file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Int/LeastGreatest.lean
|
Int.exists_greatest_of_bdd
|
[
{
"state_after": "P : ℤ → Prop\ninst✝ : DecidablePred P\nHbdd : ∃ b, ∀ (z : ℤ), P z → z ≤ b\nHinh : ∃ z, P z\nb : ℤ\nHb : ∀ (z : ℤ), P z → z ≤ b\n⊢ ∃ ub, P ub ∧ ∀ (z : ℤ), P z → z ≤ ub",
"state_before": "P : ℤ → Prop\ninst✝ : DecidablePred P\nHbdd : ∃ b, ∀ (z : ℤ), P z → z ≤ b\nHinh : ∃ z, P z\n⊢ ∃ ub, P ub ∧ ∀ (z : ℤ), P z → z ≤ ub",
"tactic": "let ⟨ b , Hb ⟩ := Hbdd"
},
{
"state_after": "P : ℤ → Prop\ninst✝ : DecidablePred P\nHbdd : ∃ b, ∀ (z : ℤ), P z → z ≤ b\nHinh : ∃ z, P z\nb : ℤ\nHb : ∀ (z : ℤ), P z → z ≤ b\nlb : ℤ\nH : P lb ∧ ∀ (z : ℤ), P z → z ≤ lb\n⊢ ∃ ub, P ub ∧ ∀ (z : ℤ), P z → z ≤ ub",
"state_before": "P : ℤ → Prop\ninst✝ : DecidablePred P\nHbdd : ∃ b, ∀ (z : ℤ), P z → z ≤ b\nHinh : ∃ z, P z\nb : ℤ\nHb : ∀ (z : ℤ), P z → z ≤ b\n⊢ ∃ ub, P ub ∧ ∀ (z : ℤ), P z → z ≤ ub",
"tactic": "let ⟨ lb , H ⟩ := greatestOfBdd b Hb Hinh"
},
{
"state_after": "no goals",
"state_before": "P : ℤ → Prop\ninst✝ : DecidablePred P\nHbdd : ∃ b, ∀ (z : ℤ), P z → z ≤ b\nHinh : ∃ z, P z\nb : ℤ\nHb : ∀ (z : ℤ), P z → z ≤ b\nlb : ℤ\nH : P lb ∧ ∀ (z : ℤ), P z → z ≤ lb\n⊢ ∃ ub, P ub ∧ ∀ (z : ℤ), P z → z ≤ ub",
"tactic": "exact ⟨ lb , H ⟩"
}
] |
[
106,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean
|
aeSeq.iSup
|
[
{
"state_after": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ ↑↑μ {a | ¬(⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a} = 0",
"state_before": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ (⨆ (n : ι), aeSeq hf p n) =ᵐ[μ] ⨆ (n : ι), f n",
"tactic": "simp_rw [Filter.EventuallyEq, ae_iff, iSup_apply]"
},
{
"state_after": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nh_ss : aeSeqSet hf p ⊆ {a | (⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a}\n⊢ ↑↑μ {a | ¬(⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a} = 0",
"state_before": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ ↑↑μ {a | ¬(⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a} = 0",
"tactic": "have h_ss : aeSeqSet hf p ⊆ { a : α | (⨆ i : ι, aeSeq hf p i a) = ⨆ i : ι, f i a } := by\n intro x hx\n congr\n exact funext fun i => aeSeq_eq_fun_of_mem_aeSeqSet hf hx i"
},
{
"state_after": "no goals",
"state_before": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nh_ss : aeSeqSet hf p ⊆ {a | (⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a}\n⊢ ↑↑μ {a | ¬(⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a} = 0",
"tactic": "exact measure_mono_null (Set.compl_subset_compl.mpr h_ss) (measure_compl_aeSeqSet_eq_zero hf hp)"
},
{
"state_after": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ x ∈ {a | (⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a}",
"state_before": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\n⊢ aeSeqSet hf p ⊆ {a | (⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a}",
"tactic": "intro x hx"
},
{
"state_after": "case e_s\nι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ (fun i => aeSeq hf p i x) = fun i => f i x",
"state_before": "ι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ x ∈ {a | (⨆ (i : ι), aeSeq hf p i a) = ⨆ (i : ι), f i a}",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_s\nι : Sort u_2\nα : Type u_3\nβ : Type u_1\nγ : Type ?u.727906\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nf : ι → α → β\nμ : MeasureTheory.Measure α\np : α → (ι → β) → Prop\ninst✝¹ : CompleteLattice β\ninst✝ : Countable ι\nhf : ∀ (i : ι), AEMeasurable (f i)\nhp : ∀ᵐ (x : α) ∂μ, p x fun n => f n x\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ (fun i => aeSeq hf p i x) = fun i => f i x",
"tactic": "exact funext fun i => aeSeq_eq_fun_of_mem_aeSeqSet hf hx i"
}
] |
[
140,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioo_subset_Ioo
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.16340\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nha : a₂ ≤ a₁\nhb : b₁ ≤ b₂\n⊢ Ioo a₁ b₁ ⊆ Ioo a₂ b₂",
"tactic": "simpa [← coe_subset] using Set.Ioo_subset_Ioo ha hb"
}
] |
[
175,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
linearIndependent_le_basis
|
[
{
"state_after": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\n⊢ (#κ) ≤ (#ι)\n\ncase inr\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Infinite ι\n⊢ (#κ) ≤ (#ι)",
"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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\n⊢ (#κ) ≤ (#ι)",
"tactic": "cases fintypeOrInfinite ι"
},
{
"state_after": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\n⊢ (#κ) ≤ ↑(Fintype.card ι)",
"state_before": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\n⊢ (#κ) ≤ (#ι)",
"tactic": "rw [Cardinal.mk_fintype ι]"
},
{
"state_after": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\nthis : Nontrivial R\n⊢ (#κ) ≤ ↑(Fintype.card ι)",
"state_before": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\n⊢ (#κ) ≤ ↑(Fintype.card ι)",
"tactic": "haveI : Nontrivial R := nontrivial_of_invariantBasisNumber R"
},
{
"state_after": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\nthis : Nontrivial R\n⊢ (#κ) ≤ ↑(Fintype.card ↑(range ↑b))",
"state_before": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\nthis : Nontrivial R\n⊢ (#κ) ≤ ↑(Fintype.card ι)",
"tactic": "rw [Fintype.card_congr (Equiv.ofInjective b b.injective)]"
},
{
"state_after": "no goals",
"state_before": "case inl\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Fintype ι\nthis : Nontrivial R\n⊢ (#κ) ≤ ↑(Fintype.card ↑(range ↑b))",
"tactic": "exact linearIndependent_le_span v i (range b) b.span_eq"
},
{
"state_after": "no goals",
"state_before": "case inr\nK : 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.369966\nR : Type u\ninst✝³ : Ring R\ninst✝² : StrongRankCondition R\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_1\nb : Basis ι R M\nκ : Type u_1\nv : κ → M\ni : LinearIndependent R v\nval✝ : Infinite ι\n⊢ (#κ) ≤ (#ι)",
"tactic": "exact linearIndependent_le_infinite_basis b v i"
}
] |
[
754,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.inverse_asEquivalence
|
[
{
"state_after": "case mk'\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nfunctor✝ : C ⥤ D\ninverse✝ : D ⥤ C\nunitIso✝ : 𝟭 C ≅ functor✝ ⋙ inverse✝\ncounitIso✝ : inverse✝ ⋙ functor✝ ≅ 𝟭 D\nfunctor_unitIso_comp✝ :\n ∀ (X : C), functor✝.map (unitIso✝.hom.app X) ≫ counitIso✝.hom.app (functor✝.obj X) = 𝟙 (functor✝.obj X)\n⊢ asEquivalence (mk' functor✝ inverse✝ unitIso✝ counitIso✝).inverse = symm (mk' functor✝ inverse✝ unitIso✝ counitIso✝)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nE : C ≌ D\n⊢ asEquivalence E.inverse = symm E",
"tactic": "cases E"
},
{
"state_after": "no goals",
"state_before": "case mk'\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nfunctor✝ : C ⥤ D\ninverse✝ : D ⥤ C\nunitIso✝ : 𝟭 C ≅ functor✝ ⋙ inverse✝\ncounitIso✝ : inverse✝ ⋙ functor✝ ≅ 𝟭 D\nfunctor_unitIso_comp✝ :\n ∀ (X : C), functor✝.map (unitIso✝.hom.app X) ≫ counitIso✝.hom.app (functor✝.obj X) = 𝟙 (functor✝.obj X)\n⊢ asEquivalence (mk' functor✝ inverse✝ unitIso✝ counitIso✝).inverse = symm (mk' functor✝ inverse✝ unitIso✝ counitIso✝)",
"tactic": "congr"
}
] |
[
594,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Topology/Order.lean
|
continuous_bot
|
[] |
[
810,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.mrange_mk'
|
[] |
[
881,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.iSup_le
|
[] |
[
883,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
881,
11
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.church_rosser
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ a b✝ c b : List (α × Bool)\nhab hac : Step a b\n⊢ ReflGen Step b b",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ a b✝ c b : List (α × Bool)\nhab hac : Step a b\n⊢ ReflTransGen Step b b",
"tactic": "rfl"
}
] |
[
244,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
SemilinearIsometryClass.antilipschitz
|
[] |
[
106,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
11
] |
Mathlib/Analysis/Normed/Group/InfiniteSum.lean
|
summable_of_norm_bounded_eventually
|
[] |
[
167,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cosh_three_mul
|
[
{
"state_after": "x y : ℝ\n⊢ ↑(cosh (3 * x)) = ↑(4 * cosh x ^ 3 - 3 * cosh x)",
"state_before": "x y : ℝ\n⊢ cosh (3 * x) = 4 * cosh x ^ 3 - 3 * cosh x",
"tactic": "rw [← ofReal_inj]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(cosh (3 * x)) = ↑(4 * cosh x ^ 3 - 3 * cosh x)",
"tactic": "simp [cosh_three_mul]"
}
] |
[
1453,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1452,
8
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_empty
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1633030\nα : Type u_1\nβ : Type ?u.1633036\nf g : Perm α\ns t : Set α\na b x y : α\n⊢ IsCycleOn f ∅",
"tactic": "simp [IsCycleOn, Set.bijOn_empty]"
}
] |
[
759,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Analysis/ODE/Gronwall.lean
|
gronwallBound_of_K_ne_0
|
[] |
[
58,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean
|
Real.pow_arith_mean_le_arith_mean_pow_of_even
|
[] |
[
