diff --git "a/random/random_our/val_increase.jsonl" "b/random/random_our/val_increase.jsonl"
new file mode 100644--- /dev/null
+++ "b/random/random_our/val_increase.jsonl"
@@ -0,0 +1,3310 @@
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+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "C : Type u_3", "inst✝¹¹ : CommSemiring R", "inst✝¹⁰ : NonUnitalSemiring A", "inst✝⁹ : StarRing R", "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 : StarRing C", "inst✝¹ : Algebra R C", "inst✝ : StarModule R C", "f : A →⋆ₙₐ[R] C", "c : StarSubalgebra R C"], "goal": "NonUnitalStarAlgHom.range f ≤ c.toNonUnitalStarSubalgebra ↔ ↑(NonUnitalStarAlgHom.range f) ⊆ ↑c"}], "premise": [1713], "state_str": "R : Type u_1\nA : Type u_2\nC : Type u_3\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : StarRing R\ninst✝⁸ : StarRing A\ninst✝⁷ : Module R A\ninst✝⁶ : SMulCommClass R A A\ninst✝⁵ : IsScalarTower R A A\ninst✝⁴ : StarModule R A\ninst✝³ : Semiring C\ninst✝² : StarRing C\ninst✝¹ : Algebra R C\ninst✝ : StarModule R C\nf : A →⋆ₙₐ[R] C\nc : StarSubalgebra R C\n⊢ NonUnitalStarAlgHom.range f ≤ c.toNonUnitalStarSubalgebra ↔ ↑(NonUnitalStarAlgHom.range f) ⊆ ↑c"}
+{"state": [{"context": ["α : Type u", "β : Type u_1", "n : ℕ", "x : α", "inst✝ : StrictOrderedRing α", "hx : x < 0", "hx' : 0 < x + 1", "hn : 1 < n"], "goal": "0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1"}], "premise": [145198], "state_str": "α : Type u\nβ : Type u_1\nn : ℕ\nx : α\ninst✝ : StrictOrderedRing α\nhx : x < 0\nhx' : 0 < x + 1\nhn : 1 < n\n⊢ 0 < ∑ i ∈ range n, x ^ i ∧ ∑ i ∈ range n, x ^ i < 1"}
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+{"state": [{"context": ["R : Type u", "inst✝² : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝¹ : Field F", "W : Jacobian F", "inst✝ : NoZeroDivisors R", "P Q : Fin 3 → R", "hP : W'.Equation P", "hQ : W'.Equation Q", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy : P y * Q z ^ 3 ≠ W'.negY Q * P z ^ 3", "hy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0"], "goal": "P y ≠ W'.negY P"}], "premise": [53688], "state_str": "R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W'.negY Q * P z ^ 3\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\n⊢ P y ≠ W'.negY P"}
+{"state": [{"context": ["R : Type u", "inst✝² : CommRing R", "W' : Jacobian R", "F : Type v", "inst✝¹ : Field F", "W : Jacobian F", "inst✝ : NoZeroDivisors R", "P Q : Fin 3 → R", "hP : W'.Equation P", "hQ : W'.Equation Q", "hx : P x * Q z ^ 2 = Q x * P z ^ 2", "hy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0", "hy : P y = W'.negY P"], "goal": "P y * Q z ^ 3 = W'.negY Q * P z ^ 3"}], "premise": [53890, 53893, 53896, 53903, 145503], "state_str": "R : Type u\ninst✝² : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\nhx : P x * Q z ^ 2 = Q x * P z ^ 2\nhy' : P y * Q z ^ 3 - Q y * P z ^ 3 = 0\nhy : P y = W'.negY P\n⊢ P y * Q z ^ 3 = W'.negY Q * P z ^ 3"}
+{"state": [{"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I : Box ι", "i✝ : ι", "x✝ : ℝ", "y : ι → ℝ", "i : ι", "x : ℝ"], "goal": "I.splitUpper i x = ⊥ ↔ I.upper i ≤ x"}], "premise": [34370, 71471], "state_str": "ι : Type u_1\nM : Type u_2\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ I.splitUpper i x = ⊥ ↔ I.upper i ≤ x"}
+{"state": [{"context": ["ι : Type u_1", "M : Type u_2", "n : ℕ", "I : Box ι", "i✝ : ι", "x✝ : ℝ", "y : ι → ℝ", "i : ι", "x : ℝ"], "goal": "(I.upper i ≤ max x (I.lower i) ∨ ∃ x, x ≠ i ∧ I.upper x ≤ I.lower x) ↔ I.upper i ≤ x"}], "premise": [34334], "state_str": "ι : Type u_1\nM : Type u_2\nn : ℕ\nI : Box ι\ni✝ : ι\nx✝ : ℝ\ny : ι → ℝ\ni : ι\nx : ℝ\n⊢ (I.upper i ≤ max x (I.lower i) ∨ ∃ x, x ≠ i ∧ I.upper x ≤ I.lower x) ↔ I.upper i ≤ x"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "s : ℕ → Set α", "H : ∀ (x : α), ∃ n, x ∈ s n", "inst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)", "n : ℕ", "x : α"], "goal": "x ∈ (fun x => Nat.find ⋯) ⁻¹' {n} ↔ x ∈ disjointed s n"}], "premise": [12881, 143303], "state_str": "case h\nα : Type u_1\nβ : Type u_2\ns : ℕ → Set α\nH : ∀ (x : α), ∃ n, x ∈ s n\ninst✝ : (x : α) → (n : ℕ) → Decidable (x ∈ s n)\nn : ℕ\nx : α\n⊢ x ∈ (fun x => Nat.find ⋯) ⁻¹' {n} ↔ x ∈ disjointed s n"}
+{"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", "x y : ⨂[R] (i : ι), s i", "p q : FreeAddMonoid (R × ((i : ι) → s i))", "hp : p ∈ x.lifts", "hq : q ∈ y.lifts"], "goal": "p + q ∈ (x + y).lifts"}], "premise": [7241, 131585], "state_str": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid E\ninst✝¹ : Module R E\nF : Type u_10\ninst✝ : AddCommMonoid F\nx y : ⨂[R] (i : ι), s i\np q : FreeAddMonoid (R × ((i : ι) → s i))\nhp : p ∈ x.lifts\nhq : q ∈ y.lifts\n⊢ p + q ∈ (x + y).lifts"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.30360, u_1} C", "inst✝ : HasZeroMorphisms C", "S S₁ S₂ S₃ : ShortComplex C", "h : S.RightHomologyData", "A : C", "hf : S.f = 0", "hg : S.g = 0"], "goal": "S.f ≫ 𝟙 S.X₂ = 0"}], "premise": [96174], "state_str": "C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ S.f ≫ 𝟙 S.X₂ = 0"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.30360, u_1} C", "inst✝ : HasZeroMorphisms C", "S S₁ S₂ S₃ : ShortComplex C", "h : S.RightHomologyData", "A : C", "hf : S.f = 0", "hg : S.g = 0"], "goal": "𝟙 S.X₂ ≫ S.g = 0"}], "premise": [93604], "state_str": "C : Type u_1\ninst✝¹ : Category.{?u.30360, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nh : S.RightHomologyData\nA : C\nhf : S.f = 0\nhg : S.g = 0\n⊢ 𝟙 S.X₂ ≫ S.g = 0"}
+{"state": [{"context": ["r : ℝ"], "goal": "0 < r.toNNReal ↔ 0 < r"}], "premise": [14279, 146644], "state_str": "r : ℝ\n⊢ 0 < r.toNNReal ↔ 0 < 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 : AddGroup α", "inst✝ : AddGroup β", "h : Antiperiodic f c", "n : ℤ"], "goal": "f (x - n • c) = ↑n.negOnePow • f x"}], "premise": [109519, 117848, 119789, 122200], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : AddGroup β\nh : Antiperiodic f c\nn : ℤ\n⊢ f (x - n • c) = ↑n.negOnePow • f x"}
+{"state": [{"context": ["α : Type u_1", "s✝ t✝ : Finset α", "inst✝ : DecidableEq α", "s t : Finset α"], "goal": "t ∈ s.ssubsets ↔ t ⊂ s"}], "premise": [1713, 1723, 136708, 138701, 138958], "state_str": "α : Type u_1\ns✝ t✝ : Finset α\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ t ∈ s.ssubsets ↔ t ⊂ 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", "c x y : P", "R : ℝ", "hR : R ≠ 0", "hy : y ≠ c"], "goal": "inversion c R ⁻¹' ↑(perpBisector c y) = sphere (inversion c R y) (R ^ 2 / dist y c) \\ {c}"}], "premise": [68673, 69018, 69020], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc x y : P\nR : ℝ\nhR : R ≠ 0\nhy : y ≠ c\n⊢ inversion c R ⁻¹' ↑(perpBisector c y) = sphere (inversion c R y) (R ^ 2 / dist y c) \\ {c}"}
+{"state": [{"context": ["G : Type u_1", "inst✝ : CommGroup G", "a b : G"], "goal": "a⁻¹ * b * a = b"}], "premise": [119707, 119829], "state_str": "G : Type u_1\ninst✝ : CommGroup G\na b : G\n⊢ a⁻¹ * b * a = b"}
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+{"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", "hs : MeasurableSet s"], "goal": "(μ.restrict s) t = μ (t ∩ s)"}], "premise": [27702, 29071, 32213], "state_str": "R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nhs : MeasurableSet s\n⊢ (μ.restrict s) t = μ (t ∩ 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✝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", "α : Type u_8", "l : Filter α", "g : α → M", "hg : ∀ᶠ (z : α) in l, g z ∈ f.source", "y : M", "hy : y ∈ f.source"], "goal": "Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y)) ↔ Tendsto g l (𝓝 y)"}], "premise": [1674, 16166, 16355, 55619, 67811], "state_str": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\n⊢ Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y)) ↔ Tendsto g l (𝓝 y)"}
+{"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", "α : Type u_8", "l : Filter α", "g : α → M", "hg : ∀ᶠ (z : α) in l, g z ∈ f.source", "y : M", "hy : y ∈ f.source", "h : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))", "u : Set M", "hu : u ∈ 𝓝 y"], "goal": "g ⁻¹' u ∈ l"}], "premise": [16355, 55619, 67821], "state_str": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\nα : Type u_8\nl : Filter α\ng : α → M\nhg : ∀ᶠ (z : α) in l, g z ∈ f.source\ny : M\nhy : y ∈ f.source\nh : Tendsto (↑(f.extend I) ∘ g) l (𝓝 (↑(f.extend I) y))\nu : Set M\nhu : u ∈ 𝓝 y\n⊢ g ⁻¹' u ∈ l"}
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+{"state": [{"context": [], "goal": "Continuous negMulLog"}], "premise": [37741, 37751, 66669], "state_str": "⊢ Continuous negMulLog"}
+{"state": [{"context": ["α : Type u", "inst✝ : Infinite α", "c : Cardinal.{u}"], "goal": "#{ t // #↑t ≤ c } ≤ #α ^ c"}], "premise": [14273, 48804, 49461], "state_str": "α : Type u\ninst✝ : Infinite α\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ #α ^ c"}
+{"state": [{"context": ["α : Type u", "inst✝ : Infinite α", "c : Cardinal.{u}"], "goal": "#{ t // #↑t ≤ c } ≤ (#α + 1) ^ c"}], "premise": [48571], "state_str": "α : Type u\ninst✝ : Infinite α\nc : Cardinal.{u}\n⊢ #{ t // #↑t ≤ c } ≤ (#α + 1) ^ c"}
+{"state": [{"context": ["α : Type u", "inst✝ : Infinite α", "β : Type u"], "goal": "#{ t // #↑t ≤ #β } ≤ (#α + 1) ^ #β"}], "premise": [48589], "state_str": "case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\n⊢ #{ t // #↑t ≤ #β } ≤ (#α + 1) ^ #β"}
+{"state": [{"context": ["α : Type u", "inst✝ : Infinite α", "β : Type u", "s : Set α", "g : ↑s ↪ β"], "goal": "∃ a, (fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) a = ⟨s, ⋯⟩"}], "premise": [1674, 2045], "state_str": "case h.hf.mk.intro\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ∃ a, (fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) a = ⟨s, ⋯⟩"}
+{"state": [{"context": ["α : Type u", "inst✝ : Infinite α", "β : Type u", "s : Set α", "g : ↑s ↪ β"], "goal": "((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y => if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) = ⟨s, ⋯⟩"}], "premise": [1751], "state_str": "case h\nα : Type u\ninst✝ : Infinite α\nβ : Type u\ns : Set α\ng : ↑s ↪ β\n⊢ ((fun f => ⟨Sum.inl ⁻¹' range f, ⋯⟩) fun y =>\n      if h : ∃ x, g x = y then Sum.inl ↑(Classical.choose h) else Sum.inr { down := 0 }) =\n    ⟨s, ⋯⟩"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "X Y Z✝ : C", "f : Y ⟶ X", "ι : Type u_1", "Z : ι → C", "g : (i : ι) → Z i ⟶ X", "j : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2", "W : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C", "k : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y"], "goal": "((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) = ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst"}], "premise": [1838], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z✝ : C\nf : Y ⟶ X\nι : Type u_1\nZ : ι → C\ng : (i : ι) → Z i ⟶ X\nj : ⦃Y : C⦄ → (f : Y ⟶ X) → ofArrows Z g f → Type u_2\nW : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → j f H → C\nk : ⦃Y : C⦄ → (f : Y ⟶ X) → (H : ofArrows Z g f) → (i : j f H) → W f H i ⟶ Y\n⊢ ((ofArrows Z g).bind fun Y f H => ofArrows (W f H) (k f H)) =\n    ofArrows (fun i => W (g i.fst) ⋯ i.snd) fun ij => k (g ij.fst) ⋯ ij.snd ≫ g ij.fst"}
+{"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", "n : ℕ", "F : L.Functions n", "i : ι", "x : Fin n → G i"], "goal": "(funMap F fun a => ⟦Structure.Sigma.mk f i (x a)⟧) = ⟦Structure.Sigma.mk f i (funMap F x)⟧"}], "premise": [24579, 127781], "state_str": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\n⊢ (funMap F fun a => ⟦Structure.Sigma.mk f i (x a)⟧) = ⟦Structure.Sigma.mk f i (funMap F x)⟧"}
+{"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", "n : ℕ", "F : L.Functions n", "i : ι", "x : Fin n → G i", "k : ι", "ik : i ≤ k", "jk : Classical.choose ⋯ ≤ k"], "goal": "(f (Classical.choose ⋯) k jk) (funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) (Classical.choose ⋯) ⋯)) = (f i k ik) (funMap F x)"}], "premise": [24277, 25036], "state_str": "case intro.intro\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝¹ : DirectedSystem G fun i j h => ⇑(f i j h)\ninst✝ : Nonempty ι\nn : ℕ\nF : L.Functions n\ni : ι\nx : Fin n → G i\nk : ι\nik : i ≤ k\njk : Classical.choose ⋯ ≤ k\n⊢ (f (Classical.choose ⋯) k jk)\n      (funMap F (unify f (fun i_1 => Structure.Sigma.mk f i (x i_1)) (Classical.choose ⋯) ⋯)) =\n    (f i k ik) (funMap F x)"}
+{"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"], "goal": "a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < a"}], "premise": [14317], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\n⊢ a⁻¹ < 1 ↔ a ≤ 0 ∨ 1 < 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 a) (κ.mutuallySingularSetSlice η a) = 0"}], "premise": [73326], "state_str": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0"}
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+{"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 : ∀ (a : α) (s : Set γ), MeasurableSet s → ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s"], "goal": "(κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0"}], "premise": [73308], "state_str": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ✝ η✝ : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\na : α\nh_eq_add : ∀ (a : α) (s : Set γ), MeasurableSet s → ((η.withDensity (κ.rnDeriv η) + κ.singularPart η) a) s = (κ a) s\n⊢ (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0"}
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+{"state": [{"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "h : 1 ∈ I ⊔ J", "i : R", "hi : i ∈ I ⊔ K"], "goal": "i ∈ I ⊔ J * K"}], "premise": [86732], "state_str": "R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\nh : 1 ∈ I ⊔ J\ni : R\nhi : i ∈ I ⊔ K\n⊢ i ∈ I ⊔ J * K"}
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+{"state": [{"context": ["R : Type u", "ι : Type u_1", "inst✝ : CommSemiring R", "I J K L : Ideal R", "i i1 : R", "hi1 : i1 ∈ I", "j : R", "hj : j ∈ J", "h : i1 + j = 1", "i' : R", "hi' : i' ∈ I", "k : R", "hk : k ∈ K", "hi : i' + k = i"], "goal": "i' + i1 * k + j * k = i"}], "premise": [119704, 119728], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\nι : Type u_1\ninst✝ : CommSemiring R\nI J K L : Ideal R\ni i1 : R\nhi1 : i1 ∈ I\nj : R\nhj : j ∈ J\nh : i1 + j = 1\ni' : R\nhi' : i' ∈ I\nk : R\nhk : k ∈ K\nhi : i' + k = i\n⊢ i' + i1 * k + j * k = i"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "δ : Type w", "s : Stream' α", "n : ℕ"], "goal": "(zip appendStream' s.inits s.tails).get n = (const s).get n"}], "premise": [127845, 127883, 127893, 127934, 127942, 127951, 127958, 127964], "state_str": "case a\nα : Type u\nβ : Type v\nδ : Type w\ns : Stream' α\nn : ℕ\n⊢ (zip appendStream' s.inits s.tails).get n = (const s).get n"}
+{"state": [{"context": ["R : Type u_1", "l : Type u_2", "m : Type u_3", "n : Type u_4", "α : Type u_5", "β : Type u_6", "ι : Type u_7", "inst✝⁵ : Fintype l", "inst✝⁴ : Fintype m", "inst✝³ : Fintype n", "inst✝² : Unique ι", "inst✝¹ : SeminormedAddCommGroup α", "inst✝ : SeminormedAddCommGroup β", "A : Matrix m n α", "f : α → β", "hf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊"], "goal": "‖A.map f‖₊ = ‖A‖₊"}], "premise": [34690, 142132], "state_str": "R : Type u_1\nl : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁵ : Fintype l\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : Unique ι\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nA : Matrix m n α\nf : α → β\nhf : ∀ (a : α), ‖f a‖₊ = ‖a‖₊\n⊢ ‖A.map f‖₊ = ‖A‖₊"}
+{"state": [{"context": ["α : Sort u_2", "inst✝¹ : DecidableEq α", "β : Sort u_1", "inst✝ : DecidableEq β", "a b : α", "e : α ≃ β", "x : β"], "goal": "((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x"}], "premise": [1717], "state_str": "α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\n⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x"}
+{"state": [{"context": ["α : Sort u_2", "inst✝¹ : DecidableEq α", "β : Sort u_1", "inst✝ : DecidableEq β", "a b : α", "e : α ≃ β", "x : β", "this : ∀ (a : α), e.symm x = a ↔ x = e a"], "goal": "((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x"}], "premise": [70750, 71875], "state_str": "α : Sort u_2\ninst✝¹ : DecidableEq α\nβ : Sort u_1\ninst✝ : DecidableEq β\na b : α\ne : α ≃ β\nx : β\nthis : ∀ (a : α), e.symm x = a ↔ x = e a\n⊢ ((e.symm.trans (swap a b)).trans e) x = (swap (e a) (e b)) x"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : DivisionMonoid α", "a✝ b c d a : α", "n : ℕ"], "goal": "a⁻¹ ^ ↑n = (a ^ ↑n)⁻¹"}], "premise": [117845, 119784], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na✝ b c d a : α\nn : ℕ\n⊢ a⁻¹ ^ ↑n = (a ^ ↑n)⁻¹"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : DivisionMonoid α", "a✝ b c d a : α", "n : ℕ"], "goal": "a⁻¹ ^ Int.negSucc n = (a ^ Int.negSucc n)⁻¹"}], "premise": [117845, 119787], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na✝ b c d a : α\nn : ℕ\n⊢ a⁻¹ ^ Int.negSucc n = (a ^ Int.negSucc n)⁻¹"}
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+{"state": [{"context": ["R : Type u", "S : Type v", "a b : R", "m n : ℕ", "ι : Type y", "inst✝ : Semiring R", "f : R[X]", "i : ℕ"], "goal": "f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)"}], "premise": [102150], "state_str": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\nf : R[X]\ni : ℕ\n⊢ f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)"}
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+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "inst✝ : MonoidalCategory C", "M : Comon_ C", "A : Mon_ Cᵒᵖ"], "goal": "A.mul.unop ≫ unop A.X ◁ A.one.unop = (ρ_ (unop A.X)).inv"}], "premise": [89633, 98602, 100051], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : MonoidalCategory C\nM : Comon_ C\nA : Mon_ Cᵒᵖ\n⊢ A.mul.unop ≫ unop A.X ◁ A.one.unop = (ρ_ (unop A.X)).inv"}
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+{"state": [{"context": ["K : Type u", "inst✝² : CommRing K", "p : ℕ", "inst✝¹ : Fact (Nat.Prime p)", "inst✝ : CharP K p", "x : ℕ × K"], "goal": "mk K p x ^ p = mk K p (x.1, x.2 ^ p)"}], "premise": [87868], "state_str": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\n⊢ mk K p x ^ p = mk K p (x.1, x.2 ^ p)"}
+{"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 : ℝ → E", "a b : ℝ", "f' : ℝ → E", "C : ℝ", "hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x", "bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C"], "goal": "∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - a)"}], "premise": [44440, 46629], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\n⊢ ∀ x ∈ Icc a b, ‖f x - f a‖ ≤ C * (x - 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", "f : ℝ → E", "a b : ℝ", "f' : ℝ → E", "C : ℝ", "hf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x", "bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C", "x : ℝ", "hx : x ∈ Ico a b"], "goal": "HasDerivWithinAt f (f' x) (Ici x) x"}], "premise": [20231, 44363, 54961], "state_str": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nf' : ℝ → E\nC : ℝ\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ C\nx : ℝ\nhx : x ∈ Ico a b\n⊢ HasDerivWithinAt f (f' x) (Ici x) x"}
+{"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✝", "s✝ : Finset β✝", "H : s✝.Nonempty", "f✝ : β✝ → α✝", "ι : Type u_7", "κ : Type u_8", "α : Type u_9", "β : Type u_10", "inst✝¹ : SemilatticeSup α", "inst✝ : SemilatticeSup β", "s : Finset ι", "t : Finset κ", "f : ι → α", "g : κ → β", "i : α × β", "a : ι", "ha : a ∈ s", "b : κ", "hb : b ∈ t"], "goal": "(s.sup' ⋯ f, t.sup' ⋯ g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i"}], "premise": [1186, 11350, 136829, 137296, 139771], "state_str": "case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ (s.sup' ⋯ f, t.sup' ⋯ g) ≤ i ↔ (s ×ˢ t).sup' ⋯ (Prod.map f g) ≤ i"}
+{"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✝", "s✝ : Finset β✝", "H : s✝.Nonempty", "f✝ : β✝ → α✝", "ι : Type u_7", "κ : Type u_8", "α : Type u_9", "β : Type u_10", "inst✝¹ : SemilatticeSup α", "inst✝ : SemilatticeSup β", "s : Finset ι", "t : Finset κ", "f : ι → α", "g : κ → β", "i : α × β", "a : ι", "ha : a ∈ s", "b : κ", "hb : b ∈ t"], "goal": "((∀ b ∈ s, f b ≤ i.1) ∧ ∀ b ∈ t, g b ≤ i.2) ↔ ∀ (a : ι) (b : κ), a ∈ s → b ∈ t → f a ≤ i.1 ∧ g b ≤ i.2"}], "premise": [2106, 2107], "state_str": "case intro.intro\nF : Type u_1\nα✝ : Type u_2\nβ✝ : Type u_3\nγ : Type u_4\nι✝ : Type u_5\nκ✝ : Type u_6\ninst✝² : SemilatticeSup α✝\ns✝ : Finset β✝\nH : s✝.Nonempty\nf✝ : β✝ → α✝\nι : Type u_7\nκ : Type u_8\nα : Type u_9\nβ : Type u_10\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns : Finset ι\nt : Finset κ\nf : ι → α\ng : κ → β\ni : α × β\na : ι\nha : a ∈ s\nb : κ\nhb : b ∈ t\n⊢ ((∀ b ∈ s, f b ≤ i.1) ∧ ∀ b ∈ t, g b ≤ i.2) ↔ ∀ (a : ι) (b : κ), a ∈ s → b ∈ t → f a ≤ i.1 ∧ g b ≤ i.2"}
+{"state": [{"context": ["ι : Type u", "γ✝ : Type w", "β : ι → Type v", "β₁ : ι → Type v₁", "β₂ : ι → Type v₂", "inst✝³ : DecidableEq ι", "γ : Type w", "α : Type x", "inst✝² : (i : ι) → AddCommMonoid (β i)", "inst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)", "inst✝ : CommMonoid γ", "s : Finset α", "g : α → Π₀ (i : ι), β i", "h : (i : ι) → β i → γ", "h_zero : ∀ (i : ι), h i 0 = 1", "h_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ * h i b₂"], "goal": "∏ i ∈ s, (g i).prod h = (∑ i ∈ s, g i).prod h"}], "premise": [138847, 149003, 149022], "state_str": "ι : Type u\nγ✝ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\nγ : Type w\nα : Type x\ninst✝² : (i : ι) → AddCommMonoid (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\ns : Finset α\ng : α → Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_zero : ∀ (i : ι), h i 0 = 1\nh_add : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ + b₂) = h i b₁ * h i b₂\n⊢ ∏ i ∈ s, (g i).prod h = (∑ i ∈ s, g i).prod h"}
+{"state": [{"context": ["α : Type u_1", "ι : Type u_2", "inst✝³ : TopologicalSpace α", "inst✝² : PolishSpace α", "inst✝¹ : MeasurableSpace α", "inst✝ : BorelSpace α"], "goal": "∀ {s : Set α}, MeasurableSet s → IsClopenable s"}], "premise": [25862], "state_str": "α : Type u_1\nι : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\n⊢ ∀ {s : Set α}, MeasurableSet s → IsClopenable s"}
+{"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", "a : Multiset α"], "goal": "Rel r a 0 ↔ a = 0"}], "premise": [138209], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\nr : α → β → Prop\np : γ → δ → Prop\na : Multiset α\n⊢ Rel r a 0 ↔ a = 0"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : DecidableEq α", "l : List α", "x : α", "xs : List α", "h : xs.Nodup", "n : ℕ", "hn : n < xs.length"], "goal": "xs.formPerm (xs.nthLe n hn) = xs.nthLe ((n + 1) % xs.length) ⋯"}], "premise": [8825], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\nl : List α\nx : α\nxs : List α\nh : xs.Nodup\nn : ℕ\nhn : n < xs.length\n⊢ xs.formPerm (xs.nthLe n hn) = xs.nthLe ((n + 1) % xs.length) ⋯"}
+{"state": [{"context": ["R : Type u_1", "L : Type u_2", "M : Type u_3", "n : Type u_4", "ι : Type u_5", "ι' : Type u_6", "ιM : Type u_7", "inst✝¹² : CommRing R", "inst✝¹¹ : AddCommGroup L", "inst✝¹⁰ : Module R L", "inst✝⁹ : AddCommGroup M", "inst✝⁸ : Module R M", "φ : L →ₗ[R] Module.End R M", "inst✝⁷ : Fintype ι", "inst✝⁶ : Fintype ι'", "inst✝⁵ : Fintype ιM", "inst✝⁴ : DecidableEq ι", "inst✝³ : DecidableEq ι'", "inst✝² : DecidableEq ιM", "b : Basis ι R L", "bₘ : Basis ιM R M", "inst✝¹ : Module.Finite R M", "inst✝ : Module.Free R M", "x : ι → R"], "goal": "Polynomial.map (MvPolynomial.eval x) (φ.polyCharpolyAux b bₘ) = charpoly (φ (b.repr.symm (Finsupp.equivFunOnFinite.symm x)))"}], "premise": [70745, 110523, 110776, 148059], "state_str": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\nι : Type u_5\nι' : Type u_6\nιM : Type u_7\ninst✝¹² : CommRing R\ninst✝¹¹ : AddCommGroup L\ninst✝¹⁰ : Module R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nφ : L →ₗ[R] Module.End R M\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : Fintype ιM\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : DecidableEq ιM\nb : Basis ι R L\nbₘ : Basis ιM R M\ninst✝¹ : Module.Finite R M\ninst✝ : Module.Free R M\nx : ι → R\n⊢ Polynomial.map (MvPolynomial.eval x) (φ.polyCharpolyAux b bₘ) =\n    charpoly (φ (b.repr.symm (Finsupp.equivFunOnFinite.symm x)))"}
+{"state": [{"context": ["M : Type u_1", "N✝ : Type u_2", "inst✝¹ : Monoid M", "inst✝ : AddMonoid N✝", "N : Submonoid M"], "goal": "FG ↥N ↔ N.FG"}], "premise": [119525, 119601], "state_str": "M : Type u_1\nN✝ : Type u_2\ninst✝¹ : Monoid M\ninst✝ : AddMonoid N✝\nN : Submonoid M\n⊢ FG ↥N ↔ N.FG"}
+{"state": [{"context": ["M : Type u_1", "N✝ : Type u_2", "inst✝¹ : Monoid M", "inst✝ : AddMonoid N✝", "N : Submonoid M"], "goal": "FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG"}], "premise": [6444, 6452, 6454, 137138], "state_str": "M : Type u_1\nN✝ : Type u_2\ninst✝¹ : Monoid M\ninst✝ : AddMonoid N✝\nN : Submonoid M\n⊢ FG ↥N ↔ (Submonoid.map N.subtype ⊤).FG"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n : ℕ"], "goal": "∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n"}], "premise": [21311, 103765], "state_str": "m : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn : ℕ\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n p : ℕ", "hp : partialSum (↑m) n = ↑p / ↑(m ^ n !)"], "goal": "∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n"}], "premise": [106249, 144421], "state_str": "case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ ∃ a b, 1 < b ∧ liouvilleNumber ↑m ≠ ↑a / ↑b ∧ |liouvilleNumber ↑m - ↑a / ↑b| < 1 / ↑b ^ n"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n p : ℕ", "hp : partialSum (↑m) n = ↑p / ↑(m ^ n !)"], "goal": "liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n"}], "premise": [142652], "state_str": "case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑(m ^ n !)\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n p : ℕ", "hp : partialSum (↑m) n = ↑p / ↑m ^ n !"], "goal": "liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n"}], "premise": [21307], "state_str": "case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ liouvilleNumber ↑m ≠ ↑p / ↑m ^ n ! ∧ |liouvilleNumber ↑m - ↑p / ↑m ^ n !| < 1 / (↑m ^ n !) ^ n"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n p : ℕ", "hp : partialSum (↑m) n = ↑p / ↑m ^ n !"], "goal": "partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧ |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n"}], "premise": [21305], "state_str": "case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n"}
+{"state": [{"context": ["m : ℕ", "hm : 2 ≤ m", "mZ1 : 1 < ↑m", "m1 : 1 < ↑m", "n p : ℕ", "hp : partialSum (↑m) n = ↑p / ↑m ^ n !", "hpos : 0 < remainder (↑m) n"], "goal": "partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧ |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n"}], "premise": [11234, 21310, 105284], "state_str": "case intro\nm : ℕ\nhm : 2 ≤ m\nmZ1 : 1 < ↑m\nm1 : 1 < ↑m\nn p : ℕ\nhp : partialSum (↑m) n = ↑p / ↑m ^ n !\nhpos : 0 < remainder (↑m) n\n⊢ partialSum (↑m) n + remainder (↑m) n ≠ partialSum (↑m) n ∧\n    |partialSum (↑m) n + remainder (↑m) n - partialSum (↑m) n| < 1 / (↑m ^ n !) ^ n"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : MeasurableSpace α", "j : JordanDecomposition α", "r : ℝ", "hr : r < 0"], "goal": "(r • j).posPart = (-r).toNNReal • j.negPart"}], "premise": [1674, 1738, 14324, 32529, 32532, 32533], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = (-r).toNNReal • j.negPart"}
+{"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 : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀"], "goal": "x₀ ∈ eigenspace ↑T ↑(T.rayleighQuotient x₀)"}], "premise": [88218], "state_str": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ x₀ ∈ eigenspace ↑T ↑(T.rayleighQuotient x₀)"}
+{"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 : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀"], "goal": "↑T x₀ = ↑(T.rayleighQuotient x₀) • x₀"}], "premise": [33831], "state_str": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsLocalExtrOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ ↑T x₀ = ↑(T.rayleighQuotient x₀) • x₀"}
+{"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 : β → ℝ≥0∞", "μ : Measure β", "a : α"], "goal": "∫⁻ (x : β), f x ∂(const α μ) a = ∫⁻ (x : β), f x ∂μ"}], "premise": [72658], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nf : β → ℝ≥0∞\nμ : Measure β\na : α\n⊢ ∫⁻ (x : β), f x ∂(const α μ) a = ∫⁻ (x : β), f x ∂μ"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommSemiring R", "𝓟 : Ideal R", "f : R[X]", "hf : f.IsEisensteinAt 𝓟", "n : ℕ", "hn : n ≠ f.natDegree"], "goal": "f.coeff n ∈ 𝓟"}], "premise": [1673, 14321], "state_str": "R : Type u\ninst✝ : CommSemiring R\n𝓟 : Ideal R\nf : R[X]\nhf : f.IsEisensteinAt 𝓟\nn : ℕ\nhn : n ≠ f.natDegree\n⊢ f.coeff n ∈ 𝓟"}
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+{"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))"], "goal": "S.LeftHomologyData"}], "premise": [94253], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nS : ShortComplex C\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ\nwπ : f' ≫ cokernel.π (kernel.ι γ) = 0\ne : Abelian.image S.f ≅ kernel γ :=\n  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))\n⊢ S.LeftHomologyData"}
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+{"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 = 0}", "o : Set α", "xo : x ∈ o", "o_open : IsOpen o", "μo : μ o < ⊤", "s : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o", "μs : μ s ≠ ⊤", "A : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s", "B : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)"], "goal": "∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, ρ s ≤ ↑c * μ s"}], "premise": [15889, 57179, 131585], "state_str": "case intro.intro.intro\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ {x | v.limRatioMeas hρ x = 0}\no : Set α\nxo : x ∈ o\no_open : IsOpen o\nμo : μ o < ⊤\ns : Set α := {x | v.limRatioMeas hρ x = 0} ∩ o\nμs : μ s ≠ ⊤\nA : ∀ (q : ℝ≥0), 0 < q → ρ s ≤ ↑q * μ s\nB : Tendsto (fun q => ↑q * μ s) (𝓝[>] 0) (𝓝 0)\n⊢ ∀ᶠ (c : ℝ≥0) in 𝓝[>] 0, ρ s ≤ ↑c * μ s"}
+{"state": [{"context": ["ι : Sort u_1", "M : Type u_2", "inst✝ : MulOneClass M", "s : Set M"], "goal": "(closure s).op = closure (MulOpposite.unop ⁻¹' s)"}], "premise": [117352, 117392, 134071], "state_str": "ι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\n⊢ (closure s).op = closure (MulOpposite.unop ⁻¹' s)"}
+{"state": [{"context": ["ι : Sort u_1", "M : Type u_2", "inst✝ : MulOneClass M", "s : Set M", "a : Submonoid Mᵐᵒᵖ"], "goal": "a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}"}], "premise": [71407, 100256], "state_str": "case e_a.h\nι : Sort u_1\nM : Type u_2\ninst✝ : MulOneClass M\ns : Set M\na : Submonoid Mᵐᵒᵖ\n⊢ a ∈ {a | s ⊆ MulOpposite.op ⁻¹' ↑a} ↔ a ∈ {S | MulOpposite.unop ⁻¹' s ⊆ ↑S}"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X : CoalgebraCat R"], "goal": "ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.ofHom Coalgebra.counit ▷ ModuleCat.of R ↑X.toModuleCat = (λ_ (ModuleCat.of R ↑X.toModuleCat)).inv"}], "premise": [79255, 112710], "state_str": "R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.ofHom Coalgebra.counit ▷ ModuleCat.of R ↑X.toModuleCat =\n    (λ_ (ModuleCat.of R ↑X.toModuleCat)).inv"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X : CoalgebraCat R"], "goal": "ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.counit = (ρ_ (ModuleCat.of R ↑X.toModuleCat)).inv"}], "premise": [79254, 112710], "state_str": "R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.counit =\n    (ρ_ (ModuleCat.of R ↑X.toModuleCat)).inv"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X : CoalgebraCat R"], "goal": "ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul = ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.ofHom Coalgebra.comul ▷ ModuleCat.of R ↑X.toModuleCat ≫ (α_ (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat)).hom"}], "premise": [112710], "state_str": "R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.of R ↑X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul =\n    ModuleCat.ofHom Coalgebra.comul ≫\n      ModuleCat.ofHom Coalgebra.comul ▷ ModuleCat.of R ↑X.toModuleCat ≫\n        (α_ (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat) (ModuleCat.of R ↑X.toModuleCat)).hom"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X : CoalgebraCat R"], "goal": "ModuleCat.ofHom Coalgebra.comul ≫ X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul = ModuleCat.ofHom Coalgebra.comul ≫ ModuleCat.ofHom Coalgebra.comul ▷ X.toModuleCat ≫ (α_ X.toModuleCat X.toModuleCat X.toModuleCat).hom"}], "premise": [2100, 79256], "state_str": "R : Type u\ninst✝ : CommRing R\nX : CoalgebraCat R\n⊢ ModuleCat.ofHom Coalgebra.comul ≫ X.toModuleCat ◁ ModuleCat.ofHom Coalgebra.comul =\n    ModuleCat.ofHom Coalgebra.comul ≫\n      ModuleCat.ofHom Coalgebra.comul ▷ X.toModuleCat ≫ (α_ X.toModuleCat X.toModuleCat X.toModuleCat).hom"}
+{"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", "h : i ≠ j"], "goal": "M.charmatrix i j = -C (M i j)"}], "premise": [117930, 142132, 142141, 142158, 142270, 142299], "state_str": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nm : Type u_3\nn : Type u_4\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nM₁₁ : Matrix m m R\nM₁₂ : Matrix m n R\nM₂₁ : Matrix n m R\nM₂₂ M : Matrix n n R\ni j : n\nh : i ≠ j\n⊢ M.charmatrix i j = -C (M i j)"}
+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝² : TopologicalSpace G", "inst✝¹ : MulOneClass G", "inst✝ : ContinuousMul G", "K U : Set G", "hK : IsCompact K", "hU : IsOpen U", "hKU : K ⊆ U"], "goal": "∃ V ∈ 𝓝 1, K * V ⊆ U"}], "premise": [58058], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∃ V ∈ 𝓝 1, K * V ⊆ U"}
+{"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", "X : Cᵒᵖ"], "goal": "(J.plusMap (J.toPlus P)).app X = (J.toPlus (J.plusObj P)).app X"}], "premise": [93401], "state_str": "case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\n⊢ (J.plusMap (J.toPlus P)).app X = (J.toPlus (J.plusObj P)).app 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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ"], "goal": "colimit.ι (J.diagram P (unop X)) S ≫ colimMap (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)) = colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)"}], "premise": [18777], "state_str": "case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)"}
+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯"], "goal": "colimit.ι (J.diagram P (unop X)) S ≫ colimMap (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)) = colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)"}], "premise": [93390, 93393, 96173], "state_str": "case w.h\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ colimit.ι (J.diagram P (unop X)) S ≫\n      colimMap\n        (J.diagramNatTrans\n          { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)) =\n    colimit.ι (J.diagram P (unop X)) S ≫\n      ⊤.toMultiequalizer (J.plusObj P) ≫ colimit.ι (J.diagram (J.plusObj P) (unop X)) (op ⊤)"}
+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯"], "goal": "(J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)).app S = (colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫ (J.diagram (J.plusObj P) (unop X)).map e.op"}], "premise": [95099], "state_str": "case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n          (unop X)).app\n      S =\n    (colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n      (J.diagram (J.plusObj P) (unop X)).map e.op"}
+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L"], "goal": "(J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ } (unop X)).app S ≫ Multiequalizer.ι ((unop S).index (J.plusObj P)) I = ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫ (J.diagram (J.plusObj P) (unop X)).map e.op) ≫ Multiequalizer.ι ((unop S).index (J.plusObj P)) I"}], "premise": [95098], "state_str": "case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ (J.diagramNatTrans { app := fun X => ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P (unop X)) (op ⊤), naturality := ⋯ }\n            (unop X)).app\n        S ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I =\n    ((colimit.ι (J.diagram P (unop X)) S ≫ ⊤.toMultiequalizer (J.plusObj P)) ≫\n        (J.diagram (J.plusObj P) (unop X)).map e.op) ≫\n      Multiequalizer.ι ((unop S).index (J.plusObj P)) I"}
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+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L"], "goal": "Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) = colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}], "premise": [18777], "state_str": "case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}
+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯"], "goal": "Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) = colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}], "premise": [91857, 93390, 93413, 96173], "state_str": "case w.h.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P ≫ colimit.ι (J.diagram P I.Y) (op ⊤) =\n    colimit.ι (J.diagram P (unop X)) S ≫ (J.plusObj P).map (Cover.Arrow.map I e.op.unop).f.op"}
+{"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", "X : Cᵒᵖ", "S : (J.Cover (unop X))ᵒᵖ", "e : unop S ⟶ ⊤ := homOfLE ⋯", "I : ((unop S).index (J.plusObj P)).L", "ee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯"], "goal": "(Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op = Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P) ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯"}], "premise": [95099], "state_str": "case w.h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nD : Type w\ninst✝² : Category.{max v u, w} D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : J.Cover X), HasMultiequalizer (S.index P)\nP : Cᵒᵖ ⥤ D\ninst✝ : ∀ (X : C), HasColimitsOfShape (J.Cover X)ᵒᵖ D\nX : Cᵒᵖ\nS : (J.Cover (unop X))ᵒᵖ\ne : unop S ⟶ ⊤ := homOfLE ⋯\nI : ((unop S).index (J.plusObj P)).L\nee : (J.pullback (Cover.Arrow.map I e).f).obj (unop S) ⟶ ⊤ := homOfLE ⋯\n⊢ (Multiequalizer.ι ((unop S).index P) I ≫ ⊤.toMultiequalizer P) ≫ (J.diagram P I.Y).map ee.op =\n    Multiequalizer.lift ((unop ((J.pullback (Cover.Arrow.map I e.op.unop).f.op.unop).op.obj S)).index P)\n      ((J.diagram P (unop X)).obj S) (fun I_1 => Multiequalizer.ι ((unop S).index P) (Cover.Arrow.base I_1)) ⋯"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "l l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "L : List (List (Option α))", "hm : m < L.length"], "goal": "take m L = take (min m L.length) (take n L) ↔ m ≤ n"}], "premise": [1793, 2134, 2581, 3503, 3679, 3731, 4576, 19701], "state_str": "α : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nL : List (List (Option α))\nhm : m < L.length\n⊢ take m L = take (min m L.length) (take n L) ↔ m ≤ n"}
+{"state": [{"context": ["𝓕 : Type u_1", "α : Type u_2", "E : Type u_3", "F : Type u_4", "inst✝¹ : SeminormedCommGroup E", "inst✝ : SeminormedCommGroup F", "a✝ a₁ a₂ b✝ b₁ b₂ : E", "r r₁ r₂ : ℝ", "a b : E"], "goal": "dist a (a * b) = ‖b‖"}], "premise": [42663, 61420, 119730], "state_str": "𝓕 : Type u_1\nα : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\na✝ a₁ a₂ b✝ b₁ b₂ : E\nr r₁ r₂ : ℝ\na b : E\n⊢ dist a (a * b) = ‖b‖"}
+{"state": [{"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "i = 𝟙 x"}], "premise": [47487], "state_str": "x : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ i = 𝟙 x"}
+{"state": [{"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "IsIso i"}], "premise": [47485], "state_str": "x : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ IsIso i"}
+{"state": [{"context": ["x : SimplexCategory", "i : x ⟶ x", "inst✝ : Epi i"], "goal": "Function.Bijective ⇑(Hom.toOrderHom i)"}], "premise": [1793, 20002, 47480, 141421], "state_str": "case hf\nx : SimplexCategory\ni : x ⟶ x\ninst✝ : Epi i\n⊢ Function.Bijective ⇑(Hom.toOrderHom i)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "b : ℝ", "f : ℕ → ℝ", "z : ℕ → E", "hfa : Monotone f", "hf0 : Tendsto f atTop (𝓝 0)"], "goal": "CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i"}], "premise": [119707], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ i ∈ Finset.range n, (-1) ^ i * f i"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "E : Type u_4", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "b : ℝ", "f : ℕ → ℝ", "z : ℕ → E", "hfa : Monotone f", "hf0 : Tendsto f atTop (𝓝 0)"], "goal": "CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x"}], "premise": [34528, 34530], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℕ → ℝ\nz : ℕ → E\nhfa : Monotone f\nhf0 : Tendsto f atTop (𝓝 0)\n⊢ CauchySeq fun n => ∑ x ∈ Finset.range n, f x * (-1) ^ x"}
+{"state": [{"context": ["it : Iterator"], "goal": "ValidFor it.s.data.reverse [] { s := it.s, i := it.s.endPos }"}], "premise": [2307, 5293], "state_str": "it : Iterator\n⊢ ValidFor it.s.data.reverse [] { s := it.s, i := it.s.endPos }"}
+{"state": [{"context": ["K : Type u_1", "inst✝¹ : Field K", "inst✝ : NumberField K", "i : Free.ChooseBasisIndex ℤ (𝓞 K)"], "goal": "(latticeBasis K) i = (canonicalEmbedding K) ((integralBasis K) i)"}], "premise": [1670, 22033, 70751, 81528], "state_str": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ni : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ (latticeBasis K) i = (canonicalEmbedding K) ((integralBasis K) i)"}
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+{"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✝", "ι : Sort u_2", "hι : Nonempty ι", "S : ι → NonUnitalSubring R", "hS : Directed (fun x x_1 => x ≤ x_1) S", "x : R", "U : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯"], "goal": "⨆ i, S i ≤ U"}], "premise": [1674, 16573, 19309], "state_str": "F : Type w\nR : Type u\nS✝ : Type v\nT : Type u_1\ninst✝⁴ : NonUnitalNonAssocRing R\ninst✝³ : NonUnitalNonAssocRing S✝\ninst✝² : NonUnitalNonAssocRing T\ninst✝¹ : FunLike F R S✝\ninst✝ : NonUnitalRingHomClass F R S✝\ng : S✝ →ₙ+* T\nf : R →ₙ+* S✝\nι : Sort u_2\nhι : Nonempty ι\nS : ι → NonUnitalSubring R\nhS : Directed (fun x x_1 => x ≤ x_1) S\nx : R\nU : NonUnitalSubring R := NonUnitalSubring.mk' (⋃ i, ↑(S i)) (⨆ i, (S i).toSubsemigroup) (⨆ i, (S i).toAddSubgroup) ⋯ ⋯\n⊢ ⨆ i, S i ≤ U"}
+{"state": [{"context": ["α : Type u", "r : α → α → Prop", "a✝ : α", "l✝ : List α", "inst✝¹ : Inhabited α", "inst✝ : Preorder α", "a : α", "l : List α", "h : Sorted (fun x x_1 => x > x_1) l", "ha : a ∈ l"], "goal": "a ≤ l.head!"}], "premise": [5037, 132430], "state_str": "α : Type u\nr : α → α → Prop\na✝ : α\nl✝ : List α\ninst✝¹ : Inhabited α\ninst✝ : Preorder α\na : α\nl : List α\nh : Sorted (fun x x_1 => x > x_1) l\nha : a ∈ l\n⊢ a ≤ l.head!"}
+{"state": [{"context": ["x : ℝ*", "h : x.Infinite"], "goal": "(-x).st = -x.st"}], "premise": [1674, 119802, 146106, 146144], "state_str": "x : ℝ*\nh : x.Infinite\n⊢ (-x).st = -x.st"}
+{"state": [{"context": ["α✝ : Type u_1", "l₁✝ l₂ : List α✝", "p : α✝ → Bool", "l₁ l₂✝ : List α✝", "a : α✝", "s : l₁ <+ l₂✝", "h : p a = true"], "goal": "eraseP p l₁ <+ l₂✝"}, {"context": ["α✝ : Type u_1", "l₁✝ l₂ : List α✝", "p : α✝ → Bool", "l₁ l₂✝ : List α✝", "a : α✝", "s : l₁ <+ l₂✝", "h : ¬p a = true"], "goal": "eraseP p l₁ <+ a :: eraseP p l₂✝"}], "premise": [1291, 1372], "state_str": "case pos\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : p a = true\n⊢ eraseP p l₁ <+ l₂✝\n\ncase neg\nα✝ : Type u_1\nl₁✝ l₂ : List α✝\np : α✝ → Bool\nl₁ l₂✝ : List α✝\na : α✝\ns : l₁ <+ l₂✝\nh : ¬p a = true\n⊢ eraseP p l₁ <+ a :: eraseP p l₂✝"}
+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝³ : TopologicalSpace β", "inst✝² : Group α", "inst✝¹ : MulAction α β", "inst✝ : ContinuousConstSMul α β", "s : Set α", "t : Set β", "ht : IsOpen t"], "goal": "IsOpen (s • t)"}], "premise": [132858], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (s • t)"}
+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝³ : TopologicalSpace β", "inst✝² : Group α", "inst✝¹ : MulAction α β", "inst✝ : ContinuousConstSMul α β", "s : Set α", "t : Set β", "ht : IsOpen t"], "goal": "IsOpen (⋃ a ∈ s, a • t)"}], "premise": [55351, 64975], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\nht : IsOpen t\n⊢ IsOpen (⋃ a ∈ s, a • t)"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : CommRing R", "W : WeierstrassCurve R", "inst✝ : Nontrivial R", "n : ℤ"], "goal": "(W.Φ n).leadingCoeff = 1"}], "premise": [129133, 129135], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : Nontrivial R\nn : ℤ\n⊢ (W.Φ n).leadingCoeff = 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : NormalizedGCDMonoid R", "p : R[X]", "h : ¬C p.content = 0"], "goal": "p.primPart.natDegree = p.natDegree"}], "premise": [74877, 74881, 102121, 103436, 119727], "state_str": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\np : R[X]\nh : ¬C p.content = 0\n⊢ p.primPart.natDegree = p.natDegree"}
+{"state": [{"context": ["a b : Prop"], "goal": "Xor' (¬a) ¬b ↔ Xor' a b"}], "premise": [1723, 1726], "state_str": "a b : Prop\n⊢ Xor' (¬a) ¬b ↔ Xor' a b"}
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+{"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", "inst✝ : Countable ι", "f : α →ₛ ℝ≥0∞", "s : ι → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) s", "μ : Measure α"], "goal": "∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (↑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (↑f ⁻¹' {x})"}], "premise": [2011, 58970], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_5\ninst✝ : Countable ι\nf : α →ₛ ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\nμ : Measure α\n⊢ ∑ x ∈ f.range, ⨆ i, x * (μ.restrict (s i)) (↑f ⁻¹' {x}) = ⨆ i, ∑ x ∈ f.range, x * (μ.restrict (s i)) (↑f ⁻¹' {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", "μ : Measure β", "inst✝¹ : SFinite μ", "ν : Measure γ", "inst✝ : SFinite ν", "x : α", "s✝ : Set (β × γ)", "a✝ : MeasurableSet s✝"], "goal": "((const α μ ×ₖ const α ν) x) s✝ = ((const α (μ.prod ν)) x) s✝"}], "premise": [72658, 74314], "state_str": "case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nμ : Measure β\ninst✝¹ : SFinite μ\nν : Measure γ\ninst✝ : SFinite ν\nx : α\ns✝ : Set (β × γ)\na✝ : MeasurableSet s✝\n⊢ ((const α μ ×ₖ const α ν) x) s✝ = ((const α (μ.prod ν)) x) s✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : MeasurableSpace α", "inst✝ : SeparatesPoints α", "x y : α", "h : x ≠ y"], "goal": "∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s"}], "premise": [53688], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : SeparatesPoints α\nx y : α\nh : x ≠ y\n⊢ ∃ s, MeasurableSet s ∧ x ∈ s ∧ y ∉ s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : MeasurableSpace α", "inst✝ : SeparatesPoints α", "x y : α", "h : ∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s"], "goal": "x = y"}], "premise": [28043], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : SeparatesPoints α\nx y : α\nh : ∀ (s : Set α), MeasurableSet s → x ∈ s → y ∈ s\n⊢ x = y"}
+{"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", "k : ℕ", "hk : a ^ k ∣ b", "hsucc : ¬a ^ (k + 1) ∣ b", "this : Finite a b"], "goal": "multiplicity a b ≤ ↑k"}], "premise": [144900], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ multiplicity a b ≤ ↑k"}
+{"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", "k : ℕ", "hk : a ^ k ∣ b", "hsucc : ¬a ^ (k + 1) ∣ b", "this : Finite a b"], "goal": "∃ (h : (multiplicity a b).Dom), (multiplicity a b).get h ≤ k"}], "premise": [143302], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\nthis : Finite a b\n⊢ ∃ (h : (multiplicity a b).Dom), (multiplicity a b).get h ≤ k"}
+{"state": [{"context": ["M : Type u_1", "N : Type u_2", "S : Type u_3", "M₀ : Type u_4", "M₁ : Type u_5", "R : Type u_6", "G : Type u_7", "G₀ : Type u_8", "inst✝⁷ : Mul M", "inst✝⁶ : Monoid N", "inst✝⁵ : Semigroup S", "inst✝⁴ : MulZeroClass M₀", "inst✝³ : MulOneClass M₁", "inst✝² : NonAssocRing R", "inst✝¹ : Group G", "inst✝ : CancelMonoidWithZero G₀", "p : R", "h : IsIdempotentElem p"], "goal": "IsIdempotentElem (1 - p)"}], "premise": [117816, 117981, 119728, 119730, 121375], "state_str": "M : Type u_1\nN : Type u_2\nS : Type u_3\nM₀ : Type u_4\nM₁ : Type u_5\nR : Type u_6\nG : Type u_7\nG₀ : Type u_8\ninst✝⁷ : Mul M\ninst✝⁶ : Monoid N\ninst✝⁵ : Semigroup S\ninst✝⁴ : MulZeroClass M₀\ninst✝³ : MulOneClass M₁\ninst✝² : NonAssocRing R\ninst✝¹ : Group G\ninst✝ : CancelMonoidWithZero G₀\np : R\nh : IsIdempotentElem p\n⊢ IsIdempotentElem (1 - p)"}
+{"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", "inst✝³ : Algebra S T", "inst✝² : IsScalarTower R S T", "inst✝¹ : IsLocalization M S", "inst✝ : IsLocalization N T", "y : ↥(localizationLocalizationSubmodule M N)"], "goal": "IsUnit ((algebraMap R T) ↑y)"}], "premise": [1673, 77398, 137122], "state_str": "R : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\n⊢ IsUnit ((algebraMap R T) ↑y)"}
+{"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", "inst✝³ : Algebra S T", "inst✝² : IsScalarTower R S T", "inst✝¹ : IsLocalization M S", "inst✝ : IsLocalization N T", "y : ↥(localizationLocalizationSubmodule M N)", "y' : ↥N", "z : ↥M", "eq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z"], "goal": "IsUnit ((algebraMap R T) ↑y)"}], "premise": [120517, 121202, 121567], "state_str": "case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap R T) ↑y)"}
+{"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", "inst✝³ : Algebra S T", "inst✝² : IsScalarTower R S T", "inst✝¹ : IsLocalization M S", "inst✝ : IsLocalization N T", "y : ↥(localizationLocalizationSubmodule M N)", "y' : ↥N", "z : ↥M", "eq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z"], "goal": "IsUnit ((algebraMap S T) ↑y') ∧ IsUnit ((algebraMap S T) ((algebraMap R S) ↑z))"}], "premise": [77573, 116782], "state_str": "case intro.intro\nR : Type u_1\ninst✝⁹ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Algebra R S\nP : Type u_3\ninst✝⁶ : CommSemiring P\nN : Submonoid S\nT : Type u_4\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : ↥(localizationLocalizationSubmodule M N)\ny' : ↥N\nz : ↥M\neq : (algebraMap R S) ↑y = ↑y' * (algebraMap R S) ↑z\n⊢ IsUnit ((algebraMap S T) ↑y') ∧ IsUnit ((algebraMap S T) ((algebraMap R S) ↑z))"}
+{"state": [{"context": ["R : Type u", "S : Type v", "A : Type w", "B : Type u_1", "inst✝² : CommSemiring R", "inst✝¹ : Ring A", "inst✝ : Algebra R A", "x : A", "r : R"], "goal": "x * (x - (algebraMap R A) r) = (x - (algebraMap R A) r) * x"}], "premise": [121168], "state_str": "R : Type u\nS : Type v\nA : Type w\nB : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\nr : R\n⊢ x * (x - (algebraMap R A) r) = (x - (algebraMap R A) r) * x"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝⁴ : OrderedRing α", "inst✝³ : OrderedAddCommGroup β", "inst✝² : Module α β", "inst✝¹ : PosSMulMono α β", "inst✝ : PosSMulReflectLE α β", "ha : a < 0"], "goal": "-a • b₂ ≤ -a • b₁ ↔ b₂ ≤ b₁"}], "premise": [104875], "state_str": "α : Type u_1\nβ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝⁴ : OrderedRing α\ninst✝³ : OrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ -a • b₂ ≤ -a • b₁ ↔ b₂ ≤ b₁"}
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+{"state": [{"context": ["α : Type u_1", "R : Type u_2", "inst✝³ : NormedRing R", "inst✝² : IsAbsoluteValue norm", "F : Type u_3", "K : Type u_4", "inst✝¹ : CommRing F", "inst✝ : NormedField K", "p : F[X]", "f : F →+* K", "i : ℕ", "h1 : p.Monic", "h2 : Splits f p", "hi : i ≤ (map f p).natDegree", "s : Multiset K", "hs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i", "B : ℝ≥0", "h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B", "z : K", "hz : z ∈ s", "hx : ↑nnnormHom z ∈ Multiset.map (⇑↑nnnormHom) s"], "goal": "↑nnnormHom z ≤ B"}], "premise": [2107, 137873], "state_str": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nR : Type u_2\ninst✝³ : NormedRing R\ninst✝² : IsAbsoluteValue norm\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\ni : ℕ\nh1 : p.Monic\nh2 : Splits f p\nhi : i ≤ (map f p).natDegree\ns : Multiset K\nhs : s ≤ (map f p).roots ∧ card s = (map f p).natDegree - i\nB : ℝ≥0\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ ↑B\nz : K\nhz : z ∈ s\nhx : ↑nnnormHom z ∈ Multiset.map (⇑↑nnnormHom) s\n⊢ ↑nnnormHom z ≤ B"}
+{"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", "m : ℤ"], "goal": "toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b"}], "premise": [105884, 119708], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nm : ℤ\n⊢ toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : Module 𝕜 E", "inst✝² : Module ℝ E", "inst✝¹ : IsScalarTower ℝ 𝕜 E", "inst✝ : ContinuousSMul ℝ E", "s : AbsConvexOpenSets 𝕜 E"], "goal": "(gaugeSeminorm ⋯ ⋯ ⋯).ball 0 1 = ↑s"}], "premise": [36275], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ (gaugeSeminorm ⋯ ⋯ ⋯).ball 0 1 = ↑s"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : Module 𝕜 E", "inst✝² : Module ℝ E", "inst✝¹ : IsScalarTower ℝ 𝕜 E", "inst✝ : ContinuousSMul ℝ E", "s : AbsConvexOpenSets 𝕜 E"], "goal": "{y | (gaugeSeminorm ⋯ ⋯ ⋯) y < 1} = ↑s"}], "premise": [35568], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | (gaugeSeminorm ⋯ ⋯ ⋯) y < 1} = ↑s"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "ι : Type u_5", "inst✝⁶ : RCLike 𝕜", "inst✝⁵ : AddCommGroup E", "inst✝⁴ : TopologicalSpace E", "inst✝³ : Module 𝕜 E", "inst✝² : Module ℝ E", "inst✝¹ : IsScalarTower ℝ 𝕜 E", "inst✝ : ContinuousSMul ℝ E", "s : AbsConvexOpenSets 𝕜 E"], "goal": "{y | gauge (↑s) y < 1} = ↑s"}], "premise": [35553, 40950, 40951, 40954], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : Module ℝ E\ninst✝¹ : IsScalarTower ℝ 𝕜 E\ninst✝ : ContinuousSMul ℝ E\ns : AbsConvexOpenSets 𝕜 E\n⊢ {y | gauge (↑s) y < 1} = ↑s"}
+{"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", "y : E", "ε : ℝ", "a : E"], "goal": "a ∈ ball y ε ↔ a ∈ {x | ‖x / y‖ < ε}"}], "premise": [42654], "state_str": "𝓕 : Type u_1\n𝕜 : Type u_2\nα : Type u_3\nι : Type u_4\nκ : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ny : E\nε : ℝ\na : E\n⊢ a ∈ ball y ε ↔ a ∈ {x | ‖x / y‖ < ε}"}
+{"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"], "goal": "∏' (x : β), a = a ^ Nat.card β"}], "premise": [141507], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : TopologicalGroup G\nf : α → G\ninst✝ : T2Space G\na : G\n⊢ ∏' (x : β), a = a ^ Nat.card β"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'"], "goal": "multiplicity (↑(d ⟨p, ⋯⟩)) n = multiplicity p m"}], "premise": [2100], "state_str": "M : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n = multiplicity p m"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'"], "goal": "multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n"}], "premise": [76126, 79579, 126197, 126301, 126302, 126303], "state_str": "case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'"], "goal": "multiplicity (Associates.mk p) (Associates.mk m) = multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩)) (Associates.mk n)"}], "premise": [76126, 77035, 125939, 126197, 126301, 126302, 126303], "state_str": "case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩))\n      (Associates.mk n)"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'", "this : Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) = ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)"], "goal": "multiplicity (Associates.mk p) (Associates.mk m) = multiplicity (↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)) (Associates.mk n)"}], "premise": [1674, 77033, 125947], "state_str": "case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ multiplicity (Associates.mk p) (Associates.mk m) =\n    multiplicity (↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)) (Associates.mk n)"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'", "this : Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) = ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)"], "goal": "Associates.mk p ∈ normalizedFactors (Associates.mk m)"}], "premise": [1674, 76109, 76114, 76126, 125839, 125939, 125947, 125951], "state_str": "case h\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁷ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁶ : CancelCommMonoidWithZero N", "inst✝⁵ : Unique Mˣ", "inst✝⁴ : Unique Nˣ", "inst✝³ : UniqueFactorizationMonoid M", "inst✝² : UniqueFactorizationMonoid N", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : M", "n : N", "hm : m ≠ 0", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : { l // l ∣ m } ≃ { l // l ∣ n }", "hd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'", "this : Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) = ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)", "q : Associates M", "hq : q ∈ normalizedFactors (Associates.mk m)", "hq' : Associated (Associates.mk p) q"], "goal": "Associates.mk p ∈ normalizedFactors (Associates.mk m)"}], "premise": [1673, 125904], "state_str": "case h.intro.intro\nM : Type u_1\ninst✝⁷ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁶ : CancelCommMonoidWithZero N\ninst✝⁵ : Unique Mˣ\ninst✝⁴ : Unique Nˣ\ninst✝³ : UniqueFactorizationMonoid M\ninst✝² : UniqueFactorizationMonoid N\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : M\nn : N\nhm : m ≠ 0\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : { l // l ∣ m } ≃ { l // l ∣ n }\nhd : ∀ (l l' : { l // l ∣ m }), ↑(d l) ∣ ↑(d l') ↔ ↑l ∣ ↑l'\nthis :\n  Associates.mk ↑(d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), ⋯⟩) =\n    ↑((mkFactorOrderIsoOfFactorDvdEquiv hd) ⟨associatesEquivOfUniqueUnits.symm p, ⋯⟩)\nq : Associates M\nhq : q ∈ normalizedFactors (Associates.mk m)\nhq' : Associated (Associates.mk p) q\n⊢ Associates.mk p ∈ normalizedFactors (Associates.mk m)"}
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+{"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", "g : β → γ"], "goal": "AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb"}], "premise": [1713, 30701, 88683], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\ninst✝ : MeasurableSpace δ\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\nf : α → β\nhf : MeasurePreserving f μa μb\nh₂ : MeasurableEmbedding f\ng : β → γ\n⊢ AEMeasurable (g ∘ f) μa ↔ AEMeasurable g μb"}
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+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [35213, 35223, 36006, 108341, 110020, 117810, 118056, 118909], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [110022], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [36003, 117714, 117810, 118076, 118909], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [36003, 117714, 117810, 118076, 118909], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
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+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [35213, 36006, 108334, 108341, 118909], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s", "h2vw : x + (2 • v + w) ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [7382, 35213, 36006, 110022, 113020, 117810, 117986, 118076], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s", "h2vw : x + (2 • v + w) ∈ interior s", "hvww : x + (v + w) + w ∈ interior s"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [46574], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s", "h2vw : x + (2 • v + w) ∈ interior s", "hvww : x + (v + w) + w ∈ interior s", "TA1 : (fun h => f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [46574], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s", "h2vw : x + (2 • v + w) ∈ interior s", "hvww : x + (v + w) + w ∈ interior s", "TA1 : (fun h => f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2", "TA2 : (fun h => f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2"], "goal": "(fun h => f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w) =o[𝓝[>] 0] fun h => h ^ 2"}], "premise": [43509], "state_str": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\n⊢ (fun h =>\n      f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n        h ^ 2 • (f'' v) w) =o[𝓝[>] 0]\n    fun h => h ^ 2"}
+{"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", "h4v : x + 4 • v ∈ interior s", "h4w : x + 4 • w ∈ interior s", "A : 1 / 2 ∈ Ioc 0 1", "B : 1 / 2 ∈ Icc 0 1", "C : ∀ (w : E), 2 • w = 2 • w", "h2v2w : x + 2 • v + 2 • w ∈ interior s", "h2vww : x + (2 • v + w) + w ∈ interior s", "h2v : x + 2 • v ∈ interior s", "h2w : x + 2 • w ∈ interior s", "hvw : x + (v + w) ∈ interior s", "h2vw : x + (2 • v + w) ∈ interior s", "hvww : x + (v + w) + w ∈ interior s", "TA1 : (fun h => f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2", "TA2 : (fun h => f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0] fun h => h ^ 2", "h : ℝ"], "goal": "f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) - h ^ 2 • (f'' v) w = f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w - (h ^ 2 / 2) • (f'' w) w - (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w - (h ^ 2 / 2) • (f'' w) w)"}], "premise": [68759, 68761, 68773, 68785, 108334, 110022, 119704, 120658], "state_str": "case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n  (fun h =>\n      f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nTA2 :\n  (fun h =>\n      f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]\n    fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n      h ^ 2 • (f'' v) w =\n    f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • (f' x) w - h ^ 2 • (f'' (2 • v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w -\n      (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • (f' x) w - h ^ 2 • (f'' (v + w)) w -\n        (h ^ 2 / 2) • (f'' w) w)"}
+{"state": [{"context": ["o : Ordinal.{u_1}", "ho : 0 < o"], "goal": "(sup fun n => o ^ ↑n) = o ^ ω"}], "premise": [1674, 14302, 49765], "state_str": "o : Ordinal.{u_1}\nho : 0 < o\n⊢ (sup fun n => o ^ ↑n) = o ^ ω"}
+{"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", "f : β →. γ", "g : α →. β", "a : Part α", "c : γ"], "goal": "c ∈ a.bind (f.comp g) ↔ c ∈ (a.bind g).bind f"}], "premise": [2038, 2039, 128217, 131108], "state_str": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ c ∈ a.bind (f.comp g) ↔ c ∈ (a.bind g).bind f"}
+{"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", "f : β →. γ", "g : α →. β", "a : Part α", "c : γ"], "goal": "(∃ a_1 x, a_1 ∈ a ∧ x ∈ g a_1 ∧ c ∈ f x) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1"}], "premise": [2049], "state_str": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ a_1 x, a_1 ∈ a ∧ x ∈ g a_1 ∧ c ∈ f x) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1"}
+{"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", "f : β →. γ", "g : α →. β", "a : Part α", "c : γ"], "goal": "(∃ b, ∃ a_1 ∈ a, b ∈ g a_1 ∧ c ∈ f b) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1"}], "premise": [1206], "state_str": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nι : Type u_6\nf✝ : α →. β\nf : β →. γ\ng : α →. β\na : Part α\nc : γ\n⊢ (∃ b, ∃ a_1 ∈ a, b ∈ g a_1 ∧ c ∈ f b) ↔ ∃ a_1 x, (x ∈ a ∧ a_1 ∈ g x) ∧ c ∈ f a_1"}
+{"state": [{"context": ["Ω : Type u_1", "ι : Type u_2", "m : MeasurableSpace Ω", "X : Ω → ℝ", "p : ℕ", "μ : Measure Ω", "t : ℝ"], "goal": "cgf (-X) μ t = cgf X μ (-t)"}], "premise": [73759], "state_str": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\n⊢ cgf (-X) μ t = cgf X μ (-t)"}
+{"state": [{"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝³ : LinearOrderedAddCommGroup 𝕜", "inst✝² : Archimedean 𝕜", "inst✝¹ : TopologicalSpace 𝕜", "inst✝ : OrderTopology 𝕜", "p : 𝕜", "hp : 0 < p", "a x : 𝕜"], "goal": "ContinuousWithinAt (toIocMod hp a) (Iic x) x"}], "premise": [1838, 2101, 105913, 119769], "state_str": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\n⊢ ContinuousWithinAt (toIocMod hp a) (Iic x) x"}
+{"state": [{"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝³ : LinearOrderedAddCommGroup 𝕜", "inst✝² : Archimedean 𝕜", "inst✝¹ : TopologicalSpace 𝕜", "inst✝ : OrderTopology 𝕜", "p : 𝕜", "hp : 0 < p", "a x : 𝕜", "this : ContinuousNeg 𝕜"], "goal": "ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x"}], "premise": [55638, 57316, 57318, 57325, 57983, 66646, 66828, 104741], "state_str": "𝕜 : Type u_1\nB : Type u_2\ninst✝³ : LinearOrderedAddCommGroup 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nthis : ContinuousNeg 𝕜\n⊢ ContinuousWithinAt ((fun x => p - x) ∘ toIcoMod hp (-a) ∘ Neg.neg) (Iic x) x"}
+{"state": [{"context": ["J : Type w", "K : Type u_1", "C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "f g : J → C", "inst✝¹ : HasBiproduct f", "inst✝ : HasBiproduct g", "p : (b : J) → f b ⟶ g b", "j✝¹ j✝ : J"], "goal": "ι f j✝ ≫ map p ≫ π g j✝¹ = ι f j✝ ≫ map' p ≫ π g j✝¹"}], "premise": [92507, 94244, 96173, 96227, 96232, 96249, 96250], "state_str": "case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ map p ≫ π g j✝¹ = ι f j✝ ≫ map' p ≫ π g j✝¹"}
+{"state": [{"context": ["J : Type w", "K : Type u_1", "C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : HasZeroMorphisms C", "f g : J → C", "inst✝¹ : HasBiproduct f", "inst✝ : HasBiproduct g", "p : (b : J) → f b ⟶ g b", "j✝¹ j✝ : J"], "goal": "ι f j✝ ≫ π f j✝¹ ≫ p j✝¹ = p j✝ ≫ ι g j✝ ≫ π g j✝¹"}], "premise": [96251], "state_str": "case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nf g : J → C\ninst✝¹ : HasBiproduct f\ninst✝ : HasBiproduct g\np : (b : J) → f b ⟶ g b\nj✝¹ j✝ : J\n⊢ ι f j✝ ≫ π f j✝¹ ≫ p j✝¹ = p j✝ ≫ ι g j✝ ≫ π g j✝¹"}
+{"state": [{"context": ["α : Type u", "β : Type v", "f : α → β", "p : α", "s : Set α", "hps : (of p).IsSupported s", "this✝ : DecidablePred s", "this : ∃ n, (lift fun a => if a ∈ s then 0 else X) (of p) = ↑n"], "goal": "p ∈ s"}], "premise": [76632], "state_str": "α : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nthis : ∃ n, (lift fun a => if a ∈ s then 0 else X) (of p) = ↑n\n⊢ p ∈ s"}
+{"state": [{"context": ["α : Type u", "β : Type v", "f : α → β", "p : α", "s : Set α", "hps : (of p).IsSupported s", "this✝ : DecidablePred s", "h : p ∉ s", "this : ∃ n, X = ↑n"], "goal": "False"}], "premise": [1690, 2198], "state_str": "case neg\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nthis : ∃ n, X = ↑n\n⊢ False"}
+{"state": [{"context": ["α : Type u", "β : Type v", "f : α → β", "p : α", "s : Set α", "hps : (of p).IsSupported s", "this : DecidablePred s", "h : p ∉ s", "w : ℤ", "H : X = ↑w"], "goal": "1 = 0"}], "premise": [101360], "state_str": "case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = ↑w\n⊢ 1 = 0"}
+{"state": [{"context": ["α : Type u", "β : Type v", "f : α → β", "p : α", "s : Set α", "hps : (of p).IsSupported s", "this✝ : DecidablePred s", "h : p ∉ s", "w : ℤ", "H : X = C ↑w", "this : X.coeff 1 = (C w).coeff 1"], "goal": "1 = 0"}], "premise": [1737, 1738, 101276, 101280, 113018], "state_str": "case neg.intro\nα : Type u\nβ : Type v\nf : α → β\np : α\ns : Set α\nhps : (of p).IsSupported s\nthis✝ : DecidablePred s\nh : p ∉ s\nw : ℤ\nH : X = C ↑w\nthis : X.coeff 1 = (C w).coeff 1\n⊢ 1 = 0"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb"], "goal": "(fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt] fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}], "premise": [27278], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this : la'.IsMeasurablyGenerated"], "goal": "(fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt] fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}], "premise": [27278], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis : la'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated"], "goal": "(fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt] fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}], "premise": [16091, 27285, 43393, 43501], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun t =>\n      ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n        (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)) =o[lt]\n    fun t => ‖∫ (x : ℝ) in ua t..va t, 1 ∂μ‖ + ‖∫ (x : ℝ) in ub t..vb t, 1 ∂μ‖"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}], "premise": [26331, 27281], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}], "premise": [16358, 16384, 26331, 27280, 27281], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}], "premise": [26331, 27281], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}], "premise": [16358, 16384, 26331, 27280, 27281], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)"], "goal": "(fun x => -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) + (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt] fun t => ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ - (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}], "premise": [15889, 131585], "state_str": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\n⊢ (fun x =>\n      -(∫ (x : ℝ) in ua x..va x, f x ∂μ - ∫ (x : ℝ) in ua x..va x, ca ∂μ) +\n        (∫ (x : ℝ) in ub x..vb x, f x ∂μ - ∫ (x : ℝ) in ub x..vb x, cb ∂μ)) =ᶠ[lt]\n    fun t =>\n    ∫ (x : ℝ) in va t..vb t, f x ∂μ - ∫ (x : ℝ) in ua t..ub t, f x ∂μ -\n      (∫ (x : ℝ) in ub t..vb t, cb ∂μ - ∫ (x : ℝ) in ua t..va t, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)", "a✝ : ι", "ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)", "a_ua : IntervalIntegrable f μ a (ua a✝)", "ub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)", "b_ub : IntervalIntegrable f μ b (ub a✝)"], "goal": "-(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) + (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) = ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ - (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)"}], "premise": [26406], "state_str": "case h\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ -(∫ (x : ℝ) in ua a✝..va a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ) +\n      (∫ (x : ℝ) in ub a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ) =\n    ∫ (x : ℝ) in va a✝..vb a✝, f x ∂μ - ∫ (x : ℝ) in ua a✝..ub a✝, f x ∂μ -\n      (∫ (x : ℝ) in ub a✝..vb a✝, cb ∂μ - ∫ (x : ℝ) in ua a✝..va a✝, ca ∂μ)"}
+{"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)", "a✝ : ι", "ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)", "a_ua : IntervalIntegrable f μ a (ua a✝)", "ub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)", "b_ub : IntervalIntegrable f μ b (ub a✝)"], "goal": "IntervalIntegrable f μ (ub a✝) (vb a✝)"}, {"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)", "a✝ : ι", "ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)", "a_ua : IntervalIntegrable f μ a (ua a✝)", "ub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)", "b_ub : IntervalIntegrable f μ b (ub a✝)"], "goal": "IntervalIntegrable f μ (ua a✝) (va a✝)"}, {"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 : 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✝³ : CompleteSpace E", "inst✝² : FTCFilter a la la'", "inst✝¹ : FTCFilter b lb lb'", "inst✝ : IsLocallyFiniteMeasure μ", "hab : IntervalIntegrable f μ a b", "hmeas_a : StronglyMeasurableAtFilter f la' μ", "hmeas_b : StronglyMeasurableAtFilter f lb' μ", "ha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)", "hb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)", "hua : Tendsto ua lt la", "hva : Tendsto va lt la", "hub : Tendsto ub lt lb", "hvb : Tendsto vb lt lb", "this✝ : la'.IsMeasurablyGenerated", "this : lb'.IsMeasurablyGenerated", "A : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)", "A' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)", "B : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)", "B' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)", "a✝ : ι", "ua_va : IntervalIntegrable f μ (ua a✝) (va a✝)", "a_ua : IntervalIntegrable f μ a (ua a✝)", "ub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)", "b_ub : IntervalIntegrable f μ b (ub a✝)"], "goal": "IntervalIntegrable f μ (ub a✝) (ua a✝)"}], "premise": [26289, 26291], "state_str": "case h.hab\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (vb a✝)\n\ncase h.hcd\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ua a✝) (va a✝)\n\ncase h.hac\nι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nA✝ : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nc ca cb : E\nl l' la la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nu v ua va ub vb : ι → ℝ\ninst✝³ : CompleteSpace E\ninst✝² : FTCFilter a la la'\ninst✝¹ : FTCFilter b lb lb'\ninst✝ : IsLocallyFiniteMeasure μ\nhab : IntervalIntegrable f μ a b\nhmeas_a : StronglyMeasurableAtFilter f la' μ\nhmeas_b : StronglyMeasurableAtFilter f lb' μ\nha_lim : Tendsto f (la' ⊓ ae μ) (𝓝 ca)\nhb_lim : Tendsto f (lb' ⊓ ae μ) (𝓝 cb)\nhua : Tendsto ua lt la\nhva : Tendsto va lt la\nhub : Tendsto ub lt lb\nhvb : Tendsto vb lt lb\nthis✝ : la'.IsMeasurablyGenerated\nthis : lb'.IsMeasurablyGenerated\nA : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ua t) (va t)\nA' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ a (ua t)\nB : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ (ub t) (vb t)\nB' : ∀ᶠ (t : ι) in lt, IntervalIntegrable f μ b (ub t)\na✝ : ι\nua_va : IntervalIntegrable f μ (ua a✝) (va a✝)\na_ua : IntervalIntegrable f μ a (ua a✝)\nub_vb : IntervalIntegrable f μ (ub a✝) (vb a✝)\nb_ub : IntervalIntegrable f μ b (ub a✝)\n⊢ IntervalIntegrable f μ (ub a✝) (ua a✝)"}
+{"state": [{"context": ["R✝ : Type u_1", "inst✝¹⁰ : CommRing R✝", "M✝ : Submonoid R✝", "S✝ : Type u_2", "inst✝⁹ : CommRing S✝", "inst✝⁸ : Algebra R✝ S✝", "P : Type u_3", "inst✝⁷ : CommRing P", "A : Type u_4", "K : Type u_5", "inst✝⁶ : CommRing A", "inst✝⁵ : IsDomain A", "R : Type u_6", "inst✝⁴ : CommRing R", "M : Submonoid R", "S : Type u_7", "inst✝³ : CommRing S", "inst✝² : Nontrivial R", "inst✝¹ : Algebra R S", "inst✝ : IsLocalization M S", "x : S"], "goal": "IsAlgebraic R x"}], "premise": [77600], "state_str": "case isAlgebraic\nR✝ : Type u_1\ninst✝¹⁰ : CommRing R✝\nM✝ : Submonoid R✝\nS✝ : Type u_2\ninst✝⁹ : CommRing S✝\ninst✝⁸ : Algebra R✝ S✝\nP : Type u_3\ninst✝⁷ : CommRing P\nA : Type u_4\nK : Type u_5\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\nR : Type u_6\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_7\ninst✝³ : CommRing S\ninst✝² : Nontrivial R\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx : S\n⊢ IsAlgebraic R x"}
+{"state": [{"context": ["M✝ : Type u_1", "N : Type u_2", "P : Type u_3", "α : Type u_4", "M : Type u_5", "inst✝² : MulOneClass M", "inst✝¹ : SMul α M", "inst✝ : IsScalarTower α M M", "c : Con M", "a : α", "w x : M", "h : c w x"], "goal": "c (a • w) (a • x)"}], "premise": [7215, 118946, 145792], "state_str": "M✝ : Type u_1\nN : Type u_2\nP : Type u_3\nα : Type u_4\nM : Type u_5\ninst✝² : MulOneClass M\ninst✝¹ : SMul α M\ninst✝ : IsScalarTower α M M\nc : Con M\na : α\nw x : M\nh : c w x\n⊢ c (a • w) (a • x)"}
+{"state": [{"context": ["α : Type u_1", "c : Set (Set α)", "hc : IsPartition c", "x : α", "t : Set α", "ht : (fun b => b ∈ c ∧ x ∈ b) t ∧ ∀ (y : Set α), (fun b => b ∈ c ∧ x ∈ b) y → y = t"], "goal": "t ∈ c ∧ x ∈ t"}], "premise": [70128], "state_str": "α : Type u_1\nc : Set (Set α)\nhc : IsPartition c\nx : α\nt : Set α\nht : (fun b => b ∈ c ∧ x ∈ b) t ∧ ∀ (y : Set α), (fun b => b ∈ c ∧ x ∈ b) y → y = t\n⊢ t ∈ c ∧ x ∈ t"}
+{"state": [{"context": ["α : Type u_1", "c : Set (Set α)", "hc : IsPartition c", "x : α", "t : Set α", "ht : (t ∈ c ∧ x ∈ t) ∧ ∀ (y : Set α), y ∈ c ∧ x ∈ y → y = t"], "goal": "t ∈ c ∧ x ∈ t"}], "premise": [1101, 1674], "state_str": "α : Type u_1\nc : Set (Set α)\nhc : IsPartition c\nx : α\nt : Set α\nht : (t ∈ c ∧ x ∈ t) ∧ ∀ (y : Set α), y ∈ c ∧ x ∈ y → y = t\n⊢ t ∈ c ∧ x ∈ t"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "inst✝² : DecidableEq ι", "inst✝¹ : Fintype ι", "inst✝ : CommMonoid α", "s : Finset ι", "f : ι → α"], "goal": "(∏ i : ι, if i ∈ s then f i else 1) = ∏ i ∈ s, f i"}], "premise": [127031, 139122, 140869], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\ninst✝ : CommMonoid α\ns : Finset ι\nf : ι → α\n⊢ (∏ i : ι, if i ∈ s then f i else 1) = ∏ i ∈ s, f i"}
+{"state": [{"context": ["R : Type u", "a b : R", "m n✝ : ℕ", "inst✝ : Semiring R", "p q : R[X]", "n : ℕ"], "goal": "(X ^ n).toFinsupp = Finsupp.single n 1"}], "premise": [101228, 101313], "state_str": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (X ^ n).toFinsupp = Finsupp.single n 1"}
+{"state": [{"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "p✝ : α → Prop", "inst✝¹ : DecidablePred p✝", "f : α → β", "s : Multiset α", "p : β → Prop", "inst✝ : DecidablePred p"], "goal": "countP p (map f s) = card (filter (fun a => p (f a)) s)"}], "premise": [137824], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\np✝ : α → Prop\ninst✝¹ : DecidablePred p✝\nf : α → β\ns : Multiset α\np : β → Prop\ninst✝ : DecidablePred p\n⊢ countP p (map f s) = card (filter (fun a => p (f a)) s)"}
+{"state": [{"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x"], "goal": "x ^ y < 1 ↔ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y"}], "premise": [1713, 37890, 37893, 39997, 106931, 149309], "state_str": "x y z : ℝ\nn : ℕ\nhx : 0 < x\n⊢ x ^ y < 1 ↔ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y"}
+{"state": [{"context": [], "goal": "∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))"}], "premise": [39168], "state_str": "⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))"}
+{"state": [{"context": ["d : ℝ", "h : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ d"], "goal": "∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))"}], "premise": [1673, 105728], "state_str": "case intro\nd : ℝ\nh : ∀ (n : ℕ), Real.log (stirlingSeq 1) - Real.log (stirlingSeq (n + 1)) ≤ d\n⊢ ∃ c, ∀ (n : ℕ), c ≤ Real.log (stirlingSeq (n + 1))"}
+{"state": [{"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "f : α → β", "s : Set β", "h : s.Finite", "hf : ∀ b ∈ s, (f ⁻¹' {b}).Finite"], "goal": "(f ⁻¹' s).Finite"}], "premise": [135520], "state_str": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : s.Finite\nhf : ∀ b ∈ s, (f ⁻¹' {b}).Finite\n⊢ (f ⁻¹' s).Finite"}
