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Mathlib/LinearAlgebra/Dimension.lean | linearIndependent_le_span | [
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.352311\nR : Type u\ninst✝⁴ : Ring R\ninst✝³ : StrongRankCondition R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ range v ≤ ↑(span R w)",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.352311\nR : Type u\ninst✝⁴ : Ring R\ninst✝³ : StrongRankCondition R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ (#ι) ≤ ↑(Fintype.card ↑w)",
"tactic": "apply linearIndependent_le_span' v i w"
},
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.352311\nR : Type u\ninst✝⁴ : Ring R\ninst✝³ : StrongRankCondition R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ range v ≤ ↑⊤",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.352311\nR : Type u\ninst✝⁴ : Ring R\ninst✝³ : StrongRankCondition R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ range v ≤ ↑(span R w)",
"tactic": "rw [s]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι✝ : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.352311\nR : Type u\ninst✝⁴ : Ring R\ninst✝³ : StrongRankCondition R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ range v ≤ ↑⊤",
"tactic": "exact le_top"
}
] | [
712,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
708,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Cardinal.IsStrongLimit.isLimit | [] | [
882,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
881,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean | MultilinearMap.mkContinuousMultilinear_norm_le | [] | [
1133,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1130,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | hasFDerivWithinAt_singleton | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.1399855\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.1399950\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf✝ f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx✝ : E\ns t : Set E\nL L₁ L₂ : Filter E\nf : E → F\nx : E\n⊢ HasFDerivWithinAt f 0 {x} x",
"tactic": "simp only [HasFDerivWithinAt, nhdsWithin_singleton, HasFDerivAtFilter, isLittleO_pure,\n ContinuousLinearMap.zero_apply, sub_self]"
}
] | [
1115,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1112,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | LiouvilleNumber.remainder_lt | [] | [
169,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Topology/Inseparable.lean | SeparationQuotient.continuousWithinAt_lift | [] | [
563,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
560,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | DoubleCentralizer.coe_snd | [] | [
506,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
505,
1
] |
Mathlib/Order/Directed.lean | IsMin.isBot | [] | [
264,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
262,
11
] |
Mathlib/Combinatorics/Additive/Behrend.lean | Behrend.map_le_of_mem_box | [] | [
170,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean | MeasureTheory.SimpleFunc.support_indicator | [] | [
272,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
270,
1
] |
Mathlib/Order/WithBot.lean | WithTop.le_coe_iff | [] | [
861,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
860,
1
] |
Mathlib/Algebra/Order/Positive/Ring.lean | Positive.coe_add | [] | [
38,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
37,
1
] |
Mathlib/Order/SymmDiff.lean | symmDiff_right_inj | [] | [
542,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.zero_apply | [] | [
183,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | HasFDerivAt.mul' | [] | [
312,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
310,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | div_le_iff_of_neg' | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.144376\nα : Type u_1\nβ : Type ?u.144382\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhc : c < 0\n⊢ b / c ≤ a ↔ c * a ≤ b",
"tactic": "rw [mul_comm, div_le_iff_of_neg hc]"
}
] | [
725,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
724,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean | CategoryTheory.Limits.Types.colimit_sound' | [
{
"state_after": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nj'' : J\nf : j ⟶ j''\nf' : j' ⟶ j''\nw : F.map f x = F.map f' x'\n⊢ (F.map f ≫ colimit.ι F j'') x = (F.map f' ≫ colimit.ι F j'') x'",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nj'' : J\nf : j ⟶ j''\nf' : j' ⟶ j''\nw : F.map f x = F.map f' x'\n⊢ colimit.ι F j x = colimit.ι F j' x'",
"tactic": "rw [← colimit.w _ f, ← colimit.w _ f']"
},
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj j' : J\nx : F.obj j\nx' : F.obj j'\nj'' : J\nf : j ⟶ j''\nf' : j' ⟶ j''\nw : F.map f x = F.map f' x'\n⊢ (F.map f ≫ colimit.ι F j'') x = (F.map f' ≫ colimit.ι F j'') x'",
"tactic": "rw [types_comp_apply, types_comp_apply, w]"
}
] | [
352,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.trim_trim | [
{
"state_after": "α : Type u_1\nβ : Type ?u.3498865\nγ : Type ?u.3498868\nδ : Type ?u.3498871\nι : Type ?u.3498874\nR : Type ?u.3498877\nR' : Type ?u.3498880\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nm₁ m₂ : MeasurableSpace α\nhm₁₂ : m₁ ≤ m₂\nhm₂ : m₂ ≤ m0\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(trim (trim μ hm₂) hm₁₂) t = ↑↑(trim μ (_ : m₁ ≤ m0)) t",
"state_before": "α : Type u_1\nβ : Type ?u.3498865\nγ : Type ?u.3498868\nδ : Type ?u.3498871\nι : Type ?u.3498874\nR : Type ?u.3498877\nR' : Type ?u.3498880\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nm₁ m₂ : MeasurableSpace α\nhm₁₂ : m₁ ≤ m₂\nhm₂ : m₂ ≤ m0\n⊢ trim (trim μ hm₂) hm₁₂ = trim μ (_ : m₁ ≤ m0)",
"tactic": "refine @Measure.ext _ m₁ _ _ (fun t ht => ?_)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3498865\nγ : Type ?u.3498868\nδ : Type ?u.3498871\nι : Type ?u.3498874\nR : Type ?u.3498877\nR' : Type ?u.3498880\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nm₁ m₂ : MeasurableSpace α\nhm₁₂ : m₁ ≤ m₂\nhm₂ : m₂ ≤ m0\nt : Set α\nht : MeasurableSet t\n⊢ ↑↑(trim (trim μ hm₂) hm₁₂) t = ↑↑(trim μ (_ : m₁ ≤ m0)) t",
"tactic": "rw [trim_measurableSet_eq hm₁₂ ht, trim_measurableSet_eq (hm₁₂.trans hm₂) ht,\n trim_measurableSet_eq hm₂ (hm₁₂ t ht)]"
}
] | [
4408,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
4404,
1
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | Asymptotics.IsEquivalent.congr_left | [] | [
116,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/Topology/MetricSpace/Baire.lean | dense_sInter_of_open | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : S = ∅\n⊢ Dense (⋂₀ S)\n\ncase inr\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\n⊢ Dense (⋂₀ S)",
"state_before": "α : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\n⊢ Dense (⋂₀ S)",
"tactic": "cases' S.eq_empty_or_nonempty with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : S = ∅\n⊢ Dense (⋂₀ S)",
"tactic": "simp [h]"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\n⊢ Dense (⋂₀ S)",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\n⊢ Dense (⋂₀ S)",
"tactic": "rcases hS.exists_eq_range h with ⟨f, hf⟩"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nF : ∀ (n : ℕ), f n ∈ S\n⊢ Dense (⋂₀ S)",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\n⊢ Dense (⋂₀ S)",
"tactic": "have F : ∀ n, f n ∈ S := fun n => by rw [hf]; exact mem_range_self _"
},
{
"state_after": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nF : ∀ (n : ℕ), f n ∈ S\n⊢ Dense (⋂ (x : ℕ), f x)",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nF : ∀ (n : ℕ), f n ∈ S\n⊢ Dense (⋂₀ S)",