72,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.nodupKeys_join
|
[
{
"state_after": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ (∀ (l : List (Sigma β)), l ∈ L → Pairwise (fun s s' => s.fst ≠ s'.fst) l) ∧\n Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L ↔\n (∀ (l : List (Sigma β)), l ∈ L → NodupKeys l) ∧ Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ NodupKeys (join L) ↔ (∀ (l : List (Sigma β)), l ∈ L → NodupKeys l) ∧ Pairwise Disjoint (map keys L)",
"tactic": "rw [nodupKeys_iff_pairwise, pairwise_join, pairwise_map]"
},
{
"state_after": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L ↔\n Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ (∀ (l : List (Sigma β)), l ∈ L → Pairwise (fun s s' => s.fst ≠ s'.fst) l) ∧\n Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L ↔\n (∀ (l : List (Sigma β)), l ∈ L → NodupKeys l) ∧ Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"tactic": "refine' and_congr (ball_congr fun l _ => by simp [nodupKeys_iff_pairwise]) _"
},
{
"state_after": "case a\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L =\n Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L ↔\n Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"tactic": "apply iff_of_eq"
},
{
"state_after": "case a.e_R.h.h.a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\nL : List (List (Sigma β))\nl₁ l₂ : List (Sigma β)\n⊢ (∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) ↔ Disjoint (keys l₁) (keys l₂)",
"state_before": "case a\nα : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\n⊢ Pairwise (fun l₁ l₂ => ∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) L =\n Pairwise (fun a b => Disjoint (keys a) (keys b)) L",
"tactic": "congr with (l₁ l₂)"
},
{
"state_after": "no goals",
"state_before": "case a.e_R.h.h.a\nα : Type u\nβ : α → Type v\nl l₁✝ l₂✝ : List (Sigma β)\nL : List (List (Sigma β))\nl₁ l₂ : List (Sigma β)\n⊢ (∀ (x : Sigma β), x ∈ l₁ → ∀ (y : Sigma β), y ∈ l₂ → x.fst ≠ y.fst) ↔ Disjoint (keys l₁) (keys l₂)",
"tactic": "simp [keys, disjoint_iff_ne]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\nL : List (List (Sigma β))\nl : List (Sigma β)\nx✝ : l ∈ L\n⊢ Pairwise (fun s s' => s.fst ≠ s'.fst) l ↔ NodupKeys l",
"tactic": "simp [nodupKeys_iff_pairwise]"
}
] |
[
153,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
nndist_pi_const
|
[] |
[
2035,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2033,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.div_mul_cancel
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.53366\nG : Type ?u.53369\nα : Type u_1\nM : Type ?u.53375\nN : Type ?u.53378\ninst✝ : DivisionMonoid α\na✝ b c : α\nh : IsUnit b\na : α\n⊢ a / b * b = a",
"tactic": "rw [div_eq_mul_inv, h.inv_mul_cancel_right]"
}
] |
[
368,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
11
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.of_isZero
|
[
{
"state_after": "ctx : Context\np : Poly\nh :\n (match p with\n | [] => true\n | x => false) =\n true\n⊢ denote ctx p = 0",
"state_before": "ctx : Context\np : Poly\nh : isZero p = true\n⊢ denote ctx p = 0",
"tactic": "simp [isZero] at h"
},
{
"state_after": "case h_1\nctx : Context\np✝ : Poly\nh : true = true\n⊢ denote ctx [] = 0\n\ncase h_2\nctx : Context\np p✝ : Poly\nx✝ : p = [] → False\nh : false = true\n⊢ denote ctx p = 0",
"state_before": "ctx : Context\np : Poly\nh :\n (match p with\n | [] => true\n | x => false) =\n true\n⊢ denote ctx p = 0",
"tactic": "split at h"
},
{
"state_after": "no goals",
"state_before": "case h_1\nctx : Context\np✝ : Poly\nh : true = true\n⊢ denote ctx [] = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h_2\nctx : Context\np p✝ : Poly\nx✝ : p = [] → False\nh : false = true\n⊢ denote ctx p = 0",
"tactic": "contradiction"
}
] |
[
628,
18
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
624,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_normalizer_eq_of_bijective
|
[] |
[
3259,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3257,
1
] |
Mathlib/Data/Multiset/Fold.lean
|
Multiset.fold_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8358\nop : α → α → α\nhc : IsCommutative α op\nha : IsAssociative α op\nb₁ b₂ : α\ns₁ s₂ : Multiset α\n⊢ fold op (op b₁ b₂) (s₁ + 0) = op (fold op b₁ s₁) (fold op b₂ 0)",
"tactic": "rw [add_zero, fold_zero, ← fold_cons'_right, ← fold_cons_right op]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8358\nop : α → α → α\nhc : IsCommutative α op\nha : IsAssociative α op\nb₁ b₂ : α\ns₁ s₂ : Multiset α\na : α\nb : Multiset α\nh : fold op (op b₁ b₂) (s₁ + b) = op (fold op b₁ s₁) (fold op b₂ b)\n⊢ fold op (op b₁ b₂) (s₁ + a ::ₘ b) = op (fold op b₁ s₁) (fold op b₂ (a ::ₘ b))",
"tactic": "rw [fold_cons_left, add_cons, fold_cons_left, h, ← ha.assoc, hc.comm a,\nha.assoc]"
}
] |
[
83,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_isPrime_of_equiv
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_3\nF : Type ?u.1964564\ninst✝³ : Ring R\ninst✝² : Ring S\nrc : RingHomClass F R S\nF' : Type u_1\ninst✝¹ : RingEquivClass F' R S\nf : F'\nI : Ideal R\ninst✝ : IsPrime I\n⊢ RingHom.ker f ≤ I",
"tactic": "simp only [RingHom.ker_equiv, bot_le]"
}
] |
[
2155,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2153,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivAt.cos
|
[] |
[
804,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
802,
1
] |
Mathlib/Topology/Connected.lean
|
IsPreconnected.eqOn_const_of_mapsTo
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\nι : Type ?u.150118\nπ : ι → Type ?u.150123\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\nT : Set β\ninst✝ : DiscreteTopology ↑T\nf : α → β\nhne : Set.Nonempty T\nhS : IsPreconnected ∅\nhc : ContinuousOn f ∅\nhTm : MapsTo f ∅ T\n⊢ ∃ y, y ∈ T ∧ EqOn f (const α y) ∅\n\ncase inr.intro\nα : Type u\nβ : Type v\nι : Type ?u.150118\nπ : ι → Type ?u.150123\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\nS : Set α\nhS : IsPreconnected S\nT : Set β\ninst✝ : DiscreteTopology ↑T\nf : α → β\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nhne : Set.Nonempty T\nx : α\nhx : x ∈ S\n⊢ ∃ y, y ∈ T ∧ EqOn f (const α y) S",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.150118\nπ : ι → Type ?u.150123\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\nS : Set α\nhS : IsPreconnected S\nT : Set β\ninst✝ : DiscreteTopology ↑T\nf : α → β\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nhne : Set.Nonempty T\n⊢ ∃ y, y ∈ T ∧ EqOn f (const α y) S",
"tactic": "rcases S.eq_empty_or_nonempty with (rfl | ⟨x, hx⟩)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nι : Type ?u.150118\nπ : ι → Type ?u.150123\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\nT : Set β\ninst✝ : DiscreteTopology ↑T\nf : α → β\nhne : Set.Nonempty T\nhS : IsPreconnected ∅\nhc : ContinuousOn f ∅\nhTm : MapsTo f ∅ T\n⊢ ∃ y, y ∈ T ∧ EqOn f (const α y) ∅",
"tactic": "exact hne.imp fun _ hy => ⟨hy, eqOn_empty _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nα : Type u\nβ : Type v\nι : Type ?u.150118\nπ : ι → Type ?u.150123\ninst✝² : TopologicalSpace α\ns t u v : Set α\ninst✝¹ : TopologicalSpace β\nS : Set α\nhS : IsPreconnected S\nT : Set β\ninst✝ : DiscreteTopology ↑T\nf : α → β\nhc : ContinuousOn f S\nhTm : MapsTo f S T\nhne : Set.Nonempty T\nx : α\nhx : x ∈ S\n⊢ ∃ y, y ∈ T ∧ EqOn f (const α y) S",
"tactic": "exact ⟨f x, hTm hx, fun x' hx' => hS.constant_of_mapsTo hc hTm hx' hx⟩"
}
] |
[
1623,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1618,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_lt_add_iff_right
|
[] |
[
786,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
11
] |
Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean
|
LiouvilleNumber.summable
|
[] |
[
84,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
11
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.mul_left_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\n⊢ q * p = 0 ↔ q = 0",
"tactic": "by_cases hq : q = 0 <;> simp [h.mul_left_ne_zero, hq]"
}
] |
[
471,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
inv_norm_smul_mem_closed_unit_ball
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.9079\nβ : Type u_1\nγ : Type ?u.9085\nι : Type ?u.9088\ninst✝² : NormedField α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nx : β\n⊢ ‖x‖⁻¹ • x ∈ closedBall 0 1",
"tactic": "simp only [mem_closedBall_zero_iff, norm_smul, norm_inv, norm_norm, ← _root_.div_eq_inv_mul,\n div_self_le_one]"
}
] |
[
80,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.coe_toInfHom
|
[] |
[
1052,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1051,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain.addMonoidHom_comp
|
[] |
[
526,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.mem_comap
|
[] |
[
308,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.Continuous.of_bundled
|
[] |
[
266,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Topology/Instances/Matrix.lean
|
HasSum.matrix_blockDiag'
|
[] |
[
449,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Std/Data/List/Lemmas.lean
|
List.disjoint_of_disjoint_append_right_right