+{"state": [{"context": ["α : Type u", "β : Type v", "ι : Sort w", "γ : Type x", "f : α → β", "s : Set β", "h : s.Finite", "hf : ∀ b ∈ s, (f ⁻¹' {b}).Finite"], "goal": "(⋃ y ∈ s, f ⁻¹' {y}).Finite"}], "premise": [135054], "state_str": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : s.Finite\nhf : ∀ b ∈ s, (f ⁻¹' {b}).Finite\n⊢ (⋃ y ∈ s, f ⁻¹' {y}).Finite"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F G : (C ⥤ D)ᵒᵖ", "f : F ⟶ G"], "goal": "f ≫ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom.op = (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom.op ≫ Quiver.Hom.op { app := fun X => f.unop.app X, naturality := ⋯ }"}], "premise": [89631], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF G : (C ⥤ D)ᵒᵖ\nf : F ⟶ G\n⊢ f ≫ (NatIso.ofComponents (fun X => Iso.refl ((unop G).obj X)) ⋯).hom.op =\n    (NatIso.ofComponents (fun X => Iso.refl ((unop F).obj X)) ⋯).hom.op ≫\n      Quiver.Hom.op { app := fun X => f.unop.app X, naturality := ⋯ }"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝³ : SemilatticeInf α", "inst✝² : OrderBot α", "inst✝¹ : SemilatticeInf β", "inst✝ : OrderBot β", "a b : α", "f : α ≃o β", "ha : Disjoint a b"], "goal": "Disjoint (f a) (f b)"}], "premise": [11122, 11127, 13482], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ Disjoint (f a) (f b)"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝³ : SemilatticeInf α", "inst✝² : OrderBot α", "inst✝¹ : SemilatticeInf β", "inst✝ : OrderBot β", "a b : α", "f : α ≃o β", "ha : Disjoint a b"], "goal": "f (a ⊓ b) ≤ f ⊥"}], "premise": [11105, 13484], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : SemilatticeInf α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeInf β\ninst✝ : OrderBot β\na b : α\nf : α ≃o β\nha : Disjoint a b\n⊢ f (a ⊓ b) ≤ f ⊥"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "F : Type u_3", "K : Type u_4", "P Q : Cubic R", "a b c d a' b' c' d' : R", "inst✝ : Semiring R", "ha : P.a = 0", "hb : P.b ≠ 0"], "goal": "P.toPoly.degree = 2"}], "premise": [102287, 112044], "state_str": "R : Type u_1\nS : Type u_2\nF : Type u_3\nK : Type u_4\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b ≠ 0\n⊢ P.toPoly.degree = 2"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * f x) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) atTop (𝓝 (f 0))"}], "premise": [26334, 28328, 38517, 117883, 148328, 148388, 148394], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\n⊢ Tendsto (fun n => (∫ (x : ℝ) in 0 ..π / 2, ↑(cos x) ^ n * f x) / ↑(∫ (x : ℝ) in 0 ..π / 2, cos x ^ n)) atTop (𝓝 (f 0))"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume) atTop (𝓝 (f 0))"}], "premise": [1690, 2106, 2107, 14271, 14298, 38607], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))", "c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume) atTop (𝓝 (f 0))"}], "premise": [20217, 38517, 38588, 104746], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))", "c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0", "c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume) atTop (𝓝 (f 0))"}], "premise": [101702, 149222], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))", "c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0", "c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x", "c_zero_pos : 0 < cos 0"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume) atTop (𝓝 (f 0))"}], "premise": [20167, 38517, 56354, 56361], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))"}
+{"state": [{"context": ["f : ℝ → ℂ", "hf : ContinuousOn f (Icc 0 (π / 2))", "c_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0", "c_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x", "c_zero_pos : 0 < cos 0", "zero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))"], "goal": "Tendsto (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume) atTop (𝓝 (f 0))"}], "premise": [26740, 38505, 63094], "state_str": "f : ℝ → ℂ\nhf : ContinuousOn f (Icc 0 (π / 2))\nc_lt : ∀ y ∈ Icc 0 (π / 2), y ≠ 0 → cos y < cos 0\nc_nonneg : ∀ x ∈ Icc 0 (π / 2), 0 ≤ cos x\nc_zero_pos : 0 < cos 0\nzero_mem : 0 ∈ closure (interior (Icc 0 (π / 2)))\n⊢ Tendsto\n    (fun n => (∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n ∂volume)⁻¹ • ∫ (x : ℝ) in Icc 0 (π / 2), cos x ^ n • f x ∂volume)\n    atTop (𝓝 (f 0))"}
+{"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'", "h : Function.Surjective ⇑f"], "goal": "f.IsIdealMorphism"}], "premise": [1674, 109361, 109377, 109390, 109391], "state_str": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : Function.Surjective ⇑f\n⊢ f.IsIdealMorphism"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "a✝ a : α", "h : 0 < a", "b c : α"], "goal": "(fun x => a * x) '' Ioo b c = Ioo (a * b) (a * c)"}], "premise": [119707, 134597], "state_str": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a : α\nh : 0 < a\nb c : α\n⊢ (fun x => a * x) '' Ioo b c = Ioo (a * b) (a * c)"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommRing R", "p q : R[X]", "hq : q.Monic"], "goal": "p %ₘ q = 0 ↔ (Ideal.Quotient.mk (Ideal.span {q})) p = 0"}], "premise": [1713, 80143, 103308], "state_str": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhq : q.Monic\n⊢ p %ₘ q = 0 ↔ (Ideal.Quotient.mk (Ideal.span {q})) p = 0"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝¹⁸ : CommSemiring R", "inst✝¹⁷ : AddCommMonoid M", "inst✝¹⁶ : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝¹⁵ : CommRing R₁", "inst✝¹⁴ : AddCommGroup M₁", "inst✝¹³ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝¹² : Field K", "inst✝¹¹ : AddCommGroup V", "inst✝¹⁰ : Module K V", "M'✝ : Type u_7", "M''✝ : Type u_8", "inst✝⁹ : AddCommMonoid M'✝", "inst✝⁸ : AddCommMonoid M''✝", "inst✝⁷ : Module R M'✝", "inst✝⁶ : Module R M''✝", "B : BilinForm R M", "B₁ : BilinForm R₁ M₁", "F : BilinForm R M", "M' : Type u_9", "inst✝⁵ : AddCommMonoid M'", "inst✝⁴ : Module R M'", "B' : BilinForm R M'", "f✝ f'✝ : M →ₗ[R] M'", "g✝ g'✝ : M' →ₗ[R] M", "M₁' : Type u_10", "inst✝³ : AddCommGroup M₁'", "inst✝² : Module R₁ M₁'", "B₁' : BilinForm R₁ M₁'", "f₁ f₁' : M₁ →ₗ[R₁] M₁'", "g₁ g₁' : M₁' →ₗ[R₁] M₁", "B₂' : BilinForm R M'", "f₂ f₂' : M →ₗ[R] M'", "g₂ g₂' : M' →ₗ[R] M", "M'' : Type u_11", "inst✝¹ : AddCommMonoid M''", "inst✝ : Module R M''", "B'' : BilinForm R M''", "f g f' g' : Module.End R M", "h : B.IsAdjointPair B f g", "h' : B.IsAdjointPair B f' g'", "x y : M"], "goal": "(B ((f * f') x)) y = (B x) ((g' * g) y)"}], "premise": [109196], "state_str": "R : Type u_1\nM : Type u_2\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : AddCommMonoid M\ninst✝¹⁶ : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹⁵ : CommRing R₁\ninst✝¹⁴ : AddCommGroup M₁\ninst✝¹³ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝¹² : Field K\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module K V\nM'✝ : Type u_7\nM''✝ : Type u_8\ninst✝⁹ : AddCommMonoid M'✝\ninst✝⁸ : AddCommMonoid M''✝\ninst✝⁷ : Module R M'✝\ninst✝⁶ : Module R M''✝\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nF : BilinForm R M\nM' : Type u_9\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : Module R M'\nB' : BilinForm R M'\nf✝ f'✝ : M →ₗ[R] M'\ng✝ g'✝ : M' →ₗ[R] M\nM₁' : Type u_10\ninst✝³ : AddCommGroup M₁'\ninst✝² : Module R₁ M₁'\nB₁' : BilinForm R₁ M₁'\nf₁ f₁' : M₁ →ₗ[R₁] M₁'\ng₁ g₁' : M₁' →ₗ[R₁] M₁\nB₂' : BilinForm R M'\nf₂ f₂' : M →ₗ[R] M'\ng₂ g₂' : M' →ₗ[R] M\nM'' : Type u_11\ninst✝¹ : AddCommMonoid M''\ninst✝ : Module R M''\nB'' : BilinForm R M''\nf g f' g' : Module.End R M\nh : B.IsAdjointPair B f g\nh' : B.IsAdjointPair B f' g'\nx y : M\n⊢ (B ((f * f') x)) y = (B x) ((g' * g) y)"}
+{"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 * B * ⅟B = A"}], "premise": [118833, 142258, 142263], "state_str": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\nA : Matrix m n α\nB : Matrix n n α\ninst✝ : Invertible B\n⊢ A * B * ⅟B = A"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "s : Finset α"], "goal": "s.card + sᶜ.card = Fintype.card α"}], "premise": [3890, 3891, 141354, 141359], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset α\n⊢ s.card + sᶜ.card = Fintype.card α"}
+{"state": [{"context": ["N✝ : ℕ", "g : ConjAct SL(2, ℤ)", "Γ : Subgroup SL(2, ℤ)", "N : ℕ+", "HN : Gamma ↑N ≤ Γ"], "goal": "Gamma ↑N ≤ g • Γ"}], "premise": [21493, 119082], "state_str": "case intro\nN✝ : ℕ\ng : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nN : ℕ+\nHN : Gamma ↑N ≤ Γ\n⊢ Gamma ↑N ≤ g • Γ"}
+{"state": [{"context": ["R : Type u", "inst✝⁴ : CommRing R", "S : Type v", "inst✝³ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝² : Algebra R S", "inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)", "inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)", "hp : p.IsMaximal"], "goal": "inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}], "premise": [24060, 86069], "state_str": "R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}
+{"state": [{"context": ["R : Type u", "inst✝⁴ : CommRing R", "S : Type v", "inst✝³ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝² : Algebra R S", "inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)", "inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)", "hp : p.IsMaximal", "a✝ : Nontrivial (S ⧸ P)"], "goal": "inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}], "premise": [80927, 115860], "state_str": "R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}
+{"state": [{"context": ["R : Type u", "inst✝⁴ : CommRing R", "S : Type v", "inst✝³ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝² : Algebra R S", "inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)", "inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)", "hp : p.IsMaximal", "a✝ : Nontrivial (S ⧸ P)", "this : comap (algebraMap R S) P = p"], "goal": "inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}], "premise": [1739], "state_str": "R : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ inertiaDeg (algebraMap R S) p P = finrank (R ⧸ p) (S ⧸ P)"}
+{"state": [{"context": ["R : Type u", "inst✝⁴ : CommRing R", "S : Type v", "inst✝³ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝² : Algebra R S", "inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)", "inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)", "hp : p.IsMaximal", "a✝ : Nontrivial (S ⧸ P)", "this : comap (algebraMap R S) P = p"], "goal": "(Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯).toAlgebra = inst✝¹"}], "premise": [121164, 127818], "state_str": "case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯).toAlgebra = inst✝¹"}
+{"state": [{"context": ["R : Type u", "inst✝⁴ : CommRing R", "S : Type v", "inst✝³ : CommRing S", "f : R →+* S", "p : Ideal R", "P : Ideal S", "inst✝² : Algebra R S", "inst✝¹ : Algebra (R ⧸ p) (S ⧸ P)", "inst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)", "hp : p.IsMaximal", "a✝ : Nontrivial (S ⧸ P)", "this : comap (algebraMap R S) P = p", "x' : R ⧸ p", "x : R"], "goal": "(Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯) ((Quotient.mk p) x) = (algebraMap (R ⧸ p) (S ⧸ P)) ((Quotient.mk p) x)"}], "premise": [80156, 81189, 121201, 121202], "state_str": "case h.e_5.h.h.e_5.h\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\nf : R →+* S\np : Ideal R\nP : Ideal S\ninst✝² : Algebra R S\ninst✝¹ : Algebra (R ⧸ p) (S ⧸ P)\ninst✝ : IsScalarTower R (R ⧸ p) (S ⧸ P)\nhp : p.IsMaximal\na✝ : Nontrivial (S ⧸ P)\nthis : comap (algebraMap R S) P = p\nx' : R ⧸ p\nx : R\n⊢ (Quotient.lift p ((Quotient.mk P).comp (algebraMap R S)) ⋯) ((Quotient.mk p) x) =\n    (algebraMap (R ⧸ p) (S ⧸ P)) ((Quotient.mk p) x)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n : ℕ", "hk : k ≠ 0", "h : k ≤ n"], "goal": "∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"}], "premise": [145198], "state_str": "α : Type u_1\ninst✝ : LinearOrderedField α\nk n : ℕ\nhk : k ≠ 0\nh : k ≤ n\n⊢ ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n✝ : ℕ", "hk : k ≠ 0", "h : k ≤ n✝", "n : ℕ", "hn : k ≤ n", "IH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"], "goal": "∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}], "premise": [124720], "state_str": "case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ i ∈ Ioc k (n + 1), (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n✝ : ℕ", "hk : k ≠ 0", "h : k ≤ n✝", "n : ℕ", "hn : k ≤ n", "IH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"], "goal": "∑ k ∈ Ioc k n, (↑k ^ 2)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}], "premise": [14272, 103917], "state_str": "case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ∑ k ∈ Ioc k n, (↑k ^ 2)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n✝ : ℕ", "hk : k ≠ 0", "h : k ≤ n✝", "n : ℕ", "hn : k ≤ n", "IH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"], "goal": "(↑k)⁻¹ - (↑n)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}], "premise": [103977, 105594, 105630, 119704, 119729, 119789, 143125, 143126], "state_str": "case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ (↑k)⁻¹ - (↑n)⁻¹ + (↑(n + 1) ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑(n + 1))⁻¹"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n✝ : ℕ", "hk : k ≠ 0", "h : k ≤ n✝", "n : ℕ", "hn : k ≤ n", "IH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹"], "goal": "((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹"}], "premise": [18824], "state_str": "case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\n⊢ ((↑n + 1) ^ 2)⁻¹ + (↑n + 1)⁻¹ ≤ (↑n)⁻¹"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedField α", "k n✝ : ℕ", "hk : k ≠ 0", "h : k ≤ n✝", "n : ℕ", "hn : k ≤ n", "IH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹", "A : 0 < ↑n", "B : 0 < ↑n + 1"], "goal": "(↑n + 1 + (↑n + 1) ^ 2) / ((↑n + 1) ^ 2 * (↑n + 1)) ≤ 1 / ↑n"}], "premise": [105706, 106079], "state_str": "case refine_2\nα : Type u_1\ninst✝ : LinearOrderedField α\nk n✝ : ℕ\nhk : k ≠ 0\nh : k ≤ n✝\nn : ℕ\nhn : k ≤ n\nIH : ∑ i ∈ Ioc k n, (↑i ^ 2)⁻¹ ≤ (↑k)⁻¹ - (↑n)⁻¹\nA : 0 < ↑n\nB : 0 < ↑n + 1\n⊢ (↑n + 1 + (↑n + 1) ^ 2) / ((↑n + 1) ^ 2 * (↑n + 1)) ≤ 1 / ↑n"}
+{"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 : (i : ι) → Set (π i)", "a : (i : ι) → π i", "hs : I.pi s ∈ 𝓝 a", "i : ι", "hi : i ∈ I"], "goal": "s i ∈ 𝓝 (a i)"}], "premise": [66542], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\nι : Type u_5\nπ : ι → Type u_6\nκ : Type u_7\ninst✝ : TopologicalSpace X\nT : (i : ι) → TopologicalSpace (π i)\nf : X → (i : ι) → π i\nI : Set ι\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\nhs : I.pi s ∈ 𝓝 a\ni : ι\nhi : i ∈ I\n⊢ s i ∈ 𝓝 (a i)"}
+{"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 : (i : ι) → Set (π i)", "a : (i : ι) → π i", "hs : I.pi s ∈ Filter.pi fun i => 𝓝 (a i)", "i : ι", "hi : i ∈ I"], "goal": "s i ∈ 𝓝 (a i)"}], "premise": [11490], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\nι : Type u_5\nπ : ι → Type u_6\nκ : Type u_7\ninst✝ : TopologicalSpace X\nT : (i : ι) → TopologicalSpace (π i)\nf : X → (i : ι) → π i\nI : Set ι\ns : (i : ι) → Set (π i)\na : (i : ι) → π i\nhs : I.pi s ∈ Filter.pi fun i => 𝓝 (a i)\ni : ι\nhi : i ∈ I\n⊢ s i ∈ 𝓝 (a i)"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ inst✝ : NumberField K", "a₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)", "h : (MonoidHom.toAdditive' ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom)) a₁✝ = (MonoidHom.toAdditive' ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom)) a₂✝"], "goal": "a₁✝ = a₂✝"}], "premise": [1670, 117181, 119966, 120084, 128619], "state_str": "K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₁✝ =\n    (MonoidHom.toAdditive'\n        ((QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))).comp\n          (QuotientGroup.quotientMulEquivOfEq ⋯).toMonoidHom))\n      a₂✝\n⊢ a₁✝ = a₂✝"}
+{"state": [{"context": ["K : Type u_1", "inst✝² : Field K", "inst✝¹ inst✝ : NumberField K", "a₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)", "h : (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))) ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) = (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K))) ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₂✝))"], "goal": "a₁✝ = a₂✝"}], "premise": [1673, 10396, 71387, 128619], "state_str": "K : Type u_1\ninst✝² : Field K\ninst✝¹ inst✝ : NumberField K\na₁✝ a₂✝ : Additive ((𝓞 K)ˣ ⧸ torsion K)\nh :\n  (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₁✝)) =\n    (QuotientGroup.kerLift (AddMonoidHom.toMultiplicative' (logEmbedding K)))\n      ((QuotientGroup.quotientMulEquivOfEq ⋯) (Additive.toMul a₂✝))\n⊢ a₁✝ = a₂✝"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ Q₀ : PresheafOfModules R₀", "N : SheafOfModules R", "f : P₀ ⟶ Q₀", "g : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N"], "goal": "((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g) = (sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g"}], "premise": [100147], "state_str": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ ((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g) =\n    (sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ Q₀ : PresheafOfModules R₀", "N : SheafOfModules R", "f : P₀ ⟶ Q₀", "g : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N"], "goal": "(SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) = (SheafOfModules.toSheaf R).map ((sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)"}], "premise": [99919], "state_str": "case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f ≫ ((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ Q₀ : PresheafOfModules R₀", "N : SheafOfModules R", "f : P₀ ⟶ Q₀", "g : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N"], "goal": "(SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) = (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)"}], "premise": [112653], "state_str": "case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) P₀ N).symm (f ≫ g)) =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      (SheafOfModules.toSheaf R).map (((fun x x_1 => sheafificationHomEquiv α) Q₀ N).symm g)"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ Q₀ : PresheafOfModules R₀", "N : SheafOfModules R", "f : P₀ ⟶ Q₀", "g : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N"], "goal": "((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N)).symm (f ≫ g).hom = (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv Q₀.presheaf ((SheafOfModules.toSheaf R).obj N)).symm g.hom"}], "premise": [95793], "state_str": "case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ Q₀ : PresheafOfModules R₀\nN : SheafOfModules R\nf : P₀ ⟶ Q₀\ng : Q₀ ⟶ (SheafOfModules.forget R ⋙ restrictScalars α).obj N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N)).symm\n      (f ≫ g).hom =\n    (SheafOfModules.toSheaf R).map ((sheafification α).map f) ≫\n      ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv Q₀.presheaf\n            ((SheafOfModules.toSheaf R).obj N)).symm\n        g.hom"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ : PresheafOfModules R₀", "M N : SheafOfModules R", "f : (sheafification α).obj P₀ ⟶ M", "g : M ⟶ N"], "goal": "((fun x x_1 => sheafificationHomEquiv α) P₀ N) (f ≫ g) = ((fun x x_1 => sheafificationHomEquiv α) P₀ M) f ≫ (SheafOfModules.forget R ⋙ restrictScalars α).map g"}], "premise": [100147], "state_str": "C : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((fun x x_1 => sheafificationHomEquiv α) P₀ N) (f ≫ g) =\n    ((fun x x_1 => sheafificationHomEquiv α) P₀ M) f ≫ (SheafOfModules.forget R ⋙ restrictScalars α).map g"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ : PresheafOfModules R₀", "M N : SheafOfModules R", "f : (sheafification α).obj P₀ ⟶ M", "g : M ⟶ N"], "goal": "((sheafificationHomEquiv α) (f ≫ g)).hom = ((sheafificationHomEquiv α) f).hom ≫ g.val.hom"}], "premise": [112652], "state_str": "case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((sheafificationHomEquiv α) (f ≫ g)).hom = ((sheafificationHomEquiv α) f).hom ≫ g.val.hom"}
+{"state": [{"context": ["C : Type u'", "inst✝⁴ : Category.{v', u'} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "inst✝¹ : J.WEqualsLocallyBijective AddCommGrp", "inst✝ : HasWeakSheafify J AddCommGrp", "P₀ : PresheafOfModules R₀", "M N : SheafOfModules R", "f : (sheafification α).obj P₀ ⟶ M", "g : M ⟶ N"], "goal": "((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N)) ((SheafOfModules.toSheaf R).map (f ≫ g)) = ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M)) ((SheafOfModules.toSheaf R).map f) ≫ g.val.hom"}], "premise": [99919], "state_str": "case a\nC : Type u'\ninst✝⁴ : Category.{v', u'} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\ninst✝¹ : J.WEqualsLocallyBijective AddCommGrp\ninst✝ : HasWeakSheafify J AddCommGrp\nP₀ : PresheafOfModules R₀\nM N : SheafOfModules R\nf : (sheafification α).obj P₀ ⟶ M\ng : M ⟶ N\n⊢ ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj N))\n      ((SheafOfModules.toSheaf R).map (f ≫ g)) =\n    ((CategoryTheory.sheafificationAdjunction J AddCommGrp).homEquiv P₀.presheaf ((SheafOfModules.toSheaf R).obj M))\n        ((SheafOfModules.toSheaf R).map f) ≫\n      g.val.hom"}
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+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L"], "goal": "1 = chainTopCoeff (⇑α) 0"}], "premise": [11251], "state_str": "K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ 1 = chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [110405], "state_str": "case hba\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [111734], "state_str": "case hba.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "h e f : L", "isSl2 : IsSl2Triple h e f", "he : e ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [111735], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\nh e f : L\nisSl2 : IsSl2Triple h e f\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e f : L", "he : e ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e f"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [2100, 108638, 108957, 110470], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e f : L", "he : e ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [1673, 87666, 107468, 108340, 108639, 108950, 110406, 111736, 111740, 119704, 119729, 119746, 119819], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e f : L", "he : e ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)", "k : K", "hk : ⁅f, k • f⁆ = ⁅f, ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1)) x⁆"], "goal": "¬1 < chainTopCoeff (⇑α) 0"}], "premise": [107474, 107708, 107712, 108328], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne f : L\nhe : e ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : ⁅f, k • f⁆ = ⁅f, ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1)) x⁆\n⊢ ¬1 < chainTopCoeff (⇑α) 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e✝ f : L", "he : e✝ ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e✝ f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)", "k : K", "hk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x", "e : 1 < chainTopCoeff (⇑α) 0"], "goal": "False"}], "premise": [2100, 107477], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ False"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e✝ f : L", "he : e✝ ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e✝ f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)", "k : K", "hk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x", "e : 1 < chainTopCoeff (⇑α) 0"], "goal": "chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}], "premise": [2100, 110471], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e✝ f : L", "he : e✝ ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e✝ f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)", "k : K", "hk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x", "e : 1 < chainTopCoeff (⇑α) 0", "this : ↑(↑(chainLength α 0) - 2 * ↑(chainTopCoeff (⇑α) 0)) = 0 (coroot α)"], "goal": "chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}], "premise": [110407, 111254, 118004, 128912, 142648, 143128], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : ↑(↑(chainLength α 0) - 2 * ↑(chainTopCoeff (⇑α) 0)) = 0 (coroot α)\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}
+{"state": [{"context": ["K : Type u_1", "L : Type u_2", "inst✝⁸ : Field K", "inst✝⁷ : CharZero K", "inst✝⁶ : LieRing L", "inst✝⁵ : LieAlgebra K L", "inst✝⁴ : IsKilling K L", "inst✝³ : FiniteDimensional K L", "H : LieSubalgebra K L", "inst✝² : H.IsCartanSubalgebra", "inst✝¹ : IsTriangularizable K (↥H) L", "α β : Weight K (↥H) L", "hα : α.IsNonZero", "inst✝ : Nontrivial L", "x : L", "hx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)", "x_ne0 : x ≠ 0", "e✝ f : L", "he : e✝ ∈ rootSpace H ⇑α", "hf : f ∈ rootSpace H (-⇑α)", "isSl2 : IsSl2Triple (↑(coroot α)) e✝ f", "prim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)", "k : K", "hk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x", "e : 1 < chainTopCoeff (⇑α) 0", "this : chainLength α 0 = 2 * chainTopCoeff (⇑α) 0"], "goal": "chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}], "premise": [3814, 103894, 122222], "state_str": "case hba.intro.intro.intro.intro.intro.intro.intro.intro\nK : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\ninst✝ : Nontrivial L\nx : L\nhx : x ∈ weightSpace L ⇑(chainTop (⇑α) 0)\nx_ne0 : x ≠ 0\ne✝ f : L\nhe : e✝ ∈ rootSpace H ⇑α\nhf : f ∈ rootSpace H (-⇑α)\nisSl2 : IsSl2Triple (↑(coroot α)) e✝ f\nprim : isSl2.HasPrimitiveVectorWith x ↑(chainLength α 0)\nk : K\nhk : 0 = ((toEnd K L L) f ^ (chainTopCoeff (⇑α) 0 + 1 + 1)) x\ne : 1 < chainTopCoeff (⇑α) 0\nthis : chainLength α 0 = 2 * chainTopCoeff (⇑α) 0\n⊢ chainTopCoeff (⇑α) 0 + 1 + 1 ≤ chainLength α 0"}
+{"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✝⁷ : AddCommGroup M", "inst✝⁶ : AddCommGroup M₂", "F : Type u_9", "inst✝⁵ : FunLike F M M₂", "inst✝⁴ : AddMonoidHomClass F M M₂", "f : F", "R : Type u_10", "S : Type u_11", "inst✝³ : Ring R", "inst✝² : Ring S", "inst✝¹ : Module R M", "inst✝ : Module S M₂", "x : ℤ", "a : M"], "goal": "f (↑x • a) = ↑x • f a"}], "premise": [110044, 117114], "state_str": "α : Type u_1\nR✝ : Type u_2\nk : Type u_3\nS✝ : Type u_4\nM : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\nι : Type u_8\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M₂\nF : Type u_9\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_10\nS : Type u_11\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nx : ℤ\na : M\n⊢ f (↑x • a) = ↑x • f a"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "r₁ r₂ : R"], "goal": "(ι 0) r₁ * (ι 0) r₂ = 0"}], "premise": [82290, 83890, 108328, 109741, 118863, 118917, 119730, 121564], "state_str": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr₁ r₂ : R\n⊢ (ι 0) r₁ * (ι 0) r₂ = 0"}
+{"state": [{"context": ["α✝ β : Type u", "α : Type u_1", "s : Set α"], "goal": "s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable ↑s)"}], "premise": [1713, 70617, 132730, 135141], "state_str": "α✝ β : Type u\nα : Type u_1\ns : Set α\n⊢ s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable ↑s)"}
+{"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", "ι : Type u", "s : ι → K[X]", "t : Finset ι"], "goal": "(∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))"}], "premise": [1674, 1713, 2106, 2107, 101117, 101136, 123871, 126900, 138715, 138847, 139166], "state_str": "R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝³ : CommRing R\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nι : Type u\ns : ι → K[X]\nt : Finset ι\n⊢ (∀ j ∈ t, s j ≠ 0) → (Splits i (∏ x ∈ t, s x) ↔ ∀ j ∈ t, Splits i (s j))"}
+{"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", "E : Set α", "ε : ℝ", "hε✝ : 0 < ε", "δ : ℝ", "hδ_pos : 0 < δ", "hε : ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε", "i : ι", "s : Set α", "hs : MeasurableSet s", "hμs : μ s ≤ ENNReal.ofReal δ"], "goal": "eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε"}], "premise": [31039, 120897, 133443], "state_str": "case intro.intro\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifIntegrable f p μ\nE : Set α\nε : ℝ\nhε✝ : 0 < ε\nδ : ℝ\nhδ_pos : 0 < δ\nhε :\n  ∀ (i : ι) (s : Set α), MeasurableSet s → μ s ≤ ENNReal.ofReal δ → eLpNorm (s.indicator (f i)) p μ ≤ ENNReal.ofReal ε\ni : ι\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≤ ENNReal.ofReal δ\n⊢ eLpNorm (s.indicator ((fun i => E.indicator (f i)) i)) p μ ≤ ENNReal.ofReal ε"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : TopologicalSpace α", "z : α", "s : Set α"], "goal": "z ∈ closure (s \\ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s"}], "premise": [55569, 57161], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : TopologicalSpace α\nz : α\ns : Set α\n⊢ z ∈ closure (s \\ {z}) ↔ ∃ᶠ (x : α) in 𝓝[≠] z, x ∈ s"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝¹ : NormedLinearOrderedField 𝕜", "P Q : 𝕜[X]", "inst✝ : OrderTopology 𝕜", "hdeg : Q.natDegree < P.natDegree", "hpos : 0 < P.leadingCoeff / Q.leadingCoeff", "hQ : Q ≠ 0"], "goal": "Tendsto (fun x => eval x P / eval x Q) atTop atTop"}], "premise": [39226, 41439, 41463], "state_str": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => eval x P / eval x Q) atTop atTop"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝¹ : NormedLinearOrderedField 𝕜", "P Q : 𝕜[X]", "inst✝ : OrderTopology 𝕜", "hdeg : Q.natDegree < P.natDegree", "hpos : 0 < P.leadingCoeff / Q.leadingCoeff", "hQ : Q ≠ 0"], "goal": "Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop"}], "premise": [15632], "state_str": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝¹ : NormedLinearOrderedField 𝕜", "P Q : 𝕜[X]", "inst✝ : OrderTopology 𝕜", "hdeg : Q.natDegree < P.natDegree", "hpos : 0 < P.leadingCoeff / Q.leadingCoeff", "hQ : Q ≠ 0"], "goal": "Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop"}], "premise": [15637], "state_str": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.natDegree < P.natDegree\nhpos : 0 < P.leadingCoeff / Q.leadingCoeff\nhQ : Q ≠ 0\n⊢ Tendsto (fun x => x ^ (↑P.natDegree - ↑Q.natDegree)) atTop atTop"}
+{"state": [{"context": ["x✝ y : ℂ", "x : ℝ"], "goal": "(starRingEnd ℂ) (cosh ↑x) = cosh ↑x"}], "premise": [148350, 149113], "state_str": "x✝ y : ℂ\nx : ℝ\n⊢ (starRingEnd ℂ) (cosh ↑x) = cosh ↑x"}
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+{"state": [{"context": ["f : Ordinal.{u} → Ordinal.{u}", "H : IsNormal f"], "goal": "deriv f = enumOrd (fixedPoints f)"}], "premise": [49023], "state_str": "f : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ deriv f = enumOrd (fixedPoints f)"}
+{"state": [{"context": ["f : Ordinal.{u} → Ordinal.{u}", "H : IsNormal f"], "goal": "fixedPoints f = ⋂ i, fixedPoints f"}], "premise": [2100, 135284], "state_str": "case h.e'_3.h.e'_1\nf : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ fixedPoints f = ⋂ i, fixedPoints f"}
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+{"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✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace γ", "μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "inst✝² : SemilatticeSup α", "inst✝¹ : NoMaxOrder α", "inst✝ : atTop.IsCountablyGenerated", "μ : Measure α", "a : α", "this : Nonempty α", "h_mono : Monotone fun x => μ (Ico a x)"], "goal": "μ (Ici a) = ⨆ i, μ (Ico a i)"}], "premise": [15757], "state_str": "case h.e'_5.h.e'_3\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)"}
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+{"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✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace ��", "μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "inst✝² : SemilatticeSup α", "inst✝¹ : NoMaxOrder α", "inst✝ : atTop.IsCountablyGenerated", "μ : Measure α", "a : α", "this : Nonempty α", "h_mono : Monotone fun x => μ (Ico a x)", "xs : ℕ → α", "hxs_mono : Monotone xs", "hxs_tendsto : Tendsto xs atTop atTop", "h_Ici : Ici a = ⋃ n, Ico a (xs n)"], "goal": "μ (Ici a) = ⨆ i, μ (Ico a i)"}], "premise": [31429, 55079], "state_str": "case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ μ (Ici a) = ⨆ i, μ (Ico a i)"}
+{"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✝⁴ : MeasurableSpace β", "inst✝³ : MeasurableSpace γ", "μ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α", "s s' t : Set α", "inst✝² : SemilatticeSup α", "inst✝¹ : NoMaxOrder α", "inst✝ : atTop.IsCountablyGenerated", "μ : Measure α", "a : α", "this : Nonempty α", "h_mono : Monotone fun x => μ (Ico a x)", "xs : ℕ → α", "hxs_mono : Monotone xs", "hxs_tendsto : Tendsto xs atTop atTop", "h_Ici : Ici a = ⋃ n, Ico a (xs n)"], "goal": "Directed (fun x x_1 => x ⊆ x_1) fun n => Ico a (xs n)"}], "premise": [16484, 20215], "state_str": "case h.e'_5.h.e'_3.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝² : SemilatticeSup α\ninst✝¹ : NoMaxOrder α\ninst✝ : atTop.IsCountablyGenerated\nμ : Measure α\na : α\nthis : Nonempty α\nh_mono : Monotone fun x => μ (Ico a x)\nxs : ℕ → α\nhxs_mono : Monotone xs\nhxs_tendsto : Tendsto xs atTop atTop\nh_Ici : Ici a = ⋃ n, Ico a (xs n)\n⊢ Directed (fun x x_1 => x ⊆ x_1) fun n => Ico a (xs n)"}
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+{"state": [{"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x", "hxy : x < y", "hz : z < 0", "this : 0 < y"], "goal": "x ^ (-z) < y ^ (-z)"}, {"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x", "hxy : x < y", "hz : z < 0", "this : 0 < y"], "goal": "0 ≤ y"}, {"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x", "hxy : x < y", "hz : z < 0", "this : 0 < y"], "goal": "0 ≤ x"}, {"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x", "hxy : x < y", "hz : z < 0", "this : 0 < y"], "goal": "0 < x ^ z"}, {"context": ["x y z : ℝ", "n : ℕ", "hx : 0 < x", "hxy : x < y", "hz : z < 0", "this : 0 < y"], "goal": "0 < y ^ z"}], "premise": [1674, 40080], "state_str": "x y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ x ^ (-z) < y ^ (-z)\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ y\n\ncase hx\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x\n\ncase ha\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < x ^ z\n\ncase hb\nx y z : ℝ\nn : ℕ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 < y ^ z"}
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+{"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'", "hf : Function.Surjective ⇑f"], "goal": "(H ⊔ f.ker).index ∣ H.index"}], "premise": [6524, 14517], "state_str": "G : Type u_1\nG' : Type u_2\ninst✝¹ : Group G\ninst✝ : Group G'\nH K L : Subgroup G\nf : G →* G'\nhf : Function.Surjective ⇑f\n⊢ (H ⊔ f.ker).index ∣ H.index"}
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+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : FirstCountableTopology X"], "goal": "T2Space (SeparationQuotient X) ↔ R1Space X"}], "premise": [1101, 1169, 16273, 61717, 61718, 61725, 64778, 64808, 71408], "state_str": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\n⊢ T2Space (SeparationQuotient X) ↔ R1Space X"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : Ring R", "A : Type u_2", "inst✝⁴ : CommRing A", "M : Type u_3", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "U U₁ U₂ : Submodule R M", "x x₁ x₂ y y₁ y₂ z z₁ z₂ : M", "N : Type u_4", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "V V₁ V₂ : Submodule R N"], "goal": "x ≡ y [SMOD U] ↔ x - y ∈ U"}], "premise": [1713, 81316, 82370], "state_str": "R : Type u_1\ninst✝⁵ : Ring R\nA : Type u_2\ninst✝⁴ : CommRing A\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nU U₁ U₂ : Submodule R M\nx x₁ x₂ y y₁ y₂ z z₁ z₂ : M\nN : Type u_4\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nV V₁ V₂ : Submodule R N\n⊢ x ≡ y [SMOD U] ↔ x - y ∈ U"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s' : Set α", "x : α", "p : Filter ι", "p' : Filter α", "g : ι → α", "c : β"], "goal": "Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'"}], "premise": [16363, 60481], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nc : β\n⊢ Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝ : UniformSpace β", "F : ι → α → β", "f : α → β", "s s' : Set α", "x : α", "p : Filter ι", "p' : Filter α", "g : ι → α", "c : β"], "goal": "Tendsto (Prod.mk c ∘ ↿F) (p ×ˢ p') (𝓤 β) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'"}], "premise": [1713], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nc : β\n⊢ Tendsto (Prod.mk c ∘ ↿F) (p ×ˢ p') (𝓤 β) ↔ TendstoUniformlyOnFilter F (fun x => c) p p'"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "p q : α → Prop", "inst✝¹ : DecidablePred p", "inst✝ : DecidablePred q", "s✝ t s : Finset α", "a : α"], "goal": "a ∈ filter q (filter p s) ↔ a ∈ filter (fun a => p a ∧ q a) s"}], "premise": [1206, 1240, 2871, 2872, 139089], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ns✝ t s : Finset α\na : α\n⊢ a ∈ filter q (filter p s) ↔ a ∈ filter (fun a => p a ∧ q a) s"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "π : ι → Type u_5", "rα : α → α → Prop", "rβ : β → β → Prop", "f : γ → α", "g : γ → β", "s : Set γ", "hα : WellFounded (rα on f)", "hβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)"], "goal": "WellFounded (Prod.Lex rα rβ on fun c => (f c, g c))"}], "premise": [1674, 13050, 15429, 15430, 20146], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nπ : ι → Type u_5\nrα : α → α → Prop\nrβ : β → β → Prop\nf : γ → α\ng : γ → β\ns : Set γ\nhα : WellFounded (rα on f)\nhβ : ∀ (a : α), (f ⁻¹' {a}).WellFoundedOn (rβ on g)\n⊢ WellFounded (Prod.Lex rα rβ on fun c => (f c, g c))"}
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+{"state": [{"context": ["a✝ b c d : ℝ≥0∞", "r p q : ℝ≥0", "a : ℝ≥0∞", "h : 0 < a"], "goal": "a ∈ {b | 1 ≤ ⊤ * b}"}], "premise": [11234, 143513], "state_str": "a✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\na : ℝ≥0∞\nh : 0 < a\n⊢ a ∈ {b | 1 ≤ ⊤ * b}"}
+{"state": [{"context": ["X Y : CommRingCat", "f : X ⟶ Y"], "goal": "(𝟭 CommRingCat).map f ≫ ((fun R => asIso (toSpecΓ R)) Y).hom = ((fun R => asIso (toSpecΓ R)) X).hom ≫ (Spec.toLocallyRingedSpace.rightOp ⋙ Γ).map f"}], "premise": [128425], "state_str": "X Y : CommRingCat\nf : X ⟶ Y\n⊢ (𝟭 CommRingCat).map f ≫ ((fun R => asIso (toSpecΓ R)) Y).hom =\n    ((fun R => asIso (toSpecΓ R)) X).hom ≫ (Spec.toLocallyRingedSpace.rightOp ⋙ Γ).map f"}
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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", "κ : Sort u_6", "r p q : α → α → Prop", "inst✝ : CompleteLattice α", "s : Set ι", "t : Set ι'", "f : ι × ι' → α", "hs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')", "ht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')", "i : ι", "i' : ι'", "hi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t", "j : ι", "j' : ι'", "hj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t", "h : (i, i') ≠ (j, j')"], "goal": "(Disjoint on f) (i, i') (j, j')"}], "premise": [70039], "state_str": "case mk.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\nκ : Sort u_6\nr p q : α → α → Prop\ninst✝ : CompleteLattice α\ns : Set ι\nt : Set ι'\nf : ι × ι' → α\nhs : s.PairwiseDisjoint fun i => ⨆ i' ∈ t, f (i, i')\nht : t.PairwiseDisjoint fun i' => ⨆ i ∈ s, f (i, i')\ni : ι\ni' : ι'\nhi : (i, i').1 ∈ s ∧ (i, i').2 ∈ t\nj : ι\nj' : ι'\nhj : (j, j').1 ∈ s ∧ (j, j').2 ∈ t\nh : (i, i') ≠ (j, j')\n⊢ (Disjoint on f) (i, i') (j, j')"}
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+{"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", "E' : Type u_5", "inst✝¹² : NormedAddCommGroup E'", "inst✝¹¹ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁰ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁹ : TopologicalSpace M'", "inst✝⁸ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝⁵ : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝⁴ : TopologicalSpace M''", "inst✝³ : ChartedSpace H'' M''", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : SmoothManifoldWithCorners I' M'", "inst✝ : SmoothManifoldWithCorners I'' M''", "e : PartialHomeomorph M H", "h : e ∈ atlas H M", "x : M", "hx : x ∈ e.source", "mem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I", "this : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I", "A : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)"], "goal": "DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)"}], "premise": [18778, 48361], "state_str": "𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)"}
+{"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", "E' : Type u_5", "inst✝¹² : NormedAddCommGroup E'", "inst✝¹¹ : NormedSpace 𝕜 E'", "H' : Type u_6", "inst✝¹⁰ : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M' : Type u_7", "inst✝⁹ : TopologicalSpace M'", "inst✝⁸ : ChartedSpace H' M'", "E'' : Type u_8", "inst✝⁷ : NormedAddCommGroup E''", "inst✝⁶ : NormedSpace 𝕜 E''", "H'' : Type u_9", "inst✝⁵ : TopologicalSpace H''", "I'' : ModelWithCorners 𝕜 E'' H''", "M'' : Type u_10", "inst✝⁴ : TopologicalSpace M''", "inst✝³ : ChartedSpace H'' M''", "inst✝² : SmoothManifoldWithCorners I M", "inst✝¹ : SmoothManifoldWithCorners I' M'", "inst✝ : SmoothManifoldWithCorners I'' M''", "e : PartialHomeomorph M H", "h : e ∈ atlas H M", "x : M", "hx : x ∈ e.source", "mem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I", "this : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I", "A : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ���¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)", "B : DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm) (↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source) ∩ range ↑I) (↑I (↑(chartAt H x) x))"], "goal": "DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)"}], "premise": [46349, 133443], "state_str": "𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nE' : Type u_5\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝¹⁰ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM' : Type u_7\ninst✝⁹ : TopologicalSpace M'\ninst✝⁸ : ChartedSpace H' M'\nE'' : Type u_8\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedSpace 𝕜 E''\nH'' : Type u_9\ninst✝⁵ : TopologicalSpace H''\nI'' : ModelWithCorners 𝕜 E'' H''\nM'' : Type u_10\ninst✝⁴ : TopologicalSpace M''\ninst✝³ : ChartedSpace H'' M''\ninst✝² : SmoothManifoldWithCorners I M\ninst✝¹ : SmoothManifoldWithCorners I' M'\ninst✝ : SmoothManifoldWithCorners I'' M''\ne : PartialHomeomorph M H\nh : e ∈ atlas H M\nx : M\nhx : x ∈ e.source\nmem : ↑I (↑(chartAt H x) x) ∈ ↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I\nthis : (chartAt H x).symm ≫ₕ e ∈ contDiffGroupoid ⊤ I\nA : ContDiffOn 𝕜 ⊤ (↑I ∘ ↑((chartAt H x).symm ≫ₕ e) ∘ ↑I.symm) (↑I.symm ⁻¹' ((chartAt H x).symm ≫ₕ e).source ∩ range ↑I)\nB :\n  DifferentiableWithinAt 𝕜 (↑I ∘ (↑e ∘ ↑(chartAt H x).symm) ∘ ↑I.symm)\n    (↑I.symm ⁻¹' (chartAt H x).target ∩ ↑I.symm ⁻¹' (↑(chartAt H x).symm ⁻¹' e.source) ∩ range ↑I)\n    (↑I (↑(chartAt H x) x))\n⊢ DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I x ↑e) (range ↑I) (↑(extChartAt I x) x)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "f : α → β", "p q r : α → α → Prop", "a b : α", "c : { x // p a x ∧ q x b }", "h : ∀ {x : α}, r (↑c) x → p a x", "x✝ : α"], "goal": "x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} ↔ x✝ ∈ {y | r (↑c) y ∧ q y b}"}], "premise": [1723], "state_str": "case h\nα : Type u_1\nβ : Type u_2\nf : α → β\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r (↑c) x → p a x\nx✝ : α\n⊢ x✝ ∈ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} ↔ x✝ ∈ {y | r (↑c) y ∧ q y b}"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]"], "goal": "(p ^ 0).degree ≤ 0 • p.degree"}], "premise": [119739, 119740], "state_str": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\n⊢ (p ^ 0).degree ≤ 0 • p.degree"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]"], "goal": "degree 1 ≤ 0"}], "premise": [102120], "state_str": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\n⊢ degree 1 ≤ 0"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]", "n : ℕ"], "goal": "(p ^ (n + 1)).degree ≤ (p ^ n).degree + p.degree"}], "premise": [119742], "state_str": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ (n + 1)).degree ≤ (p ^ n).degree + p.degree"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]", "n : ℕ"], "goal": "(p ^ n * p).degree ≤ (p ^ n).degree + p.degree"}], "premise": [102217], "state_str": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n * p).degree ≤ (p ^ n).degree + p.degree"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]", "n : ℕ"], "goal": "(p ^ n).degree + p.degree ≤ (n + 1) • p.degree"}], "premise": [119741], "state_str": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n).degree + p.degree ≤ (n + 1) • p.degree"}
+{"state": [{"context": ["R : Type u", "S : Type v", "a b c d : R", "n✝ m : ℕ", "inst✝ : Semiring R", "p✝¹ p✝ q : R[X]", "ι : Type u_1", "p : R[X]", "n : ℕ"], "goal": "(p ^ n).degree + p.degree ��� n • p.degree + p.degree"}], "premise": [103886], "state_str": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝¹ p✝ q : R[X]\nι : Type u_1\np : R[X]\nn : ℕ\n⊢ (p ^ n).degree + p.degree ≤ n • p.degree + p.degree"}
+{"state": [{"context": ["α : Type u_1", "a b c d : ℝ≥0∞", "r p q : ℝ≥0"], "goal": "⋂ n, Ici ↑n = {⊤}"}], "premise": [12072, 20351, 135287, 143268], "state_str": "α : Type u_1\na b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ ⋂ n, Ici ↑n = {⊤}"}
+{"state": [{"context": ["α : Type u", "r : α → α → Prop", "a b : α", "hwf : WellFounded r"], "goal": "hwf.rank a = sup fun b => succ (hwf.rank ↑b)"}], "premise": [52576], "state_str": "α : Type u\nr : α → α → Prop\na b : α\nhwf : WellFounded r\n⊢ hwf.rank a = sup fun b => succ (hwf.rank ↑b)"}
+{"state": [{"context": ["a b : Cardinal.{u}", "n m : ℕ"], "goal": "Prime ↑n ↔ Nat.Prime n"}], "premise": [144342], "state_str": "a b : Cardinal.{u}\nn m : ℕ\n⊢ Prime ↑n ↔ Nat.Prime n"}
+{"state": [{"context": ["a b : Cardinal.{u}", "n m : ℕ"], "goal": "(↑n ≠ 0 ∧ ¬IsUnit ↑n ∧ ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b) ↔ n ≠ 0 ∧ ¬IsUnit n ∧ ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b"}], "premise": [1963], "state_str": "a b : Cardinal.{u}\nn m : ℕ\n⊢ (↑n ≠ 0 ∧ ¬IsUnit ↑n ∧ ∀ (a b : Cardinal.{u_1}), ↑n ∣ a * b → ↑n ∣ a ∨ ↑n ∣ b) ↔\n    n ≠ 0 ∧ ¬IsUnit n ∧ ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b"}
+{"state": [{"context": ["a b✝ : Cardinal.{u}", "n m : ℕ", "h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b", "b c : Cardinal.{u_1}", "hbc : ↑n ∣ b * c"], "goal": "↑n ∣ b ∨ ↑n ∣ c"}], "premise": [14316], "state_str": "case refine_3\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c"}
+{"state": [{"context": ["a b✝ : Cardinal.{u}", "n m : ℕ", "h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b", "b c : Cardinal.{u_1}", "hbc : ↑n ∣ b * c", "h' : ℵ₀ ≤ b * c"], "goal": "↑n ∣ b ∨ ↑n ∣ c"}], "premise": [1673, 48796], "state_str": "case refine_3.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\n⊢ ↑n ∣ b ∨ ↑n ∣ c"}
+{"state": [{"context": ["a b✝ : Cardinal.{u}", "n m : ℕ", "h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b", "b c : Cardinal.{u_1}", "hbc : ↑n ∣ b * c", "h' : ℵ₀ ≤ b * c", "hb : b ≠ 0", "hc : c ≠ 0", "hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c"], "goal": "↑n ∣ b ∨ ↑n ∣ c"}], "premise": [108583, 108656], "state_str": "case refine_3.inr.intro.intro\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\n⊢ ↑n ∣ b ∨ ↑n ∣ c"}
+{"state": [{"context": ["a b✝ : Cardinal.{u}", "n m : ℕ", "h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b", "b c : Cardinal.{u_1}", "hbc : ↑n ∣ b * c", "h' : ℵ₀ ≤ b * c", "hb : b ≠ 0", "hc : c ≠ 0", "hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c", "hn : ↑n ≠ 0", "this : ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ}, (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) → ∀ (b c : Cardinal.{u_1}), ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c", "hℵ₀b : ¬ℵ₀ ≤ b"], "goal": "↑n ∣ b ∨ ↑n ∣ c"}, {"context": ["n✝ : ℕ", "a b✝ : Cardinal.{u}", "n m : ℕ", "h : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b", "b c : Cardinal.{u_1}", "hbc : ↑n ∣ b * c", "h' : ℵ₀ ≤ b * c", "hb : b ≠ 0", "hc : c ≠ 0", "hℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c", "hn : ↑n ≠ 0", "hℵ₀b : ℵ₀ ≤ b"], "goal": "↑n ∣ b ∨ ↑n ∣ c"}], "premise": [1724, 2111], "state_str": "case refine_3.inr.intro.intro.inr\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nthis :\n  ∀ {a b : Cardinal.{u}} {n : ℕ} {m : ℕ},\n    (∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b) →\n      ∀ (b c : Cardinal.{u_1}),\n        ↑n ∣ b * c → ℵ₀ ≤ b * c → b ≠ 0 → c ≠ 0 → ℵ₀ ≤ b ∨ ℵ₀ ≤ c → ↑n ≠ 0 → ℵ₀ ≤ b → ↑n ∣ b ∨ ↑n ∣ c\nhℵ₀b : ¬ℵ₀ ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c\n\nn✝ : ℕ\na b✝ : Cardinal.{u}\nn m : ℕ\nh : ∀ (a b : ℕ), n ∣ a * b → n ∣ a ∨ n ∣ b\nb c : Cardinal.{u_1}\nhbc : ↑n ∣ b * c\nh' : ℵ₀ ≤ b * c\nhb : b ≠ 0\nhc : c ≠ 0\nhℵ₀ : ℵ₀ ≤ b ∨ ℵ₀ ≤ c\nhn : ↑n ≠ 0\nhℵ₀b : ℵ��� ≤ b\n⊢ ↑n ∣ b ∨ ↑n ∣ c"}
+{"state": [{"context": ["R : Type u", "I : Type v", "inst✝ : CommSemiring R", "x y z : R", "s : I → R", "t : Finset I", "m n : ℕ"], "goal": "IsCoprime ↑m ↑n ↔ m.Coprime n"}], "premise": [1713, 74463, 128978], "state_str": "R : Type u\nI : Type v\ninst✝ : CommSemiring R\nx y z : R\ns : I → R\nt : Finset I\nm n : ℕ\n⊢ IsCoprime ↑m ↑n ↔ m.Coprime n"}
+{"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", "hf : Monotone f"], "goal": "f (max a b) = max (f a) (f b)"}], "premise": [14308], "state_str": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\ns : Set α\na b c d : α\nhf : Monotone f\n⊢ f (max a b) = max (f a) (f b)"}
+{"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_8, u_1} C", "inst✝⁸ : Category.{?u.46756, u_2} C₁", "inst✝⁷ : Category.{?u.46760, u_3} C₂", "inst✝⁶ : Category.{?u.46764, u_4} C₃", "inst✝⁵ : Category.{u_10, u_5} D₁", "inst✝⁴ : Category.{u_9, u_6} D₂", "inst✝³ : Category.{?u.46776, 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", "f : X ⟶ Y"], "goal": "(homEquiv W L₁ L₂) (L₁.map f) = L₂.map f"}], "premise": [92006], "state_str": "C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{u_8, u_1} C\ninst✝⁸ : Category.{?u.46756, u_2} C₁\ninst✝⁷ : Category.{?u.46760, u_3} C₂\ninst✝⁶ : Category.{?u.46764, u_4} C₃\ninst✝⁵ : Category.{u_10, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.46776, u_7} D₃\nW : MorphismProperty C\nL₁ : C ⥤ D₁\ninst✝² : L₁.IsLocalization W\nL₂ : C ⥤ D₂\ninst✝¹ : L₂.IsLocalization W\nL₃ : C ⥤ D₃\ninst✝ : L₃.IsLocalization W\nX Y Z : C\nf : X ⟶ Y\n⊢ (homEquiv W L₁ L₂) (L₁.map f) = L₂.map f"}
+{"state": [{"context": ["α : Type u_1", "a b : α", "s : α → α → Bool", "l r : List α", "h : ¬s a b = true"], "goal": "merge s (a :: l) (b :: r) = b :: merge s (a :: l) r"}], "premise": [1595, 1738], "state_str": "α : Type u_1\na b : α\ns : α → α → Bool\nl r : List α\nh : ¬s a b = true\n⊢ merge s (a :: l) (b :: r) = b :: merge s (a :: l) r"}
+{"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✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "htwo : Invertible 2", "B : BilinForm R M", "Q : QuadraticMap R M R", "hB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0"], "goal": "∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x"}], "premise": [83940], "state_str": "case intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x"}
+{"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✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "htwo : Invertible 2", "B : BilinForm R M", "Q : QuadraticMap R M R", "hB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0", "x : M", "hx : Q x ≠ 0"], "goal": "∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x"}], "premise": [83937], "state_str": "case intro.intro\nS : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nA : Type u_7\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nhtwo : Invertible 2\nB : BilinForm R M\nQ : QuadraticMap R M R\nhB₁✝ hB₁ : (QuadraticMap.associatedHom ℕ) Q ≠ 0\nx : M\nhx : Q x ≠ 0\n⊢ ∃ x, ¬IsOrtho ((QuadraticMap.associatedHom ℕ) Q) x x"}
+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y Z : C", "f g : X ⟶ Y", "h : Y ⟶ Z"], "goal": "inv (𝟙 X) = 𝟙 X"}], "premise": [88790], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ inv (𝟙 X) = 𝟙 X"}
+{"state": [{"context": ["z : ℂ", "t : ℝ"], "goal": "‖cexp (↑t * I)‖ = 1"}], "premise": [46164, 149348], "state_str": "z : ℂ\nt : ℝ\n⊢ ‖cexp (↑t * I)‖ = 1"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{?u.2051, u_2} D", "inst✝² : HasZeroMorphisms C", "inst✝¹ : HasZeroMorphisms D", "S S₁ S₂ : ShortComplex C", "inst✝ : S.HasHomology", "h : S.LeftHomologyData"], "goal": "S.Exact ↔ IsZero h.H"}], "premise": [114537], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.2051, u_2} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ S.Exact ↔ IsZero h.H"}
+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "inst✝⁴ : Category.{u_3, u_1} C", "inst✝³ : Category.{?u.2051, u_2} D", "inst✝² : HasZeroMorphisms C", "inst✝¹ : HasZeroMorphisms D", "S S₁ S₂ : ShortComplex C", "inst✝ : S.HasHomology", "h : S.LeftHomologyData"], "goal": "IsZero S.homology ↔ IsZero h.H"}], "premise": [94060], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{?u.2051, u_2} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\ninst✝ : S.HasHomology\nh : S.LeftHomologyData\n⊢ IsZero S.homology ↔ IsZero h.H"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "a b c : α", "f : α ↪o β", "hab : a ⩿ b", "h : (range ⇑f).OrdConnected"], "goal": "f a ⩿ f b"}], "premise": [11045, 16579], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\n⊢ f a ⩿ f b"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "a b c✝ : α", "f : α ↪o β", "hab : a ⩿ b", "h : (range ⇑f).OrdConnected", "c : α", "ha : f a < f c", "hb : f c < f b"], "goal": "False"}], "premise": [11043], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : f a < f c\nhb : f c < f b\n⊢ False"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Preorder α", "inst✝ : Preorder β", "a b c✝ : α", "f : α ↪o β", "hab : a ⩿ b", "h : (range ⇑f).OrdConnected", "c : α", "ha : a < c", "hb : c < b"], "goal": "False"}], "premise": [2106], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\na b c✝ : α\nf : α ↪o β\nhab : a ⩿ b\nh : (range ⇑f).OrdConnected\nc : α\nha : a < c\nhb : c < b\n⊢ False"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "X : C"], "goal": "(β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv"}], "premise": [88815], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (β_ X (𝟙_ C)).inv = (λ_ X).hom ≫ (ρ_ X).inv"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "X : C"], "goal": "(β_ X (𝟙_ C)).hom ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)"}], "premise": [107103], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "X : C"], "goal": "((ρ_ X).hom ≫ (λ_ X).inv) ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)"}], "premise": [96173, 99211, 99212, 99216, 99217, 99218, 99219, 99220, 99221, 99222, 99223, 99224, 99225, 99601, 99602, 99603, 99604, 99605, 99606, 99607, 99611, 99612], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((ρ_ X).hom ≫ (λ_ X).inv) ≫ (λ_ X).hom ≫ (ρ_ X).inv = 𝟙 (X ⊗ 𝟙_ C)"}
+{"state": [{"context": ["n m : ℕ", "p✝ : Fin (n + 1)", "i✝ j : Fin n", "p i : Fin (n + 1)", "h : p < i", "hi : optParam (i ≠ 0) ⋯"], "goal": "p.succAbove (i.pred hi) = i"}], "premise": [4137, 143003], "state_str": "n m : ℕ\np✝ : Fin (n + 1)\ni✝ j : Fin n\np i : Fin (n + 1)\nh : p < i\nhi : optParam (i ≠ 0) ⋯\n⊢ p.succAbove (i.pred hi) = i"}
+{"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", "o : Ordinal.{u_3}", "c : Cardinal.{u_3}"], "goal": "o.card ≤ c ↔ o < (succ c).ord"}], "premise": [1713, 17389, 49800], "state_str": "α : Type u\nβ : Type u_1\nγ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_3}\nc : Cardinal.{u_3}\n⊢ o.card ≤ c ↔ o < (succ c).ord"}
+{"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), (f i).NeBot", "I : Set ι", "h : I.pi s ∈ pi f", "i : ι", "hi : i ∈ I"], "goal": "s i ∈ f i"}], "premise": [1673, 11488], "state_str": "ι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\n⊢ s i ∈ f i"}
+{"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), (f i).NeBot", "I : Set ι", "h : I.pi s ∈ pi f", "i : ι", "hi : i ∈ I", "I' : Set ι", "t : (i : ι) → Set (α i)", "htf : ∀ (i : ι), t i ∈ f i", "hts : I'.pi t ⊆ I.pi s"], "goal": "s i ∈ f i"}], "premise": [15884], "state_str": "case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\n⊢ s i ∈ f i"}
+{"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), (f i).NeBot", "I : Set ι", "h : I.pi s ∈ pi f", "i : ι", "hi : i ∈ I", "I' : Set ι", "t : (i : ι) → Set (α i)", "htf : ∀ (i : ι), t i ∈ f i", "hts : I'.pi t ⊆ I.pi s", "x : α i", "hx : x ∈ t i"], "goal": "x ∈ s i"}], "premise": [15959], "state_str": "case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\n⊢ x ∈ s i"}
+{"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), (f i).NeBot", "I : Set ι", "h : I.pi s ∈ pi f", "i : ι", "hi : i ∈ I", "I' : Set ι", "t : (i : ι) → Set (α i)", "htf : ∀ (i : ι), t i ∈ f i", "hts : I'.pi t ⊆ I.pi s", "x : α i", "hx : x ∈ t i", "g : (i : ι) → α i", "hg : ∀ (i : ι), g i ∈ t i"], "goal": "x ∈ s i"}], "premise": [70039], "state_str": "case intro.intro.intro.intro\nι : Type u_1\nα : ι → Type u_2\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\np : (i : ι) → α i → Prop\ninst✝ : ∀ (i : ι), (f i).NeBot\nI : Set ι\nh : I.pi s ∈ pi f\ni : ι\nhi : i ∈ I\nI' : Set ι\nt : (i : ι) → Set (α i)\nhtf : ∀ (i : ι), t i ∈ f i\nhts : I'.pi t ⊆ I.pi s\nx : α i\nhx : x ∈ t i\ng : (i : ι) → α i\nhg : ∀ (i : ι), g i ∈ t i\n⊢ x ∈ s i"}
+{"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": "List.filterMap (Option.guard p) l = List.filter (fun b => decide (p b)) l"}], "premise": [5170], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ List.filterMap (Option.guard p) l = List.filter (fun b => decide (p b)) l"}
+{"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": "Option.guard p = Option.guard fun x => decide (p x) = true"}], "premise": [1838], "state_str": "case e_f\nα : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\ninst✝ : DecidablePred p\ns : Multiset α\nl : List α\n⊢ Option.guard p = Option.guard fun x => decide (p x) = true"}
+{"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": "0 ≤ ⌊a⌋ ↔ 0 ≤ a"}], "premise": [1713, 105090, 128749], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ 0 ≤ ⌊a⌋ ↔ 0 ≤ a"}
+{"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 ��✝ γ", "β : Type u_6", "inst✝ : CommMonoid β", "s : Finset α", "f : α → β"], "goal": "s.noncommProd f ⋯ = s.prod f"}], "premise": [138845], "state_str": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ✝ : Type u_4\nγ : Type u_5\nf✝ : α → β✝ → β✝\nop : α → α → α\ninst✝³ : Monoid β✝\ninst✝² : Monoid γ\ninst✝¹ : FunLike F β✝ γ\nβ : Type u_6\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\n⊢ s.noncommProd f ⋯ = s.prod f"}
+{"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", "h : 0 < (expSeries 𝕂 𝔸).radius"], "goal": "HasStrictFDerivAt (exp 𝕂) 1 0"}], "premise": [40234, 46818], "state_str": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\n⊢ HasStrictFDerivAt (exp 𝕂) 1 0"}
+{"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", "h : 0 < (expSeries 𝕂 𝔸).radius", "x : 𝔸"], "goal": "x = (expSeries 𝕂 𝔸 1) fun x_1 => x"}], "premise": [40210], "state_str": "case h.e'_10.h\n𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕂\ninst✝² : NormedRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nh : 0 < (expSeries 𝕂 𝔸).radius\nx : 𝔸\n⊢ x = (expSeries 𝕂 𝔸 1) fun x_1 => x"}
+{"state": [{"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "inst✝ : TotalComplexShape c₁ c₂ c₁₂", "i₁ i₁' : I₁", "i₂ i₂' : I₂", "h₁ : c₁.Rel i₁ i₁'", "h₂ : c₂.Rel i₂ i₂'"], "goal": "c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁, i₂) * 1 = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}], "premise": [119703, 128841], "state_str": "I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁, i₂) * 1 = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}
+{"state": [{"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "inst✝ : TotalComplexShape c₁ c₂ c₁₂", "i₁ i₁' : I₁", "i₂ i₂' : I₂", "h₁ : c₁.Rel i₁ i₁'", "h₂ : c₂.Rel i₂ i₂'"], "goal": "c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}], "premise": [114832, 119703], "state_str": "I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}
+{"state": [{"context": ["I₁ : Type u_1", "I₂ : Type u_2", "I₁₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "c₁₂ : ComplexShape I₁₂", "inst✝ : TotalComplexShape c₁ c₂ c₁₂", "i₁ i₁' : I₁", "i₂ i₂' : I₂", "h₁ : c₁.Rel i₁ i₁'", "h₂ : c₂.Rel i₂ i₂'"], "goal": "c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) = -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}], "premise": [117792, 119703, 119728, 122240, 122241, 128841], "state_str": "I₁ : Type u_1\nI₂ : Type u_2\nI₁₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nc₁₂ : ComplexShape I₁₂\ninst✝ : TotalComplexShape c₁ c₂ c₁₂\ni₁ i₁' : I₁\ni₂ i₂' : I₂\nh₁ : c₁.Rel i₁ i₁'\nh₂ : c₂.Rel i₂ i₂'\n⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (-c₁.ε₁ c₂ c₁₂ (i₁, i₂) * c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')) =\n    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')"}
+{"state": [{"context": ["G : Type u_1", "H : Type u_2", "α : Type u_3", "β : Type u_4", "E : Type u_5", "inst✝⁴ : Group G", "inst✝³ : MulAction G α", "s : Set α", "x : α", "inst✝² : Group H", "inst✝¹ : MulAction H α", "inst✝ : SMulCommClass H G α", "g : H"], "goal": "fundamentalFrontier G (g • s) = g • fundamentalFrontier G s"}], "premise": [118875, 132882, 132982], "state_str": "G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\nE : Type u_5\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ns : Set α\nx : α\ninst✝² : Group H\ninst✝¹ : MulAction H α\ninst✝ : SMulCommClass H G α\ng : H\n⊢ fundamentalFrontier G (g • s) = g • fundamentalFrontier G s"}
+{"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", "e : X ≃ Y", "he : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s", "s : Set X"], "goal": "IsOpen (⇑e.symm ⁻¹' s) ↔ IsOpen s"}], "premise": [1718], "state_str": "X : Type u_1\nY : Type u_2\nZ✝ : Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\ne : X ≃ Y\nhe : ∀ (s : Set Y), IsOpen (⇑e ⁻¹' s) ↔ IsOpen s\ns : Set X\n⊢ IsOpen (⇑e.symm ⁻¹' s) ↔ IsOpen s"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : TopologicalSpace α", "inst✝ : MeasurableSpace α", "μ ν : Measure α", "s k : Set α", "hk : IsCompact k", "h'k : k ⊆ s \\ μ.everywherePosSubset s"], "goal": "μ k = 0"}], "premise": [58058], "state_str": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns k : Set α\nhk : IsCompact k\nh'k : k ⊆ s \\ μ.everywherePosSubset s\n⊢ μ k = 0"}
+{"state": [{"context": ["ι : Type u_1", "μ : Type u_2", "μ' : Type u_3", "inst✝ : DecidableEq ι", "n k : ℕ", "a✝ : Fin k → ℕ"], "goal": "a✝ ∈ univ.piAntidiag n ↔ a✝ ∈ Nat.antidiagonalTuple k n"}], "premise": [142551], "state_str": "case a\nι : Type u_1\nμ : Type u_2\nμ' : Type u_3\ninst✝ : DecidableEq ι\nn k : ℕ\na✝ : Fin k → ℕ\n⊢ a✝ ∈ univ.piAntidiag n ↔ a✝ ∈ Nat.antidiagonalTuple k n"}
+{"state": [{"context": ["α : Type u", "k : ℕ", "i : Fin k"], "goal": "indexOf i (finRange k) = ↑i"}], "premise": [1674, 132462], "state_str": "α : Type u\nk : ℕ\ni : Fin k\n⊢ indexOf i (finRange k) = ↑i"}
+{"state": [{"context": ["α : Type u", "k : ℕ", "i : Fin k", "this : indexOf i (finRange k) < (finRange k).length"], "goal": "indexOf i (finRange k) = ↑i"}], "premise": [132487], "state_str": "α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\n⊢ indexOf i (finRange k) = ↑i"}
+{"state": [{"context": ["α : Type u", "k : ℕ", "i : Fin k", "this : indexOf i (finRange k) < (finRange k).length", "h₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i"], "goal": "indexOf i (finRange k) = ↑i"}], "premise": [129281], "state_str": "α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\n⊢ indexOf i (finRange k) = ↑i"}
+{"state": [{"context": ["α : Type u", "k : ℕ", "i : Fin k", "this : indexOf i (finRange k) < (finRange k).length", "h₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i", "h₂ : (finRange k).get ⟨↑i, ⋯⟩ = i"], "goal": "indexOf i (finRange k) = ↑i"}], "premise": [1673, 2100, 2101, 129270, 129728], "state_str": "α : Type u\nk : ℕ\ni : Fin k\nthis : indexOf i (finRange k) < (finRange k).length\nh₁ : (finRange k).get ⟨indexOf i (finRange k), this⟩ = i\nh₂ : (finRange k).get ⟨↑i, ⋯⟩ = i\n⊢ indexOf i (finRange k) = ↑i"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝³ : Nonempty β", "inst✝² : LinearOrder β", "inst✝¹ : Preorder γ", "inst✝ : NoMaxOrder γ", "g : β → γ", "f : α → β", "l : Filter α", "hg : Monotone g", "hg' : Tendsto g atTop atTop"], "goal": "IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f"}], "premise": [14706], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l (g ∘ f) ↔ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "ι : Type u_4", "ι' : Type u_5", "inst✝³ : Nonempty β", "inst✝² : LinearOrder β", "inst✝¹ : Preorder γ", "inst✝ : NoMaxOrder γ", "g : β → γ", "f : α → β", "l : Filter α", "hg : Monotone g", "hg' : Tendsto g atTop atTop", "c : γ", "hc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c"], "goal": "IsBoundedUnder (fun x x_1 => x ≤ x_1) l f"}], "premise": [16164], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Type u_5\ninst✝³ : Nonempty β\ninst✝² : LinearOrder β\ninst✝¹ : Preorder γ\ninst✝ : NoMaxOrder γ\ng : β → γ\nf : α → β\nl : Filter α\nhg : Monotone g\nhg' : Tendsto g atTop atTop\nc : γ\nhc : ∀ᶠ (x : γ) in map (g ∘ f) l, (fun x x_1 => x ≤ x_1) x c\n⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) l f"}
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+{"state": [{"context": ["M : Type u_1", "inst✝ : Mul M", "c d : Con M"], "goal": "{ toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c ≤ { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d ↔ c ≤ d"}], "premise": [7245], "state_str": "M : Type u_1\ninst✝ : Mul M\nc d : Con M\n⊢ { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } c ≤\n      { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } d ↔\n    c ≤ d"}
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+{"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}", "l : a.IsLimit", "h✝ : b < a", "c : Ordinal.{u_4}", "h : c < a - b"], "goal": "succ (b + c) < a"}], "premise": [1673, 2106, 52299], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_4}\nl : a.IsLimit\nh✝ : b < a\nc : Ordinal.{u_4}\nh : c < a - b\n⊢ succ (b + c) < a"}
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+{"state": [{"context": ["α✝ α : Type u_1", "inst✝¹ : Nonempty α", "inst✝ : DecidableEq α", "s : Finset α", "f : α → ℤ", "n : ℕ", "h : ∑ i ∈ s, (f i).natAbs ≤ n"], "goal": "∃ β x sgn g, (∀ (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"}], "premise": [146339], "state_str": "α✝ α : Type u_1\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\n⊢ ∃ 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"}
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+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)"], "goal": "(fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}], "premise": [15609, 41201, 43402], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] ((fun x => |x| ^ (-b)) ∘ Neg.neg)"], "goal": "(fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}], "premise": [1670, 105279], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    ((fun x => |x| ^ (-b)) ∘ Neg.neg)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)"], "goal": "(fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}], "premise": [43411, 43422], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\n⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b)"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)", "x : ℝ"], "goal": "‖‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖‖ ≤ ‖((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) x‖"}], "premise": [1670, 42680, 43341, 62351], "state_str": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx : ℝ\n⊢ ‖‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖‖ ≤\n    ‖((fun x =>\n            ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n          Neg.neg)\n        x‖"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)", "x✝ x : ℝ", "hx : x ∈ Icc (x✝ + R) (x✝ + S)"], "goal": "‖(ContinuousMap.restrict (Icc (x✝ + R) (x✝ + S)) f) ⟨x, hx⟩‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖"}], "premise": [62144], "state_str": "case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖(ContinuousMap.restrict (Icc (x✝ + R) (x✝ + S)) f) ⟨x, hx⟩‖ ≤\n    ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : NormedAddCommGroup E", "f : C(ℝ, E)", "b : ℝ", "hb : 0 < b", "hf : ⇑f =O[atBot] fun x => |x| ^ (-b)", "R S : ℝ", "h1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)", "h2 : ((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘ Neg.neg) =O[atBot] fun x => |x| ^ (-b)", "this : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)", "x✝ x : ℝ", "hx : x ∈ Icc (x✝ + R) (x✝ + S)"], "goal": "‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖"}], "premise": [14277, 62349], "state_str": "case mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atBot] fun x => |x| ^ (-b)\nR S : ℝ\nh1 : ⇑(f.comp { toFun := fun a => -a, continuous_toFun := ⋯ }) =O[atTop] fun x => |x| ^ (-b)\nh2 :\n  ((fun x =>\n        ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖) ∘\n      Neg.neg) =O[atBot]\n    fun x => |x| ^ (-b)\nthis : (fun x => |x| ^ (-b)) ∘ Neg.neg = fun x => |x| ^ (-b)\nx✝ x : ℝ\nhx : x ∈ Icc (x✝ + R) (x✝ + S)\n⊢ ‖f x‖ ≤ ‖ContinuousMap.restrict (Icc (-x✝ + -S) (-x✝ + -R)) (f.comp { toFun := fun a => -a, continuous_toFun := ⋯ })‖"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "a : α", "l : List α", "x : α", "n : ℕ"], "goal": "l.length ≤ (insertNth n x l).length"}], "premise": [14317], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\na : α\nl : List α\nx : α\nn : ℕ\n⊢ l.length ≤ (insertNth n x l).length"}
+{"state": [{"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)"], "goal": "S.Solution'_descent.multiplicity = S.multiplicity - 1"}], "premise": [2100, 24535, 79545], "state_str": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\n⊢ S.Solution'_descent.multiplicity = S.multiplicity - 1"}
+{"state": [{"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "k : 𝓞 K", "hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}], "premise": [119703, 119742], "state_str": "case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1 + 1) * k\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}
+{"state": [{"context": ["K : Type u_1", "inst✝⁴ : Field K", "inst✝³ : NumberField K", "inst✝² : IsCyclotomicExtension {3} ℚ K", "ζ : K", "hζ : IsPrimitiveRoot ζ ↑3", "S : Solution hζ", "S' : Solution' hζ", "inst✝¹ : DecidableRel fun a b => a ∣ b", "inst✝ : DecidableEq (𝓞 K)", "k : 𝓞 K", "hk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)"], "goal": "λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}], "premise": [1169, 1182, 1191, 24765, 108286, 108288, 125797], "state_str": "case intro\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : IsCyclotomicExtension {3} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ ↑3\nS : Solution hζ\nS' : Solution' hζ\ninst✝¹ : DecidableRel fun a b => a ∣ b\ninst✝ : DecidableEq (𝓞 K)\nk : 𝓞 K\nhk : λ ^ (S.multiplicity - 1) * FermatLastTheoremForThreeGen.Solution.X S = λ ^ (S.multiplicity - 1) * (λ * k)\n⊢ λ ∣ FermatLastTheoremForThreeGen.Solution.X S"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f : X✝ ⟶ Y", "inst✝¹ : HasKernel f", "inst✝ : HasCokernels C", "X A B : C", "g : B ⟶ X", "hg : Mono g", "i : A ≅ B", "hf : Mono (i.hom ≫ g)"], "goal": "(fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg"}], "premise": [89269, 89626], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf : X✝ ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasCokernels C\nX A B : C\ng : B ⟶ X\nhg : Mono g\ni : A ≅ B\nhf : Mono (i.hom ≫ g)\n⊢ (fun A f x => Subobject.mk (cokernel.π f).op) A (i.hom ≫ g) hf = (fun A f x => Subobject.mk (cokernel.π f).op) B g hg"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f✝ : X✝ ⟶ Y", "inst✝¹ : HasKernel f✝", "inst✝ : HasCokernels C", "X A B : C", "f : A ⟶ X", "g : B ⟶ X", "hf : Mono f", "hg : Mono g", "h : Subobject.mk f ≤ Subobject.mk g"], "goal": "Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)"}], "premise": [89253], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤\n    Subobject.lift (fun A f x => Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "X✝ Y Z : C", "inst✝² : HasZeroMorphisms C", "f✝ : X✝ ⟶ Y", "inst✝¹ : HasKernel f✝", "inst✝ : HasCokernels C", "X A B : C", "f : A ⟶ X", "g : B ⟶ X", "hf : Mono f", "hg : Mono g", "h : Subobject.mk f ≤ Subobject.mk g"], "goal": "Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op"}], "premise": [89261], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\nX✝ Y Z : C\ninst✝² : HasZeroMorphisms C\nf✝ : X✝ ⟶ Y\ninst✝¹ : HasKernel f✝\ninst✝ : HasCokernels C\nX A B : C\nf : A ⟶ X\ng : B ⟶ X\nhf : Mono f\nhg : Mono g\nh : Subobject.mk f ≤ Subobject.mk g\n⊢ Subobject.mk (cokernel.π f).op ≤ Subobject.mk (cokernel.π g).op"}
+{"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", "inst✝ : CompleteLattice β", "S : Set (Set α)", "f : α → β"], "goal": "⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x"}], "premise": [19391, 135451, 135594], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nι' : Sort u_5\nι₂ : Sort u_6\nκ : ι → Sort u_7\nκ₁ : ι → Sort u_8\nκ₂ : ι → Sort u_9\nκ' : ι' → Sort u_10\ninst✝ : CompleteLattice β\nS : Set (Set α)\nf : α → β\n⊢ ⨅ x ∈ ⋃₀ S, f x = ⨅ s ∈ S, ⨅ x ∈ s, f x"}
+{"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", "hx : 0 < x", "hn : n ≠ 0"], "goal": "f 1 + (x * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + ↑m)) ≤ f x"}], "premise": [39598, 105713, 117807, 119708], "state_str": "f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n ! - ∑ m ∈ Finset.range (n + 1), log (x + ↑m)) ≤ f x"}
+{"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", "hx : 0 < x", "hn : n ≠ 0"], "goal": "f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)"}], "premise": [14273, 14277, 39600], "state_str": "f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nhn : n ≠ 0\n⊢ f 1 + (x * log ↑n + log ↑n !) ≤ f (↑(n + 1) + x)"}
+{"state": [{"context": ["J : Type v", "inst✝ : SmallCategory J", "F : J ⥤ Cat", "X✝ Y : limit (F ⋙ objects)", "X : J"], "goal": "{ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (𝟙 X) = 𝟙 ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X)"}], "premise": [1838], "state_str": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)"}
+{"state": [{"context": ["J : Type v", "inst✝ : SmallCategory J", "F : J ⥤ Cat", "X✝ Y : limit (F ⋙ objects)", "X : J", "f : { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X", "this : Category.{v, v} (objects.obj (F.obj X)) := inferInstance"], "goal": "{ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (𝟙 X) f = 𝟙 ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y, map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj X) f"}], "premise": [97759, 99920], "state_str": "case h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX✝ Y : limit (F ⋙ objects)\nX : J\nf :\n  { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n        map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n    X\nthis : Category.{v, v} (objects.obj (F.obj X)) := inferInstance\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (𝟙 X) f =\n    𝟙\n      ({ obj := fun j => limit.π (F ⋙ objects) j X✝ ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.obj\n        X)\n      f"}
+{"state": [{"context": ["J : Type v", "inst✝ : SmallCategory J", "F : J ⥤ Cat", "X Y : limit (F ⋙ objects)", "x✝¹ x✝ Z : J", "f : x✝¹ ⟶ x✝", "g : x✝ ⟶ Z"], "goal": "{ obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map (f ≫ g) = { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map f ≫ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y, map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map g"}], "premise": [1838], "state_str": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ Cat\nX Y : limit (F ⋙ objects)\nx✝¹ x✝ Z : J\nf : x✝¹ ⟶ x✝\ng : x✝ ⟶ Z\n⊢ { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n          map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n      (f ≫ g) =\n    { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n        f ≫\n      { obj := fun j => limit.π (F ⋙ objects) j X ⟶ limit.π (F ⋙ objects) j Y,\n            map := fun {X_1 Y_1} f g => eqToHom ⋯ ≫ (F.map f).map g ≫ eqToHom ⋯ }.map\n        g"}
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+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝¹ : SemilatticeInf α", "inst✝ : SemilatticeInf β", "s✝ t✝ : Set α", "a b : α", "s : Set α", "t : Set β", "u : Finset α", "hu : u.Nonempty", "hus : ↑u ⊆ s", "v : Finset β", "hv : v.Nonempty", "hvt : ↑v ⊆ t"], "goal": "(u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t)"}], "premise": [136853], "state_str": "case mk.intro.intro.intro.intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeInf α\ninst✝ : SemilatticeInf β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Set β\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\nv : Finset β\nhv : v.Nonempty\nhvt : ↑v ⊆ t\n⊢ (u.inf' hu id, v.inf' hv id) ∈ infClosure (s ×ˢ t)"}
+{"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", "hts : t ⊆ s", "ha : a ∈ t"], "goal": "s \\ t ∪ t.erase a = s.erase a"}], "premise": [1674, 138756, 139041, 139060], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\nhts : t ⊆ s\nha : a ∈ t\n⊢ s \\ t ∪ t.erase a = s.erase a"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : PartialOrder α", "a b c : α", "h : a ⩿ b", "h2✝ : a ≤ c", "h3✝ : c ≤ b", "h2 : a < c", "h3 : c < b"], "goal": "c = a ∨ c = b"}], "premise": [2106], "state_str": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : PartialOrder α\na b c : α\nh : a ⩿ b\nh2✝ : a ≤ c\nh3✝ : c ≤ b\nh2 : a < c\nh3 : c < b\n⊢ c = a ∨ c = b"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ₀ γ₁ : Γ₀ˣ"], "goal": "∃ k, v.ltAddSubgroup k ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁"}], "premise": [1674, 2045], "state_str": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∃ k, v.ltAddSubgroup k ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ₀ γ₁ : Γ₀ˣ"], "goal": "v.ltAddSubgroup (min γ₀ γ₁) ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁"}], "premise": [1179, 1186, 1199, 1813, 2029, 12956, 14567, 103342, 103349, 122573, 133320], "state_str": "case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ v.ltAddSubgroup (min γ₀ γ₁) ≤ v.ltAddSubgroup γ₀ ⊓ v.ltAddSubgroup γ₁"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ₀ γ₁ : Γ₀ˣ"], "goal": "∀ (a : R), v a < ↑γ₀ → v a < ↑γ₁ → v a < ↑γ₀"}], "premise": [1101, 1674], "state_str": "case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ₀ γ₁ : Γ₀ˣ\n⊢ ∀ (a : R), v a < ↑γ₀ → v a < ↑γ₁ → v a < ↑γ₀"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ : Γ₀ˣ"], "goal": "∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)"}], "premise": [104110], "state_str": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ γ₀ : Γ₀ˣ", "h : γ₀ * γ₀ ≤ γ"], "goal": "∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)"}], "premise": [1674, 2045], "state_str": "case intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) * ↑(v.ltAddSubgroup j) ⊆ ↑(v.ltAddSubgroup γ)"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "γ γ₀ : Γ₀ˣ", "h : γ₀ * γ₀ ≤ γ", "r : R", "r_in : r ∈ ↑(v.ltAddSubgroup γ₀)", "s : R", "s_in : s ∈ ↑(v.ltAddSubgroup γ₀)"], "goal": "(fun x x_1 => x * x_1) r s ∈ ↑(v.ltAddSubgroup γ)"}], "premise": [75408, 102787], "state_str": "case h.intro.intro.intro.intro\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : Γ₀ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : r ∈ ↑(v.ltAddSubgroup γ₀)\ns : R\ns_in : s ∈ ↑(v.ltAddSubgroup γ₀)\n⊢ (fun x x_1 => x * x_1) r s ∈ ↑(v.ltAddSubgroup γ)"}
+{"state": [{"context": ["R : Type u", "inst✝¹ : Ring R", "Γ₀ : Type v", "inst✝ : LinearOrderedCommGroupWithZero Γ₀", "v : Valuation R Γ₀", "x : R", "γ : Γ���ˣ"], "goal": "∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)"}], "premise": [108401], "state_str": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\nγ : Γ₀ˣ\n⊢ ∃ j, ↑(v.ltAddSubgroup j) ⊆ (fun x_1 => x * x_1) ⁻¹' ↑(v.ltAddSubgroup γ)"}
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+{"state": [{"context": ["α : Type u_1", "le : α → α → Bool", "r : Nat", "a : α", "c : HeapNode α", "s : Heap α"], "goal": "Batteries.BinomialHeap.Imp.FindMin.HasSize (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1) → (cons r a c s).realSize = (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize + 1"}], "premise": [2173, 2176, 3683, 3684, 3685], "state_str": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nr : Nat\na : α\nc : HeapNode α\ns : Heap α\n⊢ Batteries.BinomialHeap.Imp.FindMin.HasSize\n      (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }) (c.realSize + s.realSize + 1) →\n    (cons r a c s).realSize =\n      (merge le (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node.toHeap\n            ((findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).before\n              (findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next)).realSize +\n        1"}