"tactic": "rw [hf, sInter_range]"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nα : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nF : ∀ (n : ℕ), f n ∈ S\n⊢ Dense (⋂ (x : ℕ), f x)",
"tactic": "exact dense_iInter_of_open_nat (fun n => ho _ (F n)) fun n => hd _ (F n)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nn : ℕ\n⊢ f n ∈ range f",
"state_before": "α : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nn : ℕ\n⊢ f n ∈ S",
"tactic": "rw [hf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12812\nγ : Type ?u.12815\nι : Type ?u.12818\ninst✝¹ : TopologicalSpace α\ninst✝ : BaireSpace α\nS : Set (Set α)\nho : ∀ (s : Set α), s ∈ S → IsOpen s\nhS : Set.Countable S\nhd : ∀ (s : Set α), s ∈ S → Dense s\nh : Set.Nonempty S\nf : ℕ → Set α\nhf : S = range f\nn : ℕ\n⊢ f n ∈ range f",
"tactic": "exact mem_range_self _"
}
] | [
206,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean | Pmf.toMeasure_apply_singleton | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.83685\nγ : Type ?u.83688\ninst✝ : MeasurableSpace α\np : Pmf α\ns t : Set α\na : α\nh : MeasurableSet {a}\n⊢ ↑↑(toMeasure p) {a} = ↑p a",
"tactic": "simp [toMeasure_apply_eq_toOuterMeasure_apply _ _ h, toOuterMeasure_apply_singleton]"
}
] | [
265,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
263,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | ContinuousLinearEquiv.uniqueDiffOn_image | [] | [
523,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
521,
1
] |
Mathlib/Topology/Algebra/Module/StrongTopology.lean | ContinuousLinearMap.strongTopology.hasBasis_nhds_zero_of_basis | [
{
"state_after": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"state_before": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"tactic": "letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F"
},
{
"state_after": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"state_before": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis : UniformSpace F := TopologicalAddGroup.toUniformSpace F\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"tactic": "haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform"
},
{
"state_after": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"state_before": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis : UniformAddGroup F\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"tactic": "letI : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖"
},
{
"state_after": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ Filter.HasBasis (Filter.comap (↑(UniformOnFun.ofFun 𝔖) ∘ FunLike.coe) (𝓝 ((↑(UniformOnFun.ofFun 𝔖) ∘ FunLike.coe) 0)))\n (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"state_before": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ Filter.HasBasis (𝓝 0) (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"tactic": "rw [(strongTopology.embedding_coeFn σ F 𝔖).toInducing.nhds_eq_comap]"
},
{
"state_after": "no goals",
"state_before": "𝕜₁ : Type u_4\n𝕜₂ : Type u_5\ninst✝¹³ : NormedField 𝕜₁\ninst✝¹² : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nE' : Type ?u.74932\nF : Type u_1\nF' : Type ?u.74938\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module 𝕜₁ E\ninst✝⁹ : AddCommGroup E'\ninst✝⁸ : Module ℝ E'\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : AddCommGroup F'\ninst✝⁴ : Module ℝ F'\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace E'\ninst✝¹ : TopologicalSpace F\ninst✝ : TopologicalAddGroup F\nι : Type u_2\n𝔖 : Set (Set E)\nh𝔖₁ : Set.Nonempty 𝔖\nh𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖\np : ι → Prop\nb : ι → Set F\nh : Filter.HasBasis (𝓝 0) p b\nthis✝¹ : UniformSpace F := TopologicalAddGroup.toUniformSpace F\nthis✝ : UniformAddGroup F\nthis : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖\n⊢ Filter.HasBasis (Filter.comap (↑(UniformOnFun.ofFun 𝔖) ∘ FunLike.coe) (𝓝 ((↑(UniformOnFun.ofFun 𝔖) ∘ FunLike.coe) 0)))\n (fun Si => Si.fst ∈ 𝔖 ∧ p Si.snd) fun Si => {f | ∀ (x : E), x ∈ Si.fst → ↑f x ∈ b Si.snd}",
"tactic": "exact (UniformOnFun.hasBasis_nhds_zero_of_basis 𝔖 h𝔖₁ h𝔖₂ h).comap FunLike.coe"
}
] | [
167,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.compl_iInf₂ | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.91020\nγ : Type ?u.91023\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : LE α\ns t : LowerSet α\na : α\nf : (i : ι) → κ i → LowerSet α\n⊢ compl (⨅ (i : ι) (j : κ i), f i j) = ⨅ (i : ι) (j : κ i), compl (f i j)",
"tactic": "simp_rw [LowerSet.compl_iInf]"
}
] | [
931,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
930,
1
] |
Mathlib/Topology/Separation.lean | Continuous.limUnder_eq | [] | [
1067,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1065,
1
] |
Mathlib/Data/Finset/NAry.lean | Finset.image₂_subset_iff_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.20278\nβ : Type u_3\nβ' : Type ?u.20284\nγ : Type u_1\nγ' : Type ?u.20290\nδ : Type ?u.20293\nδ' : Type ?u.20296\nε : Type ?u.20299\nε' : Type ?u.20302\nζ : Type ?u.20305\nζ' : Type ?u.20308\nν : Type ?u.20311\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\n⊢ image₂ f s t ⊆ u ↔ ∀ (b : β), b ∈ t → image (fun a => f a b) s ⊆ u",
"tactic": "simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_swap α]"
}
] | [
117,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Probability/Kernel/Basic.lean | ProbabilityTheory.kernel.IsSFiniteKernel.finset_sum | [
{
"state_after": "case empty\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI : Finset ι\nh✝ : ∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)\nh : ∀ (i : ι), i ∈ ∅ → IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel (∑ i in ∅, κs i)\n\ncase insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\n⊢ IsSFiniteKernel (∑ i in insert i I, κs i)",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI : Finset ι\nh : ∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel (∑ i in I, κs i)",
"tactic": "induction' I using Finset.induction with i I hi_nmem_I h_ind h"
},
{
"state_after": "case empty\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI : Finset ι\nh✝ : ∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)\nh : ∀ (i : ι), i ∈ ∅ → IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel 0",
"state_before": "case empty\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI : Finset ι\nh✝ : ∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)\nh : ∀ (i : ι), i ∈ ∅ → IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel (∑ i in ∅, κs i)",
"tactic": "rw [Finset.sum_empty]"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI : Finset ι\nh✝ : ∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)\nh : ∀ (i : ι), i ∈ ∅ → IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel 0",
"tactic": "infer_instance"
},
{
"state_after": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"state_before": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\n⊢ IsSFiniteKernel (∑ i in insert i I, κs i)",
"tactic": "rw [Finset.sum_insert hi_nmem_I]"
},
{
"state_after": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\nthis : IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"state_before": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"tactic": "haveI : IsSFiniteKernel (κs i) := h i (Finset.mem_insert_self _ _)"
},
{