|
[] |
[
1405,
32
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1404,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.ext
|
[
{
"state_after": "R : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type v₁\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\nB : Type v₂\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\nS : Type ?u.721832\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nC : Type v₃\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ng h : A ⊗[R] B →ₐ[R] C\nH : ∀ (a : A) (b : B), ↑g (a ⊗ₜ[R] b) = ↑h (a ⊗ₜ[R] b)\n⊢ AlgHom.toLinearMap g = AlgHom.toLinearMap h",
"state_before": "R : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type v₁\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\nB : Type v₂\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\nS : Type ?u.721832\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nC : Type v₃\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ng h : A ⊗[R] B →ₐ[R] C\nH : ∀ (a : A) (b : B), ↑g (a ⊗ₜ[R] b) = ↑h (a ⊗ₜ[R] b)\n⊢ g = h",
"tactic": "apply @AlgHom.toLinearMap_injective R (A ⊗[R] B) C _ _ _ _ _ _ _ _"
},
{
"state_after": "case a.h.h\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type v₁\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\nB : Type v₂\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\nS : Type ?u.721832\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nC : Type v₃\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ng h : A ⊗[R] B →ₐ[R] C\nH : ∀ (a : A) (b : B), ↑g (a ⊗ₜ[R] b) = ↑h (a ⊗ₜ[R] b)\nx✝¹ : A\nx✝ : B\n⊢ ↑(↑(AlgebraTensorModule.curry (AlgHom.toLinearMap g)) x✝¹) x✝ =\n ↑(↑(AlgebraTensorModule.curry (AlgHom.toLinearMap h)) x✝¹) x✝",
"state_before": "R : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type v₁\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\nB : Type v₂\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\nS : Type ?u.721832\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nC : Type v₃\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ng h : A ⊗[R] B →ₐ[R] C\nH : ∀ (a : A) (b : B), ↑g (a ⊗ₜ[R] b) = ↑h (a ⊗ₜ[R] b)\n⊢ AlgHom.toLinearMap g = AlgHom.toLinearMap h",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h.h\nR : Type u\ninst✝¹⁰ : CommSemiring R\nA : Type v₁\ninst✝⁹ : Semiring A\ninst✝⁸ : Algebra R A\nB : Type v₂\ninst✝⁷ : Semiring B\ninst✝⁶ : Algebra R B\nS : Type ?u.721832\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\nC : Type v₃\ninst✝¹ : Semiring C\ninst✝ : Algebra R C\ng h : A ⊗[R] B →ₐ[R] C\nH : ∀ (a : A) (b : B), ↑g (a ⊗ₜ[R] b) = ↑h (a ⊗ₜ[R] b)\nx✝¹ : A\nx✝ : B\n⊢ ↑(↑(AlgebraTensorModule.curry (AlgHom.toLinearMap g)) x✝¹) x✝ =\n ↑(↑(AlgebraTensorModule.curry (AlgHom.toLinearMap h)) x✝¹) x✝",
"tactic": "simp [H]"
}
] |
[
535,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_lt_sub_left_of_lt_add
|
[
{
"state_after": "a b c : Int\nh✝ : c < a + b\nh : -a < -c + b\n⊢ -a < b - c",
"state_before": "a b c : Int\nh : c < a + b\n⊢ -a < b - c",
"tactic": "have h := Int.lt_neg_add_of_add_lt (Int.sub_left_lt_of_lt_add h)"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : c < a + b\nh : -a < -c + b\n⊢ -a < b - c",
"tactic": "rwa [Int.add_comm] at h"
}
] |
[
1108,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1106,
11
] |
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
BoxIntegral.IntegrationParams.toFilterDistortioniUnion_neBot
|
[] |
[
542,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.JordanDecomposition.real_smul_nonneg
|
[] |
[
144,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.BisimO.imp
|
[] |
[
479,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.48941\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.48953\ninst✝³ : Bot α\ninst✝² : Bot β\ninst✝¹ : Bot γ\ninst✝ : Bot δ\ng : BotHom β γ\nf₁ f₂ : BotHom α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← BotHom.comp_apply, h, BotHom.comp_apply]"
}
] |
[
484,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Data/Real/Irrational.lean
|
one_lt_natDegree_of_irrational_root
|
[
{
"state_after": "x : ℝ\np : ℤ[X]\nhx : Irrational x\np_nonzero : p ≠ 0\nx_is_root : ↑(aeval x) p = 0\nrid : ¬1 < natDegree p\n⊢ False",
"state_before": "x : ℝ\np : ℤ[X]\nhx : Irrational x\np_nonzero : p ≠ 0\nx_is_root : ↑(aeval x) p = 0\n⊢ 1 < natDegree p",
"tactic": "by_contra rid"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nrid : ¬1 < natDegree (↑C a * X + ↑C b)\n⊢ False",
"state_before": "x : ℝ\np : ℤ[X]\nhx : Irrational x\np_nonzero : p ≠ 0\nx_is_root : ↑(aeval x) p = 0\nrid : ¬1 < natDegree p\n⊢ False",
"tactic": "rcases exists_eq_X_add_C_of_natDegree_le_one (not_lt.1 rid) with ⟨a, b, rfl⟩"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\n⊢ False",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nrid : ¬1 < natDegree (↑C a * X + ↑C b)\n⊢ False",
"tactic": "clear rid"
},
{
"state_after": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : ↑a * x = -↑b\n⊢ False",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\n⊢ False",
"tactic": "have : (a : ℝ) * x = -b := by simpa [eq_neg_iff_add_eq_zero] using x_is_root"
},
{
"state_after": "case intro.intro.inl\nx : ℝ\nhx : Irrational x\nb : ℤ\np_nonzero : ↑C 0 * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C 0 * X + ↑C b) = 0\nthis : ↑0 * x = -↑b\n⊢ False\n\ncase intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : ↑a * x = -↑b\nha : ¬a = 0\n⊢ False",
"state_before": "case intro.intro\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : ↑a * x = -↑b\n⊢ False",
"tactic": "rcases em (a = 0) with (rfl | ha)"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\n⊢ ↑a * x = -↑b",
"tactic": "simpa [eq_neg_iff_add_eq_zero] using x_is_root"
},
{
"state_after": "case intro.intro.inl\nx : ℝ\nhx : Irrational x\np_nonzero : ↑C 0 * X + ↑C 0 ≠ 0\nx_is_root : ↑(aeval x) (↑C 0 * X + ↑C 0) = 0\nthis : ↑0 * x = -↑0\n⊢ False",
"state_before": "case intro.intro.inl\nx : ℝ\nhx : Irrational x\nb : ℤ\np_nonzero : ↑C 0 * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C 0 * X + ↑C b) = 0\nthis : ↑0 * x = -↑b\n⊢ False",
"tactic": "obtain rfl : b = 0 := by simpa"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl\nx : ℝ\nhx : Irrational x\np_nonzero : ↑C 0 * X + ↑C 0 ≠ 0\nx_is_root : ↑(aeval x) (↑C 0 * X + ↑C 0) = 0\nthis : ↑0 * x = -↑0\n⊢ False",
"tactic": "simp at p_nonzero"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : Irrational x\nb : ℤ\np_nonzero : ↑C 0 * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C 0 * X + ↑C b) = 0\nthis : ↑0 * x = -↑b\n⊢ b = 0",
"tactic": "simpa"
},
{
"state_after": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : -↑b / ↑a = x\nha : ¬a = 0\n⊢ False\n\ncase intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"state_before": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : ↑a * x = -↑b\nha : ¬a = 0\n⊢ False",
"tactic": "rw [mul_comm, ← eq_div_iff_mul_eq, eq_comm] at this"
},
{
"state_after": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : -↑b / ↑a = x\nha : ¬a = 0\n⊢ ↑(-↑b / ↑a) = x\n\ncase intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"state_before": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : -↑b / ↑a = x\nha : ¬a = 0\n⊢ False\n\ncase intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"tactic": "refine' hx ⟨-b / a, _⟩"
},
{
"state_after": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"state_before": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : -↑b / ↑a = x\nha : ¬a = 0\n⊢ ↑(-↑b / ↑a) = x\n\ncase intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"tactic": "assumption_mod_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr\nx : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : ↑C a * X + ↑C b ≠ 0\nx_is_root : ↑(aeval x) (↑C a * X + ↑C b) = 0\nthis : x * ↑a = -↑b\nha : ¬a = 0\n⊢ ↑a ≠ 0",
"tactic": "assumption_mod_cast"
}
] |
[
515,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.closedEmbedding_comap_of_surjective
|
[] |
[
755,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Antitone.continuousWithinAt_Ioi_iff_rightLim_eq
|
[] |
[
364,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.map_eq_zero
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : DivisionRing R\np q : R[X]\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nf : R →+* S\n⊢ (∀ (n : ℕ), coeff (map f p) n = coeff 0 n) ↔ ∀ (n : ℕ), coeff p n = coeff 0 n",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : DivisionRing R\np q : R[X]\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nf : R →+* S\n⊢ map f p = 0 ↔ p = 0",
"tactic": "simp only [Polynomial.ext_iff]"
},
{
"state_after": "case a.h.a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : DivisionRing R\np q : R[X]\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nf : R →+* S\na✝ : ℕ\n⊢ coeff (map f p) a✝ = coeff 0 a✝ ↔ coeff p a✝ = coeff 0 a✝",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : DivisionRing R\np q : R[X]\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nf : R →+* S\n⊢ (∀ (n : ℕ), coeff (map f p) n = coeff 0 n) ↔ ∀ (n : ℕ), coeff p n = coeff 0 n",
"tactic": "congr!"