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+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "o : Type u_4", "R : Type u_5", "inst✝⁴ : Fintype n", "inst✝³ : Fintype o", "inst✝² : CommRing R", "inst✝¹ : StrongRankCondition R", "inst✝ : Fintype m", "f : n → m", "e : n ≃ m", "A : Matrix m m R"], "goal": "finrank R ↥(Submodule.map (LinearMap.funLeft R R f) (LinearMap.range A.mulVecLin)) ≤ finrank R ↥(LinearMap.range A.mulVecLin)"}], "premise": [86373], "state_str": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nR : Type u_5\ninst✝⁴ : Fintype n\ninst✝³ : Fintype o\ninst✝² : CommRing R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype m\nf : n → m\ne : n ≃ m\nA : Matrix m m R\n⊢ finrank R ↥(Submodule.map (LinearMap.funLeft R R f) (LinearMap.range A.mulVecLin)) ≤\n    finrank R ↥(LinearMap.range A.mulVecLin)"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝ : PseudoEMetricSpace α", "f g : α → ℝ", "hf : LocallyLipschitz f", "a : ℝ"], "goal": "LocallyLipschitz fun x => min a (f x)"}], "premise": [19697, 60126], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : PseudoEMetricSpace α\nf g : α → ℝ\nhf : LocallyLipschitz f\na : ℝ\n⊢ LocallyLipschitz fun x => min a (f x)"}
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+{"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", "S : Set (Set α)", "K : DirectedOn (fun x x_1 => x ⊆ x_1) S", "H : ∀ s ∈ S, IsPreconnected s", "u v : Set α", "hu : IsOpen u", "hv : IsOpen v", "Huv : ⋃₀ S ⊆ u ∪ v", "a : α", "hau : a ∈ u", "s : Set α", "hsS : s ∈ S", "has : a ∈ s", "b : α", "hbv : b ∈ v", "t : Set α", "htS : t ∈ S", "hbt : b ∈ t", "r : Set α", "hrS : r ∈ S", "hsr : s ⊆ r", "htr : t ⊆ r", "Hnuv : (r ∩ (u ∩ v)).Nonempty", "Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v)"], "goal": "(⋃₀ S ∩ (u ∩ v)).Nonempty"}], "premise": [133349], "state_str": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t✝ u✝ v✝ : Set α\nS : Set (Set α)\nK : DirectedOn (fun x x_1 => x ⊆ x_1) S\nH : ∀ s ∈ S, IsPreconnected s\nu v : Set α\nhu : IsOpen u\nhv : IsOpen v\nHuv : ⋃₀ S ⊆ u ∪ v\na : α\nhau : a ∈ u\ns : Set α\nhsS : s ∈ S\nhas : a ∈ s\nb : α\nhbv : b ∈ v\nt : Set α\nhtS : t ∈ S\nhbt : b ∈ t\nr : Set α\nhrS : r ∈ S\nhsr : s ⊆ r\nhtr : t ⊆ r\nHnuv : (r ∩ (u ∩ v)).Nonempty\nKruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v)\n⊢ (⋃₀ S ∩ (u ∩ v)).Nonempty"}
+{"state": [{"context": ["α : Type u_1", "r : Rel α α", "β : Type u_2", "s : Rel β β", "p : RelSeries r", "i : Fin p.length"], "goal": "r ((p.toFun ∘ Fin.rev) i.succ) ((p.toFun ∘ Fin.rev) i.castSucc)"}], "premise": [1670], "state_str": "α : Type u_1\nr : Rel α α\nβ : Type u_2\ns : Rel β β\np : RelSeries r\ni : Fin p.length\n⊢ r ((p.toFun ∘ Fin.rev) i.succ) ((p.toFun ∘ Fin.rev) i.castSucc)"}
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+{"state": [{"context": ["xl xr : Type u", "x : PGame"], "goal": "0 ⧏ x ↔ ∃ i, ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j"}], "premise": [50196], "state_str": "xl xr : Type u\nx : PGame\n⊢ 0 ⧏ x ↔ ∃ i, ∀ (j : (x.moveLeft i).RightMoves), 0 ⧏ (x.moveLeft i).moveRight j"}
+{"state": [{"context": ["α : Type u", "β : Type v", "inst✝ : SemilatticeSup α", "a✝ b✝ c✝ d a b c : α"], "goal": "a ⊔ b ⊔ c = a ⊔ c ⊔ b"}], "premise": [14537, 14538], "state_str": "α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na✝ b✝ c✝ d a b c : α\n⊢ a ⊔ b ⊔ c = a ⊔ c ⊔ b"}
+{"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✝⁵ : CountablyGenerated γ", "inst✝⁴ : MeasurableSpace Ω", "inst✝³ : StandardBorelSpace Ω", "inst✝² : Nonempty Ω", "inst✝¹ : Countable α", "κ : Kernel α (β × Ω)", "inst✝ : IsFiniteKernel κ", "x y : α", "h : x ∈ measurableAtom y"], "goal": "(fun a => (κ a).condKernel) x = (fun a => (κ a).condKernel) y"}], "premise": [72633], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝⁵ : CountablyGenerated γ\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : Countable α\nκ : Kernel α (β × Ω)\ninst✝ : IsFiniteKernel κ\nx y : α\nh : x ∈ measurableAtom y\n⊢ (fun a => (κ a).condKernel) x = (fun a => (κ a).condKernel) y"}
+{"state": [{"context": ["n : ℤ"], "goal": "cos (↑n * (2 * ↑π) + ↑π) = -1"}], "premise": [38709, 38710, 109478, 109550, 149150], "state_str": "n : ℤ\n⊢ cos (↑n * (2 * ↑π) + ↑π) = -1"}
+{"state": [{"context": ["t : ℝ", "ht : 0 < t"], "goal": "‖F_nat 0 0 t - 1‖ ≤ rexp (-π * t) / (1 - rexp (-π * t))"}], "premise": [21572, 101700], "state_str": "t : ℝ\nht : 0 < t\n⊢ ‖F_nat 0 0 t - 1‖ ≤ rexp (-π * t) / (1 - rexp (-π * t))"}
+{"state": [{"context": ["a b c d : ℝ≥0∞", "r p q : ℝ≥0", "ha : a ≠ ⊤", "hb : b ≠ 0"], "goal": "a < a + b"}], "premise": [103552, 119729, 143500], "state_str": "a b c d : ℝ≥0∞\nr p q : ℝ≥0\nha : a ≠ ⊤\nhb : b ≠ 0\n⊢ a < a + b"}
+{"state": [{"context": ["α : Type u_1", "t : Ordnode α", "ht : t.Sized"], "goal": "t.size = 0 ↔ t = nil"}], "premise": [2107], "state_str": "α : Type u_1\nt : Ordnode α\nht : t.Sized\n⊢ t.size = 0 ↔ t = nil"}
+{"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)", "u₁ u₂ : UniformSpace γ"], "goal": "uniformSpace α γ 𝔖 = uniformSpace α γ 𝔖 ⊓ uniformSpace α γ 𝔖"}], "premise": [19459, 60300], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\n⊢ uniformSpace α γ 𝔖 = uniformSpace α γ 𝔖 ⊓ uniformSpace α γ 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)", "u₁ u₂ : UniformSpace γ"], "goal": "⨅ i, uniformSpace α γ 𝔖 = ⨅ b, bif b then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖"}], "premise": [19283], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nu₁ u₂ : UniformSpace γ\n⊢ ⨅ i, uniformSpace α γ 𝔖 = ⨅ b, bif b then uniformSpace α γ 𝔖 else uniformSpace α γ 𝔖"}
+{"state": [{"context": ["α✝ : Type u_1", "G : Type u_2", "α : Type u_3", "inst✝⁵ : MeasurableSpace G", "inst✝⁴ : Div G", "m : MeasurableSpace α", "f✝ g✝ : α → G", "μ : Measure α", "E : Type u_4", "inst✝³ : MeasurableSpace E", "inst✝² : AddGroup E", "inst✝¹ : MeasurableSingletonClass E", "inst✝ : MeasurableSub₂ E", "f g : α → E", "hf : AEMeasurable f μ", "hg : AEMeasurable g μ"], "goal": "NullMeasurableSet {x | f x = g x} μ"}], "premise": [29107, 29169, 29177, 31225], "state_str": "α✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ NullMeasurableSet {x | f x = g x} μ"}
+{"state": [{"context": ["α✝ : Type u_1", "G : Type u_2", "α : Type u_3", "inst✝⁵ : MeasurableSpace G", "inst✝⁴ : Div G", "m : MeasurableSpace α", "f✝ g✝ : α → G", "μ : Measure α", "E : Type u_4", "inst✝³ : MeasurableSpace E", "inst✝² : AddGroup E", "inst✝¹ : MeasurableSingletonClass E", "inst✝ : MeasurableSub₂ E", "f g : α → E", "hf : AEMeasurable f μ", "hg : AEMeasurable g μ"], "goal": "{x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᶠ[ae μ] {x | f x = g x}"}], "premise": [15889, 29108, 131585], "state_str": "α✝ : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Div G\nm : MeasurableSpace α\nf✝ g✝ : α → G\nμ : Measure α\nE : Type u_4\ninst✝³ : MeasurableSpace E\ninst✝² : AddGroup E\ninst✝¹ : MeasurableSingletonClass E\ninst✝ : MeasurableSub₂ E\nf g : α → E\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ {x | AEMeasurable.mk f hf x = AEMeasurable.mk g hg x} =ᶠ[ae μ] {x | f x = g x}"}
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+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "inst✝⁸ : CommRing R", "Ms : ι → Type u_3", "inst✝⁷ : (i : ι) → AddCommGroup (Ms i)", "inst✝⁶ : (i : ι) → Module R (Ms i)", "N : Type u_4", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R N", "Ns : ι → Type u_5", "inst✝³ : (i : ι) → AddCommGroup (Ns i)", "inst✝² : (i : ι) → Module R (Ns i)", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι", "p : (i : ι) → Submodule R (Ms i)", "i : ι", "x : Ms i ⧸ p i", "x' : Ms i", "j : ι"], "goal": "(quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯) (Pi.single i (Quotient.mk'' x')) j = LinearMap.id (Pi.single i (Quotient.mk'' x')) j"}], "premise": [82367, 82397, 82693, 82694, 109728, 109758], "state_str": "ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (quotientPiLift p (fun i => (p i).mkQ) ⋯ ∘ₗ piQuotientLift p (pi Set.univ p) single ⋯)\n      (Pi.single i (Quotient.mk'' x')) j =\n    LinearMap.id (Pi.single i (Quotient.mk'' x')) j"}
+{"state": [{"context": ["ι : Type u_1", "R : Type u_2", "inst✝⁸ : CommRing R", "Ms : ι → Type u_3", "inst✝⁷ : (i : ι) → AddCommGroup (Ms i)", "inst✝⁶ : (i : ι) → Module R (Ms i)", "N : Type u_4", "inst✝⁵ : AddCommGroup N", "inst✝⁴ : Module R N", "Ns : ι → Type u_5", "inst✝³ : (i : ι) → AddCommGroup (Ns i)", "inst✝² : (i : ι) → Module R (Ns i)", "inst✝¹ : Fintype ι", "inst✝ : DecidableEq ι", "p : (i : ι) → Submodule R (Ms i)", "i : ι", "x : Ms i ⧸ p i", "x' : Ms i", "j : ι"], "goal": "(fun i_1 => (p i_1).mkQ ((single i) x' i_1)) j = Pi.single i (Quotient.mk x') j"}], "premise": [82385, 83986], "state_str": "ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type u_5\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\ni : ι\nx : Ms i ⧸ p i\nx' : Ms i\nj : ι\n⊢ (fun i_1 => (p i_1).mkQ ((single i) x' i_1)) j = Pi.single i (Quotient.mk x') j"}
+{"state": [{"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "m n : ℕ", "h : m ≤ n"], "goal": "(ZMod.castHom ⋯ (ZMod (p ^ m))).comp ((zmodEquivTrunc p n).symm.toRingHom.comp (truncate n)) = ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n)"}], "premise": [75624, 121580], "state_str": "p : ℕ\nhp : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\n⊢ (ZMod.castHom ⋯ (ZMod (p ^ m))).comp ((zmodEquivTrunc p n).symm.toRingHom.comp (truncate n)) =\n    ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n)"}
+{"state": [{"context": ["p : ℕ", "hp : Fact (Nat.Prime p)", "m n : ℕ", "h : m ≤ n"], "goal": "((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n) = (zmodEquivTrunc p m).symm.toRingHom.comp (truncate m)"}], "premise": [75884, 121580], "state_str": "p : ℕ\nhp : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\n⊢ ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n) =\n    (zmodEquivTrunc p m).symm.toRingHom.comp (truncate m)"}
+{"state": [{"context": ["x✝ : ℂ"], "goal": "LSeries 0 x✝ = 0 x✝"}], "premise": [1178, 64162, 108302, 120640], "state_str": "case h\nx✝ : ℂ\n⊢ LSeries 0 x✝ = 0 x✝"}
+{"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", "f : L.obj X ⟶ L.obj Y", "g₁ g₂ : L.obj Y ⟶ L.obj Z"], "goal": "f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}], "premise": [92953], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}
+{"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", "f : L.obj X ⟶ L.obj Y", "g₁ g₂ : L.obj Y ⟶ L.obj Z", "α : W.LeftFraction X Y", "hα : f = α.map L ⋯"], "goal": "f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}], "premise": [92129], "state_str": "case intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ��� L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}
+{"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", "f : L.obj X ⟶ L.obj Y", "g₁ g₂ : L.obj Y ⟶ L.obj Z", "α : W.LeftFraction X Y", "hα : f = α.map L ⋯", "β : W.LeftFraction₂ Y Z", "hβ₁ : g₁ = β.fst.map L ⋯", "hβ₂ : g₂ = β.snd.map L ⋯"], "goal": "f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}], "premise": [92128, 92892], "state_str": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}
+{"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", "f : L.obj X ⟶ L.obj Y", "g₁ g₂ : L.obj Y ⟶ L.obj Z", "α : W.LeftFraction X Y", "hα : f = α.map L ⋯", "β : W.LeftFraction₂ Y Z", "hβ₁ : g₁ = β.fst.map L ⋯", "hβ₂ : g₂ = β.snd.map L ⋯", "γ : W.LeftFraction₂ α.Y' β.Y'", "hγ₁ : β.f ≫ γ.s = α.s ≫ γ.f", "hγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'"], "goal": "f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}], "premise": [91598, 91599, 92122, 92187, 92952, 97269], "state_str": "case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ f ≫ add' W g₁ g₂ = add' W (f ≫ g₁) (f ≫ g₂)"}
+{"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", "f : L.obj X ⟶ L.obj Y", "g₁ g₂ : L.obj Y ⟶ L.obj Z", "α : W.LeftFraction X Y", "hα : f = α.map L ⋯", "β : W.LeftFraction₂ Y Z", "hβ₁ : g₁ = β.fst.map L ⋯", "hβ₂ : g₂ = β.snd.map L ⋯", "γ : W.LeftFraction₂ α.Y' β.Y'", "hγ₁ : β.f ≫ γ.s = α.s ≫ γ.f", "hγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'"], "goal": "(LeftFraction.mk (α.f ≫ (γ.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 ⋯"}], "premise": [91599], "state_str": "case intro.intro.intro.intro.intro\nC : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_4, u_1} C\ninst✝³ : Category.{u_3, u_2} D\ninst✝² : Preadditive C\nL : C ⥤ D\nW : MorphismProperty C\ninst✝¹ : L.IsLocalization W\ninst✝ : W.HasLeftCalculusOfFractions\nX Y Z : C\nf : L.obj X ⟶ L.obj Y\ng₁ g₂ : L.obj Y ⟶ L.obj Z\nα : W.LeftFraction X Y\nhα : f = α.map L ⋯\nβ : W.LeftFraction₂ Y Z\nhβ₁ : g₁ = β.fst.map L ⋯\nhβ₂ : g₂ = β.snd.map L ⋯\nγ : W.LeftFraction₂ α.Y' β.Y'\nhγ₁ : β.f ≫ γ.s = α.s ≫ γ.f\nhγ₂ : β.f' ≫ γ.s = α.s ≫ γ.f'\n⊢ (LeftFraction.mk (α.f ≫ (γ.f + γ.f')) (β.s ≫ γ.s) ⋯).map L ⋯ =\n    (LeftFraction.mk (α.f ≫ γ.f + α.f ≫ γ.f') (β.s ≫ γ.s) ⋯).map L ⋯"}
+{"state": [{"context": ["R✝ : Type u", "S : Type u_1", "A✝ : Type v", "inst✝⁹ : CommRing R✝", "inst✝⁸ : CommRing S", "inst✝⁷ : Ring A✝", "inst✝⁶ : Algebra R✝ A✝", "inst✝⁵ : Algebra R✝ S", "inst✝⁴ : Algebra S A✝", "inst✝³ : IsScalarTower R✝ S A✝", "R : Type u", "A : Type v", "inst✝² : DivisionRing A", "inst✝¹ : Field R", "inst✝ : Algebra R A", "n : ℚ"], "goal": "IsAlgebraic R ↑n"}], "premise": [148050], "state_str": "R✝ : Type u\nS : Type u_1\nA✝ : Type v\ninst✝⁹ : CommRing R✝\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A✝\ninst✝⁶ : Algebra R✝ A✝\ninst✝⁵ : Algebra R✝ S\ninst✝⁴ : Algebra S A✝\ninst✝³ : IsScalarTower R✝ S A✝\nR : Type u\nA : Type v\ninst✝² : DivisionRing A\ninst✝¹ : Field R\ninst✝ : Algebra R A\nn : ℚ\n⊢ IsAlgebraic R ↑n"}
+{"state": [{"context": ["R✝ : Type u", "S : Type u_1", "A✝ : Type v", "inst✝⁹ : CommRing R✝", "inst✝⁸ : CommRing S", "inst✝⁷ : Ring A✝", "inst✝⁶ : Algebra R✝ A✝", "inst✝⁵ : Algebra R✝ S", "inst✝⁴ : Algebra S A✝", "inst✝³ : IsScalarTower R✝ S A✝", "R : Type u", "A : Type v", "inst✝² : DivisionRing A", "inst✝¹ : Field R", "inst✝ : Algebra R A", "n : ℚ"], "goal": "IsAlgebraic R ((algebraMap R A) ↑n)"}], "premise": [75563], "state_str": "R✝ : Type u\nS : Type u_1\nA✝ : Type v\ninst✝⁹ : CommRing R✝\ninst✝⁸ : CommRing S\ninst✝⁷ : Ring A✝\ninst✝⁶ : Algebra R✝ A✝\ninst✝⁵ : Algebra R✝ S\ninst✝⁴ : Algebra S A✝\ninst✝³ : IsScalarTower R✝ S A✝\nR : Type u\nA : Type v\ninst✝² : DivisionRing A\ninst✝¹ : Field R\ninst✝ : Algebra R A\nn : ℚ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)"}
+{"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 →L[ℝ] F"], "goal": "Function.HasTemperateGrowth ⇑f"}], "premise": [36647, 45100], "state_str": "𝕜 : Type u_1\n𝕜' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E →L[ℝ] F\n⊢ Function.HasTemperateGrowth ⇑f"}
+{"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✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "p : X → Prop", "s : Set X", "t : Set ↑s"], "goal": "Dense t ↔ s ⊆ closure (val '' t)"}], "premise": [56029, 66494, 133298], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\ns : Set X\nt : Set ↑s\n⊢ Dense t ↔ s ⊆ closure (val '' t)"}
+{"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✝² : TopologicalSpace X", "inst✝¹ : TopologicalSpace Y", "inst✝ : TopologicalSpace Z", "p : X → Prop", "s : Set X", "t : Set ↑s"], "goal": "(∀ (x : X) (h : x ∈ s), ↑⟨x, h⟩ ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)"}], "premise": [1713], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\np : X → Prop\ns : Set X\nt : Set ↑s\n⊢ (∀ (x : X) (h : x ∈ s), ↑⟨x, h⟩ ∈ closure (val '' t)) ↔ s ⊆ closure (val '' t)"}
+{"state": [{"context": ["X✝ : Type u_1", "X : Type u_2", "x : FreeAbelianGroup X"], "goal": "toFreeAbelianGroup (toFinsupp x) = x"}], "premise": [6254, 117175, 117188], "state_str": "X✝ : Type u_1\nX : Type u_2\nx : FreeAbelianGroup X\n⊢ toFreeAbelianGroup (toFinsupp x) = x"}
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+{"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 : Monoid M", "inst✝ : ContinuousMul M", "k : ℕ"], "goal": "Continuous fun a => a ^ (k + 1)"}], "premise": [119745], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nk : ℕ\n⊢ Continuous fun a => a ^ (k + 1)"}
+{"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 : Monoid M", "inst✝ : ContinuousMul M", "k : ℕ"], "goal": "Continuous fun a => a * a ^ k"}], "premise": [55628, 65028], "state_str": "ι : Type u_1\nα : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : Monoid M\ninst✝ : ContinuousMul M\nk : ℕ\n⊢ Continuous fun a => a * a ^ k"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace ℝ E", "v✝ v : E"], "goal": "HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0"}], "premise": [69222], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv✝ v : E\n⊢ HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : InnerProductSpace ℝ E", "v✝ v : E", "this : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((Submodule.span ℝ {v})ᗮ.subtypeL 0)"], "goal": "HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0"}], "premise": [45095, 45115], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nv✝ v : E\nthis : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((Submodule.span ℝ {v})ᗮ.subtypeL 0)\n⊢ HasFDerivAt (stereoInvFunAux v ∘ Subtype.val) (Submodule.span ℝ {v})ᗮ.subtypeL 0"}
+{"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.15810, u_1} C", "inst✝⁸ : Category.{u_10, u_2} C₁", "inst✝⁷ : Category.{u_11, u_3} C₂", "inst✝⁶ : Category.{?u.15822, u_4} C₃", "inst✝⁵ : Category.{u_8, u_5} D₁", "inst✝⁴ : Category.{u_9, u_6} D₂", "inst✝³ : Category.{?u.15834, u_7} D₃", "W₁ : MorphismProperty C₁", "W₂ : MorphismProperty C₂", "W₃ : MorphismProperty C₃", "Φ : LocalizerMorphism W₁ W₂", "Ψ : LocalizerMorphism W₂ W₃", "L₁ : C₁ ⥤ D₁", "inst✝² : L₁.IsLocalization W₁", "L₂ : C₂ ⥤ D₂", "inst✝¹ : L₂.IsLocalization W₂", "L₃ : C₃ ⥤ D₃", "inst✝ : L₃.IsLocalization W₃", "X Y Z : C₁", "G : D₁ ⥤ D₂", "e : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G", "f : L₁.obj X ⟶ L₁.obj Y", "G' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂", "e' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'", "this : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }", "α : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm"], "goal": "e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y"}], "premise": [92287, 96175], "state_str": "C : Type u_1\nC₁ : Type u_2\nC₂ : Type u_3\nC₃ : Type u_4\nD₁ : Type u_5\nD₂ : Type u_6\nD₃ : Type u_7\ninst✝⁹ : Category.{?u.15810, u_1} C\ninst✝⁸ : Category.{u_10, u_2} C₁\ninst✝⁷ : Category.{u_11, u_3} C₂\ninst✝⁶ : Category.{?u.15822, u_4} C₃\ninst✝⁵ : Category.{u_8, u_5} D₁\ninst✝⁴ : Category.{u_9, u_6} D₂\ninst✝³ : Category.{?u.15834, u_7} D₃\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nW₃ : MorphismProperty C₃\nΦ : LocalizerMorphism W₁ W₂\nΨ : LocalizerMorphism W₂ W₃\nL₁ : C₁ ⥤ D₁\ninst✝² : L₁.IsLocalization W₁\nL₂ : C₂ ⥤ D₂\ninst✝¹ : L₂.IsLocalization W₂\nL₃ : C₃ ⥤ D₃\ninst✝ : L₃.IsLocalization W₃\nX Y Z : C₁\nG : D₁ ⥤ D₂\ne : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G\nf : L₁.obj X ⟶ L₁.obj Y\nG' : D₁ ⥤ D₂ := Φ.localizedFunctor L₁ L₂\ne' : Φ.functor ⋙ L₂ ≅ L₁ ⋙ G' := CatCommSq.iso Φ.functor L₁ L₂ G'\nthis : Localization.Lifting L₁ W₁ (Φ.functor ⋙ L₂) G := { iso' := e.symm }\nα : G' ≅ G := Localization.liftNatIso L₁ W₁ (L₁ ⋙ G') (Φ.functor ⋙ L₂) G' G e'.symm\n⊢ e'.hom.app X ≫ G'.map f ≫ e'.inv.app Y = e.hom.app X ≫ G.map f ≫ e.inv.app Y"}
+{"state": [{"context": ["G : Type u_1", "inst✝ : Group G", "H K : Subgroup G", "S T : Set G", "hS : S ∈ rightTransversals ↑H", "g : G"], "goal": "(g * (↑(toFun hS g))⁻¹)⁻¹ = ↑(toFun hS g) * g⁻¹"}], "premise": [119770, 119808], "state_str": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhS : S ∈ rightTransversals ↑H\ng : G\n⊢ (g * (↑(toFun hS g))⁻¹)⁻¹ = ↑(toFun hS g) * g⁻¹"}
+{"state": [{"context": ["c : ℝ", "f g : ℕ → Bool", "n : ℕ"], "goal": "¬Set.univ.Countable"}], "premise": [14324, 48787, 145877], "state_str": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ¬Set.univ.Countable"}
+{"state": [{"context": ["c : ℝ", "f g : ℕ → Bool", "n : ℕ"], "goal": "ℵ₀ < 𝔠"}], "premise": [48651], "state_str": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ ℵ₀ < 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 c : Ordinal.{u_4}"], "goal": "a + b < a + c ↔ b < c"}], "premise": [1713, 14324, 103890], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b c : Ordinal.{u_4}\n⊢ a + b < a + c ↔ b < c"}
+{"state": [{"context": [], "goal": "0 < 2 * π"}], "premise": [38514], "state_str": "⊢ 0 < 2 * π"}
+{"state": [{"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t✝ u✝ u t : Set Ω", "hs : s.Finite"], "goal": "(condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount s) t"}], "premise": [13471, 14561, 15131, 73791, 133648, 135045], "state_str": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount s) t"}
+{"state": [{"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t✝ u✝ u t : Set Ω", "hs : s.Finite", "this : (condCount s) t = (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) + (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)"], "goal": "(condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ = (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) + (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)"}], "premise": [73781], "state_str": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t✝ u✝ u t : Set Ω\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)\n⊢ (condCount (s ∩ u)) t * (condCount s) u + (condCount (s ∩ uᶜ)) t * (condCount s) uᶜ =\n    (condCount (s ∩ u)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ u) +\n      (condCount (s ∩ uᶜ)) t * (condCount (s ∩ u ∪ s ∩ uᶜ)) (s ∩ uᶜ)"}
+{"state": [{"context": ["J : Type v", "inst✝³ : SmallCategory J", "inst✝² : IsCofiltered J", "F : J ⥤ Profinite", "C : Cone F", "α : Type u_1", "inst✝¹ : Finite α", "inst✝ : Nonempty α", "hC : IsLimit C", "f : LocallyConstant (↑C.pt.toTop) α", "inhabited_h : Inhabited α"], "goal": "∃ j g, f = LocallyConstant.comap (C.π.app j) g"}], "premise": [59420], "state_str": "J : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\n⊢ ∃ j g, f = LocallyConstant.comap (C.π.app j) g"}
+{"state": [{"context": ["J : Type v", "inst✝³ : SmallCategory J", "inst✝² : IsCofiltered J", "F : J ⥤ Profinite", "C : Cone F", "α : Type u_1", "inst✝¹ : Finite α", "inst✝ : Nonempty α", "hC : IsLimit C", "f : LocallyConstant (↑C.pt.toTop) α", "inhabited_h : Inhabited α", "j : J", "gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)", "h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg", "ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1", "σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default", "x : ↑C.pt.toTop"], "goal": "f x = (LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)) x"}], "premise": [1670, 62686, 62691, 99936], "state_str": "case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\n⊢ f x = (LocallyConstant.comap (C.π.app j) (LocallyConstant.map σ gg)) x"}
+{"state": [{"context": ["J : Type v", "inst✝³ : SmallCategory J", "inst✝² : IsCofiltered J", "F : J ⥤ Profinite", "C : Cone F", "α : Type u_1", "inst✝¹ : Finite α", "inst✝ : Nonempty α", "hC : IsLimit C", "f : LocallyConstant (↑C.pt.toTop) α", "inhabited_h : Inhabited α", "j : J", "gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)", "h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg", "ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1", "σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default", "x : ↑C.pt.toTop", "h1 : ι (f x) = gg ((C.π.app j) x)", "h2 : ∃ a, ι a = gg ((C.π.app j) x)"], "goal": "f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default"}], "premise": [1739], "state_str": "case intro.intro.h\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)\nh : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (C.π.app j) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default\nx : ↑C.pt.toTop\nh1 : ι (f x) = gg ((C.π.app j) x)\nh2 : ∃ a, ι a = gg ((C.π.app j) x)\n⊢ f x = if h : ∃ a, ι a = gg ((C.π.app j) x) then h.choose else default"}
+{"state": [{"context": ["Γ : Type u_1", "Γ' : Type u_2", "R : Type u_3", "V : Type u_4", "inst✝⁶ : PartialOrder Γ", "inst✝⁵ : PartialOrder Γ'", "inst✝⁴ : VAdd Γ Γ'", "inst✝³ : IsOrderedCancelVAdd Γ Γ'", "inst✝² : AddCommMonoid V", "inst✝¹ : MulZeroClass R", "inst✝ : SMulWithZero R V", "x : HahnSeries Γ R", "y : HahnModule Γ' R V", "h : x.support +ᵥ ((of R).symm y).support = x.support +ᵥ ((of R).symm y).support"], "goal": "((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support"}], "premise": [80591], "state_str": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\ninst✝⁶ : PartialOrder Γ\ninst✝⁵ : PartialOrder Γ'\ninst✝⁴ : VAdd Γ Γ'\ninst✝³ : IsOrderedCancelVAdd Γ Γ'\ninst✝² : AddCommMonoid V\ninst✝¹ : MulZeroClass R\ninst✝ : SMulWithZero R V\nx : HahnSeries Γ R\ny : HahnModule Γ' R V\nh : x.support +ᵥ ((of R).symm y).support = x.support +ᵥ ((of R).symm y).support\n⊢ ((of R).symm (x • y)).support ⊆ x.support +ᵥ ((of R).symm y).support"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁶ : _root_.RCLike 𝕜", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : InnerProductSpace 𝕜 E", "inst✝³ : NormedAddCommGroup F", "inst✝² : InnerProductSpace ℝ F", "V : Type u_4", "inst✝¹ : NormedAddCommGroup V", "inst✝ : InnerProductSpace ℂ V", "T : V →ₗ[ℂ] V", "x y : V"], "goal": "⟪T y, x⟫_ℂ = (⟪T (x + y), x + y⟫_ℂ - ⟪T (x - y), x - y⟫_ℂ + Complex.I * ⟪T (x + Complex.I • y), x + Complex.I • y⟫_ℂ - Complex.I * ⟪T (x - Complex.I • y), x - Complex.I • y⟫_ℂ) / 4"}], "premise": [36754, 36755, 36758, 36761, 36792, 36793, 109741, 117079, 117101, 117874, 117897, 119703, 119728, 119750, 119769, 122241, 122248, 148351, 148383], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : _root_.RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\nV : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nT : V →ₗ[ℂ] V\nx y : V\n⊢ ⟪T y, x⟫_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"}
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+{"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 : MeasurableSpace X", "μ : Measure X", "s : Set X", "hs : μ s ≠ ⊤"], "goal": "ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = ENNReal.ofReal (∫ (x : X) in s, ‖1‖ ∂μ)"}], "premise": [43260], "state_str": "X✝ : Type u_1\nY : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝² : MeasurableSpace X✝\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : X✝ → E\ns✝ t : Set X✝\nμ✝ ν : Measure X✝\nl l' : Filter X✝\nX : Type u_5\nm : MeasurableSpace X\nμ : Measure X\ns : Set X\nhs : μ s ≠ ⊤\n⊢ ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = ENNReal.ofReal (∫ (x : X) in s, ‖1‖ ∂μ)"}
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+{"state": [{"context": ["X : Type u_1", "inst✝ : TopologicalSpace X", "P : Set X → Prop", "c : CU P", "h0 : 0 < 2⁻¹", "h1234 : 2⁻¹ < 3 / 4", "h1 : 3 / 4 < 1"], "goal": "Continuous c.lim"}], "premise": [1674, 12615, 55639, 61256], "state_str": "case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\n⊢ Continuous c.lim"}
+{"state": [{"context": ["X : Type u_1", "inst✝ : TopologicalSpace X", "P : Set X → Prop", "c : CU P", "h0 : 0 < 2⁻¹", "h1234 : 2⁻¹ < 3 / 4", "h1 : 3 / 4 < 1", "x : X", "n : ℕ", "x✝ : True"], "goal": "∀ᶠ (x_1 : X) in 𝓝 x, c.lim x_1 ∈ Metric.closedBall (c.lim x) ((3 / 4) ^ n)"}], "premise": [61169], "state_str": "case intro.intro\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Prop\nc : CU P\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : ℕ\nx✝ : True\n⊢ ∀ᶠ (x_1 : X) in 𝓝 x, c.lim x_1 ∈ Metric.closedBall (c.lim x) ((3 / 4) ^ n)"}
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+{"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": "s ∈ f * ⊤ → s ∈ ⊤"}], "premise": [2036, 2038, 13660, 15918], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns : Set α\n⊢ s ∈ f * ⊤ → s ∈ ⊤"}
+{"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 t : Set α", "ht : t ∈ f", "hs : t * univ ⊆ s"], "goal": "s = univ"}], "premise": [1673, 13603, 131963, 133392], "state_str": "case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nε : Type u_6\ninst✝ : Monoid α\nf g : Filter α\ns✝ : Set α\na : α\nm n : ℕ\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : t * univ ⊆ s\n⊢ s = univ"}
+{"state": [{"context": ["n : ℕ", "F : TypeVec.{u} (n + 1) → Type u", "q : MvQPF F", "α : TypeVec.{u} n", "x y : Cofix F α", "h : x.dest = y.dest"], "goal": "x = y"}], "premise": [142623], "state_str": "n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nx y : Cofix F α\nh : x.dest = y.dest\n⊢ 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", "S : AffineSpace V P", "p₁ p₂ : P", "s : Set P", "inst✝ : Nontrivial P"], "goal": "affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤"}], "premise": [133383], "state_str": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ns : Set P\ninst✝ : Nontrivial P\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{u_2, u_1} C", "inst✝ : HasShift C ℤ", "X : Cᵒᵖ"], "goal": "(shiftFunctorZero Cᵒᵖ ℤ).hom.app X = (shiftFunctorOpIso C 0 0 ⋯).hom.app X ≫ ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op"}], "premise": [91995, 92054], "state_str": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : HasShift C ℤ\nX : Cᵒᵖ\n⊢ (shiftFunctorZero Cᵒᵖ ℤ).hom.app X =\n    (shiftFunctorOpIso C 0 0 ⋯).hom.app X ≫ ((shiftFunctorZero C ℤ).inv.app (Opposite.unop X)).op"}
+{"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": "NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0"}], "premise": [70039], "state_str": "α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 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✝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", "hs : μ s ≠ 0"], "goal": "NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0"}], "premise": [70039], "state_str": "case inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 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✝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", "hs : μ s ≠ 0", "ht : ν t ≠ 0"], "goal": "NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0"}], "premise": [27001], "state_str": "case inr.inr\nα : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nE : Type u_6\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : MeasurableSpace β'\ninst✝³ : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\nhs : μ s ≠ 0\nht : ν t ≠ 0\n⊢ NullMeasurableSet (s ×ˢ t) (μ.prod ν) ↔ NullMeasurableSet s μ ∧ NullMeasurableSet t ν ∨ μ s = 0 ∨ ν t = 0"}
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+{"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", "hs : (s ∩ mulSupport f).Finite", "ht : (t ∩ mulSupport f).Finite"], "goal": "(fun i => ∏ᶠ (_ : i ∈ s ∩ t ∩ mulSupport f), f i) = fun i => ∏ᶠ (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i"}], "premise": [133440, 133444, 133445], "state_str": "case e_a.e_f\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf g : α → M\na b : α\ns t : Set α\nhs : (s ∩ mulSupport f).Finite\nht : (t ∩ mulSupport f).Finite\n⊢ (fun i => ∏ᶠ (_ : i ∈ s ∩ t ∩ mulSupport f), f i) = fun i => ∏ᶠ (_ : i ∈ s ∩ mulSupport f ∩ (t ∩ mulSupport f)), f i"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : Topology.IsUpperSet α", "s✝ s : Set α"], "goal": "closure s = ↑(lowerClosure s)"}], "premise": [19966], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns✝ s : Set α\n⊢ closure s = ↑(lowerClosure s)"}
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+{"state": [{"context": ["R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝⁶ : CommSemiring R", "inst✝⁵ : Semiring A", "inst✝⁴ : Algebra R A", "inst✝³ : Semiring B", "inst✝² : Algebra R B", "inst✝¹ : Semiring C", "inst✝ : Algebra R C", "φ✝ φ : A →ₐ[R] B", "x✝ : B"], "goal": "x✝ ∈ ↑φ.range ↔ x✝ ∈ Set.range ⇑φ"}], "premise": [122081, 128379], "state_str": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ x✝ ∈ ↑φ.range �� x✝ ∈ Set.range ⇑φ"}
+{"state": [{"context": ["R' : Type u'", "R : Type u", "A : Type v", "B : Type w", "C : Type w'", "inst✝⁶ : CommSemiring R", "inst✝⁵ : Semiring A", "inst✝⁴ : Algebra R A", "inst✝³ : Semiring B", "inst✝² : Algebra R B", "inst✝¹ : Semiring C", "inst✝ : Algebra R C", "φ✝ φ : A →ₐ[R] B", "x✝ : B"], "goal": "(∃ x, φ x = x✝) ↔ x✝ ∈ Set.range ⇑φ"}], "premise": [1713], "state_str": "case h\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Semiring B\ninst✝² : Algebra R B\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\nφ✝ φ : A →ₐ[R] B\nx✝ : B\n⊢ (∃ x, φ x = x✝) ↔ x✝ ∈ Set.range ⇑φ"}
+{"state": [{"context": ["L : Language", "ι : Type v", "inst✝³ : Preorder ι", "G : ι → Type w", "inst✝² : (i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "inst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1", "inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)", "i fst✝¹ : ι", "snd✝¹ : G fst✝¹", "hx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i", "fst✝ : ι", "snd✝ : G fst✝", "hy : ⟨fst✝, snd✝⟩.fst ≤ i", "j : ι", "w✝¹ : fst✝¹ ≤ j", "w✝ : fst✝ ≤ j", "h✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝", "k : ι", "ik : i ≤ k", "jk : j ≤ k", "h : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)"], "goal": "(f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd"}], "premise": [24286], "state_str": "case mk.mk.intro.intro.intro.intro.intro\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd = (f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd"}
+{"state": [{"context": ["L : Language", "ι : Type v", "inst✝³ : Preorder ι", "G : ι → Type w", "inst✝² : (i : ι) → L.Structure (G i)", "f : (i j : ι) → i ≤ j → G i ↪[L] G j", "inst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1", "inst✝ : DirectedSystem G fun i j h => ⇑(f i j h)", "i fst✝¹ : ι", "snd✝¹ : G fst✝¹", "hx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i", "fst✝ : ι", "snd✝ : G fst✝", "hy : ⟨fst✝, snd✝⟩.fst ≤ i", "j : ι", "w✝¹ : fst✝¹ ≤ j", "w✝ : fst✝ ≤ j", "h✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝", "k : ι", "ik : i ≤ k", "jk : j ≤ k", "h : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)"], "goal": "(f i k ik) ((f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd) = (f i k ik) ((f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd)"}], "premise": [25033], "state_str": "case mk.mk.intro.intro.intro.intro.intro.a\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirected ι fun x x_1 => x ≤ x_1\ninst✝ : DirectedSystem G fun i j h => ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy : ⟨fst✝, snd✝⟩.fst ≤ i\nj : ι\nw✝¹ : fst✝¹ ≤ j\nw✝ : fst✝ ≤ j\nh✝ : (f fst✝¹ j w✝¹) snd✝¹ = (f fst✝ j w✝) snd✝\nk : ι\nik : i ≤ k\njk : j ≤ k\nh : (f j k jk) ((f fst✝¹ j w✝¹) snd✝¹) = (f j k jk) ((f fst✝ j w✝) snd✝)\n⊢ (f i k ik) ((f ⟨fst✝¹, snd✝¹⟩.fst i hx) ⟨fst✝¹, snd✝¹⟩.snd) = (f i k ik) ((f ⟨fst✝, snd✝⟩.fst i hy) ⟨fst✝, snd✝⟩.snd)"}
+{"state": [{"context": ["α : Type u_1", "s t : Finset α", "a : α"], "goal": "IsAtom s ↔ ∃ a, s = {a}"}], "premise": [1717, 18915, 136906], "state_str": "α : Type u_1\ns t : Finset α\na : α\n⊢ IsAtom s ↔ ∃ a, s = {a}"}
+{"state": [{"context": ["R M : Type u", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I : Ideal R", "F F' : I.Filtration M", "h : F.Stable"], "goal": "Submodule.span (↥(reesAlgebra I)) (⋃ i, ⇑(single R i) '' ↑(F.N i)) = F.submodule"}], "premise": [77317, 86700, 109628], "state_str": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n⊢ Submodule.span (↥(reesAlgebra I)) (⋃ i, ⇑(single R i) '' ↑(F.N i)) = F.submodule"}
+{"state": [{"context": ["R M : Type u", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "I : Ideal R", "F F' : I.Filtration M", "h : F.Stable"], "goal": "Submodule.span ↥(reesAlgebra I) ↑F.submodule = F.submodule"}], "premise": [86688], "state_str": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n⊢ Submodule.span ↥(reesAlgebra I) ↑F.submodule = F.submodule"}
+{"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 : Continuous f"], "goal": "ClosedEmbedding f ↔ ∀ (i : ι), ClosedEmbedding ((U i).carrier.restrictPreimage f)"}], "premise": [54009], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ ClosedEmbedding f ↔ ∀ (i : ι), ClosedEmbedding ((U i).carrier.restrictPreimage 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 : Continuous f"], "goal": "Embedding f ∧ IsClosed (range f) ↔ ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsClosed (range ((U i).carrier.restrictPreimage f))"}], "premise": [2026], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    ∀ (i : ι), Embedding ((U i).carrier.restrictPreimage f) ∧ IsClosed (range ((U i).carrier.restrictPreimage 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 : Continuous f"], "goal": "Embedding f ∧ IsClosed (range f) ↔ (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧ ∀ (x : ι), IsClosed (range ((U x).carrier.restrictPreimage f))"}], "premise": [1963], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nι : Type u_3\nU : ι → Opens β\nhU : iSup U = ⊤\nh : Continuous f\n⊢ Embedding f ∧ IsClosed (range f) ↔\n    (∀ (x : ι), Embedding ((U x).carrier.restrictPreimage f)) ∧\n      ∀ (x : ι), IsClosed (range ((U x).carrier.restrictPreimage f))"}
+{"state": [{"context": ["x : ℝ", "hx : 0 < x"], "goal": "HasStrictDerivAt log x⁻¹ x"}], "premise": [11234, 14284, 16027, 37865, 37923, 39394, 43844, 55910, 149295], "state_str": "x : ℝ\nhx : 0 < x\n⊢ HasStrictDerivAt log x⁻¹ x"}
+{"state": [{"context": ["x : ℝ", "hx : 0 < x", "this : HasStrictDerivAt log (rexp (log x))⁻¹ x"], "goal": "HasStrictDerivAt log x⁻¹ x"}], "premise": [37865], "state_str": "x : ℝ\nhx : 0 < x\nthis : HasStrictDerivAt log (rexp (log x))⁻¹ x\n⊢ HasStrictDerivAt log x⁻¹ x"}
+{"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.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j) = pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)"}], "premise": [130194, 130196, 130199], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.J), HasPullback (𝒰.map i ≫ f) g\ni j k : 𝒰.J\n⊢ t' 𝒰 f g i j k ≫\n      t' 𝒰 f g j k i ≫\n        t' 𝒰 f g k i j ≫\n          pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j) =\n    pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫ pullback.snd (pullback.fst (𝒰.map i ≫ f) g ≫ 𝒰.map i) (𝒰.map j)"}
+{"state": [{"context": ["x : ℝ", "hx this : |x| ≤ 1"], "goal": "|rexp x - 1| ≤ 2 * |x|"}], "premise": [149317], "state_str": "x : ℝ\nhx this : |x| ≤ 1\n⊢ |rexp x - 1| ≤ 2 * |x|"}
+{"state": [{"context": ["x : ℝ", "hx this✝ : |x| ≤ 1", "this : Complex.abs (cexp ↑x - 1) ≤ 2 * Complex.abs ↑x"], "goal": "|rexp x - 1| ≤ 2 * |x|"}], "premise": [148226, 148293, 148387, 149093], "state_str": "x : ℝ\nhx this✝ : |x| ≤ 1\nthis : Complex.abs (cexp ↑x - 1) ≤ 2 * Complex.abs ↑x\n⊢ |rexp x - 1| ≤ 2 * |x|"}
+{"state": [{"context": ["n : ℤ"], "goal": "↑↑n.natAbs = ↑(nnabs ↑n)"}], "premise": [105577, 105582, 146639, 146780], "state_str": "case a\nn : ℤ\n⊢ ↑↑n.natAbs = ↑(nnabs ↑n)"}
+{"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", "A : Matrix m n R", "v : m → R"], "goal": "(A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0"}], "premise": [85011, 142402], "state_str": "m : Type u_1\nn : Type u_2\np : Type u_3\nR : Type u_4\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : Fintype p\ninst✝⁴ : PartialOrder R\ninst✝³ : NonUnitalRing R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nA : Matrix m n R\nv : m → R\n⊢ (A * Aᴴ) *ᵥ v = 0 ↔ Aᴴ *ᵥ v = 0"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Preorder α", "inst✝ : DecidableRel fun x x_1 => x ≤ x_1", "x : α", "t : Ordset α", "h_mem : x ∈ t"], "goal": "0 < t.size"}], "premise": [2715], "state_str": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : x ∈ t\n⊢ 0 < t.size"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Preorder α", "inst✝ : DecidableRel fun x x_1 => x ≤ x_1", "x : α", "t : Ordset α", "h_mem : Ordnode.mem x ↑t = true"], "goal": "0 < t.size"}], "premise": [2115, 146408, 146456], "state_str": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : Ordnode.mem x ↑t = true\n⊢ 0 < t.size"}
+{"state": [{"context": ["C : Type u_2", "D : Type ?u.23461", "inst✝¹ : Category.{u_1, u_2} C", "inst✝ : Category.{?u.23469, ?u.23461} D", "X : C", "S : Presieve X", "T : Sieve X", "h : S.FactorsThru T.arrows", "Y W : C", "i : Y ⟶ W", "e : W ⟶ X", "h1 : T.arrows e", "hf : i ≫ e ∈ S"], "goal": "i ≫ e ∈ T.arrows"}], "premise": [91045], "state_str": "case intro.intro.intro.intro\nC : Type u_2\nD : Type ?u.23461\ninst✝¹ : Category.{u_1, u_2} C\ninst✝ : Category.{?u.23469, ?u.23461} D\nX : C\nS : Presieve X\nT : Sieve X\nh : S.FactorsThru T.arrows\nY W : C\ni : Y ⟶ W\ne : W ⟶ X\nh1 : T.arrows e\nhf : i ≫ e ∈ S\n⊢ i ≫ e ∈ T.arrows"}
+{"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": "(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) ≫ ((↑(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)) = 𝟙 (((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).X n)"}], "premise": [91598, 91599, 91682, 93604, 93605, 96173, 99919, 113954, 119727, 119729], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (H.map ((↑(fst φ)).v n m ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) ≫\n      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map ((inl φ).v m n ⋯) +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map ((inr φ).f n)) =\n    𝟙 (((H.mapHomologicalComplex (ComplexShape.up ℤ)).obj (mappingCone φ)).X n)"}
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+{"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 φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n = 𝟙 ((mappingCone φ).X n)"}], "premise": [114192], "state_str": "case e_a\nC : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst φ)).v n m ⋯ ≫ (inl φ).v m n ⋯ + (snd φ).v n n ⋯ ≫ (inr φ).f n = 𝟙 ((mappingCone φ).X 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)"}], "premise": [91598, 91599, 96173, 96175, 99920, 99921, 113954, 114177, 114178, 114179, 114180], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map ((inl φ).v m n ⋯) +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map ((inr φ).f n)) ≫\n      (H.map ((↑(fst φ)).v n m ⋯) ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        H.map ((snd φ).v n n ⋯) ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) =\n    𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X 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 ⋯ ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ + (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫ H.map 0 ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ + ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫ H.map 0 ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n + (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map 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)"}], "premise": [114192], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{?u.266427, u_1} C\ninst✝⁵ : Category.{?u.266431, u_2} D\ninst✝⁴ : Preadditive C\ninst✝³ : Preadditive D\nF G : CochainComplex C ℤ\nφ : F ⟶ G\ninst✝² : HasHomotopyCofiber φ\nH : C ⥤ D\ninst✝¹ : H.Additive\ninst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)\nn m : ℤ\nhnm : n + 1 = m\n⊢ (↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          H.map 0 ≫ (inl ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v m n ⋯ +\n      ((↑(fst ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ))).v n m ⋯ ≫\n          H.map 0 ≫ (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n +\n        (snd ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).v n n ⋯ ≫\n          (inr ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).f n) =\n    𝟙 ((mappingCone ((H.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).X n)"}
+{"state": [{"context": ["n : ℕ", "c : Composition n"], "goal": "c.boundary 0 = 0"}], "premise": [3976], "state_str": "n : ℕ\nc : Composition n\n⊢ c.boundary 0 = 0"}
+{"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", "m n : ℕ", "a b : α"], "goal": "Odd (a + a + (2 * b + 1))"}], "premise": [119704, 122222], "state_str": "case intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : ℕ\na b : α\n⊢ Odd (a + a + (2 * b + 1))"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "ρ : Measure (α × ℝ)"], "goal": "Measurable fun a r => (preCDF ρ r a).toReal"}], "premise": [28880], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\n⊢ Measurable fun a r => (preCDF ρ r a).toReal"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "mα : MeasurableSpace α", "ρ : Measure (α × ℝ)"], "goal": "∀ (a : ℚ), Measurable fun x => (preCDF ρ a x).toReal"}], "premise": [25989, 72727], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\n⊢ ∀ (a : ℚ), Measurable fun x => (preCDF ρ a x).toReal"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "F : C ⥤ D", "X Y Z : C", "f : Y ⟶ X", "S R : Sieve X", "Y✝ : C", "f✝ : Y✝ ⟶ X"], "goal": "(sieveOfSubfunctor S.functorInclusion).arrows f✝ ↔ S.arrows f✝"}], "premise": [91117, 91119], "state_str": "case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (sieveOfSubfunctor S.functorInclusion).arrows f✝ ↔ S.arrows f✝"}
+{"state": [{"context": ["m n✝ n : ℕ"], "goal": "Even (n * (n + 1))"}], "premise": [119671, 119677], "state_str": "m n✝ n : ℕ\n⊢ Even (n * (n + 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", "β : Type u_5", "mβ : MeasurableSpace β", "f : α → β", "g✝ : β → E", "g : β → F", "hf : MeasurableEmbedding f"], "goal": "Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ"}], "premise": [28735, 29434], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nβ : Type u_5\nmβ : MeasurableSpace β\nf : α → β\ng✝ : β → E\ng : β → F\nhf : MeasurableEmbedding f\n⊢ Memℒp g p (Measure.map f μ) ↔ Memℒp (g ∘ f) p μ"}
+{"state": [{"context": ["ι : Type u_1", "inst✝ : Fintype ι", "a b : ℝ"], "goal": "volume (uIcc a b) = ofReal |b - a|"}], "premise": [18563, 30087, 105360], "state_str": "ι : Type u_1\ninst✝ : Fintype ι\na b : ℝ\n⊢ volume (uIcc a b) = ofReal |b - a|"}
+{"state": [{"context": ["n₁ n₂ d₁ d₂ : Int", "z₁ : d₁ ≠ 0", "z₂ : d₂ ≠ 0"], "goal": "n₁ /. d₁ * (n₂ /. d₂) = n₁ * n₂ /. (d₁ * d₂)"}], "premise": [713, 744, 2465, 2557, 2558, 2578, 3407], "state_str": "n₁ n₂ d₁ d₂ : Int\nz₁ : d₁ ≠ 0\nz₂ : d₂ ≠ 0\n⊢ n₁ /. d₁ * (n₂ /. d₂) = n₁ * n₂ /. (d₁ * d₂)"}
+{"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", "f : α → β", "hf : _root_.Embedding f", "s : Set α"], "goal": "IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s"}], "premise": [65544], "state_str": "α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\n⊢ IsTotallyDisconnected (f '' s) ↔ IsTotallyDisconnected s"}
+{"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", "f : α → β", "hf : _root_.Embedding f", "s : Set α", "hs : IsTotallyDisconnected s", "u : Set β", "hus : u ⊆ f '' s", "hu : IsPreconnected u"], "goal": "u.Subsingleton"}], "premise": [133448, 133451, 134148], "state_str": "α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nu : Set β\nhus : u ⊆ f '' s\nhu : IsPreconnected u\n⊢ u.Subsingleton"}
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+{"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", "f : α → β", "hf : _root_.Embedding f", "s : Set α", "hs : IsTotallyDisconnected s", "v : Set α", "hvs : v ⊆ s", "hus : f '' v ⊆ f '' s", "hu : IsPreconnected v"], "goal": "(f '' v).Subsingleton"}], "premise": [134262], "state_str": "case intro.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns✝ t u v✝ : Set α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : _root_.Embedding f\ns : Set α\nhs : IsTotallyDisconnected s\nv : Set α\nhvs : v ⊆ s\nhus : f '' v ⊆ f '' s\nhu : IsPreconnected v\n⊢ (f '' v).Subsingleton"}
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+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1"], "goal": "False"}], "premise": [152, 2623, 3059, 3082, 3090, 3840, 3886, 4989, 5180, 5303, 5323, 8934, 19710, 117715, 117721, 119704], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1"], "goal": "False"}], "premise": [8936], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1", "h₄ : t * t' = 1", "h₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))"], "goal": "False"}], "premise": [2100, 2102, 8160], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1", "h₄ : t * t' = 1", "h₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))", "h₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length"], "goal": "False"}], "premise": [103886], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1", "h₄ : t * t' = 1", "h₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))", "h₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length", "h₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1"], "goal": "False"}], "premise": [1673, 3886, 4576, 132677], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1", "h₄ : t * t' = 1", "h₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))", "h₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length", "h₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1", "h₈ : j' - 1 < (ω.eraseIdx j).length"], "goal": "False"}], "premise": [132677], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length + 1 + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False"}
+{"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", "j j' : ℕ", "j_lt_j' : j < j'", "j'_lt_length : j' < ω.length", "dup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1", "t : W := (cs.rightInvSeq ω).getD j 1", "h₁ : t = (cs.rightInvSeq ω).getD j 1", "t' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1", "h₃ : t' = (cs.rightInvSeq ω).getD j' 1", "h₄ : t * t' = 1", "h₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))", "h₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length", "h₇ : ω.length + 1 + 1 ≤ (ω.eraseIdx j).length + 1", "h₈ : j' - 1 < (ω.eraseIdx j).length"], "goal": "False"}], "premise": [132677], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.rightInvSeq ω).getD j 1\nt' : W := (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₂ : t' = (cs.rightInvSeq (ω.eraseIdx j)).getD (j' - 1) 1\nh₃ : t' = (cs.rightInvSeq ω).getD j' 1\nh₄ : t * t' = 1\nh₅ : cs.wordProd ω = cs.wordProd ((ω.eraseIdx j).eraseIdx (j' - 1))\nh₆ : ω.length ≤ ((ω.eraseIdx j).eraseIdx (j' - 1)).length\nh₇ : ω.length + 1 + 1 ≤ (ω.eraseIdx j).length + 1\nh₈ : j' - 1 < (ω.eraseIdx j).length\n⊢ False"}
+{"state": [{"context": ["n : ℕ", "E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace 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"}], "premise": [25460, 33641, 108334, 120650], "state_str": "n : ℕ\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf g : (Fin n → ℂ) → E\nc : Fin n → ℂ\nR : Fin n → ℝ\nhf : TorusIntegrable f c R\nhg : TorusIntegrable g c R\n⊢ (∯ (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": ["F✝ : Type u_1", "inst✝⁴ : AddCommGroup F✝", "inst✝³ : Module ℝ F✝", "inst✝² : TopologicalSpace F✝", "F : Type u_2", "inst✝¹ : NormedAddCommGroup F", "inst✝ : NormedSpace ℝ F", "x : F", "a✝ : x ∈ univ"], "goal": "(convexHull ℝ) (connectedComponentIn univ x) = univ"}], "premise": [34907, 65908, 65928], "state_str": "F✝ : Type u_1\ninst✝⁴ : AddCommGroup F✝\ninst✝³ : Module ℝ F✝\ninst✝² : TopologicalSpace F✝\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\na✝ : x ∈ univ\n⊢ (convexHull ℝ) (connectedComponentIn univ x) = univ"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsReduced R", "p k m : ℕ", "inst✝ : ExpChar R p", "x : R"], "goal": "x ^ (p ^ k * m) = 1 ↔ x ^ m = 1"}], "premise": [119764], "state_str": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ x ^ (p ^ k * m) = 1 ↔ x ^ m = 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsReduced R", "p k m : ℕ", "inst✝ : ExpChar R p", "x : R"], "goal": "(x ^ m) ^ p ^ k = 1 ↔ x ^ m = 1"}], "premise": [71387, 124463], "state_str": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (x ^ m) ^ p ^ k = 1 ↔ x ^ m = 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝² : CommRing R", "inst✝¹ : IsReduced R", "p k m : ℕ", "inst✝ : ExpChar R p", "x : R"], "goal": "(iterateFrobenius R p k) 1 = 1"}], "premise": [117064], "state_str": "case h.e'_1.h.e'_3\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsReduced R\np k m : ℕ\ninst✝ : ExpChar R p\nx : R\n⊢ (iterateFrobenius R p k) 1 = 1"}
+{"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 : M →* N", "hf : ∀ (y : ↥S), IsUnit (f ↑y)", "y z : ↥S", "h : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹"], "goal": "f ↑y = f ↑z"}], "premise": [1717, 9137, 119730], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : M →* N\nhf : ∀ (y : ↥S), IsUnit (f ↑y)\ny z : ↥S\nh : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹\n⊢ f ↑y = f ↑z"}
+{"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 : M →* N", "hf : ∀ (y : ↥S), IsUnit (f ↑y)", "y z : ↥S", "h : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹"], "goal": "f ↑z * ↑((IsUnit.liftRight (f.restrict S) hf) z)⁻¹ = 1"}], "premise": [120397], "state_str": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : M →* N\nhf : ∀ (y : ↥S), IsUnit (f ↑y)\ny z : ↥S\nh : ((IsUnit.liftRight (f.restrict S) hf) y)⁻¹ = ((IsUnit.liftRight (f.restrict S) hf) z)⁻¹\n⊢ f ↑z * ↑((IsUnit.liftRight (f.restrict S) hf) z)⁻¹ = 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", "F : ℕ → α → ℝ≥0∞", "f bound : α → ℝ≥0∞", "hF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ", "h_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound", "h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤", "h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))"], "goal": "Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))"}], "premise": [29108, 30260], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\n⊢ Tendsto (fun n => ∫⁻ (a : α), F n a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂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 : ℕ → α → ℝ≥0∞", "f bound : α → ℝ≥0∞", "hF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ", "h_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound", "h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤", "h_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))", "this : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ"], "goal": "Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))"}], "premise": [29107, 30345], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nF : ℕ → α → ℝ≥0∞\nf bound : α → ℝ≥0∞\nhF_meas : ∀ (n : ℕ), AEMeasurable (F n) μ\nh_bound : ∀ (n : ℕ), F n ≤ᶠ[ae μ] bound\nh_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤\nh_lim : ∀ᵐ (a : α) ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))\nthis : ∀ (n : ℕ), ∫⁻ (a : α), F n a ∂μ = ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ\n⊢ Tendsto (fun n => ∫⁻ (a : α), AEMeasurable.mk (F n) ⋯ a ∂μ) atTop (𝓝 (∫⁻ (a : α), f a ∂μ))"}
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+{"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", "f : P →ᵃ[R] P'", "hf : Function.Injective ⇑f", "fp₁ : P'", "hfp₁ : fp₁ ∈ map f s", "fp₂ : P'", "hfp₂ : fp₂ ∈ map f s", "h : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)"], "goal": "s.WOppSide x y"}], "premise": [84539], "state_str": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y : P\nf : P →ᵃ[R] P'\nhf : Function.Injective ⇑f\nfp₁ : P'\nhfp₁ : fp₁ ∈ map f s\nfp₂ : P'\nhfp₂ : fp₂ ∈ map f s\nh : SameRay R (f x -ᵥ fp₁) (fp₂ -ᵥ f y)\n⊢ s.WOppSide x y"}
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+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X Y Z✝ : C", "f : X ⟶ Y", "I : IsIso f", "Z : C", "g : Y ⟶ Z"], "goal": "inv f ≫ f ≫ g = g"}], "premise": [96173], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z✝ : C\nf : X ⟶ Y\nI : IsIso f\nZ : C\ng : Y ⟶ Z\n⊢ inv f ≫ f ≫ g = g"}
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+{"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✝ a' : α", "b : M", "f : α →₀ M", "a : α", "h : f.support ⊆ {a}", "x : α"], "goal": "f x = (single a (f a)) x"}], "premise": [1787, 148085, 148086], "state_str": "case intro.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\n⊢ f x = (single a (f a)) x"}
+{"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✝ a' : α", "b : M", "f : α →₀ M", "a : α", "h : f.support ⊆ {a}", "x : α", "hx : ¬a = x"], "goal": "f x = 0"}], "premise": [1673, 1681, 2100, 138737, 148065], "state_str": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝ : Zero M\na✝ a' : α\nb : M\nf : α →₀ M\na : α\nh : f.support ⊆ {a}\nx : α\nhx : ¬a = x\n⊢ f x = 0"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "f : α → β", "hf : ClosedEmbedding f", "inst✝ : QuasiSober β", "S✝ : Set α", "hS : IsIrreducible S✝", "hS' : IsClosed S✝", "hS'' : IsIrreducible (f '' S✝)"], "goal": "∃ x, IsGenericPoint x S✝"}], "premise": [56113, 57691], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "f : α → β", "hf : ClosedEmbedding f", "inst✝ : QuasiSober β", "S✝ : Set α", "hS : IsIrreducible S✝", "hS' : IsClosed S✝", "hS'' : IsIrreducible (f '' S✝)", "x : β", "hx : IsGenericPoint x (f '' S✝)"], "goal": "∃ x, IsGenericPoint x S✝"}], "premise": [57680], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\nx : β\nhx : IsGenericPoint x (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "f : α → β", "hf : ClosedEmbedding f", "inst✝ : QuasiSober β", "S✝ : Set α", "hS : IsIrreducible S✝", "hS' : IsClosed S✝", "hS'' : IsIrreducible (f '' S✝)", "y : α", "hx : IsGenericPoint (f y) (f '' S✝)"], "goal": "∃ x, IsGenericPoint x S✝"}], "premise": [1674, 2045], "state_str": "case intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ ∃ x, IsGenericPoint x S✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : TopologicalSpace α", "inst✝¹ : TopologicalSpace β", "f : α → β", "hf : ClosedEmbedding f", "inst✝ : QuasiSober β", "S✝ : Set α", "hS : IsIrreducible S✝", "hS' : IsClosed S✝", "hS'' : IsIrreducible (f '' S✝)", "y : α", "hx : IsGenericPoint (f y) (f '' S✝)"], "goal": "IsGenericPoint y S✝"}], "premise": [1674, 54005, 134318], "state_str": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : ClosedEmbedding f\ninst✝ : QuasiSober β\nS✝ : Set α\nhS : IsIrreducible S✝\nhS' : IsClosed S✝\nhS'' : IsIrreducible (f '' S✝)\ny : α\nhx : IsGenericPoint (f y) (f '' S✝)\n⊢ IsGenericPoint y S✝"}
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+{"state": [{"context": ["𝕜 : Type u_1", "inst✝¹¹ : NontriviallyNormedField 𝕜", "E : Type u_2", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "F : Type u_3", "inst✝⁸ : NormedAddCommGroup F", "inst✝⁷ : NormedSpace 𝕜 F", "G : Type u_4", "inst✝⁶ : NormedAddCommGroup G", "inst✝⁵ : NormedSpace 𝕜 G", "G' : Type u_5", "inst✝⁴ : NormedAddCommGroup G'", "inst✝³ : 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", "R : Type u_6", "inst✝² : NormedRing R", "inst✝¹ : NormedAlgebra 𝕜 R", "inst✝ : CompleteSpace R", "x : Rˣ", "this : (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] _root_.id"], "goal": "HasFDerivAt Ring.inverse (-((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x"}], "premise": [46303], "state_str": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nR : Type u_6\ninst✝² : NormedRing R\ninst✝¹ : NormedAlgebra 𝕜 R\ninst✝ : CompleteSpace R\nx : Rˣ\nthis : (fun t => Ring.inverse (↑x + t) - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =o[𝓝 0] _root_.id\n⊢ HasFDerivAt Ring.inverse (-((mulLeftRight 𝕜 R) ↑x⁻¹) ↑x⁻¹) ↑x"}
+{"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", "s₁ : Set α", "hs₁ : s₁ ∈ 𝒫 s", "s₂ : Set α", "hs₂ : s₂ ∈ 𝒫 s", "h' : f '' s₁ = f '' s₂"], "goal": "s₁ = s₂"}], "premise": [135819], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nπ : α → Type u_5\ns s₁✝ s₂✝ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : InjOn f s\ns₁ : Set α\nhs₁ : s₁ ∈ 𝒫 s\ns₂ : Set α\nhs₂ : s₂ ∈ 𝒫 s\nh' : f '' s₁ = f '' s₂\n⊢ s₁ = s₂"}
+{"state": [{"context": ["C : Type u_1", "inst✝² : Category.{?u.76625, u_1} C", "inst✝¹ : Preadditive C", "S : ShortComplex (CochainComplex C ℤ)", "σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting", "inst✝ : HasBinaryBiproducts C", "p : ℤ"], "goal": "((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫ ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) = (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom"}], "premise": [114599], "state_str": "C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom"}
+{"state": [{"context": ["C : Type u_1", "inst✝² : Category.{?u.76625, u_1} C", "inst✝¹ : Preadditive C", "S : ShortComplex (CochainComplex C ℤ)", "σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting", "inst✝ : HasBinaryBiproducts C", "p : ℤ", "r_f : (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f = 𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ - (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s"], "goal": "((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫ ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) = (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom"}], "premise": [114597], "state_str": "C : Type u_1\ninst✝² : Category.{?u.76625, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex (CochainComplex C ℤ)\nσ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting\ninst✝ : HasBinaryBiproducts C\np : ℤ\nr_f :\n  (σ (p + 1 + 1)).r ≫ (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).f =\n    𝟙 (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).X₂ -\n      (S.map (eval C (ComplexShape.up ℤ) (p + 1 + 1))).g ≫ (σ (p + 1 + 1)).s\n⊢ ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) p).hom ≫\n      ((CategoryTheory.shiftFunctor (HomologicalComplex C (ComplexShape.up ℤ)) 1).obj S.X₂).d p (p + 1) =\n    (mappingCone (homOfDegreewiseSplit S σ)).d p (p + 1) ≫\n      ((fun p => mappingConeHomOfDegreewiseSplitXIso S σ p (p + { as := 1 }.as) ⋯) (p + 1)).hom"}
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+{"state": [{"context": ["M : Type u_1", "m₁ m₂ : DivInvMonoid M", "h_mul : HMul.hMul = HMul.hMul", "h_inv : Inv.inv = Inv.inv"], "goal": "m₁ = m₂"}], "premise": [118434], "state_str": "M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\n⊢ m₁ = m₂"}
+{"state": [{"context": ["M : Type u_1", "m₁ m₂ : DivInvMonoid M", "h_mul : HMul.hMul = HMul.hMul", "h_inv : Inv.inv = Inv.inv", "h_mon : toMonoid = toMonoid", "h₁ : One.one = One.one", "f : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }", "this : Monoid.npow = Monoid.npow"], "goal": "m₁ = m₂"}], "premise": [117226], "state_str": "M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis : Monoid.npow = Monoid.npow\n⊢ m₁ = m₂"}
+{"state": [{"context": ["M : Type u_1", "m₁ m₂ : DivInvMonoid M", "h_mul : HMul.hMul = HMul.hMul", "h_inv : Inv.inv = Inv.inv", "h_mon : toMonoid = toMonoid", "h₁ : One.one = One.one", "f : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }", "this✝ : Monoid.npow = Monoid.npow", "this : DivInvMonoid.zpow = DivInvMonoid.zpow"], "goal": "m₁ = m₂"}], "premise": [117090], "state_str": "M : Type u_1\nm₁ m₂ : DivInvMonoid M\nh_mul : HMul.hMul = HMul.hMul\nh_inv : Inv.inv = Inv.inv\nh_mon : toMonoid = toMonoid\nh₁ : One.one = One.one\nf : M →* M := { toFun := id, map_one' := h₁, map_mul' := ⋯ }\nthis✝ : Monoid.npow = Monoid.npow\nthis : DivInvMonoid.zpow = DivInvMonoid.zpow\n⊢ m₁ = m₂"}
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+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "F✝ : C ⥤ Type w", "F : Cᵒᵖ ⥤ Type v"], "goal": "(fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 (CostructuredArrow yoneda F)"}], "premise": [97754], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nF✝ : C ⥤ Type w\nF : Cᵒᵖ ⥤ Type v\n⊢ (fromCostructuredArrow F).rightOp ⋙ toCostructuredArrow F = 𝟭 (CostructuredArrow yoneda F)"}
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+{"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 ∩ Z.baseSet j", "v : F"], "goal": "{ proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈ Z.baseSet i ∩ Z.baseSet (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩ Z.baseSet j"}, {"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 ∩ Z.baseSet j", "v : F"], "goal": "b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet"}], "premise": [2106, 2107, 60915], "state_str": "case a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj ∈\n    Z.baseSet i ∩\n        Z.baseSet\n          (Z.indexAt { proj := (b, v).1, snd := (Z.coordChange i (Z.indexAt (b, v).1) (b, v).1) (b, v).2 }.proj) ∩\n      Z.baseSet j\n\ncase hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_5\nZ : VectorBundleCore R B F ι\nb✝ : B\na : F\ni j : ι\nb : B\nhb : b ∈ Z.baseSet i ∩ Z.baseSet j\nv : F\n⊢ b ∈ (Z.localTriv i).baseSet ∩ (Z.localTriv j).baseSet"}
+{"state": [{"context": ["l : Type u_1", "R : Type u_2", "inst✝² : DecidableEq l", "inst✝¹ : Fintype l", "inst✝ : CommRing R"], "goal": "J l R ∈ symplecticGroup l R"}], "premise": [81384, 140699, 140711], "state_str": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\n⊢ J l R ∈ symplecticGroup l R"}
+{"state": [{"context": ["ι : Type u_1", "V : Type u", "inst✝³ : Category.{v, u} V", "inst✝² : HasZeroMorphisms V", "c : ComplexShape ι", "C : HomologicalComplex V c", "inst✝¹ : HasImages V", "inst✝ : HasEqualizers V", "i j : ι", "r : c.Rel i j"], "goal": "imageSubobject (C.dTo j) = imageSubobject (C.d i j)"}], "premise": [113873], "state_str": "ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject (C.dTo j) = imageSubobject (C.d i j)"}
+{"state": [{"context": ["ι : Type u_1", "V : Type u", "inst✝³ : Category.{v, u} V", "inst✝² : HasZeroMorphisms V", "c : ComplexShape ι", "C : HomologicalComplex V c", "inst✝¹ : HasImages V", "inst✝ : HasEqualizers V", "i j : ι", "r : c.Rel i j"], "goal": "imageSubobject ((C.xPrevIso r).hom ≫ C.d i j) = imageSubobject (C.d i j)"}], "premise": [89353], "state_str": "ι : Type u_1\nV : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝¹ : HasImages V\ninst✝ : HasEqualizers V\ni j : ι\nr : c.Rel i j\n⊢ imageSubobject ((C.xPrevIso r).hom ≫ C.d i j) = imageSubobject (C.d i j)"}
+{"state": [{"context": ["h : Nat.Primrec fun n => ack n n"], "goal": "False"}], "premise": [69837], "state_str": "h : Nat.Primrec fun n => ack n n\n⊢ False"}
+{"state": [{"context": ["h : Nat.Primrec fun n => ack n n", "m : ℕ", "hm : ∀ (n : ℕ), ack n n < ack m n"], "goal": "False"}], "premise": [11233], "state_str": "case intro\nh : Nat.Primrec fun n => ack n n\nm : ℕ\nhm : ∀ (n : ℕ), ack n n < ack m n\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "n : ℕ", "μ : R", "hμ : IsPrimitiveRoot μ (n + 1)"], "goal": "(-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1"}], "premise": [127178, 137642], "state_str": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\n⊢ (-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "n : ℕ", "μ : R", "hμ : IsPrimitiveRoot μ (n + 1)", "this : (-1) ^ n = ∏ k ∈ range n, -1"], "goal": "(-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1"}], "premise": [78046, 117837, 122248, 127009], "state_str": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\nthis : (-1) ^ n = ∏ k ∈ range n, -1\n⊢ (-1) ^ n * ∏ k ∈ range n, (μ ^ (k + 1) - 1) = ↑n + 1"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "l✝ l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "l : List α"], "goal": "l.tail <:+ l"}], "premise": [1271], "state_str": "α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nl : List α\n⊢ l.tail <:+ l"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "l✝ l₁ l₂ l₃ : List α", "a b : α", "m n : ℕ", "l : List α"], "goal": "drop 1 l <:+ l"}], "premise": [1521], "state_str": "α : Type u_1\nβ : Type u_2\nl✝ l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nl : List α\n⊢ drop 1 l <:+ l"}
+{"state": [{"context": ["M : Type u_1", "S✝ T : Set M", "inst✝ : Mul M", "S : Set M"], "goal": "S.centralizer.centralizer.centralizer = S.centralizer"}], "premise": [119320, 133329], "state_str": "M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\n⊢ S.centralizer.centralizer.centralizer = S.centralizer"}
+{"state": [{"context": ["M : Type u_1", "S✝ T : Set M", "inst✝ : Mul M", "S : Set M", "x : M", "hx : x ∈ S.centralizer.centralizer.centralizer"], "goal": "x ∈ S.centralizer"}], "premise": [119312], "state_str": "M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\n⊢ x ∈ S.centralizer"}
+{"state": [{"context": ["M : Type u_1", "S✝ T : Set M", "inst✝ : Mul M", "S : Set M", "x : M", "hx : x ∈ S.centralizer.centralizer.centralizer", "y : M", "hy : y ∈ S"], "goal": "y * x = x * y"}], "premise": [119312], "state_str": "M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : x ∈ S.centralizer.centralizer.centralizer\ny : M\nhy : y ∈ S\n⊢ y * x = x * y"}
+{"state": [{"context": ["M : Type u_1", "S✝ T : Set M", "inst✝ : Mul M", "S : Set M", "x : M", "hx : ∀ m ∈ S.centralizer.centralizer, m * x = x * m", "y : M", "hy : y ∈ S"], "goal": "y * x = x * y"}], "premise": [119320], "state_str": "M : Type u_1\nS✝ T : Set M\ninst✝ : Mul M\nS : Set M\nx : M\nhx : ∀ m ∈ S.centralizer.centralizer, m * x = x * m\ny : M\nhy : y ∈ S\n⊢ y * x = x * y"}
+{"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 : UpperSet α", "t : Set α", "a : α", "hts : t ⊆ ↑s", "hst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t"], "goal": "upperClosure t ⊓ s.sdiff t = s"}], "premise": [14579, 21214], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort u_4\nκ : ι → Sort u_5\ninst✝ : Preorder α\ns : UpperSet α\nt : Set α\na : α\nhts : t ⊆ ↑s\nhst : ∀ b ∈ s, ∀ c ∈ t, b ≤ c → b ∈ t\n⊢ upperClosure t ⊓ s.sdiff t = s"}
+{"state": [{"context": ["X : Type u_1", "Y✝ : Type u_2", "Z : Type u_3", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : TopologicalSpace Y✝", "inst✝⁴ : TopologicalSpace Z", "X' : Type u_4", "Y' : Type u_5", "inst✝³ : TopologicalSpace X'", "inst✝² : TopologicalSpace Y'", "ι : Type u_6", "p : ι → Prop", "Y : ι → Type u_7", "inst✝¹ : (i : ι) → TopologicalSpace (Y i)", "inst✝ : DecidablePred p"], "goal": "Continuous (Equiv.piEquivPiSubtypeProd p Y).toFun"}], "premise": [66393, 66537, 66538], "state_str": "X : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\n⊢ Continuous (Equiv.piEquivPiSubtypeProd p Y).toFun"}
+{"state": [{"context": ["X : Type u_1", "Y✝ : Type u_2", "Z : Type u_3", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : TopologicalSpace Y✝", "inst✝⁴ : TopologicalSpace Z", "X' : Type u_4", "Y' : Type u_5", "inst✝³ : TopologicalSpace X'", "inst✝² : TopologicalSpace Y'", "ι : Type u_6", "p : ι → Prop", "Y : ι → Type u_7", "inst✝¹ : (i : ι) → TopologicalSpace (Y i)", "inst✝ : DecidablePred p", "j : ι", "h✝ : p j"], "goal": "Continuous fun a => a.1 ⟨j, h✝⟩"}, {"context": ["X : Type u_1", "Y✝ : Type u_2", "Z : Type u_3", "inst✝⁶ : TopologicalSpace X", "inst✝⁵ : TopologicalSpace Y✝", "inst✝⁴ : TopologicalSpace Z", "X' : Type u_4", "Y' : Type u_5", "inst✝³ : TopologicalSpace X'", "inst✝² : TopologicalSpace Y'", "ι : Type u_6", "p : ι → Prop", "Y : ι → Type u_7", "inst✝¹ : (i : ι) → TopologicalSpace (Y i)", "inst✝ : DecidablePred p", "j : ι", "h✝ : ¬p j"], "goal": "Continuous fun a => a.2 ⟨j, h✝⟩"}], "premise": [55630, 66377, 66385, 66538], "state_str": "case pos\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : p j\n⊢ Continuous fun a => a.1 ⟨j, h✝⟩\n\ncase neg\nX : Type u_1\nY✝ : Type u_2\nZ : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace Z\nX' : Type u_4\nY' : Type u_5\ninst✝³ : TopologicalSpace X'\ninst✝² : TopologicalSpace Y'\nι : Type u_6\np : ι → Prop\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : DecidablePred p\nj : ι\nh✝ : ¬p j\n⊢ Continuous fun a => a.2 ⟨j, h✝⟩"}
+{"state": [{"context": ["α : Type u", "x y : α", "h : pure x = pure y"], "goal": "x = y"}], "premise": [16310], "state_str": "α : Type u\nx y : α\nh : pure x = pure y\n⊢ x = y"}
+{"state": [{"context": ["α : Type u", "x y : α", "h : pure x = pure y", "this : {x} ∈ pure y"], "goal": "x = y"}], "premise": [1673, 2100, 16188, 133512], "state_str": "α : Type u\nx y : α\nh : pure x = pure y\nthis : {x} ∈ pure y\n⊢ x = y"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "U : Set G", "hU : (interior U).Nonempty", "K : Set G", "h1K : IsCompact K", "h2K : (interior K).Nonempty"], "goal": "0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }"}], "premise": [104338], "state_str": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nhU : (interior U).Nonempty\nK : Set G\nh1K : IsCompact K\nh2K : (interior K).Nonempty\n⊢ 0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }"}
+{"state": [{"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : SeminormedRing β", "l : Filter α", "f g : α → β", "hf : l.BoundedAtFilter f", "hg : l.BoundedAtFilter g"], "goal": "l.BoundedAtFilter (f * g)"}], "premise": [43411, 43587], "state_str": "𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f * g)"}
+{"state": [{"context": ["𝕜 : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : SeminormedRing β", "l : Filter α", "f g : α → β", "hf : l.BoundedAtFilter f", "hg : l.BoundedAtFilter g"], "goal": "(fun x => 1 x * 1 x) =O[l] 1"}], "premise": [43424], "state_str": "𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ (fun x => 1 x * 1 x) =O[l] 1"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "x : α", "h : some x = none"], "goal": "False"}], "premise": [1695], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nx : α\nh : some x = none\n⊢ False"}
+{"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 n₂ : ℤ", "z₁ : Cochain F G 0", "z₂ : Cochain G K n₂", "m₂ : ���", "h₂ : n₂ + 1 = m₂"], "goal": "1 + n₂ = m₂"}], "premise": [119708], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m n₂ : ℤ\nz₁ : Cochain F G 0\nz₂ : Cochain G K n₂\nm₂ : ℤ\nh₂ : n₂ + 1 = m₂\n⊢ 1 + n₂ = m₂"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "M : Type u_3", "E : Type u_4", "inst✝ : One β", "l : Filter α", "f : α → β", "s : Set α", "hf : f =ᶠ[l] 1"], "goal": "s.mulIndicator 1 =ᶠ[l] 1"}], "premise": [120895], "state_str": "α : Type u_1\nβ : Type u_2\nM : Type u_3\nE : Type u_4\ninst✝ : One β\nl : Filter α\nf : α → β\ns : Set α\nhf : f =ᶠ[l] 1\n⊢ s.mulIndicator 1 =ᶠ[l] 1"}
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+{"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"}], "premise": [6962, 119703, 119707, 119745], "state_str": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Finite G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Nat.card G\nH : Subgroup G\nhH : Nat.card ↥H = p ^ n\ns : ℕ\nhs : Nat.card G = s * p ^ (n + 1)\n⊢ Nat.card (G ⧸ H) * Nat.card ↥H = s * p * Nat.card ↥H"}
+{"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 ⧸ comap H.normalizer.subtype H) % p"], "goal": "Nat.card (↥H.normalizer ⧸ comap H.normalizer.subtype H) % p = 0"}], "premise": [1717, 3528, 108897], "state_str": "G : Type u\nα : Type v\nβ : Type w\ninst✝¹ : Group G\ninst✝ : Finite G\np n : ℕ\nhp : Fact (Nat.Prime p)\nhdvd : p ^ (n + 1) ∣ Nat.card G\nH : Subgroup G\nhH : Nat.card ↥H = p ^ n\ns : ℕ\nhs : Nat.card G = s * p ^ (n + 1)\nhcard : Nat.card (G ⧸ H) = s * p\nhm : s * p % p = Nat.card (↥H.normalizer ⧸ comap H.normalizer.subtype H) % p\n⊢ Nat.card (↥H.normalizer ⧸ comap H.normalizer.subtype H) % p = 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''", "n : ℕ"], "goal": "χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f"}], "premise": [23055, 23057, 119739, 119742], "state_str": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nχ : MulChar R R'\nf : R' →+* R''\nn : ℕ\n⊢ χ.ringHomComp f ^ n = (χ ^ n).ringHomComp f"}
+{"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✝ f : α → β", "hf : LocallyLipschitz f"], "goal": "Continuous f"}], "premise": [55639], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf✝ f : α → β\nhf : LocallyLipschitz f\n⊢ Continuous f"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "G : Type u_3", "M : Type u_4", "inst✝ : DivisionMonoid α", "a b c d : α", "h : 1 / a = 1 / b"], "goal": "a = b"}], "premise": [117840], "state_str": "α : Type u_1\nβ : Type u_2\nG : Type u_3\nM : Type u_4\ninst✝ : DivisionMonoid α\na b c d : α\nh : 1 / a = 1 / b\n⊢ a = b"}
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+{"state": [{"context": ["n : ℕ", "R : Type u", "S : Type v", "T : Type w", "inst✝⁵ : CommRing R", "inst✝⁴ : CommRing S", "inst✝³ : CommRing T", "inst✝² : IsJacobson S", "inst✝¹ : Algebra R S", "inst✝ : Algebra R T", "IH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral", "v : S[X] ≃ₐ[R] T", "P : Ideal T", "hP : P.IsMaximal", "Q : Ideal S[X] := comap (↑v).toRingHom P", "hw : map v Q = P", "hQ : Q.IsMaximal", "w : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯", "Q' : Ideal S := comap Polynomial.C Q", "w' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯", "h_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))"], "goal": "(w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))).IsIntegral"}], "premise": [81940], "state_str": "n : ℕ\nR : Type u\nS : Type v\nT : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : IsJacobson S\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nIH : ∀ (Q : Ideal S), Q.IsMaximal → (algebraMap R (S ⧸ Q)).IsIntegral\nv : S[X] ≃ₐ[R] T\nP : Ideal T\nhP : P.IsMaximal\nQ : Ideal S[X] := comap (↑v).toRingHom P\nhw : map v Q = P\nhQ : Q.IsMaximal\nw : (S[X] ⧸ Q) ≃ₐ[R] T ⧸ P := Q.quotientEquivAlg P v ⋯\nQ' : Ideal S := comap Polynomial.C Q\nw' : S ⧸ Q' →ₐ[R] S[X] ⧸ Q := quotientMapₐ Q (Ideal.MvPolynomial.Cₐ R S) ⋯\nh_eq : algebraMap R (T ⧸ P) = w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))\n⊢ (w.toRingEquiv.toRingHom.comp (w'.comp (algebraMap R (S ⧸ Q')))).IsIntegral"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasFiniteProducts C", "inst✝ : HasPullbacks C", "X Y : Dial C"], "goal": "(X.tensorObj Y).rel = (Subobject.pullback (prod.map (prod.braiding X.src Y.src).hom (prod.braiding X.tgt Y.tgt).hom)).obj (Y.tensorObj X).rel"}], "premise": [14579, 89300, 90541], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasFiniteProducts C\ninst✝ : HasPullbacks C\nX Y : Dial C\n⊢ (X.tensorObj Y).rel =\n    (Subobject.pullback (prod.map (prod.braiding X.src Y.src).hom (prod.braiding X.tgt Y.tgt).hom)).obj\n      (Y.tensorObj X).rel"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "R : Type x", "inst✝ : CommSemiring α", "a✝ b✝ c a b : α"], "goal": "(a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b"}], "premise": [119704, 119708, 122236], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : CommSemiring α\na✝ b✝ c a b : α\n⊢ (a + b) ^ 2 = a ^ 2 + b ^ 2 + 2 * a * b"}
+{"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, succ o = succ a"], "goal": "(succ o).pred = o"}], "premise": [1084, 1739, 2100, 17414], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal.{u_4}\nh : ∃ a, succ o = succ a\n⊢ (succ o).pred = o"}
