "hUA : ∀ x' ∈ B.powerset.erase ∅, ↑(U * A).card / ↑U.card ≤ ↑(x' * A).card / ↑x'.card"], "goal": "↑(A * C).card ≤ ↑(A * B).card * ↑(B * C).card / ↑B.card"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "μ₁ μ₂ : MeasureTheory.Measure α"], "goal": "μ₁ = μ₂ ↔ ∀ (s : Set α), μ₁ s = μ₂ s"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "K : AddSubgroup G"], "goal": "AddSubgroup.closure ↑K = K"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "S : StarSubalgebra R A"], "goal": "Set.range ⇑(algebraMap R A) ⊆ ↑S"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Limits.HasImages C", "CategoryTheory.Limits.HasBinaryCoproducts C", "CategoryTheory.Limits.HasInitial C", "CategoryTheory.Limits.InitialMonoClass C", "I : Type u_1", "A B : C", "s : Finset I", "P : I → CategoryTheory.Subobject B", "f : A ⟶ B", "h : ∃ i ∈ s, (P i).Factors f"], "goal": "(s.sup P).Factors f"}}
+{"state": {"context": ["a✝ b c d m n✝ k : ℕ", "p q : ℕ → Prop", "a n : ℕ", "hp : 1 < a.succ", "hk : 1 < n.succ", "h : a.succ ^ n.succ ∣ a.succ"], "goal": "a.succ * a.succ ^ n ∣ a.succ * 1"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : Semiring R", "inst✝ : NoZeroDivisors R", "p✝ q✝ p q : R[X]", "h0 : q ≠ 0", "hl : q.degree < p.degree", "hcontra : p ∣ q"], "goal": "False"}}
+{"state": {"context": ["k : Type u_1", "G : Type u_2", "V : Type u_3", "inst✝⁵ : CommSemiring k", "inst✝⁴ : Monoid G", "inst✝³ : AddCommMonoid V", "inst✝² : Module k V", "ρ : Representation k G V", "M : Type u_4", "inst✝¹ : AddCommMonoid M", "inst✝ : Module (MonoidAlgebra k G) M", "g : G", "x : RestrictScalars k (MonoidAlgebra k G) ρ.asModule"], "goal": "(RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule) ((RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule).symm (MonoidAlgebra.single g 1 • (RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule) x)) = (RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule) ((RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule).symm (ρ.asModuleEquiv.symm ((ρ g) (ρ.asModuleEquiv ((RestrictScalars.addEquiv k (MonoidAlgebra k G) ρ.asModule) x)))))"}}
+{"state": {"context": ["α : Type u", "u : Ultrafilter (Ultrafilter α)", "x : Ultrafilter α"], "goal": "↑u ≤ 𝓝 x ↔ x = joinM u"}}
+{"state": {"context": ["α : Type u", "b : ULift α"], "goal": "Eq { down := b.down } b"}}
+{"state": {"context": ["𝕜 : Type u", "A : Type v", "inst✝² : Field 𝕜", "inst✝¹ : Ring A", "inst✝ : Algebra 𝕜 A", "a : Aˣ", "k : 𝕜", "hk : k ∈ σ ↑a⁻¹"], "goal": "k ∈ (σ ↑a)⁻¹"}}
+{"state": {"context": ["α : Type u", "inst✝ : DecidableEq α", "β : Type v", "n : ℕ", "j : Fin n", "mem : swap j.castSucc j.succ ∈ Submonoid.closure (Set.range fun i => swap i.castSucc i.succ)", "ih : j.castSucc ≠ j.castSucc → j.castSucc < j.castSucc → swap j.castSucc j.castSucc ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))", "ne : j.castSucc ≠ j.succ", "lt : j.castSucc < j.succ"], "goal": "swap j.castSucc j.succ ∈ ↑(Submonoid.closure (Set.range fun i => swap i.castSucc i.succ))"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedCoheytingAlgebra α", "a b c : α", "h : a ⊔ c ≤ b ⊔ c"], "goal": "a \\ c ≤ b \\ c"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : TopologicalSpace α", "x : ℝ"], "goal": "∀ᶠ (a : EReal × EReal) in 𝓝 (⊤, ⊤), ↑x < a.1 * a.2"}}
+{"state": {"context": ["R : Type u", "inst✝ : CommSemiring R", "x y z : R", "h : IsUnit x", "b : R", "hb : b * x = 1"], "goal": "b * x + 0 * x = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y : E"], "goal": "(starRingEnd 𝕜) (-⟪y, x⟫_𝕜) = -⟪x, y⟫_𝕜"}}
+{"state": {"context": ["α β : Type u", "a b : Cardinal.{u_1}", "h : ¬a * b < ℵ₀ ↔ ¬(a = 0 ∨ b = 0 ∨ a < ℵ₀ ∧ b < ℵ₀) := Iff.not mul_lt_aleph0_iff"], "goal": "ℵ₀ ≤ a * b ↔ a ≠ 0 ∧ b ≠ 0 ∧ (ℵ₀ ≤ a ∨ ℵ₀ ≤ b)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "m0 : MeasurableSpace α", "NormedAddCommGroup E", "NormedSpace ℝ E", "CompleteSpace E", "μ : MeasureTheory.Measure α", "s : Set E", "f : α → E", "g : E → ℝ", "MeasureTheory.IsFiniteMeasure μ", "NeZero μ", "hg : ConvexOn ℝ s g", "hgc : ContinuousOn g s", "hsc : IsClosed s", "hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s", "hfi : MeasureTheory.Integrable f μ", "hgi : MeasureTheory.Integrable (g ∘ f) μ"], "goal": "(⨍ (x : α), f x ∂μ, ⨍ (x : α), g (f x) ∂μ) ∈ {p | p.1 ∈ s ∧ g p.1 ≤ p.2}"}}
+{"state": {"context": ["R : Type u", "Semiring R"], "goal": "{ toFinsupp := 1 } = 1"}}
+{"state": {"context": ["α : Type u_2", "m0 : MeasurableSpace α", "μ ν : MeasureTheory.Measure α", "s t : Set α", "hs : s ⊆ t", "h : μ.restrict t = ν.restrict t"], "goal": "μ.restrict s = ν.restrict s"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "CommRing R", "LieRing L", "LieAlgebra R L", "AddCommGroup M", "Module R M", "LieRingModule L M", "LieModule R L M", "N : LieSubmodule R L M", "I : LieIdeal R L"], "goal": "↑⁅I, N⁆ = Submodule.span R {m | ∃ x ∈ I, ∃ n ∈ N, ⁅x, n⁆ = m}"}}
+{"state": {"context": ["R : Type u_2", "A : Type u_3", "CommSemiring R", "StarRing R", "Semiring A", "StarRing A", "Algebra R A", "StarModule R A", "S₁ S₂ : StarSubalgebra R A", "h : S₁ ≤ S₂"], "goal": "∀ (a : ↥S₁), (StarSubalgebra.inclusion h) a = Subtype.map id h a"}}
+{"state": {"context": ["R : Type u", "Ring R"], "goal": "Polynomial.restriction 1 = 1"}}
+{"state": {"context": ["R : Type u", "K : Type v", "L : Type z", "p : R", "inst✝¹⁰ : CommRing R", "inst✝⁹ : Field K", "inst✝⁸ : Field L", "inst✝⁷ : Algebra K L", "inst✝⁶ : Algebra R L", "inst✝⁵ : Algebra R K", "inst✝⁴ : IsScalarTower R K L", "inst✝³ : Algebra.IsSeparable K L", "inst✝² : IsDomain R", "inst✝¹ : IsFractionRing R K", "inst✝ : IsIntegrallyClosed R", "B : PowerBasis K L", "hp : _root_.Prime p", "hBint : IsIntegral R B.gen", "z : L", "hzint : IsIntegral R z", "hz : p • z ∈ adjoin R {B.gen}", "hei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})"], "goal": "z ∈ adjoin R {B.gen}"}}
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+{"state": {"context": ["R : Type u", "inst✝¹⁴ : CommSemiring R", "S : Type ?u.153874", "inst✝¹³ : Semiring S", "inst✝¹² : Algebra R S", "ι₁ : Type v₁", "ι₂ : Type v₂", "inst✝¹¹ : DecidableEq ι₁", "inst✝¹⁰ : DecidableEq ι₂", "M₁ : ι₁ → Type w₁", "M₁' : Type w₁'", "M₂ : ι₂ → Type w₂", "M₂' : Type w₂'", "inst✝⁹ : (i₁ : ι₁) → AddCommMonoid (M₁ i₁)", "inst✝⁸ : AddCommMonoid M₁'", "inst✝⁷ : (i₂ : ι₂) → AddCommMonoid (M₂ i₂)", "inst✝⁶ : AddCommMonoid M₂'", "inst✝⁵ : (i₁ : ι₁) → Module R (M₁ i₁)", "inst✝⁴ : Module R M₁'", "inst✝³ : (i₂ : ι₂) → Module R (M₂ i₂)", "inst✝² : Module R M₂'", "inst✝¹ : (i₁ : ι₁) → Module S (M₁ i₁)", "inst✝ : ∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)", "x : M₁'", "i : ι₂", "y : M₂ i"], "goal": "(directSumRight R M₁' M₂).symm ((lof R ι₂ (fun i => M₁' ⊗[R] M₂ i) i) (x ⊗ₜ[R] y)) = x ⊗ₜ[R] (lof R ι₂ M₂ i) y"}}
+{"state": {"context": ["G : Type u_3", "Group G", "G' : Type u_5", "Group G'", "Group.FG G", "Group.FG G'", "f : G →* G'", "hf : Function.Surjective ⇑f"], "goal": "Group.rank G' ≤ Group.rank G"}}
+{"state": {"context": ["A : Type u_1", "B : Type u_2", "inst✝² : Field A", "inst✝¹ : Ring B", "inst✝ : Algebra A B", "x : B", "p : A[X]", "x✝ : minpoly A x ∣ p", "q : A[X]", "hq : p = minpoly A x * q"], "goal": "(Polynomial.aeval x) p = 0"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "CommSemiring R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "TopologicalSpace A", "Ring A", "StarRing A", "Algebra R A", "ContinuousFunctionalCalculus R p", "a : A", "h_spec : spectrum R a ⊆ {0}", "ha : p a := by cfc_tac"], "goal": "a = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝¹ : UniformSpace α", "inst✝ : UniformSpace β", "f : α → β", "x : β", "h_cont : Continuous f", "hx : Tendsto f (cocompact α) (𝓝 x)", "r : Set (β × β)", "hr : r ∈ 𝓤 β"], "goal": "{x | (f x.1, f x.2) ∈ r} ∈ 𝓤 α"}}
+{"state": {"context": ["ξ : ℝ", "u v : ℤ", "hv : 2 ≤ v", "hξ₀ : 0 < fract ξ", "u' : ℤ := u - ⌊ξ⌋ * v", "hu₀ : 0 < u'", "huv : u' < v", "hu' : u' = u - ⌊ξ⌋ * v", "hu'ℝ : ↑u = ↑u' + ↑⌊ξ⌋ * ↑v", "Hu : 0 < ↑u'", "Hu' : 0 < 2 * ↑u' - 1", "Hv : 0 < ↑v", "Hv' : 0 < 2 * ↑v - 1", "H₁ : 0 < ↑v / ↑u' / fract ξ", "h : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹", "h' : |fract ξ - ↑u' / ↑v| < (↑v * (2 * ↑v - 1))⁻¹", "H : 2 * ↑u' - 1 ≤ (2 * ↑v - 1) * fract ξ", "help₁ : ∀ {a b c : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → |a⁻¹ - b / c| = |(a - c / b) * (b / c / a)|", "help₂ : ∀ {a b c d : ℝ}, a ≠ 0 → b ≠ 0 → c ≠ 0 → d ≠ 0 → (b * c)⁻¹ * (b / d / a) = (d * c * a)⁻¹"], "goal": "|(fract ξ)⁻¹ - ↑v / ↑u'| < (↑u' * (2 * ↑u' - 1))⁻¹"}}
+{"state": {"context": ["ι : Type u_1", "ι₂ : Type u_2", "ι₃ : Type u_3", "R : Type u_4", "inst✝¹⁰ : CommSemiring R", "R₁ : Type u_5", "R₂ : Type u_6", "s : ι → Type u_7", "inst✝⁹ : (i : ι) → AddCommMonoid (s i)", "inst✝⁸ : (i : ι) → Module R (s i)", "M : Type u_8", "inst✝⁷ : AddCommMonoid M", "inst✝⁶ : Module R M", "E : Type u_9", "inst✝⁵ : AddCommMonoid E", "inst✝⁴ : Module R E", "F : Type u_10", "inst✝³ : AddCommMonoid F", "t : ι → Type u_11", "inst✝² : (i : ι) → AddCommMonoid (t i)", "inst✝¹ : (i : ι) → Module R (t i)", "inst✝ : IsEmpty ι", "x : ⨂[R] (i : ι), s i"], "goal": "(fun r => r • (tprod R) isEmptyElim) ({ toFun := ⇑(lift (constOfIsEmpty R s 1)), map_add' := ⋯, map_smul' := ⋯ }.toFun x) = x"}}
+{"state": {"context": ["G : Type u_1", "inst✝⁸ : TopologicalSpace G", "inst✝⁷ : LocallyCompactSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "�� : Measure G", "inst✝² : IsFiniteMeasureOnCompacts μ", "E : Type u_2", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "g : G → E", "hg : Continuous g", "h'g : HasCompactSupport g", "k : Set G := tsupport g", "k_comp : IsCompact k", "x₀ : G", "t : Set G", "t_comp : IsCompact t", "ht : t ∈ 𝓝 x₀"], "goal": "ContinuousAt (fun x => ∫ (y : G), g (y⁻¹ * x) ∂μ) x₀"}}
+{"state": {"context": ["α : Type u_1", "ι : Sort u_4", "f : ι → α", "s : Set α"], "goal": "Set.range f ⊆ s ↔ ∀ (y : ι), f y ∈ s"}}
+{"state": {"context": ["n : Type u_2", "DecidableEq n", "Fintype n", "R : Type v", "CommRing R"], "goal": "⇑Matrix.detMonoidHom = Matrix.det"}}
+{"state": {"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "inst✝³ : MonoidalCategory C", "A B : Mon_ C", "M✝ : Bimod A B", "inst✝² : HasCoequalizers C", "inst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)", "inst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)", "X Y Z : Mon_ C", "M : Bimod X Y", "N : Bimod Y Z"], "goal": "coequalizer.π (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ≫ colimMap (parallelPairHom (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ((M.X ⊗ Y.X) ◁ 𝟙 N.X) (M.X ◁ 𝟙 N.X) ⋯ ⋯) = coequalizer.π (M.actRight ▷ N.X) ((α_ M.X Y.X N.X).hom ≫ M.X ◁ N.actLeft) ≫ 𝟙 (TensorBimod.X M N)"}}
+{"state": {"context": ["R✝ : Type u", "S : Type v", "inst✝⁶ : CommSemiring R✝", "inst✝⁵ : CommSemiring S", "R : Type u", "inst✝⁴ : CommRing R", "inst✝³ : IsNoetherianRing R", "A : Type u", "inst✝² : CommRing A", "inst✝¹ : IsDomain A", "inst✝ : IsNoetherianRing A", "h_fA : ¬IsField A", "I : Ideal A"], "goal": "I ≠ ⊥ → ∃ Z, (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥"}}
+{"state": {"context": ["R : Type u", "S : Type v", "CommSemiring R", "CommSemiring S", "f : R →+* S", "hf : Function.Surjective ⇑f"], "goal": "Function.Injective ⇑(PrimeSpectrum.comap f)"}}
+{"state": {"context": ["p : ℕ", "G : Type u_1", "Group G", "self : Sylow p G"], "goal": "IsPGroup p ↥↑self"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "inst✝ : DecidableRel r", "l : List α", "o✝ : Option α", "a✝ m✝ : α", "hr₀ : Irreflexive r", "hr₁ : Transitive r", "tl : List α", "a : α", "ih : ∀ {a m : α} {o : Option α}, a ∈ tl → m ∈ foldl (argAux r) o tl → ¬r a m", "b m : α", "o : Option α", "hb : b ∈ tl ++ [a]", "ho : m ∈ Option.casesOn (foldl (argAux r) o tl) (some a) fun c => if r a c then some a else some c", "c : α", "hf : foldl (argAux r) o tl = some c"], "goal": "¬r b m"}}
+{"state": {"context": ["ι : Type u_1", "l : Filter ι", "𝕜 : Type u_2", "inst✝² : RCLike 𝕜", "G : Type u_3", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "f : ι → 𝕜 → G", "g : 𝕜 → G", "f' : ι → 𝕜 → G", "g' : 𝕜 → G", "x : 𝕜", "hf' : UniformCauchySeqOnFilter f' l (𝓝 x)", "hfg : Cauchy (map (fun n => f n x) l)", "hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2"], "goal": "UniformCauchySeqOnFilter f l (𝓝 x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "TopologicalSpace α", "LinearOrder α", "OrderClosedTopology α", "f g : β → α", "TopologicalSpace β", "hf : Continuous f", "hg : Continuous g"], "goal": "frontier {b | f b < g b} ⊆ {b | f b = g b}"}}
+{"state": {"context": ["α : Type u_1", "ConditionallyCompleteLattice α", "Zero α"], "goal": "sSup 0 = 0"}}
+{"state": {"context": ["α : Type u_3", "AddCommMonoid α", "s : Multiset α"], "goal": "Multiplicative.ofAdd s.sum = (Multiset.map (⇑Multiplicative.ofAdd) s).prod"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "A : Type v", "Semiring A", "Algebra R A", "M N P : Submodule R A"], "goal": "(M ⊔ N) * P = M * P ⊔ N * P"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "Group G", "A B : Rep k G"], "goal": "((CategoryTheory.ihom.ev A).app B).hom =\n  (TensorProduct.uncurry k (CoeSort.coe A) (CoeSort.coe A →ₗ[k] CoeSort.coe B) (CoeSort.coe B)) LinearMap.id.flip"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "a b : α", "s s₁ s₂ t t₁ t₂ u : Set α", "hs : s.Nonempty"], "goal": "s ⊆ {a, b} ↔ s = {a} ∨ s = {b} ∨ s = {a, b}"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "s✝ : Set E", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : AddCommGroup E", "inst✝⁷ : Module 𝕜 E", "inst✝⁶ : Module ℝ E", "inst✝⁵ : SMulCommClass ℝ 𝕜 E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : LocallyConvexSpace ℝ E", "inst✝² : ContinuousSMul 𝕜 E", "inst✝¹ : ContinuousSMul ℝ E", "inst✝ : TopologicalAddGroup E", "s : Set E", "hs_zero : 0 ∈ s", "hs_open : IsOpen s", "hs_balanced : Balanced 𝕜 s", "hs_convex : Convex ℝ s"], "goal": "∃ i, (i ∈ 𝓝 0 ∧ Balanced 𝕜 i ∧ Convex ℝ i) ∧ id i ⊆ id s"}}
+{"state": {"context": ["α : Type u", "p : Prop", "Decidable p", "t : ¬p → Set α", "x : α"], "goal": "(x ∈ if h : p then Set.univ else t h) ↔ ∀ (h : ¬p), x ∈ t h"}}
+{"state": {"context": ["K : Type u_1", "V : Type u_2", "inst✝² : Field K", "inst✝¹ : AddCommGroup V", "inst✝ : Module K V"], "goal": "span Set.univ = ⊤"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "TopologicalSpace G", "K₀ : TopologicalSpace.PositiveCompacts G", "U : Set G", "K : TopologicalSpace.Compacts G"], "goal": "0 ≤ MeasureTheory.Measure.haar.addPrehaar (↑K₀) U K"}}
+{"state": {"context": ["B : Type u_1", "B' : Type u_2", "e : B ≃ B'", "W : Type u_3", "H : Type u_4", "inst✝¹ : Group W", "inst✝ : Group H", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "i i' : B", "m : ℕ", "k : ℤ"], "goal": "(if Even (2 * k + 1) then 1 else cs.simple i') * (cs.simple i * cs.simple i') ^ ((2 * k + 1) / 2) = (if Even (↑(M.M i i') * 2 - (2 * k + 1)) then 1 else cs.simple i) * (cs.simple i' * cs.simple i) ^ ((↑(M.M i i') * 2 - (2 * k + 1)) / 2)"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "G : Type u_5", "inst✝² : Group G", "inst✝¹ : Fintype G", "hG : 0 < Fintype.card G", "inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ ∅ ^ k"], "goal": "∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(∅ ^ k) = Fintype.card ↑(∅ ^ Fintype.card G)"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "a b : α"], "goal": "(Set.Icc a b).Nonempty ↔ a ≤ b"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : SeminormedAddCommGroup E", "inst✝ : SeminormedAddCommGroup F", "f : ι → E", "g : ι → ℝ", "hg : Summable g", "h : ∀ᶠ (i : ι) in cofinite, ‖f i‖ ≤ g i", "ε : ℝ", "hε : ε > 0"], "goal": "∃ s, ∀ (t : Finset ι), Disjoint t s → ‖∑ i ∈ t, f i‖ < ε"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "R : ℝ", "c w : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "hw : w ∈ ball c R \\ s", "hc : ContinuousOn f (closedBall c R)", "hd : ∀ x ∈ ball c R \\ s, DifferentiableAt ℂ f x"], "goal": "(∮ (z : ℂ) in C(c, R), (z - w)⁻¹ • f z) = (2 * ↑π * I) • f w"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "ν : MeasureTheory.Measure α"], "goal": "MeasureTheory.Measure.rnDeriv 0 ν =ᵐ[ν] 0"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : OrderedSemiring β", "l : Filter α", "f g : α → β", "hf : 0 ≤ᶠ[l] f", "hg : 0 ≤ᶠ[l] g"], "goal": "0 ≤ᶠ[l] f * g"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "TopologicalSpace Y", "h : X ≃ₜ Y"], "goal": "⇑h.toEquiv = ⇑h"}}
+{"state": {"context": ["R : Type u", "M : Type v", "inst✝ : CommRing R", "H : ∀ (P : Ideal R), P.IsPrime → IsPrincipal P"], "goal": "∀ (x : Ideal R), x ∉ nonPrincipals R"}}
+{"state": {"context": ["G : Type u_1", "TopologicalSpace G", "Group G", "TopologicalGroup G", "MeasurableSpace G", "BorelSpace G", "LocallyCompactSpace G", "μ' μ : MeasureTheory.Measure G", "μ.IsHaarMeasure", "MeasureTheory.IsFiniteMeasureOnCompacts μ'", "μ'.IsMulLeftInvariant", "s : Set G", "h's : IsCompact (closure s)"], "goal": "μ' s = μ'.haarScalarFactor μ • μ s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f fa : α → α", "fb : β → β", "x y : α", "m n : ℕ", "hm : IsPeriodicPt f m (f^[n] x)", "r : ℕ", "hr : r > 0", "hr' : IsPeriodicPt f r x"], "goal": "n ≤ (n / r + 1) * r"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "K : Type u_2", "inst✝³ : Field K", "inst✝² : Algebra R K", "inst✝¹ : IsFractionRing R K", "inst✝ : IsDedekindDomain R", "v : HeightOneSpectrum R", "I : FractionalIdeal R⁰ K", "hI : I ≠ 0", "a : R", "J : Ideal R", "haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J"], "goal": "∏ᶠ (v : HeightOneSpectrum R), ↑v.asIdeal ^ (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) - ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) = I"}}
+{"state": {"context": ["L : FirstOrder.Language", "α : Type u'", "n : ℕ", "P : {m : ℕ} → L.BoundedFormula α m → Prop", "φ : L.BoundedFormula α n", "hqf : ∀ {m : ℕ} {ψ : L.BoundedFormula α m}, ψ.IsQF → P ψ", "hall : ∀ {m : ℕ} {ψ : L.BoundedFormula α (m + 1)}, P ψ → P ψ.all", "hex : ∀ {m : ℕ} {φ : L.BoundedFormula α (m + 1)}, P φ → P φ.ex", "hse : ∀ {m : ℕ} {φ₁ φ₂ : L.BoundedFormula α m}, ∅.SemanticallyEquivalent φ₁ φ₂ → (P φ₁ ↔ P φ₂)"], "goal": "P φ"}}
+{"state": {"context": ["R : Type u_1", "CommMonoidWithZero R", "n : ℕ", "χ : DirichletCharacter R n", "d : ℕ", "h : χ.FactorsThrough d"], "goal": "χ = (DirichletCharacter.changeLevel ⋯) h.χ₀"}}
+{"state": {"context": ["X Y Z : AlgebraicGeometry.Scheme", "𝒰 : X.OpenCover", "f : X ⟶ Z", "g : Y ⟶ Z", "∀ (i : 𝒰.J), CategoryTheory.Limits.HasPullback (𝒰.map i ≫ f) g", "i : 𝒰.J"], "goal": "(AlgebraicGeometry.Scheme.Pullback.pullbackP1Iso 𝒰 f g i).inv ≫\n    CategoryTheory.Limits.pullback.fst (AlgebraicGeometry.Scheme.Pullback.p1 𝒰 f g) (𝒰.map i) =\n  (AlgebraicGeometry.Scheme.Pullback.gluing 𝒰 f g).ι i"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "A' : Type u_1", "B : Type u_2", "a b : R", "n : ℕ", "inst✝⁶ : CommSemiring R", "inst✝⁵ : Semiring A", "inst✝⁴ : Semiring B", "inst✝³ : Algebra R A", "inst✝² : Algebra R B", "p q r : R[X]", "C : Type z", "inst✝¹ : Semiring C", "inst✝ : Algebra R C", "f : A ≃ₐ[R] B", "g : B ≃ₐ[R] C"], "goal": "∀ (a : A[X]), ((mapAlgEquiv f).trans (mapAlgEquiv g)) a = (mapAlgEquiv (f.trans g)) a"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "inst✝¹ : MeasurableSpace α", "f✝ f g : α → ℚ → ℝ", "a : α", "p : α → Prop", "inst✝ : DecidablePred p", "hf : p a → IsRatStieltjesPoint f a", "hg : ¬p a → IsRatStieltjesPoint g a", "h : p a"], "goal": "∀ (t : ℚ), ⨅ r, f a ↑r = f a t"}}
+{"state": {"context": ["ι : Type u_1", "I✝ J : Box ι", "x y : ι → ℝ", "I : WithBot (Box ι)"], "goal": "⋃ J, ⋃ (_ : ↑J = I), ↑J = ↑I"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "s t : Set X", "hs : IsLindelof s", "f : Filter X", "inst✝ : CountableInterFilter f", "hf : sᶜ ∉ f"], "goal": "∃ x ∈ s, sᶜ ∉ 𝓝 x ⊓ f"}}
+{"state": {"context": ["z : ℤ", "c : ℝ≥0"], "goal": "|z| ≤ ↑⌊c⌋₊ ↔ ‖z‖₊ ≤ c"}}
+{"state": {"context": ["E : Type u_1", "ι : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "U K : Set ℂ", "z : ℂ", "M r δ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "u : ι → ℝ", "hu : Summable u", "hf : ∀ (i : ι), DifferentiableOn ℂ (F i) U", "hU : IsOpen U", "hF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i", "hz : z ∈ U", "hc : TendstoLocallyUniformlyOn (fun t x => ∑ n ∈ t, F n x) (fun x => ∑' (n : ι), F n x) atTop U"], "goal": "Tendsto (fun s => ∑ b ∈ s, deriv (F b) z) atTop (𝓝 (deriv (fun w => ∑' (i : ι), F i w) z))"}}
+{"state": {"context": ["p n : ℕ+", "A : Type w", "B : Type z", "K : Type u", "L : Type v", "C : Type w", "inst✝⁷ : CommRing A", "inst✝⁶ : CommRing B", "inst✝⁵ : Algebra A B", "inst✝⁴ : IsCyclotomicExtension {n} A B", "inst✝³ : Field L", "ζ : L", "hζ : IsPrimitiveRoot ζ ↑n", "inst✝² : Field K", "inst✝¹ : Algebra K L", "hn : IsPrimePow ↑n", "inst✝ : IsCyclotomicExtension {n} K L", "hirr : Irreducible (cyclotomic (↑n) K)", "h : n ≠ 2", "this : 2 < n"], "goal": "(Algebra.norm K) (ζ - 1) = ↑(↑n).minFac"}}
+{"state": {"context": ["α✝ : Type u_1", "β : Type u_2", "m : Type u_3", "n✝ : Type u_4", "R : Type u_5", "α : Type u_6", "n : Type u_7", "inst✝³ : Zero α", "inst✝² : One α", "inst✝¹ : DecidableEq n", "inst✝ : AddGroup n", "i j : n"], "goal": "circulant (Pi.single 0 1) i j = 1 i j"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "i : B"], "goal": "cs.length (cs.simple i) = 1"}}
+{"state": {"context": ["x : ℂ"], "goal": "deriv Complex.tan x = 1 / Complex.cos x ^ 2"}}