},
{
"state_after": "no goals",
"state_before": "case a.h.a\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝² : DivisionRing R\np q : R[X]\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nf : R →+* S\na✝ : ℕ\n⊢ coeff (map f p) a✝ = coeff 0 a✝ ↔ coeff p a✝ = coeff 0 a✝",
"tactic": "simp [map_eq_zero, coeff_map, coeff_zero]"
}
] |
[
142,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
|
MeasureTheory.Memℒp.exists_hasCompactSupport_integral_rpow_sub_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "have I : 0 < ε ^ (1 / p) := Real.rpow_pos_of_pos hε _"
},
{
"state_after": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "have A : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0 := by\n simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]"
},
{
"state_after": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "have B : ENNReal.ofReal p ≠ 0 := by simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\nhg : snorm (f - g) (↑(Real.toNNReal p)) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "rcases hf.exists_hasCompactSupport_snorm_sub_le ENNReal.coe_ne_top A with\n ⟨g, g_support, hg, g_cont, g_mem⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : snorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\nhg : snorm (f - g) (↑(Real.toNNReal p)) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "change snorm _ (ENNReal.ofReal p) _ ≤ _ at hg"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : snorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : snorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ ∃ g, HasCompactSupport g ∧ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε ∧ Continuous g ∧ Memℒp g (ENNReal.ofReal p)",
"tactic": "refine' ⟨g, g_support, _, g_cont, g_mem⟩"
},
{
"state_after": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : (∫ (a : α), ‖(f - g) a‖ ^ p ∂μ) ^ p⁻¹ ≤ ε ^ p⁻¹\n⊢ 0 ≤ ∫ (a : α), ‖(f - g) a‖ ^ p ∂μ",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : snorm (f - g) (ENNReal.ofReal p) μ ≤ ENNReal.ofReal (ε ^ (1 / p))\n⊢ (∫ (x : α), ‖f x - g x‖ ^ p ∂μ) ≤ ε",
"tactic": "rwa [(hf.sub g_mem).snorm_eq_integral_rpow_norm B ENNReal.coe_ne_top,\n ENNReal.ofReal_le_ofReal_iff I.le, one_div, ENNReal.toReal_ofReal hp.le,\n Real.rpow_le_rpow_iff _ hε.le (inv_pos.2 hp)] at hg"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\nB : ENNReal.ofReal p ≠ 0\ng : α → E\ng_support : HasCompactSupport g\ng_cont : Continuous g\ng_mem : Memℒp g ↑(Real.toNNReal p)\nhg : (∫ (a : α), ‖(f - g) a‖ ^ p ∂μ) ^ p⁻¹ ≤ ε ^ p⁻¹\n⊢ 0 ≤ ∫ (a : α), ‖(f - g) a‖ ^ p ∂μ",
"tactic": "exact integral_nonneg fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\n⊢ ENNReal.ofReal (ε ^ (1 / p)) ≠ 0",
"tactic": "simp only [Ne.def, ENNReal.ofReal_eq_zero, not_le, I]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : BorelSpace α\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\np✝ : ℝ≥0∞\ninst✝² : NormedSpace ℝ E\ninst✝¹ : LocallyCompactSpace α\ninst✝ : Measure.Regular μ\np : ℝ\nhp : 0 < p\nf : α → E\nhf : Memℒp f (ENNReal.ofReal p)\nε : ℝ\nhε : 0 < ε\nI : 0 < ε ^ (1 / p)\nA : ENNReal.ofReal (ε ^ (1 / p)) ≠ 0\n⊢ ENNReal.ofReal p ≠ 0",
"tactic": "simpa only [Ne.def, ENNReal.ofReal_eq_zero, not_le] using hp"
}
] |
[
214,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
contDiff_iff_iteratedDeriv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.124644\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\n⊢ ContDiff 𝕜 n f ↔\n (∀ (m : ℕ), ↑m ≤ n → Continuous (iteratedDeriv m f)) ∧ ∀ (m : ℕ), ↑m < n → Differentiable 𝕜 (iteratedDeriv m f)",
"tactic": "simp only [contDiff_iff_continuous_differentiable, iteratedFDeriv_eq_equiv_comp,\n LinearIsometryEquiv.comp_continuous_iff, LinearIsometryEquiv.comp_differentiable_iff]"
}
] |
[
265,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.comp_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.11378\nγ : Type ?u.11381\nδ : Type ?u.11384\nr✝ : α → β → Prop\np : β → γ → Prop\nq : γ → δ → Prop\nr : α → Prop → Prop\nthis : (fun x x_1 => x ↔ x_1) = fun x x_1 => x = x_1\n⊢ (r ∘r fun x x_1 => x ↔ x_1) = r",
"tactic": "rw [this, comp_eq]"
},
{
"state_after": "case h.h\nα : Type u_1\nβ : Type ?u.11378\nγ : Type ?u.11381\nδ : Type ?u.11384\nr✝ : α → β → Prop\np : β → γ → Prop\nq : γ → δ → Prop\nr : α → Prop → Prop\na b : Prop\n⊢ (a ↔ b) = (a = b)",
"state_before": "α : Type u_1\nβ : Type ?u.11378\nγ : Type ?u.11381\nδ : Type ?u.11384\nr✝ : α → β → Prop\np : β → γ → Prop\nq : γ → δ → Prop\nr : α → Prop → Prop\n⊢ (fun x x_1 => x ↔ x_1) = fun x x_1 => x = x_1",
"tactic": "funext a b"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\nβ : Type ?u.11378\nγ : Type ?u.11381\nδ : Type ?u.11384\nr✝ : α → β → Prop\np : β → γ → Prop\nq : γ → δ → Prop\nr : α → Prop → Prop\na b : Prop\n⊢ (a ↔ b) = (a = b)",
"tactic": "exact iff_eq_eq"
}
] |
[
156,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.contDiffAt_log
|
[] |
[
89,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.isBounded_principal
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5475\nγ : Type ?u.5478\nι : Type ?u.5481\nr : α → α → Prop\nf g : Filter α\ns : Set α\n⊢ IsBounded r (𝓟 s) ↔ ∃ t, ∀ (x : α), x ∈ s → r x t",
"tactic": "simp [IsBounded, subset_def]"
}
] |
[
87,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/Part.lean
|
Part.some_mod_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.73852\nγ : Type ?u.73855\ninst✝ : Mod α\na b : α\n⊢ some a % some b = some (a % b)",
"tactic": "simp [mod_def]"
}
] |
[
800,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/GroupTheory/GroupAction/ConjAct.lean
|
ConjAct.ofConjAct_one
|
[] |
[
119,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
LipschitzWith.compLp_zero
|
[
{
"state_after": "α : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\n⊢ ↑↑(compLp hg g0 0) =ᵐ[μ] 0",
"state_before": "α : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\n⊢ compLp hg g0 0 = 0",
"tactic": "rw [Lp.eq_zero_iff_ae_eq_zero]"
},
{
"state_after": "α : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\n⊢ g ∘ ↑↑0 =ᵐ[μ] 0",
"state_before": "α : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\n⊢ ↑↑(compLp hg g0 0) =ᵐ[μ] 0",
"tactic": "apply (coeFn_compLp _ _ _).trans"
},
{
"state_after": "case h\nα : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\na✝ : α\nha : ↑↑0 a✝ = OfNat.ofNat 0 a✝\n⊢ (g ∘ ↑↑0) a✝ = OfNat.ofNat 0 a✝",
"state_before": "α : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\n⊢ g ∘ ↑↑0 =ᵐ[μ] 0",
"tactic": "filter_upwards [Lp.coeFn_zero E p μ] with _ ha"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.2123310\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\na✝ : α\nha : ↑↑0 a✝ = OfNat.ofNat 0 a✝\n⊢ (g ∘ ↑↑0) a✝ = OfNat.ofNat 0 a✝",
"tactic": "simp only [ha, g0, Function.comp_apply, Pi.zero_apply]"
}
] |
[
913,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.dlookup_cons_ne
|
[] |
[
189,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_sub
|
[] |
[
340,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean
|
GeneralizedContinuedFraction.zeroth_convergent'_aux_eq_zero
|
[] |
[
174,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
src/lean/Init/SimpLemmas.lean
|
beq_self_eq_true'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\n⊢ (a == a) = true",
"tactic": "simp [BEq.beq]"
}
] |
[
149,
97
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
149,
9
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
Continuous.homeoOfEquivCompactToT2.t1_counterexample
|
[] |
[
509,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kronecker_smul
|
[] |
[
322,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Set.singleton_vsub_self
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nP : Type u_2\ninst✝ : AddGroup G\nT : AddTorsor G P\np : P\n⊢ {p} -ᵥ {p} = {0}",
"tactic": "rw [Set.singleton_vsub_singleton, vsub_self]"
}
] |
[
198,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.map_eq_empty
|
[] |
[
242,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isInt_mul
|
[] |
[
358,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.bit1_lt_bit0
|
[
{
"state_after": "n m : ℕ\nh : n < succ m\nthis✝ : n ≤ m\nthis : succ (n + n) ≤ succ (m + m)\n⊢ succ (n + n) ≤ succ (m + m)",
"state_before": "n m : ℕ\nh : n < succ m\nthis✝ : n ≤ m\nthis : succ (n + n) ≤ succ (m + m)\n⊢ succ (n + n) ≤ succ m + m",