+{"state": [{"context": ["α : Type u", "β : Type v", "ι : Sort w", "κ : ι → Sort w'", "minAx : MinimalAxioms α", "f : (a : ι) → κ a → α", "x✝ : CompleteLattice α := minAx.toCompleteLattice"], "goal": "let x := minAx.toCompleteLattice; ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)"}], "premise": [14296, 14950], "state_str": "α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\nminAx : MinimalAxioms α\nf : (a : ι) → κ a → α\nx✝ : CompleteLattice α := minAx.toCompleteLattice\n⊢ let x := minAx.toCompleteLattice;\n  ⨅ i, ⨆ j, f i j = ⨆ g, ⨅ i, f i (g i)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "(tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}], "premise": [96173, 99312, 99340], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫\n      (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}], "premise": [96173, 96175, 99215], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}], "premise": [96173, 99210], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}], "premise": [96173, 96175, 99215], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}], "premise": [99217, 99218], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ tensor_μ C (M.X, N.X) (M.X, N.X) ▷ (P.X ⊗ P.X) ≫ (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}], "premise": [96173, 107150], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n      tensor_μ C (M.X, N.X) (M.X, N.X) ▷ (P.X ⊗ P.X) ≫\n        (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "inst✝¹ : MonoidalCategory C", "inst✝ : BraidedCategory C", "M N P : Mon_ C"], "goal": "(((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul) = ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫ tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}], "premise": [96173], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n        tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫\n      (M.mul ⊗ N.mul ⊗ P.mul) =\n    ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n      tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n        (M.X ⊗ M.X) ◁ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (M.mul ⊗ N.mul ⊗ P.mul)"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "D : Type u'", "inst✝² : Category.{v', u'} D", "F : C ⥤ D", "G : D ⥤ C", "adj : F ⊣ G", "inst✝¹ : F.Full", "inst✝ : F.Faithful", "P : C", "hP : Projective (F.obj P)", "E✝ X✝ : C", "f : P ⟶ X✝", "g : E✝ ⟶ X✝", "x✝ : Epi g", "this : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F"], "goal": "∃ f', f' ≫ g = f"}], "premise": [91435], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nP : C\nhP : Projective (F.obj P)\nE✝ X✝ : C\nf : P ⟶ X✝\ng : E✝ ⟶ X✝\nx✝ : Epi g\nthis : PreservesColimitsOfSize.{0, 0, v, v', u, u'} F\n⊢ ∃ f', f' ≫ g = f"}
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+{"state": [{"context": ["R : Type w₁", "A : Type w₂", "B : Type w₃", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : Algebra R A", "inst✝³ : CommRing B", "inst✝² : Algebra R B", "f : A →ₐ[R] B", "hf : Surjective ⇑f", "inst✝¹ : FinitePresentation R A", "inst✝ : FinitePresentation R B"], "goal": "(RingHom.ker f.toRingHom).FG"}], "premise": [78371], "state_str": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\n⊢ (RingHom.ker f.toRingHom).FG"}
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+{"state": [{"context": ["R : Type w₁", "A : Type w₂", "B : Type w₃", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : Algebra R A", "inst✝³ : CommRing B", "inst✝² : Algebra R B", "f : A →ₐ[R] B", "hf : Surjective ⇑f", "inst✝¹ : FinitePresentation R A", "inst✝ : FinitePresentation R B", "n : ℕ", "g : MvPolynomial (Fin n) R →ₐ[R] A", "hg : Surjective ⇑g", "right✝ : (RingHom.ker g.toRingHom).FG"], "goal": "RingHom.ker f.toRingHom = Ideal.map g.toRingHom (RingHom.ker (f.comp g).toRingHom)"}], "premise": [80706, 121046, 121076], "state_str": "case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ RingHom.ker f.toRingHom = Ideal.map g.toRingHom (RingHom.ker (f.comp g).toRingHom)"}
+{"state": [{"context": ["R : Type w₁", "A : Type w₂", "B : Type w₃", "inst✝⁶ : CommRing R", "inst✝⁵ : CommRing A", "inst✝⁴ : Algebra R A", "inst✝³ : CommRing B", "inst✝² : Algebra R B", "f : A ��ₐ[R] B", "hf : Surjective ⇑f", "inst✝¹ : FinitePresentation R A", "inst✝ : FinitePresentation R B", "n : ℕ", "g : MvPolynomial (Fin n) R →ₐ[R] A", "hg : Surjective ⇑g", "right✝ : (RingHom.ker g.toRingHom).FG"], "goal": "Ideal.comap ↑f ⊥ = Ideal.map (↑g) (Ideal.comap ((↑f).comp ↑g) ⊥)"}], "premise": [80644, 80671], "state_str": "case h.e'_3\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nhf : Surjective ⇑f\ninst✝¹ : FinitePresentation R A\ninst✝ : FinitePresentation R B\nn : ℕ\ng : MvPolynomial (Fin n) R →ₐ[R] A\nhg : Surjective ⇑g\nright✝ : (RingHom.ker g.toRingHom).FG\n⊢ Ideal.comap ↑f ⊥ = Ideal.map (↑g) (Ideal.comap ((↑f).comp ↑g) ⊥)"}
+{"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", "hx : x ∈ closure s"], "goal": "infDist x s = 0"}], "premise": [61828], "state_str": "ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhx : x ∈ closure s\n⊢ infDist x s = 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", "hx : x ∈ closure s"], "goal": "infDist x (closure s) = 0"}], "premise": [61813], "state_str": "ι : Sort u_1\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\nhx : x ∈ closure s\n⊢ infDist x (closure s) = 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", "R S T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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": "actLeft P Q ▷ T.X ≫ actRight P Q = (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}], "premise": [1673, 96190], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ actLeft P Q ▷ T.X ≫ actRight P Q = (α_ R.X (X P Q) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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": "(R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫ actLeft P Q ▷ T.X ≫ actRight P Q = (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫ (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}], "premise": [96173, 99222, 106560], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      actLeft P Q ▷ T.X ≫ actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫ actRight P Q = (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫ (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}], "premise": [96173, 106563], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ ((α_ R.X P.X Q.X).inv ▷ T.X ≫\n        P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n      actRight P Q =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ P.actLeft ▷ Q.X ▷ 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) = (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫ (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}], "premise": [96173, 99258], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      P.actLeft ▷ Q.X ▷ T.X ≫\n        (α_ P.X Q.X T.X).hom ≫\n          P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) = (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫ (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫ R.X ◁ actRight P Q ≫ actLeft P Q"}], "premise": [96173, 99261], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X ≫\n      (α_ R.X (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X).hom ≫\n        R.X ◁ actRight P Q ≫ actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ 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_ R.X (P.X ⊗ Q.X) T.X).hom ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫ R.X ◁ actRight P Q) ≫ actLeft P Q"}], "premise": [96173, 99219, 106563], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (((α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n          R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X) ≫\n        R.X ◁ actRight P Q) ≫\n      actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ 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_ R.X (P.X ⊗ Q.X) T.X).hom ≫ (R.X ◁ (α_ P.X Q.X T.X).hom ≫ R.X ◁ P.X ◁ Q.actRight ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫ actLeft P Q"}], "premise": [96173, 106560], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      (R.X ◁ (α_ P.X Q.X T.X).hom ≫\n          R.X ◁ P.X ◁ Q.actRight ≫ R.X ◁ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ≫\n        actLeft P Q"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ 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_ R.X (P.X ⊗ Q.X) T.X).hom ≫ R.X ◁ (α_ P.X Q.X T.X).hom ≫ R.X ◁ P.X ◁ Q.actRight ≫ (α_ R.X P.X Q.X).inv ≫ P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}], "premise": [96173, 99265], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        R.X ◁ P.X ◁ Q.actRight ≫\n          (α_ R.X P.X Q.X).inv ≫\n            P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}
+{"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 T : Mon_ C", "P : Bimod R S", "Q : Bimod S T", "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_ R.X P.X Q.X).inv ▷ T.X ≫ (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ 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_ R.X (P.X ⊗ Q.X) T.X).hom ≫ R.X ◁ (α_ P.X Q.X T.X).hom ≫ (((α_ R.X P.X (Q.X ⊗ T.X)).inv ≫ (R.X ⊗ P.X) ◁ Q.actRight) ≫ P.actLeft ▷ Q.X) ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}], "premise": [96173, 99226], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst✝² : HasCoequalizers C\nR S T : Mon_ C\nP : Bimod R S\nQ : Bimod S T\ninst✝¹ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)\ninst✝ : (X : C) → PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)\n⊢ (α_ R.X P.X Q.X).inv ▷ T.X ≫\n      (((α_ (R.X ⊗ P.X) Q.X T.X).hom ≫ P.actLeft ▷ (Q.X ⊗ T.X)) ≫ P.X ◁ Q.actRight) ≫\n        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) =\n    (α_ R.X (P.X ⊗ Q.X) T.X).hom ≫\n      R.X ◁ (α_ P.X Q.X T.X).hom ≫\n        (((α_ R.X P.X (Q.X ⊗ T.X)).inv ≫ (R.X ⊗ P.X) ◁ Q.actRight) ≫ P.actLeft ▷ Q.X) ≫\n          coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)"}
+{"state": [{"context": ["α : Type u_1", "p : α → Prop", "inst✝ : DecidablePred p", "l : List α", "a : α"], "goal": "(l.any fun a => decide (p a)) = true ↔ ∃ a, a ∈ l ∧ p a"}], "premise": [140221], "state_str": "α : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\nl : List α\na : α\n⊢ (l.any fun a => decide (p a)) = true ↔ ∃ a, a ∈ l ∧ p a"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁵ : NormedDivisionRing 𝕜", "inst✝⁴ : SeminormedAddCommGroup E", "inst✝³ : Module 𝕜 E", "inst✝² : BoundedSMul 𝕜 E", "P : Type u_3", "inst✝¹ : PseudoMetricSpace P", "inst✝ : NormedAddTorsor E P", "c : P", "k : 𝕜", "hk : k ≠ 0", "p : P"], "goal": "(fun x => k • x +ᵥ c) ((k⁻¹ • fun x => x -ᵥ c) p) = p"}], "premise": [7383], "state_str": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\np : P\n⊢ (fun x => k • x +ᵥ c) ((k⁻¹ • fun x => x -ᵥ c) p) = p"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁵ : NormedDivisionRing 𝕜", "inst✝⁴ : SeminormedAddCommGroup E", "inst✝³ : Module 𝕜 E", "inst✝² : BoundedSMul 𝕜 E", "P : Type u_3", "inst✝¹ : PseudoMetricSpace P", "inst✝ : NormedAddTorsor E P", "c : P", "k : 𝕜", "hk : k ≠ 0", "x y : E"], "goal": "edist ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun x) ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun y) = ↑‖k‖₊ * edist x y"}], "premise": [59856], "state_str": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : BoundedSMul 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun x)\n      ({ toFun := fun x => k • x +ᵥ c, invFun := k⁻¹ • fun x => x -ᵥ c, left_inv := ⋯, right_inv := ⋯ }.toFun y) =\n    ↑‖k‖₊ * edist x y"}
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+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0"], "goal": "∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a f < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε"}], "premise": [1674, 2045], "state_str": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ∃ δ > 0, ∀ (a : ↥(Lp G 1 μ)), dist a f < δ → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε"}
+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0"], "goal": "ε / 2 / c'.toReal > 0 ∧ ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε"}], "premise": [104338, 106102, 143376], "state_str": "case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\n⊢ ε / 2 / c'.toReal > 0 ∧\n    ∀ (a : ↥(Lp G 1 μ)), dist a f < ε / 2 / c'.toReal → dist (Integrable.toL1 ↑↑a ⋯) (Integrable.toL1 ↑↑f ⋯) < ε"}
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+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0", "g : ↥(Lp G 1 μ)", "hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal"], "goal": "(eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε"}], "premise": [28465, 28568], "state_str": "case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε"}
+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0", "g : ↥(Lp G 1 μ)", "hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal", "h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' := fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')"], "goal": "(eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε"}], "premise": [16105, 28581, 28685], "state_str": "case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\n⊢ (eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ').toReal < ε"}
+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0", "g : ↥(Lp G 1 μ)", "hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal", "h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' := fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')", "this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'"], "goal": "(eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε"}], "premise": [28685, 30989, 30996], "state_str": "case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε"}
+{"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", "μ' : Measure α", "c' : ℝ≥0∞", "hc' : c' ≠ ⊤", "hμ'_le : μ' ≤ c' • μ", "hc'0 : ¬c' = 0", "f : ↥(Lp G 1 μ)", "ε : ℝ", "hε_pos : ε > 0", "g : ↥(Lp G 1 μ)", "hfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal", "h_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' := fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')", "this : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'", "h_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤"], "goal": "(eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε"}], "premise": [28696, 28704, 118863, 143517], "state_str": "case h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n𝕜 : Type u_6\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g✝ : α → E\nμ' : Measure α\nc' : ℝ≥0∞\nhc' : c' ≠ ⊤\nhμ'_le : μ' ≤ c' • μ\nhc'0 : ¬c' = 0\nf : ↥(Lp G 1 μ)\nε : ℝ\nhε_pos : ε > 0\ng : ↥(Lp G 1 μ)\nhfg : (eLpNorm (↑↑g - ↑↑f) 1 μ).toReal < ε / 2 / c'.toReal\nh_int : ∀ (f' : ↥(Lp G 1 μ)), Integrable (↑↑f') μ' :=\n  fun f' => Integrable.of_measure_le_smul c' hc' hμ'_le (L1.integrable_coeFn f')\nthis : eLpNorm (↑↑(Integrable.toL1 ↑↑g ⋯) - ↑↑(Integrable.toL1 ↑↑f ⋯)) 1 μ' = eLpNorm (↑↑g - ↑↑f) 1 μ'\nh_eLpNorm_ne_top : eLpNorm (↑↑g - ↑↑f) 1 μ ≠ ⊤\n⊢ (eLpNorm (↑↑g - ↑↑f) 1 μ').toReal < ε"}
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+{"state": [{"context": ["R : Type u", "S : Type u_1", "A : Type v", "inst✝⁷ : CommRing R", "inst✝⁶ : CommRing S", "inst✝⁵ : Ring A", "inst✝⁴ : Algebra R A", "inst✝³ : Algebra R S", "inst✝² : Algebra S A", "inst✝¹ : IsScalarTower R S A", "inst✝ : Nontrivial R", "n : ℤ"], "goal": "IsAlgebraic R ((algebraMap R A) ↑n)"}], "premise": [75563], "state_str": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R S\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : Nontrivial R\nn : ℤ\n⊢ IsAlgebraic R ((algebraMap R A) ↑n)"}
+{"state": [{"context": ["α : Type u", "a : α", "G : Type u_1", "H : Type u_2", "inst✝ : AddGroup G", "g : G", "x✝ : ℤ"], "goal": "x✝ ∈ ((zmultiplesHom G) g).ker ↔ x✝ ∈ zmultiples ↑(addOrderOf g)"}], "premise": [1717, 8442, 108873, 123017, 128923, 128946, 129975], "state_str": "case h\nα : Type u\na : α\nG : Type u_1\nH : Type u_2\ninst✝ : AddGroup G\ng : G\nx✝ : ℤ\n⊢ x✝ ∈ ((zmultiplesHom G) g).ker ↔ x✝ ∈ zmultiples ↑(addOrderOf g)"}
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+{"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 y : F"], "goal": "(starRingEnd 𝕜) (-⟪y, x⟫_𝕜) = -⟪x, y⟫_𝕜"}], "premise": [36720, 121575], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝² : _root_.RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ (starRingEnd 𝕜) (-⟪y, x⟫_𝕜) = -⟪x, y⟫_𝕜"}
+{"state": [{"context": ["α : Type u_1", "inst✝⁴ : DecidableEq α", "inst✝³ : Fintype α", "f✝ g✝ : Perm α", "β : Type u_2", "inst✝² : DecidableEq β", "inst✝¹ : Fintype β", "p : β → Prop", "inst✝ : DecidablePred p", "f : α ≃ Subtype p", "g : Perm α", "b : β"], "goal": "b ∈ (g.extendDomain f).support ↔ b ∈ map f.asEmbedding g.support"}], "premise": [2045, 8669, 70657, 70663, 70668, 137366], "state_str": "case a\nα : Type u_1\ninst✝⁴ : DecidableEq α\ninst✝³ : Fintype α\nf✝ g✝ : Perm α\nβ : Type u_2\ninst✝² : DecidableEq β\ninst✝¹ : Fintype β\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\ng : Perm α\nb : β\n⊢ b ∈ (g.extendDomain f).support ↔ b ∈ map f.asEmbedding g.support"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : 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"}], "premise": [1673, 71383], "state_str": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\n⊢ seminormFromBounded' f ≠ 0"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : 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", "x : R", "hx : f x ≠ 0 x"], "goal": "seminormFromBounded' f ≠ 0"}], "premise": [71383], "state_str": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f ≠ 0"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : 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", "x : R", "hx : f x ≠ 0 x"], "goal": "∃ a, seminormFromBounded' f a ≠ 0 a"}], "premise": [1674, 2045], "state_str": "case intro\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ ∃ a, seminormFromBounded' f a ≠ 0 a"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : 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", "x : R", "hx : f x ≠ 0 x"], "goal": "seminormFromBounded' f x ≠ 0 x"}], "premise": [1169, 43875, 120640], "state_str": "case h\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nx : R\nhx : f x ≠ 0 x\n⊢ seminormFromBounded' f x ≠ 0 x"}
+{"state": [{"context": ["a b : EReal"], "goal": "-a < b ↔ -b < a"}], "premise": [1713, 119769, 147225], "state_str": "a b : EReal\n⊢ -a < b ↔ -b < a"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v, u₁} C", "X : Scheme", "U : X.Opens", "r : ↑Γ(↑U, ⊤)"], "goal": "U.ι ''ᵁ (↑U).basicOpen r = X.basicOpen r"}], "premise": [126611, 129851], "state_str": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX : Scheme\nU : X.Opens\nr : ↑Γ(↑U, ⊤)\n⊢ U.ι ''ᵁ (↑U).basicOpen r = X.basicOpen r"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : CommSemiring α", "E : LinearRecurrence α", "init : Fin E.order → α", "n : Fin E.order"], "goal": "(if h : ↑n < E.order then init ⟨↑n, h⟩ else ∑ k : Fin E.order, let_fun x := ⋯; E.coeffs k * E.mkSol init (↑n - E.order + ↑k)) = init n"}], "premise": [1739, 3972, 3974, 3982], "state_str": "α : Type u_1\ninst✝ : CommSemiring α\nE : LinearRecurrence α\ninit : Fin E.order → α\nn : Fin E.order\n⊢ (if h : ↑n < E.order then init ⟨↑n, h⟩\n    else\n      ∑ k : Fin E.order,\n        let_fun x := ⋯;\n        E.coeffs k * E.mkSol init (↑n - E.order + ↑k)) =\n    init n"}
+{"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", "y : M", "hy : y ∈ f.source"], "goal": "map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y"}], "premise": [57201, 67819, 67826, 71106, 133443], "state_str": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ny : M\nhy : y ∈ f.source\n⊢ map (↑(f.extend I)) (𝓝[s] y) = 𝓝[↑(f.extend I).symm ⁻¹' s ∩ range ↑I] ↑(f.extend I) y"}
+{"state": [{"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I"], "goal": "(delayReflRight 0 γ) t = (γ.trans (refl y)) t"}], "premise": [1670, 67462, 67468, 67489, 67491], "state_str": "case a.h\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\n⊢ (delayReflRight 0 γ) t = (γ.trans (refl y)) t"}
+{"state": [{"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ¬↑t ≤ 1 / 2"], "goal": "γ (qRight (t, 0)) = y"}, {"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ↑t ≤ 1 / 2"], "goal": "γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩"}], "premise": [67465], "state_str": "case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = y\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩"}
+{"state": [{"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ¬↑t ≤ 1 / 2"], "goal": "γ (qRight (t, 0)) = γ 1"}, {"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ↑t ≤ 1 / 2"], "goal": "γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩"}], "premise": [56193], "state_str": "case neg\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ 1\n\ncase pos\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ γ (qRight (t, 0)) = γ ⟨2 * ↑t, ⋯⟩"}
+{"state": [{"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ¬↑t ≤ 1 / 2"], "goal": "(if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑1"}, {"context": ["X : Type u", "inst✝ : TopologicalSpace X", "x y : X", "γ : Path x y", "t : ↑I", "h : ↑t ≤ 1 / 2"], "goal": "(if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑⟨2 * ↑t, ⋯⟩"}], "premise": [1737, 1738], "state_str": "case neg.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ¬↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑1\n\ncase pos.a\nX : Type u\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\nt : ↑I\nh : ↑t ≤ 1 / 2\n⊢ (if ↑t ≤ 1 / 2 then 2 * ↑t else 1) = ↑⟨2 * ↑t, ⋯⟩"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "a : α", "f : α → Seq1 β"], "goal": "(ret a).bind f = f a"}], "premise": [147613], "state_str": "α : Type u\nβ : Type v\nγ : Type w\na : α\nf : α → Seq1 β\n⊢ (ret a).bind f = f a"}
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+{"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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ", "ε0 : 0 < ε", "hε : ⇑f ⁻¹' ball 0 ε ⊆ ball 0 1"], "goal": "∃ K, AntilipschitzWith K ⇑f"}], "premise": [42718, 131591, 133323], "state_str": "case intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ\nε0 : 0 < ε\nhε : ⇑f ⁻¹' ball 0 ε ⊆ ball 0 1\n⊢ ∃ K, AntilipschitzWith K ⇑f"}
+{"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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1"], "goal": "∃ K, AntilipschitzWith K ⇑f"}], "premise": [43327], "state_str": "case intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\n⊢ ∃ K, AntilipschitzWith K ⇑f"}
+{"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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖c‖"], "goal": "∃ K, AntilipschitzWith K ⇑f"}], "premise": [42032], "state_str": "case intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\n⊢ ∃ K, AntilipschitzWith K ⇑f"}
+{"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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x✝ y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖c‖", "x : E", "hx : ¬f x = 0"], "goal": "‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}], "premise": [1673, 42922], "state_str": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f 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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x✝ y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖c‖", "x : E", "hx : ¬f x = 0", "hc₀ : c ≠ 0"], "goal": "‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}], "premise": [43345], "state_str": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f 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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x✝ y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖σ₁₂ c‖", "x : E", "hx : ¬f x = 0", "hc₀ : c ≠ 0"], "goal": "‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}], "premise": [36366], "state_str": "case neg\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f 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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x✝ y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖σ₁₂ c‖", "x : E", "hx : ¬f x = 0", "hc₀ : c ≠ 0", "n : ℤ", "hlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε", "hle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖"], "goal": "‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}], "premise": [8241, 43345, 108204], "state_str": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖σ₁₂ c ^ n • f x‖ < ↑ε\nhle : ‖σ₁₂ c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖σ₁₂ c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f 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✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedAddCommGroup F", "inst✝⁸ : NormedAddCommGroup G", "inst✝⁷ : NormedAddCommGroup Fₗ", "inst✝⁶ : NontriviallyNormedField 𝕜", "inst✝⁵ : NontriviallyNormedField 𝕜₂", "inst✝⁴ : NontriviallyNormedField 𝕜₃", "inst✝³ : NormedSpace 𝕜 E", "inst✝² : NormedSpace 𝕜₂ F", "inst✝¹ : NormedSpace 𝕜₃ G", "inst✝ : NormedSpace 𝕜 Fₗ", "c✝ : 𝕜", "σ₁₂ : 𝕜 →+* 𝕜₂", "σ₂₃ : 𝕜₂ →+* 𝕜₃", "f✝ g : E →SL[σ₁₂] F", "x✝ y z : E", "h : RingHomIsometric σ₁₂", "f : E →ₛₗ[σ₁₂] F", "hf : comap (⇑f) (𝓝 0) ≤ 𝓝 0", "ε : ℝ≥0", "ε0 : 0 < ↑ε", "hε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1", "c : 𝕜", "hc : 1 < ‖σ₁₂ c‖", "x : E", "hx : ¬f x = 0", "hc₀ : c ≠ 0", "n : ℤ", "hlt : ‖f (c ^ n • x)‖ < ↑ε", "hle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖"], "goal": "‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}], "premise": [1674, 7382, 41371, 42680, 43297, 102621, 104331, 108446, 119730], "state_str": "case neg.intro.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup G\ninst✝⁷ : NormedAddCommGroup Fₗ\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NontriviallyNormedField 𝕜₃\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\ninst✝¹ : NormedSpace 𝕜₃ G\ninst✝ : NormedSpace 𝕜 Fₗ\nc✝ : 𝕜\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nf✝ g : E →SL[σ₁₂] F\nx✝ y z : E\nh : RingHomIsometric σ₁₂\nf : E →ₛₗ[σ₁₂] F\nhf : comap (⇑f) (𝓝 0) ≤ 𝓝 0\nε : ℝ≥0\nε0 : 0 < ↑ε\nhε : ∀ (x : E), ‖f x‖ < ↑ε → ‖x‖ < 1\nc : 𝕜\nhc : 1 < ‖σ₁₂ c‖\nx : E\nhx : ¬f x = 0\nhc₀ : c ≠ 0\nn : ℤ\nhlt : ‖f (c ^ n • x)‖ < ↑ε\nhle : ‖c ^ n‖⁻¹ ≤ (↑ε)⁻¹ * ‖c‖ * ‖f x‖\n⊢ ‖x‖ ≤ ↑(ε⁻¹ * ‖c‖₊) * ‖f x‖"}
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+{"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": "⟨k + l - s.card, ⟨s.card, ⟨Finset.map (finCongr ⋯).toEmbedding s, ⋯⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩"}], "premise": [103403], "state_str": "case mk.mk.mk\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ ⟨k + l - s.card, ⟨s.card, ⟨Finset.map (finCongr ⋯).toEmbedding s, ⋯⟩⟩⟩ = ⟨k, ⟨l, ⟨s, hs⟩⟩⟩"}
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+{"state": [{"context": ["p : ℝ", "hp_pos : 0 < p", "hp_le_one : p ≤ 1", "this : HasSum (fun i => (1 - p) ^ i * p) (p⁻¹ * p)"], "goal": "HasSum (fun n => (1 - p) ^ n * p) 1"}], "premise": [11234, 108408, 117885], "state_str": "p : ℝ\nhp_pos : 0 < p\nhp_le_one : p ≤ 1\nthis : HasSum (fun i => (1 - p) ^ i * p) (p⁻¹ * p)\n⊢ HasSum (fun n => (1 - p) ^ n * p) 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"}], "premise": [95041, 95042, 96174, 96175, 99920], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nI✝ : MultispanIndex C\nK : Multicofork I✝\nI : MultispanIndex C\nP : C\nπ : (b : I.R) → I.right b ⟶ P\nw : ∀ (a : I.L), I.fst a ≫ π (I.fstFrom a) = I.snd a ≫ π (I.sndFrom a)\n⊢ ∀ ⦃X Y : WalkingMultispan I.fstFrom I.sndFrom⦄ (f : X ⟶ Y),\n    I.multispan.map f ≫\n        (fun x =>\n            match x with\n            | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a)\n            | WalkingMultispan.right b => π b)\n          Y =\n      (fun x =>\n            match x with\n            | WalkingMultispan.left a => I.fst a ≫ π (I.fstFrom a)\n            | WalkingMultispan.right b => π b)\n          X ≫\n        ((Functor.const (WalkingMultispan I.fstFrom I.sndFrom)).obj P).map f"}
+{"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"}], "premise": [93876], "state_str": "X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.J\n⊢ (fun x y z => 𝒰.gluedCoverT' x y z) x y z ≫\n      pullback.snd ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y z)\n        ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) y x) =\n    pullback.fst ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x y)\n        ((fun x y => pullback.fst (𝒰.map x) (𝒰.map y)) x z) ≫\n      (fun x y => (pullbackSymmetry (𝒰.map x) (𝒰.map y)).hom) x y"}
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+{"state": [{"context": ["R : Type u", "S : Type v", "inst✝¹ : CommSemiring R", "inst✝ : CommSemiring S", "f g : R", "x : PrimeSpectrum R"], "goal": "x ∈ zeroLocus {f * g} ↔ x ∈ zeroLocus {f} ∪ zeroLocus {g}"}], "premise": [77738, 80420], "state_str": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf g : R\nx : PrimeSpectrum R\n⊢ x ∈ zeroLocus {f * g} ↔ x ∈ zeroLocus {f} ∪ zeroLocus {g}"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : LinearOrderedRing α", "inst✝² : LinearOrderedAddCommGroup β", "inst✝¹ : Module α β", "inst✝ : OrderedSMul α β", "s : Finset ι", "σ : Perm ι", "f : ι → α", "g : ι → β", "hfg : MonovaryOn f g ↑s", "hσ : {x | σ x ≠ x} ⊆ ↑s"], "goal": "∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s"}], "premise": [11244, 106703, 106704], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : MonovaryOn f g ↑s\nhσ : {x | σ x ≠ x} ⊆ ↑s\n⊢ ∑ i ∈ s, f (σ i) • g i < ∑ i ∈ s, f i • g i ↔ ¬MonovaryOn (f ∘ ⇑σ) g ↑s"}
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+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝ : FiniteDimensional K L", "this : IsGalois K L"], "goal": "(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s"}], "premise": [25238], "state_str": "p n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s"}
+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝ : FiniteDimensional K L", "this : IsGalois K L", "H : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac"], "goal": "(Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s"}], "premise": [117983], "state_str": "case refine_2\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) (η₁ + 1 - 1) = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ (Algebra.norm K) (IntermediateField.AdjoinSimple.gen K η) ^ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑↑p ^ ↑p ^ s"}
+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝ : FiniteDimensional K L", "this : IsGalois K L", "H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac"], "goal": "FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}], "premise": [85823], "state_str": "case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝¹ : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝ : FiniteDimensional K L\nthis : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}
+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝¹ : FiniteDimensional K L", "this✝ : IsGalois K L", "H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac", "this : FiniteDimensional.finrank K ↥K⟮η⟯ * FiniteDimensional.finrank (↥K⟮η⟯) L = FiniteDimensional.finrank K L"], "goal": "FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}], "premise": [2143, 25228, 70028, 119703, 119707, 141238, 145335], "state_str": "case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : FiniteDimensional.finrank K ↥K⟮η⟯ * FiniteDimensional.finrank (↥K⟮η⟯) L = FiniteDimensional.finrank K L\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}
+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝¹ : FiniteDimensional K L", "this✝ : IsGalois K L", "H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac", "this : (↑p - 1) * (↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L) = (↑p - 1) * ↑p ^ (k.succ - 1)"], "goal": "FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}], "premise": [1674, 11234, 70028, 103654, 108560, 144294], "state_str": "case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : (↑p - 1) * (↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L) = (↑p - 1) * ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}
+{"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", "k s : ℕ", "hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))", "hpri : Fact (Nat.Prime ↑p)", "inst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L", "hirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)", "hs : s ≤ k", "htwo : p ^ (k - s + 1) ≠ 2", "hirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)", "η : L := ζ ^ ↑p ^ s - 1", "η₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η", "this✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯", "hη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))", "this✝¹ : FiniteDimensional K L", "this✝ : IsGalois K L", "H : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac", "this : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)"], "goal": "FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}], "premise": [2100, 3715, 3886, 103416], "state_str": "case refine_2.e_a\np n : ℕ+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsCyclotomicExtension {n} A B\ninst✝³ : Field L\nζ : L\nhζ✝ : IsPrimitiveRoot ζ ↑n\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))\nhpri : Fact (Nat.Prime ↑p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (↑(p ^ (k + 1))) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ : Irreducible (cyclotomic (↑(p ^ (k - s + 1))) K)\nη : L := ζ ^ ↑p ^ s - 1\nη₁ : ↥K⟮η⟯ := IntermediateField.AdjoinSimple.gen K η\nthis✝² : IsCyclotomicExtension {p ^ (k - s + 1)} K ↥K⟮η⟯\nhη : IsPrimitiveRoot (η₁ + 1) ↑(p ^ (k - s + 1))\nthis✝¹ : FiniteDimensional K L\nthis✝ : IsGalois K L\nH : (Algebra.norm K) η₁ = ↑(↑(p ^ (k - s + 1))).minFac\nthis : ↑p ^ ((k - s).succ - 1) * FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ (k.succ - 1)\n⊢ FiniteDimensional.finrank (↥K⟮η⟯) L = ↑p ^ s"}
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+{"state": [{"context": ["α β : DistLat", "e : ↑α ≃o ↑β", "x✝ : (forget DistLat).obj β"], "goal": "({ toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 β) x✝"}], "premise": [11075], "state_str": "case w\nα β : DistLat\ne : ↑α ≃o ↑β\nx✝ : (forget DistLat).obj β\n⊢ ({ toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } ≫ { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ }) x✝ = (𝟙 β) x✝"}
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+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : LocalRing R", "inst✝¹ : CommRing S", "inst✝ : Nontrivial S", "f : R →+* S", "hf : Function.Surjective ⇑f", "a : R"], "goal": "IsUnit (f a) ∨ IsUnit (1 - f a)"}], "premise": [1217, 76991, 121184], "state_str": "case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (f a) ∨ IsUnit (1 - f a)"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝³ : CommRing R", "inst✝² : LocalRing R", "inst✝¹ : CommRing S", "inst✝ : Nontrivial S", "f : R →+* S", "hf : Function.Surjective ⇑f", "a : R"], "goal": "IsUnit (1 - a) → IsUnit (1 - f a)"}], "premise": [121565, 121576], "state_str": "case intro\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : LocalRing R\ninst✝¹ : CommRing S\ninst✝ : Nontrivial S\nf : R →+* S\nhf : Function.Surjective ⇑f\na : R\n⊢ IsUnit (1 - a) → IsUnit (1 - f a)"}
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+{"state": [{"context": ["a : Cardinal.{u_1}"], "goal": "↑(toENat a) = a ↔ a ≤ ℵ₀"}], "premise": [1717, 12787, 47752], "state_str": "a : Cardinal.{u_1}\n⊢ ↑(toENat a) = a ↔ a ≤ ℵ₀"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "S T : WSeq (Computation α)", "a : α", "h1 : ∀ (s : Computation α), s ∈ S → s ~> a", "H : WSeq.LiftRel Equiv S T", "h2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1", "a' : α", "h : a' ∈ parallel S"], "goal": "a' ∈ parallel T"}], "premise": [148430], "state_str": "α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel S\n⊢ a' ∈ parallel T"}
+{"state": [{"context": ["α : Type u", "β : Type v", "S T : WSeq (Computation α)", "a : α", "h1 : ∀ (s : Computation α), s ∈ S → s ~> a", "H : WSeq.LiftRel Equiv S T", "h2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1", "a' : α", "h : a ∈ parallel S", "aa : a = a'"], "goal": "a ∈ parallel T"}], "premise": [1673, 148427, 148431, 148713], "state_str": "α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel S\naa : a = a'\n⊢ a ∈ parallel T"}
+{"state": [{"context": ["α : Type u", "β : Type v", "S T : WSeq (Computation α)", "a : α", "h1 : ∀ (s : Computation α), s ∈ S → s ~> a", "H : WSeq.LiftRel Equiv S T", "h2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1", "a' : α", "h : a' ∈ parallel T"], "goal": "a' ∈ parallel S"}], "premise": [148430], "state_str": "α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a' ∈ parallel T\n⊢ a' ∈ parallel S"}
+{"state": [{"context": ["α : Type u", "β : Type v", "S T : WSeq (Computation α)", "a : α", "h1 : ∀ (s : Computation α), s ∈ S → s ~> a", "H : WSeq.LiftRel Equiv S T", "h2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1", "a' : α", "h : a ∈ parallel T", "aa : a = a'"], "goal": "a ∈ parallel S"}], "premise": [1674, 148427, 148431, 148714], "state_str": "α : Type u\nβ : Type v\nS T : WSeq (Computation α)\na : α\nh1 : ∀ (s : Computation α), s ∈ S → s ~> a\nH : WSeq.LiftRel Equiv S T\nh2 : ∀ (t : Computation α), t ∈ T → t ~> a := (parallel_congr_lem H).mp h1\na' : α\nh : a ∈ parallel T\naa : a = a'\n⊢ a ∈ parallel S"}
+{"state": [{"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "δ : Type u_3", "a : α", "b : β", "s : Multiset α", "t : Multiset β", "l₁ : List α", "l₂ : List β"], "goal": "(↑l₁).product ↑l₂ = ↑(l₁.product l₂)"}], "premise": [135613], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\nδ : Type u_3\na : α\nb : β\ns : Multiset α\nt : Multiset β\nl₁ : List α\nl₂ : List β\n⊢ (↑l₁).product ↑l₂ = ↑(l₁.product l₂)"}
+{"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", "k : ℕ", "hk : a ^ k ∣ b", "hsucc : ¬a ^ (k + 1) ∣ b"], "goal": "k = (multiplicity a b).get ⋯"}], "premise": [79544, 144873, 144883], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\nhk : a ^ k ∣ b\nhsucc : ¬a ^ (k + 1) ∣ b\n⊢ k = (multiplicity a b).get ⋯"}
+{"state": [{"context": ["a : ℤ", "ha0 : 0 < a"], "goal": "↑(↑a)⁻¹.den = a"}], "premise": [708, 143125, 147935, 147976, 148012], "state_str": "a : ℤ\nha0 : 0 < a\n⊢ ↑(↑a)⁻¹.den = a"}
+{"state": [{"context": ["a : ℤ", "ha0 : 0 < a"], "goal": "↑(↑1 / ↑a).den = a"}], "premise": [147909], "state_str": "a : ℤ\nha0 : 0 < a\n⊢ ↑(↑1 / ↑a).den = a"}
+{"state": [{"context": ["a : ℤ", "ha0 : 0 < a"], "goal": "(Int.natAbs 1).Coprime a.natAbs"}], "premise": [3244], "state_str": "a : ℤ\nha0 : 0 < a\n⊢ (Int.natAbs 1).Coprime a.natAbs"}
+{"state": [{"context": ["a : ℤ", "ha0 : 0 < a"], "goal": "Nat.Coprime 1 a.natAbs"}], "premise": [425], "state_str": "a : ℤ\nha0 : 0 < a\n⊢ Nat.Coprime 1 a.natAbs"}
+{"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", "s : Set M", "p : (x : M) → x ∈ closure s → Prop", "mem : ∀ (x : M) (h : x ∈ s), p x ⋯", "mul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯", "x : M", "hx : x ∈ closure s"], "goal": "p x hx"}], "premise": [1727], "state_str": "M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nmem : ∀ (x : M) (h : x ∈ s), p x ⋯\nmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯\nx : M\nhx : x ∈ closure s\n⊢ p x hx"}
+{"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", "s : Set M", "p : (x : M) → x ∈ closure s → Prop", "mem : ∀ (x : M) (h : x ∈ s), p x ⋯", "mul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯", "x : M", "hx : x ∈ closure s"], "goal": "∃ (x_1 : x ∈ closure s), p x x_1"}], "premise": [116288], "state_str": "M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\ns : Set M\np : (x : M) → x ∈ closure s → Prop\nmem : ∀ (x : M) (h : x ∈ s), p x ⋯\nmul : ∀ (x : M) (hx : x ∈ closure s) (y : M) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯\nx : M\nhx : x ∈ closure s\n⊢ ∃ (x_1 : x ∈ closure s), p x x_1"}
+{"state": [{"context": ["μ : YoungDiagram", "j : ℕ"], "goal": "μ.colLen j = (μ.col j).card"}], "premise": [49887], "state_str": "μ : YoungDiagram\nj : ℕ\n⊢ μ.colLen j = (μ.col j).card"}
+{"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", "f : E → F", "f' : E →L[𝕜] F", "x₀ : E", "hf : HasFDerivAt f f' x₀", "s : Set E", "hs : s ∈ 𝓝 x₀", "C : ℝ≥0", "hlip : LipschitzOnWith C f s"], "goal": "‖f'‖ ≤ ↑C"}], "premise": [46304, 146609], "state_str": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖f'‖ ≤ ↑C"}
+{"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", "f : E → F", "f' : E →L[𝕜] F", "x₀ : E", "hf : HasFDerivAt f f' x₀", "s : Set E", "hs : s ∈ 𝓝 x₀", "C : ℝ≥0", "hlip : LipschitzOnWith C f s"], "goal": "∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖"}], "premise": [15889, 55499, 131585], "state_str": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf'✝ f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns✝ t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nf' : E →L[𝕜] F\nx₀ : E\nhf : HasFDerivAt f f' x₀\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ∀ᶠ (x : E) in 𝓝 x₀, ‖f x - f x₀‖ ≤ ↑C * ‖x - x₀‖"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "xs : List α", "ys : List β", "x : α", "y : β"], "goal": "(x, y) ∈ xs.product ys ↔ x ∈ xs ∧ y ∈ ys"}], "premise": [1186, 1193, 2043, 5127, 5239], "state_str": "α : Type u_1\nβ : Type u_2\nxs : List α\nys : List β\nx : α\ny : β\n⊢ (x, y) ∈ xs.product ys ↔ x ∈ xs ∧ y ∈ ys"}
+{"state": [{"context": ["a✝ b✝ m n p a b : ℕ", "hb : b ≠ 0"], "goal": "a.factorization ≤ (a * b).factorization"}], "premise": [70039], "state_str": "a✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\n⊢ a.factorization ≤ (a * b).factorization"}
+{"state": [{"context": ["a✝ b✝ m n p a b : ℕ", "hb : b ≠ 0", "ha : a ≠ 0"], "goal": "a.factorization ≤ (a * b).factorization"}], "premise": [108269, 144203], "state_str": "case inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a.factorization ≤ (a * b).factorization"}
+{"state": [{"context": ["a✝ b✝ m n p a b : ℕ", "hb : b ≠ 0", "ha : a ≠ 0"], "goal": "a ∣ a * b"}], "premise": [108871], "state_str": "case inr\na✝ b✝ m n p a b : ℕ\nhb : b ≠ 0\nha : a ≠ 0\n⊢ a ∣ a * b"}
+{"state": [{"context": ["F : Type u_1", "ι : Type u_2", "α : Type u_3", "β : Type u_4", "inst✝¹ : Group α", "inst✝ : AddGroup β", "f : Multiplicative ℤ →* α", "n : Multiplicative ℤ"], "goal": "f n = f (ofAdd 1) ^ toAdd n"}], "premise": [70751, 128948, 128949], "state_str": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Group α\ninst✝ : AddGroup β\nf : Multiplicative ℤ →* α\nn : Multiplicative ℤ\n⊢ f n = f (ofAdd 1) ^ toAdd n"}
+{"state": [{"context": ["F : Type u → Type u", "q : QPF F", "α β γ : Type u", "f : α → β", "g : β → γ", "x : F α"], "goal": "(g ∘ f) <$> x = g <$> f <$> x"}], "premise": [141121], "state_str": "F : Type u → Type u\nq : QPF F\nα β γ : Type u\nf : α → β\ng : β → γ\nx : F α\n⊢ (g ∘ f) <$> x = g <$> f <$> x"}
+{"state": [{"context": ["F : Type u → Type u", "q : QPF F", "α β γ : Type u", "f✝ : α → β", "g : β → γ", "x : F α", "a : (P F).A", "f : (P F).B a → α"], "goal": "(g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩"}], "premise": [141120], "state_str": "case mk\nF : Type u → Type u\nq : QPF F\nα β γ : Type u\nf✝ : α → β\ng : β → γ\nx : F α\na : (P F).A\nf : (P F).B a → α\n⊢ (g ∘ f✝) <$> abs ⟨a, f⟩ = g <$> f✝ <$> abs ⟨a, f⟩"}
+{"state": [{"context": ["n : ℕ", "R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R"], "goal": "(algebraMap R[X] (RatFunc R)) (cyclotomic n R) = ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}], "premise": [3735], "state_str": "n : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}
+{"state": [{"context": ["n : ℕ", "R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "hpos : n > 0"], "goal": "(algebraMap R[X] (RatFunc R)) (cyclotomic n R) = ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}], "premise": [75191, 126888], "state_str": "case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}
+{"state": [{"context": ["n : ℕ", "R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "hpos : n > 0", "h : ∀ (n : ℕ), 0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)"], "goal": "(algebraMap R[X] (RatFunc R)) (cyclotomic n R) = ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}], "premise": [1673, 23969, 77057], "state_str": "case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n  ∀ (n : ℕ),\n    0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n    ∏ i ∈ n.divisorsAntidiagonal, (algebraMap R[X] (RatFunc R)) (X ^ i.2 - 1) ^ μ i.1"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₀ : PositiveCompacts G", "U : Set G", "K : Compacts G", "hU : (interior U).Nonempty"], "goal": "↑(index (↑K) U) / ↑(index (↑K₀) U) ≤ ↑(index ↑K ↑K₀)"}], "premise": [106024], "state_str": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₀ : PositiveCompacts G\nU : Set G\nK : Compacts G\nhU : (interior U).Nonempty\n⊢ ↑(index (↑K) U) / ↑(index (↑K₀) U) ≤ ↑(index ↑K ↑K₀)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k N : ℕ", "x : Fin n✝ → ℕ", "n d : ℕ"], "goal": "∃ k, ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}], "premise": [53441], "state_str": "α : Type u_1\nβ : Type u_2\nn✝ d✝ k N : ℕ\nx : Fin n✝ → ℕ\nn d : ℕ\n⊢ ∃ k, ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card"], "goal": "↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}], "premise": [3735], "state_str": "case intro.intro\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0"], "goal": "↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}], "premise": [3735], "state_str": "case intro.intro.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0", "hd : d > 0"], "goal": "↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}], "premise": [106071], "state_str": "case intro.intro.inr.inr\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(d ^ n) / ↑(n * d ^ 2) ≤ ↑(sphere n d k).card"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0", "hd : d > 0"], "goal": "↑(n * (d - 1) ^ 2) + 1 ≤ ↑(n * d ^ 2)"}], "premise": [105721, 117899, 118064, 119730, 119756, 122285, 128746, 142648, 142652, 143125], "state_str": "case intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ ↑(n * (d - 1) ^ 2) + 1 ≤ ↑(n * d ^ 2)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0", "hd : d > 0"], "goal": "1 ≤ ↑n * (2 * ↑d - 1)"}], "premise": [106807], "state_str": "case intro.intro.inr.inr.refine_3\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ ↑n * (2 * ↑d - 1)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0", "hd : d > 0"], "goal": "1 ≤ 2 * ↑d - 1"}], "premise": [105711], "state_str": "case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ 2 * ↑d - 1"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "n✝ d✝ k✝ N : ℕ", "x : Fin n✝ → ℕ", "n d k : ℕ", "hk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card", "hn : n > 0", "hd : d > 0"], "goal": "1 ≤ d"}], "premise": [1674, 142600], "state_str": "case intro.intro.inr.inr.refine_3.hb\nα : Type u_1\nβ : Type u_2\nn✝ d✝ k✝ N : ℕ\nx : Fin n✝ → ℕ\nn d k : ℕ\nhk : ↑(d ^ n) / (↑(n * (d - 1) ^ 2) + 1) ≤ ↑(sphere n d k).card\nhn : n > 0\nhd : d > 0\n⊢ 1 ≤ d"}
+{"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", "H : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t"], "goal": "IsPreconnected s"}], "premise": [133383], "state_str": "α : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nH : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\n⊢ IsPreconnected s"}
+{"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", "H : ∀ x ∈ ∅, ∀ y ∈ ∅, ∃ t ⊆ ∅, x ∈ t ∧ y ∈ t ∧ IsPreconnected t"], "goal": "IsPreconnected ∅"}, {"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", "H : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t", "x : α", "hx : x ∈ s"], "goal": "IsPreconnected s"}], "premise": [65850, 65854], "state_str": "case inl\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nH : ∀ x ∈ ∅, ∀ y ∈ ∅, ∃ t ⊆ ∅, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\n⊢ IsPreconnected ∅\n\ncase inr.intro\nα : Type u\nβ : Type v\nι : Type u_1\nπ : ι → Type u_2\ninst✝ : TopologicalSpace α\ns✝ t u v s : Set α\nH : ∀ x ∈ s, ∀ y ∈ s, ∃ t ⊆ s, x ∈ t ∧ y ∈ t ∧ IsPreconnected t\nx : α\nhx : x ∈ s\n⊢ IsPreconnected s"}
+{"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' b₁ b₂ b₃ c₁ c₂ : R", "k : ℕ"], "goal": "a ^ Nat.mul 2 k = a ^ k * a ^ k"}], "premise": [3694, 119758], "state_str": "u✝ : Lean.Level\narg : Q(Type u✝)\nsα✝ : Q(CommSemiring «$arg»)\nu : Lean.Level\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\nR : Type u_1\ninst✝ : CommSemiring R\na a' a₁ a₂ a₃ b' b₁ b₂ b₃ c₁ c₂ : R\nk : ℕ\n⊢ a ^ Nat.mul 2 k = a ^ k * a ^ k"}
+{"state": [{"context": ["α✝ : Type u_1", "β : Type u_2", "n : ℕ", "α : Type u_3", "inst✝¹ : Fintype α", "k : ℕ", "inst✝ : Fintype (Sym α k)"], "goal": "Fintype.card (Sym α k) = (Fintype.card α + k - 1).choose k"}], "premise": [127837, 143925], "state_str": "α✝ : Type u_1\nβ : Type u_2\nn : ℕ\nα : Type u_3\ninst✝¹ : Fintype α\nk : ℕ\ninst✝ : Fintype (Sym α k)\n⊢ Fintype.card (Sym α k) = (Fintype.card α + k - 1).choose k"}
+{"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", "f : α → β", "l : Filter β"], "goal": "Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f atTop l"}], "premise": [1671, 1713, 15717, 16361], "state_str": "ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeSup α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f atTop l"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "κ : Kernel α β", "η : Kernel (α × β) γ", "a : α", "inst✝¹ : IsSFiniteKernel κ", "inst✝ : IsSFiniteKernel η", "t : Set (α × β)", "ht : MeasurableSet t", "c : ℝ≥0∞"], "goal": "Measurable fun a => ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a"}], "premise": [28863, 30372], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\na : α\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun a => ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "mα : MeasurableSpace α", "mβ : MeasurableSpace β", "mγ : MeasurableSpace γ", "κ : Kernel α β", "η : Kernel (α × β) γ", "a : α", "inst✝¹ : IsSFiniteKernel κ", "inst✝ : IsSFiniteKernel η", "t : Set (α × β)", "ht : MeasurableSet t", "c : ℝ≥0∞"], "goal": "Measurable fun x => c * (κ x) (Prod.mk x ⁻¹' t)"}], "premise": [31178, 73628], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\na : α\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun x => c * (κ x) (Prod.mk x ⁻¹' t)"}
+{"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", "t : Finset ι", "hw₀ : ∀ i ∈ t, w i ≤ 0", "hws : ∑ i ∈ t, w i < 0", "hz : ∀ i ∈ t, z i ∈ s"], "goal": "t.centerMass w z ∈ (convexHull R) s"}], "premise": [39909], "state_str": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\nt : Finset ι\nhw₀ : ∀ i ∈ t, w i ≤ 0\nhws : ∑ i ∈ t, w i < 0\nhz : ∀ i ∈ t, z i ∈ s\n⊢ t.centerMass w z ∈ (convexHull R) s"}
+{"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 : �� → E", "t : Finset ι", "hw₀ : ∀ i ∈ t, w i ≤ 0", "hws : ∑ i ∈ t, w i < 0", "hz : ∀ i ∈ t, z i ∈ s"], "goal": "t.centerMass (-w) z ∈ (convexHull R) s"}], "premise": [1674, 39925], "state_str": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nι : Type u_5\nι' : Type u_6\nα : Type u_7\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : LinearOrderedField R'\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt✝ : Finset ι\nw : ι → R\nz : ι → E\nt : Finset ι\nhw₀ : ∀ i ∈ t, w i ≤ 0\nhws : ∑ i ∈ t, w i < 0\nhz : ∀ i ∈ t, z i ∈ s\n⊢ t.centerMass (-w) z ∈ (convexHull R) s"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "l : Filter α", "inst✝¹ : CountableInterFilter l", "g : Set (Set α)", "f : Filter α", "inst✝ : CountableInterFilter f", "h : g ⊆ f.sets", "s : Set α", "S : Set (Set α)", "Sct : S.Countable", "a✝ : ∀ s ∈ S, CountableGenerateSets g s", "ih : ∀ s ∈ S, s ∈ f"], "goal": "⋂₀ S ∈ f"}], "premise": [1674, 12107], "state_str": "case mpr.sInter\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝¹ : CountableInterFilter l\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\nS : Set (Set α)\nSct : S.Countable\na✝ : ∀ s ∈ S, CountableGenerateSets g s\nih : ∀ s ∈ S, s ∈ f\n⊢ ⋂₀ S ∈ f"}
+{"state": [{"context": ["α : Type u", "β : α → Type v", "inst✝ : DecidableEq α", "a : α", "s : AList β"], "goal": "⟦AList.erase a s⟧.keys = ⟦s⟧.keys.erase a"}], "premise": [131309], "state_str": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\na : α\ns : AList β\n⊢ ⟦AList.erase a s⟧.keys = ⟦s⟧.keys.erase a"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "L : C ⥤ D", "R : D ⥤ C", "h : L ⊣ R", "X : C"], "goal": "R.map (L.map (R.map (h.counit.app (L.obj X)))) ≫ R.map (h.counit.app (L.obj X)) = R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))"}], "premise": [99919], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (R.map (h.counit.app (L.obj X)))) ≫ R.map (h.counit.app (L.obj X)) =\n    R.map (h.counit.app (L.obj (R.obj (L.obj X)))) ≫ R.map (h.counit.app (L.obj X))"}
+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "D : Type u₂", "inst✝ : Category.{v₂, u₂} D", "L : C ⥤ D", "R : D ⥤ C", "h : L ⊣ R", "X : C"], "goal": "R.map (L.map (h.unit.app X)) ≫ R.map (h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))"}], "premise": [99919], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\nX : C\n⊢ R.map (L.map (h.unit.app X)) ≫ R.map (h.counit.app (L.obj X)) = 𝟙 (R.obj (L.obj X))"}
+{"state": [{"context": ["p n✝ k n : ℕ"], "goal": "∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p = n.centralBinom"}], "premise": [143660], "state_str": "p n✝ k n : ℕ\n⊢ ∏ p ∈ Finset.range (2 * n + 1), p ^ n.centralBinom.factorization p = n.centralBinom"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "G : Type u_4", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "T : E →ₗ.[𝕜] F", "S : F →ₗ.[𝕜] E", "h : T.IsFormalAdjoint S", "y : ↥S.domain", "x✝ : ↥T.domain"], "goal": "⟪↑S y, ↑x✝⟫_𝕜 = ⟪↑y, ↑T x✝⟫_𝕜"}], "premise": [36750], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ.[𝕜] F\nS : F →ₗ.[𝕜] E\nh : T.IsFormalAdjoint S\ny : ↥S.domain\nx✝ : ↥T.domain\n⊢ ⟪↑S y, ↑x✝⟫_𝕜 = ⟪↑y, ↑T x✝⟫_𝕜"}
+{"state": [{"context": ["H : Type u", "H' : Type u_1", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝ : TopologicalSpace H", "c : ChartedSpaceCore H M", "e : PartialEquiv M H", "he : e ∈ c.atlas"], "goal": "IsOpen e.target"}], "premise": [71119, 133329, 133447], "state_str": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\n⊢ IsOpen e.target"}
+{"state": [{"context": ["H : Type u", "H' : Type u_1", "M : Type u_2", "M' : Type u_3", "M'' : Type u_4", "inst✝ : TopologicalSpace H", "c : ChartedSpaceCore H M", "e : PartialEquiv M H", "he : e ∈ c.atlas", "E : e.target ∩ ↑e.symm ⁻¹' e.source = e.target"], "goal": "IsOpen e.target"}], "premise": [67312, 71145], "state_str": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\nE : e.target ∩ ↑e.symm ⁻¹' e.source = e.target\n⊢ IsOpen e.target"}
+{"state": [{"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "R : ℝ≥0", "c : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c ↑R)", "hd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z", "hR : 0 < R", "w : ℂ", "hw : w ∈ EMetric.ball 0 ↑R"], "goal": "HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))"}], "premise": [42755, 42794, 42868, 143205, 146644], "state_str": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))"}
+{"state": [{"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "R : ℝ≥0", "c : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c ↑R)", "hd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z", "hR : 0 < R", "w : ℂ", "hw : w ∈ EMetric.ball 0 ↑R", "hw' : c + w ∈ ball c ↑R"], "goal": "HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))"}], "premise": [47127], "state_str": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) (f (c + w))"}
+{"state": [{"context": ["E : Type u", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℂ E", "inst✝ : CompleteSpace E", "R : ℝ≥0", "c : ℂ", "f : ℂ → E", "s : Set ℂ", "hs : s.Countable", "hc : ContinuousOn f (closedBall c ↑R)", "hd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z", "hR : 0 < R", "w : ℂ", "hw : w ∈ EMetric.ball 0 ↑R", "hw' : c + w ∈ ball c ↑R"], "goal": "HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w) ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)"}], "premise": [2115, 25820, 25847, 35854, 57321, 61182], "state_str": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ EMetric.ball 0 ↑R\nhw' : c + w ∈ ball c ↑R\n⊢ HasSum (fun n => (cauchyPowerSeries f c (↑R) n) fun x => w)\n    ((2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - (c + w))⁻¹ • f z)"}
+{"state": [{"context": ["R : Type u_1", "inst✝⁵ : CommRing R", "inst✝⁴ : TopologicalSpace R", "inst✝³ : TopologicalRing R", "A : Type u_2", "inst✝² : CommRing A", "inst✝¹ : TopologicalSpace A", "inst✝ : TopologicalRing A"], "goal": "IsAdic ⊥ ↔ DiscreteTopology A"}], "premise": [64461], "state_str": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : TopologicalSpace R\ninst✝³ : TopologicalRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : TopologicalRing A\n⊢ IsAdic ⊥ ↔ DiscreteTopology A"}
+{"state": [{"context": ["x y : ℝ", "f : ℝ → ℝ", "hf : ContinuousOn f (Icc x y)", "hxy : x < y", "hf'_mono : StrictMonoOn (deriv f) (Ioo x y)", "h : ∀ w ∈ Ioo x y, deriv f w ≠ 0"], "goal": "∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}], "premise": [44343, 46351], "state_str": "x y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}
+{"state": [{"context": ["x y : ℝ", "f : ℝ → ℝ", "hf : ContinuousOn f (Icc x y)", "hxy : x < y", "hf'_mono : StrictMonoOn (deriv f) (Ioo x y)", "h : ∀ w ∈ Ioo x y, deriv f w ≠ 0", "A : DifferentiableOn ℝ f (Ioo x y)"], "goal": "∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}], "premise": [46667], "state_str": "x y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}
+{"state": [{"context": ["x y : ℝ", "f : ℝ → ℝ", "hf : ContinuousOn f (Icc x y)", "hxy : x < y", "hf'_mono : StrictMonoOn (deriv f) (Ioo x y)", "h : ∀ w ∈ Ioo x y, deriv f w ≠ 0", "A : DifferentiableOn ℝ f (Ioo x y)", "a : ℝ", "ha : deriv f a = (f y - f x) / (y - x)", "hxa : x < a", "hay : a < y"], "goal": "∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}], "premise": [1674, 20188], "state_str": "case intro.intro.intro\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nh : ∀ w ∈ Ioo x y, deriv f w ≠ 0\nA : DifferentiableOn ℝ f (Ioo x y)\na : ℝ\nha : deriv f a = (f y - f x) / (y - x)\nhxa : x < a\nhay : a < y\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"}
+{"state": [{"context": [], "goal": "sin π = 0"}], "premise": [38509, 108574, 113025, 115817, 122222, 149221], "state_str": "⊢ sin π = 0"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : Sym2 α", "f : α → β", "inst✝ : DecidableEq α", "s : Finset α", "x✝ y✝ : α"], "goal": "s(x✝, y✝) ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ s(x✝, y✝) ∈ image Sym2.mk s.offDiag"}], "premise": [136829, 136870, 137297, 137413, 139089], "state_str": "case a.h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\ninst✝ : DecidableEq α\ns : Finset α\nx✝ y✝ : α\n⊢ s(x✝, y✝) ∈ filter (fun a => ¬a.IsDiag) (image Sym2.mk (s ×ˢ s)) ↔ s(x✝, y✝) ∈ image Sym2.mk s.offDiag"}
+{"state": [{"context": ["M : Type u", "inst✝ : Monoid M", "X x✝¹ x✝ : Discrete M", "f : x✝¹ ⟶ x✝"], "goal": "{ as := X.as * x✝¹.as } = { as := X.as * x✝.as }"}], "premise": [92500], "state_str": "M : Type u\ninst✝ : Monoid M\nX x✝¹ x✝ : Discrete M\nf : x✝¹ ⟶ x✝\n⊢ { as := X.as * x✝¹.as } = { as := X.as * x✝.as }"}
+{"state": [{"context": ["M : Type u", "inst✝ : Monoid M", "X₁✝ X₂✝ : Discrete M", "f : X₁✝ ⟶ X₂✝", "X : Discrete M"], "goal": "{ as := X₁✝.as * X.as } = { as := X₂✝.as * X.as }"}], "premise": [92500], "state_str": "M : Type u\ninst✝ : Monoid M\nX₁✝ X₂✝ : Discrete M\nf : X₁✝ ⟶ X₂✝\nX : Discrete M\n⊢ { as := X₁✝.as * X.as } = { as := X₂✝.as * X.as }"}
+{"state": [{"context": ["M : Type u", "inst✝ : Monoid M", "X₁✝ Y₁✝ X₂✝ Y₂✝ : Discrete M", "f : X₁✝ ⟶ Y₁✝", "g : X₂✝ ⟶ Y₂✝"], "goal": "{ as := X₁✝.as * X₂✝.as } = { as := Y₁✝.as * Y₂✝.as }"}], "premise": [92500], "state_str": "M : Type u\ninst✝ : Monoid M\nX₁✝ Y₁✝ X₂✝ Y₂✝ : Discrete M\nf : X₁✝ ⟶ Y₁✝\ng : X₂✝ ⟶ Y₂✝\n⊢ { as := X₁✝.as * X₂✝.as } = { as := Y₁✝.as * Y₂✝.as }"}
+{"state": [{"context": ["A : Type u_1", "inst✝⁹ : NonUnitalNormedRing A", "inst✝⁸ : CompleteSpace A", "inst✝⁷ : PartialOrder A", "inst✝⁶ : StarRing A", "inst✝⁵ : StarOrderedRing A", "inst✝⁴ : CstarRing A", "inst✝³ : NormedSpace ℂ A", "inst✝² : StarModule ℂ A", "inst✝¹ : SMulCommClass ℂ A A", "inst✝ : IsScalarTower ℂ A A", "a b : A", "hb : autoParam (IsSelfAdjoint b) _auto✝", "h₁ : a * b * star a = star (star a) * b * star a", "h₂ : a * star a = star (star a) * star a"], "goal": "star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)"}], "premise": [34937], "state_str": "A : Type u_1\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : CompleteSpace A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : CstarRing A\ninst✝³ : NormedSpace ℂ A\ninst✝² : StarModule ℂ A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : IsScalarTower ℂ A A\na b : A\nhb : autoParam (IsSelfAdjoint b) _auto✝\nh₁ : a * b * star a = star (star a) * b * star a\nh₂ : a * star a = star (star a) * star a\n⊢ star (star a) * b * star a ≤ ‖b‖ • (star (star a) * star a)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : OrderedCancelCommMonoid α", "s t : Set α", "a✝ : α", "x y : ↑(s.mulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : α"], "goal": "(s.mulAntidiagonal t a).Finite"}], "premise": [1673, 134999], "state_str": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\n⊢ (s.mulAntidiagonal t a).Finite"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : OrderedCancelCommMonoid α", "s t : Set α", "a✝ : α", "x y : ↑(s.mulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : α", "h : (s.mulAntidiagonal t a).Infinite", "h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)"], "goal": "False"}], "premise": [2115, 13079], "state_str": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\n⊢ False"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : OrderedCancelCommMonoid α", "s t : Set α", "a✝ : α", "x y : ↑(s.mulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : α", "h : (s.mulAntidiagonal t a).Infinite", "h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻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.mulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))"], "goal": "False"}], "premise": [2115], "state_str": "case intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : OrderedCancelCommMonoid α", "s t : Set α", "a✝ : α", "x y : ↑(s.mulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : α", "h : (s.mulAntidiagonal t a).Infinite", "h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻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.mulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.mulAntidiagonal 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.mulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))"], "goal": "False"}], "premise": [14111, 70665], "state_str": "case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ False"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : OrderedCancelCommMonoid α", "s t : Set α", "a✝ : α", "x y : ↑(s.mulAntidiagonal t a✝)", "hs : s.IsPWO", "ht : t.IsPWO", "a : α", "h : (s.mulAntidiagonal t a).Infinite", "h1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)", "h2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻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.mulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.mulAntidiagonal 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.mulAntidiagonal t a) h) (g m)) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))"], "goal": "(Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)"}], "premise": [131573], "state_str": "case intro.intro.intro.intro\nα : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\na✝ : α\nx y : ↑(s.mulAntidiagonal t a✝)\nhs : s.IsPWO\nht : t.IsPWO\na : α\nh : (s.mulAntidiagonal t a).Infinite\nh1 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1)\nh2 : (s.mulAntidiagonal t a).PartiallyWellOrderedOn (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1)\ng : ℕ ↪o ℕ\nhg :\n  ∀ (m n : ℕ),\n    m ≤ n →\n      (Prod.fst ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n        ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\nm n : ℕ\nmn : m < n\nh2' :\n  (Prod.snd ⁻¹'o fun x x_1 => x ≤ x_1) ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m))\n    ↑((Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n))\n⊢ (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g m) = (Infinite.natEmbedding (s.mulAntidiagonal t a) h) (g n)"}
+{"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", "ι : Type u_4", "a : ι → Set α", "l : Filter ι", "f : α → E", "c : E", "g : ι → α → ℝ", "K : ℝ", "hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)", "f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ", "hg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)", "g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i", "g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal"], "goal": "Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}], "premise": [1201, 1673, 2107, 11212, 15889, 33635, 53688, 55915, 101702, 131585], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}
+{"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", "ι : Type u_4", "a : ι → Set α", "l : Filter ι", "f : α → E", "c : E", "g : ι → α → ℝ", "K : ℝ", "hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)", "f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ", "hg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)", "g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i", "g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal", "g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ", "I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ", "L0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)"], "goal": "Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}], "premise": [62551, 65039], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : ��), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}
+{"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", "ι : Type u_4", "a : ι → Set α", "l : Filter ι", "f : α → E", "c : E", "g : ι → α → ℝ", "K : ℝ", "hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)", "f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ", "hg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)", "g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i", "g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal", "g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ", "I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ", "L0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)", "this : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 (0 + 1 • c))"], "goal": "Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}], "premise": [118910, 119727], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 (0 + 1 • c))\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}
+{"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", "ι : Type u_4", "a : ι → Set α", "l : Filter ι", "f : α → E", "c : E", "g : ι → α → ℝ", "K : ℝ", "hf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)", "f_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ", "hg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)", "g_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i", "g_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal", "g_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ", "I : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ", "L0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)", "this : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂μ) • c) l (𝓝 c)"], "goal": "Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}], "premise": [16350], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nm0 : MeasurableSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nμ ν : Measure α\ns t : Set α\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i => ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) in l, IntegrableOn f (a i) μ\nhg : Tendsto (fun i => ∫ (y : α), g i y ∂μ) l (𝓝 1)\ng_supp : ∀ᶠ (i : ι) in l, support (g i) ⊆ a i\ng_bound : ∀ᶠ (i : ι) in l, ∀ (x : α), |g i x| ≤ K / (μ (a i)).toReal\ng_int : ∀ᶠ (i : ι) in l, Integrable (g i) μ\nI : ∀ᶠ (i : ι) in l, ∫ (y : α), g i y • (f y - c) ∂μ + (∫ (y : α), g i y ∂μ) • c = ∫ (y : α), g i y • f y ∂μ\nL0 : Tendsto (fun i => ∫ (y : α), g i y • (f y - c) ∂μ) l (𝓝 0)\nthis : Tendsto (fun x => ∫ (y : α), g x y • (f y - c) ∂μ + (∫ (y : α), g x y ∂��) • c) l (𝓝 c)\n⊢ Tendsto (fun i => ∫ (y : α), g i y • f y ∂μ) l (𝓝 c)"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "L : C ⥤ D", "R : D ⥤ C", "h : L ⊣ R", "inst✝ : ∀ (X : D), Epi (h.counit.app X)", "X Y : D", "f g : X ⟶ Y", "hfg : R.map f = R.map g"], "goal": "f = g"}], "premise": [96188], "state_str": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ f = g"}
+{"state": [{"context": ["C : Type u₁", "inst✝² : Category.{v₁, u₁} C", "D : Type u₂", "inst✝¹ : Category.{v₂, u₂} D", "L : C ⥤ D", "R : D ⥤ C", "h : L ⊣ R", "inst✝ : ∀ (X : D), Epi (h.counit.app X)", "X Y : D", "f g : X ⟶ Y", "hfg : R.map f = R.map g"], "goal": "h.counit.app X ≫ f = h.counit.app X ≫ g"}], "premise": [70736], "state_str": "case a\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\nh : L ⊣ R\ninst✝ : ∀ (X : D), Epi (h.counit.app X)\nX Y : D\nf g : X ⟶ Y\nhfg : R.map f = R.map g\n⊢ h.counit.app X ≫ f = h.counit.app X ≫ g"}
+{"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 x", "n : ℕ"], "goal": "IsPeriodicPt f (n * m) x"}], "premise": [88938, 119707], "state_str": "α : Type u_1\nβ : Type u_2\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhm : IsPeriodicPt f m x\nn : ℕ\n⊢ IsPeriodicPt f (n * m) 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✝ s s' t : Set α", "ht : MeasurableSet t"], "goal": "(μ.restrict (s ∪ s')) t ≤ (μ.restrict s + μ.restrict s') t"}], "premise": [27566, 133470], "state_str": "R : Type u_1\nα : Type u_2\nβ : Type u_3\nδ : Type u_4\nγ : Type u_5\nι : Type u_6\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s'✝ t✝ s s' t : Set α\nht : MeasurableSet t\n⊢ (μ.restrict (s ∪ s')) t ≤ (μ.restrict s + μ.restrict s') t"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x"], "goal": "HasSum (fun n => (↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) (oddKernel (↑a) x)"}], "premise": [22972, 46266], "state_str": "a x : ℝ\nhx : 0 < x\n⊢ HasSum (fun n => (↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) (oddKernel (↑a) x)"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x"], "goal": "HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x))) (cexp (-↑π * ↑a ^ 2 * ↑x) * (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}], "premise": [22216, 148280, 148320], "state_str": "a x : ℝ\nhx : 0 < x\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x", "h1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))"], "goal": "HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x))) (cexp (-↑π * ↑a ^ 2 * ↑x) * (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}], "premise": [22218, 148280, 148320], "state_str": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x", "h1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))", "h2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))"], "goal": "HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x))) (cexp (-↑π * ↑a ^ 2 * ↑x) * (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}], "premise": [63042, 63050, 64063, 64125], "state_str": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\n⊢ HasSum (fun x_1 => ↑((↑x_1 + a) * rexp (-π * (↑x_1 + a) ^ 2 * x)))\n    (cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂ (↑a * I * ↑x) (I * ↑x)))"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x", "h1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))", "h2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))", "n : ℤ"], "goal": "↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) = cexp (-↑π * ↑a ^ 2 * ↑x) * (jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x))"}], "premise": [39324, 108574, 117739, 119703, 149081], "state_str": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ ↑((↑n + a) * rexp (-π * (↑n + a) ^ 2 * x)) =\n    cexp (-↑π * ↑a ^ 2 * ↑x) *\n      (jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x) / (2 * ↑π * I) + ↑a * jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x))"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x", "h1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))", "h2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))", "n : ℤ"], "goal": "(↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) = (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))"}], "premise": [119703], "state_str": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ (↑n + ↑a) * cexp (-↑π * (↑n + ↑a) ^ 2 * ↑x) =\n    (↑n + ↑a) * cexp (-↑π * ↑a ^ 2 * ↑x + (2 * ↑π * I * ↑n * (↑a * I * ↑x) + ↑π * I * ↑n ^ 2 * (I * ↑x)))"}
+{"state": [{"context": ["a x : ℝ", "hx : 0 < x", "h1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))", "h2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))", "n : ℤ"], "goal": "-↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I"}], "premise": [117740, 119703, 148312], "state_str": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n => jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n => jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2) * I"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : NormedLinearOrderedField 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : StrictConvexSpace ℝ E", "x y z : E", "a b r : ℝ", "h : ¬SameRay ℝ x y"], "goal": "‖x + y‖ < ‖x‖ + ‖y‖"}], "premise": [1999, 41483], "state_str": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nh : ¬SameRay ℝ x y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : NormedLinearOrderedField 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : StrictConvexSpace ℝ E", "x y z : E", "a b r : ℝ", "hx : x ≠ 0", "hy : y ≠ 0", "hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y"], "goal": "‖x + y‖ < ‖x‖ + ‖y‖"}], "premise": [42922], "state_str": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : NormedLinearOrderedField 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : StrictConvexSpace ℝ E", "x y z : E", "a b r : ℝ", "hx : 0 < ‖x‖", "hy : 0 < ‖y‖", "hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y", "hxy : 0 < ‖x‖ + ‖y‖"], "goal": "‖x + y‖ < ‖x‖ + ‖y‖"}], "premise": [11234, 36593, 40547, 104338, 108408, 115817], "state_str": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝⁴ : NormedLinearOrderedField 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : StrictConvexSpace ℝ E", "x y z : E", "a b r : ℝ", "hx : 0 < ‖x‖", "hy : 0 < ‖y‖", "hne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y", "hxy : 0 < ‖x‖ + ‖y‖", "this : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1"], "goal": "‖x + y‖ < ‖x‖ + ‖y‖"}], "premise": [1674, 7383, 11234, 41371, 42718, 42904, 104330, 106089, 108334, 117883, 118866], "state_str": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedLinearOrderedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y z : E\na b r : ℝ\nhx : 0 < ‖x‖\nhy : 0 < ‖y‖\nhne : ‖x‖⁻¹ • x ≠ ‖y‖⁻¹ • y\nhxy : 0 < ‖x‖ + ‖y‖\nthis : (‖x‖ / (‖x‖ + ‖y‖)) • ‖x‖⁻¹ • x + (‖y‖ / (‖x‖ + ‖y‖)) • ‖y‖⁻¹ • y ∈ ball 0 1\n⊢ ‖x + y‖ < ‖x‖ + ‖y‖"}
+{"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 t : Set σ"], "goal": "supported R Set.univ = ⊤"}], "premise": [110157, 122122], "state_str": "σ : Type u_1\nτ : Type u_2\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ supported R Set.univ = ⊤"}
+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "ι : Type u_3", "ι' : Sort u_4", "inst✝ : TopologicalSpace X", "S : Set (Set X)", "h : ∀ s ∈ S, IsGδ s", "hS : S.Countable"], "goal": "IsGδ (⋂₀ S)"}], "premise": [55723, 135450], "state_str": "X : Type u_1\nY : Type u_2\nι : Type u_3\nι' : Sort u_4\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nh : ∀ s ∈ S, IsGδ s\nhS : S.Countable\n⊢ IsGδ (⋂₀ S)"}
+{"state": [{"context": ["x : ℝ", "n : ℕ∞"], "goal": "ContDiffOn ℝ n log {0}ᶜ"}], "premise": [18778, 48350], "state_str": "x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ n log {0}ᶜ"}
+{"state": [{"context": ["x : ℝ", "n : ℕ∞"], "goal": "ContDiffOn ℝ ⊤ log {0}ᶜ"}], "premise": [1674, 51758, 64692], "state_str": "x : ℝ\nn : ℕ∞\n⊢ ContDiffOn ℝ ⊤ log {0}ᶜ"}
+{"state": [{"context": ["x : ℝ", "n : ℕ∞"], "goal": "DifferentiableOn ℝ log {0}ᶜ ∧ ContDiffOn ℝ ⊤ (deriv log) {0}ᶜ"}], "premise": [38091, 51735], "state_str": "x : ℝ\nn : ℕ∞\n⊢ DifferentiableOn ℝ log {0}ᶜ ∧ ContDiffOn ℝ ⊤ (deriv log) {0}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'"], "goal": "I.incl.idealRange = I"}], "premise": [107978, 109255, 109267, 109329, 109375, 109404], "state_str": "R : Type u\nL : Type v\nL' : Type w₂\nM : Type w\nM' : Type w₁\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : LieRing L'\ninst✝⁸ : LieAlgebra R L'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n⊢ I.incl.idealRange = I"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : NonUnitalRing α", "a b c : α", "h : a ∣ c"], "goal": "a ∣ b - c ↔ a ∣ b"}], "premise": [1674, 119789, 121990, 121993], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ c\n⊢ a ∣ b - c ↔ a ∣ b"}
+{"state": [{"context": ["α : Type u", "inst✝ : Semiring α", "n : ℕ"], "goal": "∑ i ∈ range (n + 2), 0 ^ i = if n + 2 = 0 then 0 else 1"}], "premise": [112468], "state_str": "α : Type u\ninst✝ : Semiring α\nn : ℕ\n⊢ ∑ i ∈ range (n + 2), 0 ^ i = if n + 2 = 0 then 0 else 1"}
+{"state": [{"context": [], "goal": "sin ↑π = 0"}], "premise": [38522, 149166], "state_str": "⊢ sin ↑π = 0"}
+{"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 => c - f y) s x = -derivWithin f s x"}], "premise": [45887], "state_str": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL : Filter 𝕜\nhxs : UniqueDiffWithinAt 𝕜 s x\nc : F\n⊢ derivWithin (fun y => c - f y) s x = -derivWithin f s x"}
+{"state": [{"context": ["R : Type u", "R' : Type u'", "M M₁ : Type v", "M' : Type v'", "inst✝⁹ : Ring R", "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 : Module R' M'", "inst✝ : Module R' M₁", "f : M →ₗ[R] M₁", "h : Surjective ⇑f"], "goal": "Module.rank R M₁ ≤ Module.rank R M"}], "premise": [85747], "state_str": "R : Type u\nR' : Type u'\nM M₁ : Type v\nM' : Type v'\ninst✝⁹ : Ring R\ninst✝⁸ : Ring R'\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Module R' M'\ninst✝ : Module R' M₁\nf : M →ₗ[R] M₁\nh : Surjective ⇑f\n⊢ Module.rank R M₁ ≤ Module.rank R M"}
