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/Order/CompleteLattice.lean
|
iInf_sum
|
[] |
[
1572,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
Polynomial.coe_monomial
|
[
{
"state_after": "case h\nσ : Type ?u.4303706\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ : R[X]\nn : ℕ\na : R\nn✝ : ℕ\n⊢ ↑(PowerSeries.coeff R n✝) ↑(↑(monomial n) a) = ↑(PowerSeries.coeff R n✝) (↑(PowerSeries.monomial R n) a)",
"state_before": "σ : Type ?u.4303706\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ : R[X]\nn : ℕ\na : R\n⊢ ↑(↑(monomial n) a) = ↑(PowerSeries.monomial R n) a",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nσ : Type ?u.4303706\nR : Type u_1\ninst✝ : CommSemiring R\nφ ψ : R[X]\nn : ℕ\na : R\nn✝ : ℕ\n⊢ ↑(PowerSeries.coeff R n✝) ↑(↑(monomial n) a) = ↑(PowerSeries.coeff R n✝) (↑(PowerSeries.monomial R n) a)",
"tactic": "simp [coeff_coe, PowerSeries.coeff_monomial, Polynomial.coeff_monomial, eq_comm]"
}
] |
[
2552,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2549,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.coe_copy
|
[] |
[
217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_restrict_of_subset
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.290160\nγ : Type ?u.290163\nδ : Type ?u.290166\nι : Type ?u.290169\nR : Type ?u.290172\nR' : Type ?u.290175\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\nu : Set α\nhu : MeasurableSet u\n⊢ ↑↑(restrict (restrict μ t) s) u = ↑↑(restrict μ s) u",
"state_before": "α : Type u_1\nβ : Type ?u.290160\nγ : Type ?u.290163\nδ : Type ?u.290166\nι : Type ?u.290169\nR : Type ?u.290172\nR' : Type ?u.290175\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\n⊢ restrict (restrict μ t) s = restrict μ s",
"tactic": "ext1 u hu"
},
{
"state_after": "case h.h\nα : Type u_1\nβ : Type ?u.290160\nγ : Type ?u.290163\nδ : Type ?u.290166\nι : Type ?u.290169\nR : Type ?u.290172\nR' : Type ?u.290175\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\nu : Set α\nhu : MeasurableSet u\n⊢ u ∩ s ⊆ t",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.290160\nγ : Type ?u.290163\nδ : Type ?u.290166\nι : Type ?u.290169\nR : Type ?u.290172\nR' : Type ?u.290175\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\nu : Set α\nhu : MeasurableSet u\n⊢ ↑↑(restrict (restrict μ t) s) u = ↑↑(restrict μ s) u",
"tactic": "rw [restrict_apply hu, restrict_apply hu, restrict_eq_self]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\nβ : Type ?u.290160\nγ : Type ?u.290163\nδ : Type ?u.290166\nι : Type ?u.290169\nR : Type ?u.290172\nR' : Type ?u.290175\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nh : s ⊆ t\nu : Set α\nhu : MeasurableSet u\n⊢ u ∩ s ⊆ t",
"tactic": "exact (inter_subset_right _ _).trans h"
}
] |
[
1645,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1642,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.symm_trans_self
|
[] |
[
376,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Algebra/Group/Prod.lean
|
Prod.mk_eq_one
|
[] |
[
116,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.conj_inv
|
[] |
[
576,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.Memℒp.of_bound
|
[] |
[
567,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
idealFactorsEquivOfQuotEquiv_is_dvd_iso
|
[
{
"state_after": "R : Type u_1\nA : Type u_2\nK : Type ?u.1117270\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type ?u.1117807\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : IsDedekindDomain B\nL✝ : Ideal B\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nL M : Ideal R\nhL : L ∣ I\nhM : M ∣ I\n⊢ ↑(idealFactorsEquivOfQuotEquiv f) { val := M, property := hM } ≤\n ↑(idealFactorsEquivOfQuotEquiv f) { val := L, property := hL } ↔\n { val := M, property := hM } ≤ { val := L, property := hL }",
"state_before": "R : Type u_1\nA : Type u_2\nK : Type ?u.1117270\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type ?u.1117807\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : IsDedekindDomain B\nL✝ : Ideal B\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nL M : Ideal R\nhL : L ∣ I\nhM : M ∣ I\n⊢ ↑(↑(idealFactorsEquivOfQuotEquiv f) { val := L, property := hL }) ∣\n ↑(↑(idealFactorsEquivOfQuotEquiv f) { val := M, property := hM }) ↔\n L ∣ M",