"tactic": "rw [succ_add]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : n < succ m\nthis✝ : n ≤ m\nthis : succ (n + n) ≤ succ (m + m)\n⊢ succ (n + n) ≤ succ (m + m)",
"tactic": "assumption"
}
] |
[
195,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
11
] |
Mathlib/FieldTheory/Tower.lean
|
FiniteDimensional.trans
|
[] |
[
103,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.isOpen_preimage
|
[] |
[
307,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measure_diff_null
|
[] |
[
228,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Algebra/ModEq.lean
|
AddCommGroup.add_nsmul_modEq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\np a a₁ a₂ b b₁ b₂ c : α\nn✝ : ℕ\nz : ℤ\nn : ℕ\n⊢ a - (a + n • p) = -↑n • p",
"tactic": "simp"
}
] |
[
126,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.invFun_eq_of_injective_of_rightInverse
|
[
{
"state_after": "α : Sort u_1\nβ : Sort u_2\ninst✝ : Nonempty α\nf : α → β\na : α\nb✝ : β\ng : β → α\nhf : Injective f\nhg : RightInverse g f\nb : β\n⊢ f (invFun f b) = b",
"state_before": "α : Sort u_1\nβ : Sort u_2\ninst✝ : Nonempty α\nf : α → β\na : α\nb✝ : β\ng : β → α\nhf : Injective f\nhg : RightInverse g f\nb : β\n⊢ f (invFun f b) = f (g b)",
"tactic": "rw [hg b]"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort u_2\ninst✝ : Nonempty α\nf : α → β\na : α\nb✝ : β\ng : β → α\nhf : Injective f\nhg : RightInverse g f\nb : β\n⊢ f (invFun f b) = b",
"tactic": "exact invFun_eq ⟨g b, hg b⟩"
}
] |
[
451,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.add'_comm
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ Quotient.mk (oreEqv R S) (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) =\n Quotient.mk (oreEqv R S) (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ OreLocalization.add' r₁ s₁ (r₂ /ₒ s₂) = OreLocalization.add' r₂ s₂ (r₁ /ₒ s₁)",
"tactic": "simp only [add', oreDiv, add'', Quotient.mk', Quotient.lift_mk]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ Quotient.mk (oreEqv R S) (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) =\n Quotient.mk (oreEqv R S) (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "rw [Quotient.eq]"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nhb : ↑s₂ * ↑(oreDenom (↑s₂) s₁) = ↑s₁ * oreNum (↑s₂) s₁\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "have hb := ore_eq (↑s₂) s₁"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nhb : ↑s₂ * ↑(oreDenom (↑s₂) s₁) = ↑s₁ * rb\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * rb, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nhb : ↑s₂ * ↑(oreDenom (↑s₂) s₁) = ↑s₁ * oreNum (↑s₂) s₁\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "set rb := oreNum (↑s₂) s₁"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nhb : ↑s₂ * ↑(oreDenom (↑s₂) s₁) = ↑s₁ * rb\n⊢ (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * rb, s₂ * oreDenom (↑s₂) s₁) ≈\n (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "set sb := oreDenom (↑s₂) s₁"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nha : ↑s₁ * ↑(oreDenom (↑s₁) s₂) = ↑s₂ * oreNum (↑s₁) s₂\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "have ha := ore_eq (↑s₁) s₂"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nha : ↑s₁ * ↑(oreDenom (↑s₁) s₂) = ↑s₂ * ra\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * ra, s₁ * oreDenom (↑s₁) s₂)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nha : ↑s₁ * ↑(oreDenom (↑s₁) s₂) = ↑s₂ * oreNum (↑s₁) s₂\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "set ra := oreNum (↑s₁) s₂"
},
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nha : ↑s₁ * ↑(oreDenom (↑s₁) s₂) = ↑s₂ * ra\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * ra, s₁ * oreDenom (↑s₁) s₂)",
"tactic": "set sa := oreDenom (↑s₁) s₂"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"tactic": "rcases oreCondition ra sb with ⟨rc, sc, hc⟩"
},
{
"state_after": "case mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"state_before": "case mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"tactic": "have : (s₁ : R) * (rb * rc) = s₁ * (sa * sc) := by\n rw [← mul_assoc, ← hb, mul_assoc, ← hc, ← mul_assoc, ← ha, mul_assoc]"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"state_before": "case mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"tactic": "rcases ore_left_cancel _ _ s₁ this with ⟨sd, hd⟩"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ∃ v,\n (r₁ * ↑sa + r₂ * ra, s₁ * sa).fst * ↑(sc * sd) = (r₂ * ↑sb + r₁ * rb, s₂ * sb).fst * v ∧\n ↑(r₁ * ↑sa + r₂ * ra, s₁ * sa).snd * ↑(sc * sd) = ↑(r₂ * ↑sb + r₁ * rb, s₂ * sb).snd * v",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₂ * ↑sb + r₁ * rb, s₂ * sb) ≈ (r₁ * ↑sa + r₂ * ra, s₁ * sa)",
"tactic": "use sc * sd"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑sa + r₂ * ra, s₁ * sa).fst * ↑(sc * sd) = (r₂ * ↑sb + r₁ * rb, s₂ * sb).fst * (rc * ↑sd) ∧\n ↑(r₁ * ↑sa + r₂ * ra, s₁ * sa).snd * ↑(sc * sd) = ↑(r₂ * ↑sb + r₁ * rb, s₂ * sb).snd * (rc * ↑sd)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ∃ v,\n (r₁ * ↑sa + r₂ * ra, s₁ * sa).fst * ↑(sc * sd) = (r₂ * ↑sb + r₁ * rb, s₂ * sb).fst * v ∧\n ↑(r₁ * ↑sa + r₂ * ra, s₁ * sa).snd * ↑(sc * sd) = ↑(r₂ * ↑sb + r₁ * rb, s₂ * sb).snd * v",
"tactic": "use rc * sd"
},
{
"state_after": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂) * (↑sc * ↑sd) =\n (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁) * (rc * ↑sd) ∧\n ↑s₁ * ↑(oreDenom (↑s₁) s₂) * (↑sc * ↑sd) = ↑s₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑sa + r₂ * ra, s₁ * sa).fst * ↑(sc * sd) = (r₂ * ↑sb + r₁ * rb, s₂ * sb).fst * (rc * ↑sd) ∧\n ↑(r₁ * ↑sa + r₂ * ra, s₁ * sa).snd * ↑(sc * sd) = ↑(r₂ * ↑sb + r₁ * rb, s₂ * sb).snd * (rc * ↑sd)",
"tactic": "dsimp"
},
{
"state_after": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂) * (↑sc * ↑sd) =\n (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁) * (rc * ↑sd)\n\ncase mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ↑s₁ * ↑(oreDenom (↑s₁) s₂) * (↑sc * ↑sd) = ↑s₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd)",
"state_before": "case mk.mk.intro\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂) * (↑sc * ↑sd) =\n (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁) * (rc * ↑sd) ∧\n ↑s₁ * ↑(oreDenom (↑s₁) s₂) * (↑sc * ↑sd) = ↑s₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\n⊢ ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)",
"tactic": "rw [← mul_assoc, ← hb, mul_assoc, ← hc, ← mul_assoc, ← ha, mul_assoc]"
},
{
"state_after": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ r₂ * (↑sb * rc * ↑sd) + r₁ * (rb * rc * ↑sd) =\n r₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd) + r₁ * oreNum (↑s₂) s₁ * (rc * ↑sd)",
"state_before": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ (r₁ * ↑(oreDenom (↑s₁) s₂) + r₂ * oreNum (↑s₁) s₂) * (↑sc * ↑sd) =\n (r₂ * ↑(oreDenom (↑s₂) s₁) + r₁ * oreNum (↑s₂) s₁) * (rc * ↑sd)",
"tactic": "rw [add_mul, add_mul, add_comm, mul_assoc (a := r₁) (b := (sa : R)),\n ← mul_assoc (a := (sa : R)), ← hd, mul_assoc (a := r₂) (b := ra),\n ← mul_assoc (a := ra) (b := (sc : R)), hc]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.left\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ r₂ * (↑sb * rc * ↑sd) + r₁ * (rb * rc * ↑sd) =\n r₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd) + r₁ * oreNum (↑s₂) s₁ * (rc * ↑sd)",
"tactic": "simp only [mul_assoc]"
},
{
"state_after": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ↑s₁ * (rb * rc * ↑sd) = ↑s₁ * rb * (rc * ↑sd)",
"state_before": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ↑s₁ * ↑(oreDenom (↑s₁) s₂) * (↑sc * ↑sd) = ↑s₂ * ↑(oreDenom (↑s₂) s₁) * (rc * ↑sd)",
"tactic": "rw [mul_assoc, ← mul_assoc (sa : R), ← hd, hb]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.right\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr₁ r₂ : R\ns₁ s₂ : { x // x ∈ S }\nrb : R := oreNum (↑s₂) s₁\nsb : { x // x ∈ S } := oreDenom (↑s₂) s₁\nhb : ↑s₂ * ↑sb = ↑s₁ * rb\nra : R := oreNum (↑s₁) s₂\nsa : { x // x ∈ S } := oreDenom (↑s₁) s₂\nha : ↑s₁ * ↑sa = ↑s₂ * ra\nrc : R\nsc : { x // x ∈ S }\nhc : ra * ↑sc = ↑sb * rc\nthis : ↑s₁ * (rb * rc) = ↑s₁ * (↑sa * ↑sc)\nsd : { x // x ∈ S }\nhd : rb * rc * ↑sd = ↑sa * ↑sc * ↑sd\n⊢ ↑s₁ * (rb * rc * ↑sd) = ↑s₁ * rb * (rc * ↑sd)",
"tactic": "simp only [mul_assoc]"
}
] |
[