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+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X : Type u"], "goal": "{ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X) = 𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)"}], "premise": [124477], "state_str": "R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X) =\n    𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X : Type u"], "goal": "⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (𝟙 X)) ∘ FreeAlgebra.ι R = ⇑(𝟙 ({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj X)) ∘ FreeAlgebra.ι R"}], "premise": [124473], "state_str": "case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX : Type u\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (𝟙 X)) ∘\n      FreeAlgebra.ι R =\n    ⇑(𝟙\n          ({ obj := fun S => mk (FreeAlgebra R S),\n                map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.obj\n            X)) ∘\n      FreeAlgebra.ι R"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝"], "goal": "{ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (f✝ ≫ g✝) = { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝"}], "premise": [124477], "state_str": "R : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n      (f✝ ≫ g✝) =\n    { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫\n      { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝"], "goal": "⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map (f✝ ≫ g✝)) ∘ FreeAlgebra.ι R = ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map f✝ ≫ { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map g✝) ∘ FreeAlgebra.ι R"}], "premise": [124473], "state_str": "case w\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\n⊢ ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n          (f✝ ≫ g✝)) ∘\n      FreeAlgebra.ι R =\n    ⇑({ obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            f✝ ≫\n          { obj := fun S => mk (FreeAlgebra R S), map := fun {X Y} f => (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f) }.map\n            g✝) ∘\n      FreeAlgebra.ι R"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "(FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ = (⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R) x✝"}], "premise": [99591], "state_str": "case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    (⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝) ≫ (FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ FreeAlgebra.ι R) x✝"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "(FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ = ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘ FreeAlgebra.ι R) x✝"}], "premise": [1670, 91291, 99591], "state_str": "case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ (FreeAlgebra.ι R ∘ (f✝ ≫ g✝)) x✝ =\n    ((⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) ∘ ⇑((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝))) ∘\n        FreeAlgebra.ι R)\n      x✝"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "M N U : ModuleCat R", "X✝ Y✝ Z✝ : Type u", "f✝ : X✝ ⟶ Y✝", "g✝ : Y✝ ⟶ Z✝", "x✝ : X✝"], "goal": "FreeAlgebra.ι R (g✝ (f✝ x✝)) = ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))"}], "premise": [124474], "state_str": "case w.h\nR : Type u\ninst✝ : CommRing R\nM N U : ModuleCat R\nX✝ Y✝ Z✝ : Type u\nf✝ : X✝ ⟶ Y✝\ng✝ : Y✝ ⟶ Z✝\nx✝ : X✝\n⊢ FreeAlgebra.ι R (g✝ (f✝ x✝)) =\n    ((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ g✝)) (((FreeAlgebra.lift R) (FreeAlgebra.ι R ∘ f✝)) (FreeAlgebra.ι R x✝))"}
+{"state": [{"context": ["m n✝ n : ℤ"], "goal": "Even (n * (n + 1))"}], "premise": [117547, 121611], "state_str": "m n✝ n : ℤ\n⊢ Even (n * (n + 1))"}
+{"state": [{"context": ["R : Type u", "X : Type v", "inst✝¹ : CommRing R", "inst✝ : Nontrivial R"], "goal": "Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u, v} (Cardinal.mk (List X))"}], "premise": [2101, 48581, 48583, 85905], "state_str": "R : Type u\nX : Type v\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ Module.rank R (FreeAlgebra R X) = Cardinal.lift.{u, v} (Cardinal.mk (List X))"}
+{"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 : α → β", "Kf : ℝ≥0", "hf : LipschitzWith Kf f", "g : α → γ", "Kg : ℝ≥0", "hg : LipschitzWith Kg g", "x y : α"], "goal": "edist ((fun x => (f x, g x)) x) ((fun x => (f x, g x)) y) ≤ ↑(max Kf Kg) * edist x y"}], "premise": [13001, 58465, 143206, 143480], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ edist ((fun x => (f x, g x)) x) ((fun x => (f x, g x)) y) ≤ ↑(max Kf Kg) * edist x y"}
+{"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 : α → β", "Kf : ℝ≥0", "hf : LipschitzWith Kf f", "g : α → γ", "Kg : ℝ≥0", "hg : LipschitzWith Kg g", "x y : α"], "goal": "max (edist ((fun x => (f x, g x)) x).1 ((fun x => (f x, g x)) y).1) (edist ((fun x => (f x, g x)) x).2 ((fun x => (f x, g x)) y).2) ≤ max (↑Kf * edist x y) (↑Kg * edist x y)"}], "premise": [12960], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf✝ : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ max (edist ((fun x => (f x, g x)) x).1 ((fun x => (f x, g x)) y).1)\n      (edist ((fun x => (f x, g x)) x).2 ((fun x => (f x, g x)) y).2) ≤\n    max (↑Kf * edist x y) (↑Kg * edist x y)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : Group α", "s : Subgroup α"], "goal": "∀ (a b : α), Setoid.r a b = (b * a⁻¹ ∈ s)"}], "premise": [1792], "state_str": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b = (b * a⁻¹ ∈ s)"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : Group α", "s : Subgroup α"], "goal": "∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s"}], "premise": [6901], "state_str": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s"}
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+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝ : DivisionMonoid α", "s t : Set α"], "goal": "univ / univ = univ"}], "premise": [119790], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝ : DivisionMonoid α\ns t : Set α\n⊢ univ / univ = univ"}
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+{"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]", "hp : p ≠ 0", "s : R", "h : p.scaleRoots s = 0"], "goal": "False"}], "premise": [1673, 1681, 102195], "state_str": "R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\n⊢ False"}
+{"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]", "hp : p ≠ 0", "s : R", "h : p.scaleRoots s = 0", "this✝ : p.coeff p.natDegree ≠ 0", "this : (p.scaleRoots s).coeff p.natDegree = 0"], "goal": "False"}], "premise": [74553], "state_str": "R : Type u_1\nS : Type u_2\nA : Type u_3\nK : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nhp : p ≠ 0\ns : R\nh : p.scaleRoots s = 0\nthis✝ : p.coeff p.natDegree ≠ 0\nthis : (p.scaleRoots s).coeff p.natDegree = 0\n⊢ False"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "ι : Type u_1", "s : ι → C", "inst✝⁴ : Preadditive C", "inst✝³ : HasFiniteBiproducts C", "o : HomOrthogonal s", "α β γ : Type", "inst✝² : Finite α", "inst✝¹ : Fintype β", "inst✝ : Finite γ", "f : α → ι", "g : β → ι", "h : γ → ι", "z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)", "w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)", "c : γ", "a : α", "j_property : a ∈ f ⁻¹' {h c}"], "goal": "o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ = (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩"}], "premise": [131591, 133512], "state_str": "case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property : a ∈ f ⁻¹' {h c}\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property⟩"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "ι : Type u_1", "s : ι → C", "inst✝⁴ : Preadditive C", "inst✝³ : HasFiniteBiproducts C", "o : HomOrthogonal s", "α β γ : Type", "inst✝² : Finite α", "inst✝¹ : Fintype β", "inst✝ : Finite γ", "f : α → ι", "g : β → ι", "h : γ → ι", "z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)", "w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)", "c : γ", "a : α", "j_property✝ : a ∈ f ⁻¹' {h c}", "j_property : f a = h c"], "goal": "o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ = (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩"}], "premise": [91581, 96173, 96174, 96175, 97613, 97736, 126920, 142237], "state_str": "case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ o.matrixDecomposition (z ≫ w) (h c) ⟨c, ⋯⟩ ⟨a, j_property✝⟩ =\n    (o.matrixDecomposition w (h c) * o.matrixDecomposition z (h c)) ⟨c, ⋯⟩ ⟨a, j_property✝⟩"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "ι : Type u_1", "s : ι → C", "inst✝⁴ : Preadditive C", "inst✝³ : HasFiniteBiproducts C", "o : HomOrthogonal s", "α β γ : Type", "inst✝² : Finite α", "inst✝¹ : Fintype β", "inst✝ : Finite γ", "f : α → ι", "g : β → ι", "h : γ → ι", "z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)", "w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)", "c : γ", "a : α", "j_property✝ : a ∈ f ⁻¹' {h c}", "j_property : f a = h c"], "goal": "eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c = ∑ x : ↑(g ⁻¹' {h c}), eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}], "premise": [92735, 96175], "state_str": "case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ w ≫ biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "ι : Type u_1", "s : ι → C", "inst✝⁴ : Preadditive C", "inst✝³ : HasFiniteBiproducts C", "o : HomOrthogonal s", "α β γ : Type", "inst✝² : Finite α", "inst✝¹ : Fintype β", "inst✝ : Finite γ", "f : α → ι", "g : β → ι", "h : γ → ι", "z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)", "w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)", "c : γ", "a : α", "j_property✝ : a ∈ f ⁻¹' {h c}", "j_property : f a = h c"], "goal": "eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫ biproduct.π (fun b => s (h b)) c = ∑ x : ↑(g ⁻¹' {h c}), eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}], "premise": [91609, 91610], "state_str": "case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ eqToHom ⋯ ≫\n      biproduct.ι (fun a => s (f a)) a ≫\n        z ≫\n          ((∑ j : β, biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n            biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}
+{"state": [{"context": ["C : Type u", "inst✝⁵ : Category.{v, u} C", "ι : Type u_1", "s : ι → C", "inst✝⁴ : Preadditive C", "inst✝³ : HasFiniteBiproducts C", "o : HomOrthogonal s", "α β γ : Type", "inst✝² : Finite α", "inst✝¹ : Fintype β", "inst✝ : Finite γ", "f : α → ι", "g : β → ι", "h : γ → ι", "z : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)", "w : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)", "c : γ", "a : α", "j_property✝ : a ∈ f ⁻¹' {h c}", "j_property : f a = h c"], "goal": "∑ j : β, eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫ biproduct.π (fun b => s (h b)) c = ∑ x : ↑(g ⁻¹' {h c}), eqToHom ⋯ ≫ biproduct.ι (fun a => s (f a)) a ≫ z ≫ biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}], "premise": [127066], "state_str": "case a.mk.refl.mk\nC : Type u\ninst✝⁵ : Category.{v, u} C\nι : Type u_1\ns : ι → C\ninst✝⁴ : Preadditive C\ninst✝³ : HasFiniteBiproducts C\no : HomOrthogonal s\nα β γ : Type\ninst✝² : Finite α\ninst✝¹ : Fintype β\ninst✝ : Finite γ\nf : α → ι\ng : β → ι\nh : γ → ι\nz : (⨁ fun a => s (f a)) ⟶ ⨁ fun b => s (g b)\nw : (⨁ fun b => s (g b)) ⟶ ⨁ fun c => s (h c)\nc : γ\na : α\nj_property✝ : a ∈ f ⁻¹' {h c}\nj_property : f a = h c\n⊢ ∑ j : β,\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            ((biproduct.π (fun b => s (g b)) j ≫ biproduct.ι (fun b => s (g b)) j) ≫ w) ≫\n              biproduct.π (fun b => s (h b)) c =\n    ∑ x : ↑(g ⁻¹' {h c}),\n      eqToHom ⋯ ≫\n        biproduct.ι (fun a => s (f a)) a ≫\n          z ≫\n            biproduct.π (fun b => s (g b)) ↑x ≫ biproduct.ι (fun a => s (g a)) ↑x ≫ w ≫ biproduct.π (fun b => s (h b)) c"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁵ : MeasurableSpace G", "inst✝⁴ : Group G", "inst✝³ : MeasurableMul₂ G", "μ ν : Measure G", "inst✝² : SFinite ν", "inst✝¹ : SFinite μ", "s : Set G", "inst✝ : μ.IsMulRightInvariant"], "goal": "MeasurePreserving ?m.119417 ?m.119413 (?m.119343.prod ?m.119344)"}], "premise": [32010], "state_str": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ns : Set G\ninst✝ : μ.IsMulRightInvariant\n⊢ MeasurePreserving ?m.119417 ?m.119413 (?m.119343.prod ?m.119344)"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "κ : ι → Sort u_5", "r : α → β → Prop", "s s₁ s₂ : Set α", "t t₁ t₂ : Set β", "c d : Concept α β r"], "goal": "c.toProd.2 ⊂ d.toProd.2 ↔ d < c"}], "premise": [1713, 12387, 14274, 19976], "state_str": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nκ : ι → Sort u_5\nr : α → β → Prop\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\nc d : Concept α β r\n⊢ c.toProd.2 ⊂ d.toProd.2 ↔ d < c"}
+{"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 : DistribLattice α", "inst✝ : OrderBot α", "s✝ : Finset ι", "t : Finset κ", "f✝ : ι → α", "g : κ → α", "a✝ : α", "s : Finset ι", "f : ι → α", "a : α"], "goal": "a ⊓ s.sup f = s.sup fun i => a ⊓ f i"}], "premise": [14615, 18841, 138844, 139650, 139651], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns✝ : Finset ι\nt : Finset κ\nf✝ : ι → α\ng : κ → α\na✝ : α\ns : Finset ι\nf : ι → α\na : α\n⊢ a ⊓ s.sup f = s.sup fun i => a ⊓ f i"}
+{"state": [{"context": ["A : Type u_1", "inst✝³ : AddCommGroup A", "inst✝² : Module ℂ A", "inst✝¹ : StarAddMonoid A", "inst✝ : StarModule ℂ A", "a : ↥(skewAdjoint A)"], "goal": "I • ↑(negISMul a) = ↑a"}], "premise": [108340, 110032, 118909, 118910, 119769, 148312, 148803], "state_str": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ I • ↑(negISMul a) = ↑a"}
+{"state": [{"context": ["α : Type u", "β : Type v", "inst✝ : Finite α", "s : Set α", "t : Set β", "h1 : lift.{max v w, u} #α = lift.{max u w, v} #β", "h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t"], "goal": "lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}], "premise": [141384], "state_str": "α : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}
+{"state": [{"context": ["α : Type u", "β : Type v", "inst✝ : Finite α", "s : Set α", "t : Set β", "h1 : lift.{max v w, u} #α = lift.{max u w, v} #β", "h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t", "val✝ : Fintype α"], "goal": "lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}], "premise": [1673, 48597], "state_str": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}
+{"state": [{"context": ["α : Type u", "β : Type v", "inst✝ : Finite α", "s : Set α", "t : Set β", "h1 : lift.{max v w, u} #α = lift.{max u w, v} #β", "h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t", "val✝ : Fintype α", "e : α ≃ β", "this : Fintype β := Fintype.ofEquiv α e"], "goal": "lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}], "premise": [2100, 141343], "state_str": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh1 : lift.{max v w, u} #α = lift.{max u w, v} #β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}
+{"state": [{"context": ["α : Type u", "β : Type v", "inst✝ : Finite α", "s : Set α", "t : Set β", "h2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t", "val✝ : Fintype α", "e : α ≃ β", "this : Fintype β := Fintype.ofEquiv α e", "h1 : Fintype.card α = Fintype.card β"], "goal": "lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}], "premise": [48619, 48725, 48746, 111254, 134992, 138678, 140846, 141359, 141428], "state_str": "case intro.intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nt : Set β\nh2 : lift.{max v w, u} #↑s = lift.{max u w, v} #↑t\nval✝ : Fintype α\ne : α ≃ β\nthis : Fintype β := Fintype.ofEquiv α e\nh1 : Fintype.card α = Fintype.card β\n⊢ lift.{max v w, u} #↑sᶜ = lift.{max u w, v} #↑tᶜ"}
+{"state": [{"context": ["i j : Fin 2"], "goal": "(LinearMap.toMatrix basisOneI basisOneI) conjAe.toLinearMap i j = !![1, 0; 0, -1] i j"}], "premise": [87070], "state_str": "case a\ni j : Fin 2\n⊢ (LinearMap.toMatrix basisOneI basisOneI) conjAe.toLinearMap i j = !![1, 0; 0, -1] i j"}
+{"state": [{"context": [], "goal": "size ≤ 2 ^ 64"}], "premise": [302], "state_str": "⊢ size ≤ 2 ^ 64"}
+{"state": [{"context": [], "goal": "2 ^ System.Platform.numBits ≤ 2 ^ 64"}], "premise": [3863], "state_str": "⊢ 2 ^ System.Platform.numBits ≤ 2 ^ 64"}
+{"state": [{"context": [], "goal": "System.Platform.numBits ≤ 64"}], "premise": [2159], "state_str": "⊢ System.Platform.numBits ≤ 64"}
+{"state": [{"context": [], "goal": "Tendsto (fun s => (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1)"}], "premise": [23503], "state_str": "⊢ Tendsto (fun s => (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1)"}
+{"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.natDegree ≤ Fintype.card n"}], "premise": [86438], "state_str": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (X • A.map ⇑C + B.map ⇑C).det.natDegree ≤ Fintype.card 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": "(∑ σ : Equiv.Perm n, sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i).natDegree ≤ Fintype.card n"}], "premise": [101021], "state_str": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ (∑ σ : Equiv.Perm n, sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i).natDegree ≤ Fintype.card 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": "Finset.fold max 0 (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ ≤ Fintype.card n"}], "premise": [106537], "state_str": "n : Type u_1\nα : Type u_2\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ Finset.fold max 0 (natDegree ∘ fun σ => sign σ • ∏ i : n, (X • A.map ⇑C + B.map ⇑C) (σ i) i) Finset.univ ≤\n    Fintype.card n"}
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+{"state": [{"context": ["I : Type u", "inst✝¹ : LinearOrder I", "inst✝ : IsWellOrder I fun x x_1 => x < x_1", "C : Set (I → Bool)", "h : eval {fun x => false} nil ∈ Submodule.span ℤ (eval {fun x => false} '' {m | m < nil})"], "goal": "False"}], "premise": [63542, 86713, 109877, 113018, 124819, 134108], "state_str": "I : Type u\ninst✝¹ : LinearOrder I\ninst✝ : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I → Bool)\nh : eval {fun x => false} nil ∈ Submodule.span ℤ (eval {fun x => false} '' {m | m < nil})\n⊢ False"}
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+{"state": [{"context": ["α : Type u", "inst✝ : Ring α", "x : α", "n : ℕ"], "goal": "op ((x - 1) * ∑ i ∈ range n, x ^ i) = op (x ^ n - 1)"}], "premise": [112493], "state_str": "α : Type u\ninst✝ : Ring α\nx : α\nn : ℕ\n⊢ op ((x - 1) * ∑ i ∈ range n, x ^ i) = op (x ^ n - 1)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : CommMonoid α", "p : Finset β", "f : β → α"], "goal": "∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)"}], "premise": [1670, 1838], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoid α\np : Finset β\nf : β → α\n⊢ ∏ i ∈ p, Associates.mk (f i) = Associates.mk (∏ i ∈ p, f i)"}
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+{"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 : 𝕜 → ℝ → E", "F' : ℝ → E", "x₀ : 𝕜", "ε_pos : 0 < ε", "hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))", "hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))", "h_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)", "h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀", "hF_int : IntegrableOn (F x₀) (Ι a b) μ", "bound_integrable : IntegrableOn bound (Ι a b) μ"], "goal": "IntegrableOn F' (Ι a b) μ ∧ HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ) ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀"}], "premise": [45358], "state_str": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀"}
+{"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 : 𝕜 → ℝ → E", "F' : ℝ → E", "x₀ : 𝕜", "ε_pos : 0 < ε", "hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))", "hF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))", "h_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)", "h_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀", "hF_int : IntegrableOn (F x₀) (Ι a b) μ", "bound_integrable : IntegrableOn bound (Ι a b) μ", "this : Integrable F' (μ.restrict (Ι a b)) ∧ HasDerivAt (fun x => ∫ (a : ℝ) in Ι a b, F x a ∂μ) (∫ (a : ℝ) in Ι a b, F' a ∂μ) x₀"], "goal": "IntegrableOn F' (Ι a b) μ ∧ HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ) ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀"}], "premise": [2106, 2107, 45301], "state_str": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nμ : Measure ℝ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_3\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\na b ε : ℝ\nbound : ℝ → ℝ\nF : 𝕜 → ℝ → E\nF' : ℝ → E\nx₀ : 𝕜\nε_pos : 0 < ε\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) (μ.restrict (Ι a b))\nhF'_meas : AEStronglyMeasurable F' (μ.restrict (Ι a b))\nh_lipsch : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), LipschitzOnWith (Real.nnabs (bound x)) (fun x_1 => F x_1 x) (ball x₀ ε)\nh_diff : ∀ᵐ (x : ℝ) ∂μ.restrict (Ι a b), HasDerivAt (fun x_1 => F x_1 x) (F' x) x₀\nhF_int : IntegrableOn (F x₀) (Ι a b) μ\nbound_integrable : IntegrableOn bound (Ι a b) μ\nthis :\n  Integrable F' (μ.restrict (Ι a b)) ∧\n    HasDerivAt (fun x => ∫ (a : ℝ) in Ι a b, F x a ∂μ) (∫ (a : ℝ) in Ι a b, F' a ∂μ) x₀\n⊢ IntegrableOn F' (Ι a b) μ ∧\n    HasDerivAt (fun x => (if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F x t ∂μ)\n      ((if a ≤ b then 1 else -1) • ∫ (t : ℝ) in Ι a b, F' t ∂μ) x₀"}
+{"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", "P Q : Submodule R A", "x : A", "hx : x ∈ P * Q"], "goal": "x ∈ span R (↑P * ↑Q)"}], "premise": [86688, 122455], "state_str": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P✝ Q✝ : Submodule R A\nm n : A\nP Q : Submodule R A\nx : A\nhx : x ∈ P * Q\n⊢ x ∈ span R (↑P * ↑Q)"}
+{"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"], "goal": "(∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α"}], "premise": [1674, 2111, 55448], "state_str": "case mpr\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\n⊢ (∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ) → IsMinimal M α"}
+{"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": "IsClosed (closure (orbit M x✝))"}, {"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": "∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)"}, {"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✝) = ∅"}], "premise": [7648, 55409, 64959], "state_str": "case mpr.refine_1\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ IsClosed (closure (orbit M x✝))\n\ncase mpr.refine_2\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ∀ (c : M), c • closure (orbit M x✝) ⊆ closure (orbit M x✝)\n\ncase mpr.refine_3\nM : Type u_1\nG : Type u_2\nα : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Group G\ninst✝³ : TopologicalSpace α\ninst✝² : MulAction M α\ninst✝¹ : MulAction G α\ninst✝ : ContinuousConstSMul M α\nH : ∀ (s : Set α), IsClosed s → (∀ (c : M), c • s ⊆ s) → s = ∅ ∨ s = univ\nx✝ : α\n⊢ ¬closure (orbit M x✝) = ∅"}
+{"state": [{"context": ["C : Type u₁", "inst✝ : Category.{v₁, u₁} C", "P X Y Z : C", "fst : P ⟶ X", "snd : P ⟶ Y", "f : X ⟶ Z", "g : Y ⟶ Z", "P' : C", "fst' : P' ⟶ X", "snd' : P' ⟶ Y", "h : IsPullback fst snd f g", "h' : IsPullback fst' snd' f g"], "goal": "(isoIsPullback X Y h h').inv ≫ snd = snd'"}], "premise": [88767, 94095], "state_str": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nP X Y Z : C\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nP' : C\nfst' : P' ⟶ X\nsnd' : P' ⟶ Y\nh : IsPullback fst snd f g\nh' : IsPullback fst' snd' f g\n⊢ (isoIsPullback X Y h h').inv ≫ snd = snd'"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "inst✝ : CommMonoidWithZero α", "p : α", "hp : Prime p", "s : Multiset β", "f : β → α", "h : p ∣ (Multiset.map f s).prod"], "goal": "∃ a ∈ s, p ∣ f a"}], "premise": [2045, 70141, 123914, 137990], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝ : CommMonoidWithZero α\np : α\nhp : Prime p\ns : Multiset β\nf : β → α\nh : p ∣ (Multiset.map f s).prod\n⊢ ∃ a ∈ s, p ∣ f a"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝⁹ : UniformSpace α", "inst✝⁸ : Group α", "inst✝⁷ : UniformGroup α", "inst✝⁶ : Group β", "γ : Type u_3", "inst✝⁵ : Group γ", "inst✝⁴ : UniformSpace γ", "inst✝³ : UniformGroup γ", "inst✝² : UniformSpace β", "F : Type u_4", "inst✝¹ : FunLike F β γ", "inst✝ : MonoidHomClass F β γ", "f : F", "hf : UniformInducing ⇑f"], "goal": "UniformContinuous fun p => p.1 / p.2"}], "premise": [1671, 59530, 117102], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Group α\ninst✝⁷ : UniformGroup α\ninst✝⁶ : Group β\nγ : Type u_3\ninst✝⁵ : Group γ\ninst✝⁴ : UniformSpace γ\ninst✝³ : UniformGroup γ\ninst✝² : UniformSpace β\nF : Type u_4\ninst✝¹ : FunLike F β γ\ninst✝ : MonoidHomClass F β γ\nf : F\nhf : UniformInducing ⇑f\n⊢ UniformContinuous fun p => p.1 / p.2"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝⁹ : UniformSpace α", "inst✝⁸ : Group α", "inst✝⁷ : UniformGroup α", "inst✝⁶ : Group β", "γ : Type u_3", "inst✝⁵ : Group γ", "inst✝⁴ : UniformSpace γ", "inst✝³ : UniformGroup γ", "inst✝² : UniformSpace β", "F : Type u_4", "inst✝¹ : FunLike F β γ", "inst✝ : MonoidHomClass F β γ", "f : F", "hf : UniformInducing ⇑f"], "goal": "UniformContinuous fun x => f x.1 / f x.2"}], "premise": [59529, 60579, 60653, 67116], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝⁹ : UniformSpace α\ninst✝⁸ : Group α\ninst✝⁷ : UniformGroup α\ninst✝⁶ : Group β\nγ : Type u_3\ninst✝⁵ : Group γ\ninst✝⁴ : UniformSpace γ\ninst✝³ : UniformGroup γ\ninst✝² : UniformSpace β\nF : Type u_4\ninst✝¹ : FunLike F β γ\ninst✝ : MonoidHomClass F β γ\nf : F\nhf : UniformInducing ⇑f\n⊢ UniformContinuous fun x => f x.1 / f x.2"}
+{"state": [{"context": ["J : Type v'", "inst✝¹ : Category.{u', v'} J", "C : Type u", "inst✝ : Category.{v, u} C", "W X Y Z : C", "f : W ⟶ X", "g : W ⟶ Y", "h : X ⟶ Z", "i : Y ⟶ Z", "H : IsPushout f g h i", "H' : H.IsVanKampen", "W' X' Y' Z' : C", "f' : W' ⟶ X'", "g' : W' ⟶ Y'", "h' : X' ⟶ Z'", "i' : Y' ⟶ Z'", "αW : W' ⟶ W", "αX : X' ⟶ Y", "αY : Y' ⟶ X", "αZ : Z' ⟶ Z", "hf : IsPullback f' αW αX g", "hg : IsPullback g' αW αY f", "hh : CommSq h' αX αZ i", "hi : CommSq i' αY αZ h", "w : CommSq f' g' h' i'"], "goal": "IsPushout f' g' h' i' ↔ IsPullback h' αX αZ i ∧ IsPullback i' αY αZ h"}], "premise": [1723, 94128, 94157, 98983], "state_str": "J : Type v'\ninst✝¹ : Category.{u', v'} J\nC : Type u\ninst✝ : Category.{v, u} C\nW X Y Z : C\nf : W ⟶ X\ng : W ⟶ Y\nh : X ⟶ Z\ni : Y ⟶ Z\nH : IsPushout f g h i\nH' : H.IsVanKampen\nW' X' Y' Z' : C\nf' : W' ⟶ X'\ng' : W' ⟶ Y'\nh' : X' ⟶ Z'\ni' : Y' ⟶ Z'\nαW : W' ⟶ W\nαX : X' ⟶ Y\nαY : Y' ⟶ X\nαZ : Z' ⟶ Z\nhf : IsPullback f' αW αX g\nhg : IsPullback g' αW αY f\nhh : CommSq h' αX αZ i\nhi : CommSq i' αY αZ h\nw : CommSq f' g' h' i'\n⊢ IsPushout f' g' h' i' ↔ IsPullback h' αX αZ i ∧ IsPullback i' αY αZ h"}
+{"state": [{"context": ["K : Type u", "A : Type v", "inst✝² : Field K", "inst✝¹ : Ring A", "inst✝ : Algebra K A"], "goal": "Algebra.IsAlgebraic K A ↔ Algebra.IsIntegral K A"}], "premise": [1713, 2015, 75554, 75581, 80627], "state_str": "K : Type u\nA : Type v\ninst✝² : Field K\ninst✝¹ : Ring A\ninst✝ : Algebra K A\n⊢ Algebra.IsAlgebraic K A ↔ Algebra.IsIntegral K A"}
+{"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 : Mul H", "inst✝² : MulAction G H", "inst✝¹ : SMulCommClass G H H", "inst✝ : IsScalarTower G H H", "g : G", "a b : H"], "goal": "Commute (g • a) b ↔ Commute a b"}], "premise": [1713, 118335, 118927], "state_str": "M : Type u_1\nN : Type u_2\nG : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁷ : Monoid M\ninst✝⁶ : MulAction M α\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ng✝ : G\na✝ b✝ : α\ninst✝³ : Mul H\ninst✝² : MulAction G H\ninst✝¹ : SMulCommClass G H H\ninst✝ : IsScalarTower G H H\ng : G\na b : H\n⊢ Commute (g • a) b ↔ Commute a b"}
+{"state": [{"context": ["a b : ℕ"], "goal": "a ≤ pair a b"}], "premise": [142567], "state_str": "a b : ℕ\n⊢ a ≤ pair a b"}
+{"state": [{"context": ["G✝ : Type u_1", "H : Type u_2", "inst✝¹ : Monoid G✝", "G : Type u_3", "inst✝ : CommGroup G"], "goal": "IsTorsionFree G ↔ CommGroup.torsion G = ⊥"}], "premise": [18818, 128384], "state_str": "G✝ : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G✝\nG : Type u_3\ninst✝ : CommGroup G\n⊢ IsTorsionFree G ↔ CommGroup.torsion G = ⊥"}
+{"state": [{"context": ["G✝ : Type u_1", "H : Type u_2", "inst✝¹ : Monoid G✝", "G : Type u_3", "inst✝ : CommGroup G"], "goal": "(∀ (g : G), g ≠ 1 → ¬IsOfFinOrder g) ↔ ∀ ⦃x : G⦄, x ∈ CommGroup.torsion G → x ∈ ⊥"}], "premise": [6219, 70072], "state_str": "G✝ : Type u_1\nH : Type u_2\ninst✝¹ : Monoid G✝\nG : Type u_3\ninst✝ : CommGroup G\n⊢ (∀ (g : G), g ≠ 1 → ¬IsOfFinOrder g) ↔ ∀ ⦃x : G⦄, x ∈ CommGroup.torsion G → x ∈ ⊥"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommRing R", "n ν : ℕ", "h : ν ≤ n"], "goal": "(bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν)"}], "premise": [103621, 117740], "state_str": "R : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\nh : ν ≤ n\n⊢ (bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν)"}
+{"state": [{"context": [], "goal": "negMulLog = fun x => -(x * log x)"}], "premise": [37750], "state_str": "⊢ negMulLog = fun x => -(x * log x)"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "f g : End R M", "hf : f.IsSemisimple", "μ : R", "k : ℕ", "hk : 0 < k"], "goal": "(f.genEigenspace μ) k = f.eigenspace μ"}], "premise": [1674, 14296, 88218, 88234], "state_str": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nhf : f.IsSemisimple\nμ : R\nk : ℕ\nhk : 0 < k\n⊢ (f.genEigenspace μ) k = f.eigenspace μ"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "f g : End R M", "hf : f.IsSemisimple", "μ : R", "k : ℕ", "hk : 0 < k", "m : M", "hm : m ∈ (f.genEigenspace μ) k"], "goal": "f m = μ • m"}], "premise": [86410], "state_str": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\nhf : f.IsSemisimple\nμ : R\nk : ℕ\nhk : 0 < k\nm : M\nhm : m ∈ (f.genEigenspace μ) k\n⊢ f m = μ • m"}
+{"state": [{"context": ["X✝ X Y : LocallyRingedSpace", "e : X ≅ Y"], "goal": "e.hom.val.base ≫ e.inv.val.base = 𝟙 ↑X.toPresheafedSpace"}], "premise": [68194, 68441], "state_str": "X✝ X Y : LocallyRingedSpace\ne : X ≅ Y\n⊢ e.hom.val.base ≫ e.inv.val.base = 𝟙 ↑X.toPresheafedSpace"}
+{"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"}], "premise": [112223], "state_str": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nf : R[X] →+* MvPolynomial PUnit.{?u.2149 + 1} R := eval₂RingHom C (X PUnit.unit)\ng : MvPolynomial PUnit.{?u.2149 + 1} R →+* R[X] := eval₂Hom Polynomial.C fun x => Polynomial.X\n⊢ ∀ (p : MvPolynomial PUnit.{?u.2149 + 1} R), (f.comp g) p = p"}
+{"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", "p : R[X]", "a : R"], "goal": "eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C a)) = Polynomial.C a"}], "premise": [102828, 112309], "state_str": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na✝ a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np : R[X]\na : R\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C a)) = Polynomial.C a"}
+{"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", "p✝ p q : R[X]", "hp : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) p) = p", "hq : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) q) = q"], "goal": "eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (p + q)) = p + q"}], "premise": [102832, 112307], "state_str": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ p q : R[X]\nhp : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) p) = p\nhq : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) q) = q\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (p + q)) = p + q"}
+{"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", "p✝ : R[X]", "p : ℕ", "n : R", "x✝ : eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ p)) = Polynomial.C n * Polynomial.X ^ p"], "goal": "eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ (p + 1))) = Polynomial.C n * Polynomial.X ^ (p + 1)"}], "premise": [102828, 102829, 102852, 102856, 112309, 112311, 112314, 112315], "state_str": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ : R[X]\np : ℕ\nn : R\nx✝ :\n  eval₂ Polynomial.C (fun x => Polynomial.X) (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ p)) =\n    Polynomial.C n * Polynomial.X ^ p\n⊢ eval₂ Polynomial.C (fun x => Polynomial.X)\n      (Polynomial.eval₂ C (X PUnit.unit) (Polynomial.C n * Polynomial.X ^ (p + 1))) =\n    Polynomial.C n * Polynomial.X ^ (p + 1)"}
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+{"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", "h₀ : f₀ =ᶠ[L] f₁", "hx : f₀ x = f₁ x", "h₁ : ∀ (x : E), f₀' x = f₁' x"], "goal": "HasFDerivAtFilter f₀ f₀' x L ↔ HasFDerivAtFilter f₁ f₁' x L"}], "premise": [46289], "state_str": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : ∀ (x : E), f₀' x = f₁' x\n⊢ HasFDerivAtFilter f₀ f₀' x L ↔ HasFDerivAtFilter f₁ f₁' 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", "h₀ : f₀ =ᶠ[L] f₁", "hx : f₀ x = f₁ x", "h₁ : ∀ (x : E), f₀' x = f₁' x"], "goal": "((fun x' => f₀ x' - f₀ x - f₀' (x' - x)) =o[L] fun x' => x' - x) ↔ (fun x' => f₁ x' - f₁ x - f₁' (x' - x)) =o[L] fun x' => x' - x"}], "premise": [16027, 16091, 43392], "state_str": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type u_5\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nh₀ : f₀ =ᶠ[L] f₁\nhx : f₀ x = f₁ x\nh₁ : ∀ (x : E), f₀' x = f₁' x\n⊢ ((fun x' => f₀ x' - f₀ x - f₀' (x' - x)) =o[L] fun x' => x' - x) ↔\n    (fun x' => f₁ x' - f₁ x - f₁' (x' - x)) =o[L] fun x' => x' - x"}
+{"state": [{"context": ["N : ℕ", "hN : ¬Prime N"], "goal": "N.succ.smoothNumbers = N.smoothNumbers"}], "premise": [21706, 21715, 139149], "state_str": "N : ℕ\nhN : ¬Prime N\n⊢ N.succ.smoothNumbers = N.smoothNumbers"}
+{"state": [{"context": ["C : Type u₁", "inst✝³ : Category.{v₁, u₁} C", "inst✝² : Preadditive C", "D : Type u_1", "inst✝¹ : Category.{v₁, u_1} D", "inst✝ : Preadditive D", "ι✝¹ : Type", "fintype✝¹ : Fintype ι✝¹", "X✝¹ : ι✝¹ → C", "i✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝¹ X✝¹)).ι", "ι✝ : Type", "fintype✝ : Fintype ι✝", "X✝ : ι✝ → C", "j✝ : ((𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)).ι", "f : Mat_.mk ι✝¹ X✝¹ ⟶ Mat_.mk ι✝ X✝"], "goal": "((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom) i✝ j✝ = (((fun M => eqToIso ⋯) (Mat_.mk ι✝¹ X✝¹)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝"}], "premise": [96186, 96187], "state_str": "case H.mk.mk\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Preadditive C\nD : Type u_1\ninst✝¹ : Category.{v₁, u_1} D\ninst✝ : Preadditive D\nι✝¹ : Type\nfintype✝¹ : Fintype ι✝¹\nX✝¹ : ι✝¹ → C\ni✝ : ((𝟭 C).mapMat_.obj (Mat_.mk ι✝¹ X✝¹)).ι\nι✝ : Type\nfintype✝ : Fintype ι✝\nX✝ : ι✝ → C\nj✝ : ((𝟭 (Mat_ C)).obj (Mat_.mk ι✝ X✝)).ι\nf : Mat_.mk ι✝¹ X✝¹ ⟶ Mat_.mk ι✝ X✝\n⊢ ((𝟭 C).mapMat_.map f ≫ ((fun M => eqToIso ⋯) (Mat_.mk ι✝ X✝)).hom) i✝ j✝ =\n    (((fun M => eqToIso ⋯) (Mat_.mk ι✝¹ X✝¹)).hom ≫ (𝟭 (Mat_ C)).map f) i✝ j✝"}
+{"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", "κ : Type u", "w✝ : Fintype κ", "a : ι → κ → R", "y : κ → N", "h₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j", "h₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0"], "goal": "∑ i : ι, m i ⊗ₜ[R] n i = 0"}], "premise": [86849, 86855], "state_str": "case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ i : ι, m i ⊗ₜ[R] n i = 0"}
+{"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", "κ : Type u", "w✝ : Fintype κ", "a : ι → κ → R", "y : κ → N", "h₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j", "h₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0"], "goal": "∑ x : ι, ∑ x_1 : κ, a x x_1 • m x ⊗ₜ[R] y x_1 = 0"}], "premise": [127022], "state_str": "case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ x : ι, ∑ x_1 : κ, a x x_1 • m x ⊗ₜ[R] y x_1 = 0"}
+{"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", "κ : Type u", "w✝ : Fintype κ", "a : ι → κ → R", "y : κ → N", "h₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j", "h₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0"], "goal": "∑ y_1 : κ, ∑ x : ι, a x y_1 • m x ⊗ₜ[R] y y_1 = 0"}], "premise": [86836, 86841, 86849, 86854, 126913], "state_str": "case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁵ : CommRing R\nM : Type u\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nκ : Type u\nw✝ : Fintype κ\na : ι → κ → R\ny : κ → N\nh₁ : ∀ (i : ι), n i = ∑ j : κ, a i j • y j\nh₂ : ∀ (j : κ), ∑ i : ι, a i j • m i = 0\n⊢ ∑ y_1 : κ, ∑ x : ι, a x y_1 • m x ⊗ₜ[R] y y_1 = 0"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "δ : Type u_5", "inst✝⁷ : TopologicalSpace α", "inst✝⁶ : Preorder α", "inst✝⁵ : TopologicalSpace β", "inst✝⁴ : Preorder β", "inst✝³ : TopologicalSpace γ", "inst✝² : Preorder γ", "inst✝¹ : TopologicalSpace δ", "inst✝ : Preorder δ", "g : EsakiaHom β γ", "f₁ f₂ : EsakiaHom α β", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "a : α"], "goal": "g (f₁ a) = g (f₂ a)"}], "premise": [54346], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : Preorder α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Preorder β\ninst✝³ : TopologicalSpace γ\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Preorder δ\ng : EsakiaHom β γ\nf₁ f₂ : EsakiaHom α β\nhg : Injective ⇑g\nh : g.comp f₁ = g.comp f₂\na : α\n⊢ g (f₁ a) = g (f₂ a)"}
+{"state": [{"context": ["m n : ℕ", "x y : ZMod n"], "goal": "(x - y).cast = (x.cast - y.cast) % ↑n"}], "premise": [128757, 138295, 138339], "state_str": "m n : ℕ\nx y : ZMod n\n⊢ (x - y).cast = (x.cast - y.cast) % ↑n"}
+{"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"}], "premise": [141384], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : EMetricSpace α\nx y z : α\ns t : Set α\nC : ℝ≥0∞\nsC : Set ℝ≥0∞\ninst✝ : Finite ↑s\n⊢ 0 < s.einfsep"}
+{"state": [{"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K"], "goal": "‖reCLM‖ = 1"}], "premise": [14296, 40918, 101700], "state_str": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ ‖reCLM‖ = 1"}
+{"state": [{"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K"], "goal": "1 ≤ ‖reLm.mkContinuous 1 ⋯‖"}], "premise": [40899], "state_str": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n⊢ 1 ≤ ‖reLm.mkContinuous 1 ⋯‖"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : LinearOrderedCommMonoidWithZero α", "a b c d x y z : α", "n : ℕ", "inst✝ : NoZeroDivisors α", "hn : n ≠ 0"], "goal": "0 < a ^ n ↔ 0 < a"}], "premise": [102772, 108283], "state_str": "α : Type u_1\ninst✝¹ : LinearOrderedCommMonoidWithZero α\na b c d x y z : α\nn : ℕ\ninst✝ : NoZeroDivisors α\nhn : n ≠ 0\n⊢ 0 < a ^ n ↔ 0 < a"}
+{"state": [{"context": ["C : Type u", "inst✝³ : Category.{v, u} C", "inst✝² : Abelian C", "inst✝¹ : HasPullbacks C", "W X Y Z : C", "f : X ⟶ Z", "g : Y ⟶ Z", "inst✝ : Epi g", "s : PullbackCone f g", "hs : IsLimit s", "this : Epi ((limit.cone (cospan f g)).π.app WalkingCospan.left)"], "goal": "Epi s.fst"}], "premise": [94253, 96199], "state_str": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nW X Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi g\ns : PullbackCone f g\nhs : IsLimit s\nthis : Epi ((limit.cone (cospan f g)).π.app WalkingCospan.left)\n⊢ Epi s.fst"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "γ : Type u_4", "inst✝³ : DecidableEq β", "inst✝² : Group α", "inst✝¹ : MulAction α β", "s✝ t : Finset β", "a✝¹ : α", "b : β", "inst✝ : DecidableEq α", "a : α", "s : Finset α", "a✝ : α"], "goal": "a✝ ∈ (a • s)⁻¹ ↔ a✝ ∈ op a⁻¹ • s⁻¹"}], "premise": [141991], "state_str": "case a\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝³ : DecidableEq β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ns✝ t : Finset β\na✝¹ : α\nb : β\ninst✝ : DecidableEq α\na : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ (a • s)⁻¹ ↔ a✝ ∈ op a⁻¹ • s⁻¹"}
+{"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", "u v w : Set α", "hu : MeasurableSet u", "hv : MeasurableSet v", "hsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u", "hsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v", "hs : ↑s (u ∆ v) = 0"], "goal": "↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0"}], "premise": [32676], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ns : SignedMeasure α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhsu : VectorMeasure.restrict 0 u ≤ VectorMeasure.restrict s u\nhsv : VectorMeasure.restrict 0 v ≤ VectorMeasure.restrict s v\nhs : ↑s (u ∆ v) = 0\n⊢ ↑s (u \\ v) = 0 ∧ ↑s (v \\ u) = 0"}
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+{"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✝¹ : IsFiniteKernel κ", "inst✝ : IsFiniteKernel ν", "a : α"], "goal": "∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i"}], "premise": [29021], "state_str": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDFAux f κ ν\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel ν\na : α\n⊢ ∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i"}
+{"state": [{"context": ["S : Type u_1", "M : Type u_2", "G : Type u_3", "inst✝ : Group G", "a✝ x✝ y✝ a x y b : G"], "goal": "b * a * b⁻¹ * (b * x * b⁻¹) = b * y * b⁻¹ * (b * a * b⁻¹) ↔ a * x = y * a"}], "premise": [119703, 119833], "state_str": "S : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝ : Group G\na✝ x✝ y✝ a x y b : G\n⊢ b * a * b⁻¹ * (b * x * b⁻¹) = b * y * b⁻¹ * (b * a * b⁻¹) ↔ a * x = y * a"}
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+{"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 : FormallySmooth R A", "inst✝² : FormallySmooth A B", "C : Type u", "inst✝¹ : CommRing C", "inst✝ : Algebra R C", "I : Ideal C", "hI : I ^ 2 = ⊥", "f : B →ₐ[R] C ⧸ I", "f' : A →ₐ[R] C", "e : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)", "this : Algebra A C := f'.toAlgebra"], "goal": "∃ a, (Ideal.Quotient.mkₐ R I).comp a = f"}], "premise": [2100, 76679, 121055], "state_str": "case comp_surjective.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f"}
+{"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 : FormallySmooth R A", "inst✝² : FormallySmooth A B", "C : Type u", "inst✝¹ : CommRing C", "inst✝ : Algebra R C", "I : Ideal C", "hI : I ^ 2 = ⊥", "f : B →ₐ[R] C ⧸ I", "f' : A →ₐ[R] C", "e : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)", "this : Algebra A C := f'.toAlgebra", "f'' : B →ₐ[A] C", "e' : AlgHom.restrictScalars R ((Ideal.Quotient.mkₐ A I).comp f'') = AlgHom.restrictScalars R (let __src := f.toRingHom; { toRingHom := __src, commutes' := ⋯ })"], "goal": "∃ a, (Ideal.Quotient.mkₐ R I).comp a = f"}], "premise": [2101, 121057], "state_str": "case comp_surjective.intro.intro\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type u\ninst✝⁹ : CommSemiring A\ninst✝⁸ : Algebra R A\nB : Type u\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallySmooth R A\ninst✝² : FormallySmooth A B\nC : Type u\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\nI : Ideal C\nhI : I ^ 2 = ⊥\nf : B →ₐ[R] C ⧸ I\nf' : A →ₐ[R] C\ne : (Ideal.Quotient.mkₐ R I).comp f' = f.comp (IsScalarTower.toAlgHom R A B)\nthis : Algebra A C := f'.toAlgebra\nf'' : B →ₐ[A] C\ne' :\n  AlgHom.restrictScalars R ((Ideal.Quotient.mkₐ A I).comp f'') =\n    AlgHom.restrictScalars R\n      (let __src := f.toRingHom;\n      { toRingHom := __src, commutes' := ⋯ })\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f"}
+{"state": [{"context": ["w x✝ y✝ z y : ℝ", "hy : 0 < y", "x : ℝ≥0"], "goal": "(x ^ (1 / y)) ^ y = x"}], "premise": [1690, 39672, 39678, 108423], "state_str": "w x✝ y✝ z y : ℝ\nhy : 0 < y\nx : ℝ≥0\n⊢ (x ^ (1 / y)) ^ y = x"}
+{"state": [{"context": ["E : Type u_1", "inst✝ : SeminormedCommGroup E", "ε δ : ℝ", "s t : Set E", "x y : E"], "goal": "s / ball x δ = x⁻¹ • thickening δ s"}], "premise": [119790], "state_str": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ s / ball x δ = x⁻¹ • thickening δ s"}
+{"state": [{"context": ["x y : ℝ≥0", "hy : 1 < y"], "goal": "∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}], "premise": [103648, 107253], "state_str": "x y : ℝ≥0\nhy : 1 < y\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}
+{"state": [{"context": ["x y : ℝ≥0", "hy : 1 < y", "m : ℕ", "hm : x < (y - 1 + 1) ^ m"], "goal": "∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}], "premise": [103586], "state_str": "case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < (y - 1 + 1) ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}
+{"state": [{"context": ["x y : ℝ≥0", "hy : 1 < y", "m : ℕ", "hm : x < y ^ m"], "goal": "∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}], "premise": [1674, 15489], "state_str": "case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\n⊢ ∀ᶠ (n : ℕ) in atTop, x ^ (1 / ↑n) ≤ y"}
+{"state": [{"context": ["x y : ℝ≥0", "hy : 1 < y", "m : ℕ", "hm : x < y ^ m", "n : ℕ", "hn : n ≥ m + 1"], "goal": "x ^ (1 / ↑n) ≤ y"}], "premise": [117810], "state_str": "case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (1 / ↑n) ≤ y"}
+{"state": [{"context": ["x y : ℝ≥0", "hy : 1 < y", "m : ℕ", "hm : x < y ^ m", "n : ℕ", "hn : n ≥ m + 1"], "goal": "x ^ (↑n)⁻¹ ≤ y"}], "premise": [1674, 2140, 2143, 39688, 39707, 106246, 142640], "state_str": "case intro\nx y : ℝ≥0\nhy : 1 < y\nm : ℕ\nhm : x < y ^ m\nn : ℕ\nhn : n ≥ m + 1\n⊢ x ^ (↑n)⁻¹ ≤ y"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝¹⁴ : CommSemiring R", "inst✝¹³ : AddCommMonoid M", "inst✝¹² : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝¹¹ : CommRing R₁", "inst✝¹⁰ : AddCommGroup M₁", "inst✝⁹ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝⁸ : Field K", "inst✝⁷ : AddCommGroup V", "inst✝⁶ : Module K V", "M'✝ : Type u_7", "M'' : Type u_8", "inst✝⁵ : AddCommMonoid M'✝", "inst✝⁴ : AddCommMonoid M''", "inst✝³ : Module R M'✝", "inst✝² : Module R M''", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M' : Type u_9", "inst✝¹ : AddCommMonoid M'", "inst✝ : Module R M'", "B : BilinForm R₁ M₁", "b : B.Nondegenerate", "φ ψ : M₁ →ₗ[R₁] M₁", "h : B.compLeft φ = B.compLeft ψ", "w : M₁"], "goal": "φ w = ψ w"}], "premise": [117827], "state_str": "case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw : M₁\n⊢ φ w = ψ w"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝¹⁴ : CommSemiring R", "inst✝¹³ : AddCommMonoid M", "inst✝¹² : Module R M", "R₁ : Type u_3", "M₁ : Type u_4", "inst✝¹¹ : CommRing R₁", "inst✝¹⁰ : AddCommGroup M₁", "inst✝⁹ : Module R₁ M₁", "V : Type u_5", "K : Type u_6", "inst✝⁸ : Field K", "inst✝⁷ : AddCommGroup V", "inst✝⁶ : Module K V", "M'✝ : Type u_7", "M'' : Type u_8", "inst✝⁵ : AddCommMonoid M'✝", "inst✝⁴ : AddCommMonoid M''", "inst✝³ : Module R M'✝", "inst✝² : Module R M''", "B✝ : BilinForm R M", "B₁ : BilinForm R₁ M₁", "M' : Type u_9", "inst✝¹ : AddCommMonoid M'", "inst✝ : Module R M'", "B : BilinForm R₁ M₁", "b : B.Nondegenerate", "φ ψ : M₁ →ₗ[R₁] M₁", "h : B.compLeft φ = B.compLeft ψ", "w v : M₁"], "goal": "(B (φ w - ψ w)) v = 0"}], "premise": [82450, 82671, 117981], "state_str": "case h\nR : Type u_1\nM : Type u_2\ninst✝¹⁴ : CommSemiring R\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nR₁ : Type u_3\nM₁ : Type u_4\ninst✝¹¹ : CommRing R₁\ninst✝¹⁰ : AddCommGroup M₁\ninst✝⁹ : Module R₁ M₁\nV : Type u_5\nK : Type u_6\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nM'✝ : Type u_7\nM'' : Type u_8\ninst✝⁵ : AddCommMonoid M'✝\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M'✝\ninst✝² : Module R M''\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nM' : Type u_9\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R₁ M₁\nb : B.Nondegenerate\nφ ψ : M₁ →ₗ[R₁] M₁\nh : B.compLeft φ = B.compLeft ψ\nw v : M₁\n⊢ (B (φ w - ψ w)) v = 0"}
+{"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", "α α' β : Type u", "f : α → α'", "x : t α β"], "goal": "tfst (pure ∘ f) x = pure (fst f x)"}], "premise": [9361], "state_str": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ✝ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : LawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα α' β : Type u\nf : α → α'\nx : t α β\n⊢ tfst (pure ∘ f) x = pure (fst f x)"}
+{"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 : α"], "goal": "Ici a * Ioi b ⊆ Ioi (a * b)"}], "premise": [103726], "state_str": "α : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\n⊢ Ici a * Ioi b ⊆ Ioi (a * b)"}
+{"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 : α", "this : ∀ (M : Type ?u.12705) (N : Type ?u.12704) (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 : y ∈ Ici a", "z : α", "hzb : z ∈ Ioi b"], "goal": "(fun x x_1 => x * x_1) y z ∈ Ioi (a * b)"}], "premise": [103908], "state_str": "case intro.intro.intro.intro\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : PartialOrder α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst✝ : CovariantClass α α (Function.swap HMul.hMul) LT.lt\na b : α\nthis :\n  ∀ (M : Type ?u.12705) (N : Type ?u.12704) (μ : M → N → N) [inst : PartialOrder N]\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\ny : α\nhya : y ∈ Ici a\nz : α\nhzb : z ∈ Ioi b\n⊢ (fun x x_1 => x * x_1) y z ∈ Ioi (a * b)"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K : Set ℂ", "z : ℂ", "M r δ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [1673, 12516, 55501, 56660], "state_str": "E : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [46607], "state_str": "case intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [55381], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [15889, 46354, 131585], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ��) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U", "h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [60207], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U", "h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)", "h4 : TendstoLocallyUniformlyOn F f φ (interior K)"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [55381, 60205, 60207], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U", "h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)", "h4 : TendstoLocallyUniformlyOn F f φ (interior K)", "h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [46880, 55380, 60222], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U", "h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)", "h4 : TendstoLocallyUniformlyOn F f φ (interior K)", "h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)", "h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [44367, 46351], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\n⊢ DifferentiableWithinAt ℂ f U x"}
+{"state": [{"context": ["E : Type u_1", "ι : Type u_2", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace ℂ E", "inst✝¹ : CompleteSpace E", "U K✝ : Set ℂ", "z : ℂ", "M r δ✝ : ℝ", "φ : Filter ι", "F : ι → ℂ → E", "f g : ℂ → E", "inst✝ : φ.NeBot", "hf : TendstoLocallyUniformlyOn F f φ U", "hF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U", "hU : IsOpen U", "x : ℂ", "hx : x ∈ U", "K : Set ℂ", "hKU : K ⊆ U", "hKx : K ∈ 𝓝 x", "hK : IsCompact K", "δ : ℝ", "left✝¹ : δ > 0", "left✝ : cthickening δ K ⊆ U", "h1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K", "h2 : interior K ⊆ U", "h3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)", "h4 : TendstoLocallyUniformlyOn F f φ (interior K)", "h5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)", "h6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x", "h7 : DifferentiableOn ℂ f (interior K)"], "goal": "DifferentiableWithinAt ℂ f U x"}], "premise": [1674, 46338, 46351, 55558], "state_str": "case intro.intro.intro.intro.intro.intro\nE : Type u_1\nι : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : CompleteSpace E\nU K✝ : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\ninst✝ : φ.NeBot\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nx : ℂ\nhx : x ∈ U\nK : Set ℂ\nhKU : K ⊆ U\nhKx : K ∈ 𝓝 x\nhK : IsCompact K\nδ : ℝ\nleft✝¹ : δ > 0\nleft✝ : cthickening δ K ⊆ U\nh1 : TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K\nh2 : interior K ⊆ U\nh3 : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) (interior K)\nh4 : TendstoLocallyUniformlyOn F f φ (interior K)\nh5 : TendstoLocallyUniformlyOn (deriv ∘ F) (cderiv δ f) φ (interior K)\nh6 : ∀ x ∈ interior K, HasDerivAt f (cderiv δ f x) x\nh7 : DifferentiableOn ℂ f (interior K)\n⊢ DifferentiableWithinAt ℂ f U 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", "inst✝ : IsFiniteMeasure μ", "c : E"], "goal": "indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) c"}], "premise": [31075], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ indicatorConstLp p ⋯ ⋯ c = (Lp.const p μ) 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✝³ : NormedAddCommGroup E", "inst✝² : NormedAddCommGroup F", "inst✝¹ : NormedAddCommGroup G", "inst✝ : IsFiniteMeasure μ", "c : E"], "goal": "Memℒp.toLp (Set.univ.indicator fun x => c) ⋯ = Memℒp.toLp (fun x => c) ⋯"}], "premise": [120888], "state_str": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : IsFiniteMeasure μ\nc : E\n⊢ Memℒp.toLp (Set.univ.indicator fun x => c) ⋯ = Memℒp.toLp (fun x => c) ⋯"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a : K"], "goal": "a ∈ Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) ↔ a ∈ ↑A.nonunits"}], "premise": [76030, 128379, 131592], "state_str": "case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ a ∈ Subtype.val '' ↑(LocalRing.maximalIdeal ↥A) ↔ a ∈ ↑A.nonunits"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a : K"], "goal": "(∃ x ∈ LocalRing.maximalIdeal ↥A, ↑x = a) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A"}], "premise": [137125], "state_str": "case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ x ∈ LocalRing.maximalIdeal ↥A, ↑x = a) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A"}
+{"state": [{"context": ["K : Type u", "inst✝ : Field K", "A : ValuationSubring K", "a : K"], "goal": "(∃ a_1, ∃ (b : a_1 ∈ ↑A), ⟨a_1, b⟩ ∈ LocalRing.maximalIdeal ↥A ∧ ↑⟨a_1, b⟩ = a) ↔ ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A"}], "premise": [2037, 2039], "state_str": "case h\nK : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : K\n⊢ (∃ a_1, ∃ (b : a_1 ∈ ↑A), ⟨a_1, b⟩ ∈ LocalRing.maximalIdeal ↥A ∧ ↑⟨a_1, b⟩ = a) ↔\n    ∃ (ha : a ∈ A), ⟨a, ha⟩ ∈ LocalRing.maximalIdeal ↥A"}
+{"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", "p : α →₀ M"], "goal": "p ∈ supported M R ↑p.support"}], "premise": [85197], "state_str": "α : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\nR : Type u_5\nS : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring S\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\np : α →₀ M\n⊢ p ∈ supported M R ↑p.support"}
+{"state": [{"context": ["x y : ℝ"], "goal": "↑x ≤ ↑y ↔ x ≤ y"}], "premise": [147867], "state_str": "x y : ℝ\n⊢ ↑x ≤ ↑y ↔ x ≤ y"}
+{"state": [{"context": ["i j k : ℤ", "hi : i ∣ k", "hj : j ∣ k"], "goal": "↑(i.natAbs.lcm j.natAbs) ∣ k"}], "premise": [1674, 3593, 4202, 130034], "state_str": "i j k : ℤ\nhi : i ∣ k\nhj : j ∣ k\n⊢ ↑(i.natAbs.lcm j.natAbs) ∣ k"}
+{"state": [{"context": ["R : Type u_1", "n✝ x y : ℕ", "hxy : 2 ∣ x - y", "hx : ¬2 ∣ x", "n : ℕ", "hn : Even n"], "goal": "multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n"}], "premise": [14308], "state_str": "R : Type u_1\nn✝ x y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 n"}
+{"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", "n : ℕ", "hn : finrank R M = n.succ"], "goal": "0 < n.succ"}], "premise": [2143], "state_str": "R : Type u\nM M₁ : Type v\nM' : Type v'\nι : Type w\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup M'\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module R M\ninst✝³ : Module R M'\ninst✝² : Module R M₁\ninst✝¹ : Nontrivial R\ninst✝ : NoZeroSMulDivisors R M\nn : ℕ\nhn : finrank R M = n.succ\n⊢ 0 < n.succ"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝¹ : NormedLinearOrderedField 𝕜", "P Q : 𝕜[X]", "inst✝ : OrderTopology 𝕜"], "goal": "Tendsto (fun x => eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0"}], "premise": [15610, 39219, 102168, 102174, 103002], "state_str": "𝕜 : Type u_1\ninst✝¹ : NormedLinearOrderedField 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ Tendsto (fun x => eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0"}
+{"state": [{"context": ["X✝ : LocallyRingedSpace", "r : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))"], "goal": "(X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val = β.val"}], "premise": [128409], "state_str": "case h\nX✝ : LocallyRingedSpace\nr : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\n⊢ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val = β.val"}
+{"state": [{"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "(toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}], "premise": [68444], "state_str": "case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (toOpen (↑R) U ≫ (X.toΓSpec ≫ Spec.locallyRingedSpaceMap f).val.c.app (op U)) ≫ X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)"}
+{"state": [{"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "(CommRingCat.ofHom f ≫ toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫ X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}], "premise": [96173], "state_str": "case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ (CommRingCat.ofHom f ≫\n        toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n      X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)"}
+{"state": [{"context": ["X✝ : LocallyRingedSpace", "r✝ : ↑(Γ.obj (op X✝))", "X : LocallyRingedSpace", "R : CommRingCat", "f : R ⟶ Γ.obj (op X)", "β : X ⟶ Spec.locallyRingedSpaceObj R", "w : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base", "h : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))", "r : ↑R", "U : Opens (PrimeSpectrum ↑R) := basicOpen r"], "goal": "CommRingCat.ofHom f ≫ (toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫ X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫ X.presheaf.map (eqToHom ⋯) = toOpen (↑R) U ≫ β.val.c.app (op U)"}], "premise": [99919, 129578], "state_str": "case h\nX✝ : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X✝))\nX : LocallyRingedSpace\nR : CommRingCat\nf : R ⟶ Γ.obj (op X)\nβ : X ⟶ Spec.locallyRingedSpaceObj R\nw : X.toΓSpec.val.base ≫ (Spec.locallyRingedSpaceMap f).val.base = β.val.base\nh : ∀ (r : ↑R), f ≫ X.presheaf.map (homOfLE ⋯).op = toOpen (↑R) (basicOpen r) ≫ β.val.c.app (op (basicOpen r))\nr : ↑R\nU : Opens (PrimeSpectrum ↑R) := basicOpen r\n⊢ CommRingCat.ofHom f ≫\n      (toOpen (↑(Γ.obj (op X))) ((Opens.comap (PrimeSpectrum.comap f)) U) ≫\n          X.toΓSpec.val.c.app (op ((Opens.map (Spec.locallyRingedSpaceMap f).val.base).obj (unop (op U))))) ≫\n        X.presheaf.map (eqToHom ⋯) =\n    toOpen (↑R) U ≫ β.val.c.app (op U)"}
+{"state": [{"context": ["X : Scheme"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}], "premise": [95803], "state_str": "X : Scheme\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}
+{"state": [{"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op ≫ (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op = 𝟙 (op Γ(X, ⊤))"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}], "premise": [88802], "state_str": "X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op ≫ (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op = 𝟙 (op Γ(X, ⊤))\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}
+{"state": [{"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = 𝟙 (op Γ(X, ⊤)) ≫ inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}], "premise": [89664, 96175, 99922, 99924, 126596, 129597], "state_str": "X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = 𝟙 (op Γ(X, ⊤)) ≫ inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}
+{"state": [{"context": ["X : Scheme", "this : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op"], "goal": "Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}], "premise": [71387, 88797, 89625, 89648], "state_str": "X : Scheme\nthis : (Scheme.Hom.app (adjunction.unit.app X) ⊤).op = inv (Scheme.ΓSpecIso Γ(X, ⊤)).inv.op\n⊢ Scheme.Hom.app (adjunction.unit.app X) ⊤ = (Scheme.ΓSpecIso Γ(X, ⊤)).hom"}
+{"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 : β → ℝ≥0∞", "μ : Measure β", "a : α", "s : Set β"], "goal": "∫⁻ (x : β) in s, f x ∂(const α μ) a = ∫⁻ (x : β) in s, f x ∂μ"}], "premise": [72658], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nf : β → ℝ≥0∞\nμ : Measure β\na : α\ns : Set β\n⊢ ∫⁻ (x : β) in s, f x ∂(const α μ) a = ∫⁻ (x : β) in s, f x ∂μ"}
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+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : oddKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [43455], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : oddKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ oddKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}], "premise": [1671, 46181, 117816], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ oddKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r"}], "premise": [39208, 43365, 43411], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖oddKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖oddKernel a x‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r : ℝ"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [22987], "state_str": "a : UnitAddCircle\nr : ℝ\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : sinKernel a =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}], "premise": [43455], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : sinKernel a =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => (ofReal' ∘ sinKernel a) x - 0) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}], "premise": [1671, 46181, 117816], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖(ofReal' ∘ sinKernel a) x - 0‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "r v : ℝ", "hv : 0 < v", "hv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)"], "goal": "(fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r"}], "premise": [39208, 43365, 43411], "state_str": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x => ‖sinKernel a x‖) =O[atTop] fun x => rexp (-v * x)\n⊢ (fun x => ‖sinKernel a x‖) =O[atTop] fun x => x ^ r"}
+{"state": [{"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ∈ Ioi 0"], "goal": "(ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x"}], "premise": [1670, 14286, 22982, 40068, 117810, 118863, 119728, 119770, 148304], "state_str": "a : UnitAddCircle\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ (ofReal' ∘ oddKernel a) (1 / x) = (1 * ↑(x ^ (3 / 2))) • (ofReal' ∘ sinKernel a) x"}
+{"state": [{"context": ["m n : ℤ", "hn : n ≠ 0"], "goal": "(↑m / ↑n).den = 1 ↔ n ∣ m"}], "premise": [1673, 147952], "state_str": "m n : ℤ\nhn : n ≠ 0\n⊢ (↑m / ↑n).den = 1 ↔ n ∣ m"}
+{"state": [{"context": ["d m n : Nat", "dgt1 : 1 < d", "Hm : d ∣ m", "Hn : d ∣ n", "co : m.Coprime n"], "goal": "d ∣ 1"}], "premise": [394], "state_str": "d m n : Nat\ndgt1 : 1 < d\nHm : d ∣ m\nHn : d ∣ n\nco : m.Coprime n\n⊢ d ∣ 1"}
+{"state": [{"context": ["d m n : Nat", "dgt1 : 1 < d", "Hm : d ∣ m", "Hn : d ∣ n", "co : m.Coprime n"], "goal": "d ∣ m.gcd n"}], "premise": [3620], "state_str": "d m n : Nat\ndgt1 : 1 < d\nHm : d ∣ m\nHn : d ∣ n\nco : m.Coprime n\n⊢ d ∣ m.gcd n"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : DecidableEq α", "s✝ t u v : Finset α", "a✝¹ b a : α", "s : Finset α", "a✝ : α"], "goal": "a✝ ∈ s \\ {a} ↔ a✝ ∈ s.erase a"}], "premise": [1713, 1723, 138737, 138958, 138998], "state_str": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝¹ b a : α\ns : Finset α\na✝ : α\n⊢ a✝ ∈ s \\ {a} ↔ a✝ ∈ s.erase a"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α"], "goal": "IsTopologicalBasis (lawsonBasis α)"}], "premise": [133633], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\n⊢ IsTopologicalBasis (lawsonBasis α)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "IsTopologicalBasis (image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U})"}], "premise": [55117, 57726, 57730], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ IsTopologicalBasis\n    (image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U})"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "inst✝¹ = induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓ induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}], "premise": [54654], "state_str": "case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ inst✝¹ =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}"], "goal": "lower α ⊓ scott α = induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓ induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}], "premise": [19], "state_str": "case h.e'_2\nα : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\n⊢ lower α ⊓ scott α =\n    induced (⇑WithLower.toLower) WithLower.instTopologicalSpace ⊓\n      induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Preorder α", "inst✝¹ : TopologicalSpace α", "inst✝ : IsLawson α", "lawsonBasis_image2 : lawsonBasis α = image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α)) {U | IsOpen U}", "x✝ : TopologicalSpace α := scott α"], "goal": "scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}], "premise": [54002, 65595], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLawson α\nlawsonBasis_image2 :\n  lawsonBasis α =\n    image2 (fun x x_1 => ⇑WithLower.toLower ⁻¹' x ∩ ⇑WithScott.toScott ⁻¹' x_1) (IsLower.lowerBasis (WithLower α))\n      {U | IsOpen U}\nx✝ : TopologicalSpace α := scott α\n⊢ scott α = induced (⇑WithScott.toScott) WithScott.instTopologicalSpace"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁"}], "premise": [91601, 114861, 114910], "state_str": "C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₁ = S₁.f ≫ (h.h₁ - h'.h₁) + (h.h₀ - h'.h₀) + (φ₂ - φ₄).τ₁"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₂ = S₁.g ≫ (h.h₂ - h'.h₂) + (h.h₁ - h'.h₁) ≫ S₂.f + (φ₂ - φ₄).τ₂"}], "premise": [91600, 91601, 114862, 114912], "state_str": "C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₂ = S₁.g ≫ (h.h₂ - h'.h₂) + (h.h₁ - h'.h₁) ≫ S₂.f + (φ₂ - φ₄).τ₂"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.133780, u_1} C", "inst✝ : Preadditive C", "S₁ S₂ S₃ : ShortComplex C", "φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂", "h : Homotopy φ₁ φ₂", "h' : Homotopy φ₃ φ₄"], "goal": "(φ₁ - φ₃).τ₃ = h.h₃ - h'.h₃ + (h.h₂ - h'.h₂) ≫ S₂.g + (φ₂ - φ₄).τ₃"}], "premise": [91600, 114863, 114911], "state_str": "C : Type u_1\ninst✝¹ : Category.{?u.133780, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh : Homotopy φ₁ φ₂\nh' : Homotopy φ₃ φ₄\n⊢ (φ₁ - φ₃).τ₃ = h.h₃ - h'.h₃ + (h.h₂ - h'.h₂) ≫ S₂.g + (φ₂ - φ₄).τ₃"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "h1 : f 1 = 0", "f_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y"], "goal": "False"}], "premise": [108557, 108558, 119728], "state_str": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f (1 * y) ≤ c * f 1 * f y\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommRing R", "f : R → ℝ", "c : ℝ", "f_ne_zero : f ≠ 0", "f_nonneg : 0 ≤ f", "h1 : f 1 = 0", "f_mul : ∀ (y : R), f y ≤ 0"], "goal": "False"}], "premise": [1673, 71383], "state_str": "R : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_ne_zero : f ≠ 0\nf_nonneg : 0 ≤ f\nh1 : f 1 = 0\nf_mul : ∀ (y : R), f y ≤ 0\n⊢ False"}
+{"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 : M"], "goal": "x ∈ (extChartAt I x).source"}], "premise": [67287, 67848], "state_str": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx : M\n⊢ x ∈ (extChartAt I x).source"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}], "premise": [70039], "state_str": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}], "premise": [70039], "state_str": "case inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}], "premise": [395, 1674, 2110, 108885, 143611, 143612], "state_str": "case inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b ↔ n ∣ a ∨ n ∣ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n✝ p : R", "k n a b : ℕ", "hab : a.Coprime b", "hn : IsPrimePow n", "ha : a ≠ 0", "hb : b ≠ 0"], "goal": "n ∣ a * b → n ∣ a ∨ n ∣ b"}], "premise": [1673, 111243], "state_str": "case inr.inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn✝ p : R\nk n a b : ℕ\nhab : a.Coprime b\nhn : IsPrimePow n\nha : a ≠ 0\nhb : b ≠ 0\n⊢ n ∣ a * b → n ∣ a ∨ n ∣ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)"], "goal": "p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b"}], "premise": [108269, 120650, 144202, 144694, 148168], "state_str": "case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ p ^ k ∣ a * b → p ^ k ∣ a ∨ p ^ k ∣ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)"], "goal": "k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}], "premise": [13484, 70089, 138900, 144182, 148065], "state_str": "case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommMonoidWithZero R", "n p✝ : R", "k✝ a b : ℕ", "hab : a.Coprime b", "ha : a ≠ 0", "hb : b ≠ 0", "p k : ℕ", "hp : Prime p", "left✝ : 0 < k", "hn : IsPrimePow (p ^ k)", "this : a.factorization p = 0 ∨ b.factorization p = 0"], "goal": "k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}], "premise": [70078], "state_str": "case inr.inr.intro.intro.intro.intro\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn p✝ : R\nk✝ a b : ℕ\nhab : a.Coprime b\nha : a ≠ 0\nhb : b ≠ 0\np k : ℕ\nhp : Prime p\nleft✝ : 0 < k\nhn : IsPrimePow (p ^ k)\nthis : a.factorization p = 0 ∨ b.factorization p = 0\n⊢ k ≤ a.factorization p + b.factorization p → k ≤ a.factorization p ∨ k ≤ b.factorization p"}
+{"state": [{"context": ["s : ℂ", "hs : 0 < s.re"], "goal": "Gamma s ≠ 0"}], "premise": [40597], "state_str": "s : ℂ\nhs : 0 < s.re\n⊢ Gamma s ≠ 0"}
+{"state": [{"context": ["s : ℂ", "hs : 0 < s.re", "m : ℕ"], "goal": "s ≠ -↑m"}], "premise": [53688], "state_str": "s : ℂ\nhs : 0 < s.re\nm : ℕ\n⊢ s ≠ -↑m"}
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+{"state": [{"context": ["a k x y : ℕ", "x✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y", "a1 : 1 < a", "hx : xn a1 k = x", "hy : yn a1 k = y", "kpos : k > 0"], "goal": "k ≤ yn a1 k ∧ (xn a1 k = 1 ∧ yn a1 k = 0 ∨ ∃ u v s t b, xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 k] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])"}], "premise": [408, 410, 1673, 1674, 2107, 2528, 3850, 4467, 4470, 14281, 14286, 14287, 22497, 22511, 22512, 22513, 22514, 22516, 22518, 22526, 22527, 22543, 101702, 108876, 108897, 108900, 130033, 144515, 144516, 144518, 144519, 144524], "state_str": "a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\nkpos : k > 0\n⊢ k ≤ yn a1 k ∧\n    (xn a1 k = 1 ∧ yn a1 k = 0 ∨\n      ∃ u v s t b,\n        xn a1 k * xn a1 k - (a * a - 1) * yn a1 k * yn a1 k = 1 ∧\n          u * u - (a * a - 1) * v * v = 1 ∧\n            s * s - (b * b - 1) * t * t = 1 ∧\n              1 < b ∧\n                b ≡ 1 [MOD 4 * yn a1 k] ∧\n                  b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 k * yn a1 k ∣ v ∧ s ≡ xn a1 k [MOD u] ∧ t ≡ k [MOD 4 * yn a1 k])"}
+{"state": [{"context": ["a k x y : ℕ", "x✝ : 1 < a ∧ k ≤ y ∧ (x = 1 ∧ y = 0 ∨ ∃ u v s t b, x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])", "a1 : 1 < a", "ky : k ≤ 0", "o : x = 1 ∧ y = 0 ∨ ∃ u v s t b, x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]", "x1 : x = 1", "y0 : y = 0"], "goal": "xn a1 k = x ∧ yn a1 k = y"}], "premise": [3741], "state_str": "a k x y : ℕ\nx✝ :\n  1 < a ∧\n    k ≤ y ∧\n      (x = 1 ∧ y = 0 ∨\n        ∃ u v s t b,\n          x * x - (a * a - 1) * y * y = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])\na1 : 1 < a\nky : k ≤ 0\no :\n  x = 1 ∧ y = 0 ∨\n    ∃ u v s t b,\n      x * x - (a * a - 1) * y * y = 1 ∧\n        u * u - (a * a - 1) * v * v = 1 ∧\n          s * s - (b * b - 1) * t * t = 1 ∧\n            1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y]\nx1 : x = 1\ny0 : y = 0\n⊢ xn a1 k = x ∧ yn a1 k = y"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0"], "goal": "i = k"}], "premise": [3523, 14292, 14314, 22512, 108900], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\n��� i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n"], "goal": "i = k"}], "premise": [22519, 108892], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\n⊢ i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n", "yd : 4 * yn a1 i ∣ 4 * n"], "goal": "i = k"}], "premise": [1673, 2528, 14286, 22526, 130033, 144515, 144516, 144524], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\n⊢ i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n", "yd : 4 * yn a1 i ∣ 4 * n", "jk : j ≡ k [MOD 4 * yn a1 i]"], "goal": "i = k"}], "premise": [3854, 14288, 22512, 22514, 103917, 122222], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\n⊢ i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n", "yd : 4 * yn a1 i ∣ 4 * n", "jk : j ≡ k [MOD 4 * yn a1 i]", "ki : k + i < 4 * yn a1 i"], "goal": "i = k"}], "premise": [1673, 2107, 2112, 3523, 3748, 14287, 14291, 22542, 22543, 144515, 144516, 144518, 144524, 144533], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\n⊢ i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n", "yd : 4 * yn a1 i ∣ 4 * n", "jk : j ≡ k [MOD 4 * yn a1 i]", "ki : k + i < 4 * yn a1 i", "ji : j ≡ i [MOD 4 * n]"], "goal": "i = k"}], "premise": [144515, 144516, 144524], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\n⊢ i = k"}
+{"state": [{"context": ["a k x y : ℕ", "a1 : 1 < a", "ky✝ : k ≤ y", "u v s t b : ℕ", "b1 : 1 < b", "i n j : ℕ", "bm1 : b ≡ 1 [MOD 4 * yn a1 i]", "ba : b ≡ a [MOD xn a1 n]", "vp : 0 < yn a1 n", "yv : yn a1 i * yn a1 i ∣ yn a1 n", "sx : xn b1 j ≡ xn a1 i [MOD xn a1 n]", "tk : yn b1 j ≡ k [MOD 4 * yn a1 i]", "ky : k ≤ yn a1 i", "x✝ : 1 < a ∧ k ≤ yn a1 i ∧ (xn a1 i = 1 ∧ yn a1 i = 0 ∨ ∃ u v s t b, xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * yn a1 i] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])", "ipos : i > 0", "iln : i ≤ n", "yd : 4 * yn a1 i ∣ 4 * n", "jk : j ≡ k [MOD 4 * yn a1 i]", "ki : k + i < 4 * yn a1 i", "ji : j ≡ i [MOD 4 * n]", "this : i % (4 * yn a1 i) = k % (4 * yn a1 i)"], "goal": "i = k"}], "premise": [3747, 3748, 4467, 14288], "state_str": "a k x y : ℕ\na1 : 1 < a\nky✝ : k ≤ y\nu v s t b : ℕ\nb1 : 1 < b\ni n j : ℕ\nbm1 : b ≡ 1 [MOD 4 * yn a1 i]\nba : b ≡ a [MOD xn a1 n]\nvp : 0 < yn a1 n\nyv : yn a1 i * yn a1 i ∣ yn a1 n\nsx : xn b1 j ≡ xn a1 i [MOD xn a1 n]\ntk : yn b1 j ≡ k [MOD 4 * yn a1 i]\nky : k ≤ yn a1 i\nx✝ :\n  1 < a ∧\n    k ≤ yn a1 i ∧\n      (xn a1 i = 1 ∧ yn a1 i = 0 ∨\n        ∃ u v s t b,\n          xn a1 i * xn a1 i - (a * a - 1) * yn a1 i * yn a1 i = 1 ∧\n            u * u - (a * a - 1) * v * v = 1 ∧\n              s * s - (b * b - 1) * t * t = 1 ∧\n                1 < b ∧\n                  b ≡ 1 [MOD 4 * yn a1 i] ∧\n                    b ≡ a [MOD u] ∧ 0 < v ∧ yn a1 i * yn a1 i ∣ v ∧ s ≡ xn a1 i [MOD u] ∧ t ≡ k [MOD 4 * yn a1 i])\nipos : i > 0\niln : i ≤ n\nyd : 4 * yn a1 i ∣ 4 * n\njk : j ≡ k [MOD 4 * yn a1 i]\nki : k + i < 4 * yn a1 i\nji : j ≡ i [MOD 4 * n]\nthis : i % (4 * yn a1 i) = k % (4 * yn a1 i)\n⊢ i = k"}
+{"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"], "goal": "W.negAddY P Q / addZ P Q ^ 3 = W.toAffine.negAddY (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))"}], "premise": [117764, 117896, 145550, 145552, 145554, 145565, 145571], "state_str": "R : Type u\ninst✝¹ : CommRing R\nW' : Jacobian R\nF : Type v\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ^ 2 ≠ Q x * P z ^ 2\n⊢ W.negAddY P Q / addZ P Q ^ 3 =\n    W.toAffine.negAddY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)\n      (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3))"}
+{"state": [{"context": ["α : Type u_1", "r : α → α → Prop", "a o : Ordinal.{u}", "f : (b : Ordinal.{u}) → b < o → Ordinal.{u}"], "goal": "(succ o).IsFundamentalSequence 1 fun x x => o"}], "premise": [49684, 52489], "state_str": "α : Type u_1\nr : α → α → Prop\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\n⊢ (succ o).IsFundamentalSequence 1 fun x x => o"}
+{"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 ν : 𝕜", "hμν : μ ≠ ν", "v : E", "hv : v ∈ eigenspace T μ", "w : E", "hw : w ∈ eigenspace T ν", "hv' : ¬v = 0"], "goal": "⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0"}], "premise": [33386, 88217], "state_str": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0"}
+{"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 ν : 𝕜", "hμν : μ ≠ ν", "v : E", "hv : v ∈ eigenspace T μ", "w : E", "hw : w ∈ eigenspace T ν", "hv' : ¬v = 0", "H : (starRingEnd 𝕜) μ = μ"], "goal": "⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0"}], "premise": [88218], "state_str": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw⟩⟫_𝕜 = 0"}
+{"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 ν : 𝕜", "hμν : μ ≠ ν", "v : E", "hv✝ : v ∈ eigenspace T μ", "hv : T v = μ • v", "w : E", "hw✝ : w ∈ eigenspace T ν", "hw : T w = ν • w", "hv' : ¬v = 0", "H : (starRingEnd 𝕜) μ = μ"], "goal": "⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0"}], "premise": [1690, 2111], "state_str": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0"}
+{"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 ν : 𝕜", "hμν : μ ≠ ν", "v : E", "hv✝ : v ∈ eigenspace T μ", "hv : T v = μ • v", "w : E", "hw✝ : w ∈ eigenspace T ν", "hw : T w = ν • w", "hv' : ¬v = 0", "H : (starRingEnd 𝕜) μ = μ"], "goal": "ν = μ ∨ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0"}], "premise": [2100, 36758, 36761], "state_str": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd 𝕜) μ = μ\n⊢ ν = μ ∨ ⟪((fun μ => (eigenspace T μ).subtypeₗᵢ) μ) ⟨v, hv✝⟩, ((fun μ => (eigenspace T μ).subtypeₗᵢ) ν) ⟨w, hw✝⟩⟫_𝕜 = 0"}
+{"state": [{"context": ["I : Type u", "inst✝⁵ : AddMonoid I", "C : Type u_1", "inst✝⁴ : Category.{u_2, u_1} C", "inst✝³ : MonoidalCategory C", "inst✝² : DecidableEq I", "inst✝¹ : HasInitial C", "inst✝ : (X₂ : C) → PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)", "X X' : GradedObject I C", "φ : X ⟶ X'"], "goal": "tensorHom (𝟙 tensorUnit) φ ≫ (leftUnitor X').hom = (leftUnitor X).hom ≫ φ"}], "premise": [98495], "state_str": "I : Type u\ninst✝⁵ : AddMonoid I\nC : Type u_1\ninst✝⁴ : Category.{u_2, u_1} C\ninst✝³ : MonoidalCategory C\ninst✝² : DecidableEq I\ninst✝¹ : HasInitial C\ninst✝ : (X₂ : C) → PreservesColimit (Functor.empty C) ((curriedTensor C).flip.obj X₂)\nX X' : GradedObject I C\nφ : X ⟶ X'\n⊢ tensorHom (𝟙 tensorUnit) φ ≫ (leftUnitor X').hom = (leftUnitor X).hom ≫ φ"}
+{"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", "g₁ g₂ : G"], "goal": "(hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ RightCosetEquivalence (↑H) g₁ g₂"}], "premise": [6894], "state_str": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (↑H) T\nhSK : IsComplement S ↑K\ng₁ g₂ : G\n⊢ (hHT.equiv g₁).2 = (hHT.equiv g₂).2 ↔ RightCosetEquivalence (↑H) g₁ g₂"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝ : Category.{u_3, u_2} A", "G : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A", "adj : G ⊣ sheafToPresheaf J A", "P : Cᵒᵖ ⥤ A"], "goal": "J.W (adj.unit.app P)"}], "premise": [90739], "state_str": "C : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{u_3, u_2} A\nG : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A\nadj : G ⊣ sheafToPresheaf J A\nP : Cᵒᵖ ⥤ A\n⊢ J.W (adj.unit.app P)"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{u_4, u_1} C", "J : GrothendieckTopology C", "A : Type u_2", "inst✝ : Category.{u_3, u_2} A", "G : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A", "adj : G ⊣ sheafToPresheaf J A", "P : Cᵒᵖ ⥤ A"], "goal": "LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)"}], "premise": [91750], "state_str": "C : Type u_1\ninst✝¹ : Category.{u_4, u_1} C\nJ : GrothendieckTopology C\nA : Type u_2\ninst✝ : Category.{u_3, u_2} A\nG : (Cᵒᵖ ⥤ A) ⥤ Sheaf J A\nadj : G ⊣ sheafToPresheaf J A\nP : Cᵒᵖ ⥤ A\n⊢ LeftBousfield.W (fun x => x ∈ Set.range (sheafToPresheaf J A).obj) (adj.unit.app P)"}
+{"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)"}], "premise": [2199, 7332, 119784], "state_str": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.ofNat n) (x ^ Int.ofNat n)"}