+{"state": {"context": ["n : ℕ", "a b : Fin n"], "goal": "Finset.map Fin.valEmbedding (Finset.Ico a b) = Finset.Ico ↑a ↑b"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_4, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "I₁ : Type u_2", "I₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "K L : HomologicalComplex₂ C c₁ c₂", "f : K ⟶ L", "i₁ i₁' : I₁", "i₂ : I₂"], "goal": "(f.f i₁).f i₂ ≫ (L.d i₁ i₁').f i₂ = (K.d i₁ i₁').f i₂ ≫ (f.f i₁').f i₂"}}
+{"state": {"context": ["X : Type u_1", "PseudoEMetricSpace X", "e e' : X ≃ᵈ X"], "goal": "⇑(e * e') = ⇑e ∘ ⇑e'"}}
+{"state": {"context": ["o : Ordinal.{u_3}"], "goal": "#(Quotient.out o).α = o.card"}}
+{"state": {"context": ["V : Type v", "CategoryTheory.Category.{w, v} V", "CategoryTheory.MonoidalCategory V", "C : Type u₁", "CategoryTheory.EnrichedCategory V C", "X Y : C"], "goal": "(CategoryTheory.EnrichedFunctor.id V C).map X Y =\n  CategoryTheory.CategoryStruct.id (CategoryTheory.EnrichedCategory.Hom X Y)"}}
+{"state": {"context": ["X : Type u_1", "l : Type u_3", "R : Type u_8", "m' : l → Type u_9", "n' : l → Type u_10", "AddCommMonoid R", "TopologicalSpace R", "DecidableEq l", "f : X → (i : l) → Matrix (m' i) (n' i) R"], "goal": "(Summable fun x => Matrix.blockDiagonal' (f x)) ↔ Summable f"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "X : Type u_2", "A : Type u_3", "Semiring A", "Algebra R A", "f : X → A"], "goal": "Algebra.adjoin R (Set.range f) = ((FreeAlgebra.lift R) f).range"}}
+{"state": {"context": ["f : ℕ → ℕ", "hf : Nat.Primrec f"], "goal": "∃ m, ∀ (n : ℕ), f n < ack m n"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "RCLike 𝕜", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "x y : E"], "goal": "⟪x, y⟫_𝕜 = ↑‖x‖ * ↑‖y‖ ↔ ↑‖y‖ • x = ↑‖x‖ • y"}}
+{"state": {"context": ["n m : Nat"], "goal": "n * m.pred = n * m - n"}}
+{"state": {"context": ["σ : Type u_1", "p : ℕ", "Fact (Nat.Prime p)", "f : MvPolynomial σ (ZMod p)"], "goal": "(frobenius (MvPolynomial σ (ZMod p)) p) f = (MvPolynomial.expand p) f"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "inst✝² : Semiring α", "inst✝¹ : LinearOrder α", "a b c d : α", "inst✝ : PosMulStrictMono α", "h : ∀ (n : ℕ), 0 ≤ a * b ^ n"], "goal": "a = 0 ∨ 0 < a ∧ 0 ≤ b"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "TopologicalSpace α", "BorelSpace α", "μ : MeasureTheory.Measure α", "MeasureTheory.IsFiniteMeasure μ", "H : μ.InnerRegularWRT IsClosed IsOpen"], "goal": "μ.WeaklyRegular"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝¹ : Field R", "p q : R[X]", "inst✝ : DecidableEq R", "hp : p ≠ 0"], "goal": "↑(normUnit p) = C p.leadingCoeff⁻¹"}}
+{"state": {"context": ["R : Type u", "inst✝¹ : Semiring R", "S : Type v", "inst✝ : Semiring S", "f : R →+* S", "p : S[X]", "n : ℕ", "h : ∀ (n : ℕ), p.coeff n ∈ f.rangeS", "k : ℕ", "hk : ¬k = n", "i : R", "hi : f i = p.coeff k"], "goal": "(erase n p).coeff k ∈ f.rangeS"}}
+{"state": {"context": ["R : Type u_1", "DivisionRing R", "CharZero R", "p r : R", "z : ℤ", "hz : z ≠ 0"], "goal": "z • r ∈ AddSubgroup.zmultiples p ↔ ∃ k, r - ↑k • (p / ↑z) ∈ AddSubgroup.zmultiples p"}}
+{"state": {"context": ["α : Type u", "x y z w : α"], "goal": "{x, y} = {z, w} ↔ x = z ∧ y = w ∨ x = w ∧ y = z"}}
+{"state": {"context": ["R : Type u", "S : Type v", "Ring R", "Ring S", "f : R →+* S", "s : Set R", "hs : IsSubring s"], "goal": "IsSubring (⇑f '' s)"}}
+{"state": {"context": ["l m r : List Char", "s : Substring"], "goal": "Substring.ValidFor l m r s → s.bsize = String.utf8Len m"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "g : β → α"], "goal": "Function.RightInverse (List.map f) (List.map g) ↔ Function.RightInverse f g"}}
+{"state": {"context": ["f : CircleDeg1Lift"], "goal": "dist (f 0) f.translationNumber ≤ 1"}}
+{"state": {"context": [], "goal": "@List.enumFrom = @List.enumFromTR"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "M : Type u_4", "N : Type u_5", "P : Type u_6", "G : Type u_7", "H : Type u_8", "F : Type u_9", "inst✝⁶ : MulOneClass M", "inst✝⁵ : MulOneClass N", "inst✝⁴ : FunLike F M N", "inst✝³ : FunLike F G H", "inst✝² : DivInvMonoid G", "inst✝¹ : DivInvMonoid H", "inst✝ : MonoidHomClass F G H", "f : F", "hf : ∀ (a : G), f a⁻¹ = (f a)⁻¹", "a b : G"], "goal": "f (a / b) = f a / f b"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "F' : Type u_4", "G : Type u_5", "𝕜 : Type u_6", "p : ℝ≥0∞", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℝ E", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace ℝ F", "inst✝⁶ : NormedAddCommGroup F'", "inst✝⁵ : NormedSpace ℝ F'", "inst✝⁴ : NormedAddCommGroup G", "m : MeasurableSpace α", "μ : Measure α", "G' : Type u_7", "G'' : Type u_8", "inst✝³ : NormedLatticeAddCommGroup G''", "inst✝² : NormedSpace ℝ G''", "inst✝¹ : NormedLatticeAddCommGroup G'", "inst✝ : NormedSpace ℝ G'", "T : Set α → G' →L[ℝ] G''", "hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x", "f : α →ₛ G'", "hf : 0 ≤ f", "hfi : Integrable (↑f) μ"], "goal": "0 ≤ setToSimpleFunc T f"}}
+{"state": {"context": ["a b : EReal"], "goal": "-a ≤ b ↔ -b ≤ a"}}
+{"state": {"context": ["α : Type u_1", "I B X : Set α", "M : IndepMatroid α"], "goal": "∀ ⦃I : Set α⦄, M.Indep I ↔ ∃ B, (fun x => x ∈ maximals (fun x x_1 => x ⊆ x_1) {I | M.Indep I}) B ∧ I ⊆ B"}}
+{"state": {"context": ["α : Type u_1", "Preorder α", "s : UpperSet α", "a : α"], "goal": "s.erase a = s ↔ a ∉ s"}}
+{"state": {"context": ["a : UnitAddCircle"], "goal": "∃ p, 0 < p ∧ HurwitzZeta.oddKernel a =O[Filter.atTop] fun x => rexp (-p * x)"}}
+{"state": {"context": ["α : Type u_4", "B : Set (Set α)", "empty_mem : ∅ ∈ B", "subset_mem : ∀ s₁ ∈ B, ∀ s₂ ⊆ s₁, s₂ ∈ B", "union_mem : ∀ s₁ ∈ B, ∀ s₂ ∈ B, s₁ ∪ s₂ ∈ B", "sUnion_univ : ⋃₀ B = Set.univ"], "goal": "Bornology.cobounded α = Filter.comk (fun x => x ∈ B) empty_mem subset_mem union_mem"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "γ : Type u_5", "Lattice α", "BoundedOrder α", "Lattice β", "BoundedOrder β", "Lattice γ", "BoundedOrder γ", "g : BoundedLatticeHom βᵒᵈ γᵒᵈ", "f : BoundedLatticeHom αᵒᵈ βᵒᵈ"], "goal": "BoundedLatticeHom.dual.symm (g.comp f) = (BoundedLatticeHom.dual.symm g).comp (BoundedLatticeHom.dual.symm f)"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "MetricSpace X", "MetricSpace Y", "f : GromovHausdorff.ProdSpaceFun X Y", "fA : f ∈ GromovHausdorff.candidates X Y", "x y : X"], "goal": "f (Sum.inl x, Sum.inl y) = dist x y"}}
+{"state": {"context": ["p : ℕ", "hp_prime : Fact (Nat.Prime p)", "r : ℚ", "x✝ : ℤ_[p]", "m n : ℕ", "h : m ≤ n", "x : ℤ_[p]"], "goal": "((toZModPow n) x).cast = (toZModPow m) x"}}
+{"state": {"context": ["α : Type u_2", "MulZeroClass α", "s : Set α"], "goal": "s * 0 ⊆ 0"}}
+{"state": {"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "f : α → β → β", "op : α → α → α", "inst✝³ : Monoid α", "inst✝² : Monoid β", "inst✝¹ : FunLike F α β", "inst✝ : DecidableEq α", "s : Multiset α", "a : α", "h : a ∈ s", "comm : {x | x ∈ s}.Pairwise Commute", "comm' : optParam (∀ x ∈ {x | x ∈ s.erase a}, ∀ y ∈ {x | x ∈ s.erase a}, x ≠ y → Commute x y) ⋯"], "goal": "a * (s.erase a).noncommProd comm' = s.noncommProd comm"}}
+{"state": {"context": ["X Y : CondensedSet", "f : X ⟶ Y"], "goal": "CategoryTheory.Epi f ↔ ∀ (S : Stonean), Function.Surjective (f.val.app (Opposite.op S.compHaus))"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_4", "κ : ι → Sort u_7", "s : Set α", "t : (i : ι) → κ i → Set β"], "goal": "s ×ˢ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s ×ˢ t i j"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝⁴ : Preorder α", "inst✝³ : AddCommSemigroup α", "inst✝² : Sub α", "inst✝¹ : OrderedSub α", "a b c d : α", "inst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1"], "goal": "a ≤ c + (b - c) + (a - b)"}}
+{"state": {"context": ["a b : NONote"], "goal": "(a.cmp b).Compares a b"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "n ν : ℕ"], "goal": "Polynomial.eval 0 ((⇑Polynomial.derivative)^[ν] (bernsteinPolynomial R n ν)) =\n  Polynomial.eval (↑(n - (ν - 1))) (ascPochhammer R ν)"}}
+{"state": {"context": ["n : ℕ", "α : Fin (n + 1) → Type u_1", "(j : Fin (n + 1)) → Add (α j)", "i : Fin (n + 1)", "x y : α i", "p q : (j : Fin n) → α (i.succAbove j)"], "goal": "i.insertNth (x + y) (p + q) = i.insertNth x p + i.insertNth y q"}}
+{"state": {"context": ["b : Int"], "goal": "0 / b = 0"}}
+{"state": {"context": ["R : Type u", "Ring R", "M : ModuleCat R", "m : ↑M"], "goal": "(𝟙 M) m = m"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "x y : R", "n : ℕ"], "goal": "x ^ 2 ^ n - y ^ 2 ^ n = (∏ i ∈ Finset.range n, (x ^ 2 ^ i + y ^ 2 ^ i)) * (x - y)"}}
+{"state": {"context": ["B : Type u_1", "B' : Type u_2", "e : B ≃ B'", "W : Type u_3", "H : Type u_4", "inst✝¹ : Group W", "inst✝ : Group H", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "p : W → Prop", "w : W", "one : p 1", "mul_simple_right : ∀ (w : W) (i : B), p w → p (w * cs.simple i)", "p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop := fun w x => p w", "this : w ∈ Submonoid.closure (Set.range cs.simple)"], "goal": "p' 1 ⋯"}}
+{"state": {"context": ["α : Type u_1", "l : List α", "n : Nat", "a : α"], "goal": "l.set n a = [] ↔ l = []"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "F : Type v", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "E : Type w", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "x : 𝕜", "G : Type u_2", "NormedAddCommGroup G", "NormedSpace 𝕜 G", "c : 𝕜 → F →L[𝕜] G", "c' : F →L[𝕜] G", "d : 𝕜 → E →L[𝕜] F", "d' : E →L[𝕜] F", "hc : HasDerivAt c c' x", "hd : HasDerivAt d d' x"], "goal": "HasDerivAt (fun y => (c y).comp (d y)) (c'.comp (d x) + (c x).comp d') x"}}
+{"state": {"context": ["α : Type u_1", "GeneralizedBooleanAlgebra α"], "goal": "Booleanisation.comp ⊥ = ⊤"}}
+{"state": {"context": ["α : Type u", "LinearOrderedAddCommGroup α", "Nontrivial α"], "goal": "∃ a, 0 < a"}}
+{"state": {"context": ["α : Type u", "β : Type v", "AddZeroClass α", "MulOneClass β", "f : Multiplicative α →* β"], "goal": "⇑(MonoidHom.toAdditive'' f) = ⇑Additive.ofMul ∘ ⇑f ∘ ⇑Multiplicative.ofAdd"}}
+{"state": {"context": ["X : AlgebraicGeometry.Scheme", "U V : X.Opens", "i : U ⟶ V"], "goal": "(X.restrictFunctor.map i).left = AlgebraicGeometry.IsOpenImmersion.lift V.ι U.ι ⋯"}}
+{"state": {"context": ["α : Type u", "PartialOrder α", "s : Set α", "a b : α", "Ha : IsGLB s a", "Hb : IsGLB s b"], "goal": "a = b"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁹ : CommRing R", "M : Submonoid R", "S✝ : Type u_2", "inst✝⁸ : CommRing S✝", "S : Type u_3", "T : Type u_4", "inst✝⁷ : CommRing S", "inst✝⁶ : CommRing T", "inst✝⁵ : Algebra R S", "inst✝⁴ : Algebra R T", "inst✝³ : Algebra S T", "inst✝² : IsScalarTower R S T", "inst✝¹ : IsLocalization M S", "inst✝ : IsFractionRing R T", "hM : M ≤ nonZeroDivisors R", "this : IsLocalization (Submonoid.map (algebraMap R S) (nonZeroDivisors R)) T"], "goal": "∀ (x : ↥(nonZeroDivisors S)), ∃ m, m * ↑x ∈ Submonoid.map (algebraMap R S) (nonZeroDivisors R)"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "LinearOrderedSemifield α", "LinearOrderedSemifield β", "f : ι → α", "g : ι → β", "hf : StrongLT 0 f", "hg : StrongLT 0 g"], "goal": "Antivary f⁻¹ g⁻¹ ↔ Antivary f g"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "δ : Type u_1", "ι : Sort x", "l✝ : Filter α", "inst✝ : SemilatticeSup β", "l : Filter α", "f g h : α → β", "hf : f ≤ᶠ[l] h", "hg : g ≤ᶠ[l] h"], "goal": "f ⊔ g ≤ᶠ[l] h"}}
+{"state": {"context": ["ι : Sort u_1", "M : Type u_2", "MulOneClass M", "S : ι → Submonoid M"], "goal": "(iInf S).op = ⨅ i, (S i).op"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e e' : PartialHomeomorph X Y", "s : Set X", "t : Set Y", "(x : X) → Decidable (x ∈ s)", "(y : Y) → Decidable (y ∈ t)", "H : e.IsImage s t", "H' : e'.IsImage s t", "Hs : e.source ∩ frontier s = e'.source ∩ frontier s", "Heq : Set.EqOn (↑e) (↑e') (e.source ∩ frontier s)"], "goal": "(e.piecewise e' s t H H' Hs Heq).toPartialEquiv = e.piecewise e'.toPartialEquiv s t H H'"}}
+{"state": {"context": ["α : Type u_1", "CanonicallyLinearOrderedAddCommMonoid α", "Sub α", "OrderedSub α", "a b c : α", "hc : AddLECancellable c"], "goal": "a < b - c ↔ a + c < b"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "ι : Type x", "k : Type y", "A : Type z", "a b : R", "m n✝ : ℕ", "inst✝ : Semiring R", "p✝ q r f : R[X]", "I : Ideal R[X]", "p : Ideal R[X] := Ideal.span {g | ∃ i, g = C (f.coeff i)}", "n : ℕ", "_hn : n ∈ f.support", "this✝ : C (f.coeff n) ∈ p", "this : (monomial n) 1 • C (f.coeff n) ∈ p"], "goal": "C (f.coeff n) * X ^ n = (monomial n) (f.coeff n)"}}
+{"state": {"context": ["α : Type u", "s : Set α"], "goal": "∅ ⊆ s"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.IsCofiltered Cᵒᵖ"], "goal": "CategoryTheory.IsFiltered C"}}
+{"state": {"context": ["R : Type u", "M M₁ : Type v", "M' : Type v'", "ι : Type w", "inst✝¹⁰ : Ring R", "inst✝⁹ : AddCommGroup M", "inst✝⁸ : AddCommGroup M'", "inst✝⁷ : AddCommGroup M₁", "inst✝⁶ : Module R M", "inst✝⁵ : Module R M'", "inst✝⁴ : Module R M₁", "inst✝³ : Nontrivial R", "inst✝² : NoZeroSMulDivisors R M", "inst✝¹ : Nontrivial R", "inst✝ : NoZeroSMulDivisors R M", "S : Submodule R M", "h : S = ⊥"], "goal": "Module.rank R ↥S = 0"}}
+{"state": {"context": ["α : Type u_2", "TopologicalSpace α", "a : { x // IsClosed x ∧ IsIrreducible x }"], "goal": "TopologicalSpace.IrreducibleCloseds.equivSubtype'.symm a = { carrier := ↑a, is_irreducible' := ⋯, is_closed' := ⋯ }"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "Mul α", "s : Finset α", "a : α"], "goal": "s * {a} = Finset.image (fun x => x * a) s"}}
+{"state": {"context": ["𝕜 : Type u_1", "G : Type u_2", "H : Type u_3", "inst✝⁶ : MeasurableSpace G", "inst✝⁵ : MeasurableSpace H", "inst✝⁴ : TopologicalSpace G", "inst✝³ : BorelSpace G", "μ : Measure G", "inst✝² : Group G", "inst✝¹ : TopologicalGroup G", "inst✝ : μ.IsMulLeftInvariant", "h : μ.InnerRegularCompactLTTop", "s : Set G", "s_meas : MeasurableSet s", "μs : μ s ≠ ⊤", "r : ℝ≥0∞", "hr : r < μ s", "K : Set G", "Ks : K ⊆ s", "K_comp : IsCompact K", "hK : r < μ K"], "goal": "closure K ⊆ s"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "s : ι → Set α", "h : Pairwise (Disjoint on s)", "x : (i : ι) × ↑(s i)"], "goal": "(Function.Embedding.sigmaSet h) x = ↑x.snd"}}
+{"state": {"context": ["R : Type u", "B : Type u_2", "CommSemiring R", "Semiring B", "Algebra R B", "f g : R[X] →ₐ[R] B", "hX : f Polynomial.X = g Polynomial.X"], "goal": "f = g"}}
+{"state": {"context": ["S : LightProfinite", "n m : ℕ", "h : n ≤ m"], "goal": "Function.Surjective ⇑(S.transitionMapLE h)"}}
+{"state": {"context": ["G✝ : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝⁵ : TopologicalSpace G✝", "inst✝⁴ : Div G✝", "inst✝³ : ContinuousDiv G✝", "G : Type u_1", "inst✝² : CommGroup G", "inst✝¹ : TopologicalSpace G", "inst✝ : ContinuousDiv G", "b c : G", "f : α → G", "l : Filter α", "h : Tendsto (fun k => b / f k) l (𝓝 (b / c))"], "goal": "Tendsto f l (𝓝 c)"}}
+{"state": {"context": ["b : ℂ", "V : Type u_1", "inst✝⁵ : NormedAddCommGroup V", "inst✝⁴ : InnerProductSpace ℝ V", "inst✝³ : FiniteDimensional ℝ V", "inst✝² : MeasurableSpace V", "inst✝¹ : BorelSpace V", "ι : Type u_2", "inst✝ : Fintype ι", "hb : 0 < b.re", "c : ℂ", "w : EuclideanSpace ℝ ι", "this : MeasurePreserving (⇑(EuclideanSpace.measurableEquiv ι).symm) volume volume", "y : ι → ℝ"], "goal": "-(b * ↑‖(EuclideanSpace.measurableEquiv ι).symm y‖ ^ 2) = -b * ∑ i : ι, ↑(y i) ^ 2"}}
+{"state": {"context": ["r₁ r₂ : ℝ", "h₁ : 0 ≤ r₁", "h₂ : r₁ ≤ r₂"], "goal": "(fun n => r₁ ^ n) =O[Filter.atTop] fun n => r₂ ^ n"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Field K", "inst✝ : IsDedekindDomain R", "I : Ideal R", "hI : I.IsPrime", "a b : R", "n : ℕ", "h : b * a ∈ I ^ n"], "goal": "a ∈ I ^ n ∨ b ∈ I"}}
+{"state": {"context": ["n m : Nat"], "goal": "n + m = 0 → n = 0 ∧ m = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : LinearOrder α", "inst✝ : LinearOrder β", "f : α → β", "s : Set α", "a a₁ a₂ b b₁ b₂ c d x : α"], "goal": "a ∈ Ι a b ↔ b < a"}}
+{"state": {"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b✝ b : EReal"], "goal": "(∀ (z : ℝ), b < ↑z → limsup u f ≤ ↑z) ↔ ∀ (c : ℝ), b < ↑c → ∀ᶠ (a : α) in f, u a ≤ ↑c"}}
+{"state": {"context": ["k : Type u_1", "V1 : Type u_2", "Ring k", "AddCommGroup V1", "Module k V1", "p₀ p₁ : V1", "c : k"], "goal": "(AffineMap.lineMap p₀ p₁) c = c • (p₁ - p₀) + p₀"}}
+{"state": {"context": ["Ω : Type u_1", "m' : MeasurableSpace Ω", "mΩ : MeasurableSpace Ω", "StandardBorelSpace Ω", "Nonempty Ω", "hm' : m' ≤ mΩ", "μ : MeasureTheory.Measure Ω", "MeasureTheory.IsFiniteMeasure μ", "s t : Set Ω", "hs : MeasurableSet s", "ht : MeasurableSet t"], "goal": "ProbabilityTheory.CondIndepSets m' hm' {s} {t} μ ↔\n  μ[(s ∩ t).indicator fun ω => 1|m'] =ᵐ[μ] μ[s.indicator fun ω => 1|m'] * μ[t.indicator fun ω => 1|m']"}}
+{"state": {"context": ["R : Type u_1", "E : Type u_3", "ι : Type u_5", "LinearOrderedField R", "AddCommGroup E", "Module R E", "w : ι → R", "z : ι → E", "s t : Finset ι", "hw : ∑ i ∈ s, w i + ∑ i ∈ t, w i = 0", "hz : ∑ i ∈ s, w i • z i + ∑ i ∈ t, w i • z i = 0"], "goal": "s.centerMass w z = t.centerMass w z"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "P : Type u_3", "N : Type u_4", "inst✝⁸ : Ring R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup P", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R M", "inst✝³ : Module R P", "inst✝² : Module R N", "inst✝¹ : IsArtinian R M", "inst✝ : IsArtinian R P", "f : M →ₗ[R] N", "g : N →ₗ[R] P", "hf : Function.Injective ⇑f", "hg : Function.Surjective ⇑g", "h : LinearMap.range f = LinearMap.ker g"], "goal": "∀ (a : Submodule R N), Submodule.map f (Submodule.comap f a) = a ⊓ LinearMap.range f"}}
+{"state": {"context": [], "goal": "logTaylor 0 = fun x => 0"}}
+{"state": {"context": ["x : ℝ"], "goal": "(↑x).expMapCircle = expMapCircle x"}}
+{"state": {"context": ["a✝ b c d : Ordinal.{u}", "a : Ordinal.{u_1}"], "goal": "{c | ∀ a' < a, a' ⨳ 1 < c} = Set.Ici a"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "F : Type u_3", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace 𝕜 F", "f : E → F", "K : Set (E →L[𝕜] F)", "c : 𝕜", "hc : 1 < ‖c‖", "r ε : ℝ", "hε : 0 < ε", "hr : 0 < r", "x : E", "L₁ L₂ : E →L[𝕜] F", "h₁ : x ∈ A f L₁ r ε", "h₂ : x ∈ A f L₂ r ε", "y : E", "ley : r / 2 / ‖c‖ ≤ ‖y‖", "ylt : ‖y‖ < r / 2"], "goal": "‖(L₁ - L₂) y‖ ≤ 4 * ‖c‖ * ε * ‖y‖"}}
+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝¹⁰ : Fintype ι", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : InnerProductSpace ℝ F", "inst✝⁷ : FiniteDimensional ℝ F", "inst✝⁶ : MeasurableSpace F", "inst✝⁵ : BorelSpace F", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : InnerProductSpace ℝ E", "inst✝² : FiniteDimensional ℝ E", "inst✝¹ : MeasurableSpace E", "inst✝ : BorelSpace E", "f : E ≃ₗᵢ[ℝ] F"], "goal": "MeasurePreserving (⇑f) volume volume"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "σ : Type u_5", "Primcodable α", "Primcodable β", "Primcodable σ", "f : α → β → σ", "hf : Computable fun p => f p.1 p.2"], "goal": "Computable₂ f"}}
+{"state": {"context": ["G : Type u_1", "Group G", "A B : Subgroup G", "d : HNNExtension.NormalWord.TransversalPair G A B", "g₁ g₂ : G"], "goal": "g₁ • HNNExtension.NormalWord.ofGroup g₂ = HNNExtension.NormalWord.ofGroup (g₁ * g₂)"}}
+{"state": {"context": ["M : Type u_5", "N : Type u_6", "Add M", "Add N", "a₁ a₂ : M", "b₁ b₂ : N"], "goal": "(a₁, b₁) + (a₂, b₂) = (a₁ + a₂, b₁ + b₂)"}}
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+{"state": {"context": ["X : Type u_1", "α : Type u_2", "α' : Type u_3", "β : Type u_4", "γ : Type u_5", "δ : Type u_6", "M : Type u_7", "E : Type u_8", "R : Type u_9", "inst✝⁷ : TopologicalSpace α", "inst✝⁶ : TopologicalSpace α'", "inst✝⁵ : One β", "inst✝⁴ : One γ", "inst✝³ : One δ", "g✝ : β → γ", "f✝ : α → β", "f₂ : α → γ", "m : β → γ → δ", "x : α", "inst✝² : T2Space α'", "hf : HasCompactMulSupport f✝", "g : α → α'", "cont : Continuous g", "f : α → γ", "inst✝¹ : TopologicalSpace γ", "inst✝ : One γ", "h : HasCompactMulSupport f", "N : Set γ", "hN : N ∈ 𝓝 1", "v : α", "hv : v ∈ (mulTSupport f)ᶜ"], "goal": "1 ∈ N"}}
+{"state": {"context": ["R : Type u_1", "NonAssocRing R"], "goal": "ringChar R = 0 ↔ CharZero R"}}
+{"state": {"context": ["α : Type u", "inst✝ : CanonicallyOrderedCommMonoid α", "a b c d : α", "h : a < b"], "goal": "∃ c x, a * c = b"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {a : α} {l₁ l₂ : List α}, l₁.Disjoint (a :: l₂) → l₁.Disjoint l₂"}}
+{"state": {"context": ["c : ℝ", "hc : 1 < c", "i : ℕ"], "goal": "(1 - c⁻¹) * c ^ i ≤ ↑⌊c ^ i⌋₊"}}
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+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝² : TopologicalSpace X", "inst✝¹ : PreconnectedSpace X", "inst✝ : DiscreteTopology X", "x y : X", "hxy : x ∉ univ"], "goal": "False"}}
+{"state": {"context": ["M : Type u_1", "AddSemigroup M", "z : M"], "goal": "z ∈ AddSubsemigroup.center M ↔ ∀ (g : M), g + z = z + g"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "a b C✝ : ℝ", "f g : ℂ → E", "z : ℂ", "hd : DiffContOnCl ℂ f {z | 0 < z.re}", "hexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)", "hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x‖", "C : ℝ", "hC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C"], "goal": "∀ (z : ℂ), 0 < z.re → f z = 0"}}
