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Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.Subsingleton.stronglyMeasurable' | [
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.21646\nβ✝ : Type ?u.21649\nγ : Type ?u.21652\nι : Type ?u.21655\ninst✝³ : Countable ι\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : Subsingleton α\nf : α → β\nx y : α\n⊢ f x = f y",
"tactic": "rw [Subsingleton.elim x y]"
}
] | [
186,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.zero_ne_trop_coe | [] | [
250,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.subset_closure_withConstants | [
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\n⊢ a ∈ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"state_before": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\n⊢ A ⊆ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"tactic": "intro a ha"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\n⊢ a ∈ LowerAdjoint.toFun (closure (L[[↑A]])) s",
"state_before": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\n⊢ a ∈ ↑(LowerAdjoint.toFun (closure (L[[↑A]])) s)",
"tactic": "simp only [SetLike.mem_coe]"
},
{
"state_after": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\na' : Language.Constants (L[[↑A]]) := Sum.inr { val := a, property := ha }\n⊢ a ∈ LowerAdjoint.toFun (closure (L[[↑A]])) s",
"state_before": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\n⊢ a ∈ LowerAdjoint.toFun (closure (L[[↑A]])) s",
"tactic": "let a' : L[[A]].Constants := Sum.inr ⟨a, ha⟩"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type w\nN : Type ?u.681989\nP : Type ?u.681992\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS : Substructure L M\nA s : Set M\nh : A ⊆ ↑S\na : M\nha : a ∈ A\na' : Language.Constants (L[[↑A]]) := Sum.inr { val := a, property := ha }\n⊢ a ∈ LowerAdjoint.toFun (closure (L[[↑A]])) s",
"tactic": "exact constants_mem a'"
}
] | [
774,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
770,
1
] |
Mathlib/Algebra/BigOperators/Order.lean | Finset.one_lt_prod | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type ?u.135473\nβ : Type ?u.135476\nM : Type u_2\nN : Type ?u.135482\nG : Type ?u.135485\nk : Type ?u.135488\nR : Type ?u.135491\ninst✝ : OrderedCancelCommMonoid M\nf g : ι → M\ns t : Finset ι\nh : ∀ (i : ι), i ∈ s → 1 < f i\nhs : Finset.Nonempty s\n⊢ 1 ≤ ∏ i in s, 1",
"tactic": "rw [prod_const_one]"
}
] | [
509,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
508,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | ContDiffWithinAt.sinh | [] | [
1185,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1183,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | Differentiable.dist | [] | [
257,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/RingTheory/Congruence.lean | RingCon.quot_mk_eq_coe | [] | [
169,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.integral_comp_add_div | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14902596\n𝕜 : Type ?u.14902599\nE : Type u_1\nF : Type ?u.14902605\nA : Type ?u.14902608\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d✝ : ℝ\nf : ℝ → E\nhc : c ≠ 0\nd : ℝ\n⊢ (∫ (x : ℝ) in a..b, f (d + x / c)) = c • ∫ (x : ℝ) in d + a / c..d + b / c, f x",
"tactic": "simpa only [div_eq_inv_mul, inv_inv] using integral_comp_add_mul f (inv_ne_zero hc) d"
}
] | [
791,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
789,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean | Filter.Tendsto.zero_mul_isBoundedUnder_le | [] | [
216,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.disjoint_union_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.501362\nγ : Type ?u.501365\ninst✝ : DecidableEq α\ns t u : Multiset α\n⊢ Disjoint s (t ∪ u) ↔ Disjoint s t ∧ Disjoint s u",
"tactic": "simp [Disjoint, or_imp, forall_and]"
}
] | [
2981,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2980,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | Complex.cpow_int_cast | [
{
"state_after": "no goals",
"state_before": "x : ℂ\nn : ℕ\n⊢ x ^ ↑↑n = x ^ ↑n",
"tactic": "simp"
},
{
"state_after": "x : ℂ\nn : ℕ\n⊢ x ^ ↑-[n+1] = (x ^ (n + 1))⁻¹",
"state_before": "x : ℂ\nn : ℕ\n⊢ x ^ ↑-[n+1] = x ^ -[n+1]",
"tactic": "rw [zpow_negSucc]"
},
{
"state_after": "no goals",
"state_before": "x : ℂ\nn : ℕ\n⊢ x ^ ↑-[n+1] = (x ^ (n + 1))⁻¹",
"tactic": "simp only [Int.negSucc_coe, Int.cast_neg, Complex.cpow_neg, inv_eq_one_div, Int.cast_ofNat,\n cpow_nat_cast]"
}
] | [
141,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
] |
Mathlib/CategoryTheory/Abelian/Images.lean | CategoryTheory.Abelian.image.fac | [] | [
64,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
11
] |
Mathlib/Topology/UniformSpace/Basic.lean | mem_compRel | [] | [
158,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
156,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean | Complex.UnitDisc.coe_ne_neg_one | [
{
"state_after": "z : 𝔻\n⊢ ↑abs ↑z ≠ 1",
"state_before": "z : 𝔻\n⊢ ↑abs ↑z ≠ ↑abs (-1)",
"tactic": "rw [abs.map_neg, map_one]"
},
{
"state_after": "no goals",
"state_before": "z : 𝔻\n⊢ ↑abs ↑z ≠ 1",
"tactic": "exact z.abs_ne_one"
}
] | [
68,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.lineMap_vadd_apply | [
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.423603\nP2 : Type ?u.423606\nV3 : Type ?u.423609\nP3 : Type ?u.423612\nV4 : Type ?u.423615\nP4 : Type ?u.423618\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np : P1\nv : V1\nc : k\n⊢ ↑(lineMap p (v +ᵥ p)) c = c • v +ᵥ p",
"tactic": "rw [lineMap_apply, vadd_vsub]"
}
] | [
538,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
537,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | Filter.Tendsto.log | [] | [
383,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean | MeasureTheory.integral_inter_add_diff | [] | [
125,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/LinearAlgebra/Quotient.lean | Submodule.ker_mkQ | [
{
"state_after": "case h\nR : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.220917\nM₂ : Type ?u.220920\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nx✝ : M\n⊢ x✝ ∈ ker (mkQ p) ↔ x✝ ∈ p",
"state_before": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.220917\nM₂ : Type ?u.220920\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\n⊢ ker (mkQ p) = p",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.220917\nM₂ : Type ?u.220920\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\nx✝ : M\n⊢ x✝ ∈ ker (mkQ p) ↔ x✝ ∈ p",
"tactic": "simp"
}
] | [
381,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
381,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean | pow_two | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.11869\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Monoid M\ninst✝ : AddMonoid A\na : M\n⊢ a ^ 2 = a * a",
"tactic": "rw [pow_succ, pow_one]"
}
] | [
105,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean | hasStrictDerivAt_of_hasDerivAt_of_continuousAt | [] | [
1357,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1353,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean | is_const_of_fderiv_eq_zero | [
{
"state_after": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.251275\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx✝² y✝ : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : E), fderiv 𝕜 f x = 0\nx✝¹ y x : E\nx✝ : x ∈ univ\n⊢ fderiv 𝕜 f x = 0",
"state_before": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.251275\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx✝² y✝ : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : E), fderiv 𝕜 f x = 0\nx✝¹ y x : E\nx✝ : x ∈ univ\n⊢ fderivWithin 𝕜 f univ x = 0",
"tactic": "rw [fderivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nF : Type ?u.251275\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\n𝕜 : Type u_1\nG : Type u_3\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → G\nC : ℝ\ns : Set E\nx✝² y✝ : E\nf' g' : E → E →L[𝕜] G\nφ : E →L[𝕜] G\nhf : Differentiable 𝕜 f\nhf' : ∀ (x : E), fderiv 𝕜 f x = 0\nx✝¹ y x : E\nx✝ : x ∈ univ\n⊢ fderiv 𝕜 f x = 0",
"tactic": "exact hf' x"
}
] | [
602,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
599,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | Embedding.isSeparable_preimage | [] | [
1949,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1947,
11
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.disjoint_right | [] | [
2909,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2908,
1
] |
Mathlib/Order/Atoms.lean | IsAtom.Iic_eq | [] | [
96,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
1
] |
Std/Data/Option/Lemmas.lean | Option.map_comp_map | [
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\nx : Option α\n⊢ (Option.map g ∘ Option.map f) x = Option.map (g ∘ f) x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\n⊢ Option.map g ∘ Option.map f = Option.map (g ∘ f)",