"state_after": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\nthis✝ : IsSFiniteKernel (κs i)\nthis : IsSFiniteKernel (∑ x in I, κs x)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"state_before": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\nthis : IsSFiniteKernel (κs i)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"tactic": "have : IsSFiniteKernel (∑ x : ι in I, κs x) :=\n h_ind fun i hiI => h i (Finset.mem_insert_of_mem hiI)"
},
{
"state_after": "no goals",
"state_before": "case insert\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : { x // x ∈ kernel α β }\nκs : ι → { x // x ∈ kernel α β }\nI✝ : Finset ι\nh✝ : ∀ (i : ι), i ∈ I✝ → IsSFiniteKernel (κs i)\ni : ι\nI : Finset ι\nhi_nmem_I : ¬i ∈ I\nh_ind : (∀ (i : ι), i ∈ I → IsSFiniteKernel (κs i)) → IsSFiniteKernel (∑ i in I, κs i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i I → IsSFiniteKernel (κs i_1)\nthis✝ : IsSFiniteKernel (κs i)\nthis : IsSFiniteKernel (∑ x in I, κs x)\n⊢ IsSFiniteKernel (κs i + ∑ x in I, κs x)",
"tactic": "exact IsSFiniteKernel.add _ _"
}
] | [
315,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
306,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.IsTrail.of_append_right | [
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\nq : Walk G v w\nh : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)\n⊢ IsTrail q",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\nq : Walk G v w\nh : IsTrail (append p q)\n⊢ IsTrail q",
"tactic": "rw [isTrail_def, edges_append, List.nodup_append] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\nq : Walk G v w\nh : List.Nodup (edges p) ∧ List.Nodup (edges q) ∧ List.Disjoint (edges p) (edges q)\n⊢ IsTrail q",
"tactic": "exact ⟨h.2.1⟩"
}
] | [
952,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
949,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean | Isometry.mapsTo_ball | [] | [
257,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | StructureGroupoid.LocalInvariantProp.liftPropOn_mono | [] | [
470,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
468,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean | Antitone.pairwise_disjoint_on_Ioc_succ | [] | [
107,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformly_extend_spec | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : UniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\na : α\n⊢ Tendsto f (comap e (𝓝 a)) (𝓝 (DenseInducing.extend (_ : DenseInducing e) f a))",
"tactic": "simpa only [DenseInducing.extend] using\n tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _)"
}
] | [
463,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
461,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snormEssSup_mono_measure | [
{
"state_after": "α : Type u_1\nE : Type ?u.1987410\nF : Type u_2\nG : Type ?u.1987416\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhμν : ν ≪ μ\n⊢ essSup (fun x => ↑‖f x‖₊) ν ≤ essSup (fun x => ↑‖f x‖₊) μ",
"state_before": "α : Type u_1\nE : Type ?u.1987410\nF : Type u_2\nG : Type ?u.1987416\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhμν : ν ≪ μ\n⊢ snormEssSup f ν ≤ snormEssSup f μ",
"tactic": "simp_rw [snormEssSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1987410\nF : Type u_2\nG : Type ?u.1987416\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nhμν : ν ≪ μ\n⊢ essSup (fun x => ↑‖f x‖₊) ν ≤ essSup (fun x => ↑‖f x‖₊) μ",
"tactic": "exact essSup_mono_measure hμν"
}
] | [
582,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
580,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.union_right_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.147908\nγ : Type ?u.147911\ninst✝ : DecidableEq α\ns✝ s₁ s₂ t✝ t₁ t₂ u✝ v : Finset α\na b : α\ns t u : Finset α\nx : α\n⊢ x ∈ s ∪ t ∪ u ↔ x ∈ s ∪ u ∪ t",
"tactic": "simp only [mem_union, or_assoc, @or_comm (x ∈ t)]"
}
] | [
1428,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1427,
1
] |
Mathlib/Algebra/Ring/Defs.lean | mul_neg | [] | [
298,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
297,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.coe_singletonOneHom | [] | [
159,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean | MulChar.IsQuadratic.pow_odd | [
{
"state_after": "case intro\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nn : ℕ\n⊢ χ ^ (2 * n + 1) = χ",
"state_before": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nn : ℕ\nhn : Odd n\n⊢ χ ^ n = χ",
"tactic": "obtain ⟨n, rfl⟩ := hn"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommRing R'\nR'' : Type w\ninst✝ : CommRing R''\nχ : MulChar R R'\nhχ : IsQuadratic χ\nn : ℕ\n⊢ χ ^ (2 * n + 1) = χ",
"tactic": "rw [pow_add, pow_one, hχ.pow_even (even_two_mul _), one_mul]"
}
] | [
531,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
528,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.lt_eq_subset | [] | [
402,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
401,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.nonnegg_cases_right | [] | [
510,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
507,
1
] |
Mathlib/CategoryTheory/Bicategory/Free.lean | CategoryTheory.FreeBicategory.mk_vcomp | [] | [
246,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean | contDiffAt_inner | [] | [
68,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
67,
1
] |
Mathlib/Topology/Bases.lean | TopologicalSpace.isBasis_countableBasis | [] | [
639,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
637,
1
] |
Mathlib/Analysis/Normed/MulAction.lean | nndist_smul₀ | [] | [
119,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
1
] |
Mathlib/Tactic/Sat/FromLRAT.lean | Sat.Literal.reify_neg | [] | [
213,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.iInf_eq_generate | [] | [
582,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
581,
1
] |
Mathlib/Order/GaloisConnection.lean | GaloisCoinsertion.u_iSup_of_lu_eq_self | [] | [
825,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
823,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | ContinuousOn.stronglyMeasurableAtFilter_nhdsWithin | [] | [
608,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
603,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.trim | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ HasFiniteIntegral f",
"state_before": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ Integrable f",
"tactic": "refine' ⟨hf.aestronglyMeasurable, _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ') < ⊤\n\nα : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ Measurable fun a => ↑‖f a‖₊",
"state_before": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ HasFiniteIntegral f",
"tactic": "rw [HasFiniteIntegral, lintegral_trim hm _]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ (∫⁻ (a : α), ↑‖f a‖₊ ∂μ') < ⊤",
"tactic": "exact hf_int.2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1258403\nγ : Type ?u.1258406\nδ : Type ?u.1258409\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β\ninst✝¹ : NormedAddCommGroup γ\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nm0 : MeasurableSpace α\nμ' : Measure α\nf : α → H\nhm : m ≤ m0\nhf_int : Integrable f\nhf : StronglyMeasurable f\n⊢ Measurable fun a => ↑‖f a‖₊",
"tactic": "exact @StronglyMeasurable.ennnorm _ m _ _ f hf"
}
] | [
1166,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1161,
1
] |
Mathlib/Data/Nat/Digits.lean | Nat.digits_inj_iff | [] | [
330,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean | Matrix.det_eq_of_forall_row_eq_smul_add_const_aux | [
{
"state_after": "case empty\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\n⊢ det A = det B",