+{"state": {"context": ["G : Type u_1", "G' : Type u_2", "inst✝¹ : Group G", "inst✝ : Group G'", "H K L : Subgroup G", "h : H.index = 2", "a b : G", "ha : a ∉ H", "hb : b ∉ H", "c : G", "hc : ∀ (b : G), Xor' (b * c ∈ H) (b ∈ H)"], "goal": "a * b * c ∉ H"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "ι : Type u_3", "inst✝⁴ : NontriviallyNormedField 𝕜", "inst✝³ : AddCommGroup E", "inst✝² : Module 𝕜 E", "inst✝¹ : TopologicalSpace E", "inst✝ : ContinuousSMul 𝕜 E", "U : Set E", "hU : U ∈ 𝓝 0", "r : ℝ", "V : Set E", "hr : 0 < r", "hV : V ∈ 𝓝 0", "hrVU : ∀ (c : 𝕜) (y : E), ‖c‖ < r → y ∈ V → c • y ∈ U", "y : 𝕜", "hy₀ : y ≠ 0", "hyr : ‖y‖ < r"], "goal": "balancedCore 𝕜 U ∈ 𝓝 0"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "M : ι → Type u_4", "N : Type u_5", "inst✝³ : DecidableEq ι", "inst✝² : Ring R", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "p : ι → Submodule R N", "h : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i => (p i).subtype) (erase i v) = ↑x → x = 0", "m : Π₀ (i : ι), ↥(p i)", "hm : ((lsum ℕ) fun i => (p i).subtype) m = 0", "i : ι"], "goal": "((lsum ℕ) fun i => (p i).subtype) (erase i m) = ↑(-m i)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoMetricSpace α", "PseudoMetricSpace β", "f : α → β", "hf : Isometry f", "x : α", "r : ℝ"], "goal": "Set.MapsTo f (Metric.ball x r) (Metric.ball (f x) r)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "inst✝ : DistribLattice α", "x y z a b c : α"], "goal": "a ⊓ (a ⊓ b ⊔ c) = (a ⊔ a ⊓ b) ⊓ (a ⊓ b ⊔ c)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝¹ : CommSemiring R", "inst✝ : NoZeroDivisors R", "p q : R[X]", "hp : p.Monic", "hp1 : p ≠ 1", "h : ∀ (f g : R[X]), f.Monic → g.Monic → f * g = p → f = 1 ∨ g = 1", "f g : R[X]", "hfg : p = f * g"], "goal": "(f * C g.leadingCoeff).Monic"}}
+{"state": {"context": ["G : Type u_1", "Group G", "H : Subgroup G", "N : Type u_5", "Group N", "f : G →* N"], "goal": "Subgroup.map f H.normalizer ≤ (Subgroup.map f H).normalizer"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace ℂ E", "V : Type u_2", "W : Type u_3", "inst✝⁸ : NormedAddCommGroup V", "inst✝⁷ : NormedSpace ℝ V", "inst✝⁶ : NormedAddCommGroup W", "inst✝⁵ : NormedSpace ℝ W", "L : V →L[ℝ] W →L[ℝ] ℝ", "f : V → E", "inst✝⁴ : SecondCountableTopology V", "inst✝³ : MeasurableSpace V", "inst✝² : BorelSpace V", "μ✝ : Measure V", "inst✝¹ : FiniteDimensional ℝ V", "μ : Measure V", "inst✝ : μ.IsAddHaarMeasure", "K N : ℕ∞", "hf : ContDiff ℝ N f", "h'f : ∀ (k n : ℕ), ↑k ≤ K → ↑n ≤ N → Integrable (fun v => ‖v‖ ^ k * ‖iteratedFDeriv ℝ n f v‖) μ", "k n : ℕ", "hk : ↑k ≤ K", "hn : ↑n ≤ N", "v : V", "w : W"], "goal": "1 ≤ 2 * π"}}
+{"state": {"context": ["G : Type u_1", "α : Type u_2", "Group G", "MulAction G α", "S : Set G", "T : Set α", "a : α", "hS : ∀ g ∈ S, g⁻¹ ∈ S", "subset : T ⊆ MulAction.orbit (↥(Subgroup.closure S)) a", "not_mem : a ∉ T", "nonempty : T.Nonempty"], "goal": "∃ σ ∈ S, ∃ a ∈ T, σ • a ∉ T"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "T : CategoryTheory.Monad C", "A B : T.Algebra", "self : A.Hom B"], "goal": "CategoryTheory.CategoryStruct.comp (T.map self.f) B.a = CategoryTheory.CategoryStruct.comp A.a self.f"}}
+{"state": {"context": ["k : ℕ", "hk : k ≠ 0", "n : ℤ"], "goal": "fourierCoeff (Complex.ofReal' ∘ periodizedBernoulli k) n = -↑k ! / (2 * ↑π * Complex.I * ↑n) ^ k"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "f : α → ℝ", "hf : MeasureTheory.Integrable f μ", "h'f : 0 ≤ f", "A : ℝ"], "goal": "∫ (x : α), ProbabilityTheory.truncation f A x ∂μ ≤ ∫ (x : α), f x ∂μ"}}
+{"state": {"context": ["α : Type u_2", "β : Type u_3", "γ : Type u_4", "AddCommMonoid α", "AddCommMonoid β", "AddCommMonoid γ", "A : Set α", "B : Set β", "C : Set γ", "f : α → β", "g : β → γ", "n : ℕ", "hg : IsAddFreimanIso n B C g", "hf : IsAddFreimanIso n A B f"], "goal": "IsAddFreimanIso n A C (g ∘ f)"}}
+{"state": {"context": [], "goal": "Function.Injective Cardinal.ofENat"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "CategoryTheory.CategoryWithHomology C"], "goal": "∀ {X Y : CategoryTheory.ShortComplex C} (f : X ⟶ Y),\n  (CategoryTheory.ShortComplex.homologyFunctor C).map f = CategoryTheory.ShortComplex.homologyMap f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Computation α", "a b : α", "m n : ℕ", "h1 : s.Results a m", "h2 : s.Results b n", "this : s.Terminates"], "goal": "m = n"}}
+{"state": {"context": ["x : SetTheory.PGame", "x.Short"], "goal": "x.birthday < Ordinal.omega"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℝ", "A : ℝ", "x : α"], "goal": "|ProbabilityTheory.truncation f A x| ≤ |A|"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Preorder α", "Preorder β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "a : α", "b : β"], "goal": "a ≤ u b → l a ≤ b"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "m m0 : MeasurableSpace α", "p : ℝ≥0∞", "q : ℝ", "μ ν : Measure α", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : IsFiniteMeasure μ", "c : F", "hc_ne_zero : c ≠ 0", "hq_ne_zero : q ≠ 0"], "goal": "eLpNorm' (fun x => c) q μ = ↑‖c‖₊ * μ Set.univ ^ (1 / q)"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "e : C ≌ D", "x✝ : C"], "goal": "e.functor.map (e.unit.app x✝) ≫ e.counit.app (e.functor.obj x✝) = 𝟙 (e.functor.obj x✝)"}}
+{"state": {"context": ["α : Type u_5", "Monoid α", "a : α", "h : IsUnit a"], "goal": "h.unit⁻¹ • a = 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁸ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝⁷ : NormedAddCommGroup D", "inst✝⁶ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : NormedSpace 𝕜 E", "F : Type uF", "inst✝³ : NormedAddCommGroup F", "inst✝² : NormedSpace 𝕜 F", "G : Type uG", "inst✝¹ : NormedAddCommGroup G", "inst✝ : NormedSpace 𝕜 G", "s✝ s₁ t u : Set E", "B : E →L[𝕜] F →L[𝕜] G", "f : D → E", "g : D → F", "N : ℕ∞", "s : Set D", "x : D", "hf : ContDiffOn 𝕜 N f s", "hg : ContDiffOn 𝕜 N g s", "hs : UniqueDiffOn 𝕜 s", "hx : x ∈ s", "n : ℕ", "hn : ↑n ≤ N", "Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D", "Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E", "Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F", "Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G", "isoD : Du ≃ₗᵢ[𝕜] D", "isoE : Eu ≃ₗᵢ[𝕜] E", "isoF : Fu ≃ₗᵢ[𝕜] F", "isoG : Gu ≃ₗᵢ[𝕜] G", "fu : Du → Eu := ⇑isoE.symm ∘ f ∘ ⇑isoD", "hfu : fu = ⇑isoE.symm ∘ f ∘ ⇑isoD", "gu : Du → Fu := ⇑isoF.symm ∘ g ∘ ⇑isoD", "hgu : gu = ⇑isoF.symm ∘ g ∘ ⇑isoD", "Bu₀ : Eu →L[𝕜] Fu →L[𝕜] G := ((B.comp ↑{ toLinearEquiv := isoE.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip.comp ↑{ toLinearEquiv := isoF.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip", "hBu₀ : Bu₀ = ((B.comp ↑{ toLinearEquiv := isoE.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip.comp ↑{ toLinearEquiv := isoF.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }).flip", "Bu : Eu →L[𝕜] Fu →L[𝕜] Gu := ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ })) Bu₀", "hBu : Bu = ((compL 𝕜 Eu (Fu →L[𝕜] G) (Fu →L[𝕜] Gu)) ((compL 𝕜 Fu G Gu) ↑{ toLinearEquiv := isoG.symm.toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ })) Bu₀", "Bu_eq : (fun y => (Bu (fu y)) (gu y)) = ⇑isoG.symm ∘ (fun y => (B (f y)) (g y)) ∘ ⇑isoD", "Bu_le : ‖Bu‖ ≤ ‖B‖", "su : Set Du := ⇑isoD ⁻¹' s"], "goal": "‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤ ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "S : Type u_2", "AddCommGroup S", "Pow S ℕ", "Module R S", "p : Polynomial R", "x : S"], "goal": "(-p).smeval x = -p.smeval x"}}
+{"state": {"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝ : UniformSpace α", "p : ι → Prop", "U : ι → Set (α × α)", "h : (𝓤 α).HasBasis p U", "s : Set α", "x : α"], "goal": "x ∈ ⋂ i, ⋂ (_ : p i), ⋃ x ∈ s, ball x (U i) ↔ x ∈ closure s"}}
+{"state": {"context": ["R✝ : Type u", "S : Type v", "T : Type w", "a✝ b✝ : R✝", "n : ℕ", "inst✝² : CommRing R✝", "inst✝¹ : IsDomain R✝", "p q : R✝[X]", "R : Type u_1", "inst✝ : CommRing R", "a b : R", "h : IsUnit (a - b)"], "goal": "-C ↑h.unit⁻¹ * (X - C a) + C ↑h.unit⁻¹ * (X - C b) = 1"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "CommRing R", "IsDomain R", "IsPrincipalIdealRing R", "S : Type u_4", "CommRing S", "IsDomain S", "Algebra R S", "Finite ι", "b : Basis ι R S", "I : Ideal S", "hI : I ≠ ⊥", "i : ι"], "goal": "↑((Ideal.selfBasis b I hI) i) = Ideal.smithCoeffs b I hI i • (Ideal.ringBasis b I hI) i"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹⁰ : TopologicalSpace α", "inst✝⁹ : MeasurableSpace α", "μ✝ ν : Measure α", "s k✝ : Set α", "G : Type u_2", "inst✝⁸ : Group G", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : TopologicalGroup G", "inst✝⁵ : LocallyCompactSpace G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "μ : Measure G", "inst✝² : μ.IsMulLeftInvariant", "inst✝¹ : IsFiniteMeasureOnCompacts μ", "inst✝ : μ.InnerRegularCompactLTTop", "k : Set G", "h : μ.IsEverywherePos k", "hk : IsCompact k", "h'k : IsClosed k", "u : ℕ → ℝ≥0∞", "u_mem : ∀ (n : ℕ), u n ∈ Ioo 0 1", "u_lim : Tendsto u atTop (𝓝 0)", "W : ℕ → Set G", "W_open : ∀ (n : ℕ), IsOpen (W n)", "mem_W : ∀ (n : ℕ), 1 ∈ W n", "hW : ∀ (n : ℕ), ∀ g ∈ W n * W n, μ (g • k \\ k) < u n", "V : ℕ → Set G := fun n => ⋂ i ∈ Finset.range n, W i", "x : G", "hx : x ∈ ⋂ n, V n * k", "v : ℕ → G", "hv : ∀ (i : ℕ), v i ∈ V i", "y : ℕ → G", "hy : ∀ (i : ℕ), y i ∈ k", "hvy : ∀ (i : ℕ), (fun x x_1 => x * x_1) (v i) (y i) = x", "z : G", "zk : z ∈ k", "hz : MapClusterPt z atTop y"], "goal": "x ∈ k"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝⁴ : Monoid K", "inst✝³ : (i : ι) → Group (G i)", "inst✝² : Group H", "φ : (i : ι) → H →* G i", "d : Transversal φ", "inst✝¹ : DecidableEq ι", "inst✝ : (i : ι) → DecidableEq (G i)", "w : NormalWord d"], "goal": "w.prod • empty = w"}}
+{"state": {"context": ["R : Type u", "S : Type u₁", "Monoid R", "Monoid S", "φ : R →* S", "A : Type v", "B : Type w", "NonUnitalNonAssocSemiring A", "DistribMulAction R A", "NonUnitalNonAssocSemiring B", "DistribMulAction S B", "f : A →ₛₙₐ[φ] B"], "goal": "⇑↑f = ⇑f"}}
+{"state": {"context": ["α : Type u_1", "f : α → ℝ≥0", "hf : Summable f"], "goal": "Tendsto f cofinite (𝓝 0)"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_3", "AddMonoid M", "TopologicalSpace α", "AddAction M α", "AddAction.IsMinimal M α", "x : α"], "goal": "Dense (AddAction.orbit M x)"}}
+{"state": {"context": ["V : Type u_1", "Fintype V", "DecidableEq V", "G : SimpleGraph V", "DecidableRel G.Adj", "x : V → ℝ"], "goal": "(((Matrix.toLinearMap₂' ℝ) (SimpleGraph.lapMatrix ℝ G)) x) x = 0 ↔ ∀ (i j : V), G.Reachable i j → x i = x j"}}
+{"state": {"context": ["R : Type u", "CommRing R", "g : R[X]", "hg : g.Monic", "f : AdjoinRoot g", "i : Fin g.natDegree"], "goal": "((AdjoinRoot.powerBasisAux' hg).repr f) i = ((AdjoinRoot.modByMonicHom hg) f).coeff ↑i"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : LinearOrderedCommRing R", "M : Type u_2", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "ι : Type u_3", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι", "e : Basis ι R M", "i : ι"], "goal": "(-1 * ∏ x ∈ Finset.univ \\ {i}, 1 x)⁻¹ • e.orientation = -e.orientation"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "Countable α", "Uncountable β", "f : α → β"], "goal": "¬Function.Surjective f"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "SMul α β", "Preorder α", "Preorder β", "Zero β", "self : SMulPosReflectLE α β", "b : β", "hb : 0 < b", "a₁ a₂ : α"], "goal": "a₁ • b ≤ a₂ • b → a₁ ≤ a₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "CompleteLattice α", "s : Set β", "f : β → α"], "goal": "IsLUB (f '' s) (⨆ x ∈ s, f x)"}}
+{"state": {"context": ["α : Type u_1", "M : Matroid α", "B I J X Y : Set α", "e : α", "hX : autoParam (X ⊆ M.E) _auto✝"], "goal": "M.Basis' I X ↔ M.Basis I X"}}
+{"state": {"context": ["α : Type u", "DecidableEq α", "a b : α", "l : List α", "h : a ∈ b :: l"], "goal": "a = b ∨ a ≠ b ∧ a ∈ l"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "M : Type u_3", "M₂ : Type u_4", "M₃ : Type u_5", "M₄ : Type u_6", "inst✝⁸ : CommRing R", "inst✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup M₂", "inst✝⁵ : AddCommGroup M₃", "inst✝⁴ : AddCommGroup M₄", "inst✝³ : Module R M", "inst✝² : Module R M₂", "inst✝¹ : Module R M₃", "inst✝ : Module R M₄", "a b : R", "h1 : IsSMulRegular M a ∧ IsSMulRegular (QuotSMulTop a M) b", "h2 : torsionBy R M b = a • torsionBy R M b → torsionBy R M b = ⊥"], "goal": "IsSMulRegular M b ∧ IsSMulRegular (QuotSMulTop b M) a"}}
+{"state": {"context": ["T : Type u", "CategoryTheory.Category.{v, u} T", "f g : CategoryTheory.Arrow T", "u : f.left ⟶ g.left", "v : f.right ⟶ g.right", "w : CategoryTheory.CategoryStruct.comp u g.hom = CategoryTheory.CategoryStruct.comp f.hom v"], "goal": "(CategoryTheory.Arrow.homMk w).right = v"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocSemiring R", "s : Set R"], "goal": "NonUnitalSubsemiring.closure ↑(AddSubmonoid.closure s) = NonUnitalSubsemiring.closure s"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "s : 𝕜 → 𝕜", "U : Set 𝕜", "hs : MeromorphicOn s U", "n : ℕ"], "goal": "MeromorphicOn (s ^ n) U"}}
+{"state": {"context": ["α : Type u_1", "MetricSpace α", "C : Set α", "hC : Perfect C", "ε : ENNReal", "ε_pos : 0 < ε", "x : α", "xC : x ∈ C"], "goal": "let D := closure (EMetric.ball x (ε / 2) ∩ C);\nPerfect D ∧ D.Nonempty ∧ D ⊆ C ∧ EMetric.diam D ≤ ε"}}
+{"state": {"context": ["ι : Type u_1", "inst✝⁴ : Fintype ι", "M : Type u_2", "inst✝³ : AddCommGroup M", "R : Type u_3", "inst✝² : CommRing R", "inst✝¹ : Module R M", "I : Ideal R", "b : ι → M", "hb : Submodule.span R (Set.range b) = ⊤", "inst✝ : DecidableEq ι", "A A' : Matrix ι ι R", "f f' : Module.End R M", "h : Represents b A f", "h' : Represents b A' f'"], "goal": "((LinearMap.llcomp R (ι → R) (ι → R) M) ((Fintype.total R R) b) ∘ₗ algEquivMatrix'.symm.toLinearMap) (A * A') = (PiToModule.fromEnd R b) (f * f')"}}
+{"state": {"context": ["R : CommRingCat"], "goal": "AlgebraicGeometry.IsNoetherian (AlgebraicGeometry.Spec R) ↔ IsNoetherianRing ↑R"}}
+{"state": {"context": ["R : Type u", "Ring R", "p : R[X]", "T : Subring R", "hp : ↑p.coeffs ⊆ ↑T", "n : ℕ"], "goal": "↑((p.toSubring T hp).coeff n) = p.coeff n"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{?u.39248, u_2} D", "inst✝² : Preadditive C", "inst✝¹ : Preadditive D", "F G : CochainComplex C ℤ", "φ : F ⟶ G", "inst✝ : HasHomotopyCofiber φ", "K : CochainComplex C ℤ", "d e : ℤ", "γ : Cochain F K d", "he : 1 + d = e"], "goal": "(inl φ).comp ((↑(fst φ)).comp γ he) ⋯ = γ"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_3", "CommRing R", "Ring B", "Algebra R B"], "goal": "IsIntegral R 1"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "A B A' B' X Y : C", "i : A ⟶ B", "i' : A' ⟶ B'", "e : CategoryTheory.Arrow.mk i ≅ CategoryTheory.Arrow.mk i'", "p : X ⟶ Y", "hip : CategoryTheory.HasLiftingProperty i p"], "goal": "CategoryTheory.HasLiftingProperty i' p"}}
+{"state": {"context": ["V : Type u", "V' : Type v", "V'' : Type w", "G : SimpleGraph V", "G' : SimpleGraph V'", "G'' : SimpleGraph V''", "u✝ v✝ w✝ x y : V", "p : G.Walk u✝ v✝", "hnp : ¬p.Nil", "u v w : V", "huv : G.Adj u v", "q : G.Walk v w"], "goal": "(notNilRec (fun {u v w} x q => q) (cons huv q) ⋯).support = (cons huv q).support.tail"}}
+{"state": {"context": ["E : Type u", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℂ E", "F : Type v", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedSpace ℂ F", "inst✝ : CompleteSpace F", "f : ℂ → F", "z w : ℂ", "r : ℝ := dist w z", "hd : DiffContOnCl ℂ f (ball z r)", "hz : IsMaxOn (norm ∘ f) (closedBall z r) z", "hw : w ∈ closedBall z r"], "goal": "¬(norm ∘ f) w < (norm ∘ f) z"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "f g : CategoryTheory.Arrow C", "F : CategoryTheory.Limits.MonoFactorisation f.hom", "sq : f ⟶ g", "CategoryTheory.IsIso sq"], "goal": "(F.ofArrowIso sq).I = F.I"}}
+{"state": {"context": ["I : Type u", "f : I → Type v₁", "(i : I) → Zero (f i)"], "goal": "0 = fun x => 0"}}
+{"state": {"context": ["o : Ordinal.{u_3}", "n : ℕ", "n.AtLeastTwo"], "goal": "o.card ≤ OfNat.ofNat n ↔ o ≤ OfNat.ofNat n"}}
+{"state": {"context": ["K : Type u", "CommRing K", "IsDomain K"], "goal": "(RatFunc.toFractionRingRingEquiv K).symm = (IsLocalization.algEquiv K[X]⁰ (FractionRing K[X]) (RatFunc K)).toRingEquiv"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X Y Z : C", "f : X ⟶ Y", "g : X ⟶ Z", "CategoryTheory.Limits.HasPushout f g"], "goal": "f ≫ CategoryTheory.Limits.pushout.inl f g = g ≫ CategoryTheory.Limits.pushout.inr f g"}}
+{"state": {"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "inst✝ : DecidableEq α", "s✝ t✝ u : Multiset α", "a✝ b a : α", "s t : Multiset α", "h : t ≤ s"], "goal": "a ::ₘ s - t = a ::ₘ (s - t)"}}
+{"state": {"context": ["α : Type u₁", "β : Type u₂", "γ : Type u₃", "δ : Type u₄", "ι : Type u₅", "inst✝¹ : MulZeroClass β", "inst✝ : DecidableEq α", "g₁ g₂ : α →₀ β", "a : α", "h : g₁ a * g₂ a ≠ 0"], "goal": "a ∈ g₁.support ∩ g₂.support"}}
+{"state": {"context": ["ι : Type u_1", "κ : Type u_2", "R : Type u_3", "α : Type u_4", "inst✝³ : DivisionSemiring α", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSemiring α", "f g : ι → α", "a a₁ a₂ : α", "inst✝ : T2Space α"], "goal": "∑' (x : ι), f x / a = (∑' (x : ι), f x) / a"}}
+{"state": {"context": ["R : Type u_1", "Ring R", "I : Type u_3", "Fintype I", "e : I → R"], "goal": "CompleteOrthogonalIdempotents e ↔ OrthogonalIdempotents e ∧ ∑ i : I, e i = 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r✝ r' r : α → α → Prop", "h : ∀ (s : Set α), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬r x m", "x : α", "hx : ¬Acc r x", "m : α", "hm : m ∈ {x | ¬Acc r x}", "hm' : ∀ x ∈ {x | ¬Acc r x}, ¬r x m", "y : α", "hy : r y m", "hy' : ¬Acc r y"], "goal": "False"}}
+{"state": {"context": ["Fq : Type u_1", "inst✝¹ : Fintype Fq", "inst✝ : Field Fq", "b : Fq[X]", "hb : b ≠ 0", "ε : ℝ", "hε : 0 < ε", "A : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ → Fq[X]", "hbε : 0 < cardPowDegree b • ε", "one_lt_q : 1 < Fintype.card Fq", "one_lt_q' : 1 < ↑(Fintype.card Fq)", "q_pos : 0 < Fintype.card Fq", "q_pos' : 0 < ↑(Fintype.card Fq)", "le_b : ⌈-log ε / log ↑(Fintype.card Fq)⌉₊ < b.natDegree", "i₀ i₁ : Fin (Fintype.card Fq ^ ⌈-log ε / log ↑(Fintype.card Fq)⌉₊).succ", "i_ne : i₀ ≠ i₁", "deg_lt : (A i₁ % b - A i₀ % b).degree < ↑(b.natDegree - ⌈-log ε / log ↑(Fintype.card Fq)⌉₊)", "h : ¬A i₁ % b = A i₀ % b", "h' : A i₁ % b - A i₀ % b ≠ 0"], "goal": "↑(A i₁ % b - A i₀ % b).natDegree < ↑?m.44242"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedField α", "inst✝ : FloorRing α", "x : α"], "goal": "x - 1 / 2 < ↑(round x)"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝¹³ : CommSemiring R✝", "A✝ : Type u_2", "B✝ : Type u_3", "inst✝¹² : Semiring A✝", "inst✝¹¹ : Semiring B✝", "inst✝¹⁰ : Algebra R✝ A✝", "inst✝⁹ : Algebra R✝ B✝", "R : Type u_4", "inst✝⁸ : CommRing R", "A : Type u_5", "B : Type u_6", "C : Type u_7", "D : Type u_8", "inst✝⁷ : Ring A", "inst✝⁶ : Ring B", "inst✝⁵ : Ring C", "inst✝⁴ : Ring D", "inst✝³ : Algebra R A", "inst✝² : Algebra R B", "inst✝¹ : Algebra R C", "inst✝ : Algebra R D", "f : A →ₐ[R] B", "g : C →ₐ[R] D", "hf : Function.Surjective ⇑f", "hg : Function.Surjective ⇑g", "this : map f g = (map f (AlgHom.id R D)).comp (map (AlgHom.id R A) g)"], "goal": "Ideal.comap (↑(map (AlgHom.id R A) g)) (RingHom.ker (map f (AlgHom.id R D))) = Ideal.comap (map (AlgHom.id R A) g) (Ideal.map (map (AlgHom.id R A) g) (Ideal.map includeLeft (RingHom.ker f)))"}}
+{"state": {"context": ["G : Type uG", "E' : Type uE'", "NormedAddCommGroup E'", "g : G → E'", "MeasurableSpace G", "μ : MeasureTheory.Measure G", "SeminormedAddCommGroup G", "BorelSpace G", "SecondCountableTopology G", "μ.IsAddLeftInvariant", "MeasureTheory.SFinite μ", "NormedSpace ℝ E'", "CompleteSpace E'", "f : G → ℝ", "x₀ : G", "R ε : ℝ", "z₀ : E'", "hε : 0 ≤ ε", "hf : Function.support f ⊆ Metric.ball 0 R", "hnf : ∀ (x : G), 0 ≤ f x", "hintf : ∫ (x : G), f x ∂μ = 1", "hmg : MeasureTheory.AEStronglyMeasurable g μ", "hg : ∀ x ∈ Metric.ball x₀ R, dist (g x) z₀ ≤ ε"], "goal": "dist (MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) z₀ ≤ ε"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v}"], "goal": "Ordinal.lsub f ≤ Order.succ (Ordinal.sup f)"}}
+{"state": {"context": ["α : Type u", "PartialOrder α", "OrderBot α", "a : α"], "goal": "¬⊥ < a ↔ a = ⊥"}}
+{"state": {"context": ["G : Type u", "inst✝ : Group G", "H : Subgroup G", "g : G", "q : G ⧸ H"], "goal": "↑(H.transferFunction g q) = q"}}
+{"state": {"context": ["α : Type u_1", "S T : Set (Set α)", "hST : S ⊆ T"], "goal": "generatePiSystem S ⊆ generatePiSystem T"}}
+{"state": {"context": ["J : Type v'", "inst✝⁴ : Category.{u', v'} J", "C : Type u", "inst✝³ : Category.{v, u} C", "K✝ : Type u_1", "inst✝² : Category.{?u.138641, u_1} K✝", "D : Type u_2", "inst✝¹ : Category.{?u.138648, u_2} D", "K : Type u_3", "inst✝ : Category.{u_4, u_3} K", "e : J ≌ K", "F : K ⥤ C", "c : Cocone F", "hc : IsVanKampenColimit c", "F' : J ⥤ C", "c' : Cocone F'", "α : F' ⟶ e.functor ⋙ F", "f : c'.pt ⟶ (Cocone.whisker e.functor c).pt", "e' : α ≫ (Cocone.whisker e.functor c).ι = c'.ι ≫ (Functor.const J).map f", "hα : NatTrans.Equifibered α"], "goal": "Nonempty (IsColimit c') ↔ ∀ (j : J), IsPullback (c'.ι.app j) (α.app j) f ((Cocone.whisker e.functor c).ι.app j)"}}
+{"state": {"context": ["ι : Type u_1", "N : Type u_5", "OrderedAddCommMonoid N", "s : Finset ι", "f : ι → N", "n : N", "h : ∀ x ∈ s, n ≤ f x"], "goal": "s.card • n ≤ s.sum f"}}
+{"state": {"context": ["α : Type u_1", "PartialOrder α", "a b : α"], "goal": "Set.Iic a = Set.Iic b ↔ a = b"}}
+{"state": {"context": ["a b : Prop"], "goal": "(a ↔ b) ↔ (a → b) ∧ (b → a)"}}