"tactic": "funext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\ng : β → γ\nx : Option α\n⊢ (Option.map g ∘ Option.map f) x = Option.map (g ∘ f) x",
"tactic": "simp"
}
] | [
152,
74
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
151,
9
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.coe_zsmulRec | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.1031657\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf g : α →ᵇ β\nx : α\nC : ℝ\nn : ℕ\n⊢ ↑(zsmulRec (Int.ofNat n) f) = Int.ofNat n • ↑f",
"tactic": "rw [zsmulRec, Int.ofNat_eq_coe, coe_nsmulRec, coe_nat_zsmul]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.1031657\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf g : α →ᵇ β\nx : α\nC : ℝ\nn : ℕ\n⊢ ↑(zsmulRec (Int.negSucc n) f) = Int.negSucc n • ↑f",
"tactic": "rw [zsmulRec, negSucc_zsmul, coe_neg, coe_nsmulRec]"
}
] | [
973,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
971,
1
] |
Mathlib/Data/Ordmap/Ordset.lean | Ordnode.Valid'.rotateR | [
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ Valid' o₂ (Ordnode.dual (Ordnode.rotateR l x r)) o₁",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ Valid' o₁ (Ordnode.rotateR l x r) o₂",
"tactic": "refine' Valid'.dual_iff.2 _"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ Valid' o₂ (Ordnode.rotateL (Ordnode.dual r) x (Ordnode.dual l)) o₁",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ Valid' o₂ (Ordnode.dual (Ordnode.rotateR l x r)) o₁",
"tactic": "rw [dual_rotateR]"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ ¬size (Ordnode.dual r) + size (Ordnode.dual l) ≤ 1\n\ncase refine'_2\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ delta * size (Ordnode.dual r) < size (Ordnode.dual l)\n\ncase refine'_3\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ 2 * size (Ordnode.dual l) ≤ 9 * size (Ordnode.dual r) + 5 ∨ size (Ordnode.dual l) ≤ 3",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ Valid' o₂ (Ordnode.rotateL (Ordnode.dual r) x (Ordnode.dual l)) o₁",
"tactic": "refine' hr.dual.rotateL hl.dual _ _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ ¬size (Ordnode.dual r) + size (Ordnode.dual l) ≤ 1",
"tactic": "rwa [size_dual, size_dual, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ delta * size (Ordnode.dual r) < size (Ordnode.dual l)",
"tactic": "rwa [size_dual, size_dual]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH1 : ¬size l + size r ≤ 1\nH2 : delta * size r < size l\nH3 : 2 * size l ≤ 9 * size r + 5 ∨ size l ≤ 3\n⊢ 2 * size (Ordnode.dual l) ≤ 9 * size (Ordnode.dual r) + 5 ∨ size (Ordnode.dual l) ≤ 3",
"tactic": "rwa [size_dual, size_dual]"
}
] | [
1303,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1295,
1
] |
Mathlib/Analysis/Convex/Join.lean | convexJoin_assoc_aux | [
{
"state_after": "ι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ ∀ (x : E),\n (∃ a, (∃ a_1, a_1 ∈ s ∧ ∃ b, b ∈ t ∧ a ∈ segment 𝕜 a_1 b) ∧ ∃ b, b ∈ u ∧ x ∈ segment 𝕜 a b) →\n ∃ a, a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ x ∈ segment 𝕜 a b",
"state_before": "ι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ convexJoin 𝕜 (convexJoin 𝕜 s t) u ⊆ convexJoin 𝕜 s (convexJoin 𝕜 t u)",
"tactic": "simp_rw [subset_def, mem_convexJoin]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"state_before": "ι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx y : E\ns t u : Set E\n⊢ ∀ (x : E),\n (∃ a, (∃ a_1, a_1 ∈ s ∧ ∃ b, b ∈ t ∧ a ∈ segment 𝕜 a_1 b) ∧ ∃ b, b ∈ u ∧ x ∈ segment 𝕜 a b) →\n ∃ a, a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ x ∈ segment 𝕜 a b",
"tactic": "rintro _ ⟨z, ⟨x, hx, y, hy, a₁, b₁, ha₁, hb₁, hab₁, rfl⟩, z, hz, a₂, b₂, ha₂, hb₂, hab₂, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ + 0 = 1\n⊢ ∃ a, a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + 0 • z ∈ segment 𝕜 a b\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"tactic": "obtain rfl | hb₂ := hb₂.eq_or_lt"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"tactic": "have ha₂b₁ : 0 ≤ a₂ * b₁ := mul_nonneg ha₂ hb₁"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"tactic": "have hab : 0 < a₂ * b₁ + b₂ := add_pos_of_nonneg_of_pos ha₂b₁ hb₂"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_1\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ 0 ≤ a₂ * b₁ / (a₂ * b₁ + b₂)\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_2\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ 0 ≤ b₂ / (a₂ * b₁ + b₂)\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_3\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ a₂ * b₁ / (a₂ * b₁ + b₂) + b₂ / (a₂ * b₁ + b₂) = 1\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_4\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ a₂ * a₁ + (a₂ * b₁ + b₂) = 1\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_5\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ (a₂ * a₁) • x + (a₂ * b₁ + b₂) • ((a₂ * b₁ / (a₂ * b₁ + b₂)) • y + (b₂ / (a₂ * b₁ + b₂)) • z) =\n a₂ • (a₁ • x + b₁ • y) + b₂ • z",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ ∃ a,\n a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + b₂ • z ∈ segment 𝕜 a b",
"tactic": "refine'\n ⟨x, hx, (a₂ * b₁ / (a₂ * b₁ + b₂)) • y + (b₂ / (a₂ * b₁ + b₂)) • z,\n ⟨y, hy, z, hz, _, _, _, _, _, rfl⟩, a₂ * a₁, a₂ * b₁ + b₂, mul_nonneg ha₂ ha₁, hab.le, _, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ + 0 = 1\n⊢ a₁ • x + b₁ • y = a₂ • (a₁ • x + b₁ • y) + 0 • z",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ + 0 = 1\n⊢ ∃ a, a ∈ s ∧ ∃ b, (∃ a, a ∈ t ∧ ∃ b_1, b_1 ∈ u ∧ b ∈ segment 𝕜 a b_1) ∧ a₂ • (a₁ • x + b₁ • y) + 0 • z ∈ segment 𝕜 a b",
"tactic": "refine' ⟨x, hx, y, ⟨y, hy, z, hz, left_mem_segment 𝕜 _ _⟩, a₁, b₁, ha₁, hb₁, hab₁, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ = 1\n⊢ a₁ • x + b₁ • y = a₂ • (a₁ • x + b₁ • y) + 0 • z",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ + 0 = 1\n⊢ a₁ • x + b₁ • y = a₂ • (a₁ • x + b₁ • y) + 0 • z",
"tactic": "rw [add_zero] at hab₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inl\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂ : 0 ≤ 0\nhab₂ : a₂ = 1\n⊢ a₁ • x + b₁ • y = a₂ • (a₁ • x + b₁ • y) + 0 • z",
"tactic": "rw [hab₂, one_smul, zero_smul, add_zero]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_1\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ 0 ≤ a₂ * b₁ / (a₂ * b₁ + b₂)",
"tactic": "exact div_nonneg ha₂b₁ hab.le"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_2\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ 0 ≤ b₂ / (a₂ * b₁ + b₂)",
"tactic": "exact div_nonneg hb₂.le hab.le"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_3\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ a₂ * b₁ / (a₂ * b₁ + b₂) + b₂ / (a₂ * b₁ + b₂) = 1",
"tactic": "rw [← add_div, div_self hab.ne']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_4\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ a₂ * a₁ + (a₂ * b₁ + b₂) = 1",
"tactic": "rw [← add_assoc, ← mul_add, hab₁, mul_one, hab₂]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.inr.refine'_5\nι : Sort ?u.52859\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns✝ t✝ u✝ : Set E\nx✝ y✝ : E\ns t u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\na₂ b₂ : 𝕜\nha₂ : 0 ≤ a₂\nhb₂✝ : 0 ≤ b₂\nhab₂ : a₂ + b₂ = 1\nhb₂ : 0 < b₂\nha₂b₁ : 0 ≤ a₂ * b₁\nhab : 0 < a₂ * b₁ + b₂\n⊢ (a₂ * a₁) • x + (a₂ * b₁ + b₂) • ((a₂ * b₁ / (a₂ * b₁ + b₂)) • y + (b₂ / (a₂ * b₁ + b₂)) • z) =\n a₂ • (a₁ • x + b₁ • y) + b₂ • z",
"tactic": "simp_rw [smul_add, ← mul_smul, mul_div_cancel' _ hab.ne', add_assoc]"
}
] | [
154,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean | FiniteField.X_pow_card_pow_sub_X_ne_zero | [] | [
257,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Data/Real/Basic.lean | Real.mk_one | [
{
"state_after": "x y : ℝ\n⊢ mk 1 = { cauchy := 1 }",
"state_before": "x y : ℝ\n⊢ mk 1 = 1",
"tactic": "rw [← ofCauchy_one]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ mk 1 = { cauchy := 1 }",
"tactic": "rfl"
}
] | [
331,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/LinearAlgebra/Basic.lean | LinearMap.funLeft_comp | [] | [
2616,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2614,
1
] |
Mathlib/Data/Set/Function.lean | Set.LeftInvOn.image_image | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.60024\nι : Sort ?u.60027\nπ : α → Type ?u.60032\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\nhf : LeftInvOn f' f s\n⊢ f' '' (f '' s) = s",
"tactic": "rw [Set.image_image, image_congr hf, image_id']"
}
] | [
1088,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1087,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.comap_top | [] | [
539,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
538,
1
] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.map_rootOfSplits | [] | [
317,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/Topology/PathConnected.lean | IsPathConnected.isConnected | [
{