589,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
9
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
ContinuousOn.measurable_piecewise
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (Set.piecewise s f g ⁻¹' t)",
"state_before": "α : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\n⊢ Measurable (Set.piecewise s f g)",
"tactic": "refine' measurable_of_isOpen fun t ht => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (f ⁻¹' t ∩ s ∪ g ⁻¹' t \\ s)",
"state_before": "α : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (Set.piecewise s f g ⁻¹' t)",
"tactic": "rw [piecewise_preimage, Set.ite]"
},
{
"state_after": "case h₁\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (f ⁻¹' t ∩ s)\n\ncase h₂\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (g ⁻¹' t \\ s)",
"state_before": "α : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (f ⁻¹' t ∩ s ∪ g ⁻¹' t \\ s)",
"tactic": "apply MeasurableSet.union"
},
{
"state_after": "case h₁.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : f ⁻¹' t ∩ s = u ∩ s\n⊢ MeasurableSet (f ⁻¹' t ∩ s)",
"state_before": "case h₁\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (f ⁻¹' t ∩ s)",
"tactic": "rcases _root_.continuousOn_iff'.1 hf t ht with ⟨u, u_open, hu⟩"
},
{
"state_after": "case h₁.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : f ⁻¹' t ∩ s = u ∩ s\n⊢ MeasurableSet (u ∩ s)",
"state_before": "case h₁.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : f ⁻¹' t ∩ s = u ∩ s\n⊢ MeasurableSet (f ⁻¹' t ∩ s)",
"tactic": "rw [hu]"
},
{
"state_after": "no goals",
"state_before": "case h₁.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : f ⁻¹' t ∩ s = u ∩ s\n⊢ MeasurableSet (u ∩ s)",
"tactic": "exact u_open.measurableSet.inter hs"
},
{
"state_after": "case h₂.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ\n⊢ MeasurableSet (g ⁻¹' t \\ s)",
"state_before": "case h₂\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\n⊢ MeasurableSet (g ⁻¹' t \\ s)",
"tactic": "rcases _root_.continuousOn_iff'.1 hg t ht with ⟨u, u_open, hu⟩"
},
{
"state_after": "case h₂.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ\n⊢ MeasurableSet (u ∩ sᶜ)",
"state_before": "case h₂.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ\n⊢ MeasurableSet (g ⁻¹' t \\ s)",
"tactic": "rw [diff_eq_compl_inter, inter_comm, hu]"
},
{
"state_after": "no goals",
"state_before": "case h₂.intro.intro\nα : Type u_1\nβ : Type ?u.930878\nγ : Type u_2\nγ₂ : Type ?u.930884\nδ : Type ?u.930887\nι : Sort y\ns✝ t✝ u✝ : Set α\ninst✝¹⁵ : TopologicalSpace α\ninst✝¹⁴ : MeasurableSpace α\ninst✝¹³ : OpensMeasurableSpace α\ninst✝¹² : TopologicalSpace β\ninst✝¹¹ : MeasurableSpace β\ninst✝¹⁰ : OpensMeasurableSpace β\ninst✝⁹ : TopologicalSpace γ\ninst✝⁸ : MeasurableSpace γ\ninst✝⁷ : BorelSpace γ\ninst✝⁶ : TopologicalSpace γ₂\ninst✝⁵ : MeasurableSpace γ₂\ninst✝⁴ : BorelSpace γ₂\ninst✝³ : MeasurableSpace δ\nα' : Type ?u.930980\ninst✝² : TopologicalSpace α'\ninst✝¹ : MeasurableSpace α'\nf g : α → γ\ns : Set α\ninst✝ : (j : α) → Decidable (j ∈ s)\nhf : ContinuousOn f s\nhg : ContinuousOn g (sᶜ)\nhs : MeasurableSet s\nt : Set γ\nht : IsOpen t\nu : Set α\nu_open : IsOpen u\nhu : g ⁻¹' t ∩ sᶜ = u ∩ sᶜ\n⊢ MeasurableSet (u ∩ sᶜ)",
"tactic": "exact u_open.measurableSet.inter hs.compl"
}
] |
[
853,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
smul_left_cancel_iff
|
[] |
[
172,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
src/lean/Init/Prelude.lean
|
Or.intro_right
|
[] |
[
527,
11
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
526,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
Filter.EventuallyEq.fderivWithin_eq_nhds
|
[] |
[
962,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
960,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_of_empty
|
[] |
[
1507,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1506,
1
] |
Mathlib/Algebra/Star/Unitary.lean
|
unitary.star_mem_iff
|
[] |
[
75,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Topology/Order/Basic.lean
|
mem_nhdsWithin_Iic_iff_exists_Icc_subset
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : NoMinOrder α\ninst✝ : DenselyOrdered α\na : α\ns : Set α\n⊢ (∃ u, ↑toDual a < ↑toDual u ∧ ↑ofDual ⁻¹' Icc u a ⊆ ↑ofDual ⁻¹' s) ↔ ∃ l, l < a ∧ Icc l a ⊆ s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : NoMinOrder α\ninst✝ : DenselyOrdered α\na : α\ns : Set α\n⊢ (∃ u, ↑toDual a < ↑toDual u ∧ Icc (↑toDual a) (↑toDual u) ⊆ ↑ofDual ⁻¹' s) ↔ ∃ l, l < a ∧ Icc l a ⊆ s",
"tactic": "simp only [dual_Icc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : NoMinOrder α\ninst✝ : DenselyOrdered α\na : α\ns : Set α\n⊢ (∃ u, ↑toDual a < ↑toDual u ∧ ↑ofDual ⁻¹' Icc u a ⊆ ↑ofDual ⁻¹' s) ↔ ∃ l, l < a ∧ Icc l a ⊆ s",
"tactic": "rfl"
}
] |
[
1839,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1834,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Prepartition.exists_iUnion_eq_diff
|
[
{
"state_after": "case intro\nι : Type u_1\nM : Type ?u.56100\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ : Prepartition I\ns : Finset (ι × ℝ)\nhs : π ⊓ splitMany I s = filter (splitMany I s) fun J => ↑J ⊆ Prepartition.iUnion π\n⊢ ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π",
"state_before": "ι : Type u_1\nM : Type ?u.56100\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ : Prepartition I\n⊢ ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π",
"tactic": "rcases π.eventually_splitMany_inf_eq_filter.exists with ⟨s, hs⟩"
},
{
"state_after": "case intro\nι : Type u_1\nM : Type ?u.56100\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ : Prepartition I\ns : Finset (ι × ℝ)\nhs : π ⊓ splitMany I s = filter (splitMany I s) fun J => ↑J ⊆ Prepartition.iUnion π\n⊢ Prepartition.iUnion (filter (splitMany I s) fun J => ¬↑J ⊆ Prepartition.iUnion π) = ↑I \\ Prepartition.iUnion π",
"state_before": "case intro\nι : Type u_1\nM : Type ?u.56100\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ : Prepartition I\ns : Finset (ι × ℝ)\nhs : π ⊓ splitMany I s = filter (splitMany I s) fun J => ↑J ⊆ Prepartition.iUnion π\n⊢ ∃ π', Prepartition.iUnion π' = ↑I \\ Prepartition.iUnion π",
"tactic": "use (splitMany I s).filter fun J => ¬(J : Set (ι → ℝ)) ⊆ π.iUnion"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_1\nM : Type ?u.56100\nn : ℕ\nI J : Box ι\ni : ι\nx : ℝ\ninst✝ : Finite ι\nπ : Prepartition I\ns : Finset (ι × ℝ)\nhs : π ⊓ splitMany I s = filter (splitMany I s) fun J => ↑J ⊆ Prepartition.iUnion π\n⊢ Prepartition.iUnion (filter (splitMany I s) fun J => ¬↑J ⊆ Prepartition.iUnion π) = ↑I \\ Prepartition.iUnion π",
"tactic": "simp [← hs]"
}
] |
[
359,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csInf_one
|
[] |
[
121,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Data/Polynomial/Taylor.lean
|
Polynomial.taylor_zero'
|
[
{
"state_after": "case h.h.a\nR : Type u_1\ninst✝ : Semiring R\nr : R\nf : R[X]\nn✝¹ n✝ : ℕ\n⊢ coeff (↑(LinearMap.comp (taylor 0) (monomial n✝¹)) 1) n✝ = coeff (↑(LinearMap.comp LinearMap.id (monomial n✝¹)) 1) n✝",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nr : R\nf : R[X]\n⊢ taylor 0 = LinearMap.id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h.a\nR : Type u_1\ninst✝ : Semiring R\nr : R\nf : R[X]\nn✝¹ n✝ : ℕ\n⊢ coeff (↑(LinearMap.comp (taylor 0) (monomial n✝¹)) 1) n✝ = coeff (↑(LinearMap.comp LinearMap.id (monomial n✝¹)) 1) n✝",
"tactic": "simp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp,\n Function.comp_apply, LinearMap.coe_comp]"
}
] |
[
62,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.Nonempty.of_mul_right
|
[] |
[
392,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Data/Nat/Cast/WithTop.lean
|
Nat.cast_withBot
|
[] |
[
27,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.sbtw_of_collinear_of_dist_center_lt_radius
|
[] |
[
392,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.IsSheafFor.valid_glue
|
[] |
[
623,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean
|
HolderWith.comp_holderOnWith
|
[] |
[
212,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean
|
StrictMonoOn.continuousWithinAt_left_of_surjOn
|
[] |
[
209,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.sub_lt_sub_of_lt_of_le
|
[] |
[
1134,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1132,
11
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.extend_eq'
|
[