+{"state": {"context": ["α : Type u", "β : Type u_1", "l₁ l₂ : List α", "h : l₁ ~r l₂", "f : α → β"], "goal": "List.map f l₁ ~r List.map f l₂"}}
+{"state": {"context": ["r : ℕ+", "H : 1 < ADEInequality.sumInv {2, 3, r}"], "goal": "r < 6"}}
+{"state": {"context": ["α : Type u_1", "a : Array α", "i j : Fin a.size", "k : Nat", "hk' : k < (a.swap i j).size"], "goal": "(a.swap i j)[k] = if k = ↑i then a[j] else if k = ↑j then a[i] else a[k]"}}
+{"state": {"context": ["α : Type u_1", "s t : Set α", "Ixx : α → α → Set α"], "goal": "Filter.TendstoIxxClass Ixx (𝓟 s) (𝓟 t) ↔ ∀ x ∈ s, ∀ y ∈ s, Ixx x y ⊆ t"}}
+{"state": {"context": ["R : Type u", "σ : Type u_1", "a : R", "s : σ →₀ ℕ", "CommSemiring R", "f : σ → R"], "goal": "(MvPolynomial.eval f) ((MvPolynomial.monomial s) a) = a * s.prod fun n e => f n ^ e"}}
+{"state": {"context": ["α : Type u_1", "M : Type u_8", "N : Type u_10", "P : Type u_11", "AddCommMonoid M", "AddCommMonoid N", "AddCommMonoid P", "g : N →+ P", "f : α → M →+ N"], "goal": "g.comp (Finsupp.liftAddHom f) = Finsupp.liftAddHom fun a => g.comp (f a)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "S : Type u_2", "inst✝⁴ : CommRing S", "f : R →+* S", "I J : Ideal S", "inst✝³ : Algebra R S", "inst✝² : IsDomain S", "inst✝¹ : Algebra.IsIntegral R S", "P : Ideal R", "inst✝ : P.IsPrime", "hP : RingHom.ker (algebraMap R S) ≤ P", "hP0 : 0 ∉ Algebra.algebraMapSubmonoid S P.primeCompl", "Rₚ : Type u_1 := Localization P.primeCompl", "Sₚ : Type u_2 := Localization (Algebra.algebraMapSubmonoid S P.primeCompl)", "this : IsDomain (Localization (Algebra.algebraMapSubmonoid S P.primeCompl)) := IsLocalization.isDomain_localization (le_nonZeroDivisors_of_noZeroDivisors hP0)"], "goal": "∃ Q, Q.IsPrime ∧ comap (algebraMap R S) Q = P"}}
+{"state": {"context": ["R : Type u_2", "M : Type u_3", "Semiring R", "S : Type u_4", "Monoid S", "AddCommMonoid M", "Module R M", "DistribMulAction S M", "N : Submodule R M", "s t : Set S"], "goal": "(s ⊔ t) • N = s • N ⊔ t • N"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "CategoryTheory.BraidedCategory C", "M : Mon_ C"], "goal": "(CategoryTheory.tensor_μ C (M.X, 𝟙_ C) (M.X, 𝟙_ C) ≫ (M.mul ⊗ (λ_ (𝟙_ C)).hom)) ≫ (ρ_ M.X).hom =\n  ((ρ_ M.X).hom ⊗ (ρ_ M.X).hom) ≫ M.mul"}}
+{"state": {"context": ["α : Type u_1", "p : Option α → Prop"], "goal": "(∃ x, x ≠ none ∧ p x) ↔ ∃ x, p (some x)"}}
+{"state": {"context": ["G₀ : Type u_3", "GroupWithZero G₀", "p : G₀ → Prop"], "goal": "(∃ u, p ↑u) ↔ ∃ x, x ≠ 0 ∧ p x"}}
+{"state": {"context": ["E : Type u_6", "SeminormedAddGroup E", "a b : E", "r : ℝ"], "goal": "b ∈ Metric.closedBall a r ↔ ‖b - a‖ ≤ r"}}
+{"state": {"context": ["M : Type u_12", "N : Type u_13", "P : Type u_14", "Q : Type u_15", "Mul P", "Mul Q", "f : M ≃ N", "g : P ≃* Q", "h : M → P", "n : N"], "goal": "(MulEquiv.arrowCongr f g) h n = g (h (f.symm n))"}}
+{"state": {"context": ["a b c d m n : ℤ", "hmn : m.natAbs ∣ n.natAbs", "hnm : n.natAbs < m.natAbs"], "goal": "n.natAbs = 0"}}
+{"state": {"context": ["G : Type u_3", "H : Type u_4", "Monoid G", "a : G", "n : ℕ", "MulAction G H", "b : H"], "goal": "(fun x => a • x)^[n] b = a ^ n • b"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "inst✝² : Fintype V", "inst✝¹ : DecidableRel G.Adj", "inst✝ : Fintype (Sym2 V)", "this : DecidableEq V"], "goal": "Fintype.card G.Dart = ∑ v : V, G.degree v"}}
+{"state": {"context": ["A : Type u_1", "α : Type u_2", "AddMonoid A", "SMul α A", "Zero α", "self : DivisibleBy A α", "n : α", "a : A"], "goal": "n ≠ 0 → n • DivisibleBy.div a n = a"}}
+{"state": {"context": ["u : ℕ → ℝ", "h : Subadditive u", "hbdd : BddBelow (range fun n => u n / ↑n)"], "goal": "Tendsto (fun n => u n / ↑n) atTop (𝓝 h.lim)"}}
+{"state": {"context": ["M : Type u_6", "N : Type u_7", "Add M", "Add N", "α : Type u_12", "e : M ≃+ N", "f : N → α", "g : M → α"], "goal": "g ∘ ⇑e.symm = f ↔ g = f ∘ ⇑e"}}
+{"state": {"context": ["a : ℤ"], "goal": "(↑a)⁻¹.den = if a = 0 then 1 else a.natAbs"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "a b : R", "n : ℕ", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "p✝ q✝ : R[X]", "inst✝ : Infinite R", "p q : R[X]", "ext : ∀ (r : R), eval r p = eval r q", "x : R"], "goal": "eval x (p - q) = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p : PMF α", "f✝ : (a : α) → a ∈ p.support → PMF β", "a : α", "f : (a' : α) → a' ∈ (pure a).support → PMF β", "b : β"], "goal": "(∑' (a_1 : α), (if a_1 = a then 1 else 0) * if h : (if a_1 = a then 1 else 0) = 0 then 0 else (f a_1 h) b) = (f a ⋯) b"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.MonoidalCategory C", "X Y : C", "Z Z' : C", "f : Z ⟶ Z'"], "goal": "CategoryTheory.CategoryStruct.comp\n    (CategoryTheory.MonoidalCategory.whiskerLeft (CategoryTheory.MonoidalCategory.tensorObj X Y) f)\n    (CategoryTheory.MonoidalCategory.associator X Y Z').hom =\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.MonoidalCategory.associator X Y Z).hom\n    (CategoryTheory.MonoidalCategory.whiskerLeft X (CategoryTheory.MonoidalCategory.whiskerLeft Y f))"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "α : Type u_3", "β✝ : Type u_4", "inst✝⁶ : SMul M α", "inst✝⁵ : SMul M β✝", "inst✝⁴ : SMul N α", "inst✝³ : SMul N β✝", "a✝ : M", "x : α × β✝", "β : Type u_5", "inst✝² : Monoid M", "inst✝¹ : AddMonoid β", "inst✝ : DistribMulAction M β", "a : M", "b : α"], "goal": "a • (b, 0) = (a • b, 0)"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁵ : MeasurableSpace α", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "f : α → E", "g : α → F", "a b : α", "μ : Measure α", "l : Filter α", "inst✝² : TopologicalSpace α", "inst✝¹ : SecondCountableTopology α", "inst✝ : (cocompact α).IsMeasurablyGenerated", "hf : LocallyIntegrable f μ", "ho : f =O[cocompact α] g", "hg : IntegrableAtFilter g (cocompact α) μ"], "goal": "Integrable f μ"}}
+{"state": {"context": ["ι : Type u_1", "inst✝³ : LinearOrder ι", "inst✝² : SuccOrder ι", "inst✝¹ : IsSuccArchimedean ι", "inst✝ : PredOrder ι", "i0 i✝ i : ι", "hi : i < i0"], "goal": "pred^[Nat.find ⋯] i0 = i"}}
+{"state": {"context": ["a b : Int"], "goal": "-a % b = (b - a) % b"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "ι : Type v", "v : Basis ι R M"], "goal": "#ι = Module.rank R M"}}
+{"state": {"context": ["α : Type u_1", "π : α → Type u_11", "i : Set α", "s t : (a : α) → Set (π a)"], "goal": "i.pi s \\ i.pi t ⊆ ⋃ a ∈ i, Function.eval a ⁻¹' (s a \\ t a)"}}
+{"state": {"context": ["R : Type u_1", "CommSemiring R", "M : Submonoid R"], "goal": "IsLocalization.toLocalizationMap M (Localization M) = Localization.monoidOf M"}}
+{"state": {"context": ["a x y : ℂ", "hx : x ≠ 0", "hy : y ≠ 0", "this✝ : abs x ≠ 0", "this : abs y ≠ 0"], "goal": "(x.re * y.re + x.im * y.im) * ((abs y)⁻¹ * (abs x)⁻¹) = (x.re * (y.re * (abs y ^ 2)⁻¹) + x.im * (y.im * (abs y ^ 2)⁻¹)) * (abs y * (abs x)⁻¹)"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "X Y : C", "h : CategoryTheory.Epi 0"], "goal": "(CategoryTheory.Limits.isoZeroOfEpiZero h).hom = 0"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "S : CategoryTheory.ShortComplex C", "A : C", "k : A ⟶ S.X₂", "S.HasLeftHomology", "x : A ⟶ S.X₁", "hx : k = CategoryTheory.CategoryStruct.comp x S.f"], "goal": "CategoryTheory.CategoryStruct.comp (S.liftCycles k ⋯) S.leftHomologyπ = 0"}}
+{"state": {"context": ["σ : Type u_1", "R : Type u_2", "inst✝ : Semiring R", "m✝ n✝ : σ →₀ ℕ", "φ ψ : MvPowerSeries σ R", "a : R", "n : σ →₀ ℕ", "h : ∀ (m : σ →₀ ℕ), Commute ((coeff R m) φ) a", "m : σ →₀ ℕ"], "goal": "(coeff R m) (φ * (monomial R n) a) = (coeff R m) ((monomial R n) a * φ)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "Group α", "Group β", "f : α → β", "hf : IsGroupHom f"], "goal": "f 1 = 1"}}
+{"state": {"context": ["m n : Nat", "a : Nat", "h : m ≤ n"], "goal": "a ^ m ∣ a ^ n"}}
+{"state": {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "inst✝ : Fact (finrank ℝ E = 2)", "o : Orientation ℝ E (Fin 2)", "x y : E"], "goal": "(o.areaForm x) (o.rightAngleRotation y) = ⟪x, y⟫_ℝ"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : PseudoEMetricSpace α", "δ : ℝ", "δ_pos : 0 < δ", "E : Set α", "x : α", "x_out : x ∉ thickening δ E"], "goal": "(thickenedIndicator δ_pos E) x = 0"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "R : Type u_3", "R₂ : Type u_4", "K : Type u_5", "M : Type u_6", "M' : Type u_7", "M'' : Type u_8", "V : Type u", "V' : Type u_9", "inst✝⁸ : Semiring R", "inst✝⁷ : AddCommMonoid M", "inst✝⁶ : Module R M", "inst✝⁵ : AddCommMonoid M'", "inst✝⁴ : Module R M'", "S : Type u_10", "inst✝³ : Semiring S", "inst✝² : Module S M'", "inst✝¹ : SMulCommClass R S M'", "inst✝ : Fintype ι", "b : Basis ι R M", "f : ι → M'", "x : M"], "goal": "((b.constr S) f) x = ∑ i : ι, b.equivFun x i • f i"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a✝ b : R", "n : ℕ", "inst✝ : Ring R", "p✝ q p : R[X]", "a : R", "this : ((X - C a) ^ rootMultiplicity a p).Monic"], "goal": "(X - C a) ^ rootMultiplicity a p * (p /ₘ (X - C a) ^ rootMultiplicity a p) = 0 + (X - C a) ^ rootMultiplicity a p * (p /ₘ (X - C a) ^ rootMultiplicity a p)"}}
+{"state": {"context": ["m : ℕ", "inst✝ : NeZero m", "f : ℕ → ℝ", "hf : Antitone f", "k : ZMod m", "H : Summable ({n | ↑n = k}.indicator f)"], "goal": "Summable (∑ a : ZMod m, {n | ↑n = a}.indicator f)"}}
+{"state": {"context": ["𝕂 : Type u_1", "𝔸 : Type u_2", "RCLike 𝕂", "NormedRing 𝔸", "NormedAlgebra 𝕂 𝔸", "CompleteSpace 𝔸"], "goal": "HasStrictFDerivAt (NormedSpace.exp 𝕂) 1 0"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹² : CommSemiring R", "S : Submonoid R", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : AddCommMonoid M'", "inst✝⁹ : AddCommMonoid M''", "A : Type u_5", "inst✝⁸ : CommSemiring A", "inst✝⁷ : Algebra R A", "inst✝⁶ : Module A M'", "inst✝⁵ : IsLocalization S A", "inst✝⁴ : Module R M", "inst✝³ : Module R M'", "inst✝² : Module R M''", "inst✝¹ : IsScalarTower R A M'", "f : M →ₗ[R] M'", "g : M →ₗ[R] M''", "inst✝ : IsLocalizedModule S f", "m : M'"], "goal": "(iso S f).symm m = LocalizedModule.mk ⋯.choose.1 ⋯.choose.2"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Mul α", "inst✝² : PartialOrder α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt", "a b c d : α", "this : ∀ (M : Type ?u.3042) (N : Type ?u.3041) (μ : M → N → N) [inst : PartialOrder N] [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1", "y : α", "hya : a ≤ y", "hyb : y ≤ b", "z : α", "hzc : c ≤ z", "hzd : z < d"], "goal": "(fun x x_1 => x * x_1) y z ∈ Ico (a * c) (b * d)"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "DecidableEq V", "u v w : V", "p : G.Walk v w", "h : u ∈ p.support"], "goal": "(p.dropUntil u h).length ≤ p.length"}}
+{"state": {"context": ["a : ℝ", "ha : 0 ≤ a"], "goal": "↑⟨a, ha⟩ = a"}}
+{"state": {"context": ["k G : Type u", "CommRing k", "Group G", "A : Rep k G", "f : G × G → CoeSort.coe A", "g : G × G × G"], "goal": "(groupCohomology.dTwo A) f g =\n  (A.ρ g.1) (f (g.2.1, g.2.2)) - f (g.1 * g.2.1, g.2.2) + f (g.1, g.2.1 * g.2.2) - f (g.1, g.2.1)"}}
+{"state": {"context": ["f✝ f : ℝ → ℝ", "hf✝ : GrowsPolynomially f", "b : ℝ", "hb : b ∈ Set.Ioo 0 1", "c₁ : ℝ", "hc₁_mem : c₁ > 0", "c₂ : ℝ", "hc₂_mem : c₂ > 0", "hf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)", "x : ℝ", "hx : ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)", "u : ℝ", "hu : u ∈ Set.Icc (b * x) x"], "goal": "f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁶ : NontriviallyNormedField 𝕜", "F : Type u_2", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "R : Type u_3", "inst✝³ : Semiring R", "inst✝² : Module R F", "inst✝¹ : SMulCommClass 𝕜 R F", "inst✝ : ContinuousConstSMul R F", "x : 𝕜", "s : Set 𝕜", "hx : x ∈ s", "h : UniqueDiffOn 𝕜 s", "f g : 𝕜 → F", "c : F", "n : ℕ", "hn : 0 < n.succ", "y : 𝕜", "hy : y ∈ s", "this : UniqueDiffWithinAt 𝕜 s y"], "goal": "derivWithin (fun z => c - f z) s y = -derivWithin f s y"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_4", "CommSemiring R", "i : M", "r : R"], "goal": "Finsupp.single i r ∈ AddMonoidAlgebra.grade R i"}}
+{"state": {"context": ["p : ℕ → Prop", "hf : (setOf p).Infinite", "n : ℕ"], "goal": "Nat.nth p n = ↑((Nat.Subtype.orderIsoOfNat (setOf p)) n)"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹³ : Category.{u_5, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝¹² : Category.{u_6, u_2} A", "inst✝¹¹ : HasWeakSheafify J A", "inst✝¹⁰ : (constantSheaf J A).Faithful", "inst✝⁹ : (constantSheaf J A).Full", "t : C", "ht : IsTerminal t", "B : Type u_3", "inst✝⁸ : Category.{u_4, u_3} B", "U : A ⥤ B", "inst✝⁷ : HasWeakSheafify J A", "inst✝⁶ : HasWeakSheafify J B", "inst✝⁵ : (constantSheaf J A).Faithful", "inst✝⁴ : (constantSheaf J A).Full", "inst✝³ : (constantSheaf J B).Faithful", "inst✝² : (constantSheaf J B).Full", "inst✝¹ : J.PreservesSheafification U", "inst✝ : J.HasSheafCompose U", "F : Sheaf J A"], "goal": "((constantSheafAdj J B ht).counit.app ((sheafCompose J U).obj F)).val = ((sheafifyComposeIso J U ((const Cᵒᵖ).obj (F.val.obj (op t)))).symm ≪≫ (presheafToSheaf J B ⋙ sheafToPresheaf J B).mapIso (constComp Cᵒᵖ (F.val.obj (op t)) U)).inv ≫ ((sheafCompose J U).map ((constantSheafAdj J A ht).counit.app F)).val"}}
+{"state": {"context": ["R : Type u", "X : Type v", "Semiring R", "A : Type w", "NonUnitalNonAssocSemiring A", "Module R A", "IsScalarTower R A A", "SMulCommClass R A A", "f : X → A", "x : X"], "goal": "((FreeNonUnitalNonAssocAlgebra.lift R) f) (FreeNonUnitalNonAssocAlgebra.of R x) = f x"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "L₂ : Type w", "LieRing L₂", "LieAlgebra R L₂", "f : L →ₗ⁅R⁆ L₂", "h : Function.Injective ⇑f", "x : L"], "goal": "(f.equivRangeOfInjective h) x = ⟨f x, ⋯⟩"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "a : M", "x y : Mˣ", "h : SemiconjBy a ↑x ↑y"], "goal": "SemiconjBy a ↑x⁻¹ ↑y⁻¹"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "x : 𝕜", "𝕜' : Type u_2", "NormedField 𝕜'", "NormedAlgebra 𝕜 𝕜'", "u : 𝕜 → 𝕜'", "v : 𝕜'"], "goal": "deriv (fun y => u y * v) x = deriv u x * v"}}
+{"state": {"context": ["R : Type u", "Semiring R", "Nontrivial R", "n : ℕ", "r : R"], "goal": "((Polynomial.X + Polynomial.C r) ^ n).natDegree = n"}}
+{"state": {"context": ["a : UnitAddCircle"], "goal": "Tendsto (fun s => s * completedHurwitzZetaEven a s) (𝓝[≠] 0) (𝓝 (if a = 0 then -1 else 0))"}}
+{"state": {"context": ["α : Type u_1", "ι : Type u_2", "MeasurableSpace α", "μ : MeasureTheory.Measure α", "l : Filter ι", "l.IsCountablyGenerated", "φ : ι → Set α", "hφ : MeasureTheory.AECover μ l φ", "f : α → ℝ≥0∞", "hfm : AEMeasurable f μ"], "goal": "Filter.Tendsto (fun i => ∫⁻ (x : α) in φ i, f x ∂μ) l (𝓝 (∫⁻ (x : α), f x ∂μ))"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "Semifield R", "StarRing R", "MetricSpace R", "TopologicalSemiring R", "ContinuousStar R", "HasContinuousInv₀ R", "TopologicalSpace A", "Ring A", "StarRing A", "Algebra R A", "ContinuousFunctionalCalculus R p", "a : Aˣ", "n : ℤ", "ha : p ↑a := by cfc_tac"], "goal": "cfc (fun x => x ^ n) ↑a = ↑(a ^ n)"}}
+{"state": {"context": ["x n : ℂ", "hx : x.arg ≠ π"], "goal": "(starRingEnd ℂ) x ^ n = (starRingEnd ℂ) (x ^ (starRingEnd ℂ) n)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "B A : C", "X : CategoryTheory.Subobject B", "f : A ⟶ B", "CategoryTheory.Mono f", "g : A ⟶ CategoryTheory.Subobject.underlying.obj X", "w : CategoryTheory.CategoryStruct.comp g X.arrow = f"], "goal": "CategoryTheory.Subobject.mk f ≤ X"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "RCLike 𝕜", "E : Type u_3", "NormedAddCommGroup E", "InnerProductSpace 𝕜 E", "b : HilbertBasis ι 𝕜 E", "i : ι"], "goal": "b.repr ((fun i => b.repr.symm (lp.single 2 i 1)) i) = lp.single 2 i 1"}}
+{"state": {"context": ["ι : Type u", "γ : Type w", "β✝ : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝² : DecidableEq ι", "inst✝¹ : (i : ι) → Zero (β✝ i)", "s : Finset ι", "x : (i : ↑↑s) → β✝ ↑i", "i✝ : ι", "β : ι → Type u_1", "inst✝ : (i : ι) → AddGroup (β i)", "f : Π₀ (i : ι), β i", "i j : ι"], "goal": "(erase i f) j = (f - single i (f i)) j"}}
+{"state": {"context": ["Ω : Type u_1", "MeasureTheory.MeasureSpace Ω", "MeasureTheory.IsProbabilityMeasure ℙ", "X : ℕ → Ω → ℝ", "hint : MeasureTheory.Integrable (X 0) ℙ", "hident : ∀ (i : ℕ), ProbabilityTheory.IdentDistrib (X i) (X 0) ℙ ℙ", "hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω"], "goal": "∀ᵐ (ω : Ω),\n  (fun n =>\n      ∑ i ∈ Finset.range n, ProbabilityTheory.truncation (X i) (↑i) ω - ∑ i ∈ Finset.range n, X i ω) =o[Filter.atTop]\n    fun n => ↑n"}}
+{"state": {"context": ["C : Type u₁", "D : Type u₂", "E : Type u₃", "F : Type u₄", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.Category.{v₃, u₃} E", "CategoryTheory.Category.{v₄, u₄} F", "G : CategoryTheory.Functor C E", "H : CategoryTheory.Functor D F", "L₁ : CategoryTheory.Functor C D", "R₁ : CategoryTheory.Functor D C", "L₂ : CategoryTheory.Functor E F", "R₂ : CategoryTheory.Functor F E", "adj₁ : L₁ ⊣ R₁", "adj₂ : L₂ ⊣ R₂", "α : G.comp L₂ ⟶ L₁.comp H", "c : C"], "goal": "CategoryTheory.CategoryStruct.comp (G.map (adj₁.unit.app c)) (((CategoryTheory.mateEquiv adj₁ adj₂) α).app (L₁.obj c)) =\n  CategoryTheory.CategoryStruct.comp (adj₂.unit.app (G.obj ((CategoryTheory.Functor.id C).obj c)))\n    (R₂.map (α.app ((CategoryTheory.Functor.id C).obj c)))"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : DistribLattice α", "inst✝³ : OrderBot α", "inst✝² : Fintype α", "inst✝¹ : DecidablePred SupIrred", "inst✝ : DecidableEq α", "a b : α"], "goal": "Fintype.finsetOrderIsoSet.symm (birkhoffSet a ∩ birkhoffSet b) = Fintype.finsetOrderIsoSet.symm (birkhoffSet a) ∩ Fintype.finsetOrderIsoSet.symm (birkhoffSet b)"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "S : Type u_2", "hp : Fact (Nat.Prime p)", "inst✝² : CommRing R", "inst✝¹ : CommRing S", "inst✝ : CharP R p", "x : 𝕎 R", "n : ℕ"], "goal": "(frobenius x).coeff n = ((map (_root_.frobenius R p)) x).coeff n"}}
+{"state": {"context": ["α : Type u_1", "m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "TopologicalSpace α", "OpensMeasurableSpace α", "MeasureTheory.IsLocallyFiniteMeasure μ", "f : α → ℝ≥0", "hf : Continuous f"], "goal": "MeasureTheory.IsLocallyFiniteMeasure (μ.withDensity fun x => ↑(f x))"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_3", "M : Type u_6", "Semiring R", "AddCommMonoid M", "Module R M", "P : Submodule R M", "b : Basis ι R ↥P", "x : M"], "goal": "x ∈ P ↔ ∃ c, x = c.sum fun i x => x • ↑(b i)"}}
+{"state": {"context": ["c : Cardinal.{u}"], "goal": "Cardinal.continuum < Cardinal.lift.{v, u} c ↔ Cardinal.continuum < c"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "E : Type u₃", "CategoryTheory.Category.{v₃, u₃} E", "F₁ : CategoryTheory.Functor C D", "F₂ : CategoryTheory.Functor D E", "F₁₂ : CategoryTheory.Functor C E", "e : F₁.comp F₂ ≅ F₁₂", "J : CategoryTheory.GrothendieckTopology C", "K : CategoryTheory.GrothendieckTopology D", "L : CategoryTheory.GrothendieckTopology E", "F₁.IsContinuous J K", "F₂.IsContinuous K L"], "goal": "F₁₂.IsContinuous J L"}}
+{"state": {"context": [], "goal": "↑NNReal.pi = Real.pi"}}
+{"state": {"context": ["m : Type u_2", "n : Type u_3", "o : Type u_4", "α : Type v", "NonUnitalNonAssocSemiring α", "Fintype n", "M : Matrix m n α"], "goal": "M * 0 = 0"}}
+{"state": {"context": ["ι : Type u_1", "I : BoxIntegral.Box ι", "π₁ π₂ : BoxIntegral.TaggedPrepartition I", "Fintype ι", "h : Disjoint π₁.iUnion π₂.iUnion"], "goal": "(π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ : P"], "goal": "∡ p₁ p₂ p₃ = ↑π ↔ Sbtw ℝ p₁ p₂ p₃"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "NontriviallyNormedField 𝕜", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "ContinuousSMul 𝕜 E", "RegularSpace E"], "goal": "(𝓝 0).HasBasis (fun s => s ∈ 𝓝 0 ∧ IsClosed s ∧ Balanced 𝕜 s) id"}}
+{"state": {"context": ["c₁ c₂ : Cardinal.{u_1}", "h₁ : ℵ₀ ≤ c₁", "h₂ : 2 ≤ c₂", "h₂' : c₂ ≤ c₁"], "goal": "c₂ ^ c₁ = 2 ^ c₁"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "γ : Type u_4", "mγ : MeasurableSpace γ", "s✝ : Set (β × γ)", "κ : Kernel α β", "inst✝¹ : IsSFiniteKernel κ", "η : Kernel (α × β) γ", "inst✝ : IsSFiniteKernel η", "a✝ : α", "s : Set β", "t : Set γ", "hs : MeasurableSet s", "ht : MeasurableSet t", "a : α", "u : Set (β × γ)", "hu : MeasurableSet u", "this : ∀ (b : β), (η (a, b)) {c | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t} = s.indicator (fun b => (η (a, b)) ({c | (b, c) ∈ u} ∩ t)) b"], "goal": "∫⁻ (b : β) in s, (η (a, b)) ({c | (b, c) ∈ u} ∩ t) ∂κ a = ∫⁻ (b : β), s.indicator (fun b => (η (a, b)) ({c | (b, c) ∈ u} ∩ t)) b ∂κ a"}}
+{"state": {"context": ["α : Type u_1", "IdemSemiring α", "n : ℕ"], "goal": "n ≠ 0 → ∀ (a : α), n • a = a"}}
+{"state": {"context": ["H : Type u", "M : Type u_2", "TopologicalSpace H", "TopologicalSpace M", "ChartedSpace H M", "e e' : PartialHomeomorph M H", "G : StructureGroupoid H", "he : e ∈ atlas H M", "he' : e' ∈ atlas H M", "HasGroupoid M G", "ClosedUnderRestriction G", "s : TopologicalSpace.Opens M", "hs : Nonempty ↥s"], "goal": "(e.subtypeRestr hs).symm ≫ₕ e'.subtypeRestr hs ∈ G"}}
+{"state": {"context": ["α : Type u_1", "σ : Type u_2", "inst✝¹ : Primcodable α", "inst✝ : Primcodable σ", "f : α → Bool", "hf : Computable f", "hp : ComputablePred fun a => f a = true", "n : α", "a : Unit"], "goal": "(a ∈ bif f n then Part.some () else Part.none) ↔ a ∈ Part.assert (f n = true) fun x => Part.some ()"}}
+{"state": {"context": ["ι : Type u", "β : ι → Type v", "DecidableEq ι", "(i : ι) → Zero (β i)", "(i : ι) → (x : β i) → Decidable (x ≠ 0)", "i : ι", "b : β i", "hb : b ≠ 0"], "goal": "(DFinsupp.single i b).support = {i}"}}