"state_after": "X : Type u_1\nY : Type ?u.695184\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.695199\nF : Set X\nhF : IsPathConnected F\n⊢ ConnectedSpace ↑F",
"state_before": "X : Type u_1\nY : Type ?u.695184\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.695199\nF : Set X\nhF : IsPathConnected F\n⊢ IsConnected F",
"tactic": "rw [isConnected_iff_connectedSpace]"
},
{
"state_after": "X : Type u_1\nY : Type ?u.695184\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.695199\nF : Set X\nhF : PathConnectedSpace ↑F\n⊢ ConnectedSpace ↑F",
"state_before": "X : Type u_1\nY : Type ?u.695184\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.695199\nF : Set X\nhF : IsPathConnected F\n⊢ ConnectedSpace ↑F",
"tactic": "rw [isPathConnected_iff_pathConnectedSpace] at hF"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.695184\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.695199\nF : Set X\nhF : PathConnectedSpace ↑F\n⊢ ConnectedSpace ↑F",
"tactic": "exact @PathConnectedSpace.connectedSpace _ _ hF"
}
] | [
1165,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1162,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.coe_ennreal_ofReal | [] | [
458,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
] |
Mathlib/Algebra/Lie/Basic.lean | LieModuleHom.sub_apply | [] | [
892,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
891,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.HasZeroMorphisms.ext | [
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\n⊢ ∀ (X Y : C), Zero.zero = Zero.zero",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\n⊢ I = J",
"tactic": "apply ext_aux"
},
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\n⊢ Zero.zero = Zero.zero",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\n⊢ ∀ (X Y : C), Zero.zero = Zero.zero",
"tactic": "intro X Y"
},
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero ≫ Zero.zero = Zero.zero\n⊢ Zero.zero = Zero.zero",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\n⊢ Zero.zero = Zero.zero",
"tactic": "have : (I.Zero X Y).zero ≫ (J.Zero Y Y).zero = (I.Zero X Y).zero := by\n apply I.zero_comp X (J.Zero Y Y).zero"
},
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero ≫ Zero.zero = Zero.zero\nthat : Zero.zero ≫ Zero.zero = Zero.zero\n⊢ Zero.zero = Zero.zero",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero ≫ Zero.zero = Zero.zero\n⊢ Zero.zero = Zero.zero",
"tactic": "have that : (I.Zero X Y).zero ≫ (J.Zero Y Y).zero = (J.Zero X Y).zero := by\n apply J.comp_zero (I.Zero X Y).zero Y"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero ≫ Zero.zero = Zero.zero\nthat : Zero.zero ≫ Zero.zero = Zero.zero\n⊢ Zero.zero = Zero.zero",
"tactic": "rw[←this,←that]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\n⊢ Zero.zero ≫ Zero.zero = Zero.zero",
"tactic": "apply I.zero_comp X (J.Zero Y Y).zero"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u'\ninst✝ : Category D\nI J : HasZeroMorphisms C\nX Y : C\nthis : Zero.zero ≫ Zero.zero = Zero.zero\n⊢ Zero.zero ≫ Zero.zero = Zero.zero",
"tactic": "apply J.comp_zero (I.Zero X Y).zero Y"
}
] | [
119,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Deprecated/Subgroup.lean | Group.mclosure_subset | [] | [
616,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
615,
1
] |
Mathlib/Topology/DenseEmbedding.lean | DenseInducing.closure_range | [] | [
62,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.iInf_add | [] | [
2400,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2398,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.hasZeroObject_of_hasInitial_object | [
{
"state_after": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\nX : C\nf : X ⟶ ⊥_ C\n⊢ f = default",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\n⊢ HasZeroObject C",
"tactic": "refine' ⟨⟨⊥_ C, fun X => ⟨⟨⟨0⟩, by aesop_cat⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => _⟩⟩⟩⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\nX : C\nf : X ⟶ ⊥_ C\n⊢ f = default",
"tactic": "calc\n f = f ≫ 𝟙 _ := (Category.comp_id _).symm\n _ = f ≫ 0 := by congr!\n _ = 0 := HasZeroMorphisms.comp_zero _ _"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\nX : C\n⊢ ∀ (a : ⊥_ C ⟶ X), a = default",
"tactic": "aesop_cat"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\nX : C\nf : X ⟶ ⊥_ C\n⊢ f ≫ 𝟙 (⊥_ C) = f ≫ 0",
"tactic": "congr!"
}
] | [
552,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
546,
1
] |
Mathlib/LinearAlgebra/Ray.lean | Module.Ray.ne_neg_self | [
{
"state_after": "case h\nR : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.180935\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx✝ y : M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx ≠ -rayOfNeZero R x hx",
"state_before": "R : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.180935\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx✝ y : M\ninst✝ : NoZeroSMulDivisors R M\nx : Ray R M\n⊢ x ≠ -x",
"tactic": "induction' x using Module.Ray.ind with x hx"
},
{
"state_after": "case h\nR : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.180935\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx✝ y : M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nhx : x ≠ 0\n⊢ ¬SameRay R x (-x)",
"state_before": "case h\nR : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.180935\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx✝ y : M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nhx : x ≠ 0\n⊢ rayOfNeZero R x hx ≠ -rayOfNeZero R x hx",
"tactic": "rw [neg_rayOfNeZero, Ne.def, ray_eq_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝⁵ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.180935\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nx✝ y : M\ninst✝ : NoZeroSMulDivisors R M\nx : M\nhx : x ≠ 0\n⊢ ¬SameRay R x (-x)",
"tactic": "exact mt eq_zero_of_sameRay_self_neg hx"
}
] | [
480,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean | Submodule.toConvexCone_le_iff | [] | [
528,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
527,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.zeta_apply | [] | [
453,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
452,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.liftHom_mk | [] | [
337,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
336,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.mk'_surjective | [] | [
97,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Topology/ContinuousOn.lean | preimage_coe_mem_nhds_subtype | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.304580\nγ : Type ?u.304583\nδ : Type ?u.304586\ninst✝ : TopologicalSpace α\ns t : Set α\na : ↑s\n⊢ Subtype.val ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a",
"tactic": "rw [← map_nhds_subtype_val, mem_map]"
}
] | [
507,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
506,
1
] |
Mathlib/Topology/Basic.lean | isClosed_imp | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns s₁ s₂ t : Set α\np✝ p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\np q : α → Prop\nhp : IsOpen {x | p x}\nhq : IsClosed {x | q x}\n⊢ IsClosed {x | p x → q x}",
"tactic": "simpa only [imp_iff_not_or] using hp.isClosed_compl.union hq"
}
] | [
267,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/Data/List/Chain.lean | List.Chain'.suffix | [] | [
315,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
] |
Std/Data/RBMap/WF.lean | Std.RBNode.cmpLT.trans | [] | [
31,
32
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
30,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | MeasureTheory.ae_le_of_forall_set_integral_le | [
{
"state_after": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\n⊢ 0 ≤ᵐ[μ] g - f",
"state_before": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\n⊢ f ≤ᵐ[μ] g",
"tactic": "rw [← eventually_sub_nonneg]"
},
{
"state_after": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, (g - f) x ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\n⊢ 0 ≤ᵐ[μ] g - f",
"tactic": "refine' ae_nonneg_of_forall_set_integral_nonneg (hg.sub hf) fun s hs => _"
},
{
"state_after": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s < ⊤ → (∫ (a : α) in s, f a ∂μ) ≤ ∫ (a : α) in s, g a ∂μ",
"state_before": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s < ⊤ → 0 ≤ ∫ (x : α) in s, (g - f) x ∂μ",
"tactic": "rw [integral_sub' hg.integrableOn hf.integrableOn, sub_nonneg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.79505\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf✝ f g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nhf_le : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) ≤ ∫ (x : α) in s, g x ∂μ\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑μ s < ⊤ → (∫ (a : α) in s, f a ∂μ) ≤ ∫ (a : α) in s, g a ∂μ",
"tactic": "exact hf_le s hs"
}
] | [
303,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
297,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean | LinearIsometry.extend_apply | [
{