+{"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.negSucc n) (x ^ Int.negSucc n)"}], "premise": [7332, 7338, 119787], "state_str": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.negSucc n) (x ^ Int.negSucc n)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_3", "f : α → β", "hf : Surjective f", "s : Set β"], "goal": "∃ a, f '' a = s"}], "premise": [1674, 2045], "state_str": "α : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ ∃ a, f '' a = s"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Sort u_3", "f : α → β", "hf : Surjective f", "s : Set β"], "goal": "f '' (f ⁻¹' s) = s"}], "premise": [134276], "state_str": "case h\nα : Type u_1\nβ : Type u_2\nι : Sort u_3\nf : α → β\nhf : Surjective f\ns : Set β\n⊢ f '' (f ⁻¹' s) = s"}
+{"state": [{"context": ["a : ℍ"], "goal": "normSq a = ‖a‖ * ‖a‖"}], "premise": [36833, 43811], "state_str": "a : ℍ\n⊢ normSq a = ‖a‖ * ‖a‖"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝ : MeasurableSpace α", "p : PMF α", "s t : Set α", "hs : MeasurableSet s"], "goal": "p.toMeasure s = 0 ↔ Disjoint p.support s"}], "premise": [1713, 73711, 73718], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : MeasurableSpace α\np : PMF α\ns t : Set α\nhs : MeasurableSet s\n⊢ p.toMeasure s = 0 ↔ Disjoint p.support s"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "i j : ι", "hij : i ≠ j", "h : Set.univ = {i, j}"], "goal": "IsInternal A ↔ IsCompl (A i) (A j)"}], "premise": [1673, 133307], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\n⊢ IsInternal A ↔ IsCompl (A i) (A j)"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "i j : ι", "hij : i ≠ j", "h : Set.univ = {i, j}", "this : ∀ (k : ι), k = i ∨ k = j"], "goal": "IsInternal A ↔ IsCompl (A i) (A j)"}], "premise": [15813, 19252, 116972, 134113, 134123, 134174], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ IsInternal A ↔ IsCompl (A i) (A j)"}
+{"state": [{"context": ["R : Type u", "inst✝² : Ring R", "ι : Type v", "dec_ι : DecidableEq ι", "M : Type u_1", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "A : ι → Submodule R M", "i j : ι", "hij : i ≠ j", "h : Set.univ = {i, j}", "this : ∀ (k : ι), k = i ∨ k = j"], "goal": "Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ ↔ IsCompl (A i) (A j)"}], "premise": [1674, 13520, 13522], "state_str": "R : Type u\ninst✝² : Ring R\nι : Type v\ndec_ι : DecidableEq ι\nM : Type u_1\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : ι → Submodule R M\ni j : ι\nhij : i ≠ j\nh : Set.univ = {i, j}\nthis : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (A i) (A j) ∧ A i ⊔ A j = ⊤ ↔ IsCompl (A i) (A j)"}
+{"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": "IsClosed (Quotient.mk' '' t)"}], "premise": [7711, 56056, 66531], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (Quotient.mk' '' t)"}
+{"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": "IsClosed (⋃ g, (fun x => g • x) '' t)"}], "premise": [58141, 66867], "state_str": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ IsClosed (⋃ g, (fun x => g • x) '' t)"}
+{"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"}], "premise": [132836, 135377], "state_str": "case h.e'_3\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : ContinuousInv α\ninst✝¹ : ContinuousSMul α β\ns : Set α\nt✝ : Set β\ninst✝ : CompactSpace α\nt : Set β\nht : IsClosed t\n⊢ ⋃ g, (fun x => g • x) '' t = univ • t"}
+{"state": [{"context": ["n : ℕ", "α : Type u_1", "inst✝ : CommMonoid α", "p : ℕ", "f : ℕ → α", "h : Prime p"], "goal": "∏ x ∈ p.divisors, f x = f p * f 1"}], "premise": [21779, 21831, 126898, 144291], "state_str": "n : ℕ\nα : Type u_1\ninst✝ : CommMonoid α\np : ℕ\nf : ℕ → α\nh : Prime p\n⊢ ∏ x ∈ p.divisors, f x = f p * f 1"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [44916, 45095], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [28597], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x", "Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [28597], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x", "Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume", "F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [26829], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x", "Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume", "F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume", "A : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [1673, 14296, 15467, 15488, 18778, 25641, 27559, 30100], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x", "Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume", "F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume", "A : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))", "B : limUnder atBot (⇑F ∘ f) = F 0"], "goal": "Tendsto f atBot (𝓝 0)"}], "premise": [56040], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (limUnder atBot (⇑F ∘ f)))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Tendsto f atBot (𝓝 0)"}
+{"state": [{"context": ["E : Type u_1", "f f' : ℝ → E", "g g' : ℝ → ℝ", "a b l : ℝ", "m : E", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℝ E", "hderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x", "f'int : IntegrableOn f' (Iic a) volume", "fint : IntegrableOn f (Iic a) volume", "F : E →L[ℝ] Completion E := Completion.toComplL", "Fderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x", "Fint : IntegrableOn (⇑F ∘ f) (Iic a) volume", "F'int : IntegrableOn (⇑F ∘ f') (Iic a) volume", "A : Tendsto (⇑F ∘ f) atBot (𝓝 (F 0))", "B : limUnder atBot (⇑F ∘ f) = F 0"], "goal": "Embedding ⇑F"}], "premise": [59270, 59554], "state_str": "E : Type u_1\nf f' : ℝ → E\ng g' : ℝ → ℝ\na b l : ℝ\nm : E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\nfint : IntegrableOn f (Iic a) volume\nF : E →L[ℝ] Completion E := Completion.toComplL\nFderiv : ∀ x ∈ Iic a, HasDerivAt (⇑F ∘ f) (F (f' x)) x\nFint : IntegrableOn (⇑F ∘ f) (Iic a) volume\nF'int : IntegrableOn (⇑F ∘ f') (Iic a) volume\nA : Tendsto (⇑F ∘ f) atBot (𝓝 (F 0))\nB : limUnder atBot (⇑F ∘ f) = F 0\n⊢ Embedding ⇑F"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)"], "goal": "index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V"}], "premise": [14296, 29543], "state_str": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)"], "goal": "index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V"}], "premise": [29535, 55281, 58110], "state_str": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\n⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card"], "goal": "index K₁.carrier V + index K₂.carrier V ≤ s.card"}], "premise": [14273, 103917, 133327, 133419, 133420], "state_str": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ index K₁.carrier V + index K₂.carrier V ≤ s.card"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card"], "goal": "(Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s).card + (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s).card ≤ s.card"}], "premise": [139121], "state_str": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s).card +\n      (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s).card ≤\n    s.card"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card"], "goal": "Disjoint (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s) (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s)"}], "premise": [1674, 139113], "state_str": "case intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\n⊢ Disjoint (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₁.carrier).Nonempty) s)\n    (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K₂.carrier).Nonempty) s)"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card", "g₁ : G", "a✝ : g₁ ∈ s", "g₂ : G", "h1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V", "h2g₂ : g₂ ∈ K₁.carrier", "g₃ : G", "h1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V", "h2g₃ : g₃ ∈ K₂.carrier"], "goal": "False"}], "premise": [131591], "state_str": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V\nh2g₃ : g₃ ∈ K₂.carrier\n⊢ False"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card", "g₁ : G", "a✝ : g₁ ∈ s", "g₂ : G", "h2g₂ : g₂ ∈ K₁.carrier", "g₃ : G", "h2g₃ : g₃ ∈ K₂.carrier", "h1g₃ : g₁ * g₃ ∈ V", "h1g₂ : g₁ * g₂ ∈ V"], "goal": "False"}], "premise": [13484], "state_str": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ False"}
+{"state": [{"context": ["G : Type u_1", "inst✝² : Group G", "inst✝¹ : TopologicalSpace G", "inst✝ : TopologicalGroup G", "K₁ K₂ : Compacts G", "V : Set G", "hV : (interior V).Nonempty", "h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)", "s : Finset G", "h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V", "h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V", "this : ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V, index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card", "g₁ : G", "a✝ : g₁ ∈ s", "g₂ : G", "h2g₂ : g₂ ∈ K₁.carrier", "g₃ : G", "h2g₃ : g₃ ∈ K₂.carrier", "h1g₃ : g₁ * g₃ ∈ V", "h1g₂ : g₁ * g₂ ∈ V"], "goal": "g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹"}], "premise": [2038, 70141, 131762, 131803], "state_str": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : TopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ K₂.carrier) V\nthis :\n  ∀ K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V,\n    index K V ≤ (Finset.filter (fun g => ((fun h => g * h) ⁻¹' V ∩ K).Nonempty) s).card\ng₁ : G\na✝ : g₁ ∈ s\ng₂ : G\nh2g₂ : g₂ ∈ K₁.carrier\ng₃ : G\nh2g₃ : g₃ ∈ K₂.carrier\nh1g₃ : g₁ * g₃ ∈ V\nh1g₂ : g₁ * g₂ ∈ V\n⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹"}
+{"state": [{"context": ["α : Type u", "n : ℕ"], "goal": "(finRange n).length = n"}], "premise": [1562, 132619], "state_str": "α : Type u\nn : ℕ\n⊢ (finRange n).length = 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"], "goal": "Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁"}], "premise": [19704, 20470, 133443], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\na a₁ a₂ b b₁ b₂ c d : α\nh : b₁ ≤ b₂\n⊢ Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁"}
+{"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 : InnerProductSpace 𝕜 E", "inst✝⁴ : NormedAddCommGroup F", "inst✝³ : InnerProductSpace ℝ F", "ι : Type u_4", "inst✝² : DecidableEq ι", "α : ι → Type u_5", "inst✝¹ : (i : ι) → AddZeroClass (α i)", "inst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)", "f : (i : ι) → α i → E", "l : Π₀ (i : ι), α i", "x : E"], "goal": "⟪l.sum f, x⟫_𝕜 = l.sum fun i a => ⟪f i a, x⟫_𝕜"}], "premise": [36766, 118863], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : _root_.RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace ℝ F\nι : Type u_4\ninst✝² : DecidableEq ι\nα : ι → Type u_5\ninst✝¹ : (i : ι) → AddZeroClass (α i)\ninst✝ : (i : ι) → (x : α i) → Decidable (x ≠ 0)\nf : (i : ι) → α i → E\nl : Π₀ (i : ι), α i\nx : E\n⊢ ⟪l.sum f, x⟫_𝕜 = l.sum fun i a => ⟪f i a, x⟫_𝕜"}
+{"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 : AddZeroClass M", "f : α →₀ M", "p : α → Prop", "inst✝ : DecidablePred p"], "goal": "(fun f => ⇑f) (filter p f + filter (fun a => ¬p a) f) = (fun f => ⇑f) f"}], "premise": [148168, 148533], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M\nf : α →₀ M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ (fun f => ⇑f) (filter p f + filter (fun a => ¬p a) f) = (fun f => ⇑f) 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✝¹ : AddZeroClass M", "f : α →₀ M", "p : α → Prop", "inst✝ : DecidablePred p"], "goal": "{x | p x}.indicator ⇑f + {a | ¬p a}.indicator ⇑f = ⇑f"}], "premise": [120946], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst✝¹ : AddZeroClass M\nf : α →₀ M\np : α → Prop\ninst✝ : DecidablePred p\n⊢ {x | p x}.indicator ⇑f + {a | ¬p a}.indicator ⇑f = ⇑f"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "inst✝³ : EMetricSpace X", "inst✝² : EMetricSpace Y", "inst✝¹ : MeasurableSpace X", "inst✝ : BorelSpace X", "m₁ m₂ : ℝ≥0∞ → ℝ≥0∞", "hle : m₁ ≤ᶠ[𝓝[≥] 0] m₂"], "goal": "mkMetric m₁ ≤ mkMetric m₂"}], "premise": [30792, 113018, 143201], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝³ : EMetricSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nhle : m₁ ≤ᶠ[𝓝[≥] 0] m₂\n⊢ mkMetric m₁ ≤ mkMetric m₂"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "M' : Type u_3", "inst✝⁴ : CommRing R", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup M'", "inst✝ : Module R M'", "W : Submodule R M", "φ : Dual R M"], "goal": "φ ∈ LinearMap.range W.mkQ.dualMap ↔ φ ∈ W.dualAnnihilator"}], "premise": [109931], "state_str": "case h\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup M'\ninst✝ : Module R M'\nW : Submodule R M\nφ : Dual R M\n⊢ φ ∈ LinearMap.range W.mkQ.dualMap ↔ φ ∈ W.dualAnnihilator"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "M₀ : PresheafOfModules R₀", "A : Sheaf J AddCommGrp", "φ : M₀.presheaf ⟶ A.val", "inst✝¹ : Presheaf.IsLocallyInjective J φ", "inst✝ : Presheaf.IsLocallySurjective J φ", "X Y : Cᵒᵖ", "π : X ⟶ Y", "r r' : ↑(R.val.obj X)", "m m' : ↑(A.val.obj X)", "S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m"], "goal": "smul α φ (r * r') m = smul α φ r (smul α φ r' m)"}], "premise": [89783, 90673], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)"}
+{"state": [{"context": ["C : Type u₁", "inst✝⁴ : Category.{v₁, u₁} C", "J : GrothendieckTopology C", "R₀ : Cᵒᵖ ⥤ RingCat", "R : Sheaf J RingCat", "α : R₀ ⟶ R.val", "inst✝³ : Presheaf.IsLocallyInjective J α", "inst✝² : Presheaf.IsLocallySurjective J α", "M₀ : PresheafOfModules R₀", "A : Sheaf J AddCommGrp", "φ : M₀.presheaf ⟶ A.val", "inst✝¹ : Presheaf.IsLocallyInjective J φ", "inst✝ : Presheaf.IsLocallySurjective J φ", "X Y : Cᵒᵖ", "π : X ⟶ Y", "r r' : ↑(R.val.obj X)", "m m' : ↑(A.val.obj X)", "S : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m", "hS : S ∈ J.sieves (Opposite.unop X)"], "goal": "smul α φ (r * r') m = smul α φ r (smul α φ r' m)"}], "premise": [90737], "state_str": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nJ : GrothendieckTopology C\nR₀ : Cᵒᵖ ⥤ RingCat\nR : Sheaf J RingCat\nα : R₀ ⟶ R.val\ninst✝³ : Presheaf.IsLocallyInjective J α\ninst✝² : Presheaf.IsLocallySurjective J α\nM₀ : PresheafOfModules R₀\nA : Sheaf J AddCommGrp\nφ : M₀.presheaf ⟶ A.val\ninst✝¹ : Presheaf.IsLocallyInjective J φ\ninst✝ : Presheaf.IsLocallySurjective J φ\nX Y : Cᵒᵖ\nπ : X ⟶ Y\nr r' : ↑(R.val.obj X)\nm m' : ↑(A.val.obj X)\nS : Sieve (Opposite.unop X) := Presheaf.imageSieve α r ⊓ Presheaf.imageSieve α r' ⊓ Presheaf.imageSieve φ m\nhS : S ∈ J.sieves (Opposite.unop X)\n⊢ smul α φ (r * r') m = smul α φ r (smul α φ r' m)"}
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+{"state": [{"context": ["G : Type w", "H : Type x", "α : Type u", "β : Type v", "inst✝³ : TopologicalSpace β", "inst✝² : Group α", "inst✝¹ : MulAction α β", "inst✝ : ContinuousConstSMul α β", "s : Set α", "t : Set β", "a : α", "x : β", "ht : t ∈ 𝓝 x", "u : Set β", "ut : u ⊆ t", "u_open : IsOpen u", "hu : x ∈ u"], "goal": "a • t ∈ 𝓝 (a • x)"}], "premise": [1674, 55494, 64975, 132865, 132875], "state_str": "case intro.intro.intro\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Group α\ninst✝¹ : MulAction α β\ninst✝ : ContinuousConstSMul α β\ns : Set α\nt : Set β\na : α\nx : β\nht : t ∈ 𝓝 x\nu : Set β\nut : u ⊆ t\nu_open : IsOpen u\nhu : x ∈ u\n⊢ a • t ∈ 𝓝 (a • x)"}
+{"state": [{"context": ["ι : Type u_1", "c : ComplexShape ι", "hc : ∀ (j : ι), ∃ i, c.Rel i j", "C : Type u_2", "inst✝³ : Category.{u_3, u_2} C", "inst✝² : Preadditive C", "inst✝¹ : HasBinaryBiproducts C", "inst✝ : CategoryWithHomology C", "K L : HomologicalComplex C c", "f g : K ⟶ L", "h : Homotopy f g", "this : DecidableRel c.Rel"], "goal": "AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g"}], "premise": [92274, 97516, 115554, 115595], "state_str": "ι : Type u_1\nc : ComplexShape ι\nhc : ∀ (j : ι), ∃ i, c.Rel i j\nC : Type u_2\ninst✝³ : Category.{u_3, u_2} C\ninst✝² : Preadditive C\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : CategoryWithHomology C\nK L : HomologicalComplex C c\nf g : K ⟶ L\nh : Homotopy f g\nthis : DecidableRel c.Rel\n⊢ AreEqualizedByLocalization (HomologicalComplex.quasiIso C c) f g"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : Abelian C", "inst✝ : EnoughInjectives C", "X Y : C", "f : X ⟶ Y", "α : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ := { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂, τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "this✝¹ : Epi α.τ₁", "this✝ : IsIso α.τ₂", "this : Mono α.τ₃"], "goal": "(ShortComplex.mk f (d f) ⋯).Exact"}], "premise": [114550], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (d f) ⋯).Exact"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : Abelian C", "inst✝ : EnoughInjectives C", "X Y : C", "f : X ⟶ Y", "α : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ := { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂, τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "this✝¹ : Epi α.τ₁", "this✝ : IsIso α.τ₂", "this : Mono α.τ₃"], "goal": "(ShortComplex.mk f (cokernel.π f) ⋯).Exact"}], "premise": [114589], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : EnoughInjectives C\nX Y : C\nf : X ⟶ Y\nα : ShortComplex.mk f (cokernel.π f) ⋯ ⟶ ShortComplex.mk f (d f) ⋯ :=\n  { τ₁ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₁, τ₂ := 𝟙 (ShortComplex.mk f (cokernel.π f) ⋯).X₂,\n    τ₃ := ι (ShortComplex.mk f (cokernel.π f) ⋯).X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nthis✝¹ : Epi α.τ₁\nthis✝ : IsIso α.τ₂\nthis : Mono α.τ₃\n⊢ (ShortComplex.mk f (cokernel.π f) ⋯).Exact"}
+{"state": [{"context": ["Ω : Type u_1", "E : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f✝ : ℕ → Ω → E", "ℱ✝ : Filtration ℕ m0", "n : ℕ", "R : ℝ≥0", "f : ℕ → Ω → ℝ", "ℱ : Filtration ℕ m0", "hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R"], "goal": "∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ ↑R"}], "premise": [123837, 124747], "state_str": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ✝ : Filtration ℕ m0\nn : ℕ\nR : ℝ≥0\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |predictablePart f ℱ μ (i + 1) ω - predictablePart f ℱ μ i ω| ≤ ↑R"}
+{"state": [{"context": ["Ω : Type u_1", "E : Type u_2", "m0 : MeasurableSpace Ω", "μ : Measure Ω", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f✝ : ℕ → Ω → E", "ℱ✝ : Filtration ℕ m0", "n : ℕ", "R : ℝ≥0", "f : ℕ → Ω → ℝ", "ℱ : Filtration ℕ m0", "hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R"], "goal": "∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(μ[f (i + 1) - f i|↑ℱ i]) ω| ≤ ↑R"}], "premise": [1673, 1674, 27606, 28088], "state_str": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst���² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ : ℕ → Ω → E\nℱ✝ : Filtration ℕ m0\nn : ℕ\nR : ℝ≥0\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |(μ[f (i + 1) - f i|↑ℱ i]) ω| ≤ ↑R"}
+{"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"], "goal": "Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop"}], "premise": [15650, 119707], "state_str": "ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : LinearOrderedField α\nl : Filter β\nf : β → α\nr : α\ninst✝ : l.NeBot\n⊢ Tendsto (fun x => f x * r) l atBot ↔ 0 < r ∧ Tendsto f l atBot ∨ r < 0 ∧ Tendsto f l atTop"}
+{"state": [{"context": ["X Y B : CompHaus", "f : X ⟶ B", "g : Y ⟶ B"], "goal": "pullback.fst f g = ((pullback.isLimit f g).conePointUniqueUpToIso (limit.isLimit (cospan f g))).hom ≫ Limits.pullback.fst f g"}], "premise": [58650, 58651, 93344], "state_str": "X Y B : CompHaus\nf : X ⟶ B\ng : Y ⟶ B\n⊢ pullback.fst f g =\n    ((pullback.isLimit f g).conePointUniqueUpToIso (limit.isLimit (cospan f g))).hom ≫ Limits.pullback.fst f g"}
+{"state": [{"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℂ E", "V : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : FiniteDimensional ℝ V", "inst✝¹ : MeasurableSpace V", "inst✝ : BorelSpace V", "f : V → E", "hf : Integrable f volume", "h'f : Differentiable ℝ f", "hf' : Integrable (fderiv ℝ f) volume"], "goal": "𝓕 (fderiv ℝ f) = fourierSMulRight (-innerSL ℝ) (𝓕 f)"}], "premise": [36923], "state_str": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : V → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) volume\n⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-innerSL ℝ) (𝓕 f)"}
+{"state": [{"context": ["E : Type u_1", "inst✝⁶ : NormedAddCommGroup E", "inst✝⁵ : NormedSpace ℂ E", "V : Type u_2", "inst✝⁴ : NormedAddCommGroup V", "inst✝³ : InnerProductSpace ℝ V", "inst✝² : FiniteDimensional ℝ V", "inst✝¹ : MeasurableSpace V", "inst✝ : BorelSpace V", "f : V → E", "hf : Integrable f volume", "h'f : Differentiable ℝ f", "hf' : Integrable (fderiv ℝ f) volume"], "goal": "𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)"}], "premise": [46507], "state_str": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : V → E\nhf : Integrable f volume\nh'f : Differentiable ℝ f\nhf' : Integrable (fderiv ℝ f) volume\n⊢ 𝓕 (fderiv ℝ f) = fourierSMulRight (-(innerSL ℝ).flip) (𝓕 f)"}
+{"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": "(think ⟨f, al⟩).tail = ⟨f, al⟩"}], "premise": [1751], "state_str": "case mk\nα : Type u\nβ : Type v\nγ : Type w\nf : Stream' (Option α)\nal : ∀ ⦃n : ℕ⦄ ⦃a : α⦄, f n = some a → f (n + 1) = some a\n⊢ (think ⟨f, al⟩).tail = ⟨f, al⟩"}
+{"state": [{"context": ["n a b : ℕ", "H : unpair n = (a, b)"], "goal": "pair a b = n"}], "premise": [142558], "state_str": "n a b : ℕ\nH : unpair n = (a, b)\n⊢ pair a b = n"}
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+{"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", "inst✝ : FiniteDimensional 𝕜 E", "f : E →ₗ[𝕜] F'", "b : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E", "A : Continuous ⇑b.equivFun"], "goal": "Continuous ⇑f"}], "premise": [68738], "state_str": "𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\n⊢ Continuous ⇑f"}
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+{"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", "inst✝ : FiniteDimensional 𝕜 E", "f : E →ₗ[𝕜] F'", "b : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E", "A : Continuous ⇑b.equivFun", "B : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)", "this : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)", "x : E"], "goal": "f x = f (b.equivFun.symm ⇑(b.repr x))"}], "premise": [87365, 87369], "state_str": "case h.e'_5.h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁷ : AddCommGroup E\ninst✝¹⁶ : Module 𝕜 E\ninst✝¹⁵ : TopologicalSpace E\ninst✝¹⁴ : TopologicalAddGroup E\ninst✝¹³ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : Module 𝕜 F\ninst✝¹⁰ : TopologicalSpace F\ninst✝⁹ : TopologicalAddGroup F\ninst✝⁸ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁷ : AddCommGroup F'\ninst✝⁶ : Module 𝕜 F'\ninst✝⁵ : TopologicalSpace F'\ninst✝⁴ : TopologicalAddGroup F'\ninst✝³ : ContinuousSMul 𝕜 F'\ninst✝² : CompleteSpace 𝕜\ninst✝¹ : T2Space E\ninst✝ : FiniteDimensional 𝕜 E\nf : E →ₗ[𝕜] F'\nb : Basis (↑(Basis.ofVectorSpaceIndex 𝕜 E)) 𝕜 E := Basis.ofVectorSpace 𝕜 E\nA : Continuous ⇑b.equivFun\nB : Continuous ⇑(f ∘ₗ ↑b.equivFun.symm)\nthis : Continuous (⇑(f ∘ₗ ↑b.equivFun.symm) ∘ ⇑b.equivFun)\nx : E\n⊢ f x = f (b.equivFun.symm ⇑(b.repr x))"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "f : α → β", "x : α", "r : ℝ", "hr : 0 < r", "K : ℝ", "h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x"], "goal": "ContinuousAt f x"}], "premise": [1674, 15884, 16021, 59429, 61135, 61259, 61315], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ ContinuousAt f x"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Type x", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "f : α → β", "x : α", "r : ℝ", "hr : 0 < r", "K : ℝ", "h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x"], "goal": "Tendsto (fun a => K * dist a x) (𝓝 x) (𝓝 0)"}], "premise": [55628, 55637, 55641, 59781, 65028], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nf : α → β\nx : α\nr : ℝ\nhr : 0 < r\nK : ℝ\nh : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x\n⊢ Tendsto (fun a => K * dist a x) (𝓝 x) (𝓝 0)"}
+{"state": [{"context": ["a x y : ℂ", "hx : x ≠ 0"], "goal": "angle x 1 = |x.arg|"}], "premise": [46473], "state_str": "a x y : ℂ\nhx : x ≠ 0\n⊢ angle x 1 = |x.arg|"}
+{"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"], "goal": "ae μ = ⊥ ↔ μ = 0"}], "premise": [1713, 15958, 27599, 31491, 133601], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\n⊢ ae μ = ⊥ ↔ μ = 0"}
+{"state": [{"context": ["α : Type u", "β : Type u_1", "inst✝⁵ : Ring α", "inst✝⁴ : LinearOrder α", "a b : α", "inst✝³ : PosMulStrictMono α", "inst✝² : MulPosStrictMono α", "inst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1", "inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1"], "goal": "a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)"}], "premise": [106909, 122241], "state_str": "α : Type u\nβ : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\na b : α\ninst✝³ : PosMulStrictMono α\ninst✝² : MulPosStrictMono α\ninst✝¹ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\n⊢ a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)"}
+{"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", "hb : 0 < b"], "goal": "a / b ≤ c ↔ a ≤ b * c"}], "premise": [1713, 106024, 119707], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhb : 0 < b\n⊢ a / b ≤ c ↔ a ≤ b * c"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝⁴ : TopologicalSpace X", "inst✝³ : UniformSpace α", "inst✝² : UniformSpace β", "F : ι → X → α", "G : ι → β → α", "inst✝¹ : TopologicalSpace ι", "inst✝ : CompactSpace X", "F_eqcont : Equicontinuous F"], "goal": "Inducing (⇑UniformFun.ofFun ∘ F) ↔ Inducing F"}], "premise": [54003], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ Inducing (⇑UniformFun.ofFun ∘ F) ↔ Inducing F"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "Y : Type u_3", "α : Type u_4", "β : Type u_5", "inst✝⁴ : TopologicalSpace X", "inst✝³ : UniformSpace α", "inst✝² : UniformSpace β", "F : ι → X → α", "G : ι → β → α", "inst✝¹ : TopologicalSpace ι", "inst✝ : CompactSpace X", "F_eqcont : Equicontinuous F"], "goal": "inst✝¹ = UniformSpace.toTopologicalSpace ↔ inst✝¹ = UniformSpace.toTopologicalSpace"}], "premise": [1713, 59633], "state_str": "ι : Type u_1\nX : Type u_2\nY : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nF : ι → X → α\nG : ι → β → α\ninst✝¹ : TopologicalSpace ι\ninst✝ : CompactSpace X\nF_eqcont : Equicontinuous F\n⊢ inst✝¹ = UniformSpace.toTopologicalSpace ↔ inst✝¹ = UniformSpace.toTopologicalSpace"}
+{"state": [{"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "w : W"], "goal": "∃ ω, cs.IsReduced ω ∧ w = cs.wordProd ω"}], "premise": [8159], "state_str": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\n⊢ ∃ ω, cs.IsReduced ω ∧ w = cs.wordProd ω"}
+{"state": [{"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "hω : ω.length = cs.length (cs.wordProd ω)"], "goal": "∃ ω_1, cs.IsReduced ω_1 ∧ cs.wordProd ω = cs.wordProd ω_1"}], "premise": [1674, 2045], "state_str": "case intro.intro\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ ∃ ω_1, cs.IsReduced ω_1 ∧ cs.wordProd ω = cs.wordProd ω_1"}
+{"state": [{"context": ["B : Type u_1", "W : Type u_2", "inst✝ : Group W", "M : CoxeterMatrix B", "cs : CoxeterSystem M W", "ω : List B", "hω : ω.length = cs.length (cs.wordProd ω)"], "goal": "cs.IsReduced ω ∧ cs.wordProd ω = cs.wordProd ω"}], "premise": [1101, 1674, 1680, 2102, 2104], "state_str": "case h\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nhω : ω.length = cs.length (cs.wordProd ω)\n⊢ cs.IsReduced ω ∧ cs.wordProd ω = cs.wordProd ω"}
+{"state": [{"context": ["R✝ : Type u_1", "R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₁ : ∃ p, Irreducible p", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q"], "goal": "DiscreteValuationRing R"}], "premise": [78845], "state_str": "R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ DiscreteValuationRing R"}
+{"state": [{"context": ["R✝ : Type u_1", "R : Type u", "inst✝² : CommRing R", "inst✝¹ : IsDomain R", "inst✝ : UniqueFactorizationMonoid R", "h₁ : ∃ p, Irreducible p", "h₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q"], "goal": "IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime"}], "premise": [78850], "state_str": "R✝ : Type u_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime"}
+{"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", "X Y : Cᵒᵖ", "f : X ⟶ Y"], "goal": "∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C"}], "premise": [99373], "state_str": "C : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\n⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C"}
+{"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", "X Y : Cᵒᵖ", "f : X ⟶ Y", "Z : C", "g : Z ⟶ Opposite.unop Y", "h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z", "H : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles"], "goal": "Triangle.mk f g.op ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op) ∈ distinguishedTriangles C"}], "premise": [99820], "state_str": "case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Triangle.mk f g.op\n      ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op) ∈\n    distinguishedTriangles C"}
+{"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", "X Y : Cᵒᵖ", "f : X ⟶ Y", "Z : C", "g : Z ⟶ Opposite.unop Y", "h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z", "H : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles"], "goal": "Opposite.unop ((triangleOpEquivalence C).inverse.obj (Triangle.mk f g.op ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ∈ Pretriangulated.distinguishedTriangles"}], "premise": [99351], "state_str": "case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ∈\n    Pretriangulated.distinguishedTriangles"}
+{"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", "X Y : Cᵒᵖ", "f : X ⟶ Y", "Z : C", "g : Z ⟶ Opposite.unop Y", "h : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z", "H : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles"], "goal": "Opposite.unop ((triangleOpEquivalence C).inverse.obj (Triangle.mk f g.op ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ≅ Triangle.mk g f.unop h"}], "premise": [89625, 99798], "state_str": "case intro.intro.intro\nC : Type u_1\ninst✝⁵ : Category.{u_2, u_1} C\ninst✝⁴ : HasShift C ℤ\ninst✝³ : HasZeroObject C\ninst✝² : Preadditive C\ninst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive\ninst✝ : Pretriangulated C\nX Y : Cᵒᵖ\nf : X ⟶ Y\nZ : C\ng : Z ⟶ Opposite.unop Y\nh : Opposite.unop X ⟶ (shiftFunctor C 1).obj Z\nH : Triangle.mk g f.unop h ∈ Pretriangulated.distinguishedTriangles\n⊢ Opposite.unop\n      ((triangleOpEquivalence C).inverse.obj\n        (Triangle.mk f g.op\n          ((opShiftFunctorEquivalence C 1).counitIso.inv.app (Opposite.op Z) ≫ (shiftFunctor Cᵒᵖ 1).map h.op))) ≅\n    Triangle.mk g f.unop h"}
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+{"state": [{"context": ["α : Type u", "σ σ' : Type v", "M : NFA α σ", "inst✝ : Fintype σ", "x : List α", "hx : x ∈ M.toDFA.accepts", "hlen : Fintype.card (Set σ) ≤ x.length"], "goal": "∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.toDFA.accepts"}], "premise": [69391], "state_str": "α : Type u\nσ σ' : Type v\nM : NFA α σ\ninst✝ : Fintype σ\nx : List α\nhx : x ∈ M.toDFA.accepts\nhlen : Fintype.card (Set σ) ≤ x.length\n⊢ ∃ a b c, x = a ++ b ++ c ∧ a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.toDFA.accepts"}
+{"state": [{"context": ["J : Type u₁", "inst✝³ : Category.{v₁, u₁} J", "K : Type u₂", "inst✝² : Category.{v₂, u₂} K", "C : Type u₃", "inst✝¹ : Category.{v₃, u₃} C", "D : Type u₄", "inst✝ : Category.{v₄, u₄} D", "F : J ⥤ C", "e : K ≌ J", "s : Cone (e.functor ⋙ F)", "k : K"], "goal": "(((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app k = (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫ ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app k"}], "premise": [89843, 95537], "state_str": "J : Type u₁\ninst✝³ : Category.{v₁, u₁} J\nK : Type u₂\ninst✝² : Category.{v₂, u₂} K\nC : Type u₃\ninst✝¹ : Category.{v₃, u₃} C\nD : Type u₄\ninst✝ : Category.{v₄, u₄} D\nF : J ⥤ C\ne : K ≌ J\ns : Cone (e.functor ⋙ F)\nk : K\n⊢ (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).π.app k =\n    (Iso.refl (((whiskering e.inverse ⋙ postcompose (e.invFunIdAssoc F).hom) ⋙ whiskering e.functor).obj s).pt).hom ≫\n      ((𝟭 (Cone (e.functor ⋙ F))).obj s).π.app k"}
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+{"state": [{"context": ["α : Type u_1", "inst✝³ : CompleteLattice α", "inst✝² : Group 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", "s✝ t s : Set α"], "goal": "⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹"}], "premise": [2100, 19344], "state_str": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ ⨅ a ∈ s, a⁻¹ = (sSup s)⁻¹"}
+{"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", "hs : x ∉ s"], "goal": "HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f')"}], "premise": [1713, 44155, 133683], "state_str": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nhs : x ∉ s\n⊢ HasDerivWithinAt f f' s x ↔ Tendsto (slope f x) (𝓝[s] x) (𝓝 f')"}
+{"state": [{"context": ["𝕜 : Type u", "inst✝⁸ : NontriviallyNormedField 𝕜", "F : Type v", "inst✝⁷ : NormedAddCommGroup F", "inst✝⁶ : NormedSpace 𝕜 F", "E : Type w", "inst✝⁵ : NormedAddCommGroup E", "inst✝⁴ : 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", "𝕜' : Type u_1", "inst✝³ : NontriviallyNormedField 𝕜'", "inst✝² : NormedAlgebra 𝕜 𝕜'", "inst✝¹ : NormedSpace 𝕜' F", "inst✝ : IsScalarTower 𝕜 𝕜' F", "s' t' : Set 𝕜'", "h : 𝕜 → 𝕜'", "h₁ : 𝕜 → 𝕜", "h₂ : 𝕜' → 𝕜'", "h' h₂' : 𝕜'", "h₁' : 𝕜", "g₁ : 𝕜' → F", "g₁' : F", "L' : Filter 𝕜'", "y : 𝕜'", "hg : HasDerivAt g₁ g₁' (h x)", "hh : HasDerivWithinAt h h' s x", "hy : y = h x"], "goal": "HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x"}], "premise": [44878], "state_str": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\ny : 𝕜'\nhg : HasDerivAt g₁ g₁' (h x)\nhh : HasDerivWithinAt h h' s x\nhy : y = h x\n⊢ HasDerivWithinAt (g₁ ∘ h) (h' • g₁') s x"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : IsDomain R", "inst✝³ : CommRing S", "K : Type u_3", "L : Type u_4", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Algebra K L", "A : Subalgebra K L", "x : ↥A", "p : K[X]", "aeval_eq : (aeval x) p = 0", "coeff_zero_ne : p.coeff 0 ≠ 0"], "goal": "(↑x)⁻¹ ∈ A"}], "premise": [2115, 122022], "state_str": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ (↑x)⁻¹ ∈ A"}
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+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "inst✝⁵ : CommRing R", "inst✝⁴ : IsDomain R", "inst✝³ : CommRing S", "K : Type u_3", "L : Type u_4", "inst✝² : Field K", "inst✝¹ : Field L", "inst✝ : Algebra K L", "A : Subalgebra K L", "x : ↥A", "p : K[X]", "aeval_eq : (aeval x) p = 0", "coeff_zero_ne : p.coeff 0 ≠ 0", "this : (aeval ↑x) p = 0"], "goal": "-(((algebraMap K L) (p.coeff 0))⁻¹ * (aeval ↑x) p.divX) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)"}], "premise": [74168, 119464, 122012, 122054, 124155], "state_str": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : CommRing S\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\nthis : (aeval ↑x) p = 0\n⊢ -(((algebraMap K L) (p.coeff 0))⁻¹ * (aeval ↑x) p.divX) = (algebraMap K L) (-p.coeff 0)⁻¹ * ↑((aeval x) p.divX)"}
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+{"state": [{"context": ["G : Type w", "inst✝¹ : TopologicalSpace G", "μ : Content G", "inst✝ : R1Space G", "A : Set G"], "goal": "MeasurableSet A ↔ ∀ (U : Opens G), μ.outerMeasure (↑U ∩ A) + μ.outerMeasure (↑U \\ A) ≤ μ.outerMeasure ↑U"}], "premise": [55746], "state_str": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ MeasurableSet A ↔ ∀ (U : Opens G), μ.outerMeasure (↑U ∩ A) + μ.outerMeasure (↑U \\ A) ≤ μ.outerMeasure ↑U"}
+{"state": [{"context": ["G : Type w", "inst✝¹ : TopologicalSpace G", "μ : Content G", "inst✝ : R1Space G", "A : Set G"], "goal": "MeasurableSet A ↔ ∀ (U : Set G) (hU : IsOpen U), μ.outerMeasure (↑{ carrier := U, is_open' := hU } ∩ A) + μ.outerMeasure (↑{ carrier := U, is_open' := hU } \\ A) ≤ μ.outerMeasure ↑{ carrier := U, is_open' := hU }"}], "premise": [28156], "state_str": "G : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nA : Set G\n⊢ MeasurableSet A ↔\n    ∀ (U : Set G) (hU : IsOpen U),\n      μ.outerMeasure (↑{ carrier := U, is_open' := hU } ∩ A) + μ.outerMeasure (↑{ carrier := U, is_open' := hU } \\ A) ≤\n        μ.outerMeasure ↑{ carrier := U, is_open' := hU }"}
+{"state": [{"context": ["α : Type u_2", "p✝ q : α → Bool", "β : Type u_1", "p : β → Bool", "f : α → β", "a : α", "l : List α"], "goal": "countP p (map f (a :: l)) = countP (p ∘ f) (a :: l)"}], "premise": [348, 2611], "state_str": "α : Type u_2\np✝ q : α → Bool\nβ : Type u_1\np : β → Bool\nf : α → β\na : α\nl : List α\n⊢ countP p (map f (a :: l)) = countP (p ∘ f) (a :: l)"}
+{"state": [{"context": ["α : Type u_1", "N : Type u_2", "inst✝ : Zero N", "r : α → α → Prop", "s : N → N → Prop", "hbot : ∀ ⦃n : N⦄, ¬s n 0", "hs : WellFounded s", "x : α →₀ N", "h : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a"], "goal": "Acc (Finsupp.Lex r s) x"}], "premise": [147479], "state_str": "α : Type u_1\nN : Type u_2\ninst✝ : Zero N\nr : α → α → Prop\ns : N → N → Prop\nhbot : ∀ ⦃n : N⦄, ¬s n 0\nhs : WellFounded s\nx : α →₀ N\nh : ∀ a ∈ x.support, Acc (rᶜ ⊓ fun x x_1 => x ≠ x_1) a\n⊢ Acc (Finsupp.Lex r s) x"}
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+{"state": [{"context": ["C : Type u_1", "ι✝ : Type u_2", "inst✝² : Category.{u_3, u_1} C", "inst✝¹ : Preadditive C", "c : ComplexShape ι✝", "K : HomologicalComplex C c", "i j : ι✝", "hij : c.Rel i j", "inst✝ : CategoryWithHomology C", "S : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯", "S' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯", "ι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hS : S.Exact", "T : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯", "T' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯", "π : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }", "hT : T.Exact"], "goal": "(composableArrows₃ K i j).Exact"}], "premise": [115189], "state_str": "C : Type u_1\nι✝ : Type u_2\ninst✝² : Category.{u_3, u_1} C\ninst✝¹ : Preadditive C\nc : ComplexShape ι✝\nK : HomologicalComplex C c\ni j : ι✝\nhij : c.Rel i j\ninst✝ : CategoryWithHomology C\nS : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.opcyclesToCycles i j) ⋯\nS' : ShortComplex C := ShortComplex.mk (K.homologyι i) (K.fromOpcycles i j) ⋯\nι : S ⟶ S' := { τ₁ := 𝟙 S.X₁, τ₂ := 𝟙 S.X₂, τ₃ := K.iCycles j, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhS : S.Exact\nT : ShortComplex C := ShortComplex.mk (K.opcyclesToCycles i j) (K.homologyπ j) ⋯\nT' : ShortComplex C := ShortComplex.mk (K.toCycles i j) (K.homologyπ j) ⋯\nπ : T' ⟶ T := { τ₁ := K.pOpcycles i, τ₂ := 𝟙 T'.X₂, τ₃ := 𝟙 T'.X₃, comm₁₂ := ⋯, comm₂₃ := ⋯ }\nhT : T.Exact\n⊢ (composableArrows₃ K i j).Exact"}
+{"state": [{"context": ["V : Type u", "V' : Type v", "G : SimpleGraph V", "G' : SimpleGraph V'", "u v w : V", "p : G.Walk u v"], "goal": "(p.toSubgraph.neighborSet w).Finite"}], "premise": [52795, 52845, 52852, 134992, 135040, 135041], "state_str": "V : Type u\nV' : Type v\nG : SimpleGraph V\nG' : SimpleGraph V'\nu v w : V\np : G.Walk u v\n⊢ (p.toSubgraph.neighborSet w).Finite"}
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+{"state": [{"context": ["C : Type u_1", "D : Type u_2", "E : Type u_3", "J : Type u_4", "inst✝² : Category.{u_5, u_1} C", "inst✝¹ : Category.{u_7, u_2} D", "inst✝ : Category.{u_6, u_3} E", "X Y✝ : GradedObject J C", "e : X ≅ Y✝", "F : C ⥤ D ⥤ E", "j : J", "Y : D"], "goal": "(F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)"}], "premise": [97575, 99919, 99920, 100030, 100031], "state_str": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst✝² : Category.{u_5, u_1} C\ninst✝¹ : Category.{u_7, u_2} D\ninst✝ : Category.{u_6, u_3} E\nX Y✝ : GradedObject J C\ne : X ≅ Y✝\nF : C ⥤ D ⥤ E\nj : J\nY : D\n⊢ (F.map (e.hom j)).app Y ≫ (F.map (e.inv j)).app Y = 𝟙 ((F.obj (X j)).obj Y)"}
+{"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✝⁶ : CommSemiring R", "inst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)", "inst✝⁴ : AddCommMonoid M'", "inst✝³ : AddCommMonoid M₂", "inst✝² : (i : Fin n✝.succ) → Module R (M i)", "inst✝¹ : Module R M'", "inst✝ : Module R M₂", "ι' : Type u_1", "k l n : ℕ", "s : Finset (Fin n)", "hk : s.card = k", "hl : sᶜ.card = l", "f : MultilinearMap R (fun x => M') M₂", "x y : M'"], "goal": "((((curryFinFinset R M₂ M' hk hl) f) fun x_1 => x) fun x => y) = f (s.piecewise (fun x_1 => x) fun x => y)"}], "premise": [2100, 2101, 86202], "state_str": "R : Type uR\nS : Type uS\nι : Type uι\nn✝ : ℕ\nM : Fin n✝.succ → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : Fin n✝.succ) → AddCommMonoid (M i)\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M₂\ninst✝² : (i : Fin n✝.succ) → Module R (M i)\ninst✝¹ : Module R M'\ninst✝ : Module R M₂\nι' : Type u_1\nk l n : ℕ\ns : Finset (Fin n)\nhk : s.card = k\nhl : sᶜ.card = l\nf : MultilinearMap R (fun x => M') M₂\nx y : M'\n⊢ ((((curryFinFinset R M₂ M' hk hl) f) fun x_1 => x) fun x => y) = f (s.piecewise (fun x_1 => x) fun x => y)"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "κ : ι → Sort u_1", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝¹ : CompleteLattice α", "inst✝ : CompleteLattice β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "f : ι → α"], "goal": "IsLUB (range (l ∘ f)) (l (iSup f))"}], "premise": [19267, 134180], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort u_1\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (range (l ∘ f)) (l (iSup f))"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "ι : Sort x", "κ : ι → Sort u_1", "a a₁ a₂ : α", "b b₁ b₂ : β", "inst✝¹ : CompleteLattice α", "inst✝ : CompleteLattice β", "l : α → β", "u : β → α", "gc : GaloisConnection l u", "f : ι → α"], "goal": "IsLUB (l '' range f) (l (sSup (range f)))"}], "premise": [12771, 19204], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort u_1\na a₁ a₂ : α\nb b₁ b₂ : β\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nf : ι → α\n⊢ IsLUB (l '' range f) (l (sSup (range f)))"}
+{"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", "t : Set X", "ht : t ∈ inst✝⁵.1"], "goal": "IsOpen t ∧ IsOpen univ ∧ Prod.fst ⁻¹' t = t ×ˢ univ"}], "premise": [131599], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nt : Set X\nht : t ∈ inst✝⁵.1\n⊢ IsOpen t ∧ IsOpen univ ∧ Prod.fst ⁻¹' t = t ×ˢ univ"}
+{"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", "t : Set Y", "ht : t ∈ inst✝⁴.1"], "goal": "IsOpen univ ∧ IsOpen t ∧ Prod.snd ⁻¹' t = univ ×ˢ t"}], "premise": [131599], "state_str": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\nε : Type u_3\nζ : Type u_4\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace W\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\nt : Set Y\nht : t ∈ inst✝⁴.1\n⊢ IsOpen univ ∧ IsOpen t ∧ Prod.snd ⁻¹' t = univ ×ˢ t"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "r : ℝ", "h₁ : 0 ≤ r", "h₂ : r < 1", "this✝ : r ≠ 1", "this : Tendsto (fun n => (r ^ n - 1) * (r - 1)⁻¹) atTop (𝓝 ((0 - 1) * (r - 1)⁻¹))"], "goal": "Tendsto (fun n => ∑ i ∈ Finset.range n, r ^ i) atTop (𝓝 (1 - r)⁻¹)"}], "premise": [112501, 115837, 119790], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nr : ℝ\nh₁ : 0 ≤ r\nh₂ : r < 1\nthis✝ : r ≠ 1\nthis : Tendsto (fun n => (r ^ n - 1) * (r - 1)⁻¹) atTop (𝓝 ((0 - 1) * (r - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i ∈ Finset.range n, r ^ i) atTop (𝓝 (1 - r)⁻¹)"}
+{"state": [{"context": ["a : ℕ", "a1 : 1 < a", "n : ℕ", "h : 2 * xn a1 n = xn a1 (n + 1)"], "goal": "a = 2 ∧ n = 0"}], "premise": [22482, 119707], "state_str": "a : ℕ\na1 : 1 < a\nn : ℕ\nh : 2 * xn a1 n = xn a1 (n + 1)\n⊢ a = 2 ∧ n = 0"}
+{"state": [{"context": ["a : ℕ", "a1 : 1 < a", "n : ℕ", "h : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n"], "goal": "a = 2 ∧ n = 0"}], "premise": [2112, 3735, 3785, 3850, 14283, 14288, 22478, 22512], "state_str": "a : ℕ\na1 : 1 < a\nn : ℕ\nh : xn a1 n * 2 = xn a1 n * a + Pell.d a1 * yn a1 n\n⊢ a = 2 ∧ n = 0"}
+{"state": [{"context": ["a : ℕ", "a1 : 1 < a", "h : 2 = a"], "goal": "a = 2 ∧ 0 = 0"}], "premise": [2100], "state_str": "case refl\na : ℕ\na1 : 1 < a\nh : 2 = a\n⊢ a = 2 ∧ 0 = 0"}
+{"state": [{"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re"], "goal": "HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}], "premise": [23515, 63937], "state_str": "a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}
+{"state": [{"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "this : HasSum (fun n => cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 + cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|(-↑n)| ^ s / 2) (cosZeta (↑a) s + cexp (2 * ↑π * I * ↑a * ↑0) / ↑|0| ^ s / 2)"], "goal": "HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}], "premise": [39263, 105279, 105293, 108302, 108303, 115817, 117895, 119729, 122241, 128749, 128750, 128753, 142644, 148270], "state_str": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n  HasSum (fun n => cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 + cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|(-↑n)| ^ s / 2)\n    (cosZeta (↑a) s + cexp (2 * ↑π * I * ↑a * ↑0) / ↑|0| ^ s / 2)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}
+{"state": [{"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "this : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)"], "goal": "HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}], "premise": [122240], "state_str": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => ↑(Real.cos (2 * π * a * ↑n)) / ↑n ^ s) (cosZeta (↑a) s)"}
+{"state": [{"context": ["a : ℝ", "s : ℂ", "hs : 1 < s.re", "this : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)"], "goal": "HasSum (fun n => (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)"}], "premise": [64063], "state_str": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n => (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n => (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)"}
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+{"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"], "goal": "(∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b"}], "premise": [108887, 119739], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\n⊢ (∀ (n : ℕ), a ^ (n + 1) ∣ b) ↔ ∀ (n : ℕ), a ^ n ∣ b"}
+{"state": [{"context": ["a b✝ c b x : ℕ"], "goal": "x ∈ Iio b ↔ x ∈ range b"}], "premise": [1713, 19603, 139145], "state_str": "case h.a\na b✝ c b x : ℕ\n⊢ x ∈ Iio b ↔ x ∈ range b"}
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+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "α : Type u_4", "inst✝⁷ : Fintype l", "inst✝⁶ : Fintype m", "inst✝⁵ : Fintype n", "inst✝⁴ : DecidableEq l", "inst✝³ : DecidableEq m", "inst✝² : DecidableEq n", "inst✝¹ : CommRing α", "ι : Type u_5", "inst✝ : Unique ι", "u v : m → α"], "goal": "(1 + col ι u * row ι v).det = 1 + v ⬝ᵥ u"}], "premise": [86013, 86446, 120650, 137767, 142183], "state_str": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁷ : Fintype l\ninst✝⁶ : Fintype m\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nι : Type u_5\ninst✝ : Unique ι\nu v : m → α\n⊢ (1 + col ι u * row ι v).det = 1 + v ⬝ᵥ u"}
+{"state": [{"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t u : Set Ω", "hs : s.Finite"], "goal": "(condCount s) (s ∩ t) = (condCount s) t"}], "premise": [27992, 73025], "state_str": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\n⊢ (condCount s) (s ∩ t) = (condCount s) t"}
+{"state": [{"context": ["a : UnitAddCircle", "x : ℝ", "hx : x ≤ 0"], "goal": "oddKernel a x = 0"}], "premise": [6915], "state_str": "a : UnitAddCircle\nx : ℝ\nhx : x ≤ 0\n⊢ oddKernel a x = 0"}
+{"state": [{"context": ["x : ℝ", "hx : x ≤ 0", "a' : ℝ"], "goal": "oddKernel (↑a') x = 0"}], "premise": [22219, 22221, 22972, 108302, 108558, 119727, 148280, 148289, 148320], "state_str": "case H\nx : ℝ\nhx : x ≤ 0\na' : ℝ\n⊢ oddKernel (↑a') x = 0"}
+{"state": [{"context": ["k : ℕ", "h : k ≠ 0", "hn : ℕ"], "goal": "(hn + 1).primeFactorsList <+ ((hn + 1) * k).primeFactorsList"}], "premise": [130810, 143719], "state_str": "case succ\nk : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+ ((hn + 1) * k).primeFactorsList"}
+{"state": [{"context": ["k : ℕ", "h : k ≠ 0", "hn : ℕ"], "goal": "(hn + 1).primeFactorsList <+~ ((hn + 1) * k).primeFactorsList"}], "premise": [802, 3849, 143731], "state_str": "k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ ((hn + 1) * k).primeFactorsList"}
+{"state": [{"context": ["k : ℕ", "h : k ≠ 0", "hn : ℕ"], "goal": "(hn + 1).primeFactorsList <+~ hn.succ.primeFactorsList ++ k.primeFactorsList"}], "premise": [804, 1294], "state_str": "k : ℕ\nh : k ≠ 0\nhn : ℕ\n⊢ (hn + 1).primeFactorsList <+~ hn.succ.primeFactorsList ++ k.primeFactorsList"}
+{"state": [{"context": ["T : ℝ", "n : ℤ", "x : AddCircle T"], "goal": "↑(-(n • x)).toCircle = (starRingEnd ℂ) ((fourier n) x)"}], "premise": [41724, 110032], "state_str": "T : ℝ\nn : ℤ\nx : AddCircle T\n⊢ ↑(-(n • x)).toCircle = (starRingEnd ℂ) ((fourier n) x)"}
+{"state": [{"context": ["T : ℝ", "n : ℤ", "x : AddCircle T"], "goal": "(fourier (-n)) x = (starRingEnd ℂ) ((fourier n) x)"}], "premise": [41731], "state_str": "T : ℝ\nn : ℤ\nx : AddCircle T\n⊢ (fourier (-n)) x = (starRingEnd ℂ) ((fourier n) 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", "mα : MeasurableSpace α", "inst✝¹ : MeasurableSpace β", "inst✝ : MeasurableSpace γ", "f : α → β", "g✝ : β → γ", "hf : MeasurableEmbedding f", "g : α → γ", "g' : β → γ", "hg : Measurable g", "hg' : Measurable g'"], "goal": "Measurable (extend f g g')"}], "premise": [28192, 28847], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng✝ : β → γ\nhf : MeasurableEmbedding f\ng : α → γ\ng' : β → γ\nhg : Measurable g\nhg' : Measurable g'\n⊢ Measurable (extend f g g')"}
+{"state": [{"context": ["s : Finset ℕ", "n✝ : ℕ", "hsub : s ⊆ (n✝ + 1).properDivisors"], "goal": "∑ x ∈ s, x = ∑ x ∈ (n✝ + 1).properDivisors, x → s = (n✝ + 1).properDivisors"}], "premise": [3744, 21800, 53688, 103903, 107019, 119704, 119727, 119729, 126913, 126952, 138688, 138727, 139024, 139035], "state_str": "case succ\ns : Finset ℕ\nn✝ : ℕ\nhsub : s ⊆ (n✝ + 1).properDivisors\n⊢ ∑ x ∈ s, x = ∑ x ∈ (n✝ + 1).properDivisors, x → s = (n✝ + 1).properDivisors"}
+{"state": [{"context": ["a b : ℝ", "n : ℕ"], "goal": "(∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) = cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n"}], "premise": [1673, 40794, 118010], "state_str": "a b : ℝ\nn : ℕ\n⊢ (∫ (x : ℝ) in a..b, cos x ^ (n + 2)) * (↑n + 2) =\n    cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a + (↑n + 1) * ∫ (x : ℝ) in a..b, cos x ^ n"}
+{"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'", "S : Set X"], "goal": "∀ {x y : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel x S"}], "premise": [56782], "state_str": "F : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\nι : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nS : Set X\n⊢ ∀ {x y : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel x S"}
+{"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'", "S : Set X"], "goal": "∀ {x y z : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel z S → x.HomotopicRel z S"}], "premise": [56783], "state_str": "F : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\nι : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nS : Set X\n⊢ ∀ {x y z : C(X, Y)}, x.HomotopicRel y S → y.HomotopicRel z S → x.HomotopicRel z S"}
+{"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∞", "hs : s.Nonempty"], "goal": "egauge 𝕜 s 0 = 0"}], "premise": [132957], "state_str": "𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nα : Type u_2\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns t : Set E\nx y : E\nr : ℝ≥0∞\nhs : s.Nonempty\n⊢ egauge 𝕜 s 0 = 0"}
+{"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∞", "hs : s.Nonempty", "this : 0 ∈ 0 • s"], "goal": "egauge 𝕜 s 0 = 0"}], "premise": [35024], "state_str": "𝕜 : Type u_1\ninst✝² : NormedDivisionRing 𝕜\nα : Type u_2\nE : Type u_3\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc : 𝕜\ns t : Set E\nx y : E\nr : ℝ≥0∞\nhs : s.Nonempty\nthis : 0 ∈ 0 • s\n⊢ egauge 𝕜 s 0 = 0"}
+{"state": [{"context": ["α : Type u", "a : α", "s t : Set α", "x : α", "H : {x}.Nontrivial"], "goal": "False"}], "premise": [132293, 133516], "state_str": "α : Type u\na : α\ns t : Set α\nx : α\nH : {x}.Nontrivial\n⊢ False"}
+{"state": [{"context": ["α : Type u", "a : α", "s t : Set α", "x : α", "H : ∃ y ∈ {x}, y ≠ x", "y : α", "hy : y ∈ {x}", "hya : y ≠ x"], "goal": "False"}], "premise": [1673, 133512], "state_str": "α : Type u\na : α\ns t : Set α\nx : α\nH : ∃ y ∈ {x}, y ≠ x\ny : α\nhy : y ∈ {x}\nhya : y ≠ x\n⊢ False"}
+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "N : Type u_3", "inst✝⁴ : CommRing R", "inst✝³ : AddCommGroup M", "inst✝² : Module R M", "inst✝¹ : AddCommGroup N", "inst✝ : Module R N", "p : PerfectPairing R M N", "x : M"], "goal": "(↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft) x = (Dual.eval R M) x"}], "premise": [85596], "state_str": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : PerfectPairing R M N\nx : M\n⊢ (↑p.toDualRight.symm.dualMap ∘ₗ ↑p.toDualLeft) x = (Dual.eval R M) x"}
+{"state": [{"context": ["α : Type u", "β : Type v", "ι : Sort w", "κ : ι → Sort w'", "inst✝ : Frame α", "s t : Set α", "a✝ b : α", "f : (i : ι) → κ i → α", "a : α"], "goal": "a ⊓ ⨆ i, ⨆ j, f i j = ⨆ i, ⨆ j, a ⊓ f i j"}], "premise": [14969], "state_str": "α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\ninst✝ : Frame α\ns t : Set α\na✝ b : α\nf : (i : ι) → κ i → α\na : α\n⊢ a ⊓ ⨆ i, ⨆ j, f i j = ⨆ i, ⨆ j, a ⊓ f i j"}
+{"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✝ f g : Π₀ (a : α), N a", "a : α"], "goal": "a ∈ f.neLocus g ↔ f a ≠ g a"}], "premise": [1214, 70132, 138858, 139089, 148948], "state_str": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf✝ g✝ f g : Π₀ (a : α), N a\na : α\n⊢ a ∈ f.neLocus g ↔ f a ≠ g a"}
+{"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", "inst✝ : IsFiniteKernel κ", "hf : IsRatCondKernelCDF f κ ν", "a : α", "x : ℝ", "s : Set β", "hs : MeasurableSet s"], "goal": "∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (s ×ˢ Iic x)).toReal"}], "premise": [143160, 143382], "state_str": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a = ((κ a) (s ×ˢ Iic x)).toReal"}
+{"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", "inst✝ : IsFiniteKernel κ", "hf : IsRatCondKernelCDF f κ ν", "a : α", "x : ℝ", "s : Set β", "hs : MeasurableSet s"], "goal": "ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)"}], "premise": [33675, 73864], "state_str": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ ENNReal.ofReal (∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x ∂ν a) = (κ a) (s ×ˢ Iic x)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : HeytingAlgebra α", "a b c : α"], "goal": "Disjoint a bᶜᶜ ↔ Disjoint a b"}], "premise": [15127, 15150], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ Disjoint a bᶜᶜ ↔ Disjoint a b"}
+{"state": [{"context": ["G : Type u_1", "inst✝¹ : Group G", "X : Type u_2", "inst✝ : MulAction G X", "hGX : IsPretransitive G X", "B : Set X", "hB : IsBlock G B", "hBe : B.Nonempty"], "goal": "∅ ∉ Set.range fun g => g • B"}, {"context": ["G : Type u_1", "inst✝¹ : Group G", "X : Type u_2", "inst✝ : MulAction G X", "hGX : IsPretransitive G X", "B : Set X", "hB : IsBlock G B", "hBe : B.Nonempty"], "goal": "∀ (a : X), ∃! b, (b ∈ Set.range fun g => g • B) ∧ a ∈ b"}, {"context": ["G : Type u_1", "inst✝¹ : Group G", "X : Type u_2", "inst✝ : MulAction G X", "hGX : IsPretransitive G X", "B : Set X", "hB : IsBlock G B", "hBe : B.Nonempty"], "goal": "∀ b ∈ Set.range fun g => g • B, IsBlock G b"}], "premise": [7803], "state_str": "case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∅ ∉ Set.range fun g => g • B\n\ncase cover\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ (a : X), ∃! b, (b ∈ Set.range fun g => g • B) ∧ a ∈ b\n\ncase mem_blocks\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nhGX : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nhBe : B.Nonempty\n⊢ ∀ b ∈ Set.range fun g => g • B, IsBlock G b"}
+{"state": [{"context": ["n✝ b n : ℕ", "n0 : 0 < n", "nb : n < b"], "goal": "b.digits n = [n] ∧ 1 < b ∧ 0 < n"}], "premise": [1673, 1674, 14273, 103886, 145041], "state_str": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\n⊢ b.digits n = [n] ∧ 1 < b ∧ 0 < n"}
+{"state": [{"context": ["n✝ b n : ℕ", "n0 : 0 < n", "nb : n < b", "b2 : 1 < b"], "goal": "b.digits n = [n]"}], "premise": [1674, 2134, 4467, 145214, 145889, 145897], "state_str": "n✝ b n : ℕ\nn0 : 0 < n\nnb : n < b\nb2 : 1 < b\n⊢ b.digits 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", "s : Simplex ℝ P 1"], "goal": "s.circumcenter = centroid ℝ univ s.points"}], "premise": [42504, 84251, 110036, 115882, 118910], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\n⊢ s.circumcenter = centroid ℝ univ s.points"}
+{"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 1", "hr : Set.univ.Pairwise fun i j => dist (s.points i) (centroid ℝ univ s.points) = dist (s.points j) (centroid ℝ univ s.points)"], "goal": "s.circumcenter = centroid ℝ univ s.points"}], "premise": [133095], "state_str": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nhr :\n  Set.univ.Pairwise fun i j =>\n    dist (s.points i) (centroid ℝ univ s.points) = dist (s.points j) (centroid ℝ univ s.points)\n⊢ s.circumcenter = centroid ℝ univ s.points"}
+{"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 1", "r : ℝ", "hr : ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = r"], "goal": "s.circumcenter = centroid ℝ univ s.points"}], "premise": [2100, 72787, 84272, 131586, 141365], "state_str": "case intro\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\nr : ℝ\nhr : ∀ x ∈ Set.univ, dist (s.points x) (centroid ℝ univ s.points) = r\n⊢ s.circumcenter = centroid ℝ univ s.points"}
+{"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", "x : F (α ::: Fix F α)"], "goal": "rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)"}], "premise": [1829, 1838, 140807, 140812], "state_str": "n : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)"}
+{"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", "x : F (α ::: Fix F α)", "this : recF g ∘ fixToW = rec g", "a : (P F).A", "f : (P F).B a ⟹ α ::: Fix F α", "h : repr x = ⟨a, f⟩"], "goal": "recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)"}], "premise": [128094, 128367, 140806], "state_str": "case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ recF g ((P F).wMk' ((TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)"}
+{"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", "x : F (α ::: Fix F α)", "this : recF g ∘ fixToW = rec g", "a : (P F).A", "f : (P F).B a ⟹ α ::: Fix F α", "h : repr x = ⟨a, f⟩"], "goal": "g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)"}], "premise": [128096, 137028, 137046, 140325, 140326], "state_str": "case mk\nn : ℕ\nF : TypeVec.{u} (n + 1) → Type u\nq : MvQPF F\nα : TypeVec.{u} n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : (P F).B a ⟹ α ::: Fix F α\nh : repr x = ⟨a, f⟩\n⊢ g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> ⟨a, f⟩)) = g ((TypeVec.id ::: rec g) <$$> x)"}
+{"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", "π : ι → Set (Set Ω)", "hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ"], "goal": "(∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔ ∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}], "premise": [72771], "state_str": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}
+{"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", "π : ι → Set (Set Ω)", "hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "h_eq' : ∀ (s : Finset ι) (f : ι → Set Ω), (∀ i ∈ s, f i ∈ π i) → ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']"], "goal": "(∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔ ∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}], "premise": [27609, 132775, 138668], "state_str": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}
+{"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", "π : ι → Set (Set Ω)", "hπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s", "μ : Measure Ω", "inst✝ : IsFiniteMeasure μ", "h_eq' : ∀ (s : Finset ι) (f : ι → Set Ω), (∀ i ∈ s, f i ∈ π i) → ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']", "h_eq : ∀ (s : Finset ι) (f : ι → Set Ω), (∀ i ∈ s, f i ∈ π i) → ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω"], "goal": "(∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔ ∀ (s : Finset ι) {f : ι → Set Ω}, (∀ i ∈ s, f i ∈ π i) → μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}], "premise": [27966, 72771, 132775], "state_str": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ i ∈ s, (fun ω => (((condexpKernel μ m') ω) (f i)).toReal) =ᶠ[ae μ] μ[(f i).indicator fun ω => 1|m']\nh_eq :\n  ∀ (s : Finset ι) (f : ι → Set Ω),\n    (∀ i ∈ s, f i ∈ π i) →\n      ∀ᵐ (ω : Ω) ∂μ, ∀ i ∈ s, (((condexpKernel μ m') ω) (f i)).toReal = (μ[(f i).indicator fun ω => 1|m']) ω\n⊢ (∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        ∀ᵐ (a : Ω) ∂μ.trim hm', ((condexpKernel μ m') a) (⋂ i ∈ s, f i) = ∏ i ∈ s, ((condexpKernel μ m') a) (f i)) ↔\n    ∀ (s : Finset ι) {f : ι → Set Ω},\n      (∀ i ∈ s, f i ∈ π i) →\n        μ[(⋂ i ∈ s, f i).indicator fun ω => 1|m'] =ᶠ[ae μ] ∏ i ∈ s, μ[(f i).indicator fun ω => 1|m']"}
+{"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"], "goal": "range m ∈ map m f"}], "premise": [134174], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f : Filter α\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ range m ∈ map m f"}
+{"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"], "goal": "m '' univ ∈ map m f"}], "premise": [15883, 16168], "state_str": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type u_1\nι : Sort x\nl f : Filter α\nm : α → β\nm' : β → γ\ns : Set α\nt : Set β\n⊢ m '' univ ∈ map m f"}
+{"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]"], "goal": "Splits (RingHom.id L) (map i f) ↔ Splits i f"}], "premise": [1713, 101116, 121584], "state_str": "R : Type u_1\nF : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\n⊢ Splits (RingHom.id L) (map i f) ↔ Splits i f"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : NonPreadditiveAbelian C", "X Y : C", "a : X ⟶ Y"], "goal": "a + -a = 0"}], "premise": [95333, 95339], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a + -a = 0"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : LinearOrderedAddCommGroup α", "a✝ b✝ c✝ a b c d : α"], "goal": "|min a b - min c d| ≤ max |a - c| |b - d|"}], "premise": [104508, 104518, 105280, 117888], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c✝ a b c d : α\n⊢ |min a b - min c d| ≤ max |a - c| |b - d|"}
+{"state": [{"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b : EReal", "ha : limsup u f < a", "hb : limsup v f < b"], "goal": "limsup (u + v) f ≤ a + b"}], "premise": [15928], "state_str": "α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\n⊢ limsup (u + v) f ≤ a + b"}
+{"state": [{"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b : EReal", "ha : limsup u f < a", "hb : limsup v f < b", "h✝ : f.NeBot"], "goal": "limsup (u + v) f ≤ a + b"}], "premise": [14797], "state_str": "case inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ a + b"}
+{"state": [{"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b : EReal", "ha : limsup u f < a", "hb : limsup v f < b", "h✝ : f.NeBot"], "goal": "limsup (u + v) f ≤ limsup (fun x => a + b) f"}], "premise": [14897, 16019, 16021, 16026, 57005], "state_str": "case inr\nα✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ limsup (u + v) f ≤ limsup (fun x => a + b) f"}
+{"state": [{"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b : EReal", "ha : limsup u f < a", "hb : limsup v f < b", "h✝ : f.NeBot"], "goal": "∀ (x : α), u x < a ∧ v x < b → (u + v) x ≤ a + b"}], "premise": [1186, 120650], "state_str": "α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\n⊢ ∀ (x : α), u x < a ∧ v x < b → (u + v) x ≤ a + b"}
+{"state": [{"context": ["α✝ : Type u_1", "inst✝ : TopologicalSpace α✝", "α : Type u_2", "f : Filter α", "u v : α → EReal", "a b : EReal", "ha : limsup u f < a", "hb : limsup v f < b", "h✝ : f.NeBot", "x : α"], "goal": "u x < a → v x < b → u x + v x ≤ a + b"}], "premise": [14286, 103917], "state_str": "α✝ : Type u_1\ninst✝ : TopologicalSpace α✝\nα : Type u_2\nf : Filter α\nu v : α → EReal\na b : EReal\nha : limsup u f < a\nhb : limsup v f < b\nh✝ : f.NeBot\nx : α\n⊢ u x < a → v x < b → u x + v x ≤ a + b"}
+{"state": [{"context": ["G : Type u", "H : Type v", "inst✝² : Mul G", "inst✝¹ : Mul H", "f : H →ₙ* G", "hf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d", "inst✝ : UniqueProds G", "A B : Finset H", "A0 : A.Nonempty", "B0 : B.Nonempty"], "goal": "∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0"}], "premise": [1673, 122367, 122382, 137413, 137425], "state_str": "G : Type u\nH : Type v\ninst✝² : Mul G\ninst✝¹ : Mul H\nf : H →ₙ* G\nhf : ∀ ⦃a b c d : H⦄, a * b = c * d → f a = f c ∧ f b = f d → a = c ∧ b = d\ninst✝ : UniqueProds G\nA B : Finset H\nA0 : A.Nonempty\nB0 : B.Nonempty\n⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0"}
+{"state": [{"context": ["c : ℝ", "f g : ℕ → Bool", "n : ℕ"], "goal": "#ℝ = 𝔠"}], "premise": [14296], "state_str": "c : ℝ\nf g : ℕ → Bool\nn : ℕ\n⊢ #ℝ = 𝔠"}
+{"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"], "goal": "LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x"}], "premise": [57165], "state_str": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type u_2\ninst✝ : Preorder β\nf g : α → β\nx : α\ns t : Set α\ny z : β\n⊢ LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsJacobson R", "P : Ideal R[X]", "hP : P.IsMaximal", "P' : Ideal R := comap C P"], "goal": "((Quotient.mk P).comp C).IsIntegral"}], "premise": [80663], "state_str": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsJacobson R\nP : Ideal R[X]\nhP : P.IsMaximal\nP' : Ideal R := comap C P\n⊢ ((Quotient.mk P).comp C).IsIntegral"}
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+{"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", "c : V", "hc : c ∈ s.direction"], "goal": "⟪c, p -ᵥ ↑((orthogonalProjection s) p)⟫_ℝ = 0"}], "premise": [36789, 69902, 115880, 119802], "state_str": "case hv\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : HasOrthogonalProjection s.direction\np : P\nc : V\nhc : c ∈ s.direction\n⊢ ⟪c, p -ᵥ ↑((orthogonalProjection s) p)⟫_ℝ = 0"}
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+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "K : Type u_3", "L : Type u_4", "inst✝¹⁷ : EuclideanDomain R", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : IsDomain S", "inst✝¹⁴ : Field K", "inst✝¹³ : Field L", "inst✝¹² : Algebra R K", "inst✝¹¹ : IsFractionRing R K", "inst✝¹⁰ : Algebra K L", "inst✝⁹ : FiniteDimensional K L", "inst✝⁸ : Algebra.IsSeparable K L", "algRL : Algebra R L", "inst✝⁷ : IsScalarTower R K L", "inst✝⁶ : Algebra R S", "inst✝⁵ : Algebra S L", "ist : IsScalarTower R S L", "iic : IsIntegralClosure S R L", "abv : AbsoluteValue R ℤ", "ι : Type u_5", "inst✝⁴ : DecidableEq ι", "inst✝³ : Fintype ι", "bS : Basis ι R S", "adm : abv.IsAdmissible", "inst✝² : Infinite R", "inst✝¹ : DecidableEq R", "inst✝ : Algebra.IsAlgebraic R L", "a b : S", "hb : b ≠ 0", "inj : Function.Injective ⇑(algebraMap R L)", "a' : S", "b' : R", "hb' : b' ≠ 0", "h : b' • a = b * a'", "q : S", "r : R", "hr : r ∈ finsetApprox bS adm", "hqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))"], "goal": "abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) < abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) b)"}], "premise": [1674, 14272, 14277, 14288, 78560, 104377, 104382], "state_str": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) <\n    abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) b)"}
+{"state": [{"context": ["R : Type u_1", "S : Type u_2", "K : Type u_3", "L : Type u_4", "inst✝¹⁷ : EuclideanDomain R", "inst✝¹⁶ : CommRing S", "inst✝¹⁵ : IsDomain S", "inst✝¹⁴ : Field K", "inst✝¹³ : Field L", "inst✝¹² : Algebra R K", "inst✝¹¹ : IsFractionRing R K", "inst✝¹⁰ : Algebra K L", "inst✝⁹ : FiniteDimensional K L", "inst✝⁸ : Algebra.IsSeparable K L", "algRL : Algebra R L", "inst✝⁷ : IsScalarTower R K L", "inst✝⁶ : Algebra R S", "inst✝⁵ : Algebra S L", "ist : IsScalarTower R S L", "iic : IsIntegralClosure S R L", "abv : AbsoluteValue R ℤ", "ι : Type u_5", "inst✝⁴ : DecidableEq ι", "inst✝³ : Fintype ι", "bS : Basis ι R S", "adm : abv.IsAdmissible", "inst✝² : Infinite R", "inst✝¹ : DecidableEq R", "inst✝ : Algebra.IsAlgebraic R L", "a b : S", "hb : b ≠ 0", "inj : Function.Injective ⇑(algebraMap R L)", "a' : S", "b' : R", "hb' : b' ≠ 0", "h : b' • a = b * a'", "q : S", "r : R", "hr : r ∈ finsetApprox bS adm", "hqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))"], "goal": "abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) = abv ((Algebra.norm R) (r • a' - b' • q)) * abv ((Algebra.norm R) b)"}], "premise": [104380, 108341, 117166, 118875, 119707, 121165, 121175], "state_str": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable K L\nalgRL : Algebra R L\ninst✝⁷ : IsScalarTower R K L\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra S L\nist : IsScalarTower R S L\niic : IsIntegralClosure S R L\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algebra.IsAlgebraic R L\na b : S\nhb : b ≠ 0\ninj : Function.Injective ⇑(algebraMap R L)\na' : S\nb' : R\nhb' : b' ≠ 0\nh : b' • a = b * a'\nq : S\nr : R\nhr : r ∈ finsetApprox bS adm\nhqr : abv ((Algebra.norm R) (r • a' - b' • q)) < abv ((Algebra.norm R) ((algebraMap R S) b'))\n⊢ abv ((Algebra.norm R) ((algebraMap R S) b')) * abv ((Algebra.norm R) (r • a - q * b)) =\n    abv ((Algebra.norm R) (r • a' - b' • q)) * abv ((Algebra.norm R) b)"}
+{"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' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'"}], "premise": [108476, 109326], "state_str": "R : Type u\nL : Type v\nM : Type w\nM₂ : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : LieRingModule L M₂\ninst✝ : LieModule R L M₂\nN N' : LieSubmodule R L M\nI J : LieIdeal R L\nN₂ : LieSubmodule R L M₂\n⊢ ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N'"}
+{"state": [{"context": ["x y : ℂ", "h : cexp x = 0"], "goal": "0 = 1"}], "premise": [119826, 149080, 149081], "state_str": "x y : ℂ\nh : cexp x = 0\n⊢ 0 = 1"}
+{"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", "p q : R[X]", "hp : p.Monic", "hdeg : q.natDegree ≤ p.natDegree", "r : R[X]", "hr : q = p * r"], "goal": "q = C q.leadingCoeff * p"}], "premise": [70039], "state_str": "case intro\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\n⊢ q = C q.leadingCoeff * p"}
+{"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", "p q : R[X]", "hp : p.Monic", "hdeg : q.natDegree ≤ p.natDegree", "r : R[X]", "hr : q = p * r", "hq : q ≠ 0", "rzero : r ≠ 0"], "goal": "q = C q.leadingCoeff * p"}], "premise": [102242], "state_str": "case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhdeg : q.natDegree ≤ p.natDegree\nr : R[X]\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p"}
+{"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", "p q : R[X]", "hp : p.Monic", "r : R[X]", "hdeg : p.natDegree + r.natDegree ≤ p.natDegree", "hr : q = p * r", "hq : q ≠ 0", "rzero : r ≠ 0"], "goal": "q = C q.leadingCoeff * p"}], "premise": [102269, 119707], "state_str": "case intro.inr\nR✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nr : R[X]\nhdeg : p.natDegree + r.natDegree ≤ p.natDegree\nhr : q = p * r\nhq : q ≠ 0\nrzero : r ≠ 0\n⊢ q = C q.leadingCoeff * p"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "s t : Set α"], "goal": "s.encard ≤ 1 ↔ s = ∅ ∨ ∃ x, s = {x}"}], "premise": [1095, 1713, 11260, 14303, 136129, 136322, 136370], "state_str": "α : Type u_1\nβ : Type u_2\ns t : Set α\n⊢ s.encard ≤ 1 ↔ s = ∅ ∨ ∃ x, s = {x}"}
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+{"state": [{"context": ["J : Type v'", "inst✝⁴ : Category.{u', v'} J", "C : Type u", "inst✝³ : Category.{v, u} C", "D : Type u''", "inst✝² : Category.{v'', u''} D", "X✝ Y✝ : C", "inst✝¹ : FinitaryPreExtensive C", "α : Type", "inst✝ : Finite α", "X : C", "Z : α → C", "π : (a : α) → Z a ⟶ X", "Y : C", "f : Y ⟶ X", "hπ : IsIso (Sigma.desc π)", "this : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso"], "goal": "IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))"}], "premise": [90445, 98752], "state_str": "J : Type v'\ninst✝⁴ : Category.{u', v'} J\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u''\ninst✝² : Category.{v'', u''} D\nX✝ Y✝ : C\ninst✝¹ : FinitaryPreExtensive C\nα : Type\ninst✝ : Finite α\nX : C\nZ : α → C\nπ : (a : α) → Z a ⟶ X\nY : C\nf : Y ⟶ X\nhπ : IsIso (Sigma.desc π)\nthis : IsColimit (Cofan.mk X π) := (coproductIsCoproduct Z).ofPointIso\n⊢ IsColimit (Cofan.mk Y fun x => pullback.fst f (π x))"}
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+{"state": [{"context": ["α : Type u", "inst✝¹ : LinearOrderedCommGroup α", "a b c : α", "inst✝ : Nontrivial α", "y : α", "hy : y ≠ 1"], "goal": "∃ a, 1 < a"}], "premise": [11261], "state_str": "case intro\nα : Type u\ninst✝¹ : LinearOrderedCommGroup α\na b c : α\ninst✝ : Nontrivial α\ny : α\nhy : y ≠ 1\n⊢ ∃ a, 1 < a"}
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+{"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✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedAddCommGroup E'", "inst✝⁹ : NormedAddCommGroup E''", "inst✝⁸ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝�� : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedSpace 𝕜 E'", "inst✝⁴ : NormedSpace 𝕜 E''", "inst✝³ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝² : MeasurableSpace G", "μ ν : Measure G", "inst✝¹ : NormedSpace ℝ F", "inst✝ : AddGroup G", "x : G", "h2x : x ∈ support (f ⋆[L, μ] g)", "hx : x ∉ support g + support f"], "goal": "(f ⋆[L, μ] g) x = 0"}], "premise": [2025, 2038, 70089, 117256, 131804], "state_str": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : x ∉ support g + support f\n⊢ (f ⋆[L, μ] g) x = 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✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedAddCommGroup E'", "inst✝⁹ : NormedAddCommGroup E''", "inst✝⁸ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedSpace 𝕜 E'", "inst✝⁴ : NormedSpace 𝕜 E''", "inst✝³ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝² : MeasurableSpace G", "μ ν : Measure G", "inst✝¹ : NormedSpace ℝ F", "inst✝ : AddGroup G", "x : G", "h2x : x ∈ support (f ⋆[L, μ] g)", "hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x"], "goal": "(f ⋆[L, μ] g) x = 0"}], "premise": [43953], "state_str": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ (f ⋆[L, μ] g) x = 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✝¹¹ : NormedAddCommGroup E", "inst✝¹⁰ : NormedAddCommGroup E'", "inst✝⁹ : NormedAddCommGroup E''", "inst✝⁸ : NormedAddCommGroup F", "f f' : G → E", "g g' : G → E'", "x✝ x' : G", "y y' : E", "inst✝⁷ : NontriviallyNormedField 𝕜", "inst✝⁶ : NormedSpace 𝕜 E", "inst✝⁵ : NormedSpace 𝕜 E'", "inst✝⁴ : NormedSpace 𝕜 E''", "inst✝³ : NormedSpace 𝕜 F", "L : E →L[𝕜] E' →L[𝕜] F", "inst✝² : MeasurableSpace G", "μ ν : Measure G", "inst✝¹ : NormedSpace ℝ F", "inst✝ : AddGroup G", "x : G", "h2x : x ∈ support (f ⋆[L, μ] g)", "hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x"], "goal": "∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0"}], "premise": [33638], "state_str": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nE'' : Type uE''\nF : Type uF\nF' : Type uF'\nF'' : Type uF''\nP : Type uP\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedAddCommGroup F\nf f' : G → E\ng g' : G → E'\nx✝ x' : G\ny y' : E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 E''\ninst✝³ : NormedSpace 𝕜 F\nL : E →L[𝕜] E' →L[𝕜] F\ninst✝² : MeasurableSpace G\nμ ν : Measure G\ninst✝¹ : NormedSpace ℝ F\ninst✝ : AddGroup G\nx : G\nh2x : x ∈ support (f ⋆[L, μ] g)\nhx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x\n⊢ ∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0"}
+{"state": [{"context": ["ι : Type u_1", "V : Type u_2", "inst✝¹ : Category.{?u.20862, u_2} V", "c : ComplexShape ι", "inst✝ : HasZeroMorphisms V", "X : HomologicalComplex Vᵒᵖ c", "i j : ι", "hij : ¬c.symm.Rel i j"], "goal": "(X.d j i).unop = 0"}], "premise": [93609, 113829], "state_str": "ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\ni j : ι\nhij : ¬c.symm.Rel i j\n⊢ (X.d j i).unop = 0"}
+{"state": [{"context": ["ι : Type u_1", "V : Type u_2", "inst✝¹ : Category.{?u.20862, u_2} V", "c : ComplexShape ι", "inst✝ : HasZeroMorphisms V", "X : HomologicalComplex Vᵒᵖ c", "x✝⁴ x✝³ x✝² : ι", "x✝¹ : c.symm.Rel x✝⁴ x✝³", "x✝ : c.symm.Rel x✝³ x✝²"], "goal": "(fun i j => (X.d j i).unop) x✝⁴ x✝³ ≫ (fun i j => (X.d j i).unop) x✝³ x✝² = 0"}], "premise": [89633, 93609, 113830], "state_str": "ι : Type u_1\nV : Type u_2\ninst✝¹ : Category.{?u.20862, u_2} V\nc : ComplexShape ι\ninst✝ : HasZeroMorphisms V\nX : HomologicalComplex Vᵒᵖ c\nx✝⁴ x✝³ x✝² : ι\nx✝¹ : c.symm.Rel x✝⁴ x✝³\nx✝ : c.symm.Rel x✝³ x✝²\n⊢ (fun i j => (X.d j i).unop) x✝⁴ x✝³ ≫ (fun i j => (X.d j i).unop) x✝³ x✝² = 0"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : SemilatticeInf α"], "goal": "Tendsto (fun a => (a, a)) atBot atBot"}], "premise": [15698], "state_str": "ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeInf α\n⊢ Tendsto (fun a => (a, a)) atBot atBot"}
+{"state": [{"context": ["ι : Type u_1", "ι' : Type u_2", "α : Type u_3", "β : Type u_4", "γ : Type u_5", "inst✝ : SemilatticeInf α"], "goal": "Tendsto (fun a => (a, a)) atBot (atBot ×ˢ atBot)"}], "premise": [12176, 16354], "state_str": "ι : Type u_1\nι' : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝ : SemilatticeInf α\n⊢ Tendsto (fun a => (a, a)) atBot (atBot ×ˢ atBot)"}
+{"state": [{"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α"], "goal": "∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b"}], "premise": [57778], "state_str": "α : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\n⊢ ∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b"}
+{"state": [{"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α", "b : Set (Set α)", "hb₁ : b.Countable", "hb₂ : t = generateFrom b"], "goal": "∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b"}], "premise": [57704, 57705, 133630], "state_str": "case intro.intro\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∃ b, b.Countable ∧ ∅ ∉ b ∧ IsTopologicalBasis b"}
+{"state": [{"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α", "b : Set (Set α)", "hb₁ : b.Countable", "hb₂ : t = generateFrom b"], "goal": "((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ b} \\ {∅}).Countable"}, {"context": ["α : Type u", "t : TopologicalSpace α", "inst✝ : SecondCountableTopology α", "b : Set (Set α)", "hb₁ : b.Countable", "hb₂ : t = generateFrom b"], "goal": "∅ ∈ {∅}"}], "premise": [132739, 132748, 132769, 133635], "state_str": "case intro.intro.refine_1\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ((fun f => ⋂₀ f) '' {f | f.Finite ∧ f ⊆ b} \\ {∅}).Countable\n\ncase intro.intro.refine_2\nα : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nb : Set (Set α)\nhb₁ : b.Countable\nhb₂ : t = generateFrom b\n⊢ ∅ ∈ {∅}"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : LinearOrder α", "a a₁ a₂ b b₁ b₂ c d : α", "h₁ : c ≤ b", "h₂ : a ≤ d", "x : α"], "goal": "x ∈ Icc a b ∪ Icc c d ↔ x ∈ Icc (min a c) (max b d)"}], "premise": [12953, 12954, 20160, 133407], "state_str": "case h\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh₁ : c ≤ b\nh₂ : a ≤ d\nx : α\n⊢ x ∈ Icc a b ∪ Icc c d ↔ x ∈ Icc (min a c) (max b d)"}