+{"state": {"context": ["C : Type u", "inst✝ : Category.{v, u} C", "D : GlueData C", "i j k : D.J", "S : Set ↑↑(D.V (i, j))", "eq₁ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.fst ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.fst (D.f i j) (D.f i k)).base", "eq₂ : (PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom ≫ pullback.snd ((forget C).map (D.f i j)) ((forget C).map (D.f i k)) = (pullback.snd (D.f i j) (D.f i k)).base"], "goal": "Function.Surjective ⇑(PreservesPullback.iso (forget C) (D.f i j) (D.f i k)).hom"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "Zero α", "Preorder α", "Preorder β", "Preorder γ", "SMul α β", "SMul α γ", "f : β → γ", "PosSMulStrictMono α γ", "hf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂", "smul : ∀ (a : α) (b : β), f (a • b) = a • f b"], "goal": "PosSMulStrictMono α β"}}
+{"state": {"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : Abelian C", "S : ShortComplex (CochainComplex C ℤ)", "hS : S.ShortExact", "n₀ n₁ : ℤ", "h : n₀ + 1 = n₁", "A : C", "x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)", "A' : C", "π : A' ⟶ A", "w✝ : Epi π", "x' : A' ⟶ (mappingCone S.f).X n₀", "w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0", "hx' : π ≫ x = (mappingCone S.f).liftCycles x' n₁ ⋯ w ≫ (mappingCone S.f).homologyπ n₀"], "goal": "x ≫ (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ ≫ 𝟙 ((HomologicalComplex.homologyFunctor C (up ℤ) n₁).obj S.X₁) = x ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h"}}
+{"state": {"context": ["α : Type u", "DecidableEq α", "Fintype α", "x y : α"], "goal": "Equiv.Perm.sign (Equiv.swap x y) = if x = y then 1 else -1"}}
+{"state": {"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "f g : PadicSeq p", "hfgne : f.norm ≠ g.norm", "hfg : ¬f + g ≈ 0", "hf : f ≈ 0", "this : (f - 0).LimZero"], "goal": "(f + g).norm = max f.norm g.norm"}}
+{"state": {"context": ["R : Type uR", "A : Type uA", "CommSemiring R", "Semiring A", "Algebra R A", "s : Set A", "x : A", "hx : x ∈ Algebra.adjoin R s", "p : (x : A) → x ∈ Algebra.adjoin R s → Prop", "mem : ∀ (x : A) (h : x ∈ s), p x ⋯", "algebraMap : ∀ (r : R), p ((algebraMap R A) r) ⋯", "add : ∀ (x : A) (hx : x ∈ Algebra.adjoin R s) (y : A) (hy : y ∈ Algebra.adjoin R s), p x hx → p y hy → p (x + y) ⋯", "mul : ∀ (x : A) (hx : x ∈ Algebra.adjoin R s) (y : A) (hy : y ∈ Algebra.adjoin R s), p x hx → p y hy → p (x * y) ⋯"], "goal": "p x hx"}}
+{"state": {"context": ["α : Type u", "β : Type v", "γ : Type w", "s : Seq α", "n : ℕ"], "goal": "((↑s).get? n).get = ↑s n"}}
+{"state": {"context": ["n : ℕ"], "goal": "∏ x ∈ Finset.range n, (x + 1).factorial = n.superFactorial"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedRing α", "FloorRing α", "a : α"], "goal": "Int.fract (Int.fract a) = Int.fract a"}}
+{"state": {"context": ["R : Type u_2", "V : Type u_3", "W : Type u_4", "P : Type u_6", "Q : Type u_7", "NormedAddCommGroup V", "MetricSpace P", "NormedAddTorsor V P", "NormedAddCommGroup W", "MetricSpace Q", "NormedAddTorsor W Q", "NormedField R", "NormedSpace R V", "NormedSpace R W", "f : P →ᴬ[R] Q", "p₁ p₂ : P"], "goal": "f.contLinear (p₁ -ᵥ p₂) = f p₁ -ᵥ f p₂"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "t : TopologicalSpace β", "f : α → β", "s : Set α"], "goal": "IsOpen s ↔ s ∈ Set.preimage f '' {s | IsOpen s}"}}
+{"state": {"context": ["α : Type u_1", "LE α", "MulOneClass α", "i : IsSymmOp α α fun x x_1 => x * x_1", "CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1", "a b : α", "ha : MulLECancellable a"], "goal": "b * a ≤ a ↔ b ≤ 1"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "rω : cs.IsReduced ω"], "goal": "(cs.rightInvSeq ω.reverse).Nodup"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "K L L' M : Set V", "Gc : G.Preconnected", "hK : K.Nonempty", "v : V", "vnK : v ∉ K"], "goal": "∃ ck, ck.1 ∈ G.componentComplMk vnK ∧ ck.2 ∈ K ∧ G.Adj ck.1 ck.2"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "Preorder α", "Preorder β", "e : α ≃o β"], "goal": "Filter.Tendsto (⇑e) Filter.atBot Filter.atBot"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "σ : Type u_3", "τ : Type u_4", "a : Set α", "σ✝ : Type u_5", "F✝ : CFilter (Set α) σ✝"], "goal": "a ∈ F✝.toFilter ↔ ∃ b, { σ := σ✝, F := F✝, eq := ⋯ }.F.f b ⊆ a"}}
+{"state": {"context": ["α : Type u", "Preorder α", "WellFoundedLT α"], "goal": "StrictMono ⇑toWellOrderExtension"}}
+{"state": {"context": ["k : Type u_1", "V : Type u_2", "P : Type u_3", "inst✝⁴ : DivisionRing k", "inst✝³ : AddCommGroup V", "inst✝² : Module k V", "inst✝¹ : AffineSpace V P", "ι : Type u_4", "s : Finset ι", "ι₂ : Type u_5", "s₂ : Finset ι₂", "inst✝ : CharZero k", "h : s.card ≠ 0"], "goal": "↑s.card ≠ 0"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "A : Type z", "a b : R", "n : ℕ", "inst✝ : CommRing R", "p q : R[X]"], "goal": "X - C a ∣ p - C (eval a p)"}}
+{"state": {"context": ["ι : Type u", "f : ι → Ordinal.{max u v}", "H : ∀ (i : ι), f i < Ordinal.sup f"], "goal": "(Ordinal.sup f).cof ≤ Cardinal.lift.{v, u} #ι"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Ring R", "AddCommGroup M", "Module R M", "StrongRankCondition R", "Module.Finite R M", "ι : Type w", "h : Basis ι R M"], "goal": "↑(FiniteDimensional.finrank R M) = #ι"}}
+{"state": {"context": ["α : Type u_1", "s : Finset α", "f : α → ℝ", "hf : ∀ i ∈ s, 0 ≤ f i"], "goal": "ENNReal.ofReal (∏ i ∈ s, f i) = ∏ i ∈ s, ENNReal.ofReal (f i)"}}
+{"state": {"context": ["α : Type u", "self : UniformSpace.Core α"], "goal": "(self.uniformity.lift' fun s => s ○ s) ≤ self.uniformity"}}
+{"state": {"context": ["x y z : ℤ", "h : PythagoreanTriple x y z"], "goal": "↑(x.gcd y) ∣ z"}}
+{"state": {"context": ["𝕜 : Type u_1", "A : Type u_2", "inst✝⁹ : RCLike 𝕜", "p : A → Prop", "inst✝⁸ : PartialOrder A", "inst✝⁷ : NormedRing A", "inst✝⁶ : StarRing A", "inst✝⁵ : StarOrderedRing A", "inst✝⁴ : TopologicalRing A", "inst✝³ : NormedAlgebra 𝕜 A", "inst✝² : CompleteSpace A", "inst✝¹ : ContinuousFunctionalCalculus 𝕜 p", "inst✝ : UniqueContinuousFunctionalCalculus 𝕜 A", "a : A", "ha : autoParam (p a) _auto✝", "h : Continuous ⇑(cfcHom ⋯)", "x✝ : ContinuousOn (exp 𝕜) (spectrum 𝕜 a)", "a✝ : ↑(spectrum 𝕜 a)"], "goal": "{ toFun := (spectrum 𝕜 a).restrict (exp 𝕜), continuous_toFun := ⋯ } a✝ = (exp 𝕜 { toFun := (spectrum 𝕜 a).restrict id, continuous_toFun := ⋯ }) a✝"}}
+{"state": {"context": ["s : ℂ"], "goal": "LSeriesSummable (fun n => ↑(ArithmeticFunction.moebius n)) s ↔ 1 < s.re"}}
+{"state": {"context": ["𝕜 : Type u", "NontriviallyNormedField 𝕜", "E : Type uE", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type uF", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "x : E", "n : ℕ"], "goal": "iteratedFDeriv 𝕜 (n + 1) f x =\n  (⇑(continuousMultilinearCurryRightEquiv' 𝕜 n E F) ∘ iteratedFDeriv 𝕜 n fun y => fderiv 𝕜 f y) x"}}
+{"state": {"context": ["M : Type u_2", "N : Type u_3", "R : Type u_5", "Semiring R", "AddCommMonoid M", "Module R M", "AddCommMonoid N", "Module R N", "ι : Type u_7", "κ : Type u_8", "e₁ : ι ≃ κ", "e₂ : M ≃ₗ[R] N", "k : κ", "n : N"], "goal": "(Finsupp.lcongr e₁ e₂).symm (Finsupp.single k n) = Finsupp.single (e₁.symm k) (e₂.symm n)"}}
+{"state": {"context": ["M : Type u_1", "A : Type u_2", "B : Type u_3", "inst✝ : MulOneClass M", "ι : Sort u_4", "S : ι → Submonoid M", "C : (x : M) → x ∈ ⨆ i, S i → Prop", "mem : ∀ (i : ι) (x : M) (hxS : x ∈ S i), C x ⋯", "one : C 1 ⋯", "mul : ∀ (x y : M) (hx : x ∈ ⨆ i, S i) (hy : y ∈ ⨆ i, S i), C x hx → C y hy → C (x * y) ⋯", "x : M", "hx : x ∈ ⨆ i, S i"], "goal": "∃ (Hx : x ∈ ⨆ i, S i), C x Hx"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommRing R", "hR : 0 < ringChar R", "φ : AddChar R ℂ", "a : R"], "goal": "(starRingEnd ℂ) (φ a) = (φ a)⁻¹"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : LinearOrderedRing α", "inst✝ : FloorRing α", "x : α", "y : ℕ"], "goal": "round (↑y + x) = ↑y + round x"}}
+{"state": {"context": ["n : ℕ", "a b : Fin n"], "goal": "Fintype.card ↑(Set.Ioo a b) = ↑b - ↑a - 1"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "β : Type u_4", "OrderedSemiring 𝕜", "AddCommMonoid E", "LE β", "SMul 𝕜 E", "s : Set E", "f : E → β", "hs : Convex 𝕜 s", "h : ∀ (r : β), Convex 𝕜 {x | r ≤ f x}"], "goal": "QuasiconcaveOn 𝕜 s f"}}
+{"state": {"context": ["α : Type u", "β : Type v", "PseudoMetricSpace α", "f : β → α", "a : α", "hf : Filter.Tendsto f Filter.cofinite (𝓝 a)"], "goal": "Bornology.IsBounded (Set.range f)"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "AddZeroClass M", "AddZeroClass N", "s : Set M", "hs : AddSubmonoid.closure s = ⊤", "f g : M →+ N", "h : Set.EqOn (⇑f) (⇑g) s"], "goal": "f = g"}}
+{"state": {"context": ["ι : Type u'", "ι' : Type u_1", "R : Type u_2", "K : Type u_3", "M : Type u_4", "M' : Type u_5", "M'' : Type u_6", "V : Type u", "V' : Type u_7", "v : ι → M", "inst✝⁷ : Ring R", "inst✝⁶ : AddCommGroup M", "inst✝⁵ : AddCommGroup M'", "inst✝⁴ : AddCommGroup M''", "inst✝³ : Module R M", "inst✝² : Module R M'", "inst✝¹ : Module R M''", "a b : R", "x y : M", "inst✝ : Nontrivial R", "s t u : Set M", "hl : LinearIndependent (ι := { x // x ∈ u }) R Subtype.val", "hsu : s ⊆ u", "htu : t ⊆ u", "hst : span R s ≤ span R t", "this : s ∪ t = t"], "goal": "s ⊆ t"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "x✝ y z : X", "ι : Type u_3", "F : Set X", "x : X", "x_in : x ∈ F", "h : pathComponentIn x F = F"], "goal": "∀ {y : X}, y ∈ F → JoinedIn F x y"}}
+{"state": {"context": ["G : Type u_1", "AddGroup G", "H N : AddSubgroup G", "h : H.normalizer ≤ N"], "goal": "H.normalizer.addSubgroupOf N = (H.addSubgroupOf N).normalizer"}}
+{"state": {"context": ["α : Type u_1", "M✝ : Matroid α", "I B✝ X✝ : Set α", "M : Matroid α", "X I' : Set α", "hI'E : I' ⊆ M.E", "B : Set α", "hB : M.Base B", "hI'B : Disjoint I' B", "hI'X : I' ⊆ X"], "goal": "(maximals (fun x x_1 => x ⊆ x_1) {Y | (fun I => I ⊆ M.E ∧ ∃ B, M.Base B ∧ Disjoint I B) Y ∧ I' ⊆ Y ∧ Y ⊆ X}).Nonempty"}}
+{"state": {"context": ["A : Type u_4", "CommRing A", "IsDomain A", "UniqueFactorizationMonoid A", "K : Type u_5", "Field K", "Algebra A K", "IsFractionRing A K", "x : K"], "goal": "∃ a b, IsRelPrime a ↑b ∧ IsLocalization.mk' K a b = x"}}
+{"state": {"context": ["R : Type u", "L : Type v", "M : Type w", "inst✝⁶ : CommRing R", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra R L", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : LieRingModule L M", "inst✝ : LieModule R L M", "N : LieSubmodule R L M"], "goal": "comap N.incl N = ⊤"}}
+{"state": {"context": ["C : Type u_1", "ι : Type u_2", "inst✝⁵ : Category.{u_3, u_1} C", "inst✝⁴ : Abelian C", "c : ComplexShape ι", "S : ShortComplex (HomologicalComplex C c)", "hS : S.Exact", "inst✝³ : Epi S.g", "i : ι", "inst✝² : S.X₁.HasHomology i", "inst✝¹ : S.X₂.HasHomology i", "inst✝ : S.X₃.HasHomology i", "this : Epi (S.map (eval C c i)).g", "hj : IsColimit (CokernelCofork.ofπ (S.map (eval C c i)).g ⋯)"], "goal": "(ShortComplex.mk (opcyclesMap S.f i) (opcyclesMap S.g i) ⋯).Exact"}}
+{"state": {"context": ["α : Type u", "β : Type v", "s : Set (Filter β)", "m : α → β"], "goal": "Filter.comap m (sSup s) = ⨆ f ∈ s, Filter.comap m f"}}
+{"state": {"context": ["G : Type u_3", "AddGroup G"], "goal": "AddGroup.FG G ↔ AddMonoid.FG G"}}
+{"state": {"context": ["𝒮 : Type u₁", "𝒳 : Type u₂", "CategoryTheory.Category.{v₁, u₂} 𝒳", "CategoryTheory.Category.{v₂, u���} 𝒮", "p : 𝒳 ⥤ 𝒮", "R S : 𝒮", "a b : 𝒳", "f : R ⟶ S", "φ : a ⟶ b", "p.IsHomLift f φ"], "goal": "p.obj a = R"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : CategoryTheory.SimplicialObject C", "n : ℕ", "j : Fin (n + 2)", "i : Fin (n + 1)", "H : j = i.castSucc"], "goal": "X.σ i ≫ X.δ j = 𝟙 (X.obj (Opposite.op (SimplexCategory.mk n)))"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "K L M : CochainComplex C ℤ", "n : ℤ", "γ γ₁ γ₂ : Cochain K L n", "a : ℤ"], "goal": "(shiftAddHom K L n a) 0 = 0"}}
+{"state": {"context": ["n m : ℕ", "p : Fin (n + 1)", "i✝ j i : Fin n"], "goal": "succAbove 0 i = i.succ"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.HasShift C ℤ", "CategoryTheory.Preadditive C", "CategoryTheory.Limits.HasZeroObject C", "∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive", "CategoryTheory.Pretriangulated C", "W : CategoryTheory.MorphismProperty C", "self : W.IsCompatibleWithTriangulation", "T₁ T₂ : CategoryTheory.Pretriangulated.Triangle C"], "goal": "T₁ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →\n  T₂ ∈ CategoryTheory.Pretriangulated.distinguishedTriangles →\n    ∀ (a : T₁.obj₁ ⟶ T₂.obj₁) (b : T₁.obj₂ ⟶ T₂.obj₂),\n      W a →\n        W b →\n          CategoryTheory.CategoryStruct.comp T₁.mor₁ b = CategoryTheory.CategoryStruct.comp a T₂.mor₁ →\n            ∃ c,\n              ∃ (_ : W c),\n                CategoryTheory.CategoryStruct.comp T₁.mor₂ c = CategoryTheory.CategoryStruct.comp b T₂.mor₂ ∧\n                  CategoryTheory.CategoryStruct.comp T₁.mor₃ ((CategoryTheory.shiftFunctor C 1).map a) =\n                    CategoryTheory.CategoryStruct.comp c T₂.mor₃"}}
+{"state": {"context": ["α : Type u_1", "E : Type u_2", "G : Type u_3", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedSpace ℝ E", "inst✝² : NormedAddCommGroup G", "inst✝¹ : NormedSpace ℝ G", "f g : α → E", "m : MeasurableSpace α", "μ : Measure α", "ι : Type u_4", "inst✝ : Countable ι", "F : ι → α → E", "hF_int : ∀ (i : ι), Integrable (F i) μ", "hF_sum : Summable fun i => ∫ (a : α), ‖F i a‖ ∂μ", "hE : CompleteSpace E", "this : ∀ (i : ι), ∫⁻ (a : α), ↑‖F i a‖₊ ∂μ = ↑‖∫ (a : α), ‖F i a‖ ∂μ‖₊"], "goal": "Summable fun x => ‖∫ (a : α), ‖F x a‖ ∂μ‖₊"}}
+{"state": {"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "M₀ : Type u_6", "inst✝⁴ : MulZeroOneClass α", "inst✝³ : MulZeroOneClass β", "inst✝² : MulZeroOneClass γ", "inst✝¹ : MulZeroOneClass δ", "inst✝ : FunLike F α β", "g : β →*₀ γ", "f₁ f₂ : α →*₀ β", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "x : α"], "goal": "g (f₁ x) = g (f₂ x)"}}
+{"state": {"context": ["α : Type u_1", "cut : α → Ordering", "t : Batteries.RBNode α", "path : Batteries.RBNode.Path α"], "goal": "Batteries.RBNode.Path.fill' (Batteries.RBNode.zoom cut t path) = path.fill t"}}
+{"state": {"context": ["S X₁ X₂ : Type u", "f : S ⟶ X₁", "g : S ⟶ X₂", "c : PushoutCocone f g", "hc : IsColimit c", "h₁ : Function.Injective f", "x₁ : X₁", "x₂ : X₂"], "goal": "(Pushout.cocone f g).inl x₁ = (Pushout.cocone f g).inr x₂ ↔ ∃ s, f s = x₁ ∧ g s = x₂"}}
+{"state": {"context": ["f : CircleDeg1Lift", "x : ℝ"], "goal": "f 0 + ↑⌊x⌋ ≤ f x"}}
+{"state": {"context": ["α : Type u", "PseudoEMetricSpace α", "x : α", "s t : Set α", "r : ℝ≥0∞", "h : x ∈ s", "H : EMetric.hausdorffEdist s t < r"], "goal": "∃ y ∈ t, edist x y < r"}}
+{"state": {"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "inst✝⁴ : PartialOrder α", "inst✝³ : PartialOrder β", "inst✝² : OrderBot α", "inst✝¹ : IsAtomic α", "inst✝ : OrderBot β", "l : α → β", "u : β → α", "gi : GaloisInsertion l u", "hbot : u ⊥ = ⊥", "h_atom : ∀ (a : α), IsAtom a → u (l a) = a", "a : α", "hla : IsAtom (l a)", "a' : α", "ha' : IsAtom a'", "hab' : a' ≤ u (l a)", "this : l a' = l a"], "goal": "IsAtom a"}}
+{"state": {"context": ["α : Type u_1", "Fintype α", "DecidableEq α", "σ τ : ↥(alternatingGroup α)", "hc : IsConj ↑σ ↑τ", "hσ : (↑σ).support.card + 2 ≤ Fintype.card α"], "goal": "IsConj σ τ"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝⁸ : NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "G : Type u_5", "H : Type u_6", "inst✝⁷ : NormedAddCommGroup E", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace 𝕜 F", "inst✝³ : NormedAddCommGroup G", "inst✝² : NormedSpace 𝕜 G", "inst✝¹ : NormedAddCommGroup H", "inst✝ : NormedSpace 𝕜 H", "f : E → F", "s : Set E", "x : E", "h : {x} ∈ 𝓝[s] x", "fc : ContinuousWithinAt f s x", "t : Set E", "ot : IsOpen t", "xt : x ∈ t", "st : t ∩ s ⊆ {x}", "r : ℝ", "r0 : r > 0", "rt : Metric.ball x r ⊆ t"], "goal": "AnalyticWithinAt 𝕜 f s x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "AddCommGroup α", "TopologicalSpace α", "TopologicalAddGroup α", "f : β → α", "a : α", "h : HasSum f a"], "goal": "HasSum (fun b => -f b) (-a)"}}
+{"state": {"context": ["α✝ β : Type u", "α : Type u_1", "n : ℕ", "h : ↑n ≤ #α"], "goal": "∃ s, n ≤ s.card"}}
+{"state": {"context": ["X : Type u_1", "TopologicalSpace X", "x y : X"], "goal": "x ∈ pathComponent y ↔ y ∈ pathComponent x"}}
+{"state": {"context": ["x : ℝ", "h : x < 0"], "goal": "Real.arctan x⁻¹ = -(Real.pi / 2) - Real.arctan x"}}
+{"state": {"context": ["R : Type uR", "S : Type uS", "ι : Type uι", "n : ℕ", "M : Fin n.succ → Type v", "M₁ : ι → Type v₁", "M₂ : Type v₂", "M₃ : Type v₃", "M' : Type v'", "inst✝⁸ : CommSemiring R", "inst✝⁷ : (i : ι) → AddCommMonoid (M₁ i)", "inst✝⁶ : (i : Fin n.succ) → AddCommMonoid (M i)", "inst✝⁵ : AddCommMonoid M₂", "inst✝⁴ : (i : Fin n.succ) → Module R (M i)", "inst✝³ : (i : ι) → Module R (M₁ i)", "inst✝² : Module R M₂", "f f' : MultilinearMap R M₁ M₂", "inst✝¹ : DecidableEq ι", "inst✝ : Fintype ι", "m : (i : ι) → M₁ i", "i : ι", "c : R", "x : M₁ i"], "goal": "f (update (c • m) i x) = c ^ (Fintype.card ι - 1) • f (update m i x)"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "M : Type u_3", "H : Type u_4", "E' : Type u_5", "M' : Type u_6", "H' : Type u_7", "inst✝⁹ : NontriviallyNormedField 𝕜", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "inst✝⁶ : TopologicalSpace H", "inst✝⁵ : TopologicalSpace M", "f f' : PartialHomeomorph M H", "I : ModelWithCorners 𝕜 E H", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "inst✝² : TopologicalSpace H'", "inst✝¹ : TopologicalSpace M'", "I' : ModelWithCorners 𝕜 E' H'", "s t : Set M", "inst✝ : I.Boundaryless", "x : M", "hx : x ∈ f.source"], "goal": "map (↑(f.extend I)) (𝓝 x) = 𝓝 (↑(f.extend I) x)"}}
+{"state": {"context": ["α₁ : Type u_1", "α₂ : Type u_2", "β : Type u_3", "LE β", "One β", "v₁ : α₁ → β", "v₂ : α₂ → β"], "goal": "Sum.elim v₁ v₂ ≤ 1 ↔ v₁ ≤ 1 ∧ v₂ ≤ 1"}}
+{"state": {"context": ["α : Type u_1", "Ω : Type u_2", "ι : Type u_3", "_mα : MeasurableSpace α", "s : ι → Set (Set Ω)", "s' : Set (Set Ω)", "_mΩ : MeasurableSpace Ω", "κ : Kernel α Ω", "μ : Measure α", "u : Set ι", "hyp : ∀ n ∈ u, IndepSets (s n) s' κ μ", "t1 t2 : Set Ω", "ht1 : t1 ∈ ⋃ n ∈ u, s n", "ht2 : t2 ∈ s'"], "goal": "∀ᵐ (a : α) ∂μ, (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2"}}
+{"state": {"context": ["J : Type u'", "CategoryTheory.Category.{v', u'} J", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasColimitsOfShape J TopCat", "∀ (X : TopCat), CategoryTheory.Limits.HasLimitsOfShape Jᵒᵖ (TopCat.Presheaf C X)", "F : J ⥤ AlgebraicGeometry.PresheafedSpace C"], "goal": "↑(AlgebraicGeometry.PresheafedSpace.colimit F) =\n  CategoryTheory.Limits.colimit (F ⋙ AlgebraicGeometry.PresheafedSpace.forget C)"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "hα : Archimedean α", "p : α", "hp : 0 < p", "a✝ b✝ c : α", "n : ℤ", "a b : α"], "goal": "toIocMod hp (a + p) b = toIocMod hp a b + p"}}
+{"state": {"context": ["α : Type u_1", "x : α", "l : List α", "n : Nat", "h : n < l.length"], "goal": "l[n] = x ↔ ∃ h, l[n] = x"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝ : MeasurableSpace α", "f : α → α", "s : Set α", "μ : Measure α", "hf : Conservative f μ", "hs : NullMeasurableSet s μ", "h0 : μ s ≠ 0", "t : ℕ → Set α := fun n => s ∩ f^[n] ⁻¹' s", "H : ¬∃ᶠ (m : ℕ) in atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0", "N : ℕ", "hN : μ (t N) ≠ 0", "hmax : ∀ n > N, μ (t n) = 0"], "goal": "False"}}
+{"state": {"context": ["ι : Type u_1", "R : Type u_2", "S : Type u_3", "inst✝⁵ : Ring R", "inst✝⁴ : Ring S", "M : Type u_4", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "m✝ : Submodule R M", "N : Type u_5", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "x✝¹ : Nontrivial M ∧ ∀ (x : M), x ≠ 0 → Function.Surjective ⇑(LinearMap.toSpanSingleton R M x)", "left✝ : Nontrivial M", "h : ∀ (x : M), x ≠ 0 → Function.Surjective ⇑(LinearMap.toSpanSingleton R M x)", "m : Submodule R M", "ne_bot : ¬m = ⊥", "x : M", "hxm : x ∈ m", "hx0 : x ≠ 0", "y : R", "x✝ : (LinearMap.toSpanSingleton R M x) y ∈ ⊤"], "goal": "(LinearMap.toSpanSingleton R M x) y ∈ m"}}
+{"state": {"context": ["m n a b c d : ℤ", "h : a ≡ b [ZMOD n]"], "goal": "c * a ≡ c * b [ZMOD c * n]"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "D : Type u₂", "CategoryTheory.Category.{v₂, u₂} D", "CategoryTheory.MonoidalCategory D", "X : C ⥤ Comon_ D"], "goal": "∀ (X_1 : C),\n  ((CategoryTheory.Monoidal.ComonFunctorCategoryEquivalence.counitIso.hom.app X).app X_1).hom = 𝟙 (X.obj X_1).X"}}
+{"state": {"context": ["R : Type u", "L : Type v", "CommRing R", "LieRing L", "LieAlgebra R L", "K K' : LieSubalgebra R L", "h : K ≤ K'"], "goal": "Function.Injective ⇑(LieSubalgebra.inclusion h)"}}
+{"state": {"context": ["m : Type u_1", "n : Type u_2", "p : Type u_3", "R : Type u_4", "inst✝⁷ : Fintype m", "inst✝⁶ : Fintype n", "inst✝⁵ : Fintype p", "inst✝⁴ : PartialOrder R", "inst✝³ : NonUnitalRing R", "inst✝² : StarRing R", "inst✝¹ : StarOrderedRing R", "inst✝ : NoZeroDivisors R", "v : n → R"], "goal": "(∀ i ∈ Finset.univ, v i * star (v i) = 0) ↔ v = 0"}}
+{"state": {"context": ["α : Type u_2", "Sub α", "f g : Filter α"], "goal": "f - g = ⊥ ↔ f = ⊥ ∨ g = ⊥"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : α → α → Prop", "s : β → β → Prop", "t : γ → γ → Prop", "ι : Type u", "f : ι → Ordinal.{u}", "h : (succ #ι).ord ≤ mex f"], "goal": "False"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "n : ℕ", "hc : G.Colorable n"], "goal": "G.chromaticNumber ≤ ↑n"}}