"state_before": "case empty\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\n⊢ ∀ (c : n → R),\n (∀ (i : n), ¬i ∈ ∅ → c i = 0) → ∀ (k : n), ¬k ∈ ∅ → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B",
"tactic": "rintro c hs k - A_eq"
},
{
"state_after": "case empty\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\n⊢ det A = det B",
"state_before": "case empty\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\n⊢ det A = det B",
"tactic": "have : ∀ i, c i = 0 := by\n intro i\n specialize hs i\n contrapose! hs\n simp [hs]"
},
{
"state_after": "case empty.e_M\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\n⊢ A = B",
"state_before": "case empty\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\n⊢ det A = det B",
"tactic": "congr"
},
{
"state_after": "case empty.e_M.a.h\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\ni j : n\n⊢ A i j = B i j",
"state_before": "case empty.e_M\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\n⊢ A = B",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case empty.e_M.a.h\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nthis : ∀ (i : n), c i = 0\ni j : n\n⊢ A i j = B i j",
"tactic": "rw [A_eq, this, MulZeroClass.zero_mul, add_zero]"
},
{
"state_after": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\n⊢ c i = 0",
"state_before": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\n⊢ ∀ (i : n), c i = 0",
"tactic": "intro i"
},
{
"state_after": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : ¬i ∈ ∅ → c i = 0\n⊢ c i = 0",
"state_before": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nhs : ∀ (i : n), ¬i ∈ ∅ → c i = 0\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\n⊢ c i = 0",
"tactic": "specialize hs i"
},
{
"state_after": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : c i ≠ 0\n⊢ ¬i ∈ ∅ ∧ c i ≠ 0",
"state_before": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : ¬i ∈ ∅ → c i = 0\n⊢ c i = 0",
"tactic": "contrapose! hs"
},
{
"state_after": "no goals",
"state_before": "m : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA B : Matrix n n R\nc : n → R\nk : n\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\ni : n\nhs : c i ≠ 0\n⊢ ¬i ∈ ∅ ∧ c i ≠ 0",
"tactic": "simp [hs]"
},
{
"state_after": "case insert\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\n⊢ det A = det B",
"state_before": "case insert\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\n⊢ ∀ (c : n → R),\n (∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0) →\n ∀ (k : n), ¬k ∈ insert i s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B",
"tactic": "intro c hs k hk A_eq"
},
{
"state_after": "case insert\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ det A = det B",
"state_before": "case insert\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\n⊢ det A = det B",
"tactic": "have hAi : A i = B i + c i • B k := funext (A_eq i)"
},
{
"state_after": "case insert.hij\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ i ≠ k\n\ncase insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ ∀ (i_1 : n), ¬i_1 ∈ s → update c i 0 i_1 = 0\n\ncase insert.k\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ n\n\ncase insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ ¬?insert.k ∈ s\n\ncase insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ ∀ (i_1 j : n), A i_1 j = updateRow B i (A i) i_1 j + update c i 0 i_1 * updateRow B i (A i) ?insert.k j",
"state_before": "case insert\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ det A = det B",
"tactic": "rw [@ih (updateRow B i (A i)) (Function.update c i 0), hAi, det_updateRow_add_smul_self]"
},
{
"state_after": "no goals",
"state_before": "case insert.hij\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ i ≠ k",
"tactic": "exact mt (fun h => show k ∈ insert i s from h ▸ Finset.mem_insert_self _ _) hk"
},
{
"state_after": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\n⊢ update c i 0 i' = 0",
"state_before": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\n⊢ ∀ (i_1 : n), ¬i_1 ∈ s → update c i 0 i_1 = 0",
"tactic": "intro i' hi'"
},
{
"state_after": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\n⊢ (if i' = i then 0 else c i') = 0",
"state_before": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\n⊢ update c i 0 i' = 0",
"tactic": "rw [Function.update_apply]"
},
{
"state_after": "case insert.x.inl\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\nhi'i : i' = i\n⊢ 0 = 0\n\ncase insert.x.inr\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\nhi'i : ¬i' = i\n⊢ c i' = 0",
"state_before": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\n⊢ (if i' = i then 0 else c i') = 0",
"tactic": "split_ifs with hi'i"
},
{
"state_after": "no goals",
"state_before": "case insert.x.inl\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' : n\nhi' : ¬i' ∈ s\nhi'i : i' = i\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
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"tactic": "exact hs i' fun h => hi' ((Finset.mem_insert.mp h).resolve_left hi'i)"
},
{
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"tactic": "exact k"
},
{
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"tactic": "exact fun h => hk (Finset.mem_insert_of_mem h)"
},
{
"state_after": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' j' : n\n⊢ A i' j' = updateRow B i (A i) i' j' + update c i 0 i' * updateRow B i (A i) k j'",
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{
"state_after": "case insert.x\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' j' : n\n⊢ A i' j' = (if i' = i then A i j' else B i' j') + (if i' = i then 0 else c i') * updateRow B i (A i) k j'",
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"tactic": "rw [updateRow_apply, Function.update_apply]"
},
{
"state_after": "case insert.x.inl\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' j' : n\nhi'i : i' = i\n⊢ A i' j' = A i j' + 0 * updateRow B i (A i) k j'\n\ncase insert.x.inr\nm : Type ?u.1702450\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\ni : n\ns : Finset n\n_hi : ¬i ∈ s\nih :\n ∀ {B : Matrix n n R} (c : n → R),\n (∀ (i : n), ¬i ∈ s → c i = 0) → ∀ (k : n), ¬k ∈ s → (∀ (i j : n), A i j = B i j + c i * B k j) → det A = det B\nB : Matrix n n R\nc : n → R\nhs : ∀ (i_1 : n), ¬i_1 ∈ insert i s → c i_1 = 0\nk : n\nhk : ¬k ∈ insert i s\nA_eq : ∀ (i j : n), A i j = B i j + c i * B k j\nhAi : A i = B i + c i • B k\ni' j' : n\nhi'i : ¬i' = i\n⊢ A i' j' = B i' j' + c i' * updateRow B i (A i) k j'",
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{
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"tactic": "rw [A_eq, updateRow_ne fun h : k = i => hk <| h ▸ Finset.mem_insert_self k s]"
},
{
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] | [
502,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
472,
1
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Mathlib/Data/Matrix/Basic.lean | Matrix.mul_mul_right | [] | [
1290,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1288,
1
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Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | NonUnitalSubsemiring.mem_closure_iff | [] | [
662,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
660,
1
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Mathlib/Topology/Algebra/InfiniteSum/Order.lean | HasSum.nonpos | [] | [
133,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
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Mathlib/Data/Real/Basic.lean | Real.add_neg_lt_sSup | [] | [
776,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
774,
1
] |