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.8090\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝² : TopologicalSpace δ\nf✝ : γ → α\ng : γ → δ\nh : δ → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\ndi : DenseInducing i\nhf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)\na : α\nb : γ\nhb : Tendsto f (comap i (𝓝 (i a))) (𝓝 b)\n⊢ extend di f (i a) = f a",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.8090\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝² : TopologicalSpace δ\nf✝ : γ → α\ng : γ → δ\nh : δ → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\ndi : DenseInducing i\nhf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)\na : α\n⊢ extend di f (i a) = f a",
"tactic": "rcases hf (i a) with ⟨b, hb⟩"
},
{
"state_after": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.8090\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝² : TopologicalSpace δ\nf✝ : γ → α\ng : γ → δ\nh : δ → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\ndi : DenseInducing i\nhf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)\na : α\nb : γ\nhb : Tendsto f (comap i (𝓝 (i a))) (𝓝 b)\n⊢ Tendsto f (𝓝 a) (𝓝 b)",
"state_before": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.8090\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝² : TopologicalSpace δ\nf✝ : γ → α\ng : γ → δ\nh : δ → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\ndi : DenseInducing i\nhf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)\na : α\nb : γ\nhb : Tendsto f (comap i (𝓝 (i a))) (𝓝 b)\n⊢ extend di f (i a) = f a",
"tactic": "refine' di.extend_eq_at' b _"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\nδ : Type ?u.8090\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ni : α → β\ndi✝ : DenseInducing i\ninst✝² : TopologicalSpace δ\nf✝ : γ → α\ng : γ → δ\nh : δ → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nf : α → γ\ndi : DenseInducing i\nhf : ∀ (b : β), ∃ c, Tendsto f (comap i (𝓝 b)) (𝓝 c)\na : α\nb : γ\nhb : Tendsto f (comap i (𝓝 (i a))) (𝓝 b)\n⊢ Tendsto f (𝓝 a) (𝓝 b)",
"tactic": "rwa [← di.toInducing.nhds_eq_comap] at hb"
}
] |
[
177,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.C_inj
|
[] |
[
254,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Order/CompactlyGenerated.lean
|
CompleteLattice.finset_sup_compact_of_compact
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\n⊢ IsCompactElement (Finset.sup s f)",
"tactic": "classical\n rw [isCompactElement_iff_le_of_directed_sSup_le]\n intro d hemp hdir hsup\n rw [← Function.comp.left_id f]\n rw [← Finset.sup_image]\n apply Finset.sup_le_of_le_directed d hemp hdir\n rintro x hx\n obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx\n specialize h p hps\n rw [isCompactElement_iff_le_of_directed_sSup_le] at h\n specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)\n simpa only [exists_prop]"
},
{
"state_after": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\n⊢ ∀ (s_1 : Set α),\n Set.Nonempty s_1 →\n DirectedOn (fun x x_1 => x ≤ x_1) s_1 → Finset.sup s f ≤ sSup s_1 → ∃ x, x ∈ s_1 ∧ Finset.sup s f ≤ x",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\n⊢ IsCompactElement (Finset.sup s f)",
"tactic": "rw [isCompactElement_iff_le_of_directed_sSup_le]"
},
{
"state_after": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup s f ≤ x",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\n⊢ ∀ (s_1 : Set α),\n Set.Nonempty s_1 →\n DirectedOn (fun x x_1 => x ≤ x_1) s_1 → Finset.sup s f ≤ sSup s_1 → ∃ x, x ∈ s_1 ∧ Finset.sup s f ≤ x",
"tactic": "intro d hemp hdir hsup"
},
{
"state_after": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup s (id ∘ f) ≤ x",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup s f ≤ x",
"tactic": "rw [← Function.comp.left_id f]"
},
{
"state_after": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup (Finset.image f s) id ≤ x",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup s (id ∘ f) ≤ x",
"tactic": "rw [← Finset.sup_image]"
},
{
"state_after": "case a\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∀ (x : α), x ∈ Finset.image f s → ∃ y, y ∈ d ∧ x ≤ y",
"state_before": "ι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∃ x, x ∈ d ∧ Finset.sup (Finset.image f s) id ≤ x",
"tactic": "apply Finset.sup_le_of_le_directed d hemp hdir"
},
{
"state_after": "case a\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\nx : α\nhx : x ∈ Finset.image f s\n⊢ ∃ y, y ∈ d ∧ x ≤ y",
"state_before": "case a\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\n⊢ ∀ (x : α), x ∈ Finset.image f s → ∃ y, y ∈ d ∧ x ≤ y",
"tactic": "rintro x hx"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"state_before": "case a\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\nx : α\nhx : x ∈ Finset.image f s\n⊢ ∃ y, y ∈ d ∧ x ≤ y",
"tactic": "obtain ⟨p, ⟨hps, rfl⟩⟩ := Finset.mem_image.mp hx"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : IsCompactElement (f p)\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"state_before": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nh : ∀ (x : β), x ∈ s → IsCompactElement (f x)\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"tactic": "specialize h p hps"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → f p ≤ sSup s → ∃ x, x ∈ s ∧ f p ≤ x\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"state_before": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : IsCompactElement (f p)\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"tactic": "rw [isCompactElement_iff_le_of_directed_sSup_le] at h"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : ∃ x, x ∈ d ∧ f p ≤ x\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"state_before": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → f p ≤ sSup s → ∃ x, x ∈ s ∧ f p ≤ x\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"tactic": "specialize h d hemp hdir (le_trans (Finset.le_sup hps) hsup)"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nι : Sort ?u.14463\nα✝ : Type ?u.14466\ninst✝¹ : CompleteLattice α✝\nf✝ : ι → α✝\nα : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\nf : β → α\ns : Finset β\nd : Set α\nhemp : Set.Nonempty d\nhdir : DirectedOn (fun x x_1 => x ≤ x_1) d\nhsup : Finset.sup s f ≤ sSup d\np : β\nhps : p ∈ s\nhx : f p ∈ Finset.image f s\nh : ∃ x, x ∈ d ∧ f p ≤ x\n⊢ ∃ y, y ∈ d ∧ f p ≤ y",
"tactic": "simpa only [exists_prop]"
}
] |
[
200,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.lt_of_sub_pos
|
[
{
"state_after": "a b : Int\nh✝ : 0 < a - b\nh : 0 + b < a - b + b\n⊢ b < a",
"state_before": "a b : Int\nh : 0 < a - b\n⊢ b < a",
"tactic": "have h := Int.add_lt_add_right h b"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nh✝ : 0 < a - b\nh : 0 + b < a - b + b\n⊢ b < a",
"tactic": "rwa [Int.sub_add_cancel, Int.zero_add] at h"
}
] |
[
940,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
938,
11
] |
Mathlib/Topology/Semicontinuous.lean
|
upperSemicontinuousAt_ciInf
|
[] |
[
1021,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1018,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.continuousWithinAt_Iio_iff_leftLim_eq
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x = ⊥\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"tactic": "rcases eq_or_ne (𝓝[<] x) ⊥ with (h' | h')"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis : NeBot (𝓝[Iio x] x)\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"tactic": "haveI : (𝓝[Iio x] x).NeBot := neBot_iff.2 h'"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis : NeBot (𝓝[Iio x] x)\nh : leftLim f x = f x\n⊢ ContinuousWithinAt f (Iio x) x",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis : NeBot (𝓝[Iio x] x)\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"tactic": "refine' ⟨fun h => tendsto_nhds_unique (hf.tendsto_leftLim x) h.tendsto, fun h => _⟩"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis✝ : NeBot (𝓝[Iio x] x)\nh : leftLim f x = f x\nthis : Tendsto f (𝓝[Iio x] x) (𝓝 (leftLim f x))\n⊢ ContinuousWithinAt f (Iio x) x",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis : NeBot (𝓝[Iio x] x)\nh : leftLim f x = f x\n⊢ ContinuousWithinAt f (Iio x) x",
"tactic": "have := hf.tendsto_leftLim x"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x ≠ ⊥\nthis✝ : NeBot (𝓝[Iio x] x)\nh : leftLim f x = f x\nthis : Tendsto f (𝓝[Iio x] x) (𝓝 (leftLim f x))\n⊢ ContinuousWithinAt f (Iio x) x",