"state_after": "ι : Type ?u.1751792\nι' : Type ?u.1751795\n𝕜 : Type u_1\ninst✝¹² : IsROrC 𝕜\nE : Type ?u.1751804\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : InnerProductSpace 𝕜 E\nE' : Type ?u.1751824\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : InnerProductSpace 𝕜 E'\nF : Type ?u.1751842\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type ?u.1751862\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fintype ι\nV : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nS : Submodule 𝕜 V\nL✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V\ns : { x // x ∈ S }\nthis : CompleteSpace { x // x ∈ S }\n⊢ ↑(extend L) ↑s = ↑L s",
"state_before": "ι : Type ?u.1751792\nι' : Type ?u.1751795\n𝕜 : Type u_1\ninst✝¹² : IsROrC 𝕜\nE : Type ?u.1751804\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : InnerProductSpace 𝕜 E\nE' : Type ?u.1751824\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : InnerProductSpace 𝕜 E'\nF : Type ?u.1751842\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type ?u.1751862\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fintype ι\nV : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nS : Submodule 𝕜 V\nL✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V\ns : { x // x ∈ S }\n⊢ ↑(extend L) ↑s = ↑L s",
"tactic": "haveI : CompleteSpace S := FiniteDimensional.complete 𝕜 S"
},
{
"state_after": "ι : Type ?u.1751792\nι' : Type ?u.1751795\n𝕜 : Type u_1\ninst✝¹² : IsROrC 𝕜\nE : Type ?u.1751804\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : InnerProductSpace 𝕜 E\nE' : Type ?u.1751824\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : InnerProductSpace 𝕜 E'\nF : Type ?u.1751842\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type ?u.1751862\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fintype ι\nV : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nS : Submodule 𝕜 V\nL✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V\ns : { x // x ∈ S }\nthis : CompleteSpace { x // x ∈ S }\n⊢ ↑(LinearMap.comp L.toLinearMap ↑(orthogonalProjection S) +\n LinearMap.comp\n (comp (subtypeₗᵢ (LinearMap.range L.toLinearMap)ᗮ)\n (LinearIsometryEquiv.toLinearIsometry\n (LinearIsometryEquiv.trans (stdOrthonormalBasis 𝕜 { x // x ∈ Sᗮ }).repr\n (LinearIsometryEquiv.symm\n (OrthonormalBasis.reindex (stdOrthonormalBasis 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ })\n (finCongr\n (_ :\n finrank 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ } =\n finrank 𝕜 { x // x ∈ Sᗮ }))).repr)))).toLinearMap\n ↑(orthogonalProjection Sᗮ))\n ↑s =\n ↑L.toLinearMap s",
"state_before": "ι : Type ?u.1751792\nι' : Type ?u.1751795\n𝕜 : Type u_1\ninst✝¹² : IsROrC 𝕜\nE : Type ?u.1751804\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : InnerProductSpace 𝕜 E\nE' : Type ?u.1751824\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : InnerProductSpace 𝕜 E'\nF : Type ?u.1751842\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type ?u.1751862\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fintype ι\nV : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nS : Submodule 𝕜 V\nL✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V\ns : { x // x ∈ S }\nthis : CompleteSpace { x // x ∈ S }\n⊢ ↑(extend L) ↑s = ↑L s",
"tactic": "simp only [LinearIsometry.extend, ← LinearIsometry.coe_toLinearMap]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1751792\nι' : Type ?u.1751795\n𝕜 : Type u_1\ninst✝¹² : IsROrC 𝕜\nE : Type ?u.1751804\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : InnerProductSpace 𝕜 E\nE' : Type ?u.1751824\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : InnerProductSpace 𝕜 E'\nF : Type ?u.1751842\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type ?u.1751862\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fintype ι\nV : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : FiniteDimensional 𝕜 V\nS : Submodule 𝕜 V\nL✝ L : { x // x ∈ S } →ₗᵢ[𝕜] V\ns : { x // x ∈ S }\nthis : CompleteSpace { x // x ∈ S }\n⊢ ↑(LinearMap.comp L.toLinearMap ↑(orthogonalProjection S) +\n LinearMap.comp\n (comp (subtypeₗᵢ (LinearMap.range L.toLinearMap)ᗮ)\n (LinearIsometryEquiv.toLinearIsometry\n (LinearIsometryEquiv.trans (stdOrthonormalBasis 𝕜 { x // x ∈ Sᗮ }).repr\n (LinearIsometryEquiv.symm\n (OrthonormalBasis.reindex (stdOrthonormalBasis 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ })\n (finCongr\n (_ :\n finrank 𝕜 { x // x ∈ (LinearMap.range L.toLinearMap)ᗮ } =\n finrank 𝕜 { x // x ∈ Sᗮ }))).repr)))).toLinearMap\n ↑(orthogonalProjection Sᗮ))\n ↑s =\n ↑L.toLinearMap s",
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] | [
963,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
954,
1
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Mathlib/Algebra/Lie/SkewAdjoint.lean | skewAdjointMatricesLieSubalgebraEquivTranspose_apply | [] | [
170,
6
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166,
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Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousAt.inv₀ | [] | [
131,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
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Mathlib/Data/Real/ENNReal.lean | ENNReal.toReal_div | [
{
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}
] | [
2326,
62
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2325,
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Mathlib/Computability/TuringMachine.lean | Turing.Reaches₀.refl | [] | [
788,
14
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Std/Data/Nat/Lemmas.lean | Nat.lt_add_right | [] | [
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46
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86,
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Mathlib/RingTheory/Finiteness.lean | Submodule.FG.sup | [
{
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185,
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Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.map_roots_le | [
{
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{
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28
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Mathlib/Topology/UniformSpace/Completion.lean | CauchyFilter.compRel_gen_gen_subset_gen_compRel | [] | [
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Mathlib/Topology/SubsetProperties.lean | isIrreducible_irreducibleComponent | [] | [
1801,
71
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1800,
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Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.deleteEdges_empty_eq | [
{
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1141,
7
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1139,
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Mathlib/LinearAlgebra/Matrix/Determinant.lean | Matrix.det_conjTranspose | [] | [
354,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
353,
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Mathlib/Data/Polynomial/Laurent.lean | LaurentPolynomial.induction_on' | [
{
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},
{
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"state_before": "case refine'_1\nR : Type u_1\ninst✝ : Semiring R\nM : R[T;T⁻¹] → Prop\np : R[T;T⁻¹]\nh_add : ∀ (p q : R[T;T⁻¹]), M p → M q → M (p + q)\nh_C_mul_T : ∀ (n : ℤ) (a : R), M (↑C a * T n)\na : R\n⊢ M (↑C a)",
"tactic": "convert h_C_mul_T 0 a"
},
{
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},
{
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"tactic": "exact fun n f _ => h_C_mul_T _ f"
}
] | [
316,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
310,
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Mathlib/Algebra/Group/Basic.lean | eq_div_iff_mul_eq'' | [
{
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"tactic": "rw [eq_div_iff_mul_eq', mul_comm]"
}
] | [
953,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
953,
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Mathlib/Algebra/Algebra/Pi.lean | Pi.constAlgHom_eq_algebra_ofId | [] | [
95,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
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Mathlib/Topology/MetricSpace/Basic.lean | Metric.uniformity_eq_comap_nhds_zero | [
{
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"state_before": "α : Type u\nβ : Type v\nX : Type ?u.203876\nι : Type ?u.203879\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 α = comap (fun p => dist p.fst p.snd) (𝓝 0)",
"tactic": "ext s"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nX : Type ?u.203876\nι : Type ?u.203879\ninst✝ : PseudoMetricSpace α\ns : Set (α × α)\n⊢ (∃ ε, ε > 0 ∧ ∀ {a b : α}, dist a b < ε → (a, b) ∈ s) ↔ ∃ i, 0 < i ∧ (fun p => dist p.fst p.snd) ⁻¹' ball 0 i ⊆ s",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.203876\nι : Type ?u.203879\ninst✝ : PseudoMetricSpace α\ns : Set (α × α)\n⊢ s ∈ 𝓤 α ↔ s ∈ comap (fun p => dist p.fst p.snd) (𝓝 0)",