Mathlib/Analysis/InnerProductSpace/PiL2.lean | OrthonormalBasis.orthonormal | [
{
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"tactic": "classical\n rw [orthonormal_iff_ite]\n intro i j\n rw [← b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j,\n EuclideanSpace.inner_single_left, EuclideanSpace.single_apply, map_one, one_mul]"
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{
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},
{
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},
{
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"tactic": "rw [← b.repr.inner_map_map (b i) (b j), b.repr_self i, b.repr_self j,\n EuclideanSpace.inner_single_left, EuclideanSpace.single_apply, map_one, one_mul]"
}
] | [
409,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
11
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Mathlib/RingTheory/Localization/AsSubring.lean | Localization.mapToFractionRing_apply | [] | [
49,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
46,
1
] |
Mathlib/Data/Nat/Factorial/Basic.lean | Nat.succ_descFactorial_succ | [
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ descFactorial (n + 1) (0 + 1) = (n + 1) * descFactorial n 0",
"tactic": "rw [descFactorial_zero, descFactorial_one, mul_one]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\n⊢ descFactorial (n + 1) (succ k + 1) = (n + 1) * descFactorial n (succ k)",
"tactic": "rw [descFactorial_succ, succ_descFactorial_succ _ k, descFactorial_succ, succ_sub_succ,\n mul_left_comm]"
}
] | [
381,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
376,
1
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Mathlib/Topology/OmegaCompletePartialOrder.lean | Scott.isOpen_sUnion | [
{
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"tactic": "simp only [IsOpen] at hs⊢"
},
{
"state_after": "case h.e'_5.h.a\nα : Type u\ninst✝ : OmegaCompletePartialOrder α\ns : Set (Set α)\nhs : ∀ (t : Set α), t ∈ s → Continuous' fun x => x ∈ t\nx✝ : α\n⊢ x✝ ∈ ⋃₀ s ↔ sSup (setOf ⁻¹' s) x✝",
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"tactic": "convert CompleteLattice.sSup_continuous' (setOf ⁻¹' s) hs"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.a\nα : Type u\ninst✝ : OmegaCompletePartialOrder α\ns : Set (Set α)\nhs : ∀ (t : Set α), t ∈ s → Continuous' fun x => x ∈ t\nx✝ : α\n⊢ x✝ ∈ ⋃₀ s ↔ sSup (setOf ⁻¹' s) x✝",
"tactic": "simp only [sSup_apply, setOf_bijective.surjective.exists, exists_prop, mem_preimage,\n SetCoe.exists, iSup_Prop_eq, mem_setOf_eq, mem_sUnion]"
}
] | [
68,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
64,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean | CategoryTheory.Limits.PullbackCone.unop_inr | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nJ : Type u₂\ninst✝ : Category J\nX✝ : Type v₂\nX Y Z : Cᵒᵖ\nf : X ⟶ Z\ng : Y ⟶ Z\nc : PullbackCone f g\n⊢ PushoutCocone.inr (unop c) = (snd c).unop",
"tactic": "aesop_cat"
}
] | [
529,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
528,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | MeasureTheory.locallyIntegrableOn_of_locallyIntegrable_restrict | [
{
"state_after": "X : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"state_before": "X : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\n⊢ LocallyIntegrableOn f s",
"tactic": "intro x _"
},
{
"state_after": "case intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"state_before": "X : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"tactic": "obtain ⟨t, ht_mem, ht_int⟩ := hf x"
},
{
"state_after": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u ⊆ t\nhu_o : IsOpen u\nhu_mem : x ∈ u\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"state_before": "case intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"tactic": "obtain ⟨u, hu_sub, hu_o, hu_mem⟩ := mem_nhds_iff.mp ht_mem"
},
{
"state_after": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u ⊆ t\nhu_o : IsOpen u\nhu_mem : x ∈ u\n⊢ IntegrableOn f (s ∩ u)",
"state_before": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u ⊆ t\nhu_o : IsOpen u\nhu_mem : x ∈ u\n⊢ IntegrableAtFilter f (𝓝[s] x)",
"tactic": "refine' ⟨_, inter_mem_nhdsWithin s (hu_o.mem_nhds hu_mem), _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nX : Type u_1\nY : Type ?u.122795\nE : Type u_2\nR : Type ?u.122801\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nhf : LocallyIntegrable f\nx : X\na✝ : x ∈ s\nt : Set X\nht_mem : t ∈ 𝓝 x\nht_int : IntegrableOn f t\nu : Set X\nhu_sub : u ⊆ t\nhu_o : IsOpen u\nhu_mem : x ∈ u\n⊢ IntegrableOn f (s ∩ u)",
"tactic": "simpa only [IntegrableOn, Measure.restrict_restrict hu_o.measurableSet, inter_comm] using\n ht_int.mono_set hu_sub"
}
] | [
150,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.chain_adj_support | [] | [
648,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
645,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.coe_add_eq_ite | [
{
"state_after": "case zero\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\na b : ZMod Nat.zero\n⊢ ↑(a + b) = ↑(if ↑Nat.zero ≤ ↑a + ↑b then a + b - ↑Nat.zero else a + b)\n\ncase succ\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\n⊢ ↑(a + b) = ↑(if ↑(Nat.succ n✝) ≤ ↑a + ↑b then a + b - ↑(Nat.succ n✝) else a + b)",
"state_before": "n✝ : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn : ℕ\na b : ZMod n\n⊢ ↑(a + b) = ↑(if ↑n ≤ ↑a + ↑b then a + b - ↑n else a + b)",
"tactic": "cases n"
},
{
"state_after": "case succ\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\n⊢ ↑(a + b) = ↑(if ↑(Nat.succ n✝) ≤ ↑a + ↑b then a + b - ↑(Nat.succ n✝) else a + b)",
"state_before": "case succ\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\n⊢ ↑(a + b) = ↑(if ↑(Nat.succ n✝) ≤ ↑a + ↑b then a + b - ↑(Nat.succ n✝) else a + b)",
"tactic": "simp only [Fin.val_add_eq_ite, ← Int.ofNat_add, ← Int.ofNat_succ, Int.ofNat_le]"
},
{
"state_after": "case succ.inl\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b - ↑(Nat.succ n✝))\n\ncase succ.inr\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ¬↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b)",
"state_before": "case succ\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\n⊢ ↑(a + b) = ↑(if ↑(Nat.succ n✝) ≤ ↑a + ↑b then a + b - ↑(Nat.succ n✝) else a + b)",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case zero\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\na b : ZMod Nat.zero\n⊢ ↑(a + b) = ↑(if ↑Nat.zero ≤ ↑a + ↑b then a + b - ↑Nat.zero else a + b)",
"tactic": "simp"
},
{
"state_after": "case succ.inl\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b - ↑(Nat.succ n✝))",
"state_before": "case succ.inl\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b - ↑(Nat.succ n✝))",
"tactic": "norm_cast"
},
{
"state_after": "case succ.inl.e_a\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ a + b = a + b - ↑(Nat.succ n✝)",
"state_before": "case succ.inl\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b - ↑(Nat.succ n✝))",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case succ.inl.e_a\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ a + b = a + b - ↑(Nat.succ n✝)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ.inr\nn : ℕ\nR : Type ?u.34539\ninst✝ : Ring R\nn✝ : ℕ\na b : ZMod (Nat.succ n✝)\nh : ¬↑(Nat.succ n✝) ≤ ↑a + ↑b\n⊢ ↑(a + b) = ↑(a + b)",
"tactic": "rfl"
}
] | [
277,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
] |
Mathlib/CategoryTheory/Idempotents/Karoubi.lean | CategoryTheory.Idempotents.Karoubi.id_eq | [
{
"state_after": "no goals",