+{"state": [{"context": ["X : Type u", "Y : Type v", "Z : Type w", "inst✝¹ : MetricSpace X", "inst✝ : MetricSpace Y", "Φ✝ : Z → X", "Ψ✝ : Z → Y", "ε✝ : ℝ", "Φ : Z → X", "Ψ : Z → Y", "ε : ℝ", "x : Y", "y : X"], "goal": "ε ≤ glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y)"}], "premise": [61937], "state_str": "X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : Y\ny : X\n⊢ ε ≤ glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y)"}
+{"state": [{"context": ["X : Type u", "Y : Type v", "Z : Type w", "inst✝¹ : MetricSpace X", "inst✝ : MetricSpace Y", "Φ✝ : Z → X", "Ψ✝ : Z → Y", "ε✝ : ℝ", "Φ : Z → X", "Ψ : Z → Y", "ε : ℝ", "x : Y", "y : X"], "goal": "ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)"}], "premise": [61939], "state_str": "X : Type u\nY : Type v\nZ : Type w\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\nΦ : Z �� X\nΨ : Z → Y\nε : ℝ\nx : Y\ny : X\n⊢ ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁴ : Finite α", "p q : α → Prop", "inst✝³ : Fintype { x // p x }", "inst✝² : Fintype { x // ¬p x }", "inst✝¹ : Fintype { x // q x }", "inst✝ : Fintype { x // ¬q x }", "h : card { x // p x } = card { x // q x }"], "goal": "card { x // ¬p x } = card { x // ¬q x }"}], "premise": [141384], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : Finite α\np q : α → Prop\ninst✝³ : Fintype { x // p x }\ninst✝² : Fintype { x // ¬p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // ¬q x }\nh : card { x // p x } = card { x // q x }\n⊢ card { x // ¬p x } = card { x // ¬q x }"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝⁴ : Finite α", "p q : α → Prop", "inst✝³ : Fintype { x // p x }", "inst✝² : Fintype { x // ¬p x }", "inst✝¹ : Fintype { x // q x }", "inst✝ : Fintype { x // ¬q x }", "h : card { x // p x } = card { x // q x }", "val✝ : Fintype α"], "goal": "card { x // ¬p x } = card { x // ¬q x }"}], "premise": [141446], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : Finite α\np q : α → Prop\ninst✝³ : Fintype { x // p x }\ninst✝² : Fintype { x // ¬p x }\ninst✝¹ : Fintype { x // q x }\ninst✝ : Fintype { x // ¬q x }\nh : card { x // p x } = card { x // q x }\nval✝ : Fintype α\n⊢ card { x // ¬p x } = card { x // ¬q x }"}
+{"state": [{"context": ["α : Type u", "inst✝ : Monoid α", "a✝ b c u : αˣ", "a : α", "h : ↑u * a = 1"], "goal": "↑u⁻¹ = ↑u⁻¹ * 1"}], "premise": [119730], "state_str": "α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ = ↑u⁻¹ * 1"}
+{"state": [{"context": ["α : Type u", "inst✝ : Monoid α", "a✝ b c u : αˣ", "a : α", "h : ↑u * a = 1"], "goal": "↑u⁻¹ * 1 = a"}], "premise": [120412], "state_str": "α : Type u\ninst✝ : Monoid α\na✝ b c u : αˣ\na : α\nh : ↑u * a = 1\n⊢ ↑u⁻¹ * 1 = a"}
+{"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", "t : Set β", "cs : Set α", "cs_count : cs.Countable", "hcs : s ⊆ closure cs", "ct : Set β", "ct_count : ct.Countable", "hct : t ⊆ closure ct"], "goal": "IsSeparable (s ×ˢ t)"}], "premise": [132772], "state_str": "case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ IsSeparable (s ×ˢ t)"}
+{"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", "t : Set β", "cs : Set α", "cs_count : cs.Countable", "hcs : s ⊆ closure cs", "ct : Set β", "ct_count : ct.Countable", "hct : t ⊆ closure ct"], "goal": "s ×ˢ t ⊆ closure (cs ×ˢ ct)"}], "premise": [66447], "state_str": "case intro.intro.intro.intro\nα : Type u\nβ✝ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns✝ : Set α\nβ : Type u_2\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\ncs : Set α\ncs_count : cs.Countable\nhcs : s ⊆ closure cs\nct : Set β\nct_count : ct.Countable\nhct : t ⊆ closure ct\n⊢ s ×ˢ t ⊆ closure (cs ×ˢ ct)"}
+{"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✝⁴ : Monoid M", "inst✝³ : MulAction M α", "inst✝² : Mul α", "r s : M", "x y : α", "inst✝¹ : IsScalarTower M α α", "inst✝ : SMulCommClass M α α"], "goal": "r • x * s • y = (r * s) • (x * y)"}], "premise": [118863, 118884, 118897, 118899], "state_str": "M : Type u_1\nN : Type u_2\nG : Type u_3\nH : Type u_4\nA : Type u_5\nB : Type u_6\nα : Type u_7\nβ : Type u_8\nγ : Type u_9\nδ : Type u_10\ninst✝⁴ : Monoid M\ninst✝³ : MulAction M α\ninst✝² : Mul α\nr s : M\nx y : α\ninst✝¹ : IsScalarTower M α α\ninst✝ : SMulCommClass M α α\n⊢ r • x * s • y = (r * s) • (x * y)"}
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+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : NormalSpace X", "u : ι → Set X", "s : Set X", "hs : IsClosed s", "uo : ∀ (i : ι), IsOpen (u i)", "uf : ∀ x ∈ s, {i | x ∈ u i}.Finite", "us : s ⊆ ⋃ i, u i", "this✝ : Nonempty (PartialRefinement u s)", "this : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub", "v : PartialRefinement u s", "hv : ∀ (a : PartialRefinement u s), v ≤ a → a = v"], "goal": "∀ (i : ι), i ∈ v.carrier"}], "premise": [53688], "state_str": "case intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\nhv : ∀ (a : PartialRefinement u s), v ≤ a → a = v\n⊢ ∀ (i : ι), i ∈ v.carrier"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : NormalSpace X", "u : ι → Set X", "s : Set X", "hs : IsClosed s", "uo : ∀ (i : ι), IsOpen (u i)", "uf : ∀ x ∈ s, {i | x ∈ u i}.Finite", "us : s ⊆ ⋃ i, u i", "this✝ : Nonempty (PartialRefinement u s)", "this : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub", "v : PartialRefinement u s", "i : ι", "hi : i ∉ v.carrier"], "goal": "∃ a, v ≤ a ∧ a ≠ v"}], "premise": [54504], "state_str": "case intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\n⊢ ∃ a, v ≤ a ∧ a ≠ v"}
+{"state": [{"context": ["ι : Type u_1", "X : Type u_2", "inst✝¹ : TopologicalSpace X", "inst✝ : NormalSpace X", "u : ι → Set X", "s : Set X", "hs : IsClosed s", "uo : ∀ (i : ι), IsOpen (u i)", "uf : ∀ x ∈ s, {i | x ∈ u i}.Finite", "us : s ⊆ ⋃ i, u i", "this✝ : Nonempty (PartialRefinement u s)", "this : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub", "v : PartialRefinement u s", "i : ι", "hi : i ∉ v.carrier", "v' : PartialRefinement u s", "hlt : v < v'"], "goal": "∃ a, v ≤ a ∧ a ≠ v"}], "premise": [11234], "state_str": "case intro.intro.intro\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu : ι → Set X\ns : Set X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s)\nthis : ∀ (c : Set (PartialRefinement u s)), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ∃ ub, ∀ v ∈ c, v ≤ ub\nv : PartialRefinement u s\ni : ι\nhi : i ∉ v.carrier\nv' : PartialRefinement u s\nhlt : v < v'\n⊢ ∃ a, v ≤ a ∧ a ≠ v"}
+{"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 ∆ c ≤ a ∆ b ⊔ b ∆ c"}], "premise": [14533, 15112], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a ∆ c ≤ a ∆ b ⊔ b ∆ c"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "e : PartialEquiv α β", "e'✝ : PartialEquiv β γ", "s : Set α", "t : Set β", "x : α", "y : β", "e' : PartialEquiv α β", "h : e.IsImage s t", "hs : e.source ∩ s = e'.source ∩ s", "heq : EqOn (↑e) (↑e') (e.source ∩ s)"], "goal": "EqOn (↑e.symm) (↑e'.symm) (e.target ∩ t)"}], "premise": [71091], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ EqOn (↑e.symm) (↑e'.symm) (e.target ∩ t)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "e : PartialEquiv α β", "e'✝ : PartialEquiv β γ", "s : Set α", "t : Set β", "x✝ : α", "y : β", "e' : PartialEquiv α β", "h : e.IsImage s t", "hs : e.source ∩ s = e'.source ∩ s", "heq : EqOn (↑e) (↑e') (e.source ∩ s)", "x : α", "hx : x ∈ e.source ∩ s", "hx' : x ∈ e'.source ∩ s"], "goal": "↑e.symm (↑e x) = ↑e'.symm (↑e x)"}], "premise": [2107, 71050], "state_str": "case intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ns : Set α\nt : Set β\nx✝ : α\ny : β\ne' : PartialEquiv α β\nh : e.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\nx : α\nhx : x ∈ e.source ∩ s\nhx' : x ∈ e'.source ∩ s\n⊢ ↑e.symm (↑e x) = ↑e'.symm (↑e x)"}
+{"state": [{"context": ["U : Type u_1", "inst✝² : Quiver U", "V : Type u_2", "inst✝¹ : Quiver V", "φ : U ⥤q V", "W : Type ?u.1952", "inst✝ : Quiver W", "ψ : V ⥤q W", "hφ : φ.IsCovering", "u v : U", "f g : u ⟶ v", "he : (fun f => φ.map f) f = (fun f => φ.map f) g", "this : φ.star u (Star.mk f) = φ.star u (Star.mk g)"], "goal": "f = g"}], "premise": [2107, 49202], "state_str": "U : Type u_1\ninst✝² : Quiver U\nV : Type u_2\ninst✝¹ : Quiver V\nφ : U ⥤q V\nW : Type ?u.1952\ninst✝ : Quiver W\nψ : V ⥤q W\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f => φ.map f) f = (fun f => φ.map f) g\nthis : φ.star u (Star.mk f) = φ.star u (Star.mk g)\n⊢ f = g"}
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+{"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", "f : G", "hxi : f ∈ Set.range ⇑(ϕ i)", "hxp : f ∈ ↑(⨆ j, ⨆ (_ : ¬j = i), (ϕ j).range)"], "goal": "f ∈ ⊥"}], "premise": [6988, 19388], "state_str": "case intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxi : f ∈ Set.range ⇑(ϕ i)\nhxp : f ∈ ↑(⨆ j, ⨆ (_ : ¬j = i), (ϕ j).range)\n⊢ f ∈ ⊥"}
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+{"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", "f : G", "hxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)", "g : (i_1 : { j // ¬j = i }) → H ↑i_1", "hgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f", "g' : H i", "hg'f : (ϕ i) g' = f", "hxi : orderOf f ∣ Fintype.card (H i)"], "goal": "f ∈ ⊥"}], "premise": [8367, 8510, 140594], "state_str": "case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\n⊢ f ∈ ⊥"}
+{"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", "f : G", "hxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)", "g : (i_1 : { j // ¬j = i }) → H ↑i_1", "hgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f", "g' : H i", "hg'f : (ϕ i) g' = f", "hxi : orderOf f ∣ Fintype.card (H i)", "hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)"], "goal": "f = 1"}], "premise": [8356, 119743], "state_str": "case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ f = 1"}
+{"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", "f : G", "hxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)", "g : (i_1 : { j // ¬j = i }) → H ↑i_1", "hgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f", "g' : H i", "hg'f : (ϕ i) g' = f", "hxi : orderOf f ∣ Fintype.card (H i)", "hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)"], "goal": "orderOf f ∣ 1"}], "premise": [3620], "state_str": "case intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\n⊢ orderOf f ∣ 1"}
+{"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", "f : G", "hxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)", "g : (i_1 : { j // ¬j = i }) → H ↑i_1", "hgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f", "g' : H i", "hg'f : (ϕ i) g' = f", "hxi : orderOf f ∣ Fintype.card (H i)", "hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)", "c : ℕ", "hc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c"], "goal": "orderOf f ∣ 1"}], "premise": [1674, 2045], "state_str": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ orderOf f ∣ 1"}
+{"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", "f : G", "hxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)", "g : (i_1 : { j // ¬j = i }) → H ↑i_1", "hgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f", "g' : H i", "hg'f : (ϕ i) g' = f", "hxi : orderOf f ∣ Fintype.card (H i)", "hxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)", "c : ℕ", "hc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c"], "goal": "(∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = 1"}], "premise": [393, 137123, 143470], "state_str": "case h\nG : Type u_1\ninst✝³ : Group G\nι : Type u_2\nhdec : DecidableEq ι\nhfin : Fintype ι\nH : ι → Type u_3\ninst✝² : (i : ι) → Group (H i)\nϕ : (i : ι) → H i →* G\nf✝ g✝ : (i : ι) → H i\nhcomm : Pairwise fun i j => ∀ (x : H i) (y : H j), Commute ((ϕ i) x) ((ϕ j) y)\ninst✝¹ : Finite ι\ninst✝ : (i : ι) → Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval✝ : Fintype ι\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\nf : G\nhxp✝ : f ∈ ↑(⨆ x, (ϕ ↑x).range)\ng : (i_1 : { j // ¬j = i }) → H ↑i_1\nhgf : (noncommPiCoprod (fun x => ϕ ↑x) ⋯) g = f\ng' : H i\nhg'f : (ϕ i) g' = f\nhxi : orderOf f ∣ Fintype.card (H i)\nhxp : orderOf f ∣ ∏ j : { j // j ≠ i }, Fintype.card (H ↑j)\nc : ℕ\nhc : (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = orderOf f * c\n⊢ (∏ j : { j // j ≠ i }, Fintype.card (H ↑j)).gcd (Fintype.card (H i)) = 1"}
+{"state": [{"context": ["a b c d m n k : ℕ", "p q : ℕ → Prop", "h : m % k = n % k"], "goal": "(m - n) % k = 0"}], "premise": [3890, 3908, 4476, 4492], "state_str": "a b c d m n k : ℕ\np q : ℕ → Prop\nh : m % k = n % k\n⊢ (m - n) % k = 0"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : NonUnitalNonAssocRing R", "ι : Type u_2", "I : ι → TwoSidedIdeal R"], "goal": "(⨅ i, I i).ringCon = ⨅ i, (I i).ringCon"}], "premise": [77522], "state_str": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nι : Type u_2\nI : ι → TwoSidedIdeal R\n⊢ (⨅ i, I i).ringCon = ⨅ i, (I i).ringCon"}
+{"state": [{"context": ["α : Type ua", "β : Type ub", "γ : Type uc", "δ : Type ud", "ι : Sort u_1", "inst✝ : UniformSpace α", "g : Set (α × α) → Filter β", "f : Filter β", "hg : Monotone g", "h : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f"], "goal": "(map Prod.swap (𝓤 α)).lift g ≤ f"}], "premise": [12695, 134234], "state_str": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\ng : Set (α × α) → Filter β\nf : Filter β\nhg : Monotone g\nh : ((𝓤 α).lift fun s => g (Prod.swap ⁻¹' s)) ≤ f\n⊢ (map Prod.swap (𝓤 α)).lift g ≤ f"}
+{"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 : α → M", "s : Set α"], "goal": "∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f), f i = ∏ᶠ (i : α) (_ : i ∈ s), f i"}], "premise": [120900, 125615], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nG : Type u_4\nM : Type u_5\nN : Type u_6\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf : α → M\ns : Set α\n⊢ ∏ᶠ (i : α) (_ : i ∈ s ∩ mulSupport f), f i = ∏ᶠ (i : α) (_ : i ∈ s), f i"}
+{"state": [{"context": ["α : Type u_1", "l : Ordnode α", "x : α", "r : Ordnode α"], "goal": "(l.rotateR x r).dual = r.dual.rotateL x l.dual"}], "premise": [146350, 146366], "state_str": "α : Type u_1\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ (l.rotateR x r).dual = r.dual.rotateL x l.dual"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "r : Rel α β"], "goal": "∀ (x : α), x ∈ r.core Set.univ ↔ x ∈ Set.univ"}], "premise": [146931], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : Rel α β\n⊢ ∀ (x : α), x ∈ r.core Set.univ ↔ x ∈ 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"], "goal": "∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂μ"}], "premise": [42784], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ↑‖f x‖₊ ∂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 : α → ℝ"], "goal": "∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ENNReal.ofReal ‖f x‖ ∂μ"}], "premise": [30232, 143377], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ENNReal.ofReal ‖f x‖ ∂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 : α → ℝ", "x : α"], "goal": "f x ≤ ‖f x‖"}], "premise": [42903], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ ‖f 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", "f : α → ℝ", "x : α"], "goal": "f x ≤ |f x|"}], "premise": [105272], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\nx : α\n⊢ f x ≤ |f x|"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [31650], "state_str": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "this : ProperSpace E", "T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [71951], "state_str": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "this : ProperSpace E", "T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint", "x : E", "hx : x ≠ 0"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [59923], "state_str": "case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "this : ProperSpace E", "T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint", "x : E", "hx : x ≠ 0", "H₁ : IsCompact (sphere 0 ‖x‖)", "H₂ : (sphere 0 ‖x‖).Nonempty"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [36957, 57324, 63111], "state_str": "case intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "this✝ : ProperSpace E", "T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint", "x : E", "hx : x ≠ 0", "H₁ : IsCompact (sphere 0 ‖x‖)", "H₂ : (sphere 0 ‖x‖).Nonempty", "x₀ : E", "hx₀' : x₀ ∈ sphere 0 ‖x‖", "hTx₀ : IsMinOn (���T').reApplyInnerSelf (sphere 0 ‖x‖) x₀", "hx₀ : ‖x₀‖ = ‖x‖", "this : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [1787, 42918], "state_str": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝³ : RCLike 𝕜", "E : Type u_2", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace 𝕜 E", "inst✝ : FiniteDimensional 𝕜 E", "_i : Nontrivial E", "T : E →ₗ[𝕜] E", "hT : T.IsSymmetric", "this✝ : ProperSpace E", "T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint", "x : E", "hx : x ≠ 0", "H₁ : IsCompact (sphere 0 ‖x‖)", "H₂ : (sphere 0 ‖x‖).Nonempty", "x₀ : E", "hx₀' : x₀ ∈ sphere 0 ‖x‖", "hTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀", "hx₀ : ‖x₀‖ = ‖x‖", "this : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀", "hx₀_ne : x₀ ≠ 0"], "goal": "HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}], "premise": [33834, 88217, 137122], "state_str": "case intro.intro.intro\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\n_i : Nontrivial E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nthis✝ : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ : IsCompact (sphere 0 ‖x‖)\nH₂ : (sphere 0 ‖x‖).Nonempty\nx₀ : E\nhx₀' : x₀ ∈ sphere 0 ‖x‖\nhTx₀ : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀\nhx₀ : ‖x₀‖ = ‖x‖\nthis : IsMinOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀_ne : x₀ ≠ 0\n⊢ HasEigenvalue T ↑(⨅ x, RCLike.re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)"}
+{"state": [{"context": ["l : Type u_1", "R : Type u_2", "inst✝² : DecidableEq l", "inst✝¹ : Fintype l", "inst✝ : CommRing R", "A : Matrix (l ⊕ l) (l ⊕ l) R", "hA : Aᵀ ∈ symplecticGroup l R"], "goal": "A ∈ symplecticGroup l R"}], "premise": [81389], "state_str": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : Aᵀ ∈ symplecticGroup l R\n⊢ A ∈ symplecticGroup l R"}
+{"state": [{"context": ["Ω : Type u_1", "inst✝¹ : MeasurableSpace Ω", "inst✝ : MeasurableSingletonClass Ω", "s t u : Set Ω", "hs : s.Finite", "hs' : s.Nonempty"], "goal": "(condCount s) s = 1"}], "premise": [27992, 73023, 133440, 143757], "state_str": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t u : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ (condCount s) s = 1"}
+{"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", "p : ℝ", "hp : 0 < p", "f : α → E", "hf : Memℒp f (ENNReal.ofReal p) μ", "ε : ℝ", "hε : 0 < ε"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}], "premise": [40008], "state_str": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}
+{"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", "p : ℝ", "hp : 0 < p", "f : α → E", "hf : Memℒp f (ENNReal.ofReal p) μ", "ε : ℝ", "hε : 0 < ε", "I : 0 < ε ^ (1 / p)"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}], "premise": [14324, 143386], "state_str": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}
+{"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", "p : ℝ", "hp : 0 < p", "f : α → E", "hf : Memℒp f (ENNReal.ofReal p) μ", "ε : ℝ", "hε : 0 < ε", "I : 0 < ε ^ (1 / p)", "A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}], "premise": [14324, 143386], "state_str": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}
+{"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", "p : ℝ", "hp : 0 < p", "f : α → E", "hf : Memℒp f (ENNReal.ofReal p) μ", "ε : ℝ", "hε : 0 < ε", "I : 0 < ε ^ (1 / p)", "A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0", "B : ENNReal.ofReal p ≠ 0"], "goal": "∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}], "premise": [27499, 143191], "state_str": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p) μ"}
+{"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", "p : ℝ", "hp : 0 < p", "f : α → E", "hf : Memℒp f (ENNReal.ofReal p) μ", "ε : ℝ", "hε : 0 < ε", "I : 0 < ε ^ (1 / p)", "A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0", "B : ENNReal.ofReal p ≠ 0", "g : α → E", "g_support : HasCompactSupport g", "g_cont : Continuous g", "g_mem : Memℒp g (↑p.toNNReal) μ", "hg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))"], "goal": "∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε"}], "premise": [1674, 27230, 33695, 40087, 104330, 117810, 143161, 143191, 143379], "state_str": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : R1Space α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst��� : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g (↑p.toNNReal) μ\nhg : eLpNorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∫ (x : α), ‖f x - g x‖ ^ p ∂μ ≤ ε"}
+{"state": [{"context": ["R✝ : Type u", "S : Type v", "T : Type w", "a b : R✝", "n✝ : ℕ", "inst✝³ : CommRing R✝", "inst✝² : IsDomain R✝", "p q : R✝[X]", "R : Type u_1", "inst✝¹ : CommRing R", "inst✝ : IsDomain R", "n : ℕ"], "goal": "nthRoots n 0 = replicate n 0"}], "premise": [102540, 102549, 117816, 121564, 137946], "state_str": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : IsDomain R✝\np q : R✝[X]\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\n⊢ nthRoots n 0 = replicate n 0"}
+{"state": [{"context": ["α : Type u_1", "inst✝ : Preorder α", "ls : ℕ", "ll : Ordnode α", "lx : α", "lr : Ordnode α", "l_ih✝ : ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α}, Valid' o₁ ll o₂ → Valid' o₁ r o₂ → All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size", "IHlr : ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α}, Valid' o₁ lr o₂ → Valid' o₁ r o₂ → All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size", "rs : ℕ", "rl : Ordnode α", "rx : α", "rr : Ordnode α", "IHrl : ∀ {o₁ : WithBot α} {o₂ : WithTop α}, Valid' o₁ (Ordnode.node ls ll lx lr) o₂ → Valid' o₁ rl o₂ → All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) → Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧ ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size", "r_ih✝ : ∀ {o₁ : WithBot α} {o₂ : WithTop α}, Valid' o₁ (Ordnode.node ls ll lx lr) o₂ → Valid' o₁ rr o₂ → All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) → Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧ ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size", "o₁ : WithBot α", "o₂ : WithTop α", "hl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂", "hr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂", "sep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)"], "goal": "Valid' o₁ ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)) o₂ ∧ ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)).size = (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size"}], "premise": [146421], "state_str": "case node.node\nα : Type u_1\ninst✝ : Preorder α\nls : ℕ\nll : Ordnode α\nlx : α\nlr : Ordnode α\nl_ih✝ :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ ll o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) ll → Valid' o₁ (ll.merge r) o₂ ∧ (ll.merge r).size = ll.size + r.size\nIHlr :\n  ∀ {r : Ordnode α} {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ lr o₂ →\n      Valid' o₁ r o₂ →\n        All (fun x => All (fun y => x < y) r) lr → Valid' o₁ (lr.merge r) o₂ ∧ (lr.merge r).size = lr.size + r.size\nrs : ℕ\nrl : Ordnode α\nrx : α\nrr : Ordnode α\nIHrl :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rl o₂ →\n        All (fun x => All (fun y => x < y) rl) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rl) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rl).size = (Ordnode.node ls ll lx lr).size + rl.size\nr_ih✝ :\n  ∀ {o₁ : WithBot α} {o₂ : WithTop α},\n    Valid' o₁ (Ordnode.node ls ll lx lr) o₂ →\n      Valid' o₁ rr o₂ →\n        All (fun x => All (fun y => x < y) rr) (Ordnode.node ls ll lx lr) →\n          Valid' o₁ ((Ordnode.node ls ll lx lr).merge rr) o₂ ∧\n            ((Ordnode.node ls ll lx lr).merge rr).size = (Ordnode.node ls ll lx lr).size + rr.size\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ (Ordnode.node ls ll lx lr) o₂\nhr : Valid' o₁ (Ordnode.node rs rl rx rr) o₂\nsep : All (fun x => All (fun y => x < y) (Ordnode.node rs rl rx rr)) (Ordnode.node ls ll lx lr)\n⊢ Valid' o₁ ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)) o₂ ∧\n    ((Ordnode.node ls ll lx lr).merge (Ordnode.node rs rl rx rr)).size =\n      (Ordnode.node ls ll lx lr).size + (Ordnode.node rs rl rx rr).size"}
+{"state": [{"context": ["α✝ : Type u_1", "β✝ : Type v", "γ : Type u_2", "inst✝¹ : DecidableEq α✝", "s✝ : Multiset α✝", "α : Type u_3", "β : Type u_4", "f : α → β", "s : Multiset α", "inst✝ : DecidableEq β", "b : β"], "goal": "count b (map f s) = card (filter (fun a => b = f a) s)"}], "premise": [1717, 2785, 138150], "state_str": "α✝ : Type u_1\nβ✝ : Type v\nγ : Type u_2\ninst✝¹ : DecidableEq α✝\ns✝ : Multiset α✝\nα : Type u_3\nβ : Type u_4\nf : α → β\ns : Multiset α\ninst✝ : DecidableEq β\nb : β\n⊢ count b (map f s) = card (filter (fun a => b = f a) s)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "f g : Perm α", "p : α → Prop", "x y z : α", "inst✝ : Finite α"], "goal": "f.SameCycle x y → ∃ i < orderOf f, (f ^ i) x = y"}], "premise": [1674, 2045, 3160, 3252, 4275, 4276, 8451, 8480, 11234, 119784, 129970], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nf g : Perm α\np : α → Prop\nx y z : α\ninst✝ : Finite α\n⊢ f.SameCycle x y → ∃ i < orderOf f, (f ^ i) x = y"}
+{"state": [{"context": ["α : Type u_1", "p : α → Bool", "l₁ l₂ : List α", "s : l₁ <+ l₂"], "goal": "List.filter p l₁ <+ List.filter p l₂"}], "premise": [5170], "state_str": "α : Type u_1\np : α → Bool\nl₁ l₂ : List α\ns : l₁ <+ l₂\n⊢ List.filter p l₁ <+ List.filter p l₂"}
+{"state": [{"context": ["α : Type u_1", "p : α → Bool", "l₁ l₂ : List α", "s : l₁ <+ l₂"], "goal": "filterMap (Option.guard fun x => p x = true) l₁ <+ filterMap (Option.guard fun x => p x = true) l₂"}], "premise": [1393], "state_str": "α : Type u_1\np : α → Bool\nl₁ l₂ : List α\ns : l₁ <+ l₂\n⊢ filterMap (Option.guard fun x => p x = true) l₁ <+ filterMap (Option.guard fun x => p x = true) l₂"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "inst✝¹ : Fintype α", "s✝ t : Finset α", "inst✝ : DecidableEq α", "a : α", "s : Finset α"], "goal": "sᶜ ≠ univ ↔ s.Nonempty"}], "premise": [140824], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : Fintype α\ns✝ t : Finset α\ninst✝ : DecidableEq α\na : α\ns : Finset α\n⊢ sᶜ ≠ univ ↔ s.Nonempty"}
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+{"state": [{"context": ["R : Type u_1", "M : Type u_2", "inst✝² : CommRing R", "inst✝¹ : AddCommGroup M", "inst✝ : Module R M", "f g : End R M"], "goal": "(-f).IsSemisimple ↔ f.IsSemisimple"}], "premise": [82423], "state_str": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf g : End R M\n⊢ (-f).IsSemisimple ↔ f.IsSemisimple"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "ι : Sort u_5", "inst✝ : MeasurableSpace α", "μ✝ μ₁ μ₂ : Measure α", "s✝ s₁ s₂ t : Set α", "μ : Measure α", "s : Set α"], "goal": "μ s = ⨅ t, μ ↑t"}], "premise": [19387, 19415, 29073], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nμ : Measure α\ns : Set α\n⊢ μ s = ⨅ t, μ ↑t"}
+{"state": [{"context": ["C : Type u_1", "inst✝¹ : Category.{?u.12556, u_1} C", "inst✝ : HasZeroMorphisms C", "I₁ : Type u_2", "I₂ : Type u_3", "c₁ : ComplexShape I₁", "c₂ : ComplexShape I₂", "K : HomologicalComplex₂ C c₁ c₂", "i i' : I₂", "w : ¬c₂.Rel i i'", "j : I₁"], "goal": "((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j"}], "premise": [113829], "state_str": "case h\nC : Type u_1\ninst✝¹ : Category.{?u.12556, u_1} C\ninst✝ : HasZeroMorphisms C\nI₁ : Type u_2\nI₂ : Type u_3\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK : HomologicalComplex₂ C c₁ c₂\ni i' : I₂\nw : ¬c₂.Rel i i'\nj : I₁\n⊢ ((fun i i' => { f := fun j => (K.X j).d i i', comm' := ⋯ }) i i').f j = Hom.f 0 j"}
+{"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"}], "premise": [1734], "state_str": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\n⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1"], "goal": "False"}], "premise": [1673, 144231], "state_str": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : p ∣ (m ^ 2 - n ^ 2).natAbs", "hp2 : p ∣ (2 * m * n).natAbs"], "goal": "False"}], "premise": [130034], "state_str": "case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : p ∣ (m ^ 2 - n ^ 2).natAbs\nhp2 : p ∣ (2 * m * n).natAbs\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n"], "goal": "False"}], "premise": [1673, 1681, 3541, 144296], "state_str": "case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(m.gcd n)"], "goal": "False"}], "premise": [75111], "state_str": "case intro.intro.intro\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(m.gcd n)", "hpn : p ∣ n.natAbs"], "goal": "False"}], "premise": [128984], "state_str": "case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(m.gcd n)\nhpn : p ∣ n.natAbs\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(n.gcd m)", "hpn : p ∣ n.natAbs"], "goal": "False"}], "premise": [1674, 1681, 128982, 130034], "state_str": "case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ False"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(n.gcd m)", "hpn : p ∣ n.natAbs"], "goal": "↑p ∣ m"}], "premise": [1673, 1978], "state_str": "case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(n.gcd m)", "hpn : p ∣ n.natAbs"], "goal": "↑p ∣ m ∨ ↑p ∣ m"}], "premise": [75112], "state_str": "case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ∨ ↑p ∣ m"}
+{"state": [{"context": ["m n : ℤ", "h : m.gcd n = 1", "hm : m % 2 = 0", "hn : n % 2 = 1", "H : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1", "p : ℕ", "hp : Nat.Prime p", "hp1 : ↑p ∣ m ^ 2 - n ^ 2", "hp2 : ↑p ∣ 2 * m * n", "hnp : ¬↑p ∣ ↑(n.gcd m)", "hpn : p ∣ n.natAbs"], "goal": "↑p ∣ m ^ 2 - n ^ 2 + n * n"}], "premise": [121987], "state_str": "case intro.intro.intro.inr\nm n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ 2 * m * n\nhnp : ¬↑p ∣ ↑(n.gcd m)\nhpn : p ∣ n.natAbs\n⊢ ↑p ∣ m ^ 2 - n ^ 2 + n * n"}
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+{"state": [{"context": ["𝕜 : Type u", "inst✝⁴ : NontriviallyNormedField 𝕜", "E : Type v", "inst✝³ : NormedAddCommGroup E", "inst✝² : NormedSpace 𝕜 E", "inst✝�� : LocallyCompactSpace 𝕜", "inst✝ : FiniteDimensional 𝕜 E", "this : ProperSpace 𝕜", "e : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 := ContinuousLinearEquiv.ofFinrankEq ⋯"], "goal": "ProperSpace E"}], "premise": [40711, 60858, 68917, 68938], "state_str": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : LocallyCompactSpace 𝕜\ninst✝ : FiniteDimensional 𝕜 E\nthis : ProperSpace 𝕜\ne : E ≃L[𝕜] Fin (finrank 𝕜 E) → 𝕜 := ContinuousLinearEquiv.ofFinrankEq ⋯\n⊢ ProperSpace E"}
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+{"state": [{"context": ["X : Type u_1", "Y : Type u_2", "Z : Type u_3", "inst✝¹ : PseudoMetricSpace X", "inst✝ : PseudoMetricSpace Y", "C r : ℝ≥0", "f : X → Y", "hf : HolderWith C r f", "x y : X", "d : ℝ≥0", "hd : nndist x y ≤ d"], "goal": "edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r"}], "premise": [60745], "state_str": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝¹ : PseudoMetricSpace X\ninst✝ : PseudoMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\nx y : X\nd : ℝ≥0\nhd : nndist x y ≤ d\n⊢ edist (f x) (f y) ≤ ↑C * ↑d ^ ↑r"}
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+{"state": [{"context": ["α : Type u_1", "M : Matroid α", "B B₁ B₂ : Set α", "hB₁ : M.Base B₁", "hB₂ : M.Base B₂"], "goal": "B₁.encard = B₂.encard"}], "premise": [139463, 139480], "state_str": "α : Type u_1\nM : Matroid α\nB B₁ B₂ : Set α\nhB₁ : M.Base B₁\nhB₂ : M.Base B₂\n⊢ B₁.encard = B₂.encard"}
+{"state": [{"context": ["C : Type u", "inst✝² : Category.{v, u} C", "inst✝¹ : HasStrictInitialObjects C", "I X : C", "inst✝ : HasBinaryProduct X I", "hI : IsInitial I"], "goal": "X ⨯ I ≅ I"}], "premise": [93088], "state_str": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasStrictInitialObjects C\nI X : C\ninst✝ : HasBinaryProduct X I\nhI : IsInitial I\n⊢ X ⨯ I ≅ I"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : SemilatticeInf α", "inst✝² : SemilatticeInf β", "ι : Sort u_4", "S : Set (Set α)", "f✝ : ι → Set α", "s t : Set α", "a✝ : α", "inst✝¹ : FunLike F α β", "inst✝ : InfHomClass F α β", "hs : InfClosed s", "f : F", "a : α", "ha : a ∈ s", "b : α", "hb : b ∈ s"], "goal": "f a ⊓ f b ∈ ⇑f '' s"}], "premise": [11638], "state_str": "case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f a ⊓ f b ∈ ⇑f '' s"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝³ : SemilatticeInf α", "inst✝² : SemilatticeInf β", "ι : Sort u_4", "S : Set (Set α)", "f✝ : ι → Set α", "s t : Set α", "a✝ : α", "inst✝¹ : FunLike F α β", "inst✝ : InfHomClass F α β", "hs : InfClosed s", "f : F", "a : α", "ha : a ∈ s", "b : α", "hb : b ∈ s"], "goal": "f (a ⊓ b) ∈ ⇑f '' s"}], "premise": [131593], "state_str": "case intro.intro.intro.intro\nF : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\nι : Sort u_4\nS : Set (Set α)\nf✝ : ι → Set α\ns t : Set α\na✝ : α\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nhs : InfClosed s\nf : F\na : α\nha : a ∈ s\nb : α\nhb : b ∈ s\n⊢ f (a ⊓ b) ∈ ⇑f '' s"}
+{"state": [{"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : InnerProductSpace ℝ E", "n : ℕ", "inst✝ : Fact (finrank ℝ E = n + 1)", "v : ↑(sphere 0 1)", "x : EuclideanSpace ℝ (Fin n)"], "goal": "↑(↑(stereographic' n v).symm x) = let U := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr; (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • 4 • ↑(U.symm x) + (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • (‖↑(U.symm x)‖ ^ 2 - 4) • ↑v"}], "premise": [36751, 42168], "state_str": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\nn : ℕ\ninst✝ : Fact (finrank ℝ E = n + 1)\nv : ↑(sphere 0 1)\nx : EuclideanSpace ℝ (Fin n)\n⊢ ↑(↑(stereographic' n v).symm x) =\n    let U := (OrthonormalBasis.fromOrthogonalSpanSingleton n ⋯).repr;\n    (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • 4 • ↑(U.symm x) + (‖↑(U.symm x)‖ ^ 2 + 4)⁻¹ • (‖↑(U.symm x)‖ ^ 2 - 4) • ↑v"}
+{"state": [{"context": ["α : Type u_1", "P : α → Prop", "size✝ : ℕ", "l : Ordnode α", "x : α", "r : Ordnode α"], "goal": "All P (node size✝ l x r) ↔ ∀ (x_1 : α), Emem x_1 (node size✝ l x r) → P x_1"}], "premise": [1998, 2026], "state_str": "α : Type u_1\nP : α → Prop\nsize✝ : ℕ\nl : Ordnode α\nx : α\nr : Ordnode α\n⊢ All P (node size✝ l x r) ↔ ∀ (x_1 : α), Emem x_1 (node size✝ l x r) → P x_1"}
+{"state": [{"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDenom_eq : (of v).partDens.get? n = some b"], "goal": "b * (of v).dens n ≤ (of v).dens (n + 1)"}], "premise": [116137, 116139], "state_str": "K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDenom_eq : (of v).partDens.get? n = some b\n⊢ b * (of v).dens n ≤ (of v).dens (n + 1)"}
+{"state": [{"context": ["K : Type u_1", "v : K", "n : ℕ", "inst✝¹ : LinearOrderedField K", "inst✝ : FloorRing K", "b : K", "nth_partDenom_eq : (of v).partDens.get? n = some b"], "goal": "b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)"}], "premise": [118585], "state_str": "K : Type u_1\nv : K\nn : ℕ\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nb : K\nnth_partDenom_eq : (of v).partDens.get? n = some b\n⊢ b * ((of v).contsAux (n + 1)).b ≤ (of v).dens (n + 1)"}
+{"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''", "f : R' →+* R''", "hf : Function.Injective ⇑f"], "goal": "Function.Injective fun x => x.ringHomComp f"}], "premise": [23020, 23024, 117118, 117125], "state_str": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommRing R'\nR'' : Type u_3\ninst✝ : CommRing R''\nf : R' →+* R''\nhf : Function.Injective ⇑f\n⊢ Function.Injective fun x => x.ringHomComp f"}
+{"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", "ha : fract a ≠ 0"], "goal": "↑⌈a⌉ - a = 1 - fract a"}], "premise": [1673, 1990, 105210], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\nha : fract a ≠ 0\n⊢ ↑⌈a⌉ - a = 1 - fract a"}
+{"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✝ S S' : IntermediateField K L", "h : S.toSubalgebra = S'.toSubalgebra", "x✝ : L"], "goal": "x✝ ∈ S ↔ x✝ ∈ S'"}], "premise": [1713, 88419], "state_str": "case h\nK : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nS✝ S S' : IntermediateField K L\nh : S.toSubalgebra = S'.toSubalgebra\nx✝ : L\n⊢ x✝ ∈ S ↔ x✝ ∈ S'"}
+{"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", "f : α → Multiset β", "s : Finset α", "b : β"], "goal": "b ∈ ∑ x ∈ s, f x ↔ ∃ a ∈ s, b ∈ f a"}], "premise": [1995, 2027, 126901, 138847], "state_str": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\nf : α → Multiset β\ns : Finset α\nb : β\n⊢ b ∈ ∑ x ∈ s, f x ↔ ∃ a ∈ s, b ∈ f a"}
+{"state": [{"context": ["m n : ℕ+"], "goal": "m.gcd n = 1 → n.gcd m = 1"}], "premise": [141185], "state_str": "m n : ℕ+\n⊢ m.gcd n = 1 → n.gcd m = 1"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "a x y : 𝕜", "ha : a ∈ s", "hx : x ∈ s", "hy : y ∈ s", "hxa : x ≠ a", "hya : y ≠ a", "hxy : x ≤ y"], "goal": "(f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}], "premise": [11248], "state_str": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy : x ≤ y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}
+{"state": [{"context": ["𝕜 : Type u_1", "inst✝ : LinearOrderedField 𝕜", "s : Set 𝕜", "f : 𝕜 → 𝕜", "hf : ConvexOn 𝕜 s f", "a x y : 𝕜", "ha : a ∈ s", "hx : x ∈ s", "hy : y ∈ s", "hxa : x ≠ a", "hya : y ≠ a", "hxy✝ : x ≤ y", "hxy : x < y"], "goal": "(f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}], "premise": [14320], "state_str": "case inr\n𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\na x y : 𝕜\nha : a ∈ s\nhx : x ∈ s\nhy : y ∈ s\nhxa : x ≠ a\nhya : y ≠ a\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁵ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero N", "inst✝³ : UniqueFactorizationMonoid N", "inst✝² : UniqueFactorizationMonoid M", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : Associates M", "n : Associates N", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)"], "goal": "multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n"}], "premise": [14296, 77032], "state_str": "M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity p m = multiplicity (↑(d ⟨p, ⋯⟩)) n"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁵ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero N", "inst✝³ : UniqueFactorizationMonoid N", "inst✝² : UniqueFactorizationMonoid M", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : Associates M", "n : Associates N", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)"], "goal": "multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m"}], "premise": [11076, 76126, 137134], "state_str": "M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity p m"}
+{"state": [{"context": ["M : Type u_1", "inst✝⁵ : CancelCommMonoidWithZero M", "N : Type u_2", "inst✝⁴ : CancelCommMonoidWithZero N", "inst✝³ : UniqueFactorizationMonoid N", "inst✝² : UniqueFactorizationMonoid M", "inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "m p : Associates M", "n : Associates N", "hn : n ≠ 0", "hp : p ∈ normalizedFactors m", "d : ↑(Set.Iic m) ≃o ↑(Set.Iic n)", "this : DecidableEq (Associates N) := Classical.decEq (Associates N)"], "goal": "multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m"}], "premise": [77031, 77032, 137133], "state_str": "M : Type u_1\ninst✝⁵ : CancelCommMonoidWithZero M\nN : Type u_2\ninst✝⁴ : CancelCommMonoidWithZero N\ninst✝³ : UniqueFactorizationMonoid N\ninst✝² : UniqueFactorizationMonoid M\ninst✝¹ : DecidableRel fun x x_1 => x ∣ x_1\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic n)\nthis : DecidableEq (Associates N) := Classical.decEq (Associates N)\n⊢ multiplicity (↑(d ⟨p, ⋯⟩)) n ≤ multiplicity (↑(d.symm (d ⟨p, ⋯⟩))) m"}
+{"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' : HasDerivWithinAt f f' (Iio y) y"], "goal": "f' ≤ slope f x y"}], "premise": [35087, 44476, 82753, 104741, 119769, 120673], "state_str": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : ConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n⊢ f' ≤ slope f x y"}
+{"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]", "x : R", "S : Type u_1", "inst✝ : Semiring S", "f : R →+* S", "r : R"], "goal": "eval₂ f (f r) p = f (eval r p)"}], "premise": [102824, 102860, 126889], "state_str": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ eval₂ f (f r) p = f (eval r p)"}
+{"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]", "x : R", "S : Type u_1", "inst✝ : Semiring S", "f : R →+* S", "r : R"], "goal": "∑ n ∈ p.support, f (p.coeff n) * f r ^ n = ∑ x ∈ p.support, f (p.coeff x * r ^ x)"}], "premise": [121567, 121592], "state_str": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\nr : R\n⊢ ∑ n ∈ p.support, f (p.coeff n) * f r ^ n = ∑ x ∈ p.support, f (p.coeff x * r ^ x)"}
+{"state": [{"context": ["ι : Type u_1", "𝕜 : Type u_2", "inst✝¹² : NontriviallyNormedField 𝕜", "H : Type u_3", "inst✝¹¹ : TopologicalSpace H", "E : Type u_4", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "inst✝⁸ : CommMonoid G", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : ChartedSpace H G", "inst✝⁵ : SmoothMul I G", "E' : Type u_6", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "H' : Type u_7", "inst✝² : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "inst✝¹ : TopologicalSpace M", "inst✝ : ChartedSpace H' M", "s : Set M", "x x₀ : M", "t : Finset ι", "f : ι → M → G", "n : ℕ∞", "p : ι → Prop", "h : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀"], "goal": "ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x₀"}], "premise": [69543], "state_str": "ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\ninst✝⁵ : SmoothMul I G\nE' : Type u_6\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\ns : Set M\nx x₀ : M\nt : Finset ι\nf : ι → M → G\nn : ℕ∞\np : ι → Prop\nh : ∀ i ∈ t, ContMDiffAt I' I n (f i) x₀\n⊢ ContMDiffAt I' I n (fun x => ∏ i ∈ t, f i x) x₀"}
+{"state": [{"context": ["ι : Type u_1", "𝕜 : Type u_2", "inst✝¹² : NontriviallyNormedField 𝕜", "H : Type u_3", "inst✝¹¹ : TopologicalSpace H", "E : Type u_4", "inst✝¹⁰ : NormedAddCommGroup E", "inst✝⁹ : NormedSpace 𝕜 E", "I : ModelWithCorners 𝕜 E H", "G : Type u_5", "inst✝⁸ : CommMonoid G", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : ChartedSpace H G", "inst✝⁵ : SmoothMul I G", "E' : Type u_6", "inst✝⁴ : NormedAddCommGroup E'", "inst✝³ : NormedSpace 𝕜 E'", "H' : Type u_7", "inst✝² : TopologicalSpace H'", "I' : ModelWithCorners 𝕜 E' H'", "M : Type u_8", "inst✝¹ : TopologicalSpace M", "inst✝ : ChartedSpace H' M", "s : Set M", "x x₀ : M", "t : Finset ι", "f : ι → M → G", "n : ℕ∞", "p : ι → Prop", "h : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) Set.univ x₀"], "goal": "ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) Set.univ x₀"}], "premise": [67621], "state_str": "ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\ninst✝⁵ : SmoothMul I G\nE' : Type u_6\ninst✝⁴ : NormedAddCommGroup E'\ninst✝³ : NormedSpace 𝕜 E'\nH' : Type u_7\ninst✝² : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nM : Type u_8\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H' M\ns : Set M\nx x₀ : M\nt : Finset ι\nf : ι → M → G\nn : ℕ∞\np : ι → Prop\nh : ∀ i ∈ t, ContMDiffWithinAt I' I n (f i) Set.univ x₀\n⊢ ContMDiffWithinAt I' I n (fun x => ∏ i ∈ t, f i x) Set.univ x₀"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ"], "goal": "μ' s = ν s"}], "premise": [15884, 54090, 55409, 55410, 64785], "state_str": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\n⊢ μ' s = ν s"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ", "k : Set G", "k_comp : IsCompact k", "k_closed : IsClosed k", "k_mem : k ∈ 𝓝 1"], "goal": "μ' s = ν s"}], "premise": [55499], "state_str": "case intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\n⊢ μ' s = ν s"}
+{"state": [{"context": ["G : Type u_1", "inst✝⁷ : TopologicalSpace G", "inst✝⁶ : Group G", "inst✝⁵ : TopologicalGroup G", "inst✝⁴ : MeasurableSpace G", "inst✝³ : BorelSpace G", "inst✝² : LocallyCompactSpace G", "μ' μ : Measure G", "inst✝¹ : μ.IsHaarMeasure", "inst✝ : μ'.IsHaarMeasure", "s : Set G", "hs : MeasurableSet s", "h's : μ.IsEverywherePos s", "ν : Measure G := μ'.haarScalarFactor μ • μ", "k : Set G", "k_comp : IsCompact k", "k_closed : IsClosed k", "k_mem : k ∈ 𝓝 1", "one_k : 1 ∈ k", "A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}", "m : Set G", "m_max : ∀ a ∈ A, m ⊆ a → a = m", "mA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k", "sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)"], "goal": "μ' s = ν s"}], "premise": [133383], "state_str": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : TopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\nν : Measure G := μ'.haarScalarFactor μ • μ\nk : Set G\nk_comp : IsCompact k\nk_closed : IsClosed k\nk_mem : k ∈ 𝓝 1\none_k : 1 ∈ k\nA : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x => x • k}\nm : Set G\nm_max : ∀ a ∈ A, m ⊆ a → a = m\nmA : m ⊆ s ∧ m.PairwiseDisjoint fun x => x • k\nsm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)\n⊢ μ' s = ν s"}
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+{"state": [{"context": ["𝕜 : Type u_1", "B : Type u_2", "inst✝¹⁷ : NontriviallyNormedField 𝕜", "inst✝¹⁶ : TopologicalSpace B", "F₁ : Type u_3", "inst✝¹⁵ : NormedAddCommGroup F₁", "inst✝¹⁴ : NormedSpace 𝕜 F₁", "E₁ : B → Type u_4", "inst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)", "F₂ : Type u_5", "inst✝¹² : NormedAddCommGroup F₂", "inst✝¹¹ : NormedSpace 𝕜 F₂", "E₂ : B → Type u_6", "inst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)", "inst✝⁹ : (x : B) → AddCommMonoid (E₁ x)", "inst✝⁸ : (x : B) → Module 𝕜 (E₁ x)", "inst✝⁷ : (x : B) → AddCommMonoid (E₂ x)", "inst✝⁶ : (x : B) → Module 𝕜 (E₂ x)", "inst✝⁵ : (x : B) → TopologicalSpace (E₁ x)", "inst✝⁴ : (x : B) → TopologicalSpace (E₂ x)", "inst✝³ : FiberBundle F₁ E₁", "inst✝² : FiberBundle F₂ E₂", "e₁ : Trivialization F₁ TotalSpace.proj", "e₂ : Trivialization F₂ TotalSpace.proj", "inst✝¹ : Trivialization.IsLinear 𝕜 e₁", "inst✝ : Trivialization.IsLinear 𝕜 e₂", "x : B", "hx : x ∈ (e₁.prod e₂).baseSet", "v₁ : E₁ x", "v₂ : E₂ x"], "goal": "(fun y => (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂, source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ, map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯, continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet, open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ } { proj := x, snd := y }).2) (v₁, v₂) = ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)"}], "premise": [2106, 2107, 59747], "state_str": "case h.h.mk\n𝕜 : Type u_1\nB : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : TopologicalSpace B\nF₁ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup F₁\ninst✝¹⁴ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_4\ninst✝¹³ : TopologicalSpace (TotalSpace F₁ E₁)\nF₂ : Type u_5\ninst✝¹² : NormedAddCommGroup F₂\ninst✝¹¹ : NormedSpace 𝕜 F₂\nE₂ : B → Type u_6\ninst✝¹⁰ : TopologicalSpace (TotalSpace F₂ E₂)\ninst✝⁹ : (x : B) → AddCommMonoid (E₁ x)\ninst✝⁸ : (x : B) → Module 𝕜 (E₁ x)\ninst✝⁷ : (x : B) → AddCommMonoid (E₂ x)\ninst✝⁶ : (x : B) → Module 𝕜 (E₂ x)\ninst✝⁵ : (x : B) → TopologicalSpace (E₁ x)\ninst✝⁴ : (x : B) → TopologicalSpace (E₂ x)\ninst✝³ : FiberBundle F₁ E₁\ninst✝² : FiberBundle F₂ E₂\ne₁ : Trivialization F₁ TotalSpace.proj\ne₂ : Trivialization F₂ TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear 𝕜 e₁\ninst✝ : Trivialization.IsLinear 𝕜 e₂\nx : B\nhx : x ∈ (e₁.prod e₂).baseSet\nv₁ : E₁ x\nv₂ : E₂ x\n⊢ (fun y =>\n        (↑{ toFun := Prod.toFun' e₁ e₂, invFun := Prod.invFun' e₁ e₂,\n                source := TotalSpace.proj ⁻¹' (e₁.baseSet ∩ e₂.baseSet), target := (e₁.baseSet ∩ e₂.baseSet) ×ˢ univ,\n                map_source' := ⋯, map_target' := ⋯, left_inv' := ⋯, right_inv' := ⋯, open_source := ⋯, open_target := ⋯,\n                continuousOn_toFun := ⋯, continuousOn_invFun := ⋯, baseSet := e₁.baseSet ∩ e₂.baseSet,\n                open_baseSet := ⋯, source_eq := ⋯, target_eq := ⋯, proj_toFun := ⋯ }\n            { proj := x, snd := y }).2)\n      (v₁, v₂) =\n    ((continuousLinearEquivAt 𝕜 e₁ x ⋯).prod (continuousLinearEquivAt 𝕜 e₂ x ⋯)) (v₁, v₂)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "e : Sym2 α", "f : α → β", "x y : α", "h : diag x = diag y"], "goal": "x = y"}], "premise": [128242], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : Sym2 α\nf : α → β\nx y : α\nh : diag x = diag y\n⊢ x = y"}
+{"state": [{"context": ["a b c p q✝ q : ℚ"], "goal": "q < 1 ↔ q.num < ↑q.den"}], "premise": [105559], "state_str": "a b c p q✝ q : ℚ\n⊢ q < 1 ↔ q.num < ↑q.den"}
+{"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", "α : Type u_3", "n : ℕ", "φ : L.BoundedFormula α n", "v : α → ↥⊤", "xs : Fin n → ↥⊤"], "goal": "φ.Realize v xs ↔ φ.Realize (Subtype.val ∘ v) (Subtype.val ∘ xs)"}], "premise": [25344], "state_str": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst✝² : L.Structure M\ninst✝¹ : L.Structure N\ninst✝ : L.Structure P\nS : L.Substructure M\ns : Set M\nα : Type u_3\nn : ℕ\nφ : L.BoundedFormula α n\nv : α → ↥⊤\nxs : Fin n → ↥⊤\n⊢ φ.Realize v xs ↔ φ.Realize (Subtype.val ∘ v) (Subtype.val ∘ xs)"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p : Perm α", "x : α", "n : ℕ"], "goal": "(p ^ n) x ∈ p.toList x ↔ x ∈ p.support"}], "premise": [1214, 8871], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\n⊢ (p ^ n) x ∈ p.toList x ↔ x ∈ p.support"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p : Perm α", "x : α", "n : ℕ"], "goal": "x ∈ p.support → p.SameCycle x ((p ^ n) x)"}], "premise": [9935], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\n⊢ x ∈ p.support → p.SameCycle x ((p ^ n) x)"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Fintype α", "inst✝ : DecidableEq α", "p : Perm α", "x : α", "n : ℕ", "x✝ : x ∈ p.support"], "goal": "p.SameCycle ((p ^ n) x) x"}], "premise": [9952], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nn : ℕ\nx✝ : x ∈ p.support\n⊢ p.SameCycle ((p ^ n) x) x"}
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+{"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"], "goal": "(∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔ ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0"}], "premise": [42108, 108557, 143177], "state_str": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\n⊢ (∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (0 * (y - x))) ↔\n    ∀ x ∈ s, ∀ y ∈ s, edist (f x) (f y) = 0"}
+{"state": [{"context": ["α β : Type u", "c : Cardinal.{u_1}", "h : ∀ (n : ℕ), ↑n ≤ c", "hn : c < ℵ₀"], "goal": "False"}], "premise": [1673, 48770], "state_str": "α β : Type u\nc : Cardinal.{u_1}\nh : ∀ (n : ℕ), ↑n ≤ c\nhn : c < ℵ₀\n⊢ False"}
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+{"state": [{"context": [], "goal": "3.141592 < π"}], "premise": [146588, 146589], "state_str": "⊢ 3.141592 < π"}
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+{"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 : AddSemigroup α", "inst✝ : Neg β", "h : Antiperiodic f c", "a x : α"], "goal": "(fun x => f (a + x)) (x + c) = -(fun x => f (a + x)) x"}], "premise": [119704], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x✝ : α\ninst✝¹ : AddSemigroup α\ninst✝ : Neg β\nh : Antiperiodic f c\na x : α\n⊢ (fun x => f (a + x)) (x + c) = -(fun x => f (a + x)) x"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "o : Part α", "inst✝ : Decidable o.Dom", "a : α"], "goal": "o.toOption = Option.some a ↔ a ∈ o"}], "premise": [1713, 3120, 131082], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\no : Part α\ninst✝ : Decidable o.Dom\na : α\n⊢ o.toOption = Option.some a ↔ a ∈ o"}
+{"state": [{"context": ["n : ℕ", "v : Vector ℕ n.succ"], "goal": "v.get 0 = v.head"}], "premise": [134773], "state_str": "n : ℕ\nv : Vector ℕ n.succ\n⊢ v.get 0 = v.head"}
+{"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", "inst✝ : IsFiniteKernel κ", "hf : IsRatCondKernelCDF f κ ν", "a : α", "x : ℝ"], "goal": "∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)"}], "premise": [27956, 30288, 73864], "state_str": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\n⊢ ∫⁻ (b : β), ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a = (κ a) (univ ×ˢ Iic x)"}
+{"state": [{"context": ["R : Type u_1", "A✝ : Type u_2", "B : Type u_3", "A : Type u_4", "inst✝⁴ : CommSemiring R", "inst✝³ : Semiring A", "inst✝² : Semiring B", "inst✝¹ : Algebra R A", "inst✝ : Algebra R B", "f g : R[ε] →ₐ[R] A", "hε : f ε = g ε"], "goal": "f (1 • ε) = g (1 • ε)"}], "premise": [118910], "state_str": "case hinr.h\nR : Type u_1\nA✝ : Type u_2\nB : Type u_3\nA : Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf g : R[ε] →ₐ[R] A\nhε : f ε = g ��\n⊢ f (1 • ε) = g (1 • ε)"}
+{"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 (-b) = -(toIocDiv hp (-a) b + 1)"}], "premise": [105865, 105878, 117878, 119789], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1)"}
+{"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 (-b) = -toIocDiv hp (-(a + p)) b"}], "premise": [1717, 117795], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIcoDiv hp a (-b) = -toIocDiv hp (-(a + p)) b"}
+{"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 = -toIcoDiv hp a (-b)"}], "premise": [105823], "state_str": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ toIocDiv hp (-(a + p)) b = -toIcoDiv hp a (-b)"}
+{"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 - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}], "premise": [105820], "state_str": "case h\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + 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", "hc : a ≤ -b - toIcoDiv hp a (-b) • p", "ho : -b - toIcoDiv hp a (-b) • p < a + p"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}], "premise": [105664, 110032, 117880, 119769], "state_str": "case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -b - toIcoDiv hp a (-b) • p < a + p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + 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", "hc : a ≤ -b - toIcoDiv hp a (-b) • p", "ho : -(a + p) < b - -toIcoDiv hp a (-b) • p"], "goal": "b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + p)"}], "premise": [105657, 110032, 117880, 119769], "state_str": "case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : a ≤ -b - toIcoDiv hp a (-b) • p\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ b - -toIcoDiv hp a (-b) • p ∈ Set.Ioc (-(a + p)) (-(a + p) + 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", "hc : b - -toIcoDiv hp a (-b) • p ≤ -a", "ho : -(a + p) < b - -toIcoDiv hp a (-b) • p"], "goal": "-a = -(a + p) + p"}], "premise": [117878, 119834], "state_str": "case h.intro\nα : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c : α\nn : ℤ\na b : α\nhc : b - -toIcoDiv hp a (-b) • p ≤ -a\nho : -(a + p) < b - -toIcoDiv hp a (-b) • p\n⊢ -a = -(a + p) + p"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : GeneralizedCoheytingAlgebra α", "a b✝ c d b : α"], "goal": "a \\ ⊥ ≤ b ↔ a ≤ b"}], "premise": [1713, 15060, 18834], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedCoheytingAlgebra α\na b✝ c d b : α\n⊢ a \\ ⊥ ≤ b ↔ a ≤ b"}
+{"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", "a b : α"], "goal": "Icc a b \\ {a, b} = Ioo a b"}], "premise": [138674], "state_str": "ι : Type u_1\nα : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na✝ b✝ c : α\ninst✝ : DecidableEq α\na b : α\n⊢ Icc a b \\ {a, b} = Ioo a b"}
+{"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", "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"}], "premise": [127075, 139122, 139132], "state_str": "ι : Type u_1\nκ : Type u_2\nα : Type u_3\nβ : Type u_4\nγ : Type u_5\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf g : α → β\n⊢ ∏ x ∈ s, t.piecewise f g x = (∏ x ∈ s ∩ t, f x) * ∏ x ∈ s \\ t, g x"}
+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "A : Cᵒᵖ ⥤ Type v", "F : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v", "X : C", "G : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v", "η : F ⟶ G", "p : YonedaCollection F X"], "goal": "(map₁ η p).yonedaEquivFst = p.yonedaEquivFst"}], "premise": [96845, 96849], "state_str": "C : Type u\ninst✝ : Category.{v, u} C\nA : Cᵒᵖ ⥤ Type v\nF : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nX : C\nG : (CostructuredArrow yoneda A)ᵒᵖ ⥤ Type v\nη : F ⟶ G\np : YonedaCollection F X\n⊢ (map₁ η p).yonedaEquivFst = p.yonedaEquivFst"}
+{"state": [{"context": ["F : Type u_1", "α : Type u_2", "β : Type u_3", "R : Type u_4", "m n✝ n : ℕ"], "goal": "∃ k, n = 2 * k ∨ n = 2 * k + 1"}], "premise": [2027, 121817, 122222], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nm n✝ n : ℕ\n⊢ ∃ k, n = 2 * k ∨ n = 2 * k + 1"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y"], "goal": "colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } } ≫ colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op = (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫ colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf) { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } }"}], "premise": [93401], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\n⊢ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n        { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } } ≫\n      colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n        colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n          colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n      colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n        { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n          ι :=\n            {\n              app := fun U =>\n                colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n              naturality := ⋯ } }"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y", "j : OpenNhds (f.base y)"], "goal": "colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫ colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } } ≫ colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op = colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫ (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫ colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫ colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫ colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf) { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf), ι := { app := fun U => colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }), naturality := ⋯ } }"}], "premise": [93392, 93413, 93414, 93425, 95551, 95617, 96173, 96175, 97705, 97706, 99920, 100030, 100031], "state_str": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      colimit.desc ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf)\n          { pt := colimit ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion (f.base x)).op ⋙ Y.presheaf)\n                    (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } } ≫\n        colimMap (whiskerLeft (OpenNhds.inclusion (f.base x)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion x).op ⋙ X.presheaf) (OpenNhds.map f.base x).op =\n    colimit.ι ((OpenNhds.inclusion (f.base y)).op ⋙ Y.presheaf) (op j) ≫\n      (colimMap (whiskerLeft (OpenNhds.inclusion (f.base y)).op f.c) ≫\n          colimMap (whiskerRight (𝟙 (OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op) X.presheaf) ≫\n            colimit.pre ((OpenNhds.inclusion y).op ⋙ X.presheaf) (OpenNhds.map f.base y).op) ≫\n        colimit.desc ((OpenNhds.inclusion y).op ⋙ X.presheaf)\n          { pt := colimit ((OpenNhds.inclusion x).op ⋙ X.presheaf),\n            ι :=\n              {\n                app := fun U =>\n                  colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop U).obj, property := ⋯ }),\n                naturality := ⋯ } }"}
+{"state": [{"context": ["C : Type u", "inst✝¹ : Category.{v, u} C", "inst✝ : HasColimits C", "X Y : PresheafedSpace C", "f : X ⟶ Y", "x y : ↑↑X", "h : x ⤳ y", "j : OpenNhds (f.base y)"], "goal": "f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫ X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) = f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫ X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫ colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf) (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })"}], "premise": [96175, 99920], "state_str": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\nj : OpenNhds (f.base y)\n⊢ f.c.app ((OpenNhds.inclusion (f.base x)).op.obj (op { obj := j.obj, property := ⋯ })) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base x ⋙ OpenNhds.inclusion x).op.obj (op { obj := j.obj, property := ⋯ }))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          ((OpenNhds.map f.base x).op.obj (op { obj := j.obj, property := ⋯ })) =\n    f.c.app ((OpenNhds.inclusion (f.base y)).op.obj (op j)) ≫\n      X.presheaf.map (𝟙 ((OpenNhds.map f.base y ⋙ OpenNhds.inclusion y).op.obj (op j))) ≫\n        colimit.ι ((OpenNhds.inclusion x).op ⋙ X.presheaf)\n          (op { obj := (unop ((OpenNhds.map f.base y).op.obj (op j))).obj, property := ⋯ })"}
+{"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", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "P : α → Prop", "h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x"], "goal": "∀ᵐ (x : α) ∂μ, P x"}], "premise": [1673, 31792, 31793, 32300, 103529, 103530], "state_str": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ✝ ν✝ ν₁ ν₂ : Measure α\ns t : Set α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nP : α → Prop\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x\n⊢ ∀ᵐ (x : α) ∂μ, P x"}
+{"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", "μ ν : Measure α", "inst✝¹ : SigmaFinite μ", "inst✝ : SigmaFinite ν", "P : α → Prop", "h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x", "this : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x"], "goal": "∀ᵐ (x : α) ∂μ, P x"}], "premise": [1674, 15889, 27606, 31800, 131585], "state_str": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ✝ ν✝ ν₁ ν₂ : Measure α\ns t : Set α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nP : α → Prop\nh : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → ν s < ⊤ → ∀ᵐ (x : α) ∂μ.restrict s, P x\nthis : ∀ (n : ℕ), ∀ᵐ (x : α) ∂μ, x ∈ spanningSets (μ + ν) n → P x\n⊢ ∀ᵐ (x : α) ∂μ, P x"}
+{"state": [{"context": ["a✝ b✝ c d : ℝ≥0∞", "r p q : ℝ≥0", "a b : ℝ≥0∞", "h : b ≤ a", "ha : a ≠ ⊤"], "goal": "(a - b).toReal = a.toReal - b.toReal"}], "premise": [18796], "state_str": "a✝ b✝ c d : ℝ≥0∞\nr p q : ℝ≥0\na b : ℝ≥0∞\nh : b ≤ a\nha : a ≠ ⊤\n⊢ (a - b).toReal = a.toReal - b.toReal"}
+{"state": [{"context": ["a✝ b✝ c d : ℝ≥0∞", "r p q b a : ℝ≥0", "h : ↑b ≤ ↑a"], "goal": "(↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal"}], "premise": [1673, 143174, 143204, 143540, 146620], "state_str": "case intro.intro\na✝ b✝ c d : ℝ≥0∞\nr p q b a : ℝ≥0\nh : ↑b ≤ ↑a\n⊢ (↑a - ↑b).toReal = (↑a).toReal - (↑b).toReal"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "f : α → Option β", "l₁ l₂ : List α", "p : l₁ ~ l₂"], "goal": "List.filterMap f l₁ ~ List.filterMap f l₂"}], "premise": [2614], "state_str": "α : Type u_1\nβ : Type u_2\nf : α → Option β\nl₁ l₂ : List α\np : l₁ ~ l₂\n⊢ List.filterMap f l₁ ~ List.filterMap f 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✝¹ : AddGroup α", "inst✝ : InvolutiveNeg β", "h1 : Periodic f c₁", "h2 : Antiperiodic f c₂"], "goal": "Antiperiodic f (c₁ - c₂)"}], "premise": [109515, 109548, 119789], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh1 : Periodic f c₁\nh2 : Antiperiodic f c₂\n⊢ Antiperiodic f (c₁ - c₂)"}
+{"state": [{"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ"], "goal": "∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in s..s + T, f x"}], "premise": [26334, 103972], "state_str": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\n⊢ ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in s..s + T, f x"}
+{"state": [{"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ"], "goal": "∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume"}], "premise": [32394], "state_str": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ", "this : VAddInvariantMeasure ↥(zmultiples T) ℝ volume"], "goal": "∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume"}], "premise": [33057], "state_str": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∫ (x : ℝ) in Ioc t (t + T), f x ∂volume = ∫ (x : ℝ) in Ioc s (s + T), f x ∂volume"}
+{"state": [{"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ", "this : VAddInvariantMeasure ↥(zmultiples T) ℝ volume"], "goal": "IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) volume"}, {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ", "this : VAddInvariantMeasure ↥(zmultiples T) ℝ volume"], "goal": "IsAddFundamentalDomain (↥(zmultiples T)) (Ioc s (s + T)) volume"}, {"context": ["E : Type u_1", "inst✝² : NormedAddCommGroup E", "inst✝¹ : NormedSpace ℝ E", "inst✝ : CompleteSpace E", "f : ℝ → E", "T : ℝ", "hf : Periodic f T", "hT : 0 < T", "t s : ℝ", "this : VAddInvariantMeasure ↥(zmultiples T) ℝ volume"], "goal": "∀ (g : ↥(zmultiples T)) (x : ℝ), f (g +ᵥ x) = f x"}], "premise": [25523, 109496], "state_str": "case hs\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc t (t + T)) volume\n\ncase ht\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ IsAddFundamentalDomain (↥(zmultiples T)) (Ioc s (s + T)) volume\n\ncase hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E\nT : ℝ\nhf : Periodic f T\nhT : 0 < T\nt s : ℝ\nthis : VAddInvariantMeasure ↥(zmultiples T) ℝ volume\n⊢ ∀ (g : ↥(zmultiples T)) (x : ℝ), f (g +ᵥ x) = f 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", "inst✝ : MeasurableSpace α", "s : ℕ → Set α", "p : ℕ → Prop", "h : ∀ (n : ℕ), p n → MeasurableSet (s n)"], "goal": "MeasurableSet (bliminf s atTop p)"}], "premise": [14835, 16577, 16578], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (bliminf s atTop p)"}
+{"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", "s : ℕ → Set α", "p : ℕ → Prop", "h : ∀ (n : ℕ), p n → MeasurableSet (s n)"], "goal": "MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)"}], "premise": [2107, 27959, 27965], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (⋃ i, ⋂ j, ⋂ (_ : p j ∧ i ≤ j), s j)"}
+{"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", "inst✝ : (i : ι) → SeminormedAddCommGroup (β i)", "f : PiLp ⊤ β"], "goal": "↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)"}], "premise": [42313, 146662], "state_str": "case a\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nf : PiLp ⊤ β\n⊢ ↑‖f‖₊ = ↑(⨆ i, ‖f i‖₊)"}
+{"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 i' : ι"], "goal": "(stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0"}], "premise": [71466, 83525, 120640], "state_str": "R : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (stdBasis R (fun _x => R) i) 1 i' = if i = i' then 1 else 0"}
+{"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 i' : ι"], "goal": "(i' = i) = (i = i')"}], "premise": [1713, 1717, 1792], "state_str": "case e_c\nR : Type u_1\nι : Type u_2\ninst✝³ : Semiring R\nφ : ι → Type u_3\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\ni i' : ι\n⊢ (i' = i) = (i = i')"}
+{"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 : Seminorm 𝕜 E", "c : 𝕜", "hc : 1 < ‖c‖", "ε : ℝ", "εpos : 0 < ε", "x : E", "hx : p x ≠ 0", "xεpos : 0 < p x / ε"], "goal": "∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x"}], "premise": [107259], "state_str": "R : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x"}
+{"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 : Seminorm 𝕜 E", "c : 𝕜", "hc : 1 < ‖c‖", "ε : ℝ", "εpos : 0 < ε", "x : E", "hx : p x ≠ 0", "xεpos : 0 < p x / ε", "n : ℤ", "hn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))", "cpos : 0 < ‖c‖"], "goal": "∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x"}], "premise": [2106, 43299], "state_str": "case intro\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nG : Type u_11\nι : Type u_12\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nε : ℝ\nεpos : 0 < ε\nx : E\nhx : p x ≠ 0\nxεpos : 0 < p x / ε\nn : ℤ\nhn : p x / ε ∈ Ico (‖c‖ ^ n) (‖c‖ ^ (n + 1))\ncpos : 0 < ‖c‖\n⊢ ∃ n, c ^ n ≠ 0 ∧ p (c ^ n • x) < ε ∧ ε / ‖c‖ ≤ p (c ^ n • x) ∧ ‖c ^ n‖⁻¹ ≤ ε⁻¹ * ‖c‖ * p x"}
+{"state": [{"context": ["K : Type uK", "inst✝⁴ : Field K", "V₁ : Type uV₁", "V₂ : Type uV₂", "inst✝³ : AddCommGroup V₁", "inst✝² : Module K V₁", "inst✝¹ : AddCommGroup V₂", "inst✝ : Module K V₂", "f : V₁ →ₗ[K] V₂"], "goal": "finrank K ↥(range f.dualMap) = finrank K ↥(range f)"}], "premise": [1674, 85537, 88018, 88129, 88155, 109850, 109935, 109937, 109958], "state_str": "K : Type uK\ninst✝⁴ : Field K\nV₁ : Type uV₁\nV₂ : Type uV₂\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ finrank K ↥(range f.dualMap) = finrank K ↥(range f)"}
+{"state": [{"context": ["K : Type uK", "inst✝⁴ : Field K", "V₁ : Type uV₁", "V₂ : Type uV₂", "inst✝³ : AddCommGroup V₁", "inst✝² : Module K V₁", "inst✝¹ : AddCommGroup V₂", "inst✝ : Module K V₂", "f : V₁ →ₗ[K] V₂"], "goal": "Function.Injective ⇑f.rangeRestrict.dualMap"}], "premise": [1673, 88019, 109937, 109973], "state_str": "K : Type uK\ninst✝⁴ : Field K\nV₁ : Type uV₁\nV₂ : Type uV₂\ninst✝³ : AddCommGroup V₁\ninst✝² : Module K V₁\ninst✝¹ : AddCommGroup V₂\ninst✝ : Module K V₂\nf : V₁ →ₗ[K] V₂\n⊢ Function.Injective ⇑f.rangeRestrict.dualMap"}
+{"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✝⁴ : FunLike F α β", "inst✝³ : CompleteLattice α", "inst✝² : CompleteLattice β", "inst✝¹ : CompleteLattice γ", "inst✝ : CompleteLattice δ", "g : FrameHom β γ", "f₁ f₂ : FrameHom α β", "hg : Injective ⇑g", "h : g.comp f₁ = g.comp f₂", "a : α"], "goal": "g (f₁ a) = g (f₂ a)"}], "premise": [10817], "state_str": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\nι : Sort u_6\nκ : ι → Sort u_7\ninst✝⁴ : FunLike F α β\ninst✝³ : CompleteLattice α\ninst✝² : CompleteLattice β\ninst✝¹ : CompleteLattice γ\ninst✝ : CompleteLattice δ\ng : FrameHom β γ\nf₁ f₂ : FrameHom α β\nhg : Injective ⇑g\nh : g.comp f₁ = g.comp f₂\na : α\n⊢ g (f₁ a) = g (f₂ a)"}
+{"state": [{"context": ["α : Type u_1", "inst✝¹ : DecidableEq α", "inst✝ : Fintype α", "t : Finset α", "a : α", "r k : ℕ", "ih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card", "𝒜 : Finset (Finset α)", "s : Finset α"], "goal": "s ∈ ∂⁺ ^[k + 1] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card"}], "premise": [1670, 2045, 71250], "state_str": "case succ\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nt : Finset α\na : α\nr k : ℕ\nih : ∀ {𝒜 : Finset (Finset α)} {s : Finset α}, s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ s ∈ ∂⁺ ^[k + 1] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + (k + 1) = s.card"}
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+{"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)"}], "premise": [95337, 95339, 95342, 95344], "state_str": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na b c : X ⟶ Y\n⊢ a - -b + c = a + (b + c)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "R✝ : Type u_3", "S✝ : Type u_4", "F✝ : Type u_5", "inst✝⁵ : NonAssocSemiring R✝", "inst✝⁴ : NonAssocSemiring S✝", "R : Type u_6", "S : Type u_7", "inst✝³ : Semiring R", "inst✝² : Semiring S", "F : Type u_8", "inst✝¹ : FunLike F R S", "inst✝ : RingHomClass F R S", "f : F", "n : ℕ", "hn : NeZero ↑n"], "goal": "NeZero (f ↑n)"}], "premise": [142663], "state_str": "α : Type u_1\nβ : Type u_2\nR✝ : Type u_3\nS✝ : Type u_4\nF✝ : Type u_5\ninst✝⁵ : NonAssocSemiring R✝\ninst✝⁴ : NonAssocSemiring S✝\nR : Type u_6\nS : Type u_7\ninst✝³ : Semiring R\ninst✝² : Semiring S\nF : Type u_8\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nn : ℕ\nhn : NeZero ↑n\n⊢ NeZero (f ↑n)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Monoid α", "n : ℕ", "G : Type u_3", "inst✝ : Group G", "f : Fin n → G", "i : ℕ", "hn : i < n"], "goal": "(partialProd f ⟨i, hn⟩.castSucc)⁻¹ * partialProd f ⟨i, hn⟩.succ = f ⟨i, hn⟩"}], "premise": [3738, 4052, 4087, 14281, 117774, 119703, 119808, 119818, 124585, 142951], "state_str": "case mk\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Monoid α\nn : ℕ\nG : Type u_3\ninst✝ : Group G\nf : Fin n → G\ni : ℕ\nhn : i < n\n⊢ (partialProd f ⟨i, hn⟩.castSucc)⁻¹ * partialProd f ⟨i, hn⟩.succ = f ⟨i, hn⟩"}
+{"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", "hb : b ≠ 0"], "goal": "b = a * b ↔ a = 1"}], "premise": [1713, 1717, 108292], "state_str": "α : Type u_1\nM₀ : Type u_2\nG₀ : Type u_3\nM₀' : Type u_4\nG₀' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\nhb : b ≠ 0\n⊢ b = a * b ↔ a = 1"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'"], "goal": "((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom = (μ R X X').hom ≫ (free R).map (f ⊗ g)"}], "premise": [86868], "state_str": "R : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ ((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom = (μ R X X').hom ≫ (free R).map (f ⊗ g)"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'"], "goal": "(TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) = (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g))"}], "premise": [85184], "state_str": "case H\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n      (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g))"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'", "x : X"], "goal": "(TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ Finsupp.lsingle x = (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ Finsupp.lsingle x"}], "premise": [109754], "state_str": "case H.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n        (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n      Finsupp.lsingle x =\n    (TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n      Finsupp.lsingle x"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'", "x : X"], "goal": "((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ Finsupp.lsingle x) 1 = ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ Finsupp.lsingle x) 1"}], "premise": [85184], "state_str": "case H.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n          (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n        Finsupp.lsingle x)\n      1 =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n        Finsupp.lsingle x)\n      1"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'", "x : X", "x' : X'"], "goal": "((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ Finsupp.lsingle x) 1 ∘ₗ Finsupp.lsingle x' = ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ Finsupp.lsingle x) 1 ∘ₗ Finsupp.lsingle x'"}], "premise": [109754], "state_str": "case H.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n            (((free R).map f �� (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x' =\n    ((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n          Finsupp.lsingle x)\n        1 ∘ₗ\n      Finsupp.lsingle x'"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'", "x : X", "x' : X'"], "goal": "(((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ Finsupp.lsingle x) 1 ∘ₗ Finsupp.lsingle x') 1 = (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ Finsupp.lsingle x) 1 ∘ₗ Finsupp.lsingle x') 1"}], "premise": [148057], "state_str": "case H.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\n⊢ (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂\n              (((free R).map f ⊗ (free R).map g) ≫ (μ R Y Y').hom) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1 =\n    (((TensorProduct.mk R ↑((free R).obj X) ↑((free R).obj X')).compr₂ ((μ R X X').hom ≫ (free R).map (f ⊗ g)) ∘ₗ\n            Finsupp.lsingle x)\n          1 ∘ₗ\n        Finsupp.lsingle x')\n      1"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "X Y X' Y' : Type u", "f : X ⟶ Y", "g : X' ⟶ Y'", "x : X", "x' : X'", "y : Y", "y' : Y'"], "goal": "((finsuppTensorFinsupp' R Y Y') (Finsupp.mapDomain f (Finsupp.single x 1) ⊗ₜ[R] Finsupp.mapDomain g (Finsupp.single x' 1))) (y, y') = (Finsupp.mapDomain (f ⊗ g) ((finsuppTensorFinsupp' R X X') (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single x' 1))) (y, y')"}], "premise": [86269, 98510, 119730, 148486], "state_str": "case H.h.h.h.h.h\nR : Type u\ninst✝ : CommRing R\nX Y X' Y' : Type u\nf : X ⟶ Y\ng : X' ⟶ Y'\nx : X\nx' : X'\ny : Y\ny' : Y'\n⊢ ((finsuppTensorFinsupp' R Y Y')\n        (Finsupp.mapDomain f (Finsupp.single x 1) ⊗ₜ[R] Finsupp.mapDomain g (Finsupp.single x' 1)))\n      (y, y') =\n    (Finsupp.mapDomain (f ⊗ g) ((finsuppTensorFinsupp' R X X') (Finsupp.single x 1 ⊗ₜ[R] Finsupp.single x' 1))) (y, y')"}
+{"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", "h : a / b ≤ c / d", "he : 0 ≤ e"], "goal": "a / (b * e) ≤ c / (d * e)"}], "premise": [117915], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nh : a / b ≤ c / d\nhe : 0 ≤ e\n⊢ a / (b * e) ≤ c / (d * e)"}
+{"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", "h : a / b ≤ c / d", "he : 0 ≤ e"], "goal": "a / b * (1 / e) ≤ c / d * (1 / e)"}], "premise": [1674, 102622, 104336], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nh : a / b ≤ c / d\nhe : 0 ≤ e\n⊢ a / b * (1 / e) ≤ c / d * (1 / e)"}
+{"state": [{"context": ["x✝ y x : ℝ", "hx : 0 < cos x"], "goal": "tan x / √(1 + tan x ^ 2) = sin x"}], "premise": [11234, 119790, 149233, 149257], "state_str": "x✝ y x : ℝ\nhx : 0 < cos x\n⊢ tan x / √(1 + tan x ^ 2) = sin x"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : CancelCommMonoidWithZero R", "α : Type u_2", "inst✝ : DecidableEq α", "x y a : R", "s : Finset α", "p : α → R", "hp : ∀ i ∈ s, Prime (p i)", "hx : x * y = a * ∏ i ∈ s, p i"], "goal": "∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i"}], "premise": [138846], "state_str": "R : Type u_1\ninst✝¹ : CancelCommMonoidWithZero R\nα : Type u_2\ninst✝ : DecidableEq α\nx y a : R\ns : Finset α\np : α → R\nhp : ∀ i ∈ s, Prime (p i)\nhx : x * y = a * ∏ i ∈ s, p i\n⊢ ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i"}
+{"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", "p : FormalMultilinearSeries 𝕜 E F", "x : E"], "goal": "HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x"}], "premise": [32792], "state_str": "α : Type u_1\n𝕜 : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\n⊢ HasFPowerSeriesWithinAt f p univ x ↔ HasFPowerSeriesAt f p x"}
+{"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", "z : S"], "goal": "(algebraMap R S) (sec M z).1 = (algebraMap R S) ↑(sec M z).2 * z"}], "premise": [77582, 119707], "state_str": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\n⊢ (algebraMap R S) (sec M z).1 = (algebraMap R S) ↑(sec M z).2 * z"}