"tactic": "simp only [mem_uniformity_dist, (nhds_basis_ball.comap _).mem_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nX : Type ?u.203876\nι : Type ?u.203879\ninst✝ : PseudoMetricSpace α\ns : Set (α × α)\n⊢ (∃ ε, ε > 0 ∧ ∀ {a b : α}, dist a b < ε → (a, b) ∈ s) ↔ ∃ i, 0 < i ∧ (fun p => dist p.fst p.snd) ⁻¹' ball 0 i ⊆ s",
"tactic": "simp [subset_def, Real.dist_0_eq_abs]"
}
] | [
1455,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1451,
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Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.514960\n⊢ leadingCoeff p = 0 ↔ degree p = ⊥",
"tactic": "rw [leadingCoeff_eq_zero, degree_eq_bot]"
}
] | [
672,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
671,
1
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Mathlib/Analysis/Seminorm.lean | Seminorm.continuousAt_zero' | [
{
"state_after": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\n⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall (↑p 0) i",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\n⊢ ContinuousAt (↑p) 0",
"tactic": "refine' Metric.nhds_basis_closedBall.tendsto_right_iff.mpr _"
},
{
"state_after": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall (↑p 0) ε",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\n⊢ ∀ (i : ℝ), 0 < i → ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall (↑p 0) i",
"tactic": "intro ε hε"
},
{
"state_after": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall 0 ε",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall (↑p 0) ε",
"tactic": "rw [map_zero]"
},
{
"state_after": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall 0 ε",
"tactic": "suffices p.closedBall 0 ε ∈ (𝓝 0 : Filter E) by\n rwa [Seminorm.closedBall_zero_eq_preimage_closedBall] at this"
},
{
"state_after": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"tactic": "rcases exists_norm_lt 𝕜 (div_pos hε hr) with ⟨k, hk0, hkε⟩"
},
{
"state_after": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"state_before": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"tactic": "have hk0' := norm_pos_iff.mp hk0"
},
{
"state_after": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"state_before": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"tactic": "have := (set_smul_mem_nhds_zero_iff hk0').mpr hp"
},
{
"state_after": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ k • x ∈ closedBall p 0 ε",
"state_before": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\n⊢ closedBall p 0 ε ∈ 𝓝 0",
"tactic": "refine' Filter.mem_of_superset this (smul_set_subset_iff.mpr fun x hx => _)"
},
{
"state_after": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ ‖k‖ * ↑p x ≤ ε / r * r",
"state_before": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ k • x ∈ closedBall p 0 ε",
"tactic": "rw [mem_closedBall_zero, map_smul_eq_mul, ← div_mul_cancel ε hr.ne.symm]"
},
{
"state_after": "case intro.intro.h₂\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ ↑p x ≤ r",
"state_before": "case intro.intro\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ ‖k‖ * ↑p x ≤ ε / r * r",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h₂\nR : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nk : 𝕜\nhk0 : 0 < ‖k‖\nhkε : ‖k‖ < ε / r\nhk0' : k ≠ 0\nthis : k • closedBall p 0 r ∈ 𝓝 0\nx : E\nhx : x ∈ closedBall p 0 r\n⊢ ↑p x ≤ r",
"tactic": "exact p.mem_closedBall_zero.mp hx"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1359761\nR' : Type ?u.1359764\n𝕜 : Type u_2\n𝕜₂ : Type ?u.1359770\n𝕜₃ : Type ?u.1359773\n𝕝 : Type ?u.1359776\nE : Type u_1\nE₂ : Type ?u.1359782\nE₃ : Type ?u.1359785\nF : Type ?u.1359788\nG : Type ?u.1359791\nι : Type ?u.1359794\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedRing 𝕝\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕝 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\nhr : 0 < r\nhp : closedBall p 0 r ∈ 𝓝 0\nε : ℝ\nhε : 0 < ε\nthis : closedBall p 0 ε ∈ 𝓝 0\n⊢ ∀ᶠ (x : E) in 𝓝 0, ↑p x ∈ Metric.closedBall 0 ε",
"tactic": "rwa [Seminorm.closedBall_zero_eq_preimage_closedBall] at this"
}
] | [
1125,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1112,
1
] |
Mathlib/Combinatorics/SetFamily/Kleitman.lean | Finset.card_biUnion_le_of_intersecting | [
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"tactic": "have : DecidableEq ι := by\n classical\n infer_instance"
},
{
"state_after": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\n\ncase inr\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : card s ≤ Fintype.card α\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"tactic": "obtain hs | hs := le_total (Fintype.card α) s.card"
},
{
"state_after": "case inr.empty\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s ≤ Fintype.card α\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ ∅ → Set.Intersecting ↑(f i)\nhs : card ∅ ≤ Fintype.card α\n⊢ card (Finset.biUnion ∅ f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card ∅)\n\ncase inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : card s ≤ Fintype.card α\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"tactic": "induction' s using Finset.cons_induction with i s hi ih generalizing f"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "set f' : ι → Finset (Finset α) :=\n fun j ↦ if hj : j ∈ cons i s hi then (hf j hj).exists_card_eq.choose else ∅"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "have hf₁ : ∀ j, j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * (f' j).card =\n 2 ^ Fintype.card α ∧ (f' j : Set (Finset α)).Intersecting := by\n rintro j hj\n simp_rw [dif_pos hj, ← Fintype.card_finset]\n exact Classical.choose_spec (hf j hj).exists_card_eq"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "have hf₂ : ∀ j, j ∈ cons i s hi → IsUpperSet (f' j : Set (Finset α)) := by\n refine' fun j hj ↦ (hf₁ _ hj).2.2.isUpperSet' ((hf₁ _ hj).2.2.is_max_iff_card_eq.2 _)\n rw [Fintype.card_finset]\n exact (hf₁ _ hj).2.1"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "refine' (card_le_of_subset <| biUnion_mono fun j hj ↦ (hf₁ _ hj).1).trans _"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (insert i s) fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (cons i s hi) fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "nth_rw 1 [cons_eq_insert i]"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i ∪ Finset.biUnion s fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (Finset.biUnion (insert i s) fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "rw [biUnion_insert]"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i) + card ((f' i ∪ Finset.biUnion s fun j => f' j) \\ f' i) ≤\n 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i ∪ Finset.biUnion s fun j => f' j) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "refine' (card_mono <| @le_sup_sdiff _ _ _ <| f' i).trans ((card_union_le _ _).trans _)"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i) + card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ) ≤\n 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i) + card ((f' i ∪ Finset.biUnion s fun j => f' j) \\ f' i) ≤\n 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "rw [union_sdiff_left, sdiff_eq_inter_compl]"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ (Fintype.card α + 1) * (card (f' i) + card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ)) ≤\n 2 ^ (Fintype.card α + 1) * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ card (f' i) + card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ) ≤\n 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))",
"tactic": "refine' le_of_mul_le_mul_left _ (pow_pos (zero_lt_two' ℕ) <| Fintype.card α + 1)"
},
{
"state_after": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * (2 * card (f' i)) + 2 * (2 ^ Fintype.card α * card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ)) ≤\n 2 * 2 ^ Fintype.card α * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ (Fintype.card α + 1) * (card (f' i) + card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ)) ≤\n 2 ^ (Fintype.card α + 1) * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"tactic": "rw [pow_succ', mul_add, mul_assoc, mul_comm _ 2, mul_assoc]"
},
{
"state_after": "case inr.cons.refine'_1\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsUpperSet ↑(Finset.biUnion s fun j => f' j)\n\ncase inr.cons.refine'_2\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsLowerSet ↑(f' iᶜ)\n\ncase inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * Fintype.card (Finset α) + 2 * (card (Finset.biUnion s fun j => f' j) * card (f' iᶜ)) ≤\n 2 * 2 ^ Fintype.card α * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"state_before": "case inr.cons\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * (2 * card (f' i)) + 2 * (2 ^ Fintype.card α * card ((Finset.biUnion s fun j => f' j) ∩ f' iᶜ)) ≤\n 2 * 2 ^ Fintype.card α * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"tactic": "refine' (add_le_add\n ((mul_le_mul_left <| pow_pos (zero_lt_two' ℕ) _).2\n (hf₁ _ <| mem_cons_self _ _).2.2.card_le) <|\n (mul_le_mul_left <| zero_lt_two' ℕ).2 <| IsUpperSet.card_inter_le_finset _ _).trans _"