"tactic": "rwa [h] at this"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh' : 𝓝[Iio x] x = ⊥\n⊢ ContinuousWithinAt f (Iio x) x ↔ leftLim f x = f x",
"tactic": "simp [leftLim_eq_of_eq_bot f h', ContinuousWithinAt, h']"
}
] |
[
196,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Init/Logic.lean
|
InvImage.irreflexive
|
[] |
[
500,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Control/Bitraversable/Lemmas.lean
|
Bitraversable.id_tsnd
|
[] |
[
74,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.hasSum_sub
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.416023\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\ny : E\nhy : y ∈ EMetric.ball x r\nthis : y - x ∈ EMetric.ball 0 r\n⊢ HasSum (fun n => ↑(p n) fun x_1 => y - x) (f y)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.416023\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\ny : E\nhy : y ∈ EMetric.ball x r\n⊢ HasSum (fun n => ↑(p n) fun x_1 => y - x) (f y)",
"tactic": "have : y - x ∈ EMetric.ball (0 : E) r := by simpa [edist_eq_coe_nnnorm_sub] using hy"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.416023\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\ny : E\nhy : y ∈ EMetric.ball x r\nthis : y - x ∈ EMetric.ball 0 r\n⊢ HasSum (fun n => ↑(p n) fun x_1 => y - x) (f y)",
"tactic": "simpa only [add_sub_cancel'_right] using hf.hasSum this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.416023\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f p x r\ny : E\nhy : y ∈ EMetric.ball x r\n⊢ y - x ∈ EMetric.ball 0 r",
"tactic": "simpa [edist_eq_coe_nnnorm_sub] using hy"
}
] |
[
451,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.isUnit_iff_isUnit_det
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.86247\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ IsUnit A ↔ IsUnit (det A)",
"tactic": "simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivDetInvertible A).nonempty_congr]"
}
] |
[
202,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Algebra/Category/GroupCat/Basic.lean
|
GroupCat.ofHom_apply
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.mem_def
|
[] |
[
180,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Combinatorics/SimpleGraph/Coloring.lean
|
SimpleGraph.colorable_of_fintype
|
[] |
[
211,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Rel.lean
|
Rel.image_id
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.15405\nγ : Type ?u.15408\nr : Rel α β\ns : Set α\nx : α\n⊢ x ∈ image Eq s ↔ x ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.15405\nγ : Type ?u.15408\nr : Rel α β\ns : Set α\n⊢ image Eq s = s",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.15405\nγ : Type ?u.15408\nr : Rel α β\ns : Set α\nx : α\n⊢ x ∈ image Eq s ↔ x ∈ s",
"tactic": "simp [mem_image]"
}
] |
[
160,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.homogeneousSubmodule_mul
|
[
{
"state_after": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\n⊢ ∀ (m_1 : MvPolynomial σ R),\n m_1 ∈ homogeneousSubmodule σ R m →\n ∀ (n_1 : MvPolynomial σ R), n_1 ∈ homogeneousSubmodule σ R n → m_1 * n_1 ∈ homogeneousSubmodule σ R (m + n)",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\n⊢ homogeneousSubmodule σ R m * homogeneousSubmodule σ R n ≤ homogeneousSubmodule σ R (m + n)",
"tactic": "rw [Submodule.mul_le]"
},
{
"state_after": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : coeff c (φ * ψ) ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\n⊢ ∀ (m_1 : MvPolynomial σ R),\n m_1 ∈ homogeneousSubmodule σ R m →\n ∀ (n_1 : MvPolynomial σ R), n_1 ∈ homogeneousSubmodule σ R n → m_1 * n_1 ∈ homogeneousSubmodule σ R (m + n)",
"tactic": "intro φ hφ ψ hψ c hc"
},
{
"state_after": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : coeff c (φ * ψ) ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "rw [coeff_mul] at hc"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "obtain ⟨⟨d, e⟩, hde, H⟩ := Finset.exists_ne_zero_of_sum_ne_zero hc"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "have aux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0 := by\n contrapose! H\n by_cases h : coeff d φ = 0 <;>\n simp_all only [Ne.def, not_false_iff, MulZeroClass.zero_mul, MulZeroClass.mul_zero]"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "specialize hφ aux.1"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "specialize hψ aux.2"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "rw [Finsupp.mem_antidiagonal] at hde"
},
{
"state_after": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff d φ ≠ 0 → coeff e ψ = 0\n⊢ coeff d φ * coeff e ψ = 0",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\n⊢ coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0",
"tactic": "contrapose! H"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ : MvPolynomial σ R\nhφ : φ ∈ homogeneousSubmodule σ R m\nψ : MvPolynomial σ R\nhψ : ψ ∈ homogeneousSubmodule σ R n\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e) ∈ Finsupp.antidiagonal c\nH : coeff d φ ≠ 0 → coeff e ψ = 0\n⊢ coeff d φ * coeff e ψ = 0",
"tactic": "by_cases h : coeff d φ = 0 <;>\n simp_all only [Ne.def, not_false_iff, MulZeroClass.zero_mul, MulZeroClass.mul_zero]"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "have hd' : d.support ⊆ d.support ∪ e.support := Finset.subset_union_left _ _"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∑ i in c.support, ↑c i = m + n",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "have he' : e.support ⊆ d.support ∪ e.support := Finset.subset_union_right _ _"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∑ x in (d, e).fst.support ∪ (d, e).snd.support, ↑((d, e).fst + (d, e).snd) x =\n ∑ x in d.support ∪ e.support, (↑d x + ↑e x)\n\ncase intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ), x ∈ d.support ∪ e.support → ¬x ∈ e.support → ↑e x = 0\n\ncase intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ), x ∈ d.support ∪ e.support → ¬x ∈ d.support → ↑d x = 0\n\ncase intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ),\n x ∈ (d, e).fst.support ∪ (d, e).snd.support →\n ¬x ∈ ((d, e).fst + (d, e).snd).support → ↑((d, e).fst + (d, e).snd) x = 0",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∑ i in c.support, ↑c i = m + n",
"tactic": "rw [← hde, ← hφ, ← hψ, Finset.sum_subset Finsupp.support_add, Finset.sum_subset hd',\n Finset.sum_subset he', ← Finset.sum_add_distrib]"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ), x ∈ d.support ∪ e.support → ¬x ∈ e.support → ↑e x = 0\n\ncase intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ), x ∈ d.support ∪ e.support → ¬x ∈ d.support → ↑d x = 0\n\ncase intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ),\n x ∈ (d, e).fst.support ∪ (d, e).snd.support →\n ¬x ∈ ((d, e).fst + (d, e).snd).support → ↑((d, e).fst + (d, e).snd) x = 0",
"tactic": "all_goals intro i hi; apply Finsupp.not_mem_support_iff.mp"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∑ x in (d, e).fst.support ∪ (d, e).snd.support, ↑((d, e).fst + (d, e).snd) x =\n ∑ x in d.support ∪ e.support, (↑d x + ↑e x)",
"tactic": "congr"
},
{
"state_after": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\ni : σ\nhi : i ∈ (d, e).fst.support ∪ (d, e).snd.support\n⊢ ¬i ∈ ((d, e).fst + (d, e).snd).support → ↑((d, e).fst + (d, e).snd) i = 0",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\n⊢ ∀ (x : σ),\n x ∈ (d, e).fst.support ∪ (d, e).snd.support →\n ¬x ∈ ((d, e).fst + (d, e).snd).support → ↑((d, e).fst + (d, e).snd) x = 0",
"tactic": "intro i hi"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro\nσ : Type u_2\nτ : Type ?u.119978\nR : Type u_1\nS : Type ?u.119984\ninst✝ : CommSemiring R\nm n : ℕ\nφ ψ : MvPolynomial σ R\nc : σ →₀ ℕ\nhc : ∑ x in Finsupp.antidiagonal c, coeff x.fst φ * coeff x.snd ψ ≠ 0\nd e : σ →₀ ℕ\nhde : (d, e).fst + (d, e).snd = c\nH : coeff (d, e).fst φ * coeff (d, e).snd ψ ≠ 0\naux : coeff d φ ≠ 0 ∧ coeff e ψ ≠ 0\nhφ : ∑ i in d.support, ↑d i = m\nhψ : ∑ i in e.support, ↑e i = n\nhd' : d.support ⊆ d.support ∪ e.support\nhe' : e.support ⊆ d.support ∪ e.support\ni : σ\nhi : i ∈ (d, e).fst.support ∪ (d, e).snd.support\n⊢ ¬i ∈ ((d, e).fst + (d, e).snd).support → ↑((d, e).fst + (d, e).snd) i = 0",
"tactic": "apply Finsupp.not_mem_support_iff.mp"
}
] |
[
110,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
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