"state_before": "C : Type ?u.15232\ninst✝ : Category C\nP : Karoubi C\n⊢ P.p = P.p ≫ P.p ≫ P.p",
"tactic": "repeat' rw [P.idem]"
},
{
"state_after": "no goals",
"state_before": "C : Type ?u.15232\ninst✝ : Category C\nP : Karoubi C\n⊢ P.p = P.p ≫ P.p",
"tactic": "rw [P.idem]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nP : Karoubi C\n⊢ 𝟙 P = Hom.mk P.p",
"tactic": "rfl"
}
] | [
128,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Data/Nat/Interval.lean | Nat.card_Iic | [
{
"state_after": "no goals",
"state_before": "a b c : ℕ\n⊢ card (Iic b) = b + 1",
"tactic": "rw [Iic_eq_Icc, card_Icc, bot_eq_zero, tsub_zero]"
}
] | [
117,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/CategoryTheory/PathCategory.lean | CategoryTheory.Prefunctor.mapPath_comp' | [] | [
149,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/LinearAlgebra/Matrix/DotProduct.lean | Matrix.dotProduct_eq_zero_iff | [] | [
73,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | isNilpotent_of_product_of_sylow_group | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\n⊢ Group.IsNilpotent G",
"tactic": "classical\n let ps := (Fintype.card G).factorization.support\n have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by\n intro p P\n haveI : Fact (Nat.Prime ↑p) := Fact.mk (Nat.prime_of_mem_factorization (Finset.coe_mem p))\n exact P.isPGroup'.isNilpotent\n exact nilpotent_of_mulEquiv e"
},
{
"state_after": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\n⊢ Group.IsNilpotent G",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\n⊢ Group.IsNilpotent G",
"tactic": "let ps := (Fintype.card G).factorization.support"
},
{
"state_after": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\nthis : ∀ (p : { x // x ∈ ps }) (P : Sylow (↑p) G), Group.IsNilpotent { x // x ∈ ↑P }\n⊢ Group.IsNilpotent G",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\n⊢ Group.IsNilpotent G",
"tactic": "have : ∀ (p : ps) (P : Sylow p G), IsNilpotent (↑P : Subgroup G) := by\n intro p P\n haveI : Fact (Nat.Prime ↑p) := Fact.mk (Nat.prime_of_mem_factorization (Finset.coe_mem p))\n exact P.isPGroup'.isNilpotent"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\nthis : ∀ (p : { x // x ∈ ps }) (P : Sylow (↑p) G), Group.IsNilpotent { x // x ∈ ↑P }\n⊢ Group.IsNilpotent G",
"tactic": "exact nilpotent_of_mulEquiv e"
},
{
"state_after": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\np : { x // x ∈ ps }\nP : Sylow (↑p) G\n⊢ Group.IsNilpotent { x // x ∈ ↑P }",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\n⊢ ∀ (p : { x // x ∈ ps }) (P : Sylow (↑p) G), Group.IsNilpotent { x // x ∈ ↑P }",
"tactic": "intro p P"
},
{
"state_after": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\np : { x // x ∈ ps }\nP : Sylow (↑p) G\nthis : Fact (Nat.Prime ↑p)\n⊢ Group.IsNilpotent { x // x ∈ ↑P }",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\np : { x // x ∈ ps }\nP : Sylow (↑p) G\n⊢ Group.IsNilpotent { x // x ∈ ↑P }",
"tactic": "haveI : Fact (Nat.Prime ↑p) := Fact.mk (Nat.prime_of_mem_factorization (Finset.coe_mem p))"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nhG : Group G\ninst✝ : Fintype G\ne : ((p : { x // x ∈ (Nat.factorization (card G)).support }) → (P : Sylow (↑p) G) → { x // x ∈ ↑P }) ≃* G\nps : Finset ℕ := (Nat.factorization (card G)).support\np : { x // x ∈ ps }\nP : Sylow (↑p) G\nthis : Fact (Nat.Prime ↑p)\n⊢ Group.IsNilpotent { x // x ∈ ↑P }",
"tactic": "exact P.isPGroup'.isNilpotent"
}
] | [
892,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
883,
1
] |
Mathlib/Topology/Spectral/Hom.lean | IsSpectralMap.continuous | [] | [
53,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Init/Algebra/Order.lean | gt_of_gt_of_ge | [] | [
134,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Algebra/Order/Module.lean | inv_smul_lt_iff_of_neg | [
{
"state_after": "no goals",
"state_before": "k : Type u_1\nM : Type u_2\nN : Type ?u.96377\ninst✝³ : LinearOrderedField k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : c < 0\n⊢ c⁻¹ • a < b ↔ c • b < a",
"tactic": "rw [← smul_lt_smul_iff_of_neg h, smul_inv_smul₀ h.ne]"
}
] | [
171,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
170,
1
] |
Mathlib/Topology/Maps.lean | openEmbedding_id | [] | [
610,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
609,
1
] |
Mathlib/Algebra/Lie/Submodule.lean | LieHom.idealRange_eq_lieSpan_range | [] | [
907,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
906,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.inducedOuterMeasure_caratheodory | [
{
"state_after": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) ↔\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ MeasurableSet s ↔\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t",
"tactic": "rw [isCaratheodory_iff_le]"
},
{
"state_after": "case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) →\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t\n\ncase mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) →\n ∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) ↔\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nt : Set α\n_ht : P t\n⊢ ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t",
"state_before": "case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) →\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t",
"tactic": "intro h t _ht"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nt : Set α\n_ht : P t\n⊢ ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t",
"tactic": "exact h t"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) u",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\n⊢ (∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) t) →\n ∀ (t : Set α),\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t",
"tactic": "intro h u"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤\n ⨅ (t : Set α) (ht : P t) (_ : u ⊆ t), m t ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) u",
"tactic": "conv_rhs => rw [inducedOuterMeasure_eq_iInf _ msU m_mono]"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ∀ (i : Set α),\n ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (ht : P i) (_ : u ⊆ i), m i ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤\n ⨅ (t : Set α) (ht : P t) (_ : u ⊆ t), m t ht",
"tactic": "refine' le_iInf _"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (ht : P t) (_ : u ⊆ t), m t ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu : Set α\n⊢ ∀ (i : Set α),\n ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (ht : P i) (_ : u ⊆ i), m i ht",
"tactic": "intro t"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\n⊢ ∀ (i : P t), ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (_ : u ⊆ t), m t i",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (ht : P t) (_ : u ⊆ t), m t ht",
"tactic": "refine' le_iInf _"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (_ : u ⊆ t), m t ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\n⊢ ∀ (i : P t), ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (_ : u ⊆ t), m t i",
"tactic": "intro ht"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\n⊢ u ⊆ t → ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ m t ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ ⨅ (_ : u ⊆ t), m t ht",
"tactic": "refine' le_iInf _"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\nh2t : u ⊆ t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ m t ht",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\n⊢ u ⊆ t → ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ m t ht",