},
{
"state_after": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * (2 ^ Fintype.card α + card (Finset.biUnion s fun j => f' j)) ≤\n 2 ^ Fintype.card α * (2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))))",
"state_before": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * Fintype.card (Finset α) + 2 * (card (Finset.biUnion s fun j => f' j) * card (f' iᶜ)) ≤\n 2 * 2 ^ Fintype.card α * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"tactic": "rw [mul_tsub, card_compl, Fintype.card_finset, mul_left_comm, mul_tsub,\n (hf₁ _ <| mem_cons_self _ _).2.1, two_mul, add_tsub_cancel_left, ← mul_tsub, ← mul_two,\n mul_assoc, ← add_mul, mul_comm]"
},
{
"state_after": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α + card (Finset.biUnion s fun j => f' j) ≤\n 2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"state_before": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α * (2 ^ Fintype.card α + card (Finset.biUnion s fun j => f' j)) ≤\n 2 ^ Fintype.card α * (2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi))))",
"tactic": "refine' mul_le_mul_left' _ _"
},
{
"state_after": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α + (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)) ≤\n 2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"state_before": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α + card (Finset.biUnion s fun j => f' j) ≤\n 2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"tactic": "refine' (add_le_add_left\n (ih _ (fun i hi ↦ (hf₁ _ <| subset_cons _ hi).2.2)\n ((card_le_of_subset <| subset_cons _).trans hs)) _).trans _"
},
{
"state_after": "no goals",
"state_before": "case inr.cons.refine'_3\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ 2 ^ Fintype.card α + (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)) ≤\n 2 * (2 ^ Fintype.card α - 2 ^ (Fintype.card α - card (cons i s hi)))",
"tactic": "rw [mul_tsub, two_mul, ← pow_succ,\n ← add_tsub_assoc_of_le (pow_le_pow' (one_le_two : (1 : ℕ) ≤ 2) tsub_le_self),\n tsub_add_eq_add_tsub hs, card_cons, add_tsub_add_eq_tsub_right]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\n⊢ DecidableEq ι",
"tactic": "classical\ninfer_instance"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\n⊢ DecidableEq ι",
"tactic": "infer_instance"
},
{
"state_after": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 1",
"state_before": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)",
"tactic": "rw [tsub_eq_zero_of_le hs, pow_zero]"
},
{
"state_after": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card ({⊥}ᶜ) = 2 ^ Fintype.card α - 1",
"state_before": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 1",
"tactic": "refine' (card_le_of_subset <| biUnion_subset.2 fun i hi a ha ↦\n mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq _"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)\nthis : DecidableEq ι\nhs : Fintype.card α ≤ card s\n⊢ card ({⊥}ᶜ) = 2 ^ Fintype.card α - 1",
"tactic": "rw [card_compl, Fintype.card_finset, card_singleton]"
},
{
"state_after": "no goals",
"state_before": "case inr.empty\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s ≤ Fintype.card α\nf : ι → Finset (Finset α)\nhf : ∀ (i : ι), i ∈ ∅ → Set.Intersecting ↑(f i)\nhs : card ∅ ≤ Fintype.card α\n⊢ card (Finset.biUnion ∅ f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card ∅)",
"tactic": "simp"
},
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nj : ι\nhj : j ∈ cons i s hi\n⊢ f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\n⊢ ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)",
"tactic": "rintro j hj"
},
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nj : ι\nhj : j ∈ cons i s hi\n⊢ f j ⊆ Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t) ∧\n 2 * card (Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)) =\n Fintype.card (Finset α) ∧\n Set.Intersecting ↑(Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t))",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nj : ι\nhj : j ∈ cons i s hi\n⊢ f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)",
"tactic": "simp_rw [dif_pos hj, ← Fintype.card_finset]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nj : ι\nhj : j ∈ cons i s hi\n⊢ f j ⊆ Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t) ∧\n 2 * card (Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)) =\n Fintype.card (Finset α) ∧\n Set.Intersecting ↑(Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t))",
"tactic": "exact Classical.choose_spec (hf j hj).exists_card_eq"
},
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nj : ι\nhj : j ∈ cons i s hi\n⊢ 2 * card (f' j) = Fintype.card (Finset α)",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\n⊢ ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)",
"tactic": "refine' fun j hj ↦ (hf₁ _ hj).2.2.isUpperSet' ((hf₁ _ hj).2.2.is_max_iff_card_eq.2 _)"
},
{
"state_after": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nj : ι\nhj : j ∈ cons i s hi\n⊢ 2 * card (f' j) = 2 ^ Fintype.card α",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nj : ι\nhj : j ∈ cons i s hi\n⊢ 2 * card (f' j) = Fintype.card (Finset α)",
"tactic": "rw [Fintype.card_finset]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nj : ι\nhj : j ∈ cons i s hi\n⊢ 2 * card (f' j) = 2 ^ Fintype.card α",
"tactic": "exact (hf₁ _ hj).2.1"
},
{
"state_after": "case inr.cons.refine'_1\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsUpperSet (⋃ (x : ι) (_ : x ∈ ↑s), ↑(f' x))",
"state_before": "case inr.cons.refine'_1\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsUpperSet ↑(Finset.biUnion s fun j => f' j)",
"tactic": "rw [coe_biUnion]"
},
{
"state_after": "no goals",
"state_before": "case inr.cons.refine'_1\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsUpperSet (⋃ (x : ι) (_ : x ∈ ↑s), ↑(f' x))",
"tactic": "exact isUpperSet_iUnion₂ fun i hi ↦ hf₂ _ <| subset_cons _ hi"
},
{
"state_after": "case inr.cons.refine'_2\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsLowerSet (↑(f' i)ᶜ)",
"state_before": "case inr.cons.refine'_2\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsLowerSet ↑(f' iᶜ)",
"tactic": "rw [coe_compl]"
},
{
"state_after": "no goals",
"state_before": "case inr.cons.refine'_2\nι : Type u_1\nα : Type u_2\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns✝ : Finset ι\nf✝ : ι → Finset (Finset α)\nhf✝ : ∀ (i : ι), i ∈ s✝ → Set.Intersecting ↑(f✝ i)\nthis : DecidableEq ι\nhs✝ : card s✝ ≤ Fintype.card α\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nih :\n ∀ (f : ι → Finset (Finset α)),\n (∀ (i : ι), i ∈ s → Set.Intersecting ↑(f i)) →\n card s ≤ Fintype.card α → card (Finset.biUnion s f) ≤ 2 ^ Fintype.card α - 2 ^ (Fintype.card α - card s)\nf : ι → Finset (Finset α)\nhf : ∀ (i_1 : ι), i_1 ∈ cons i s hi → Set.Intersecting ↑(f i_1)\nhs : card (cons i s hi) ≤ Fintype.card α\nf' : ι → Finset (Finset α) :=\n fun j =>\n if hj : j ∈ cons i s hi then\n Exists.choose (_ : ∃ t, f j ⊆ t ∧ 2 * card t = Fintype.card (Finset α) ∧ Set.Intersecting ↑t)\n else ∅\nhf₁ : ∀ (j : ι), j ∈ cons i s hi → f j ⊆ f' j ∧ 2 * card (f' j) = 2 ^ Fintype.card α ∧ Set.Intersecting ↑(f' j)\nhf₂ : ∀ (j : ι), j ∈ cons i s hi → IsUpperSet ↑(f' j)\n⊢ IsLowerSet (↑(f' i)ᶜ)",
"tactic": "exact (hf₂ _ <| mem_cons_self _ _).compl"
}
] | [
88,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
1
] |
Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean | Matrix.SpecialLinearGroup.coeToGL_det | [] | [
172,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
] |
Mathlib/Analysis/Calculus/DiffContOnCl.lean | DiffContOnCl.mono | [] | [
90,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
11
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | HasDerivWithinAt.congr_of_eventuallyEq | [] | [
590,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
588,
1
] |
Mathlib/Computability/Primrec.lean | Primrec.succ | [] | [
268,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.inf_eq_inter | [] | [
1316,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1315,
1
] |
Mathlib/Data/Matrix/Kronecker.lean | Matrix.one_kronecker | [] | [
356,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
src/lean/Init/Data/List/Basic.lean | List.of_concat_eq_concat | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b : α\nh : concat as a = concat bs b\n⊢ as = bs ∧ a = b",