+{"state": {"context": ["α : Type u_1", "α' : Type u_2", "β : Type u_3", "β' : Type u_4", "γ : Type u_5", "γ' : Type u_6", "δ : Type u_7", "δ' : Type u_8", "ε : Type u_9", "ε' : Type u_10", "ζ : Type u_11", "ζ' : Type u_12", "ν : Type u_13", "inst✝⁷ : DecidableEq α'", "inst✝⁶ : DecidableEq β'", "inst✝⁵ : DecidableEq γ", "inst✝⁴ : DecidableEq γ'", "inst✝³ : DecidableEq δ", "inst✝² : DecidableEq δ'", "inst✝¹ : DecidableEq ε", "inst✝ : DecidableEq ε'", "f f'✝ : α → β → γ", "g✝ g'✝ : α → β → γ → δ", "s s' : Finset α", "t t' : Finset β", "u u' : Finset γ", "a a' : α", "b b' : β", "c : γ", "g : γ → δ", "f' : β' → α → δ", "g' : β → β'", "h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' (g' b) a"], "goal": "g '' image2 f ↑s ↑t = image2 f' (g' '' ↑t) ↑s"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "ι' : Sort u_5", "ι₂ : Sort u_6", "κ : ι → Sort u_7", "κ₁ : ι → Sort u_8", "κ₂ : ι → Sort u_9", "κ' : ι' → Sort u_10", "s : Set α", "t : (i : ι) → κ i → Set α"], "goal": "s ∩ ⋃ i, ⋃ j, t i j = ⋃ i, ⋃ j, s ∩ t i j"}}
+{"state": {"context": ["z : ℍ", "x : Fin 2 → ℤ", "hx : x ≠ 0"], "goal": "EisensteinSeries.r z * ‖x‖ ≤ Complex.abs (↑(x 0) * ↑z + ↑(x 1))"}}
+{"state": {"context": ["C : Type u", "X Y Z : CategoryTheory.FreeMonoidalCategory C", "f : X.Hom Y", "g : Y.Hom Z"], "goal": "⟦f.comp g⟧ = CategoryTheory.CategoryStruct.comp ⟦f⟧ ⟦g⟧"}}
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+{"state": {"context": ["α : Type u_4", "β : Type u_5", "Fintype α", "Fintype β", "f : α ≃ β"], "goal": "Fintype.card α = Fintype.card β"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "ConditionallyCompleteLattice α", "f : Filter β", "u : β → α"], "goal": "(Filter.bliminf u f fun x => True) = Filter.liminf u f"}}
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+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "W : Type u_4", "TopologicalSpace X", "TopologicalSpace Y", "TopologicalSpace Z", "TopologicalSpace W", "f : X → Y", "g : Z → W", "hf : IsProperMap f", "hg : IsProperMap g"], "goal": "IsProperMap (Prod.map f g)"}}
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+{"state": {"context": ["X : FinPartOrd"], "goal": "FinPartOrd.dual.obj X = FinPartOrd.of (↑X.toPartOrd)ᵒᵈ"}}
+{"state": {"context": ["α : Type u_2", "DecidableEq α", "InvolutiveNeg α", "s : Finset α", "a : α"], "goal": "a ∈ -s ↔ -a ∈ s"}}
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+{"state": {"context": ["M : Type u_1", "Add M", "s : AddSubsemigroup M"], "goal": "(AddMemClass.subtype s).srange = s"}}
+{"state": {"context": ["α β : Type u", "n c : ℕ", "h : ↑c ≤ ↑n"], "goal": "∃ m ≤ n, ↑c = ↑m"}}
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+{"state": {"context": ["K : Type uK", "V : Type uV", "CommRing K", "AddCommGroup V", "Module K V", "Module.Free K V"], "goal": "Subsingleton (Module.Dual K V) ↔ Subsingleton V"}}
+{"state": {"context": ["ι : Type u", "s : Finset ι", "w z : ι → ℝ", "x : ℝ", "hw : ∀ i ∈ s, 0 ≤ w i", "hw' : ∑ i ∈ s, w i = 1", "hz : ∀ i ∈ s, 0 ≤ z i", "hx : ∀ i ∈ s, w i ≠ 0 → z i = x"], "goal": "∏ i ∈ s, z i ^ w i = ∏ i ∈ s, x ^ w i"}}
+{"state": {"context": ["F : Type u", "E : Type v", "inst✝⁸ : Field F", "inst✝⁷ : Field E", "inst✝⁶ : Algebra F E", "K : Type w", "inst✝⁵ : Field K", "inst✝⁴ : Algebra F K", "inst✝³ : Algebra E K", "inst✝² : IsScalarTower F E K", "inst✝¹ : IsPurelyInseparable F E", "inst✝ : Algebra.IsSeparable E K", "S : IntermediateField F K := separableClosure F K", "h : Module.rank E K ≤ Module.rank F ↥S"], "goal": "sepDegree F K = Module.rank E K"}}
+{"state": {"context": ["Ω : Type u_1", "inst✝⁶ : MeasurableSpace Ω", "R : Type u_2", "inst✝⁵ : SMul R ℝ≥0", "inst✝⁴ : SMul R ℝ≥0∞", "inst✝³ : IsScalarTower R ℝ≥0 ℝ≥0∞", "inst✝² : IsScalarTower R ℝ≥0∞ ℝ≥0∞", "inst✝¹ : TopologicalSpace Ω", "inst✝ : OpensMeasurableSpace Ω", "γ : Type u_3", "F : Filter γ", "μs : γ → FiniteMeasure Ω", "mass_lim : Tendsto (fun i => (μs i).mass) F (𝓝 0)", "f : Ω →ᵇ ℝ≥0"], "goal": "Tendsto (fun i => (μs i).testAgainstNN f) F (𝓝 (testAgainstNN 0 f))"}}
+{"state": {"context": ["x : ℝ"], "goal": "ContinuousAt (fun x => x * log x) x"}}
+{"state": {"context": ["R : Type u_1", "K : Type u_2", "inst✝ : Field K", "s : ℤ", "this : Valued.v 1 = 1"], "goal": "Valued.v ((single s) 1) = ↑(Multiplicative.ofAdd (-s))"}}
+{"state": {"context": ["R : Type u_3", "S : Type u_4", "F : Type u_5", "NonAssocSemiring R", "NonAssocSemiring S", "FunLike F R S", "n : ℕ", "h : NeZero ↑n", "RingHomClass F R S", "f : F", "hf : Function.Injective ⇑f"], "goal": "NeZero ↑n"}}
+{"state": {"context": ["Ω : Type u_1", "β : Type u_2", "ι : Type u_3", "m : MeasurableSpace Ω", "f : Filtration ℕ m", "u : ℕ → Ω → β", "τ π : Ω → ℕ", "inst✝ : AddCommMonoid β", "n i : ℕ"], "goal": "i ∈ Finset.Iio n ↔ i ∈ Finset.range n"}}
+{"state": {"context": ["α : Type u", "inst✝ : Semiring α", "x : α", "n : ℕ"], "goal": "∑ i ∈ range (n + 1), x ^ i = x * ∑ i ∈ range n, x ^ i + 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "N : Matroid β", "B : Set α"], "goal": "(N.comap f).Base B ↔ N.Basis (f '' B) (f '' (f ⁻¹' N.E)) ∧ Set.InjOn f B ∧ B ⊆ f ⁻¹' N.E"}}
+{"state": {"context": ["R : Type u", "M : Type v", "Sub R", "Sub M", "x₁ x₂ : tsze R M"], "goal": "(x₁ - x₂).snd = x₁.snd - x₂.snd"}}
+{"state": {"context": ["M : Type u_1", "R : Type u_2", "inst✝² : Group M", "inst✝¹ : CommRing R", "inst✝ : MulSemiringAction M R", "a : M", "S : Ideal R", "ha : Function.Surjective fun r => a • r", "x✝ : R"], "goal": "x✝ ∈ (a • S).toAddSubmonoid ↔ x✝ ∈ a • S.toAddSubmonoid"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "α : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : TopologicalSpace X", "inst✝² : TopologicalSpace Y", "inst✝¹ : TopologicalSpace Z", "inst✝ : (i : ι) → TopologicalSpace (π i)", "x✝ y z : X", "s : Set X", "f g : X → Y", "hf : Continuous f", "h : SpecializingMap f", "x : X"], "goal": "IsClosed (f '' closure {x})"}}
+{"state": {"context": ["a b : ℝ", "t : ℂ", "ht : t ≠ -1"], "goal": "∫ (x : ℝ) in a..b, ↑x * (1 + ↑x ^ 2) ^ t =\n  (1 + ↑b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + ↑a ^ 2) ^ (t + 1) / (2 * (t + 1))"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : _root_.RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace ℝ F", "x y : E"], "goal": "im ⟪x, y⟫_𝕜 = -im ⟪y, x⟫_𝕜"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : OrderedCommGroup α", "a b : α", "m n : ℤ", "hmn : m ≠ n", "x : α", "hx : x ∈ Ioc (a * b ^ m) (a * b ^ (m + 1)) ∩ Ioc (a * b ^ n) (a * b ^ (n + 1))", "hb : 1 < b", "i1 : a * b ^ m < a * b ^ (n + 1)"], "goal": "m = n"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : CommRing R", "I J : Ideal R", "M : Type u_2", "inst✝⁵ : AddCommGroup M", "inst✝⁴ : Module R M", "M' : Type u_3", "inst✝³ : AddCommGroup M'", "inst✝² : Module R M'", "f : M →ₗ[R] M'", "inst✝¹ : IsNoetherianRing R", "inst✝ : Nontrivial M"], "goal": "(associatedPrimes R M).Nonempty"}}
+{"state": {"context": ["Ω : Type u_1", "MeasurableSpace Ω", "TopologicalSpace Ω", "OpensMeasurableSpace Ω", "μ : MeasureTheory.FiniteMeasure Ω", "f₁ f₂ : Ω →ᵇ ℝ≥0"], "goal": "μ.testAgainstNN (f₁ + f₂) = μ.testAgainstNN f₁ + μ.testAgainstNN f₂"}}
+{"state": {"context": ["R : Type u", "S : Type v", "σ : Type u_1", "τ : Type u_2", "r : R", "e : ℕ", "n m : σ", "s : σ →₀ ℕ", "inst✝² : CommSemiring R", "p q : MvPolynomial σ R", "A : Type u_3", "inst✝¹ : CommRing A", "inst✝ : IsDomain A", "a : A", "ha : a ≠ 0", "φ : MvPolynomial σ A", "i : σ"], "goal": "(∃ d, coeff d (C a * φ) ≠ 0 ∧ i ∈ d.support) ↔ ∃ d, coeff d φ ≠ 0 ∧ i ∈ d.support"}}
+{"state": {"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝² : LinearOrderedField 𝕜", "inst✝¹ : TopologicalSpace 𝕜", "inst✝ : OrderTopology 𝕜", "p q : 𝕜", "hp : Fact (0 < p)", "u : AddCircle p", "n : ℕ", "h : 0 < n", "k : 𝕜", "hk : addOrderOf ↑k = n", "a : ℤ", "ha : a • p = n • k", "h0 : ↑n ≠ 0"], "goal": "∃ m < n, m.gcd n = 1 ∧ ↑(↑m / ↑n * p) = ↑k"}}
+{"state": {"context": ["α : Type u_2", "InvolutiveNeg α", "s : Set α", "hs : s.Finite"], "goal": "(-s).Finite"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.HasShift C ℤ", "A : CategoryTheory.Pretriangulated.Triangle C"], "goal": "(𝟙 A).hom₃ = 𝟙 A.obj₃"}}
+{"state": {"context": ["α : Type u_1", "𝒜 : Set (Set α)", "h𝒜 : MeasureTheory.IsSetAlgebra ����"], "goal": "MeasureTheory.generateSetAlgebra 𝒜 ⊆ 𝒜"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "S₁ S₂ : CategoryTheory.ShortComplex C", "φ₁ φ₂ : S₁ ⟶ S₂", "h : φ₁ = φ₂"], "goal": "(CategoryTheory.ShortComplex.Homotopy.ofEq h).h₃ = 0"}}
+{"state": {"context": ["X Y : FinPartOrd", "a : ↑X.toPartOrd →o ↑Y.toPartOrd"], "goal": "FinPartOrd.dual.map a = OrderHom.dual a"}}
+{"state": {"context": ["R : Type u_1", "Semiring R", "x : PowerSeries R", "n : ℕ"], "goal": "((HahnSeries.ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff R n) x"}}
+{"state": {"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p : R", "k : ℕ", "h : Nat.minFac 0 ^ (Nat.factorization 0) (Nat.minFac 0) = 0", "hn : 0 ≠ 1"], "goal": "IsPrimePow 0"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "n ν : ℕ", "h : ν ≤ n"], "goal": "bernsteinPolynomial R n ν = (bernsteinPolynomial R n (n - ν)).comp (1 - Polynomial.X)"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "n : ℕ", "hyp : ∀ m < n, card ↑(derangements (Fin m)) = numDerangements m"], "goal": "card ↑(derangements (Fin n)) = numDerangements n"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "E : Type u_3", "F : Type u_4", "G : Type u_5", "E' : Type u_6", "F' : Type u_7", "G' : Type u_8", "E'' : Type u_9", "F'' : Type u_10", "G'' : Type u_11", "E''' : Type u_12", "R : Type u_13", "R' : Type u_14", "𝕜 : Type u_15", "𝕜' : Type u_16", "inst✝¹⁴ : Norm E", "inst✝¹³ : Norm F", "inst✝¹² : Norm G", "inst✝¹¹ : SeminormedAddCommGroup E'", "inst✝¹⁰ : SeminormedAddCommGroup F'", "inst✝⁹ : SeminormedAddCommGroup G'", "inst✝⁸ : NormedAddCommGroup E''", "inst✝⁷ : NormedAddCommGroup F''", "inst✝⁶ : NormedAddCommGroup G''", "inst✝⁵ : SeminormedRing R", "inst✝⁴ : SeminormedAddGroup E'''", "inst✝³ : SeminormedRing R'", "inst✝² : NormedDivisionRing 𝕜", "inst✝¹ : NormedDivisionRing 𝕜'", "c c' c₁ c₂ : ℝ", "f : α → E", "g✝ : α → F", "k : α → G", "f' : α → E'", "g'✝ : α → F'", "k' : α → G'", "f'' : α → E''", "g'' : α → F''", "k'' : α → G''", "l l' : Filter α", "inst✝ : TopologicalSpace α", "x : α", "s : Set α", "g : α → E'", "g' : α → F'", "h : g x = 0"], "goal": "g =o[𝓝[insert x s] x] g' ↔ g =o[𝓝[s] x] g'"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommRing R", "AddCommGroup M", "Module R M", "x : M", "f : Module.Dual R M", "y : M", "g : Module.Dual R M", "h : f x = 2"], "goal": "Module.preReflection ((Module.preReflection x f) y) ((Module.preReflection f ((Module.Dual.eval R M) x)) g) =\n  Module.preReflection x f ∘ₗ Module.preReflection y g ∘ₗ Module.preReflection x f"}}
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+{"state": {"context": ["a b : ℤ", "ha : a ≤ 0", "hb : b ≤ 0"], "goal": "a.natAbs = b.natAbs ↔ a = b"}}
+{"state": {"context": ["α : Sort u", "β : Sort v", "Uncountable α", "Countable β", "f : α → β"], "goal": "¬Function.Injective f"}}
+{"state": {"context": ["f : CircleDeg1Lift"], "goal": "f.transnumAuxSeq = fun n => (f ^ 2 ^ n) 0 / 2 ^ n"}}
+{"state": {"context": ["α : Type u_1", "a : α", "l : List α", "p : α → Bool", "h : p a = true"], "goal": "eraseP p (a :: l) = l"}}
+{"state": {"context": ["x y : UInt64"], "goal": "x = y ↔ x ≤ y ∧ y ≤ x"}}
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+{"state": {"context": ["a b k✝ m n✝ p n k : ℕ"], "goal": "(n ^ (k + 1)).primeFactors = n.primeFactors"}}
+{"state": {"context": ["ι : Type u_1", "G₀ : Type u_3", "GroupWithZero G₀", "f : ι → G₀"], "goal": "Function.support f⁻¹ = Function.support f"}}
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+{"state": {"context": ["M : Type u_1", "AddMonoid M", "c : AddCon M", "f : M → M", "hf : ∀ (x : M), c (f x + x) 0", "x y : M", "h : c x y"], "goal": "c (f x) (f y)"}}
+{"state": {"context": ["M : Type u_1", "α : Type u_7", "Monoid M", "MulAction M α", "a₁ a₂ : M"], "goal": "((fun x => a₁ • x) ∘ fun x => a₂ • x) = fun x => (a₁ * a₂) • x"}}
+{"state": {"context": ["a : UnitAddCircle", "s : ℂ", "hs : s ≠ 1"], "goal": "DifferentiableAt ℂ (HurwitzZeta.hurwitzZeta a) s"}}
+{"state": {"context": ["p n : ℕ", "hp : Nat.Prime p", "hdiv : ¬p ∣ n", "R : Type u_1", "inst✝ : CommRing R"], "goal": "(expand R p) (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R"}}
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+{"state": {"context": ["ι : Type u_1", "E : Type u_2", "μ : Measure ℝ", "l : Filter ι", "inst✝² : l.NeBot", "inst✝¹ : l.IsCountablyGenerated", "inst✝ : NormedAddCommGroup E", "a b✝ : ι → ℝ", "f : ℝ → E", "I b : ℝ", "hfi✝ : ∀ (i : ι), IntegrableOn f (Ioc (a i) b) μ", "ha : Tendsto a l atBot", "h : ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I", "hφ : AECover (μ.restrict (Iic b)) l fun i => Ioi (a i)", "hfi : ∀ (i : ι), IntegrableOn f (Ioi (a i)) (μ.restrict (Iic b))", "i : ι", "hai : a i ≤ b"], "goal": "∫ (x : ℝ) in a i..b, ‖f x‖ ∂μ ≤ I → ∫ (x : ℝ) in Ioi (a i), ‖f x‖ ∂μ.restrict (Iic b) ≤ I"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "Semiring R", "StarMul R", "TrivialStar R", "AddCommGroup A", "Module R A", "StarAddMonoid A", "StarModule R A", "Invertible 2"], "goal": "skewAdjointPart R ∘ₗ (skewAdjoint.submodule R A).subtype = LinearMap.id"}}
+{"state": {"context": ["n k m : ℕ"], "goal": "n - m + (m + k - (n + k)) = n - m + (m - n)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "inst✝ : CountablyGenerated γ", "κ : Kernel α (γ × β)", "ν : Kernel α γ", "hκν : κ.fst ≤ ν", "hν : IsFiniteKernel ν", "n : ℕ", "a : α", "s : Set β", "hs : MeasurableSet s", "u : Set γ", "hu : u ∈ countablePartition γ n", "this✝ : IsFiniteKernel κ", "hu_meas : MeasurableSet u", "this : ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a = ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a", "h0 : ¬(ν a) u = 0"], "goal": "(ν a) u ≠ ⊤"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "P : Type u_3", "MulOneClass M", "MulOneClass N", "Monoid P", "f : M →* P", "g : N →* P"], "goal": "(Monoid.Coprod.lift f g).comp Monoid.Coprod.inr = g"}}
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+{"state": {"context": ["α : Type u_2", "β : Type u_4", "TopologicalSpace α", "Zero β", "f : α → β"], "goal": "HasCompactSupport f ↔ IsCompact (closure (Function.support f))"}}
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+{"state": {"context": ["p : ℕ", "inst✝ : Fact (Nat.Prime p)", "F : Polynomial ℤ_[p]", "a : ℤ_[p]", "hnorm : ‖Polynomial.eval a F‖ < ‖Polynomial.eval a (Polynomial.derivative F)‖ ^ 2", "hnsol : Polynomial.eval a F ≠ 0", "n k : ℕ", "this : 2 ^ n ≤ 2 ^ (n + k)"], "goal": "‖newton_seq (n + (k + 1)) - newton_seq n‖ = ‖newton_seq (n + k + 1) - newton_seq n‖"}}
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+{"state": {"context": ["C : Type u", "X : CategoryTheory.FreeMonoidalCategory C", "Y₁ Y₂ : CategoryTheory.FreeMonoidalCategory C", "f : Y₁.Hom Y₂"], "goal": "⟦CategoryTheory.FreeMonoidalCategory.Hom.whiskerLeft X f⟧ = CategoryTheory.MonoidalCategory.whiskerLeft X ⟦f⟧"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "TopologicalSpace α", "TopologicalSpace β", "b : β", "a : α"], "goal": "(ContinuousMap.const α b) a = b"}}
+{"state": {"context": ["N i : ℕ", "H : i ≤ N"], "goal": "(Polynomial.revAt N) i = N - i"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝⁴ : RCLike 𝕜", "n : Type u_2", "inst✝³ : LinearOrder n", "inst✝² : IsWellOrder n fun x x_1 => x < x_1", "inst✝¹ : LocallyFiniteOrderBot n", "S : Matrix n n 𝕜", "inst✝ : Fintype n", "hS : S.PosDef"], "goal": "lower hS * diag hS * (lower hS)ᴴ = S"}}
+{"state": {"context": ["R S : Type u_1", "inst✝³ : CommRing R", "inst✝² : CommRing S", "inst✝¹ : Algebra R S", "r : R", "inst✝ : IsLocalization.Away r S"], "goal": "(algebraMap R S).FinitePresentation"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝³ : NormedField 𝕜", "inst✝² : SeminormedAddCommGroup E", "inst✝¹ : NormedSpace 𝕜 E", "inst✝ : NormedSpace ℝ E", "x✝ y z : E", "δ ε : ℝ", "hε : 0 < ε", "hδ : 0 < δ", "s : Set E", "x : E"], "goal": "(∃ z ∈ s, dist x z < ε + δ) → ∃ z, (∃ z_1 ∈ s, dist z z_1 < δ) ∧ dist x z < ε"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "X : C", "S : CategoryTheory.Sieve X", "J : CategoryTheory.GrothendieckTopology C"], "goal": "S = ⊤ → S ∈ J.sieves X"}}
+{"state": {"context": ["α : Type u_3", "β : Type u_4", "CommMonoid α", "CommMonoid β", "s : Multiset α", "F : Type u_6", "FunLike F α β", "MonoidHomClass F α β", "f : F"], "goal": "(Multiset.map (⇑f) s).prod = f s.prod"}}
+{"state": {"context": ["a b : Nat"], "goal": "¬b ≥ a → b < a"}}
+{"state": {"context": ["R : Type u", "NonAssocSemiring R", "p p' : Subsemiring R"], "goal": "↑(p ⊓ p') = ↑p ∩ ↑p'"}}
+{"state": {"context": ["X : Type u", "Y✝ : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y✝", "s t : Set X", "b : ι → Set X", "hb : IsTopologicalBasis (range b)", "U : Set X", "hUc : IsCompact U", "hUo : IsOpen U", "Y : Type u", "f' : Y → ι", "e : U = ⋃ i, (b ∘ f') i", "hf' : ∀ (i : Y), b (f' i) = (b ∘ f') i"], "goal": "∃ s, s.Finite ∧ U = ⋃ i ∈ s, b i"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹⁰ : Fintype ι", "inst✝⁹ : Fintype ι'", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace ℝ E", "inst✝⁵ : NormedSpace ℝ F", "inst✝⁴ : MeasurableSpace E", "inst✝³ : BorelSpace E", "inst✝² : MeasurableSpace F", "inst✝¹ : BorelSpace F", "inst✝ : SecondCountableTopologyEither E F", "v : Basis ι ℝ E", "w : Basis ι' ℝ F", "this : FiniteDimensional ℝ E"], "goal": "(v.prod w).addHaar = v.addHaar.prod w.addHaar"}}
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+{"state": {"context": ["f : ℕ → ℂ", "h : f 1 = 1"], "goal": "f * LSeries.delta = LSeries.delta"}}
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+{"state": {"context": ["b m n : ℕ"], "goal": "Nat.ofDigits b (b.digits n ++ b.digits m) = n + b ^ (b.digits n).length * m"}}
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+{"state": {"context": ["α : Type u_1", "BooleanAlgebra α", "s : Finset α", "hs : (↑s).Intersecting"], "goal": "Disjoint s (Finset.map { toFun := compl, inj' := ⋯ } s)"}}
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+{"state": {"context": ["R : Type u_1", "A : Type u_2", "K : Type u_3", "inst✝³ : CommRing R", "inst✝² : CommRing A", "inst✝¹ : Field K", "inst✝ : IsDedekindDomain R", "ι : Type u_4", "s : Finset ι", "f : ι → Ideal R", "P : Ideal R", "hP : P.IsPrime", "hP0 : ¬P = ⊥"], "goal": "P ∣ ∏ i ∈ s, f i ↔ ∃ i ∈ s, P ∣ f i"}}
+{"state": {"context": ["a k n : ℕ", "a_pos : 0 < a"], "goal": "(filter a.Coprime (Ico k (k + (n % a + a * (n / a))))).card ≤ φ a * (n / a + 1)"}}
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+{"state": {"context": ["α : Type u_1", "m n : Num"], "goal": "(m.cmp n).swap = Ordering.lt ↔ m.cmp n = Ordering.gt"}}
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+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "inst✝² : Lattice α", "inst✝¹ : Zero α", "inst✝ : LocallyFiniteOrder α", "f g : ι →₀ α"], "goal": "(uIcc f g).card = ∏ i ∈ f.support ∪ g.support, (uIcc (f i) (g i)).card"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Type u_5", "R : Type u_6", "R' : Type u_7", "m0 : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "f✝¹ : α → β", "μa μa' : Measure α", "μb μb' : Measure β", "μc : Measure γ", "f✝ : α → β", "f : α → α", "hf : QuasiMeasurePreserving f μ μ", "hs : f ⁻¹' sᶜ =ᵐ[μ] sᶜ", "x✝ : ℕ", "n : α"], "goal": "n ∈ (compl ∘ fun n => (preimage f)^[n] s) x✝ ↔ n ∈ (preimage f)^[x✝] sᶜ"}}
+{"state": {"context": ["G : Type u_1", "Group G", "G₂ : Type u_2", "Group G₂", "p₁ p₂ : ℕ", "hp₁ : Fact (Nat.Prime p₁)", "hp₂ : Fact (Nat.Prime p₂)", "hne : p₁ ≠ p₂", "H₁ : Subgroup G", "H₂ : Subgroup G₂", "Finite ↥H₁", "Finite ↥H₂", "hH₁ : IsPGroup p₁ ↥H₁", "hH₂ : IsPGroup p₂ ↥H₂"], "goal": "(Nat.card ↥H₁).Coprime (Nat.card ↥H₂)"}}
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+{"state": {"context": ["ιA : Type u_1", "ιM : Type u_3", "A : ιA → Type u_4", "M : ιM → Type u_5", "AddMonoid ιA", "VAdd ιA ιM", "GradedMonoid.GMonoid A", "self : GradedMonoid.GMulAction A M", "b : GradedMonoid M"], "goal": "1 • b = b"}}
+{"state": {"context": ["α : Type u", "s : Computation α", "h : s.Terminates", "a : α"], "goal": "a ∈ s → s.get = a"}}
+{"state": {"context": ["α : Type u_2", "LinearOrderedSemifield α", "a b : α", "ha : 0 < a", "hb : 0 < b"], "goal": "a⁻¹ < b⁻¹ ↔ b < a"}}
+{"state": {"context": ["α : Type u_1", "Primcodable α"], "goal": "Primrec₂ Option.getD"}}
+{"state": {"context": ["n : Type u_1", "𝕜 : Type u_3", "Field 𝕜", "DecidableEq n", "Fintype n", "L : List (Matrix.TransvectionStruct n 𝕜)"], "goal": "(List.map Matrix.TransvectionStruct.toMatrix L).prod.det = 1"}}
+{"state": {"context": ["x y : ℝ≥0∞", "z : ℝ", "hz : 0 < z"], "goal": "x ≤ y ^ (1 / z) ↔ x ^ z ≤ y"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MetricSpace α", "C : Set α", "hC : Perfect C", "ε : ℝ≥0∞", "hnonempty : C.Nonempty", "ε_pos : 0 < ε", "D₀ D₁ : Set α", "perf0 : Perfect D₀", "sub0 : D₀ ⊆ C", "hdisj : Disjoint D₀ D₁", "perf1 : Perfect D₁", "sub1 : D₁ ⊆ C", "x₀ : α", "hx₀ : x₀ ∈ D₀", "x₁ : α", "hx₁ : x₁ ∈ D₁", "perf0' : Perfect (closure (EMetric.ball x₀ (ε / 2) ∩ D₀))", "non0' : (closure (EMetric.ball x₀ (ε / 2) ∩ D₀)).Nonempty", "sub0' : closure (EMetric.ball x₀ (ε / 2) ∩ D₀) ⊆ D₀", "diam0 : EMetric.diam (closure (EMetric.ball x₀ (ε / 2) ∩ D₀)) ≤ ε", "perf1' : Perfect (closure (EMetric.ball x₁ (ε / 2) ∩ D₁))", "non1' : (closure (EMetric.ball x₁ (ε / 2) ∩ D₁)).Nonempty", "sub1' : closure (EMetric.ball x₁ (ε / 2) ∩ D₁) ⊆ D₁", "diam1 : EMetric.diam (closure (EMetric.ball x₁ (ε / 2) ∩ D₁)) ≤ ε"], "goal": "∃ C₀ C₁, (Perfect C₀ ∧ C₀.Nonempty ∧ C₀ ⊆ C ∧ EMetric.diam C₀ ≤ ε) ∧ (Perfect C₁ ∧ C₁.Nonempty ∧ C₁ ⊆ C ∧ EMetric.diam C₁ ≤ ε) ∧ Disjoint C₀ C₁"}}