"tactic": "intro h2t"
},
{
"state_after": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\nh2t : u ⊆ t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s)",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\nh2t : u ⊆ t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤ m t ht",
"tactic": "refine' le_trans _ (le_trans (h t ht) <| le_of_eq <| inducedOuterMeasure_eq' _ msU m_mono ht)"
},
{
"state_after": "no goals",
"state_before": "case mpr\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns : Set α\nh :\n ∀ (t : Set α),\n P t →\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s) ≤ ↑(inducedOuterMeasure m P0 m0) t\nu t : Set α\nht : P t\nh2t : u ⊆ t\n⊢ ↑(inducedOuterMeasure m P0 m0) (u ∩ s) + ↑(inducedOuterMeasure m P0 m0) (u \\ s) ≤\n ↑(inducedOuterMeasure m P0 m0) (t ∩ s) + ↑(inducedOuterMeasure m P0 m0) (t \\ s)",
"tactic": "refine'\n add_le_add (mono' _ <| Set.inter_subset_inter_left _ h2t)\n (mono' _ <| diff_subset_diff_left h2t)"
}
] | [
1541,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1520,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.copy_eq | [] | [
159,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
158,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.restr_univ | [] | [
726,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
725,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.stalkToFiberRingHom_germ' | [] | [
548,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/RingTheory/Localization/Basic.lean | IsLocalization.map_eq_zero_iff | [
{
"state_after": "case mp\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ ↑(algebraMap R S) r = 0 → ∃ m, ↑m * r = 0\n\ncase mpr\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ (∃ m, ↑m * r = 0) → ↑(algebraMap R S) r = 0",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ ↑(algebraMap R S) r = 0 ↔ ∃ m, ↑m * r = 0",
"tactic": "constructor"
},
{
"state_after": "case mp\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nh : ↑(algebraMap R S) r = 0\n⊢ ∃ m, ↑m * r = 0\n\ncase mpr\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ (∃ m, ↑m * r = 0) → ↑(algebraMap R S) r = 0",
"state_before": "case mp\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ ↑(algebraMap R S) r = 0 → ∃ m, ↑m * r = 0\n\ncase mpr\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ (∃ m, ↑m * r = 0) → ↑(algebraMap R S) r = 0",
"tactic": "intro h"
},
{
"state_after": "case mp.intro\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nh : ↑(algebraMap R S) r = 0\nm : { x // x ∈ M }\nhm : ↑m * 0 = ↑m * r\n⊢ ∃ m, ↑m * r = 0",
"state_before": "case mp\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nh : ↑(algebraMap R S) r = 0\n⊢ ∃ m, ↑m * r = 0",
"tactic": "obtain ⟨m, hm⟩ := (IsLocalization.eq_iff_exists M S).mp ((algebraMap R S).map_zero.trans h.symm)"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nh : ↑(algebraMap R S) r = 0\nm : { x // x ∈ M }\nhm : ↑m * 0 = ↑m * r\n⊢ ∃ m, ↑m * r = 0",
"tactic": "exact ⟨m, by simpa using hm.symm⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nh : ↑(algebraMap R S) r = 0\nm : { x // x ∈ M }\nhm : ↑m * 0 = ↑m * r\n⊢ ↑m * r = 0",
"tactic": "simpa using hm.symm"
},
{
"state_after": "case mpr.intro\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nm : { x // x ∈ M }\nhm : ↑m * r = 0\n⊢ ↑(algebraMap R S) r = 0",
"state_before": "case mpr\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\n⊢ (∃ m, ↑m * r = 0) → ↑(algebraMap R S) r = 0",
"tactic": "rintro ⟨m, hm⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nR : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.395998\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nr : R\nm : { x // x ∈ M }\nhm : ↑m * r = 0\n⊢ ↑(algebraMap R S) r = 0",
"tactic": "rw [← (IsLocalization.map_units S m).mul_right_inj, mul_zero, ← RingHom.map_mul, hm,\n RingHom.map_zero]"
}
] | [
236,
24
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229,
1
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Mathlib/Logic/Equiv/Set.lean | Equiv.Set.insert_symm_apply_inl | [] | [
290,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
1
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Mathlib/RingTheory/Ideal/Operations.lean | RingHom.liftOfRightInverse_comp_apply | [] | [
2280,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2277,
1
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Mathlib/Algebra/Order/Field/Power.lean | Odd.zpow_neg_iff | [
{
"state_after": "case intro\nα : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn k : ℤ\nhk : n = 2 * k + 1\n⊢ a ^ n < 0 ↔ a < 0",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhn : Odd n\n⊢ a ^ n < 0 ↔ a < 0",
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},
{
"state_after": "no goals",
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"tactic": "simpa only [hk, two_mul] using zpow_bit1_neg_iff"
}
] | [
190,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
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Mathlib/Data/Quot.lean | Quotient.out_eq | [] | [
376,
16
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375,
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Mathlib/Data/Set/Intervals/Instances.lean | Set.Ico.coe_nonneg | [] | [
212,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
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Mathlib/Algebra/Order/Sub/Canonical.lean | tsub_add_min | [
{
"state_after": "case h\nα : Type u_1\ninst✝² : CanonicallyLinearOrderedAddMonoid α\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\n⊢ min a b ≤ a",
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"tactic": "rw [← tsub_min, @tsub_add_cancel_of_le]"
},
{
"state_after": "no goals",
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}
] | [
512,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
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Mathlib/Analysis/Calculus/FDeriv/Basic.lean | HasFDerivAtFilter.congr_of_eventuallyEq | [] | [
844,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
842,
1
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Mathlib/Data/Set/Function.lean | Set.MapsTo.subset_preimage | [] | [
395,
5
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
393,
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Mathlib/Data/Fin/Basic.lean | Fin.eq_succ_of_ne_zero | [] | [
436,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
435,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.range_subtype | [
{
"state_after": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ x ∈ Hom.range (Embedding.toHom (subtype S)) ↔ x ∈ S",
"state_before": "L : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\n⊢ Hom.range (Embedding.toHom (subtype S)) = S",
"tactic": "ext x"
},
{
"state_after": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ (∃ y, ↑y = x) ↔ x ∈ S",
"state_before": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ x ∈ Hom.range (Embedding.toHom (subtype S)) ↔ x ∈ S",
"tactic": "simp only [Hom.mem_range, Embedding.coe_toHom, coeSubtype]"
},
{
"state_after": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ (∃ y, ↑y = x) → x ∈ S",
"state_before": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ (∃ y, ↑y = x) ↔ x ∈ S",
"tactic": "refine' ⟨_, fun h => ⟨⟨x, h⟩, rfl⟩⟩"
},
{
"state_after": "case h.intro.mk\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\ny : M\nhy : y ∈ S\n⊢ ↑{ val := y, property := hy } ∈ S",
"state_before": "case h\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\nx : M\n⊢ (∃ y, ↑y = x) → x ∈ S",