"tactic": "match as, bs with\n| [], [] => simp [concat] at h; simp [h]\n| [_], [] => simp [concat] at h\n| _::_::_, [] => simp [concat] at h\n| [], [_] => simp [concat] at h\n| [], _::_::_ => simp [concat] at h\n| _::_, _::_ => simp [concat] at h; simp [h]; apply of_concat_eq_concat h.2"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b : α\nh : a = b\n⊢ nil = nil ∧ a = b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b : α\nh : concat nil a = concat nil b\n⊢ nil = nil ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b : α\nh : a = b\n⊢ nil = nil ∧ a = b",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝ : α\nh : concat (head✝ :: nil) a = concat nil b\n⊢ head✝ :: nil = nil ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ head✝ : α\ntail✝ : List α\nh : concat (head✝¹ :: head✝ :: tail✝) a = concat nil b\n⊢ head✝¹ :: head✝ :: tail✝ = nil ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝ : α\nh : concat nil a = concat (head✝ :: nil) b\n⊢ nil = head✝ :: nil ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ head✝ : α\ntail✝ : List α\nh : concat nil a = concat (head✝¹ :: head✝ :: tail✝) b\n⊢ nil = head✝¹ :: head✝ :: tail✝ ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ : α\ntail✝¹ : List α\nhead✝ : α\ntail✝ : List α\nh : head✝¹ = head✝ ∧ concat tail✝¹ a = concat tail✝ b\n⊢ head✝¹ :: tail✝¹ = head✝ :: tail✝ ∧ a = b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ : α\ntail✝¹ : List α\nhead✝ : α\ntail✝ : List α\nh : concat (head✝¹ :: tail✝¹) a = concat (head✝ :: tail✝) b\n⊢ head✝¹ :: tail✝¹ = head✝ :: tail✝ ∧ a = b",
"tactic": "simp [concat] at h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ : α\ntail✝¹ : List α\nhead✝ : α\ntail✝ : List α\nh : head✝¹ = head✝ ∧ concat tail✝¹ a = concat tail✝ b\n⊢ tail✝¹ = tail✝ ∧ a = b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ : α\ntail✝¹ : List α\nhead✝ : α\ntail✝ : List α\nh : head✝¹ = head✝ ∧ concat tail✝¹ a = concat tail✝ b\n⊢ head✝¹ :: tail✝¹ = head✝ :: tail✝ ∧ a = b",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas bs : List α\na b head✝¹ : α\ntail✝¹ : List α\nhead✝ : α\ntail✝ : List α\nh : head✝¹ = head✝ ∧ concat tail✝¹ a = concat tail✝ b\n⊢ tail✝¹ = tail✝ ∧ a = b",
"tactic": "apply of_concat_eq_concat h.2"
}
] | [
848,
78
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
841,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.to_of_int | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.1062538\nn : ℤ\n⊢ ↑↑0 = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1062538\nn : ℤ\n⊢ ∀ (k : ℤ), 0 ≤ k → ↑↑k = k → ↑↑(k + 1) = k + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.1062538\nn : ℤ\n⊢ ∀ (k : ℤ), k ≤ 0 → ↑↑k = k → ↑↑(k - 1) = k - 1",
"tactic": "simp"
}
] | [
1556,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1555,
1
] |
Std/Data/Nat/Gcd.lean | Nat.gcd_assoc | [] | [
79,
71
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
70,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | MeasureTheory.Measure.withDensity_rnDeriv_le | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ\n\ncase neg\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : ¬HaveLebesgueDecomposition μ ν\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"state_before": "α : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"tactic": "by_cases hl : HaveLebesgueDecomposition μ ν"
},
{
"state_after": "case pos.intro\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\nleft✝ : singularPart μ ν ⟂ₘ ν\nh : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"tactic": "cases' (haveLebesgueDecomposition_spec μ ν).2 with _ h"
},
{
"state_after": "case pos.intro\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\nleft✝ : singularPart μ ν ⟂ₘ ν\nh : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)\n⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν)",
"state_before": "case pos.intro\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\nleft✝ : singularPart μ ν ⟂ₘ ν\nh : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"tactic": "conv_rhs => rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : HaveLebesgueDecomposition μ ν\nleft✝ : singularPart μ ν ⟂ₘ ν\nh : μ = singularPart μ ν + withDensity ν (rnDeriv μ ν)\n⊢ withDensity ν (rnDeriv μ ν) ≤ singularPart μ ν + withDensity ν (rnDeriv μ ν)",
"tactic": "exact Measure.le_add_left le_rfl"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : ¬HaveLebesgueDecomposition μ ν\n⊢ 0 ≤ μ",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : ¬HaveLebesgueDecomposition μ ν\n⊢ withDensity ν (rnDeriv μ ν) ≤ μ",
"tactic": "rw [rnDeriv, dif_neg hl, withDensity_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.34335\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\nhl : ¬HaveLebesgueDecomposition μ ν\n⊢ 0 ≤ μ",
"tactic": "exact Measure.zero_le μ"
}
] | [
156,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.restrictScalars_apply | [] | [
465,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
464,
1
] |
Mathlib/Algebra/Regular/SMul.lean | IsSMulRegular.of_smul_eq_one | [
{
"state_after": "R : Type u_2\nS : Type u_1\nM : Type u_3\na b : R\ns : S\ninst✝⁴ : Monoid S\ninst✝³ : SMul R M\ninst✝² : SMul R S\ninst✝¹ : MulAction S M\ninst✝ : IsScalarTower R S M\nh : a • s = 1\n⊢ IsSMulRegular M 1",
"state_before": "R : Type u_2\nS : Type u_1\nM : Type u_3\na b : R\ns : S\ninst✝⁴ : Monoid S\ninst✝³ : SMul R M\ninst✝² : SMul R S\ninst✝¹ : MulAction S M\ninst✝ : IsScalarTower R S M\nh : a • s = 1\n⊢ IsSMulRegular M (a • s)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\nM : Type u_3\na b : R\ns : S\ninst✝⁴ : Monoid S\ninst✝³ : SMul R M\ninst✝² : SMul R S\ninst✝¹ : MulAction S M\ninst✝ : IsScalarTower R S M\nh : a • s = 1\n⊢ IsSMulRegular M 1",
"tactic": "exact one M"
}
] | [
180,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
] |
Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.CompactOpens.map_comp | [] | [
595,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
592,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean | Matrix.equiv_block_det | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.40762\nβ : Type ?u.40765\nm : Type u_1\nn : Type ?u.40771\no : Type ?u.40774\nm' : α → Type ?u.40779\nn' : α → Type ?u.40784\nR : Type v\ninst✝⁶ : CommRing R\nM✝ N : Matrix m m R\nb : m → α\ninst✝⁵ : DecidableEq m\ninst✝⁴ : Fintype m\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nM : Matrix m m R\np q : m → Prop\ninst✝¹ : DecidablePred p\ninst✝ : DecidablePred q\ne : ∀ (x : m), q x ↔ p x\n⊢ det (toSquareBlockProp M p) = det (toSquareBlockProp M q)",
"tactic": "convert Matrix.det_reindex_self (Equiv.subtypeEquivRight e) (toSquareBlockProp M q)"
}
] | [
157,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.restrict_eq_self_of_ae_mem | [] | [
1794,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1790,
1
] |
Mathlib/Order/Zorn.lean | zorn_nonempty_preorder | [] | [
115,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
113,
1
] |
Mathlib/Data/Finset/Prod.lean | Finset.subset_product_image_snd | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.6733\ns s' : Finset α\nt t' : Finset β\na : α\nb : β\ninst✝ : DecidableEq β\ni : β\n⊢ i ∈ image Prod.snd (s ×ˢ t) → i ∈ t",
"tactic": "simp (config := { contextual := true }) [mem_image]"
}
] | [
75,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/Data/List/Forall2.lean | List.sublistForall₂_iff | [
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh : SublistForall₂ R l₁ l₂\n⊢ ∃ l, Forall₂ R l₁ l ∧ l <+ l₂\n\ncase mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh : ∃ l, Forall₂ R l₁ l ∧ l <+ l₂\n⊢ SublistForall₂ R l₁ l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\n⊢ SublistForall₂ R l₁ l₂ ↔ ∃ l, Forall₂ R l₁ l ∧ l <+ l₂",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ l✝ : List β\n⊢ ∃ l, Forall₂ R [] l ∧ l <+ l✝\n\ncase mp.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\na : α\nb : β\nl1 : List α\nl2 : List β\nrab : R a b\na✝ : SublistForall₂ R l1 l2\nih : ∃ l, Forall₂ R l1 l ∧ l <+ l2\n⊢ ∃ l, Forall₂ R (a :: l1) l ∧ l <+ b :: l2\n\ncase mp.cons_right\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nb : β\nl1 : List α\nl2 : List β\na✝ : SublistForall₂ R l1 l2\nih : ∃ l, Forall₂ R l1 l ∧ l <+ l2\n⊢ ∃ l, Forall₂ R l1 l ∧ l <+ b :: l2",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh : SublistForall₂ R l₁ l₂\n⊢ ∃ l, Forall₂ R l₁ l ∧ l <+ l₂",
"tactic": "induction' h with _ a b l1 l2 rab _ ih b l1 l2 _ ih"
},
{
"state_after": "no goals",
"state_before": "case mp.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ l✝ : List β\n⊢ ∃ l, Forall₂ R [] l ∧ l <+ l✝",
"tactic": "exact ⟨nil, Forall₂.nil, nil_sublist _⟩"
},
{
"state_after": "case mp.cons.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\na : α\nb : β\nl1 : List α\nl2 : List β\nrab : R a b\na✝ : SublistForall₂ R l1 l2\nl : List β\nhl1 : Forall₂ R l1 l\nhl2 : l <+ l2\n⊢ ∃ l, Forall₂ R (a :: l1) l ∧ l <+ b :: l2",
"state_before": "case mp.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\na : α\nb : β\nl1 : List α\nl2 : List β\nrab : R a b\na✝ : SublistForall₂ R l1 l2\nih : ∃ l, Forall₂ R l1 l ∧ l <+ l2\n⊢ ∃ l, Forall₂ R (a :: l1) l ∧ l <+ b :: l2",
"tactic": "obtain ⟨l, hl1, hl2⟩ := ih"