+{"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": "∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ‖↑(r i n) - b i * ↑n‖ ≤ ↑n / log ↑n ^ 2"}], "premise": [16032], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\n⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), ‖↑(r i n) - b i * ↑n‖ ≤ ↑n / log ↑n ^ 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 : α → ℕ → ℕ", "R : AkraBazziRecurrence T g a b r", "i : α"], "goal": "∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2"}], "premise": [43362, 76697], "state_str": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\nR : AkraBazziRecurrence T g a b r\ni : α\n⊢ ∀ᶠ (x : ℕ) in atTop, ‖↑(r i x) - b i * ↑x‖ ≤ ↑x / log ↑x ^ 2"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n"], "goal": "∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}], "premise": [22072], "state_str": "ξ : ℝ\nn : ℕ\nn_pos : 0 < n\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)"], "goal": "∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}], "premise": [1674, 105573], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)", "hk₀' : 0 < ↑k"], "goal": "∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}], "premise": [4210, 147888, 148013], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)", "hk₀' : 0 < ↑k", "hden : ↑(↑j / ↑k).den ≤ k"], "goal": "∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}], "premise": [1673, 142597], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ ∃ q, |ξ - ↑q| ≤ 1 / ((↑n + 1) * ↑q.den) ∧ q.den ≤ n"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)", "hk₀' : 0 < ↑k", "hden : ↑(↑j / ↑k).den ≤ k"], "goal": "|ξ - ↑(↑j / ↑k)| ≤ 1 / ((↑n + 1) * ↑(↑j / ↑k).den)"}], "premise": [1674, 106022, 117896, 142640, 147933], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| ≤ 1 / ((↑n + 1) * ↑(↑j / ↑k).den)"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)", "hk₀' : 0 < ↑k", "hden : ↑(↑j / ↑k).den ≤ k"], "goal": "|ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)"}], "premise": [1674, 102621, 105294, 105569], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑(↑j / ↑k).den ≤ 1 / (↑n + 1)"}
+{"state": [{"context": ["ξ : ℝ", "n : ℕ", "n_pos : 0 < n", "j k : ℤ", "hk₀ : 0 < k", "hk₁ : k ≤ ↑n", "h : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)", "hk₀' : 0 < ↑k", "hden : ↑(↑j / ↑k).den ≤ k"], "goal": "|ξ - ↑(↑j / ↑k)| * ↑k ≤ 1 / (↑n + 1)"}], "premise": [11234, 105284, 106300, 108421, 119707, 147512, 148032], "state_str": "case intro.intro.intro.intro\nξ : ℝ\nn : ℕ\nn_pos : 0 < n\nj k : ℤ\nhk₀ : 0 < k\nhk₁ : k ≤ ↑n\nh : |↑k * ξ - ↑j| ≤ 1 / (↑n + 1)\nhk₀' : 0 < ↑k\nhden : ↑(↑j / ↑k).den ≤ k\n⊢ |ξ - ↑(↑j / ↑k)| * ↑k ≤ 1 / (↑n + 1)"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "π : ι → Type u_4", "inst✝ : GeneralizedHeytingAlgebra α", "a✝ b✝ c d a b : α", "h : b ≤ a"], "goal": "a ⇔ b = a ⇨ b"}], "premise": [1674, 15035, 18837], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝ : GeneralizedHeytingAlgebra α\na✝ b✝ c d a b : α\nh : b ≤ a\n⊢ a ⇔ b = a ⇨ b"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : NonAssocSemiring R"], "goal": "↑(ringChar R) = 0"}], "premise": [124637], "state_str": "R : Type u_1\ninst✝ : NonAssocSemiring R\n⊢ ↑(ringChar R) = 0"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u_2", "β : Type u_3", "inst✝ : GeneralizedCoheytingAlgebra α", "a b c d : α", "h : b ≤ a"], "goal": "a \\ b ⊔ b = a"}], "premise": [14537, 15078], "state_str": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\nh : b ≤ a\n⊢ a \\ b ⊔ b = a"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "W : Affine R", "x y : R"], "goal": "W.Nonsingular x (W.negY x y) ↔ W.Nonsingular x y"}], "premise": [1691, 132179, 132186, 132190], "state_str": "R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Nonsingular x (W.negY x y) ↔ W.Nonsingular x y"}
+{"state": [{"context": ["R : Type u", "inst✝ : CommRing R", "W : Affine R", "x y : R"], "goal": "W.Equation x y ∧ (W.a₁ * W.negY x y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ W.negY x y) ↔ W.Equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃)"}], "premise": [1157, 1205, 1674, 1713, 1718, 1773, 1965, 70089], "state_str": "R : Type u\ninst✝ : CommRing R\nW : Affine R\nx y : R\n⊢ W.Equation x y ∧ (W.a₁ * W.negY x y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ W.negY x y) ↔\n    W.Equation x y ∧ (W.a₁ * y ≠ 3 * x ^ 2 + 2 * W.a₂ * x + W.a₄ ∨ y ≠ -y - W.a₁ * x - W.a₃)"}
+{"state": [{"context": ["X : Type u", "Y : Type v", "ι : Type u_1", "inst✝¹ : TopologicalSpace X", "inst✝ : TopologicalSpace Y", "s t U : Set X"], "goal": "Continuous U.boolIndicator ↔ IsClopen U"}], "premise": [1713, 66464, 131687], "state_str": "X : Type u\nY : Type v\nι : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns t U : Set X\n⊢ Continuous U.boolIndicator ↔ IsClopen U"}
+{"state": [{"context": ["F : PFunctor.{u}", "X : Type u_1", "f✝ : X → ↑F X", "inst✝¹ : DecidableEq F.A", "inst✝ : Inhabited F.M", "ps : Path F", "a : F.A", "f : F.B a → F.M", "i : F.B a"], "goal": "isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = isubtree ps (f i)"}], "premise": [1739, 128876, 128882], "state_str": "F : PFunctor.{u}\nX : Type u_1\nf✝ : X → ↑F X\ninst✝¹ : DecidableEq F.A\ninst✝ : Inhabited F.M\nps : Path F\na : F.A\nf : F.B a → F.M\ni : F.B a\n⊢ isubtree (⟨a, i⟩ :: ps) (M.mk ⟨a, f⟩) = isubtree ps (f i)"}
+{"state": [{"context": ["m c : ℤ"], "goal": "2 * ((2 * c + 1) / 2) + 1 = 2 * c + 1"}], "premise": [119707], "state_str": "case intro\nm c : ℤ\n⊢ 2 * ((2 * c + 1) / 2) + 1 = 2 * c + 1"}
+{"state": [{"context": ["m c : ℤ"], "goal": "(2 * c + 1) / 2 * 2 + 1 = 2 * c + 1"}], "premise": [4239], "state_str": "case intro\nm c : ℤ\n⊢ (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1"}
+{"state": [{"context": ["m c : ℤ"], "goal": "1 = (2 * c + 1) % 2"}], "premise": [4287], "state_str": "case h.e'_2.h.e'_6\nm c : ℤ\n⊢ 1 = (2 * c + 1) % 2"}
+{"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", "p : M → Prop", "x : M", "s : Set M", "hs : closure s = ⊤", "mem : ∀ x ∈ s, p x", "mul : ∀ (x y : M), p x → p y → p (x * y)"], "goal": "p x"}], "premise": [116288], "state_str": "M : Type u_1\nN : Type u_2\nA : Type u_3\ninst✝¹ : Mul M\ns✝ : Set M\ninst✝ : Add A\nt : Set A\nS : Subsemigroup M\np : M → Prop\nx : M\ns : Set M\nhs : closure s = ⊤\nmem : ∀ x ∈ s, p x\nmul : ∀ (x y : M), p x → p y → p (x * y)\n⊢ p x"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0"], "goal": "∃ r, (eval r) F ≠ 0"}], "premise": [53688], "state_str": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\n⊢ ∃ r, (eval r) F ≠ 0"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0", "hF₀ : F ≠ 0"], "goal": "∃ r, (eval r) F ≠ 0"}], "premise": [78204, 111161, 111315], "state_str": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\n⊢ ∃ r, (eval r) F ≠ 0"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0", "hF₀ : F ≠ 0", "hdeg : ((finSuccEquiv R N) F).natDegree < n + 1"], "goal": "∃ r, (eval r) F ≠ 0"}], "premise": [1674, 2045], "state_str": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ ∃ r, (eval r) F ≠ 0"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0", "hF₀ : F ≠ 0", "hdeg : ((finSuccEquiv R N) F).natDegree < n + 1"], "goal": "(eval (Fin.cons 1 0)) F ≠ 0"}], "premise": [1690, 3913, 78187, 78207, 139145, 147079], "state_str": "case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\n⊢ (eval (Fin.cons 1 0)) F ≠ 0"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0", "hF₀ : F ≠ 0", "hdeg : ((finSuccEquiv R N) F).natDegree < n + 1", "aux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0"], "goal": "(eval (Fin.cons 1 0)) F ≠ 0"}], "premise": [102862, 102956, 111307, 112387, 119727, 119730, 119756, 126889, 126923, 127135], "state_str": "case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ (eval (Fin.cons 1 0)) F ≠ 0"}
+{"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 : ℕ", "F : MvPolynomial (Fin N.succ) R", "n : ℕ", "hF : F.IsHomogeneous n", "hFn : ((finSuccEquiv R N) F).coeff n ≠ 0", "hF₀ : F ≠ 0", "hdeg : ((finSuccEquiv R N) F).natDegree < n + 1", "aux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0"], "goal": "constantCoeff (((finSuccEquiv R N) F).coeff n) ≠ 0"}], "premise": [53688], "state_str": "case h\nσ : Type u_1\nτ : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² inst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nφ ψ : MvPolynomial σ R\nm n✝ N : ℕ\nF : MvPolynomial (Fin N.succ) R\nn : ℕ\nhF : F.IsHomogeneous n\nhFn : ((finSuccEquiv R N) F).coeff n ≠ 0\nhF₀ : F ≠ 0\nhdeg : ((finSuccEquiv R N) F).natDegree < n + 1\naux : ∀ i ∈ Finset.range n, constantCoeff (((finSuccEquiv R N) F).coeff i) = 0\n⊢ constantCoeff (((finSuccEquiv R N) F).coeff n) ≠ 0"}
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+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : Scheme", "x : ↑↑X.toPresheafedSpace", "U : X.Opens", "hxU : x ∈ U", "V : TopologicalSpace.Opens ↑X.toTopCat", "hxV : x ∈ V", "R : CommRingCat", "e : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)", "this : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U", "r : ↑R", "hr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)", "hr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)", "f : Spec (CommRingCat.of (Localization.Away r)) ⟶ X := Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯"], "goal": "x ∈ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ∧ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ⊆ ↑U"}], "premise": [126859], "state_str": "case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈\n      ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) ''\n        Set.range ⇑(Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r)))).val.base ⊆\n      ↑U"}
+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "X : Scheme", "x : ↑↑X.toPresheafedSpace", "U : X.Opens", "hxU : x ∈ U", "V : TopologicalSpace.Opens ↑X.toTopCat", "hxV : x ∈ V", "R : CommRingCat", "e : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)", "this : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U", "r : ↑R", "hr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)", "hr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)", "f : Spec (CommRingCat.of (Localization.Away r)) ⟶ X := Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯"], "goal": "x ∈ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ∧ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ⊆ ↑U"}], "premise": [1674, 88743, 134140], "state_str": "case intro.mk.intro.intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nX : Scheme\nx : ↑↑X.toPresheafedSpace\nU : X.Opens\nhxU : x ∈ U\nV : TopologicalSpace.Opens ↑X.toTopCat\nhxV : x ∈ V\nR : CommRingCat\ne : X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R)\nthis : e.hom.val.base ⟨x, hxV⟩ ∈ (Opens.map (e.inv.val.base ≫ V.inclusion)).obj U\nr : ↑R\nhr : e.hom.val.base ⟨x, hxV⟩ ∈ ↑(PrimeSpectrum.basicOpen r)\nhr' : ↑(PrimeSpectrum.basicOpen r) ⊆ ↑((Opens.map (e.inv.val.base ≫ V.inclusion)).obj U)\nf : Spec (CommRingCat.of (Localization.Away r)) ⟶ X :=\n  Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) ≫ e.inv ≫ X.ofRestrict ⋯\n⊢ x ∈ ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ∧\n    ⇑(e.inv.val.base ≫ (X.ofRestrict ⋯).val.base) '' ↑(PrimeSpectrum.basicOpen r) ⊆ ↑U"}
+{"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": "↑q < 0 ↔ q < 0"}], "premise": [1713], "state_str": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\np q : ℚ\nK : Type u_5\ninst✝ : LinearOrderedField K\n⊢ ↑q < 0 ↔ q < 0"}
+{"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)", "μ : Kernel α β", "κ η : Kernel (α × β) γ", "inst✝² : IsSFiniteKernel μ", "inst✝¹ : IsSFiniteKernel κ", "inst✝ : IsSFiniteKernel η", "a : α", "s : Set (β × γ)", "hs : MeasurableSet s"], "goal": "((μ ⊗ₖ (κ + η)) a) s = ((μ ⊗ₖ κ + μ ⊗ₖ η) a) s"}], "premise": [31458, 72613, 74222, 120650], "state_str": "case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ((μ ⊗ₖ (κ + η)) a) s = ((μ ⊗ₖ κ + μ ⊗ₖ η) a) s"}
+{"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)", "μ : Kernel α β", "κ η : Kernel (α × β) γ", "inst✝² : IsSFiniteKernel μ", "inst✝¹ : IsSFiniteKernel κ", "inst✝ : IsSFiniteKernel η", "a : α", "s : Set (β × γ)", "hs : MeasurableSet s"], "goal": "∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} + (η (a, b)) {c | (b, c) ∈ s} ∂μ a = ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} ∂μ a + ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂μ a"}], "premise": [30275], "state_str": "case h.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} + (η (a, b)) {c | (b, c) ∈ s} ∂μ a =\n    ∫⁻ (b : β), (κ (a, b)) {c | (b, c) ∈ s} ∂μ a + ∫⁻ (b : β), (η (a, b)) {c | (b, c) ∈ s} ∂μ a"}
+{"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)", "μ : Kernel α β", "κ η : Kernel (α × β) γ", "inst✝² : IsSFiniteKernel μ", "inst✝¹ : IsSFiniteKernel κ", "inst✝ : IsSFiniteKernel η", "a : α", "s : Set (β × γ)", "hs : MeasurableSet s"], "goal": "Measurable fun b => (κ (a, b)) {c | (b, c) ∈ s}"}], "premise": [73629], "state_str": "case h.h.hf\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\ns✝ : Set (β × γ)\nμ : Kernel α β\nκ η : Kernel (α × β) γ\ninst✝² : IsSFiniteKernel μ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\ns : Set (β × γ)\nhs : MeasurableSet s\n⊢ Measurable fun b => (κ (a, b)) {c | (b, c) ∈ 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", "hab : a < b"], "goal": "upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧ lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n"}], "premise": [1674, 105739], "state_str": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n"}
+{"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", "hab : a < b", "hab' : 0 < b - a", "hf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω"], "goal": "upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧ lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n"}], "premise": [1713, 105484, 105709], "state_str": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n m : ℕ\nω : Ω\nℱ : Filtration ℕ m0\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\n⊢ upperCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n    lowerCrossingTime 0 (b - a) (fun n ω => (f n ω - a)⁺) N n = lowerCrossingTime a b f N n"}
+{"state": [{"context": ["M : Type u", "N : Type v", "inst✝¹ : Monoid M", "inst✝ : Monoid N", "e : M ≃* N"], "goal": "𝟭 (SingleObj M) = e.toMonoidHom.toFunctor ⋙ e.symm.toMonoidHom.toFunctor"}], "premise": [99131, 99132], "state_str": "M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ 𝟭 (SingleObj M) = e.toMonoidHom.toFunctor ⋙ e.symm.toMonoidHom.toFunctor"}
+{"state": [{"context": ["M : Type u", "N : Type v", "inst✝¹ : Monoid M", "inst✝ : Monoid N", "e : M ≃* N"], "goal": "e.symm.toMonoidHom.toFunctor ⋙ e.toMonoidHom.toFunctor = 𝟭 (SingleObj N)"}], "premise": [99131, 99132], "state_str": "M : Type u\nN : Type v\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\n⊢ e.symm.toMonoidHom.toFunctor ⋙ e.toMonoidHom.toFunctor = 𝟭 (SingleObj N)"}
+{"state": [{"context": ["α : Type u_1", "β : Type v", "γ : Type u_2", "a b : α"], "goal": "card {a, b} = 2"}], "premise": [137819, 137910, 137911], "state_str": "α : Type u_1\nβ : Type v\nγ : Type u_2\na b : α\n⊢ card {a, b} = 2"}
+{"state": [{"context": ["Γ : Type u_1", "inst✝¹ : Inhabited Γ", "inst✝ : Finite Γ"], "goal": "∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}], "premise": [141498], "state_str": "Γ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}
+{"state": [{"context": ["Γ : Type u_1", "inst✝¹ : Inhabited Γ", "inst✝ : Finite Γ", "n : ℕ", "e : Γ ≃ Fin n", "this : DecidableEq Γ := e.decidableEq"], "goal": "∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}], "premise": [2101, 140170, 140171], "state_str": "case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}
+{"state": [{"context": ["Γ : Type u_1", "inst✝¹ : Inhabited Γ", "inst✝ : Finite Γ", "n : ℕ", "e : Γ ≃ Fin n", "this : DecidableEq Γ := e.decidableEq", "G : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }", "H : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding", "enc : Γ ↪ Vector Bool n := H.setValue default (Vector.replicate n false)"], "goal": "∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}], "premise": [70654, 70672, 71449], "state_str": "case intro.intro\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\ninst✝ : Finite Γ\nn : ℕ\ne : Γ ≃ Fin n\nthis : DecidableEq Γ := e.decidableEq\nG : Fin n ↪ Fin n → Bool := { toFun := fun a b => decide (a = b), inj' := ⋯ }\nH : Γ ↪ Vector Bool n := (e.toEmbedding.trans G).trans (Equiv.vectorEquivFin Bool n).symm.toEmbedding\nenc : Γ ↪ Vector Bool n := H.setValue default (Vector.replicate n false)\n⊢ ∃ n enc dec, enc default = Vector.replicate n false ∧ ∀ (a : Γ), dec (enc a) = a"}
+{"state": [{"context": ["a✝ b✝ m n p✝ a b : ℕ", "hb0 : b ≠ 0", "hab : a ∣ b", "p : ℕ"], "goal": "p ^ a.factorization p ∣ p ^ b.factorization p"}], "premise": [70035], "state_str": "a✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p"}
+{"state": [{"context": ["a✝ b✝ m n p✝ a b : ℕ", "hb0 : b ≠ 0", "hab : a ∣ b", "p : ℕ", "pp : Prime p"], "goal": "p ^ a.factorization p ∣ p ^ b.factorization p"}], "premise": [70039], "state_str": "case inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p"}
+{"state": [{"context": ["a✝ b✝ m n p✝ a b : ℕ", "hb0 : b ≠ 0", "hab : a ∣ b", "p : ℕ", "pp : Prime p", "ha0 : a ≠ 0"], "goal": "p ^ a.factorization p ∣ p ^ b.factorization p"}], "premise": [4731, 144294], "state_str": "case inr.inr\na✝ b✝ m n p✝ a b : ℕ\nhb0 : b ≠ 0\nhab : a ∣ b\np : ℕ\npp : Prime p\nha0 : a ≠ 0\n⊢ p ^ a.factorization p ∣ p ^ b.factorization p"}
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+{"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", "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 : Sheaf J A"], "goal": "Presheaf.IsSheaf K (G.op.ran.obj ℱ.val)"}], "premise": [91192], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\n⊢ Presheaf.IsSheaf K (G.op.ran.obj ℱ.val)"}
+{"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", "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 : Sheaf J A", "X : D", "S : K.Cover X"], "goal": "Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))"}], "premise": [91174], "state_str": "C : Type u_1\nD : Type u_2\ninst✝⁴ : Category.{u_3, u_1} C\ninst✝³ : Category.{u_4, u_2} D\nG : C ⥤ D\nA : Type w\ninst✝² : Category.{w', w} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝¹ : G.IsCocontinuous J K\ninst✝ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasPointwiseRightKanExtension F\nℱ : Sheaf J A\nX : D\nS : K.Cover X\n⊢ Nonempty (IsLimit (S.multifork (G.op.ran.obj ℱ.val)))"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "f : E →ₗᵢ[𝕜] F", "U V : Submodule 𝕜 E", "h : U ⟂ V"], "goal": "Submodule.map f U ⟂ Submodule.map f V"}], "premise": [33527], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : U ⟂ V\n⊢ Submodule.map f U ⟂ Submodule.map f V"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "F : Type u_3", "inst✝⁴ : RCLike 𝕜", "inst✝³ : NormedAddCommGroup E", "inst✝² : InnerProductSpace 𝕜 E", "inst✝¹ : NormedAddCommGroup F", "inst✝ : InnerProductSpace 𝕜 F", "f : E →ₗᵢ[𝕜] F", "U V : Submodule 𝕜 E", "h : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0"], "goal": "∀ u ∈ Submodule.map f U, ∀ v ∈ Submodule.map f V, ⟪u, v⟫_𝕜 = 0"}], "premise": [1186, 2010, 2052, 36860, 110262], "state_str": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E →ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫_𝕜 = 0\n⊢ ∀ u ∈ Submodule.map f U, ∀ v ∈ Submodule.map f V, ⟪u, v⟫_𝕜 = 0"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "f : α → Option α", "l : List α"], "goal": "(lookmap f l).length = l.length"}], "premise": [5122, 132639], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\nf : α → Option α\nl : List α\n⊢ (lookmap f l).length = l.length"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁵ : DecidableEq l", "inst✝⁴ : DecidableEq m", "inst✝³ : DecidableEq n", "inst✝² : Semiring α", "inst✝¹ : Finite m", "inst✝ : Finite n", "P : Matrix m n α → Prop", "M : Matrix m n α", "h_zero : P 0", "h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)", "h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)"], "goal": "P M"}], "premise": [141384], "state_str": "l : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\n⊢ P M"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁵ : DecidableEq l", "inst✝⁴ : DecidableEq m", "inst✝³ : DecidableEq n", "inst✝² : Semiring α", "inst✝¹ : Finite m", "inst✝ : Finite n", "P : Matrix m n α → Prop", "M : Matrix m n α", "h_zero : P 0", "h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)", "h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)", "val✝ : Fintype m"], "goal": "P M"}], "premise": [141384], "state_str": "case intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝ : Fintype m\n⊢ P M"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁵ : DecidableEq l", "inst✝⁴ : DecidableEq m", "inst✝³ : DecidableEq n", "inst✝² : Semiring α", "inst✝¹ : Finite m", "inst✝ : Finite n", "P : Matrix m n α → Prop", "M : Matrix m n α", "h_zero : P 0", "h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)", "h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)", "val✝¹ : Fintype m", "val✝ : Fintype n"], "goal": "P M"}], "premise": [127015, 139285], "state_str": "case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P M"}
+{"state": [{"context": ["l : Type u_1", "m : Type u_2", "n : Type u_3", "R : Type u_4", "α : Type u_5", "inst✝⁵ : DecidableEq l", "inst✝⁴ : DecidableEq m", "inst✝³ : DecidableEq n", "inst✝² : Semiring α", "inst✝¹ : Finite m", "inst✝ : Finite n", "P : Matrix m n α → Prop", "M : Matrix m n α", "h_zero : P 0", "h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)", "h_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)", "val✝¹ : Fintype m", "val✝ : Fintype n"], "goal": "P (∑ x ∈ Finset.univ ×ˢ Finset.univ, stdBasisMatrix x.1 x.2 (M x.1 x.2))"}], "premise": [127164], "state_str": "case intro.intro\nl : Type u_1\nm : Type u_2\nn : Type u_3\nR : Type u_4\nα : Type u_5\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : Semiring α\ninst✝¹ : Finite m\ninst✝ : Finite n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_zero : P 0\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m) (j : n) (x : α), P (stdBasisMatrix i j x)\nval✝¹ : Fintype m\nval✝ : Fintype n\n⊢ P (∑ x ∈ Finset.univ ×ˢ Finset.univ, stdBasisMatrix x.1 x.2 (M x.1 x.2))"}
+{"state": [{"context": ["m n : ℕ"], "goal": "(m + n)# = m# * ∏ p ∈ filter Nat.Prime (Ico (m + 1) (m + n + 1)), p"}], "premise": [19112, 20654, 126931, 139120], "state_str": "m n : ℕ\n⊢ (m + n)# = m# * ∏ p ∈ filter Nat.Prime (Ico (m + 1) (m + n + 1)), p"}
+{"state": [{"context": ["m n : ℕ"], "goal": "0 ≤ m + 1"}, {"context": ["m n : ℕ"], "goal": "m + 1 ≤ m + n + 1"}, {"context": ["m n : ℕ"], "goal": "Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))"}], "premise": [2134, 3747, 20613, 103886, 139114], "state_str": "case hab\nm n : ℕ\n⊢ 0 ≤ m + 1\n\ncase hbc\nm n : ℕ\n⊢ m + 1 ≤ m + n + 1\n\nm n : ℕ\n⊢ Disjoint (filter Nat.Prime (Ico 0 (m + 1))) (filter Nat.Prime (Ico (m + 1) (m + n + 1)))"}
+{"state": [{"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "g : SL(2, R)", "hg : ↑g 1 0 = 0"], "goal": "∃ a b, ∃ (h : a ≠ 0), g = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}], "premise": [85121], "state_str": "n : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\ng : SL(2, R)\nhg : ↑g 1 0 = 0\n⊢ ∃ a b, ∃ (h : a ≠ 0), g = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}
+{"state": [{"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0"], "goal": "∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}], "premise": [108558, 117816], "state_str": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}
+{"state": [{"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0", "had : a * d = 1"], "goal": "∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}], "premise": [108274], "state_str": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ∃ a_1 b_1, ∃ (h : a_1 ≠ 0), ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a_1, b_1; 0, a_1⁻¹], ⋯⟩"}
+{"state": [{"context": ["n : Type u", "inst✝⁴ : DecidableEq n", "inst✝³ : Fintype n", "R✝ : Type v", "inst✝² : CommRing R✝", "S : Type u_1", "inst✝¹ : CommRing S", "R : Type u_2", "inst✝ : Field R", "a b c d : R", "h_det : a * d - b * c = 1", "hg : c = 0", "had : a * d = 1"], "goal": "⟨!![a, b; c, d], ⋯⟩ = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}], "premise": [117820], "state_str": "case h\nn : Type u\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\nR✝ : Type v\ninst✝² : CommRing R✝\nS : Type u_1\ninst✝¹ : CommRing S\nR : Type u_2\ninst✝ : Field R\na b c d : R\nh_det : a * d - b * c = 1\nhg : c = 0\nhad : a * d = 1\n⊢ ⟨!![a, b; c, d], ⋯⟩ = ⟨!![a, b; 0, a⁻¹], ⋯⟩"}
+{"state": [{"context": ["α β : Type", "n : ℕ", "f g : (α → ℕ) → ℕ", "df : DiophFn f", "dg : DiophFn g"], "goal": "DiophFn fun v => f v ^ g v"}], "premise": [22658, 22660, 22661, 22672, 22678, 22679, 22682, 22683, 22684, 22685, 22686, 22688, 22691, 22693], "state_str": "α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\n⊢ DiophFn fun v => f v ^ g v"}
+{"state": [{"context": ["α β : Type", "n : ℕ", "f g : (α → ℕ) → ℕ", "df : DiophFn f", "dg : DiophFn g", "proof : Dioph (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩ (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪ {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | ∃ x, (x :: v) ∈ {v | (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v => ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩ ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩ ((fun v => const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 = v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩ ({v | v &6 < v &3} ∩ ({v | v &7 ≤ v &5} ∩ ({v | v &8 ≤ v &5} ∩ fun v => v &4 * v &4 - ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) - const (Fin2 (succ 8) → ℕ) 1 v) * (v &5 * v &2) * (v &5 * v &2) = const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))", "this : Dioph {v | v &2 = 0 ∧ v &0 = 1 ∨ 0 < v &2 ∧ (v &1 = 0 ∧ v &0 = 0 ∨ 0 < v &1 ∧ ∃ w a t z x y, (∃ (a1 : 1 < a), xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧ x ≡ y * (a - v &1) + v &0 [MOD t] ∧ 2 * a * v &1 = t + (v &1 * v &1 + 1) ∧ v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)}"], "goal": "DiophFn fun v => f v ^ g v"}], "premise": [1674, 1713, 1715, 1718, 1963, 1980, 22546, 22654, 22673, 22681], "state_str": "α β : Type\nn : ℕ\nf g : (α → ℕ) → ℕ\ndf : DiophFn f\ndg : DiophFn g\nproof :\n  Dioph\n    (((fun v => v &2 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 1 v) ∪\n      {v | const (Fin2 3 → ℕ) 0 v < v &2} ∩\n        (((fun v => v &1 = const (Fin2 3 → ℕ) 0 v) ∩ fun v => v &0 = const (Fin2 3 → ℕ) 0 v) ∪\n          {v | const (Fin2 3 → ℕ) 0 v < v &1} ∩\n            {v |\n              ∃ x,\n                (x :: v) ∈\n                  {v |\n                    ∃ x,\n                      (x :: v) ∈\n                        {v |\n                          ∃ x,\n                            (x :: v) ∈\n                              {v |\n                                ∃ x,\n                                  (x :: v) ∈\n                                    {v |\n                                      ∃ x,\n                                        (x :: v) ∈\n                                          {v |\n                                            ∃ x,\n                                              (x :: v) ∈\n                                                {v |\n                                                    (v ∘ &4 :: &8 :: &1 :: &0 :: []) ∈ fun v =>\n                                                      ∃ (h : 1 < v &0), xn h (v &1) = v &2 ∧ yn h (v &1) = v &3} ∩\n                                                  ((fun v => v &1 ≡ v &0 * (v &4 - v &7) + v &6 [MOD v &3]) ∩\n                                                    ((fun v =>\n                                                        const (Fin2 (succ 8) → ℕ) 2 v * v &4 * v &7 =\n                                                          v &3 + (v &7 * v &7 + const (Fin2 (succ 8) → ℕ) 1 v)) ∩\n                                                      ({v | v &6 < v &3} ∩\n                                                        ({v | v &7 ≤ v &5} ∩\n                                                          ({v | v &8 ≤ v &5} ∩ fun v =>\n                                                            v &4 * v &4 -\n                                                                ((v &5 + const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                        (v &5 + const (Fin2 (succ 8) → ℕ) 1 v) -\n                                                                      const (Fin2 (succ 8) → ℕ) 1 v) *\n                                                                    (v &5 * v &2) *\n                                                                  (v &5 * v &2) =\n                                                              const (Fin2 (succ 8) → ℕ) 1 v)))))}}}}}}))\nthis :\n  Dioph\n    {v |\n      v &2 = 0 ∧ v &0 = 1 ∨\n        0 < v &2 ∧\n          (v &1 = 0 ∧ v &0 = 0 ∨\n            0 < v &1 ∧\n              ∃ w a t z x y,\n                (∃ (a1 : 1 < a), xn a1 (v &2) = x ∧ yn a1 (v &2) = y) ∧\n                  x ≡ y * (a - v &1) + v &0 [MOD t] ∧\n                    2 * a * v &1 = t + (v &1 * v &1 + 1) ∧\n                      v &0 < t ∧ v &1 ≤ w ∧ v &2 ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)}\n⊢ DiophFn fun v => f v ^ g v"}
+{"state": [{"context": ["α : Type u_1", "l : Filter α", "f : α → ℝ"], "goal": "Tendsto (fun x => (f x).toNNReal) l atTop ↔ Tendsto f l atTop"}], "premise": [1671, 1713, 16363, 56432], "state_str": "α : Type u_1\nl : Filter α\nf : α → ℝ\n⊢ Tendsto (fun x => (f x).toNNReal) l atTop ↔ Tendsto f l atTop"}
+{"state": [{"context": ["z : ↥circle"], "goal": "normSq ↑z = 1"}], "premise": [148258], "state_str": "z : ↥circle\n⊢ normSq ↑z = 1"}
+{"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", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J"], "goal": "(pullbackCoverAffineRefinementObjIso f 𝒰 i).inv ≫ 𝒰.affineRefinement.openCover.pullbackHom f i = (𝒰.obj i.fst).affineCover.pullbackHom (𝒰.pullbackHom f i.fst) i.snd"}], "premise": [2098, 88746, 88755, 88789, 88799, 93340, 93753, 93754, 93898, 96173, 96174, 128130, 128141, 128142], "state_str": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\n⊢ (pullbackCoverAffineRefinementObjIso f 𝒰 i).inv ≫ 𝒰.affineRefinement.openCover.pullbackHom f i =\n    (𝒰.obj i.fst).affineCover.pullbackHom (𝒰.pullbackHom f i.fst) i.snd"}
+{"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", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J"], "goal": "(pullbackSymmetry ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)).inv ≫ (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫ pullback.fst (𝒰.affineRefinement.map i) f = pullback.snd (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd)"}], "premise": [93897], "state_str": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\n⊢ (pullbackSymmetry ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)).inv ≫\n      (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n        pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.snd (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd)"}
+{"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", "f : X ⟶ Y", "𝒰 : Y.OpenCover", "i : (𝒰.affineRefinement.openCover.pullbackCover f).J", "e_1✝ : (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ Spec (𝒰.affineRefinement.obj i)) = (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ (𝒰.obj i.fst).affineCover.obj i.snd)", "e_5✝ : Spec (𝒰.affineRefinement.obj i) = (𝒰.obj i.fst).affineCover.obj i.snd"], "goal": "(pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫ pullback.fst (𝒰.affineRefinement.map i) f = pullback.fst ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)"}], "premise": [94200], "state_str": "case h.e'_2.h.h.e'_7.h\nX Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.J), HasPullback (𝒰✝.map x ≫ f✝) g\nf : X ⟶ Y\n𝒰 : Y.OpenCover\ni : (𝒰.affineRefinement.openCover.pullbackCover f).J\ne_1✝ :\n  (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶ Spec (𝒰.affineRefinement.obj i)) =\n    (pullback (pullback.fst (𝒰.map i.fst) f) ((𝒰.obj i.fst).affineCover.map i.snd) ⟶\n      (𝒰.obj i.fst).affineCover.obj i.snd)\ne_5✝ : Spec (𝒰.affineRefinement.obj i) = (𝒰.obj i.fst).affineCover.obj i.snd\n⊢ (pullbackRightPullbackFstIso (𝒰.map i.fst) f ((𝒰.obj i.fst).affineCover.map i.snd)).hom ≫\n      pullback.fst (𝒰.affineRefinement.map i) f =\n    pullback.fst ((𝒰.obj i.fst).affineCover.map i.snd) (pullback.fst (𝒰.map i.fst) f)"}
+{"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", "c : α"], "goal": "c ∈ p.support"}], "premise": [375, 3786], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nG : SimpleGraph α\na b : α\np : G.Walk a b\nhp : p.IsHamiltonian\nc : α\n⊢ c ∈ p.support"}
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+{"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'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'", "p q : ℤ", "hpq : p + t' = q", "h' : n' + m = t'"], "goal": "((γ.comp γ' h).leftShift a t' ht').v p q hpq = (a * m).negOnePow • ((γ.leftShift a n' hn').comp γ' ⋯).v p q hpq"}], "premise": [96173, 97903, 114452, 115392, 118909, 119703, 122189], "state_str": "case h\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ ((γ.comp γ' h).leftShift a t' ht').v p q hpq = (a * m).negOnePow • ((γ.leftShift a n' hn').comp γ' ⋯).v p q hpq"}
+{"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'", "m t t' : ℤ", "γ' : Cochain L M m", "h : n + m = t", "ht' : t + a = t'", "p q : ℤ", "hpq : p + t' = q", "h' : n' + m = t'"], "goal": "(a * (n' + m)).negOnePow = (a * m).negOnePow * (a * n').negOnePow"}], "premise": [119708, 122189], "state_str": "case h.e_a.e_a\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nK L M : CochainComplex C ℤ\nn : ℤ\nγ γ₁ γ₂ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\nm t t' : ℤ\nγ' : Cochain L M m\nh : n + m = t\nht' : t + a = t'\np q : ℤ\nhpq : p + t' = q\nh' : n' + m = t'\n⊢ (a * (n' + m)).negOnePow = (a * m).negOnePow * (a * n').negOnePow"}
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+{"state": [{"context": ["G : Type u_1", "M : Type u_2", "inst✝ : Monoid M", "a✝ b c : M", "m n✝ : ℕ", "a : M", "n : ℕ"], "goal": "a ^ n * a = a * a ^ n"}], "premise": [119742, 119745], "state_str": "G : Type u_1\nM : Type u_2\ninst✝ : Monoid M\na✝ b c : M\nm n✝ : ℕ\na : M\nn : ℕ\n⊢ a ^ n * a = a * a ^ n"}
+{"state": [{"context": ["R : Type u_1", "A : Type u_2", "M : Type u_3", "inst✝⁶ : CommSemiring R", "inst✝⁵ : CommSemiring A", "inst✝⁴ : Algebra R A", "inst✝³ : AddCommMonoid M", "inst✝² : Module A M", "inst✝¹ : Module R M", "inst✝ : IsScalarTower R A M", "d : Derivation R A M", "a : A"], "goal": "∀ (a_1 b : R[X]), { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } (a_1 * b) = a_1 • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } b + b • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } a_1"}], "premise": [79729, 101165], "state_str": "R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid M\ninst✝² : Module A M\ninst✝¹ : Module R M\ninst✝ : IsScalarTower R A M\nd : Derivation R A M\na : A\n⊢ ∀ (a_1 b : R[X]),\n    { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } (a_1 * b) =\n      a_1 • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } b +\n        b • { toFun := fun f => (AEval.of R M a) (d ((aeval a) f)), map_add' := ⋯, map_smul' := ⋯ } a_1"}
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+{"state": [{"context": ["C : Type u₁", "inst✝¹ : Category.{v₁, u₁} C", "X Y : C", "f : X ⟶ Y", "inst✝ : IsIso f"], "goal": "(inv f).op = inv f.op"}], "premise": [88792], "state_str": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ (inv f).op = inv f.op"}
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+{"state": [{"context": ["α β : Type u", "f : Type u → Type v", "inst✝¹ : Functor f", "inst✝ : LawfulFunctor f", "x : f α"], "goal": "(mapEquiv f (Equiv.refl α)) x = (Equiv.refl (f α)) x"}], "premise": [70748, 70952], "state_str": "case H\nα β : Type u\nf : Type u → Type v\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nx : f α\n⊢ (mapEquiv f (Equiv.refl α)) x = (Equiv.refl (f α)) x"}
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+{"state": [{"context": ["t : Type u → Type u", "inst✝⁵ : Traversable t", "inst✝⁴ : LawfulTraversable t", "F G : Type u → Type u", "inst✝³ : Applicative F", "inst✝² : LawfulApplicative F", "inst✝¹ : Applicative G", "inst✝ : LawfulApplicative G", "α β γ : Type u", "g : α → F β", "h : β → G γ", "f : β → γ", "x : t β", "x✝ : t α"], "goal": "traverse pure x✝ = pure x✝"}], "premise": [9734], "state_str": "case h\nt : Type u → Type u\ninst✝⁵ : Traversable t\ninst✝⁴ : LawfulTraversable t\nF G : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nα β γ : Type u\ng : α → F β\nh : β → G γ\nf : β → γ\nx : t β\nx✝ : t α\n⊢ traverse pure x✝ = pure x✝"}
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+{"state": [{"context": ["C : Type u_1", "inst✝⁶ : Category.{?u.130934, u_1} C", "inst✝⁵ : 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✝⁴ : DecidableEq I₁₂", "inst✝³ : TotalComplexShape c₁ c₂ c₁₂", "inst✝² : K.HasTotal c₁₂", "inst✝¹ : L.HasTotal c₁₂", "inst✝ : M.HasTotal c₁₂"], "goal": "map e.inv c₁₂ ≫ map e.hom c₁₂ = 𝟙 (L.total c₁₂)"}], "premise": [88742, 115958, 115959], "state_str": "C : Type u_1\ninst✝⁶ : Category.{?u.130934, u_1} C\ninst✝⁵ : Preadditive C\nI₁ : Type u_2\nI₂ : Type u_3\nI₁₂ : Type u_4\nc₁ : ComplexShape I₁\nc₂ : ComplexShape I₂\nK L M : HomologicalComplex₂ C c₁ c₂\nφ : K ⟶ L\ne : K ≅ L\nψ : L ⟶ M\nc₁₂ : ComplexShape I₁₂\ninst✝⁴ : DecidableEq I₁₂\ninst✝³ : TotalComplexShape c₁ c₂ c₁₂\ninst✝² : K.HasTotal c₁₂\ninst✝¹ : L.HasTotal c₁₂\ninst✝ : M.HasTotal c₁₂\n⊢ map e.inv c₁₂ ≫ map e.hom c₁₂ = 𝟙 (L.total c₁₂)"}
+{"state": [{"context": ["R : Type u_1", "inst✝¹ : LinearOrderedSemifield R", "inst✝ : FloorSemiring R", "b : ℕ"], "goal": "log b 1 = 0"}], "premise": [2200, 14272, 105015, 129044, 142694], "state_str": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\n⊢ log b 1 = 0"}
+{"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 : C", "f : W ⟶ X", "g : Y ⟶ Z"], "goal": "(𝟙 Y ⊗ f) ≫ (g ⊗ 𝟙 X) = g ⊗ f"}], "premise": [99215], "state_str": "C✝ : Type u\n𝒞 : Category.{v, u} C✝\ninst✝² : MonoidalCategory C✝\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : MonoidalCategory C\nU V W X Y Z : C\nf : W ⟶ X\ng : Y ⟶ Z\n⊢ (𝟙 Y ⊗ f) ≫ (g ⊗ 𝟙 X) = g ⊗ f"}
+{"state": [{"context": ["α : Type u_1", "m : Set α → ℝ≥0∞", "β : Type u_2", "f : β → α", "h : (Monotone fun s => m ↑s) ∨ Surjective f"], "goal": "(comap f) (boundedBy m) = boundedBy fun s => m (f '' s)"}], "premise": [2101, 28102], "state_str": "α : Type u_1\nm : Set α → ℝ≥0∞\nβ : Type u_2\nf : β → α\nh : (Monotone fun s => m ↑s) ∨ Surjective f\n⊢ (comap f) (boundedBy m) = boundedBy fun s => m (f '' s)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "F : Type u_3", "inst✝¹ : FunLike F (Set α) ℝ≥0∞", "inst✝ : OuterMeasureClass F α", "μ : F", "s✝ t✝ s t : Set α"], "goal": "μ (s ∆ t) = 0 ↔ s =ᶠ[ae μ] t"}], "premise": [17200, 27622], "state_str": "α : Type u_1\nβ : Type u_2\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns✝ t✝ s t : Set α\n⊢ μ (s ∆ t) = 0 ↔ s =ᶠ[ae μ] t"}
+{"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'", "l₁ l₂ : (sheafify J G).toPresheaf ⟶ F'", "e : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂", "U : Cᵒᵖ", "s : F.obj U", "hs : s ∈ (sheafify J G).obj U"], "goal": "l₁.app U ⟨s, hs⟩ = l₂.app U ⟨s, hs⟩"}], "premise": [90222, 90232], "state_str": "case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\n⊢ l₁.app U ⟨s, hs⟩ = l₂.app U ⟨s, hs⟩"}
+{"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'", "l₁ l₂ : (sheafify J G).toPresheaf ⟶ F'", "e : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂", "U : Cᵒᵖ", "s : F.obj U", "hs : s ∈ (sheafify J G).obj U", "V : C", "i : V ⟶ unop U", "hi : F.map i.op s ∈ G.obj (op V)"], "goal": "F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)"}], "premise": [91300], "state_str": "case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ F'.map i.op (l₁.app U ⟨s, hs⟩) = F'.map i.op (l₂.app U ⟨s, hs⟩)"}
+{"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'", "l₁ l₂ : (sheafify J G).toPresheaf ⟶ F'", "e : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂", "U : Cᵒᵖ", "s : F.obj U", "hs : s ∈ (sheafify J G).obj U", "V : C", "i : V ⟶ unop U", "hi : F.map i.op s ∈ G.obj (op V)"], "goal": "l₁.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩) = l₂.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩)"}], "premise": [97889], "state_str": "case w.h.h.mk\nC : Type u\ninst✝ : Category.{v, u} C\nJ : GrothendieckTopology C\nF F' F'' : Cᵒᵖ ⥤ Type w\nG G' : Subpresheaf F\nh : Presieve.IsSheaf J F'\nl₁ l₂ : (sheafify J G).toPresheaf ⟶ F'\ne : homOfLe ⋯ ≫ l₁ = homOfLe ⋯ ≫ l₂\nU : Cᵒᵖ\ns : F.obj U\nhs : s ∈ (sheafify J G).obj U\nV : C\ni : V ⟶ unop U\nhi : F.map i.op s ∈ G.obj (op V)\n⊢ l₁.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩) =\n    l₂.app (op V) ((sheafify J G).toPresheaf.map i.op ⟨s, hs⟩)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "l✝ m : Language α", "a b x : List α", "l : Language α"], "goal": "1 + l * l∗ = l∗"}], "premise": [69669, 69673, 119739, 119745], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ 1 + l * l∗ = l∗"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "l✝ m : Language α", "a b x : List α", "l : Language α"], "goal": "l ^ 0 + ⨆ i, l ^ (i + 1) = ⨆ i, l ^ i"}], "premise": [19488], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ l ^ 0 + ⨆ i, l ^ (i + 1) = ⨆ i, l ^ i"}
+{"state": [{"context": ["k G : Type u", "inst✝¹ : CommRing k", "n : ℕ", "inst✝ : Monoid G"], "goal": "((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫ (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫ (HomotopyEquiv.ofIso ((ChainComplex.single₀ (ModuleCat k)).mapIso (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f 0 = (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)"}], "premise": [113853], "state_str": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ ((HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom ≫\n          (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom ≫\n            (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom).f\n      0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)"}
+{"state": [{"context": ["k G : Type u", "inst✝¹ : CommRing k", "n : ℕ", "inst✝ : Monoid G"], "goal": "(HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫ (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫ (HomotopyEquiv.ofIso ((ChainComplex.single₀ (ModuleCat k)).mapIso (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f 0 = (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)"}], "premise": [96175], "state_str": "k G : Type u\ninst✝¹ : CommRing k\nn : ℕ\ninst✝ : Monoid G\n⊢ (HomotopyEquiv.ofIso (compForgetAugmentedIso k G).symm).hom.f 0 ≫\n      (extraDegeneracyCompForgetAugmentedToModule k G).homotopyEquiv.hom.f 0 ≫\n        (HomotopyEquiv.ofIso\n                ((ChainComplex.single₀ (ModuleCat k)).mapIso\n                  (Finsupp.LinearEquiv.finsuppUnique k k (⊤_ Type u)).toModuleIso)).hom.f\n          0 =\n    (forget₂ (Rep k G) (ModuleCat k)).map (ε k G)"}
+{"state": [{"context": ["d : ℤ", "a b : Solution₁ d"], "goal": "(a * b).x = a.x * b.x + d * (a.y * b.y)"}], "premise": [119703], "state_str": "d : ℤ\na b : Solution₁ d\n⊢ (a * b).x = a.x * b.x + d * (a.y * b.y)"}
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+{"state": [{"context": ["n✝ b n : ℕ"], "goal": "(b.digits n).length ≤ (b.digits (n + 1)).length"}], "premise": [70037], "state_str": "n✝ b n : ℕ\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length"}
+{"state": [{"context": ["n✝ b n : ℕ", "hn : n ≠ 0"], "goal": "(b.digits n).length ≤ (b.digits (n + 1)).length"}], "premise": [14317], "state_str": "case inr\nn✝ b n : ℕ\nhn : n ≠ 0\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length"}
+{"state": [{"context": ["n✝ b n : ℕ", "hn : n ≠ 0", "hb : 1 < b"], "goal": "(b.digits n).length ≤ (b.digits (n + 1)).length"}], "premise": [2140, 142711, 145920], "state_str": "case inr.inr\nn✝ b n : ℕ\nhn : n ≠ 0\nhb : 1 < b\n⊢ (b.digits n).length ≤ (b.digits (n + 1)).length"}
+{"state": [{"context": ["R : Type u_1", "inst✝ : CommSemiring R", "p : R[X]", "hp : p.Monic", "r : R", "q : R[X]", "h : p = C r * q"], "goal": "IsUnit r"}], "premise": [101414, 120490], "state_str": "case intro\nR : Type u_1\ninst✝ : CommSemiring R\np : R[X]\nhp : p.Monic\nr : R\nq : R[X]\nh : p = C r * q\n⊢ IsUnit r"}
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+{"state": [{"context": ["α : Type u_1", "inst✝¹ : Lattice α", "inst✝ : IsModularLattice α", "x y z a b : α", "c : ↑(Ioo b (a ⊔ b))"], "goal": "a ⊓ ↑c ⊔ b = ↑c"}], "premise": [1674, 2106, 2107, 14568, 14579, 17614, 137122], "state_str": "α : Type u_1\ninst✝¹ : Lattice α\ninst✝ : IsModularLattice α\nx y z a b : α\nc : ↑(Ioo b (a ⊔ b))\n⊢ a ⊓ ↑c ⊔ b = ↑c"}
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+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝² : RCLike 𝕜", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f : E →ₗ[𝕜] E", "hf : LipschitzWith 1 ⇑f", "g : E →L[𝕜] ↥(eqLocus f 1)", "hg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x", "hg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))", "y : E", "hy : g y = 0", "z : E", "hz : IsFixedPt (⇑f) z", "this : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}"], "goal": "Tendsto (fun x => birkhoffAverage 𝕜 (⇑f) _root_.id x y) atTop (𝓝 0)"}], "premise": [1674, 55412, 134168], "state_str": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\n⊢ Tendsto (fun x => birkhoffAverage 𝕜 (⇑f) _root_.id x y) atTop (𝓝 0)"}
+{"state": [{"context": ["𝕜 : Type u_1", "E : Type u_2", "inst✝² : RCLike 𝕜", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace 𝕜 E", "f : E →ₗ[𝕜] E", "hf : LipschitzWith 1 ⇑f", "g : E →L[𝕜] ↥(eqLocus f 1)", "hg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x", "hg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))", "y : E", "hy : g y = 0", "z : E", "hz : IsFixedPt (⇑f) z", "this✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}", "x : E", "this : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))"], "goal": "(f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}"}], "premise": [115674], "state_str": "case intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(ker g) ⊆ closure ↑(range (f - 1))\ny : E\nhy : g y = 0\nz : E\nhz : IsFixedPt (⇑f) z\nthis✝ : IsClosed {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}\nx : E\nthis : Bornology.IsBounded (Set.range fun x_1 => _root_.id ((⇑f)^[x_1] x))\n⊢ (f - 1) x ∈ {x | Tendsto (fun x_1 => birkhoffAverage 𝕜 (⇑f) _root_.id x_1 x) atTop (𝓝 0)}"}
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+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Finite α", "p q : α → Prop", "inst✝¹ : DecidablePred p", "inst✝ : DecidablePred q", "e : { x // p x } ≃ { x // q x }", "x : α", "hx : p x"], "goal": "(e.subtypeCongr e.toCompl) x = ↑(e ⟨x, hx⟩)"}], "premise": [70750, 71740], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (e.subtypeCongr e.toCompl) x = ↑(e ⟨x, hx⟩)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝² : Finite α", "p q : α → Prop", "inst✝¹ : DecidablePred p", "inst✝ : DecidablePred q", "e : { x // p x } ≃ { x // q x }", "x : α", "hx : p x"], "goal": "(sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e ⟨x, hx⟩)"}], "premise": [138, 71772, 71774], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : Finite α\np q : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : { x // p x } ≃ { x // q x }\nx : α\nhx : p x\n⊢ (sumCompl q) (Sum.map (⇑e) (⇑e.toCompl) ((sumCompl p).symm x)) = ↑(e ⟨x, hx⟩)"}
+{"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)", "j : N →* P", "x✝ : N"], "goal": "(f.lift ⋯) x✝ = j x✝"}], "premise": [9211], "state_str": "case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ (f.lift ⋯) x✝ = j x✝"}
+{"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)", "j : N →* P", "x✝ : N"], "goal": "j (f.toMap (f.sec x✝).1) = j (f.toMap ↑(f.sec x✝).2) * j x✝"}], "premise": [9134, 117166], "state_str": "case h\nM : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nj : N →* P\nx✝ : N\n⊢ j (f.toMap (f.sec x✝).1) = j (f.toMap ↑(f.sec x✝).2) * j x✝"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝ : CommMonoid β", "n : ℕ", "f : Fin n → β"], "goal": "(List.ofFn f).prod = ∏ i : Fin n, f i"}], "premise": [126882], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝ : CommMonoid β\nn : ℕ\nf : Fin n → β\n⊢ (List.ofFn f).prod = ∏ i : Fin n, f i"}
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+{"state": [{"context": ["α : Type u", "a : α", "inst✝¹ : Group α", "G : Type u_1", "inst✝ : Group G", "x✝ : Fintype G", "p : ℕ", "hp : Fact (Nat.Prime p)", "h : Fintype.card G = p", "g g' : G", "hg : g ≠ 1"], "goal": "g' ∈ zpowers g"}], "premise": [7941, 122655], "state_str": "α : Type u\na : α\ninst✝¹ : Group α\nG : Type u_1\ninst✝ : Group G\nx✝ : Fintype G\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card G = p\ng g' : G\nhg : g ≠ 1\n⊢ g' ∈ zpowers g"}
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+{"state": [{"context": ["V : Type u_1", "inst✝² : Fintype V", "inst✝¹ : DecidableEq V", "G : SimpleGraph V", "inst✝ : DecidableRel G.Adj", "n r : ℕ", "s t u : V", "h : G.IsTuranMaximal r"], "goal": "∀ {x y : V}, ¬G.Adj x y → ¬G.Adj y x"}], "premise": [50396], "state_str": "V : Type u_1\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn r : ℕ\ns t u : V\nh : G.IsTuranMaximal r\n⊢ ∀ {x y : V}, ¬G.Adj x y → ¬G.Adj y x"}
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+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "a : α", "r₁ r₂ : List (List α)"], "goal": "foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r₂) (toArray r₁).data).toList"}], "premise": [1444], "state_str": "α : Type u\nβ : Type v\nγ : Type w\na : α\nr₁ r₂ : List (List α)\n⊢ foldl (fun r l => r ++ [a :: l]) r₂ r₁ = (foldl (fun r l => r.push (a :: l)) (toArray r��) (toArray r₁).data).toList"}
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+{"state": [{"context": ["s : ℝ", "hs : 0 < s"], "goal": "∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1"}], "premise": [47085], "state_str": "s : ℝ\nhs : 0 < s\n⊢ ∃ M, 𝓝[stolzCone s] 1 ≤ 𝓝[stolzSet M] 1"}
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+{"state": [{"context": ["C : Type u", "inst✝ : Category.{v, u} C", "G : Cᵒᵖ ⥤ Grp", "X : C", "I : Type w'", "U : I → C", "γ₁ γ₂✝ γ₂ : OneCocycle G U", "α : ZeroCochain G U", "h : OneCohomologyRelation γ₁.toOneCochain γ₂✝.toOneCochain α", "β : ZeroCochain G U", "h' : OneCohomologyRelation γ₂✝.toOneCochain γ₂.toOneCochain β"], "goal": "γ₁.IsCohomologous γ₂"}], "premise": [90425], "state_str": "case intro.intro\nC : Type u\ninst✝ : Category.{v, u} C\nG : Cᵒᵖ ⥤ Grp\nX : C\nI : Type w'\nU : I → C\nγ₁ γ₂✝ γ₂ : OneCocycle G U\nα : ZeroCochain G U\nh : OneCohomologyRelation γ₁.toOneCochain γ₂✝.toOneCochain α\nβ : ZeroCochain G U\nh' : OneCohomologyRelation γ₂✝.toOneCochain γ₂.toOneCochain β\n⊢ γ₁.IsCohomologous γ₂"}
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+{"state": [{"context": ["K : Subfield ℂ", "hc : IsClosed ↑K"], "goal": "range ofReal' ⊆ closure (ofReal' '' range Rat.cast)"}], "premise": [19963, 46226, 55655], "state_str": "K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ closure (ofReal' '' range Rat.cast)"}
+{"state": [{"context": ["K : Subfield ℂ", "hc : IsClosed ↑K"], "goal": "range ofReal' ⊆ ofReal' '' closure (range Rat.cast)"}], "premise": [54388, 55665, 56331], "state_str": "K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' closure (range Rat.cast)"}
+{"state": [{"context": ["K : Subfield ℂ", "hc : IsClosed ↑K"], "goal": "range ofReal' ⊆ ofReal' '' univ"}], "premise": [134174], "state_str": "K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal' ⊆ ofReal' '' univ"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "l₁ l₂ : List α", "p : α → Bool", "xs ys : List α", "ls : List (List α)", "f : List α → List α", "x : α", "inst✝ : DecidableEq α", "hd : α", "tl hd' : List α", "tl' : List (List α)", "ih : (intersperse [x] (hd' :: tl')).join = tl", "h' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'"], "goal": "(intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl"}], "premise": [132597], "state_str": "case cons.cons\nι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\nl₁ l₂ : List α\np : α → Bool\nxs ys : List α\nls : List (List α)\nf : List α → List α\nx : α\ninst✝ : DecidableEq α\nhd : α\ntl hd' : List α\ntl' : List (List α)\nih : (intersperse [x] (hd' :: tl')).join = tl\nh' : splitOnP (fun x_1 => x_1 == x) tl = hd' :: tl'\n⊢ (intersperse [x] (splitOnP (fun x_1 => x_1 == x) (hd :: tl))).join = hd :: tl"}
+{"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.2 / x.1) (𝓝 1)"}], "premise": [16208, 60481], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (fun x => x.2 / x.1) (𝓝 1)"}
+{"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 (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α)"}], "premise": [1673, 14296, 16224], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\n⊢ 𝓤 α = comap (Prod.mk 1 ∘ fun x => x.2 / x.1) (𝓤 α)"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "X : Type u_4", "Y : Type u_5", "f : ι → Filter α"], "goal": "𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i)"}], "premise": [55683], "state_str": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nf : ι → Filter α\n⊢ 𝓝 (⨅ i, f i) = ⨅ i, 𝓝 (f i)"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "X : Type u_4", "Y : Type u_5", "f : ι → Filter α"], "goal": "(⨅ i, f i).lift' (Iic ∘ 𝓟) = ⨅ i, (f i).lift' (Iic ∘ 𝓟)"}], "premise": [12741], "state_str": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nX : Type u_4\nY : Type u_5\nf : ι → Filter α\n⊢ (⨅ i, f i).lift' (Iic ∘ 𝓟) = ⨅ i, (f i).lift' (Iic ∘ 𝓟)"}
+{"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✝ : MeasurableSpace α", "f : α → ℝ≥0", "μ : Measure α", "h : AEMeasurable (fun x => ↑(f x)) μ"], "goal": "AEMeasurable f μ"}], "premise": [25964, 146652], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0\nμ : Measure α\nh : AEMeasurable (fun x => ↑(f x)) μ\n⊢ AEMeasurable f μ"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "σ : α → Type u_3", "head✝ : α", "l : List α"], "goal": "((head✝ :: l).sigma fun a => []) = []"}], "premise": [130484], "state_str": "α : Type u_1\nβ : Type u_2\nσ : α → Type u_3\nhead✝ : α\nl : List α\n⊢ ((head✝ :: l).sigma fun a => []) = []"}
+{"state": [{"context": ["ι : Type u_1", "α : Type u", "β : Type v", "γ : Type w", "inst✝¹ : PseudoMetricSpace α", "inst✝ : PseudoMetricSpace β", "h✝ h : α ≃ᵢ β", "x : α", "r : ℝ"], "goal": "⇑h '' Metric.closedBall x r = Metric.closedBall (h x) r"}], "premise": [59867, 59877, 59911], "state_str": "ι : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nh✝ h : α ≃ᵢ β\nx : α\nr : ℝ\n⊢ ⇑h '' Metric.closedBall x r = Metric.closedBall (h x) r"}
+{"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 (0 ≤ a) _auto✝"], "goal": "‖a‖ ∈ spectrum ℝ a"}], "premise": [34934], "state_str": "A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CstarRing A\ninst✝⁵ : CompleteSpace A\ninst✝⁴ : NormedAlgebra ℂ A\ninst✝³ : StarModule ℂ A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nontrivial A\na : A\nha : autoParam (0 ≤ a) _auto✝\n⊢ ‖a‖ ∈ spectrum ℝ a"}
+{"state": [{"context": ["this : eulerMascheroniSeq 6 = 49 / 20 - log 7"], "goal": "1 / 2 < eulerMascheroniSeq 6"}], "premise": [37887, 105742, 105755], "state_str": "this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 1 / 2 < eulerMascheroniSeq 6"}
+{"state": [{"context": ["this : eulerMascheroniSeq 6 = 49 / 20 - log 7"], "goal": "7 < rexp (49 / 20 - 1 / 2)"}], "premise": [14287, 149289], "state_str": "this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < rexp (49 / 20 - 1 / 2)"}
+{"state": [{"context": ["this : eulerMascheroniSeq 6 = 49 / 20 - log 7"], "goal": "7 < ∑ i ∈ Finset.range 7, (49 / 20 - 1 / 2) ^ i / ↑i.factorial"}], "premise": [127135, 144416], "state_str": "this : eulerMascheroniSeq 6 = 49 / 20 - log 7\n⊢ 7 < ∑ i ∈ Finset.range 7, (49 / 20 - 1 / 2) ^ i / ↑i.factorial"}
+{"state": [{"context": ["α : Type u", "β : Type v", "R✝ r : α → α → Prop", "l l₁ l₂ : List α", "a✝ b✝ : α", "R : α → α → Prop", "a b : α", "t : List α"], "goal": "Chain R a (b :: t) ↔ (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧ ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)"}], "premise": [1532], "state_str": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ Chain R a (b :: t) ↔\n    (∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)) ∧\n      ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)"}
+{"state": [{"context": ["α : Type u", "β : Type v", "R✝ r : α → α → Prop", "l l₁ l₂ : List α", "a✝ b✝ : α", "R : α → α → Prop", "a b : α", "t : List α", "h0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)", "h : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)", "i : ℕ", "w : i < t.length - 1"], "goal": "R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)"}], "premise": [2581], "state_str": "case mpr.intro.right.right\nα : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\nh0 : ∀ (h : 0 < (b :: t).length), R a ((b :: t).get ⟨0, h⟩)\nh : ∀ (i : ℕ) (h : i < (b :: t).length - 1), R ((b :: t).get ⟨i, ⋯⟩) ((b :: t).get ⟨i + 1, ⋯⟩)\ni : ℕ\nw : i < t.length - 1\n⊢ R (t.get ⟨i, ⋯⟩) (t.get ⟨i + 1, ⋯⟩)"}
+{"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"], "goal": "𝟭 (W.Localization ⥤ D) = functor W D ⋙ inverse W D"}], "premise": [97754], "state_str": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nD : Type uD\ninst✝ : Category.{uD', uD} D\nG : C ⥤ D\nhG : W.IsInvertedBy G\n⊢ 𝟭 (W.Localization ⥤ D) = functor W D ⋙ inverse W D"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Semiring α", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "p a b : α", "h : multiplicity p a ≤ multiplicity p b"], "goal": "min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)"}], "premise": [19701, 79561], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≤ multiplicity p b\n⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Semiring α", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "p a b : α", "h : multiplicity p a ≤ multiplicity p b"], "goal": "∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b"}], "premise": [1673, 79561, 121987], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p a ≤ multiplicity p b\n⊢ ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ a + b"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "inst✝¹ : Semiring α", "inst✝ : DecidableRel fun x x_1 => x ∣ x_1", "p a b : α", "h : multiplicity p b ≤ multiplicity p a"], "goal": "min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)"}], "premise": [19702, 79561], "state_str": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Semiring α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\np a b : α\nh : multiplicity p b ≤ multiplicity p a\n⊢ min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b)"}
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+{"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", "g : F → G", "f : E → F", "q : FormalMultilinearSeries 𝕜 F G", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "rg : ℝ≥0∞", "Hg : HasFPowerSeriesOnBall g q (f x) rg", "rf : ℝ≥0∞", "Hf : HasFPowerSeriesOnBall f p x rf", "r : ℝ≥0", "r_pos : 0 < r", "hr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg", "rf' : ℝ≥0∞ := min rf δ", "min_pos : 0 < min rf' ↑r"], "goal": "HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x (min rf' ↑r)"}], "premise": [14273, 19691, 34732], "state_str": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\n⊢ HasFPowerSeriesOnBall (g ∘ f) (q.comp p) x (min rf' ↑r)"}
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+{"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", "g : F → G", "f : E → F", "q : FormalMultilinearSeries 𝕜 F G", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "rg : ℝ≥0∞", "Hg : HasFPowerSeriesOnBall g q (f x) rg", "rf : ℝ≥0∞", "Hf : HasFPowerSeriesOnBall f p x rf", "r : ℝ≥0", "r_pos : 0 < r", "hr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg", "rf' : ℝ≥0∞ := min rf δ", "min_pos : 0 < min rf' ↑r", "y : E", "hy : y ∈ EMetric.ball 0 (min rf' ↑r)", "y_mem : y ∈ EMetric.ball 0 rf"], "goal": "HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}], "premise": [14273, 19690, 19691, 42787, 42789, 58482], "state_str": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}
+{"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", "g : F → G", "f : E → F", "q : FormalMultilinearSeries 𝕜 F G", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "rg : ℝ≥0∞", "Hg : HasFPowerSeriesOnBall g q (f x) rg", "rf : ℝ≥0∞", "Hf : HasFPowerSeriesOnBall f p x rf", "r : ℝ≥0", "r_pos : 0 < r", "hr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg", "rf' : ℝ≥0∞ := min rf δ", "min_pos : 0 < min rf' ↑r", "y : E", "hy : y ∈ EMetric.ball 0 (min rf' ↑r)", "y_mem : y ∈ EMetric.ball 0 rf", "fy_mem : f (x + y) ∈ EMetric.ball (f x) rg"], "goal": "HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}], "premise": [15489, 16350, 19114, 35854, 35899, 55522, 63868, 66817, 118073, 124715], "state_str": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}
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+{"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", "g : F → G", "f : E → F", "q : FormalMultilinearSeries 𝕜 F G", "p : FormalMultilinearSeries 𝕜 E F", "x : E", "rg : ℝ≥0∞", "Hg : HasFPowerSeriesOnBall g q (f x) rg", "rf : ℝ≥0∞", "Hf : HasFPowerSeriesOnBall f p x rf", "r : ℝ≥0", "r_pos : 0 < r", "hr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst", "δ : ℝ≥0∞", "δpos : 0 < δ", "hδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg", "rf' : ℝ≥0∞ := min rf δ", "min_pos : 0 < min rf' ↑r", "y : E", "hy : y ∈ EMetric.ball 0 (min rf' ↑r)", "y_mem : y ∈ EMetric.ball 0 rf", "fy_mem : f (x + y) ∈ EMetric.ball (f x) rg", "A : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))", "B : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))"], "goal": "HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}], "premise": [34740], "state_str": "case intro.intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\ng : F → G\nf : E → F\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMultilinearSeries 𝕜 E F\nx : E\nrg : ℝ≥0∞\nHg : HasFPowerSeriesOnBall g q (f x) rg\nrf : ℝ≥0∞\nHf : HasFPowerSeriesOnBall f p x rf\nr : ℝ≥0\nr_pos : 0 < r\nhr : Summable fun i => ‖q.compAlongComposition p i.snd‖₊ * r ^ i.fst\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : ∀ {z : E}, z ∈ EMetric.ball x δ → f z ∈ EMetric.ball (f x) rg\nrf' : ℝ≥0∞ := min rf δ\nmin_pos : 0 < min rf' ↑r\ny : E\nhy : y ∈ EMetric.ball 0 (min rf' ↑r)\ny_mem : y ∈ EMetric.ball 0 rf\nfy_mem : f (x + y) ∈ EMetric.ball (f x) rg\nA : Tendsto (fun n => ∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y) atTop (𝓝 (f (x + y) - f x))\nB : Tendsto (fun n => q.partialSum n (∑ a ∈ Finset.Ico 1 n, (p a) fun _b => y)) atTop (𝓝 (g (f (x + y))))\n⊢ HasSum (fun n => (q.comp p n) fun x => y) ((g ∘ f) (x + y))"}
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+{"state": [{"context": ["n : ℕ", "P : MvPFunctor.{u} (n + 1)", "α : TypeVec.{u} n", "β : Type u", "g : β → ↑P (α ::: β)", "x : β", "a : P.A", "f : P.B a ⟹ α ::: β"], "goal": "⟨a, splitFun (dropFun f) ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ = (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩"}], "premise": [128094], "state_str": "case mk\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ ⟨a,\n      splitFun (dropFun f)\n        ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f)⟩ =\n    (TypeVec.id ::: corec P g) <$$> ⟨a, f⟩"}
+{"state": [{"context": ["n : ℕ", "P : MvPFunctor.{u} (n + 1)", "α : TypeVec.{u} n", "β : Type u", "g : β → ↑P (α ::: β)", "x : β", "a : P.A", "f : P.B a ⟹ α ::: β"], "goal": "splitFun (dropFun f) ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) = (TypeVec.id ::: corec P g) ⊚ f"}], "premise": [137041, 137045], "state_str": "case mk.e_snd\nn : ℕ\nP : MvPFunctor.{u} (n + 1)\nα : TypeVec.{u} n\nβ : Type u\ng : β → ↑P (α ::: β)\nx : β\na : P.A\nf : P.B a ⟹ α ::: β\n⊢ splitFun (dropFun f)\n      ((corec' P (fun b => (g b).fst) (fun b => dropFun (g b).snd) fun b => lastFun (g b).snd) ∘ lastFun f) =\n    (TypeVec.id ::: corec P g) ⊚ f"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "a b : ℝ", "f : ℝ → E", "s : ℂ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s.re < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s.re"], "goal": "MellinConvergent f s"}], "premise": [26447, 27429, 34454, 133326], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s.re\n⊢ MellinConvergent f s"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "inst✝ : NormedSpace ℂ E", "a b : ℝ", "f : ℝ → E", "s : ℂ", "hfc : LocallyIntegrableOn f (Ioi 0) volume", "hf_top : f =O[atTop] fun x => x ^ (-a)", "hs_top : s.re < a", "hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)", "hs_bot : b < s.re"], "goal": "IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) (Ioi 0) volume"}], "premise": [27422, 34457], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x => x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)\nhs_bot : b < s.re\n⊢ IntegrableOn (fun t => t ^ (s.re - 1) * ‖f t‖) (Ioi 0) volume"}
+{"state": [{"context": ["ι : Sort u_1", "α : Type u_2", "β : Type u_3", "l : Filter α", "inst✝ : CountableInterFilter l", "g : Set (Set α)", "s : Set α", "S : Set (Set α)", "Sg : S ⊆ g", "Sct : S.Countable", "hS : ⋂₀ S ⊆ s"], "goal": "s ∈ countableGenerate g"}], "premise": [1674, 12107, 15884], "state_str": "case mpr.intro.intro.intro\nι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝ : CountableInterFilter l\ng : Set (Set α)\ns : Set α\nS : Set (Set α)\nSg : S ⊆ g\nSct : S.Countable\nhS : ⋂₀ S ⊆ s\n⊢ s ∈ countableGenerate g"}
+{"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 : V →ₗ[K] V"], "goal": "IsUnit f ↔ range f = ⊤"}], "premise": [1713, 87647, 87657], "state_str": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V\n⊢ IsUnit f ↔ range f = ⊤"}
+{"state": [{"context": ["α : Type u_1", "σ : Type u_2", "inst✝¹ : Primcodable α", "inst✝ : Primcodable σ", "p : α → Prop"], "goal": "ComputablePred p ↔ RePred p ∧ RePred fun a => ¬p a"}], "premise": [69856], "state_str": "α : Type u_1\nσ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\np : α → Prop\n⊢ ComputablePred p ↔ RePred p ∧ RePred fun a => ¬p a"}
+{"state": [{"context": ["l : Type u_1", "m : Type u", "n : Type u'", "α : Type v", "inst✝² : Fintype n", "inst✝¹ : DecidableEq n", "inst✝ : CommRing α", "A✝ B A : Matrix n n α", "b : n → α", "h : IsUnit A.det"], "goal": "A.det • b ᵥ* A⁻¹ = Aᵀ.cramer b"}], "premise": [85382, 85389, 85454, 86457, 142362, 142368], "state_str": "l : Type u_1\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA✝ B A : Matrix n n α\nb : n → α\nh : IsUnit A.det\n⊢ A.det • b ᵥ* A⁻¹ = Aᵀ.cramer b"}
+{"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 : H = ⊤"], "goal": "Nat.card ↥H = Nat.card G"}], "premise": [118543], "state_str": "G : Type u_1\ninst✝² : Group G\nA : Type u_2\ninst✝¹ : AddGroup A\nH K : Subgroup G\ninst✝ : Finite ↥H\nh : H = ⊤\n⊢ Nat.card ↥H = Nat.card G"}
+{"state": [{"context": ["L : Language", "T : L.Theory", "α : Type w", "n : ℕ", "T' : L.Theory", "h : ∀ φ ∈ T', T ⊨ᵇ φ", "φ : L.Formula α", "hφ : T' ⊨ᵇ φ"], "goal": "T ⊨ᵇ φ"}], "premise": [24831], "state_str": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nh : ∀ φ ∈ T', T ⊨ᵇ φ\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\n⊢ T ⊨ᵇ φ"}
+{"state": [{"context": ["L : Language", "T : L.Theory", "α : Type w", "n : ℕ", "T' : L.Theory", "φ : L.Formula α", "hφ : T' ⊨ᵇ φ", "h : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ", "M : T.ModelType"], "goal": "∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs"}], "premise": [1674, 25325], "state_str": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nT' : L.Theory\nφ : L.Formula α\nhφ : T' ⊨ᵇ φ\nh : ∀ φ ∈ T', ∀ (M : T.ModelType), ↑M ⊨ φ\nM : T.ModelType\n⊢ ∀ (v : α → ↑M) (xs : Fin 0 → ↑M), BoundedFormula.Realize φ v xs"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "f g : ℂ → E", "c : ℂ", "R : ℝ", "inst✝ : NormedSpace ℂ E", "hf : CircleIntegrable f c R"], "goal": "IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)"}], "premise": [25808, 26278], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : CircleIntegrable f c R\n⊢ IntervalIntegrable (fun θ => deriv (circleMap c R) θ • f (circleMap c R θ)) volume 0 (2 * π)"}
+{"state": [{"context": ["E : Type u_1", "inst✝¹ : NormedAddCommGroup E", "f g : ℂ → E", "c : ℂ", "R : ℝ", "inst✝ : NormedSpace ℂ E", "hf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume"], "goal": "IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume"}], "premise": [28454, 28502, 28544], "state_str": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf g : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : NormedSpace ℂ E\nhf : IntegrableOn (fun θ => f (circleMap c R θ)) (Ι 0 (2 * π)) volume\n⊢ IntegrableOn (fun θ => (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume"}
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+{"state": [{"context": ["Ω : Type u_1", "ι : Type u_2", "m : MeasurableSpace Ω", "X✝ : Ω → ℝ", "p : ℕ", "μ : Measure Ω", "t : ℝ", "X Y : Ω → ℝ", "h_indep : IndepFun X Y μ", "h_int_X : Integrable (fun ω => rexp (t * X ω)) μ", "h_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ"], "goal": "Integrable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ"}], "premise": [73761, 74030], "state_str": "Ω : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\nX✝ : Ω → ℝ\np : ℕ\nμ : Measure Ω\nt : ℝ\nX Y : Ω → ℝ\nh_indep : IndepFun X Y μ\nh_int_X : Integrable (fun ω => rexp (t * X ω)) μ\nh_int_Y : Integrable (fun ω => rexp (t * Y ω)) μ\n⊢ Integrable (fun ω => rexp (t * X ω) * rexp (t * Y ω)) μ"}
+{"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", "inst✝ : Countable ι", "s : ι → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) s"], "goal": "μ (⋃ i, s i) = ⨆ i, μ (s i)"}], "premise": [72049], "state_str": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝ : Countable ι\ns : ι → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) s\n⊢ μ (⋃ i, s i) = ⨆ i, μ (s i)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s s₁ s₂ t✝ : Set α", "t : ℕ → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) t"], "goal": "μ (⋃ n, t n) = ⨆ n, μ (t n)"}], "premise": [14296, 19309, 27559, 135255], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\n⊢ μ (⋃ n, t n) = ⨆ n, μ (t n)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s s₁ s₂ t✝ : Set α", "t : ℕ → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) t", "T : ℕ → Set α := fun n => toMeasurable μ (t n)", "Td : ℕ → Set α := disjointed T"], "goal": "μ (⋃ n, t n) ≤ ⨆ n, μ (t n)"}], "premise": [27980, 29104], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "γ : Type u_3", "δ : Type u_4", "R : Type u_6", "R' : Type u_7", "m : MeasurableSpace α", "μ μ₁ μ₂ : Measure α", "s s₁ s₂ t✝ : Set α", "t : ℕ → Set α", "hd : Directed (fun x x_1 => x ⊆ x_1) t", "T : ℕ → Set α := fun n => toMeasurable μ (t n)", "Td : ℕ → Set α := disjointed T", "hm : ∀ (n : ℕ), MeasurableSet (Td n)"], "goal": "μ (⋃ n, t n) ≤ ⨆ n, μ (t n)"}], "premise": [2100, 12871, 12879, 12880, 19295, 19309, 27559, 27562, 29102, 31390, 31420, 58992, 71985, 132775, 135248, 135263, 135265, 136516], "state_str": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t✝ : Set α\nt : ℕ → Set α\nhd : Directed (fun x x_1 => x ⊆ x_1) t\nT : ℕ → Set α := fun n => toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\nhm : ∀ (n : ℕ), MeasurableSet (Td n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "ι : Type u_3", "inst✝ : MeasurableSpace α", "f : α → ℚ → ℝ", "a : α"], "goal": "(if IsRatStieltjesPoint (fun p => f p.2) ((), a) then (fun p => f p.2) ((), a) else defaultRatCDF) = if IsRatStieltjesPoint f a then f a else defaultRatCDF"}], "premise": [72890], "state_str": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nf : α → ℚ → ℝ\na : α\n⊢ (if IsRatStieltjesPoint (fun p => f p.2) ((), a) then (fun p => f p.2) ((), a) else defaultRatCDF) =\n    if IsRatStieltjesPoint f a then f a else defaultRatCDF"}
+{"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", "N' : LieSubmodule R L ↥↑N"], "goal": "map N.incl N' ≤ N"}], "premise": [109427], "state_str": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥↑N\n⊢ map N.incl N' ≤ N"}
+{"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", "N' : LieSubmodule R L ↥↑N"], "goal": "map N.incl N' ≤ map N.incl ⊤"}], "premise": [18778, 109352], "state_str": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥↑N\n⊢ map N.incl N' ≤ map N.incl ⊤"}
+{"state": [{"context": ["K : Type u_1", "E : Type u_2", "inst✝ : RCLike K", "z : K"], "goal": "(∃ r, ↑r = z) ↔ ∃ r, z = ↑r"}], "premise": [1717], "state_str": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\nz : K\n⊢ (∃ r, ↑r = z) ↔ ∃ r, z = ↑r"}
+{"state": [{"context": ["α : Type u", "β : Type v", "γ : Type w", "inst✝² : TopologicalSpace α", "inst✝¹ : Preorder α", "inst✝ : ClosedIciTopology α", "f : β → α", "a b : α", "s : Set α"], "goal": "BddBelow (closure s) ↔ BddBelow s"}], "premise": [54926], "state_str": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : ClosedIciTopology α\nf : β → α\na b : α\ns : Set α\n⊢ BddBelow (closure s) ↔ BddBelow s"}
+{"state": [{"context": ["Γ : Type u_1", "Γ' : Type u_2", "inst✝¹ : Inhabited Γ", "inst✝ : Inhabited Γ'", "f : PointedMap Γ Γ'", "d : Dir", "head✝ : Γ", "left✝ right✝ : ListBlank Γ"], "goal": "map f (move d { head := head✝, left := left✝, right := right✝ }) = move d (map f { head := head✝, left := left✝, right := right✝ })"}], "premise": [1793, 1957, 73491, 73492, 73493], "state_str": "case mk\nΓ : Type u_1\nΓ' : Type u_2\ninst✝¹ : Inhabited Γ\ninst✝ : Inhabited Γ'\nf : PointedMap Γ Γ'\nd : Dir\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ map f (move d { head := head✝, left := left✝, right := right✝ }) =\n    move d (map f { head := head✝, left := left✝, right := right✝ })"}
+{"state": [{"context": ["ι : Type u_1", "α : ι → Type u_2", "s₁ s₂ : Finset ι", "S₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)", "S₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)", "inst✝ : DecidableEq ι", "f : (i : ι) → α i"], "goal": "f ∈ cylinder s₁ S₁ ∪ cylinder s₂ S₂ ↔ f ∈ cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)"}], "premise": [26016, 131585, 133407], "state_str": "case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ f ∈ cylinder s₁ S₁ ∪ cylinder s₂ S₂ ↔\n    f ∈ cylinder (s₁ ∪ s₂) ((fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∪ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂)"}
+{"state": [{"context": ["ι : Type u_1", "α : ι → Type u_2", "s₁ s₂ : Finset ι", "S₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)", "S₂ : Set ((i : { x // x ∈ s₂ }) → α ↑i)", "inst✝ : DecidableEq ι", "f : (i : ι) → α i"], "goal": "(fun i => f ↑i) ∈ S₁ ∨ (fun i => f ↑i) ∈ S₂ ↔ (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∨ (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂"}], "premise": [1713], "state_str": "case h\nι : Type u_1\nα : ι → Type u_2\ns₁ s₂ : Finset ι\nS₁ : Set ((i : { x // x ∈ s₁ }) → α ↑i)\nS₂ : Set ((i : { x // x �� s₂ }) → α ↑i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\n⊢ (fun i => f ↑i) ∈ S₁ ∨ (fun i => f ↑i) ∈ S₂ ↔\n    (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₁ ∨ (fun i => f ↑i) ∈ (fun f j => f ⟨↑j, ⋯⟩) ⁻¹' S₂"}
+{"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 ≪ ν", "hνν' : ν ⟂ₘ ν'", "t : Set α := hνν'.nullSet"], "goal": "μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν"}], "premise": [30039], "state_str": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν"}
+{"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 ≪ ν", "hνν' : ν ⟂ₘ ν'", "t : Set α := hνν'.nullSet", "ht : MeasurableSet t"], "goal": "μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν"}], "premise": [32306], "state_str": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae ν] μ.rnDeriv ν"}
+{"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 ≪ ν", "hνν' : ν ⟂ₘ ν'", "t : Set α := hνν'.nullSet", "ht : MeasurableSet t"], "goal": "μ.rnDeriv (ν + ν') =ᶠ[ae (ν.restrict tᶜ)] μ.rnDeriv ν"}], "premise": [29033, 29106, 34822], "state_str": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\n⊢ μ.rnDeriv (ν + ν') =ᶠ[ae (ν.restrict tᶜ)] μ.rnDeriv ν"}
+{"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 ≪ ν", "hνν' : ν ⟂ₘ ν'", "t : Set α := hνν'.nullSet", "ht : MeasurableSet t", "this : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))"], "goal": "(ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)"}], "premise": [27953, 31336, 31568, 32750], "state_str": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ ν ν' : Measure α\ninst✝² : SigmaFinite μ\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite ν'\nhμν : μ ≪ ν\nhνν' : ν ⟂ₘ ν'\nt : Set α := hνν'.nullSet\nht : MeasurableSet t\nthis : (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = ((ν + ν').restrict tᶜ).withDensity (μ.rnDeriv (ν + ν'))\n⊢ (ν.restrict tᶜ).withDensity (μ.rnDeriv (ν + ν')) = (ν.restrict tᶜ).withDensity (μ.rnDeriv ν)"}
+{"state": [{"context": ["α : Type u_1", "n : ℕ", "s t : Multiset α", "l : List α"], "goal": "s ∈ powersetCard n ⟦l⟧ ↔ s ≤ ⟦l⟧ ∧ card s = n"}], "premise": [134949], "state_str": "α : Type u_1\nn : ℕ\ns t : Multiset α\nl : List α\n⊢ s ∈ powersetCard n ⟦l⟧ ↔ s ≤ ⟦l⟧ ∧ card s = n"}
+{"state": [{"context": ["α : Type u_1", "G : Type u_2", "M : Type u_3", "R : Type u_4", "A : Type u_5", "inst✝⁴ : Monoid M", "inst✝³ : AddMonoid A", "s✝ t u : Set M", "N : Type u_6", "inst✝² : CommMonoid N", "inst✝¹ : MulAction M N", "inst✝ : IsScalarTower M N N", "r : M", "s : Set N", "x : N", "hx : x ∈ closure s"], "goal": "∃ n, r ^ n • x ∈ closure (r • s)"}], "premise": [117660], "state_str": "α : Type u_1\nG : Type u_2\nM : Type u_3\nR : Type u_4\nA : Type u_5\ninst✝⁴ : Monoid M\ninst✝³ : AddMonoid A\ns✝ t u : Set M\nN : Type u_6\ninst✝² : CommMonoid N\ninst✝¹ : MulAction M N\ninst✝ : IsScalarTower M N N\nr : M\ns : Set N\nx : N\nhx : x ∈ closure s\n⊢ ∃ n, r ^ n • x ∈ closure (r • s)"}
+{"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": "x ∉ closure s ↔ 𝓝[s] x = ⊥"}], "premise": [1713, 15922, 55574], "state_str": "X : Type u\nY : Type v\nι : Sort w\nα : Type u_1\nβ : Type u_2\nx : X\ns s₁ s₂ t : Set X\np p₁ p₂ : X → Prop\ninst✝ : TopologicalSpace X\n⊢ x ∉ closure s ↔ 𝓝[s] x = ⊥"}
+{"state": [{"context": ["α : Type u_1", "β : Type u_2", "s t : Set α"], "goal": "s.encard = 0 ↔ s = ∅"}], "premise": [1713, 47622, 70308, 70736, 70752, 71387, 133373, 144972], "state_str": "α : Type u_1\nβ : Type u_2\ns t : Set α\n⊢ s.encard = 0 ↔ s = ∅"}
+{"state": [{"context": ["a✝ b✝ c p q : ℚ", "a : ℤ", "ha : 0 ≤ a", "b : ℕ"], "goal": "0 ≤ mkRat a b"}], "premise": [105548, 129973], "state_str": "a✝ b✝ c p q : ℚ\na : ℤ\nha : 0 ≤ a\nb : ℕ\n⊢ 0 ≤ mkRat a b"}