"tactic": "rintro ⟨⟨y, hy⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.intro.mk\nL : Language\nM : Type w\nN : Type ?u.1144938\nP : Type ?u.1144941\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nS✝ S : Substructure L M\ny : M\nhy : y ∈ S\n⊢ ↑{ val := y, property := hy } ∈ S",
"tactic": "exact hy"
}
] | [
1016,
11
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1011,
1
] |
Mathlib/Order/BooleanAlgebra.lean | compl_eq_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.54315\nw x y z : α\ninst✝ : BooleanAlgebra α\n⊢ xᶜ = y ↔ yᶜ = x",
"tactic": "rw [eq_comm, compl_eq_iff_isCompl, eq_compl_iff_isCompl]"
}
] | [
628,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
627,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean | CategoryTheory.CartesianClosed.curry_eq_iff | [] | [
215,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
] |
Mathlib/Logic/Equiv/Basic.lean | Equiv.subtypeQuotientEquivQuotientSubtype_symm_mk | [] | [
1503,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1499,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.VectorMeasure.coe_neg | [] | [
351,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean | Ordinal.IsNormal.cof_le | [
{
"state_after": "case inl\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\n⊢ cof 0 ≤ cof (f 0)\n\ncase inr.inl.intro\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\nb : Ordinal\n⊢ cof (succ b) ≤ cof (f (succ b))\n\ncase inr.inr\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\na : Ordinal\nha : IsLimit a\n⊢ cof a ≤ cof (f a)",
"state_before": "α : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\na : Ordinal\n⊢ cof a ≤ cof (f a)",
"tactic": "rcases zero_or_succ_or_limit a with (rfl | ⟨b, rfl⟩ | ha)"
},
{
"state_after": "case inl\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\n⊢ 0 ≤ cof (f 0)",
"state_before": "case inl\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\n⊢ cof 0 ≤ cof (f 0)",
"tactic": "rw [cof_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\n⊢ 0 ≤ cof (f 0)",
"tactic": "exact zero_le _"
},
{
"state_after": "case inr.inl.intro\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\nb : Ordinal\n⊢ 0 < f (succ b)",
"state_before": "case inr.inl.intro\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\nb : Ordinal\n⊢ cof (succ b) ≤ cof (f (succ b))",
"tactic": "rw [cof_succ, Cardinal.one_le_iff_ne_zero, cof_ne_zero, ← Ordinal.pos_iff_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.intro\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\nb : Ordinal\n⊢ 0 < f (succ b)",
"tactic": "exact (Ordinal.zero_le (f b)).trans_lt (hf.1 b)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type ?u.99458\nr : α → α → Prop\nf : Ordinal → Ordinal\nhf : IsNormal f\na : Ordinal\nha : IsLimit a\n⊢ cof a ≤ cof (f a)",
"tactic": "rw [hf.cof_eq ha]"
}
] | [
695,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
689,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean | AffineBasis.surjective_coord | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\n⊢ Function.Surjective ↑(coord b i)",
"tactic": "classical\n intro x\n obtain ⟨j, hij⟩ := exists_ne i\n let s : Finset ι := {i, j}\n have hi : i ∈ s := by simp\n have _ : j ∈ s := by simp\n let w : ι → k := fun j' => if j' = i then x else 1 - x\n have hw : s.sum w = 1 := by\n simp [Finset.sum_ite, Finset.filter_insert, hij]\n erw [Finset.filter_eq']\n simp [hij.symm]\n use s.affineCombination k b w\n simp [b.coord_apply_combination_of_mem hi hw]"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\n⊢ Function.Surjective ↑(coord b i)",
"tactic": "intro x"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "obtain ⟨j, hij⟩ := exists_ne i"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "let s : Finset ι := {i, j}"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "have hi : i ∈ s := by simp"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "have _ : j ∈ s := by simp"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "let w : ι → k := fun j' => if j' = i then x else 1 - x"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n⊢ ∃ a, ↑(coord b i) a = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "have hw : s.sum w = 1 := by\n simp [Finset.sum_ite, Finset.filter_insert, hij]\n erw [Finset.filter_eq']\n simp [hij.symm]"
},
{
"state_after": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n⊢ ↑(coord b i) (↑(Finset.affineCombination k s ↑b) w) = x",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n⊢ ∃ a, ↑(coord b i) a = x",
"tactic": "use s.affineCombination k b w"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\nhw : Finset.sum s w = 1\n⊢ ↑(coord b i) (↑(Finset.affineCombination k s ↑b) w) = x",
"tactic": "simp [b.coord_apply_combination_of_mem hi hw]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\n⊢ i ∈ s",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\n⊢ j ∈ s",
"tactic": "simp"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ↑(Finset.card (insert i (Finset.filter (fun x => x = i) {j}))) * x + (1 - x) = 1",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ Finset.sum s w = 1",
"tactic": "simp [Finset.sum_ite, Finset.filter_insert, hij]"
},
{
"state_after": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ↑(Finset.card (insert i (if i ∈ {j} then {i} else ∅))) * x + (1 - x) = 1",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ↑(Finset.card (insert i (Finset.filter (fun x => x = i) {j}))) * x + (1 - x) = 1",
"tactic": "erw [Finset.filter_eq']"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type ?u.141682\nk : Type u_3\nV : Type u_4\nP : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ns✝ : Finset ι\ni✝ j✝ : ι\ne : ι ≃ ι'\ninst✝ : Nontrivial ι\ni : ι\nx : k\nj : ι\nhij : j ≠ i\ns : Finset ι := {i, j}\nhi : i ∈ s\nx✝ : j ∈ s\nw : ι → k := fun j' => if j' = i then x else 1 - x\n⊢ ↑(Finset.card (insert i (if i ∈ {j} then {i} else ∅))) * x + (1 - x) = 1",
"tactic": "simp [hij.symm]"
}
] | [
273,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Algebra/Support.lean | Function.mulSupport_min | [] | [
180,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.sup_eq_union | [] | [
1311,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1310,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.biproduct.hom_ext' | [] | [
472,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
470,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.frequently_const | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.163070\nι : Sort x\nf : Filter α\ninst✝ : NeBot f\np : Prop\nh : p\n⊢ (∃ᶠ (x : α) in f, p) ↔ p",
"tactic": "simpa [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.163070\nι : Sort x\nf : Filter α\ninst✝ : NeBot f\np : Prop\nh : ¬p\n⊢ (∃ᶠ (x : α) in f, p) ↔ p",
"tactic": "simp [h]"
}
] | [
1336,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1335,
1
] |
Mathlib/Algebra/CharP/Basic.lean | CharP.intCast_eq_intCast | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na b : ℤ\n⊢ ↑a = ↑b ↔ a ≡ b [ZMOD ↑p]",
"tactic": "rw [eq_comm, ← sub_eq_zero, ← Int.cast_sub, CharP.int_cast_eq_zero_iff R p, Int.modEq_iff_dvd]"
}
] | [
146,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Algebra/Order/Archimedean.lean | eq_of_forall_rat_lt_iff_lt | [] | [
298,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
] |
Mathlib/CategoryTheory/Endofunctor/Algebra.lean | CategoryTheory.Endofunctor.Algebra.mono_of_mono | [] | [
172,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
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