},
{
"state_after": "no goals",
"state_before": "case mp.cons.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\na : α\nb : β\nl1 : List α\nl2 : List β\nrab : R a b\na✝ : SublistForall₂ R l1 l2\nl : List β\nhl1 : Forall₂ R l1 l\nhl2 : l <+ l2\n⊢ ∃ l, Forall₂ R (a :: l1) l ∧ l <+ b :: l2",
"tactic": "refine' ⟨b :: l, Forall₂.cons rab hl1, hl2.cons_cons b⟩"
},
{
"state_after": "case mp.cons_right.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nb : β\nl1 : List α\nl2 : List β\na✝ : SublistForall₂ R l1 l2\nl : List β\nhl1 : Forall₂ R l1 l\nhl2 : l <+ l2\n⊢ ∃ l, Forall₂ R l1 l ∧ l <+ b :: l2",
"state_before": "case mp.cons_right\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nb : β\nl1 : List α\nl2 : List β\na✝ : SublistForall₂ R l1 l2\nih : ∃ l, Forall₂ R l1 l ∧ l <+ l2\n⊢ ∃ l, Forall₂ R l1 l ∧ l <+ b :: l2",
"tactic": "obtain ⟨l, hl1, hl2⟩ := ih"
},
{
"state_after": "no goals",
"state_before": "case mp.cons_right.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nb : β\nl1 : List α\nl2 : List β\na✝ : SublistForall₂ R l1 l2\nl : List β\nhl1 : Forall₂ R l1 l\nhl2 : l <+ l2\n⊢ ∃ l, Forall₂ R l1 l ∧ l <+ b :: l2",
"tactic": "exact ⟨l, hl1, hl2.trans (Sublist.cons _ (Sublist.refl _))⟩"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ l : List β\nhl1 : Forall₂ R l₁ l\nhl2 : l <+ l₂\n⊢ SublistForall₂ R l₁ l₂",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh : ∃ l, Forall₂ R l₁ l ∧ l <+ l₂\n⊢ SublistForall₂ R l₁ l₂",
"tactic": "obtain ⟨l, hl1, hl2⟩ := h"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nhl2 : l <+ l₂\n⊢ ∀ {l₁ : List α}, Forall₂ R l₁ l → SublistForall₂ R l₁ l₂",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ l : List β\nhl1 : Forall₂ R l₁ l\nhl2 : l <+ l₂\n⊢ SublistForall₂ R l₁ l₂",
"tactic": "revert l₁"
},
{
"state_after": "case mpr.intro.intro.slnil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nl₁ : List α\nhl1 : Forall₂ R l₁ []\n⊢ SublistForall₂ R l₁ []\n\ncase mpr.intro.intro.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝ l₂✝ : List β\na✝¹ : β\na✝ : l₁✝ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝ → SublistForall₂ R l₁ l₂✝\nl₁ : List α\nhl1 : Forall₂ R l₁ l₁✝\n⊢ SublistForall₂ R l₁ (a✝¹ :: l₂✝)\n\ncase mpr.intro.intro.cons₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝ l₂✝ : List β\na✝¹ : β\na✝ : l₁✝ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝ → SublistForall₂ R l₁ l₂✝\nl₁ : List α\nhl1 : Forall₂ R l₁ (a✝¹ :: l₁✝)\n⊢ SublistForall₂ R l₁ (a✝¹ :: l₂✝)",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nhl2 : l <+ l₂\n⊢ ∀ {l₁ : List α}, Forall₂ R l₁ l → SublistForall₂ R l₁ l₂",
"tactic": "induction' hl2 with _ _ _ _ ih _ _ _ _ ih <;> intro l₁ hl1"
},
{
"state_after": "case mpr.intro.intro.slnil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nl₁ : List α\nhl1 : Forall₂ R l₁ []\n⊢ SublistForall₂ R [] []",
"state_before": "case mpr.intro.intro.slnil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nl₁ : List α\nhl1 : Forall₂ R l₁ []\n⊢ SublistForall₂ R l₁ []",
"tactic": "rw [forall₂_nil_right_iff.1 hl1]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.slnil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l : List β\nl₁ : List α\nhl1 : Forall₂ R l₁ []\n⊢ SublistForall₂ R [] []",
"tactic": "exact SublistForall₂.nil"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝ l₂✝ : List β\na✝¹ : β\na✝ : l₁✝ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝ → SublistForall₂ R l₁ l₂✝\nl₁ : List α\nhl1 : Forall₂ R l₁ l₁✝\n⊢ SublistForall₂ R l₁ (a✝¹ :: l₂✝)",
"tactic": "exact SublistForall₂.cons_right (ih hl1)"
},
{
"state_after": "case mpr.intro.intro.cons₂.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝¹ l₂✝ : List β\na✝² : β\na✝¹ : l₁✝¹ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝¹ → SublistForall₂ R l₁ l₂✝\na✝ : α\nl₁✝ : List α\nhr : R a✝ a✝²\nhl : Forall₂ R l₁✝ l₁✝¹\n⊢ SublistForall₂ R (a✝ :: l₁✝) (a✝² :: l₂✝)",
"state_before": "case mpr.intro.intro.cons₂\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝ l₂✝ : List β\na✝¹ : β\na✝ : l₁✝ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝ → SublistForall₂ R l₁ l₂✝\nl₁ : List α\nhl1 : Forall₂ R l₁ (a✝¹ :: l₁✝)\n⊢ SublistForall₂ R l₁ (a✝¹ :: l₂✝)",
"tactic": "cases' hl1 with _ _ _ _ hr hl _"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.cons₂.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.158809\nδ : Type ?u.158812\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₂ l l₁✝¹ l₂✝ : List β\na✝² : β\na✝¹ : l₁✝¹ <+ l₂✝\nih : ∀ {l₁ : List α}, Forall₂ R l₁ l₁✝¹ → SublistForall₂ R l₁ l₂✝\na✝ : α\nl₁✝ : List α\nhr : R a✝ a✝²\nhl : Forall₂ R l₁✝ l₁✝¹\n⊢ SublistForall₂ R (a✝ :: l₁✝) (a✝² :: l₂✝)",
"tactic": "exact SublistForall₂.cons hr (ih hl)"
}
] | [
349,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
333,
1
] |
Mathlib/Order/Bounds/Basic.lean | not_bddAbove_univ | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ns t : Set α\na b : α\ninst✝ : NoMaxOrder α\n⊢ ¬BddAbove univ",
"tactic": "simp [BddAbove]"
}
] | [
850,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
850,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean | linearDependent_comp_subtype' | [
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.174501\nR : Type u_1\nK : Type ?u.174507\nM : Type u_2\nM' : Type ?u.174513\nM'' : Type ?u.174516\nV : Type u\nV' : Type ?u.174521\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\n⊢ ¬LinearIndependent R (v ∘ Subtype.val) ↔ ∃ f, f ∈ Finsupp.supported R R s ∧ ↑(Finsupp.total ι M R v) f = 0 ∧ f ≠ 0",
"tactic": "simp [linearIndependent_comp_subtype, and_left_comm]"
}
] | [
371,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.sup_le_of_range_subset | [] | [
1328,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1324,
1
] |
Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | CategoryTheory.Groupoid.Free.of_eq | [] | [
152,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/Algebra/Module/Equiv.lean | LinearEquiv.coe_toEquiv_symm | [] | [
308,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/Data/Set/Basic.lean | Set.Nonempty.eq_univ | [
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝ : Subsingleton α\nx : α\nhx : x ∈ s\n⊢ s = univ",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝ : Subsingleton α\n⊢ Set.Nonempty s → s = univ",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝ : Subsingleton α\nx : α\nhx : x ∈ s\n⊢ s = univ",
"tactic": "refine' eq_univ_of_forall fun y => by rwa [Subsingleton.elim y x]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\ninst✝ : Subsingleton α\nx : α\nhx : x ∈ s\ny : α\n⊢ y ∈ s",
"tactic": "rwa [Subsingleton.elim y x]"
}
] | [
699,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
697,
1
] |
Mathlib/Data/List/Sigma.lean | List.nodupKeys_singleton | [] | [
133,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
] |
Mathlib/LinearAlgebra/Span.lean | Submodule.mem_sup' | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.100784\nK : Type ?u.100787\nM : Type u_1\nM₂ : Type ?u.100793\nV : Type ?u.100796\nS : Type ?u.100799\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\n⊢ (∃ y, y ∈ p ∧ ∃ z, z ∈ p' ∧ y + z = x) ↔ ∃ y z, ↑y + ↑z = x",
"tactic": "simp only [Subtype.exists, exists_prop]"
}
] | [
363,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
362,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.ringHom_ext | [] | [
542,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
540,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean | Filter.Tendsto.div_atTop | [
{
"state_after": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\na : 𝕜\nh : Tendsto f l (𝓝 a)\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => f x * (g x)⁻¹) l (𝓝 0)",
"state_before": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\na : 𝕜\nh : Tendsto f l (𝓝 a)\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => f x / g x) l (𝓝 0)",
"tactic": "simp only [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nα : Type u_1\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\na : 𝕜\nh : Tendsto f l (𝓝 a)\nhg : Tendsto g l atTop\n⊢ Tendsto (fun x => f x * (g x)⁻¹) l (𝓝 0)",
"tactic": "exact mul_zero a ▸ h.mul (tendsto_inv_atTop_zero.comp hg)"
}
] | [
146,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.eval_nodal_not_at_node | [
{
"state_after": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhx : ∀ (i : ι), i ∈ s → x ≠ v i\n⊢ ∀ (a : ι), a ∈ s → x ≠ v a",
"state_before": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhx : ∀ (i : ι), i ∈ s → x ≠ v i\n⊢ eval x (nodal s v) ≠ 0",
"tactic": "simp_rw [nodal, eval_prod, prod_ne_zero_iff, eval_sub, eval_X, eval_C, sub_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhx : ∀ (i : ι), i ∈ s → x ≠ v i\n⊢ ∀ (a : ι), a ∈ s → x ≠ v a",
"tactic": "exact hx"
}
] | [
515,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
513,
1
] |
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