+{"state": {"context": ["M : Type u_1", "N : Type u_2", "inst✝¹ : SeminormedAddCommGroup M", "inst✝ : SeminormedAddCommGroup N", "S : AddSubgroup M", "x y : M", "a b : ↑↑S"], "goal": "⨅ a, ‖x + y + ↑a‖ ≤ ‖x + ↑a‖ + ‖y + ↑b‖"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Preadditive C", "CategoryTheory.HasShift C ℤ", "T : CategoryTheory.Pretriangulated.Triangle C"], "goal": "T.invRotate.obj₂ = T.obj₁"}}
+{"state": {"context": ["α : Type u_1", "LT α", "s : Set α", "IsEmpty α"], "goal": "s.chainHeight = 0"}}
+{"state": {"context": ["F : Type v'", "R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝¹⁶ : CommSemiring R", "inst✝¹⁵ : StarRing R", "inst✝¹⁴ : NonUnitalSemiring A", "inst✝¹³ : StarRing A", "inst✝¹² : Module R A", "inst✝¹¹ : IsScalarTower R A A", "inst✝¹⁰ : SMulCommClass R A A", "inst✝⁹ : StarModule R A", "inst✝⁸ : NonUnitalSemiring B", "inst✝⁷ : StarRing B", "inst✝⁶ : Module R B", "inst✝⁵ : IsScalarTower R B B", "inst✝⁴ : SMulCommClass R B B", "inst✝³ : StarModule R B", "inst✝² : FunLike F A B", "inst✝¹ : NonUnitalAlgHomClass F R A B", "inst✝ : NonUnitalStarAlgHomClass F R A B", "ι : Sort u_1", "S : ι → NonUnitalStarSubalgebra R A", "x : A"], "goal": "x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i"}}
+{"state": {"context": ["M : Type u_1", "Monoid M", "s t : Set M", "ht : IsSubmonoid t", "h : s ⊆ t"], "goal": "Monoid.Closure s ⊆ t"}}
+{"state": {"context": ["k : Type u_1", "inst✝⁷ : Field k", "K : Type u_2", "inst✝⁶ : Field K", "F : Type u_3", "inst✝⁵ : Field F", "inst✝⁴ : NumberField K", "inst✝³ : Algebra k K", "inst✝² : Algebra k F", "inst✝¹ : Algebra K F", "inst✝ : IsScalarTower k K F", "σ : K ≃ₐ[k] K", "w : InfinitePlace K"], "goal": "IsUnramified k w ↔ w.mult ≤ (w.comap (algebraMap k K)).mult"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝³ : NormedAddCommGroup V", "inst✝² : InnerProductSpace ℝ V", "inst✝¹ : MetricSpace P", "inst✝ : NormedAddTorsor V P", "t₁ t₂ : Triangle ℝ P", "i₁ i₂ i₃ j₁ j₂ j₃ : Fin 3", "hi₁₂ : i₁ ≠ i₂", "hi₁₃ : i₁ ≠ i₃", "hi₂₃ : i₂ ≠ i₃", "hj₁₂ : j₁ ≠ j₂", "hj₁₃ : j₁ ≠ j₃", "hj₂₃ : j₂ ≠ j₃", "h₁ : t₂.points j₁ = t₁.orthocenter", "h₂ : t₂.points j₂ = t₁.points i₂", "h₃ : t₂.points j₃ = t₁.points i₃"], "goal": "affineSpan ℝ {t₁.points i₁, t₁.points i₂} = altitude t₂ j₂"}}
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+{"state": {"context": ["α : Type u_1", "Sub α", "a b : Part α", "hab : (a - b).Dom"], "goal": "(a - b).get hab = a.get ⋯ - b.get ⋯"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "f : ℝ → F", "K : Set F", "r ε : ℝ", "L : F", "x y z : ℝ", "hy : y ∈ Icc x (x + r / 2)", "hz : z ∈ Icc x (x + r / 2)", "r' : ℝ", "r'mem : r' ∈ Ioc (r / 2) r", "hr' : ∀ y ∈ Icc x (x + r'), ∀ z ∈ Icc x (x + r'), ‖f z - f y - (z - y) • L‖ ≤ ε * r", "A : x + r / 2 ≤ x + r'"], "goal": "‖f z - f y - (z - y) • L‖ ≤ ε * r"}}
+{"state": {"context": ["ι : Type u_1", "G : ι → Type u_2", "H : Type u_3", "K : Type u_4", "inst✝² : Monoid K", "inst✝¹ : (i : ι) → Group (G i)", "inst✝ : Group H", "φ : (i : ι) → H →* G i", "hφ : ∀ (i : ι), Injective ⇑(φ i)", "i j : ι", "hij : i ≠ j", "x : PushoutI φ", "g₁ : G i", "hg₁ : (of i) g₁ = x", "g₂ : G j", "hg₂ : (of j) g₂ = x", "hx : ¬x ∈ (base φ).range", "hx1 : x ≠ 1", "hg₁1 : g₁ ≠ 1", "hg₂1 : g₂ ≠ 1", "hg₁r : g₁ ∉ (φ i).range", "hg₂r : g₂ ∉ (φ j).range"], "goal": "False"}}
+{"state": {"context": [], "goal": "∀ {α : Type u_1} {a : α} {as : List α} {i : Nat}, List.take (i + 1) (a :: as) = a :: List.take i as"}}
+{"state": {"context": ["R : Type u", "S : Type v", "k : Type y", "A : Type z", "a b : R", "n : ℕ", "inst✝ : Field R", "p q : R[X]", "this : p.degree ≤ 0", "hc : p.coeff 0 = 0", "h : degree 0 = 0"], "goal": "False"}}
+{"state": {"context": ["α : Sort u_1", "p : α → Prop", "s : Subtype p"], "goal": "sizeOf s = sizeOf s.val + 1"}}
+{"state": {"context": ["R : Type u_1", "V : Type u_2", "P : Type u_4", "StrictOrderedCommRing R", "AddCommGroup V", "Module R V", "AddTorsor V P", "s : AffineSubspace R P", "x y : P"], "goal": "s.WOppSide x y ↔ s.WOppSide y x"}}
+{"state": {"context": ["G : Type w", "α : Type u", "TopologicalSpace G", "Neg G", "ContinuousNeg G", "TopologicalSpace α", "f : α → G", "x : α", "hf : ContinuousAt f x"], "goal": "ContinuousAt (fun x => -f x) x"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "Z : Type u_1", "W : Type u_2", "ε : Type u_3", "ζ : Type u_4", "ι : Type u_5", "κ : Type u_6", "σ : ι → Type u_7", "τ : κ → Type u_8", "inst✝² : (i : ι) → TopologicalSpace (σ i)", "inst✝¹ : (k : κ) → TopologicalSpace (τ k)", "inst✝ : TopologicalSpace X", "fst✝ : ι", "snd✝ : σ fst✝"], "goal": "𝓝 ⟨fst✝, snd✝⟩ = Filter.map (mk ⟨fst✝, snd✝⟩.fst) (𝓝 ⟨fst✝, snd✝⟩.snd)"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "D : Type u_1", "inst✝⁵ : Category.{u_2, u_1} D", "inst✝⁴ : Abelian C", "inst✝³ : HasProjectiveResolutions C", "inst✝² : Abelian D", "F G : C ⥤ D", "inst✝¹ : F.Additive", "inst✝ : G.Additive", "α : F ⟶ G", "X : C", "P : ProjectiveResolution X"], "goal": "(HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app ((projectiveResolutions C).obj X).as) ≫ (G.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((G.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex)) = ((F.mapHomotopyCategory (ComplexShape.down ℕ)).map P.iso.hom ≫ 𝟙 ((HomotopyCategory.quotient D (ComplexShape.down ℕ)).obj ((F.mapHomologicalComplex (ComplexShape.down ℕ)).obj P.complex))) ≫ (HomotopyCategory.quotient D (ComplexShape.down ℕ)).map ((NatTrans.mapHomologicalComplex α (ComplexShape.down ℕ)).app P.complex)"}}
+{"state": {"context": ["α : Type u", "l : List α", "n : Fin l.length"], "goal": "l.get n :: List.drop (↑n + 1) l = List.drop (↑n) l"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "x : E", "s : Set E", "h : IsOpen s", "hx : x ∈ s"], "goal": "HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x"}}
+{"state": {"context": ["k : ℕ", "hk : k ≠ 0", "n : ℤ", "this : ofReal' ∘ periodizedBernoulli k = AddCircle.liftIco 1 0 (ofReal' ∘ bernoulliFun k)"], "goal": "fourierCoeff (ofReal' ∘ periodizedBernoulli k) n = -↑k ! / (2 * ↑π * I * ↑n) ^ k"}}
+{"state": {"context": ["R : Type u_1", "A : Type u_2", "p : A → Prop", "inst✝¹³ : CommRing R", "inst✝¹² : Nontrivial R", "inst✝¹¹ : StarRing R", "inst✝¹⁰ : MetricSpace R", "inst✝⁹ : TopologicalRing R", "inst✝⁸ : ContinuousStar R", "inst✝⁷ : TopologicalSpace A", "inst✝⁶ : NonUnitalRing A", "inst✝⁵ : StarRing A", "inst✝⁴ : Module R A", "inst✝³ : IsScalarTower R A A", "inst✝² : SMulCommClass R A A", "inst✝¹ : NonUnitalContinuousFunctionalCalculus R p", "f g : R → R", "a : A", "hf✝ : autoParam (ContinuousOn f (σₙ R a)) _auto✝", "hf0 : autoParam (f 0 = 0) _auto✝", "hg : autoParam (ContinuousOn g (σₙ R a)) _auto✝", "hg0 : autoParam (g 0 = 0) _auto✝", "inst✝ : UniqueNonUnitalContinuousFunctionalCalculus R A", "hf : autoParam (ContinuousOn f ((fun x => -x) '' σₙ R a)) _auto✝", "h0 : autoParam (f 0 = 0) _auto✝", "ha : autoParam (p a) _auto✝"], "goal": "cfcₙ (fun x => f (-x)) a = cfcₙ f (-a)"}}
+{"state": {"context": ["y z : ℝ", "f : ℝ → ℝ≥0∞", "hzy : z ≤ y"], "goal": "∫⁻ (x : ℝ) in Set.Iic y, f x = (∫⁻ (x : ℝ) in Set.Iio z, f x) + ∫⁻ (x : ℝ) in Set.Icc z y, f x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝¹ : Fintype α", "s : Set α", "inst✝ : Fintype ↑s"], "goal": "Fintype.card ↑s = Fintype.card α ↔ s = Set.univ"}}
+{"state": {"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "e : C ≌ D", "Y : D"], "goal": "e.unit.app (e.inverse.obj Y) = e.inverse.map (e.counitInv.app Y)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "Finite α", "f : α → β"], "goal": "Nat.card ↑(Set.range f) ≤ Nat.card α"}}
+{"state": {"context": ["E : Type u_1", "NormedAddCommGroup E", "NormedSpace ℝ E", "MeasurableSpace E", "BorelSpace E", "FiniteDimensional ℝ E", "μ : MeasureTheory.Measure E", "μ.IsAddHaarMeasure", "f : E ≃ₗ[ℝ] E", "s : Set E"], "goal": "μ (⇑f ⁻¹' s) = ENNReal.ofReal |LinearMap.det ↑f.symm| * μ s"}}
+{"state": {"context": ["F : Type u_1", "inst✝¹ : Field F", "ι : Type u_2", "inst✝ : DecidableEq ι", "s t : Finset ι", "i j : ι", "v r r' : ι → F", "hvs : Set.InjOn v ↑s", "hs : s.Nonempty"], "goal": "∑ x ∈ s, Lagrange.basis s v x = 1"}}
+{"state": {"context": ["n k : ℕ", "x : Fin k → ℕ"], "goal": "x ∈ List.Nat.antidiagonalTuple k n ↔ ∑ i : Fin k, x i = n"}}
+{"state": {"context": ["C : Type u", "inst✝⁶ : Category.{v, u} C", "D : Type u_1", "inst✝⁵ : Category.{u_2, u_1} D", "inst✝⁴ : Abelian C", "inst✝³ : HasInjectiveResolutions C", "inst✝² : Abelian D", "F G : C ⥤ D", "inst✝¹ : F.Additive", "inst✝ : G.Additive", "α : F ⟶ G", "X : C", "P : InjectiveResolution X", "n : ℕ"], "goal": "(HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map ((NatTrans.rightDerivedToHomotopyCategory α).app X) = ((HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map (P.isoRightDerivedToHomotopyCategoryObj F).hom ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) n).hom.app ((F.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex)) ≫ HomologicalComplex.homologyMap ((NatTrans.mapHomologicalComplex α (ComplexShape.up ℕ)).app P.cocomplex) n ≫ (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up ℕ) n).inv.app ((G.mapHomologicalComplex (ComplexShape.up ℕ)).obj P.cocomplex) ≫ (HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n).map (P.isoRightDerivedToHomotopyCategoryObj G).inv"}}
+{"state": {"context": ["α : Type u_1", "DecidableEq α", "s : Finset α", "a b : α", "h : a ∈ s"], "goal": "a ∈ insert b s"}}
+{"state": {"context": ["ι : Type u_5", "α : ι → Type u_6", "(i : ι) → Lattice (α i)", "s : Set ι", "t : (i : ι) → Set (α i)", "ht : ∀ i ∈ s, IsSublattice (t i)"], "goal": "IsSublattice (s.pi t)"}}
+{"state": {"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝³ : TopologicalSpace X", "inst✝² : UniformSpace α", "inst✝¹ : UniformSpace β", "F : ι → X → α", "G : ι → β → α", "inst✝ : CompactSpace X", "F_eqcont : Equicontinuous F", "U : Set (α × α)", "hU : U ∈ 𝓤 α", "V : Set (α × α)", "hV : V ∈ 𝓤 α", "Vsymm : SymmetricRel V", "hVU : V ○ V ○ V ⊆ U"], "goal": "Prod.map F F ⁻¹' UniformFun.gen X α U ∈ ⨅ i, comap (fun x => (F x.1 i, F x.2 i)) (𝓤 α)"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "c : ComplexShape ι", "c' : ComplexShape ι'", "C : Type u_3", "CategoryTheory.Category.{u_4, u_3} C", "CategoryTheory.Limits.HasZeroObject C", "CategoryTheory.Limits.HasZeroMorphisms C", "K : HomologicalComplex C c", "e : c.Embedding c'", "i' j' : ι'", "j : ι", "hj : e.f j = j'", "hj' : ¬c.Rel (c.prev j) j"], "goal": "(K.extend e).d i' j' = 0"}}
+{"state": {"context": ["a b d m n k : ℕ", "p q : ℕ → Prop", "h : b ∣ a", "e : ℕ"], "goal": "a ∣ a * e"}}
+{"state": {"context": ["α : Type u", "Add α", "a b : WithBot α", "x : α"], "goal": "a + b = ↑x ↔ ∃ a' b', ↑a' = a ∧ ↑b' = b ∧ a' + b' = x"}}
+{"state": {"context": ["α : Type u_1", "m : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "c : ℝ≥0∞"], "goal": "∫⁻ (x : α) in s, c ∂μ = c * μ s"}}
+{"state": {"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "m n : ℕ", "h : m ≤ n", "k : ℕ", "hk : n = m + k"], "goal": "I ^ n ≤ I ^ m"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "TopologicalSpace α", "OrderClosedTopology α", "a b c : α", "h : b < c"], "goal": "Set.Iic c ∈ 𝓝ˢ (Set.Ioc a b)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : Filter α", "c : β → α", "h : Set.range c ∈ f", "W : Set β", "W_in : W ∈ Filter.comap c f"], "goal": "c '' W ∈ f"}}
+{"state": {"context": ["R✝ : Type u_1", "inst✝¹³ : CommSemiring R✝", "A✝ : Type u_2", "B✝ : Type u_3", "inst✝¹² : Semiring A✝", "inst✝¹¹ : Semiring B✝", "inst✝¹⁰ : Algebra R✝ A✝", "inst✝⁹ : Algebra R✝ B✝", "R : Type u_4", "inst✝⁸ : CommRing R", "A : Type u_5", "B : Type u_6", "C : Type u_7", "D : Type u_8", "inst✝⁷ : Ring A", "inst✝⁶ : Ring B", "inst✝⁵ : Ring C", "inst✝⁴ : Ring D", "inst✝³ : Algebra R A", "inst✝² : Algebra R B", "inst✝¹ : Algebra R C", "inst✝ : Algebra R D", "f : A →ₐ[R] B", "g : C →ₐ[R] D", "hf : Function.Surjective ⇑f", "this : Submodule.restrictScalars R (RingHom.ker (map f (AlgHom.id R C))) = LinearMap.ker (LinearMap.rTensor C f.toLinearMap)"], "goal": "LinearMap.range (LinearMap.rTensor C (LinearMap.ker f.toLinearMap).subtype) = LinearMap.range (LinearMap.rTensor C (Submodule.restrictScalars R (RingHom.ker f)).subtype)"}}
+{"state": {"context": ["α : Type u_1", "inst✝⁴ : CancelCommMonoidWithZero α", "inst✝³ : UniqueFactorizationMonoid α", "inst✝² : DecidableEq (Associates α)", "inst✝¹ : (p : Associates α) → Decidable (Irreducible p)", "inst✝ : DecidableEq (Associates α)", "a : Associates α", "ha : a ≠ 0", "k : ℕ", "hk : ∀ (p : Associates α), Irreducible p → k ∣ p.count a.factors", "a✝ : Nontrivial α"], "goal": "∃ b, a = b ^ k"}}
+{"state": {"context": ["B : Type u_1", "W : Type u_2", "Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B"], "goal": "cs.IsReduced ω.reverse ↔ cs.IsReduced ω"}}
+{"state": {"context": ["R : Type u_1", "S : Type u_2", "CommSemiring R", "Semiring S", "Algebra R S", "f : S ≃ₐ[R] S"], "goal": "↑((AlgEquiv.algHomUnitsEquiv R S).symm f) = ↑f"}}
+{"state": {"context": ["R : Type u", "NonUnitalNonAssocRing R", "ι : Sort u_2", "S : ι → NonUnitalSubring R", "x : R"], "goal": "x ∈ ⨅ i, S i ↔ ∀ (i : ι), x ∈ S i"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_2", "CommSemiring R", "σ : Type u_3", "CanonicallyLinearOrderedAddCommMonoid M", "w : σ → M", "p : MvPolynomial σ R"], "goal": "MvPolynomial.IsWeightedHomogeneous w p 0 ↔ MvPolynomial.weightedTotalDegree w p = 0"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝⁴ : TopologicalSpace α", "ι✝ : Type u_5", "π : ι✝ → Type u_6", "inst✝³ : (i : ι✝) → TopologicalSpace (π i)", "inst✝² : TopologicalSpace β", "inst✝¹ : TopologicalSpace γ", "inst✝ : TopologicalSpace δ", "ι : Sort u_7", "f : α → β", "s : ι → Set α", "hs : ∀ (x : α), ∃ i, s i ∈ 𝓝 x", "hf : ∀ (i : ι), ContinuousOn f (s i)", "x : α", "i : ι", "hi : s i ∈ 𝓝 x"], "goal": "Tendsto f (𝓝[s i] x) (𝓝 (f x))"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "F : CategoryTheory.Functor Cᵒᵖ (Type v)", "X : F.Elementsᵒᵖ"], "goal": "(CategoryTheory.CategoryOfElements.costructuredArrowYonedaEquivalence F).functor.obj X =\n  CategoryTheory.CostructuredArrow.mk (CategoryTheory.yonedaEquiv.symm (Opposite.unop X).snd)"}}
+{"state": {"context": ["𝕜 : Type u_1", "inst✝¹¹ : NontriviallyNormedField 𝕜", "D : Type uD", "inst✝¹⁰ : NormedAddCommGroup D", "inst✝⁹ : NormedSpace 𝕜 D", "E : Type uE", "inst✝⁸ : NormedAddCommGroup E", "inst✝⁷ : NormedSpace 𝕜 E", "F : Type uF", "inst✝⁶ : NormedAddCommGroup F", "inst✝⁵ : NormedSpace 𝕜 F", "G : Type uG", "inst✝⁴ : NormedAddCommGroup G", "inst✝³ : NormedSpace 𝕜 G", "X : Type u_2", "inst✝² : NormedAddCommGroup X", "inst✝¹ : NormedSpace 𝕜 X", "s s₁ t u : Set E", "f✝ f₁ : E → F", "g : F → G", "x x₀ : E", "c : F", "b : E × F → G", "m n : ℕ∞", "p : E → FormalMultilinearSeries 𝕜 E F", "inst✝ : CompleteSpace E", "f : PartialHomeomorph E F", "f₀' : E ≃L[𝕜] F", "a : F", "ha : a ∈ f.target", "hf₀' : HasFDerivAt (↑f) (↑f₀') (↑f.symm a)", "Itop : ∀ (n : ℕ), ContDiffAt 𝕜 (↑n) (↑f) (↑f.symm a) → ContDiffAt 𝕜 (↑n) (↑f.symm) a", "hf : ContDiffAt 𝕜 ⊤ (↑f) (↑f.symm a)"], "goal": "ContDiffAt 𝕜 ⊤ (↑f.symm) a"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝⁵ : Fintype ι", "inst✝⁴ : Fintype ι'", "inst✝³ : AddCommGroup E", "inst✝² : Module ℝ E", "inst✝¹ : AddCommGroup F", "inst✝ : Module ℝ F", "v : ι → E"], "goal": "Convex ℝ (parallelepiped v)"}}
+{"state": {"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "p : R[X]", "h : ¬p = 0"], "goal": "p.primPart.IsPrimitive"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "inst✝¹⁰ : Category.{u_6, u_1} C", "inst✝⁹ : Category.{u_7, u_2} D", "inst✝⁸ : Category.{?u.89019, u_3} E", "F : C ⥤ D", "G : D ⥤ E", "A : Type u_4", "B : Type u_5", "inst✝⁷ : AddMonoid A", "inst✝⁶ : AddCommMonoid B", "inst✝⁵ : HasShift C A", "inst✝⁴ : HasShift D A", "inst✝³ : HasShift E A", "inst✝² : HasShift C B", "inst✝¹ : HasShift D B", "inst✝ : F.CommShift A", "X : C", "a b : A", "h : a + b = 0"], "goal": "𝟙 (F.obj ((shiftFunctor C a ⋙ shiftFunctor C b).obj X)) = ((F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) ≫ (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) ≫ (shiftFunctorCompIsoId D a b h).hom.app (F.obj X)) ≫ (shiftFunctorCompIsoId D a b h).inv.app (F.obj X) ≫ (shiftFunctor D b).map ((F.commShiftIso a).inv.app X) ≫ (F.commShiftIso b).inv.app ((shiftFunctor C a).obj X)"}}
+{"state": {"context": ["a : ↥circle", "h : rotation a = conjLIE"], "goal": "False"}}
+{"state": {"context": ["α : Type u", "DecidableEq α", "s : Set α", "a : α", "hs : (insert a s).Finite"], "goal": "hs.toFinset = insert a ⋯.toFinset"}}
+{"state": {"context": ["a b c d : ℝ≥0∞"], "goal": "c ≠ ⊤ → a < b → c ≤ d → a + c < b + d"}}
+{"state": {"context": ["R : Type u_1", "inst✝¹ : EuclideanDomain R", "abv : AbsoluteValue R ℤ", "ι : Type u_2", "inst✝ : Fintype ι", "ε : ℝ", "hε : 0 < ε", "b : R", "hb : b ≠ 0", "h✝ : abv.IsAdmissible", "A : Fin (h✝.card ε ^ Fintype.card ι).succ → ι → R", "e : ι ≃ Fin (Fintype.card ι) := Fintype.equivFin ι", "i₀ i₁ : Fin (h✝.card ε ^ Fintype.card ι).succ", "ne : i₀ ≠ i₁", "h : ∀ (k : Fin (Fintype.card ι)), ↑(abv (A i₁ (e.symm k) % b - A i₀ (e.symm k) % b)) < abv b • ε", "k : ι"], "goal": "↑(abv (A i₁ k % b - A i₀ k % b)) < abv b • ε"}}
+{"state": {"context": ["R : Type u₁", "L : Type u₂", "inst✝² : CommRing R", "inst✝¹ : LieRing L", "inst✝ : LieAlgebra R L", "x y : L"], "goal": "__src✝.toFun ⁅x, y⁆ = ⁅__src✝.toFun x, __src✝.toFun y⁆"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_3", "TopologicalSpace X", "TopologicalSpace Y", "e : PartialHomeomorph X Y", "s : Set X", "t : Set Y", "h : e.IsImage s t"], "goal": "↑e '' (e.source ∩ s) = e.target ∩ t"}}
+{"state": {"context": ["n : Type u_2", "R : Type u_3", "Fintype n", "CommRing R", "PartialOrder R", "StarRing R", "DecidableEq n", "M : Matrix n n R", "hM : M.PosSemidef"], "goal": "M⁻¹.PosSemidef"}}
+{"state": {"context": ["R : Type u", "CommSemiring R", "p : R[X]"], "goal": "p.coeff 0 = (Polynomial.aeval 0) p"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "r : α → α → Prop", "s : β → β → Prop", "f g : r ↪r s"], "goal": "f = g ↔ ∀ (x : α), f x = g x"}}
+{"state": {"context": ["w : Nat", "x y z : BitVec w"], "goal": "x ^^^ y ^^^ z = x ^^^ (y ^^^ z)"}}
+{"state": {"context": ["α : Type u", "m n : ℕ", "a : Fin m → α", "b : Fin n → α"], "goal": "ofFn (Fin.append a b) = ofFn a ++ ofFn b"}}
+{"state": {"context": ["α : Type u_1", "r : Rel α α", "p q : RelSeries r", "connect : r p.last q.head", "i : Fin (q.length + 1)"], "goal": "(p.append q connect).toFun (↑↑(Fin.natAdd p.length i) + 1) = q.toFun i"}}
+{"state": {"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ inst✝ : NumberField K", "x ζ : (𝓞 K)ˣ", "f : Fin (rank K) → ℤ", "hζ : ζ ∈ torsion K", "h : x = ζ * ∏ i : Fin (rank K), fundSystem K i ^ f i"], "goal": "f = ⇑((basisModTorsion K).repr (Additive.ofMul ↑x))"}}
+{"state": {"context": ["R : Type u", "Semiring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : ι → Type w", "(i : ι) → AddCommMonoid (M i)", "(i : ι) → Module R (M i)", "i : ι", "b : M i"], "goal": "DFinsupp.single i b = (DirectSum.lof R ι M i) b"}}
+{"state": {"context": ["K : Type u_8", "M : Type u_9", "L : Type u_10", "CommRing K", "Semiring M", "Algebra K M", "CommRing L", "IsDomain L", "Algebra K L", "NoZeroSMulDivisors K L"], "goal": "LinearIndependent K AlgHom.toLinearMap"}}
+{"state": {"context": ["α : Type u_1", "p : α → Prop", "DecidablePred p", "a : α", "s : Multiset α"], "goal": "Multiset.filter p (a ::ₘ s) = (if p a then {a} else 0) + Multiset.filter p s"}}
+{"state": {"context": ["G : Type u_1", "M : Type u_2", "S : Type u_3", "inst✝ : Monoid M", "a b : M", "h : Commute a b", "n : ℕ"], "goal": "(a * b) ^ (n + 1) = a ^ (n + 1) * b ^ (n + 1)"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "l : Filter α", "k f g : α → β", "TopologicalSpace β", "LinearOrderedCommRing β", "OrderTopology β", "hf : Asymptotics.SuperpolynomialDecay l k f", "hfg : ∀ (x : α), |g x| ≤ |f x|"], "goal": "Asymptotics.SuperpolynomialDecay l k g"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Sort u_4", "π : α → Type u_5", "fa : α → α", "fb : β → β", "f : α → β", "g : β → γ", "s t : Set α", "h : Semiconj f fa fb", "ha : Surjective fa"], "goal": "SurjOn fb (range f) (range f)"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁶ : Ring R", "E : Type u_2", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : Module R E", "F : Type u_3", "inst✝³ : AddCommGroup F", "inst✝² : Module R F", "G : Type u_4", "inst✝¹ : AddCommGroup G", "inst✝ : Module R G", "f : E →ₗ.[R] F", "x : E"], "goal": "(∃ x_1, ∃ (h : x ∈ f.domain), ↑f ⟨x, ⋯⟩ = x_1) ↔ x ∈ f.domain"}}
+{"state": {"context": ["α : Type u_1", "LinearOrder α", "a b : α"], "goal": "(cmp a b).Compares a b"}}
+{"state": {"context": ["R : Type u_1", "M : Type u_4", "S : Type u_7", "Semiring R", "AddCommMonoid M", "Module R M", "SetLike S M", "AddSubmonoidClass S M", "SMulMemClass S R M", "s : S"], "goal": "↑(Submodule.span R ↑s) = ↑s"}}
+{"state": {"context": ["T : ℝ", "hT : Fact (0 < T)", "f : ↥(MeasureTheory.Lp ℂ 2 AddCircle.haarAddCircle)", "i : ℤ"], "goal": "↑(fourierBasis.repr f) i = fourierCoeff (↑↑f) i"}}
+{"state": {"context": ["α : Type u_1", "inst✝ : MeasurableSpace α", "f : α → ℝ", "μ : Measure α", "f_nn : 0 ≤ᶠ[ae μ] f", "f_mble : AEMeasurable f μ", "p : ℝ", "p_pos : 0 < p", "t : ℝ", "ht : μ {a | t ≤ f a} = μ {a | t < f a}"], "goal": "μ {a | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) = μ {a | t < f a} * ENNReal.ofReal (t ^ (p - 1))"}}
+{"state": {"context": ["C : Type u_1", "D : Type u_2", "inst✝⁶ : Category.{u_3, u_1} C", "inst✝⁵ : Category.{u_4, u_2} D", "inst✝⁴ : HasZeroMorphisms C", "inst✝³ : HasZeroMorphisms D", "S S₁ S₂ : ShortComplex C", "h : S.Exact", "F : C ⥤ D", "inst✝² : F.PreservesZeroMorphisms", "inst✝¹ : F.PreservesLeftHomologyOf S", "inst✝ : F.PreservesRightHomologyOf S", "this : S.HasHomology"], "goal": "(S.map F).Exact"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : NormedSpace ℝ V", "inst✝² : StrictConvexSpace ℝ V", "inst✝¹ : PseudoMetricSpace P", "inst✝ : NormedAddTorsor V P", "p p₁ p₂ p₃ : P", "r : ℝ", "h : Collinear ℝ {p₁, p₂, p₃}", "hp₁ : dist p₁ p = r", "hp₂ : dist p₂ p < r", "hp₃ : dist p₃ p = r", "hp₁p₃ : p₁ ≠ p₃"], "goal": "p₂ ≠ p₁"}}
+{"state": {"context": ["𝕜 : Type u_1", "NontriviallyNormedField 𝕜", "E : Type u_2", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "F : Type u_3", "NormedAddCommGroup F", "NormedSpace 𝕜 F", "f : E → F", "f' : E →L[𝕜] F", "x : E", "L : Filter E", "hf : HasFDerivAtFilter f f' x L", "C : ℝ≥0", "hf' : AntilipschitzWith C ⇑f'"], "goal": "(fun x' => x' - x) =O[L] fun x' => f x' - f x"}}
+{"state": {"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "π : ι → Type u_6", "inst✝³ : OrderedCommRing 𝕜", "inst✝² : NoZeroDivisors 𝕜", "inst✝¹ : AddCommGroup E", "inst✝ : Module 𝕜 E", "x y : E", "h : LinearIndependent 𝕜 ![x, y]", "s t : 𝕜", "c : E", "hs : -1 * s - t * -1 = 0"], "goal": "t = s"}}
+{"state": {"context": ["α : Type u", "Preorder α", "a b : α", "h : a ≤ b"], "goal": "lowerBounds (Set.Icc a b) = Set.Iic a"}}
+{"state": {"context": ["p : ℕ", "R : Type u_1", "hp : Fact (Nat.Prime p)", "CommRing R", "x : WittVector p R", "n k : ℕ"], "goal": "((⇑WittVector.verschiebung)^[n] x).coeff (k + n) = x.coeff k"}}
+{"state": {"context": ["α : Type u_1", "NatCast α", "n : ℕ", "n.AtLeastTwo"], "goal": "AddOpposite.unop (OfNat.ofNat n) = OfNat.ofNat n"}}
+{"state": {"context": ["I : Type u", "inst✝² : LinearOrder I", "inst✝¹ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "l : Products I", "J : I → Prop", "h : ∀ a ∈ ↑l, J a", "inst✝ : (j : I) → Decidable (J j)", "x : ↑C"], "goal": "(∀ i ∈ ↑l, ↑(Set.MapsTo.restrict (Proj J) C (π C J) ⋯ x) i = true) = ∀ i ∈ ↑l, ↑x i = true"}}
+{"state": {"context": ["a n : Nat", "h : 0 < a"], "goal": "0 < a ^ n"}}
+{"state": {"context": ["α✝ : Type u", "α' : Type u_1", "β : Type v", "β' : Type u_2", "γ : Type u_3", "φ : Type u_4", "inst✝⁵ : OmegaCompletePartialOrder α✝", "inst✝⁴ : OmegaCompletePartialOrder β", "inst✝³ : OmegaCompletePartialOrder γ", "inst✝² : OmegaCompletePartialOrder φ", "inst✝¹ : OmegaCompletePartialOrder α'", "inst✝ : OmegaCompletePartialOrder β'", "α : Type u_5", "f : α → β →𝒄 γ", "c✝ : Chain β", "x : α"], "goal": "ωSup (c✝.map ↑(f x)) = ωSup (c✝.map { toFun := fun x y => (f y) x, monotone' := ⋯ }) x"}}
+{"state": {"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "a : ℝ", "f g : ℂ → E", "l : Filter ℂ", "u : ℂ → ℝ", "this : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖", "cf : ℝ", "hcf : cf < a", "Bf : ℝ", "hOf : f =O[l] fun z => expR (Bf * expR (cf * |u z|))", "cg : ℝ", "hcg : cg < a", "Bg : ℝ", "hOg : g =O[l] fun z => expR (Bg * expR (cg * |u z|))"], "goal": "(f - g) =O[l] fun z => expR (max 0 (max Bf Bg) * expR (max cf cg * |u z|))"}}
+{"state": {"context": ["α : Type u", "β : Type v", "f : α → β", "l₁ : Filter α", "l₂ : Filter β", "p : β → Prop", "hf : Filter.Tendsto f l₁ l₂", "h : ∀ᶠ (y : β) in l₂, p y"], "goal": "∀ᶠ (x : α) in l₁, p (f x)"}}
+{"state": {"context": ["σ K : Type u", "inst✝ : Field K"], "goal": "Module.rank K (MvPolynomial σ K) = Cardinal.mk (σ →₀ ℕ)"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "TopologicalSpace X", "DiscreteTopology X", "f : X → Y"], "goal": "IsLocallyConstant f"}}
+{"state": {"context": ["α : Type u_1", "r : α → α → Prop", "a b c : α", "hab : Relation.ReflTransGen r a b", "hbc : Relation.ReflTransGen r b c"], "goal": "Relation.ReflTransGen r a c"}}
+{"state": {"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "c : ℂ", "R : ℝ", "h0 : 0 < R", "f : ℂ → E", "y : E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c R \\ {c})", "hd : ∀ z ∈ (ball c R \\ {c}) \\ s, DifferentiableAt ℂ f z", "hy : Tendsto f (𝓝[≠] c) (𝓝 y)"], "goal": "(∮ (z : ℂ) in C(c, R), (z - c)⁻¹ • f z) = (2 * ↑π * I) • y"}}
+{"state": {"context": ["ι : Type u_1", "𝕜 : Type u_2", "NontriviallyNormedField 𝕜", "H : Type u_3", "TopologicalSpace H", "E : Type u_4", "NormedAddCommGroup E", "NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "CommMonoid G", "TopologicalSpace G", "ChartedSpace H G", "SmoothMul I G", "E' : Type u_6", "NormedAddCommGroup E'", "NormedSpace 𝕜 E'", "H' : Type u_7", "TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "TopologicalSpace M", "ChartedSpace H' M", "s : Set M", "t : Finset ι", "f : ι → M → G", "n : ℕ∞", "h : ∀ i ∈ t, ContMDiffOn I' I n (f i) s"], "goal": "ContMDiffOn I' I n (fun x => ∏ i ∈ t, f i x) s"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_3, u_1} C", "CategoryTheory.Limits.HasZeroMorphisms C", "ι : Type u_2", "c : ComplexShape ι", "K : HomologicalComplex C c", "i j : ι", "hj : c.next i = j", "h : K.d i j = 0", "K.HasHomology i"], "goal": "(K.iCyclesIso i j hj h).inv ≫ K.iCycles i = 𝟙 (K.X i)"}}
+{"state": {"context": ["p : ℕ", "Fact (Nat.Prime p)"], "goal": "↑(p - 1)! = -1"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "S : Submonoid R", "P : Type u_2", "CommRing P", "Algebra R P", "I : FractionalIdeal S P"], "goal": "↑I ≠ ⊥ ↔ I ≠ 0"}}
+{"state": {"context": ["R : Type u", "ι : Type w", "s : Finset ι", "inst✝ : CommSemiring R", "f : ι → R[X]", "t : Multiset R[X]"], "goal": "t.prod.coeff 0 = (Multiset.map (fun f => f.coeff 0) t).prod"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "t : α", "ts ys : List α", "f : List α → β"], "goal": "(permutationsAux2 (id t) ts (map f []) ys ?f').2 = (permutationsAux2 t ts [] ys f).2"}}
+{"state": {"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : Nonempty α", "T : ℕ → ℝ", "g : ℝ → ℝ", "a b : α → ℝ", "r : �� → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "n : ℕ", "hn : R.n₀ ≤ n"], "goal": "∀ (i : α), r i n < n"}}
+{"state": {"context": ["α : Type u_1", "m m0 : MeasurableSpace α", "μ : MeasureTheory.Measure α", "s : Set α", "hs : MeasurableSet s", "f : α → ℝ"], "goal": "∫ (x : α) in s, |(μ[f|m]) x| ∂μ ≤ ∫ (x : α) in s, |f x| ∂μ"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "a b : ℤ"], "goal": "Finite a b ↔ Finite a.natAbs b.natAbs"}}
+{"state": {"context": ["α : Type u_1", "inst✝³ : Mul α", "inst✝² : PartialOrder α", "inst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1", "inst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt", "a b c d : α", "this : ∀ (M : Type ?u.4635) (N : Type ?u.4634) (μ : M → N → N) [inst : PartialOrder N] [inst_1 : CovariantClass M N μ fun x x_1 => x < x_1], CovariantClass M N μ fun x x_1 => x ≤ x_1"], "goal": "Ico a b * Icc c d ⊆ Ico (a * c) (b * d)"}}
+{"state": {"context": ["s : List Char", "i e n : Nat"], "goal": "go₂ s { byteIdx := i + n } { byteIdx := e + n } = go₂ s { byteIdx := i } { byteIdx := e }"}}
+{"state": {"context": ["R : Type u", "S : Type v", "T : Type w", "inst✝² : NonAssocSemiring R", "M : Submonoid R", "inst✝¹ : NonAssocSemiring S", "inst✝ : NonAssocSemiring T", "s : Set R", "x : R", "hx : x ∈ closure ↑(AddSubmonoid.closure s)"], "goal": "x ∈ closure s"}}
+{"state": {"context": ["𝕜 : Type u", "hnorm : NontriviallyNormedField 𝕜", "E : Type v", "AddCommGroup E", "Module 𝕜 E", "TopologicalSpace E", "TopologicalAddGroup E", "ContinuousSMul 𝕜 E", "F' : Type x", "AddCommGroup F'", "Module 𝕜 F'", "TopologicalSpace F'", "TopologicalAddGroup F'", "ContinuousSMul 𝕜 F'", "CompleteSpace 𝕜", "T2Space E", "FiniteDimensional 𝕜 E"], "goal": "⇑LinearMap.toContinuousLinearMap.symm = ContinuousLinearMap.toLinearMap"}}
+{"state": {"context": ["α : Type u_1", "MeasurableSpace α", "s : Set α", "s_mble : MeasurableSet s", "hsc : MeasureTheory.Measure.count s = 0"], "goal": "s = ∅"}}
+{"state": {"context": ["z : ℍ", "x : Fin 2 → ℤ"], "goal": "‖x‖₊ = max ↑(x 0).natAbs ↑(x 1).natAbs"}}
+{"state": {"context": ["R : Type u_1", "CommRing R", "I : Ideal R", "M : Type u_2", "AddCommGroup M", "Module R M", "α : Type u_4", "s : Finset α", "f : α → AdicCompletion I M", "n : ℕ"], "goal": "↑(s.sum f) n = ∑ a ∈ s, ↑(f a) n"}}
+{"state": {"context": ["𝕜 : Type u_1", "V : Type u_2", "W : Type u_3", "Q : Type u_4", "P : Type u_5", "inst✝⁴ : Ring 𝕜", "inst✝³ : AddCommGroup V", "inst✝² : Module 𝕜 V", "inst✝¹ : TopologicalSpace P", "inst✝ : AddTorsor V P", "s t : Set P", "x✝ x : P"], "goal": "intrinsicInterior 𝕜 {x} = {x}"}}
+{"state": {"context": ["M : Type u_4", "N : Type u_5", "MulOneClass M", "MulOneClass N", "x : M"], "goal": "1 x = 1"}}
+{"state": {"context": ["ι : Type u_1", "ι' : Type u_2", "κ : Type u_3", "κ' : Type u_4", "R : Type u_5", "M : Type u_6", "inst✝¹² : CommSemiring R", "inst✝¹¹ : AddCommMonoid M", "inst✝¹⁰ : Module R M", "R₂ : Type u_7", "M₂ : Type u_8", "inst✝⁹ : CommRing R₂", "inst✝⁸ : AddCommGroup M₂", "inst✝⁷ : Module R₂ M₂", "e✝ : Basis ι R M", "v✝ : ι' → M", "i : ι", "j : ι'", "N : Type u_9", "inst✝⁶ : AddCommMonoid N", "inst✝⁵ : Module R N", "b✝ : Basis ι R M", "b' : Basis ι' R M", "c : Basis κ R N", "c' : Basis κ' R N", "f : M →ₗ[R] N", "inst✝⁴ : Fintype ι'", "inst✝³ : Finite κ", "inst✝² : Fintype ι", "inst✝¹ : DecidableEq ι", "inst✝ : DecidableEq ι'", "b : Basis ι R M", "v : ι' → M", "e : ι ≃ ι'", "i✝ j✝ : ι'"], "goal": "(b.reindex e).toMatrix v i✝ j✝ = (reindexAlgEquiv R R e) (b.toMatrix (v ∘ ⇑e)) i✝ j✝"}}
+{"state": {"context": ["α : Type u_2", "GeneralizedHeytingAlgebra α", "a b : α", "h : Codisjoint a b"], "goal": "b ⇨ a = a"}}
+{"state": {"context": ["X : Type u", "Y : Type v", "TopologicalSpace X", "s : Set Y", "hs : s.Finite", "f : Y → Set X", "h : ∀ i ∈ s, IsClopen (f i)"], "goal": "IsClopen (⋂ i ∈ s, f i)"}}
+{"state": {"context": ["R : Type u", "ι : Type v", "M₁ : ι → Type w₁", "M₂ : Type w₂", "Semiring R", "(i : ι) → AddCommMonoid (M₁ i)", "AddCommMonoid M₂", "(i : ι) → Module R (M₁ i)", "Module R M₂", "(i : ι) → TopologicalSpace (M₁ i)", "TopologicalSpace M₂", "f : ContinuousMultilinearMap R M₁ M₂", "ContinuousAdd M₂", "DecidableEq ι", "Fintype ι", "x y : (i : ι) → M₁ i"], "goal": "(f.linearDeriv x) y = ∑ i : ι, f (Function.update x i (y i))"}}
+{"state": {"context": ["R : Type u_1", "B : Type u_2", "F : Type u_3", "E : B → Type u_4", "inst✝⁸ : NontriviallyNormedField R", "inst✝⁷ : (x : B) → AddCommMonoid (E x)", "inst✝⁶ : (x : B) → Module R (E x)", "inst✝⁵ : NormedAddCommGroup F", "inst✝⁴ : NormedSpace R F", "inst✝³ : TopologicalSpace B", "inst✝² : TopologicalSpace (TotalSpace F E)", "inst✝¹ : (x : B) → TopologicalSpace (E x)", "inst✝ : FiberBundle F E", "ι : Type u_5", "Z : VectorBundleCore R B F ι", "i j k : ι", "x : B", "hx : x ∈ Z.baseSet i ∩ Z.baseSet j ∩ Z.baseSet k", "v : F"], "goal": "((Z.coordChange j k x).comp (Z.coordChange i j x)) v = (Z.coordChange i k x) v"}}
+{"state": {"context": ["R : Type u", "A : Type v", "CommRing R", "Ring A", "Algebra R A"], "goal": "Algebra.Transcendental R A ↔ ¬Algebra.IsAlgebraic R A"}}
+{"state": {"context": ["N : ℕ", "inst✝¹ : NeZero N", "R : Type u_1", "inst✝ : CommRing R", "e : AddChar (ZMod N) R", "χ : DirichletCharacter R N", "d : ℕ", "hd : d ∣ N", "he : e.mulShift ↑d = 1", "u : (ZMod N)ˣ", "hu : (ZMod.unitsMap hd) u = 1"], "goal": "χ ↑u * gaussSum χ e = gaussSum χ e"}}
+{"state": {"context": ["n : ℕ+", "x : ℂˣ", "hn0 : ↑↑n ≠ 0"], "goal": "↑x ^ ↑n = 1 ↔ ∃ i < ↑n, cexp (2 * ↑π * I * (↑i / ↑↑n)) = ↑x"}}
+{"state": {"context": ["V : Type u", "G : SimpleGraph V", "G' : G.Subgraph", "v : V", "Fintype ↑(G'.neighborSet v)"], "goal": "(G'.neighborSet v).toFinset.card = G'.degree v"}}
+{"state": {"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : CommSemiring α", "s✝ : Multiset (Multiset α)", "a : Multiset α", "s : Multiset (Multiset α)", "ih : (map sum s).prod = (map prod s.Sections).sum"], "goal": "(map sum (a ::ₘ s)).prod = (map prod (a ::ₘ s).Sections).sum"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "W✝ X Y Z : C", "D : Type u₂", "inst✝² : Category.{v₂, u₂} D", "G : C ⥤ D", "f : X ⟶ Y", "g : X ⟶ Z", "inst✝¹ : HasPushout f g", "inst✝ : HasPushout (G.map f) (G.map g)", "W : C", "h : Y ⟶ W", "k : Z ⟶ W", "w : f ≫ h = g ≫ k"], "goal": "pushoutComparison G f g ≫ G.map (pushout.desc h k w) = pushout.desc (G.map h) (G.map k) ⋯"}}
+{"state": {"context": ["n : ℕ", "h₁ : 1 < n", "h₂ : ¬Nat.Prime n"], "goal": "n.FermatPsp 1"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "LinearOrder α", "TopologicalSpace γ", "a b : α", "h : a ≤ b", "TopologicalSpace α", "OrderTopology α", "TopologicalSpace β", "f : γ → ↑(Set.Icc a b) → β", "g : γ → α", "hf : Continuous ↿f", "hg : Continuous g"], "goal": "Continuous fun a_1 => Set.IccExtend h (f a_1) (g a_1)"}}
+{"state": {"context": ["C : Type u_1", "CategoryTheory.Category.{u_2, u_1} C", "CategoryTheory.Preadditive C", "CategoryTheory.HasShift C ℤ", "A : Cᵒᵖ", "T : CategoryTheory.Pretriangulated.Triangle C", "n₀ n₁ : ℤ", "h : n₀ + 1 = n₁", "x : Opposite.unop A ⟶ (CategoryTheory.shiftFunctor C n₀).obj T.obj₃"], "goal": "((CategoryTheory.preadditiveCoyoneda.obj A).homologySequenceδ T n₀ n₁ h) x =\n  CategoryTheory.CategoryStruct.comp x\n    (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor C n₀).map T.mor₃)\n      ((CategoryTheory.shiftFunctorAdd' C 1 n₀ n₁ ⋯).inv.app T.obj₁))"}}
+{"state": {"context": ["X Y Z : Scheme", "f : X ⟶ Y", "g : Y ⟶ Z", "inst✝¹ : IsClosedImmersion g", "inst✝ : IsClosedImmersion (f ≫ g)", "x : ↑↑X.toPresheafedSpace", "h : Function.Surjective ⇑(Scheme.Hom.stalkMap g (f.val.base x) ≫ Scheme.Hom.stalkMap f x)"], "goal": "Function.Surjective ⇑(Scheme.Hom.stalkMap f x)"}}
+{"state": {"context": ["α : Type u", "β : Type v", "ι✝ : Sort w", "γ : Type x", "ι : Type u_1", "s : ι → Set α", "t : Set α", "tfin : t.Finite", "h : t ⊆ ⋃ i, s i", "I : Set ι", "Ifin : I.Finite", "hI : t ⊆ ⋃ i ∈ I, s i", "x : α"], "goal": "x ∈ t ↔ x ∈ ⋃ i, (fun x => s ↑x ∩ t) i"}}
+{"state": {"context": ["z : ℂ", "τ : ℂ", "hτ : 0 < τ.im"], "goal": "HasSum (fun n => jacobiTheta₂_term_fderiv n z τ) (jacobiTheta₂_fderiv z τ)"}}
+{"state": {"context": ["C : Type u₁", "CategoryTheory.Category.{v₁, u₁} C", "CategoryTheory.MonoidalCategory C", "X Y : C", "self : CategoryTheory.ExactPairing X Y"], "goal": "Y ◁ CategoryTheory.ExactPairing.coevaluation' ≫ (α_ Y X Y).inv ≫ CategoryTheory.ExactPairing.evaluation' ▷ Y =\n  (ρ_ Y).hom ≫ (λ_ Y).inv"}}
+{"state": {"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "R R₁ R₂ : ℝ", "f : ℂ → E", "c z z₀ : ℂ", "hd : DifferentiableOn ℂ f (ball c R₁)", "h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)", "hz : z ∈ ball c R₁"], "goal": "‖dslope f c z‖ ≤ R₂ / R₁"}}
+{"state": {"context": ["J : Type w", "C : Type u", "CategoryTheory.Category.{v, u} C", "CategoryTheory.Limits.HasZeroMorphisms C", "Unique J", "f : J → C"], "goal": "(CategoryTheory.Limits.biproductUniqueIso f).hom =\n  CategoryTheory.Limits.biproduct.desc (CategoryTheory.Limits.limitBiconeOfUnique f).bicone.ι"}}
+{"state": {"context": ["L : Language", "T✝ : L.Theory", "α : Type w", "n : ℕ", "T' : L.Theory", "ι : Type u_1", "inst✝ : Nonempty ι", "T : ι → L.Theory", "h : Directed (fun x x_1 => x ⊆ x_1) T", "h' : ∀ (i : ι), (T i).IsSatisfiable"], "goal": "∀ (T0 : Finset L.Sentence), ↑T0 ⊆ ⋃ i, T i → IsSatisfiable ↑T0"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : Preadditive C", "F : C ⥤ C", "inst✝ : F.Additive", "A₁ A₂ : Coalgebra F", "a✝ b✝ : A₁ ⟶ A₂"], "goal": "(a✝ - b✝).f = (a✝ + -b✝).f"}}
+{"state": {"context": ["l : List ℕ", "h : ∀ p ∈ l, Nat.Prime p"], "goal": "↑(PrimeMultiset.ofNatList l h).prod = l.prod"}}
+{"state": {"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "Z X✝ Y P : C", "f : Z ⟶ X✝", "g : Z ⟶ Y", "inl : X✝ ⟶ P", "inr : Y ⟶ P", "inst✝¹ : HasZeroObject C", "inst✝ : HasZeroMorphisms C", "X : C", "s : Cocone (span 0 (𝟙 X))", "c : ∀ (j : WalkingSpan), s.ι.app j ≫ 0 = s.ι.app j ≫ 𝟙 s.pt"], "goal": "∀ (j : WalkingSpan), (PushoutCocone.mk 0 0 ⋯).ι.app j ≫ (fun s => 0) s = s.ι.app j"}}
+{"state": {"context": ["α✝ : Type u_1", "G : Type u_2", "α : Type u_3", "inst✝⁴ : MeasurableSpace G", "inst✝³ : Div G", "m✝ : MeasurableSpace α", "f✝ g✝ : α → G", "μ : Measure α", "m : MeasurableSpace α", "E : Type u_4", "inst✝² : MeasurableSpace E", "inst✝¹ : MeasurableSingletonClass E", "inst✝ : Countable E", "f g : α → E", "hf : Measurable f", "hg : Measurable g", "this : {x | f x = g x} = ⋃ j, {x | f x = j} ∩ {x | g x = j}", "j : E"], "goal": "MeasurableSet {x | g x = j}"}}
+{"state": {"context": ["K : Type u_1", "L : Type u_2", "inst✝⁷ : Field K", "inst✝⁶ : CharZero K", "inst✝⁵ : LieRing L", "inst✝⁴ : LieAlgebra K L", "inst✝³ : IsKilling K L", "inst✝² : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝¹ : H.IsCartanSubalgebra", "inst✝ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα✝ : α.IsNonZero", "hα : ¬α.IsZero"], "goal": "chainBotCoeff (⇑α) β + chainTopCoeff (⇑α) β ≤ chainLength α β"}}
+{"state": {"context": ["V : Type u_1", "P : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : MetricSpace P", "inst✝¹ : NormedAddTorsor V P", "hd2 : Fact (finrank ℝ V = 2)", "inst✝ : Module.Oriented ℝ V (Fin 2)", "p₁ p₂ p₃ p₄ : P", "h : 2 • ∡ p₁ p₂ p₄ = 2 • ∡ p₁ p₃ p₄", "hn : ¬Collinear ℝ {p₁, p₂, p₄}"], "goal": "Cospherical {p₁, p₂, p₃, p₄}"}}
+{"state": {"context": ["α : Type u_1", "a b c d : ℝ≥0∞", "r p q : ℝ≥0", "x : ℝ≥0∞"], "goal": "x.toReal = 1 ↔ x = 1"}}
+{"state": {"context": ["C : Type u", "CategoryTheory.Category.{v, u} C", "β : Type u_1", "X Y : CategoryTheory.GradedObject β C", "e : (i : β) → X i ≅ Y i", "i : β"], "goal": "(X.isoMk Y e).hom i = (e i).hom"}}
+{"state": {"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Preadditive C", "R : Type u_1", "inst✝¹ : Ring R", "inst✝ : Linear R C", "F G K L : CochainComplex C ℤ", "n m n₁ : ℤ", "z₁ : Cochain F G n₁", "z₂ : Cochain G K 0", "m₁ : ℤ", "h₁ : n₁ + 1 = m₁"], "goal": "δ n₁ m₁ (z₁.comp z₂ ⋯) = z₁.comp (δ 0 1 z₂) h₁ + (δ n₁ m₁ z₁).comp z₂ ⋯"}}
+{"state": {"context": ["α : Type u_1", "𝕜 : Type u_2", "inst✝¹⁰ : NontriviallyNormedField 𝕜", "E : Type u_3", "F : Type u_4", "G : Type u_5", "H : Type u_6", "inst✝⁹ : NormedAddCommGroup E", "inst✝⁸ : NormedSpace 𝕜 E", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace 𝕜 F", "inst✝⁵ : NormedAddCommGroup G", "inst✝⁴ : NormedSpace 𝕜 G", "inst✝³ : NormedAddCommGroup H", "inst✝² : NormedSpace 𝕜 H", "inst✝¹ : CompleteSpace F", "inst✝ : CompleteSpace G", "f : F → G", "g : E → F", "s : Set F", "t : Set E", "x : E", "hf : AnalyticWithinAt 𝕜 f s (g x)", "hg : AnalyticWithinAt 𝕜 g t x", "h : MapsTo g t s"], "goal": "AnalyticWithinAt 𝕜 (f ∘ g) t x"}}
+{"state": {"context": ["α : Type u_1", "β : Type u_2", "m : MeasurableSpace α", "M : Type u_3", "inst✝² : TopologicalSpace M", "inst✝¹ : OrderedAddCommMonoid M", "inst✝ : OrderClosedTopology M", "v w : VectorMeasure α M", "i j : Set α", "f : ℕ → Set α", "hf₁ : ∀ (n : ℕ), MeasurableSet (f n)", "hf₂ : ∀ (n : ℕ), v.restrict (f n) ≤ w.restrict (f n)", "a : Set α", "ha₁ : MeasurableSet a", "ha₂ : a ⊆ ⋃ n, f n", "ha₃ : ⋃ n, a ∩ disjointed f n = a", "ha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)"], "goal": "↑v a ≤ ↑w a"}}
+{"state": {"context": ["R : Type u_1", "inst✝⁴ : CommRing R", "C : Type u", "inst✝³ : Category.{v, u} C", "D : Type u", "inst✝² : Category.{v, u} D", "inst✝¹ : Preadditive D", "inst✝ : Linear R D", "F : C ⥤ D", "X Y Z : Free R C", "f : X ⟶ Y", "g : Y ⟶ Z"], "goal": "∀ (a : X ⟶ Y) (b : R), { obj := fun X => F.obj X, map := fun {X Y} f => sum f fun f' r => r • F.map f' }.map (single a b ≫ g) = { obj := fun X => F.obj X, map := fun {X Y} f => sum f fun f' r => r • F.map f' }.map (single a b) ≫ { obj := fun X => F.obj X, map := fun {X Y} f => sum f fun f' r => r • F.map f' }.map g"}}
+{"state": {"context": ["n i : Nat"], "goal": "OfNat.ofNat i = BitVec.ofNat n i"}}
+{"state": {"context": ["α : Type u_2", "Preorder α", "LocallyFiniteOrder α", "a b : α"], "goal": "Finset.Ico a b ⊆ Finset.Icc a b"}}
+{"state": {"context": ["X : Type u_1", "Y : Type u_2", "E : Type u_3", "F : Type u_4", "R : Type u_5", "inst✝⁸ : MeasurableSpace X", "inst✝⁷ : TopologicalSpace X", "inst✝⁶ : MeasurableSpace Y", "inst✝⁵ : TopologicalSpace Y", "inst✝⁴ : NormedAddCommGroup E", "inst✝³ : NormedAddCommGroup F", "f g✝ : X → E", "μ : Measure X", "s : Set X", "inst✝² : OpensMeasurableSpace X", "A K : Set X", "inst✝¹ : NormedRing R", "inst✝ : SecondCountableTopologyEither X R", "g g' : X → R", "hg : Memℒp g 1 (μ.restrict A)", "hg' : ContinuousOn g' K", "hA : MeasurableSet A", "hK : IsCompact K", "hAK : A ⊆ K", "C : ℝ", "hC : ∀ x ∈ K, ‖g' x‖ ≤ C"], "goal": "Memℒp (fun x => g x * g' x) 1 (μ.restrict A)"}}
+{"state": {"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝ p q : R[X]", "ι : Type u_1", "n : ℕ", "ih : p.leadingCoeff ^ n ≠ 0 → (p ^ n).leadingCoeff = p.leadingCoeff ^ n", "h : p.leadingCoeff ^ n.succ ≠ 0"], "goal": "(p ^ n.succ).leadingCoeff = p.leadingCoeff ^ n.succ"}}