"tactic": "suffices\n idealFactorsEquivOfQuotEquiv f ⟨M, hM⟩ ≤ idealFactorsEquivOfQuotEquiv f ⟨L, hL⟩ ↔\n (⟨M, hM⟩ : { p : Ideal R | p ∣ I }) ≤ ⟨L, hL⟩\n by rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nK : Type ?u.1117270\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type ?u.1117807\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : IsDedekindDomain B\nL✝ : Ideal B\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nL M : Ideal R\nhL : L ∣ I\nhM : M ∣ I\n⊢ ↑(idealFactorsEquivOfQuotEquiv f) { val := M, property := hM } ≤\n ↑(idealFactorsEquivOfQuotEquiv f) { val := L, property := hL } ↔\n { val := M, property := hM } ≤ { val := L, property := hL }",
"tactic": "exact (idealFactorsEquivOfQuotEquiv f).le_iff_le"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\nK : Type ?u.1117270\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Field K\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nB : Type ?u.1117807\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : IsDedekindDomain B\nL✝ : Ideal B\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nf : R ⧸ I ≃+* A ⧸ J\nL M : Ideal R\nhL : L ∣ I\nhM : M ∣ I\nthis :\n ↑(idealFactorsEquivOfQuotEquiv f) { val := M, property := hM } ≤\n ↑(idealFactorsEquivOfQuotEquiv f) { val := L, property := hL } ↔\n { val := M, property := hM } ≤ { val := L, property := hL }\n⊢ ↑(↑(idealFactorsEquivOfQuotEquiv f) { val := L, property := hL }) ∣\n ↑(↑(idealFactorsEquivOfQuotEquiv f) { val := M, property := hM }) ↔\n L ∣ M",
"tactic": "rw [dvd_iff_le, dvd_iff_le, Subtype.coe_le_coe, this, Subtype.mk_le_mk]"
}
] |
[
1128,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1121,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
AffineIndependent.range
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\n⊢ AffineIndependent k fun x => ↑x",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\n⊢ AffineIndependent k fun x => ↑x",
"tactic": "let f : Set.range p → ι := fun x => x.property.choose"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\n⊢ AffineIndependent k fun x => ↑x",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\n⊢ AffineIndependent k fun x => ↑x",
"tactic": "have hf : ∀ x, p (f x) = x := fun x => x.property.choose_spec"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\n⊢ AffineIndependent k fun x => ↑x",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\n⊢ AffineIndependent k fun x => ↑x",
"tactic": "let fe : Set.range p ↪ ι := ⟨f, fun x₁ x₂ he => Subtype.ext (hf x₁ ▸ hf x₂ ▸ he ▸ rfl)⟩"
},
{
"state_after": "case h.e'_9.h.h.e\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\nx✝ : ↑(Set.range p)\n⊢ Subtype.val = p ∘ ↑fe",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\n⊢ AffineIndependent k fun x => ↑x",
"tactic": "convert ha.comp_embedding fe"
},
{
"state_after": "case h.e'_9.h.h.e.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\nx✝¹ : ↑(Set.range p)\nx✝ : { x // x ∈ Set.range p }\n⊢ ↑x✝ = (p ∘ ↑fe) x✝",
"state_before": "case h.e'_9.h.h.e\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\nx✝ : ↑(Set.range p)\n⊢ Subtype.val = p ∘ ↑fe",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.e'_9.h.h.e.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nha : AffineIndependent k p\nf : ↑(Set.range p) → ι := fun x => Exists.choose (_ : ↑x ∈ Set.range p)\nhf : ∀ (x : ↑(Set.range p)), p (f x) = ↑x\nfe : ↑(Set.range p) ↪ ι := { toFun := f, inj' := (_ : ∀ (x₁ x₂ : ↑(Set.range p)), f x₁ = f x₂ → x₁ = x₂) }\nx✝¹ : ↑(Set.range p)\nx✝ : { x // x ∈ Set.range p }\n⊢ ↑x✝ = (p ∘ ↑fe) x✝",
"tactic": "simp [hf]"
}
] |
[
328,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
11
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_le_card_of_inj_on
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : α → β\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nf_inj : ∀ (a₁ : α), a₁ ∈ s → ∀ (a₂ : α), a₂ ∈ s → f a₁ = f a₂ → a₁ = a₂\n⊢ card s ≤ card t",
"tactic": "classical calc\n s.card = (s.image f).card := (card_image_of_injOn f_inj).symm\n _ ≤ t.card := card_le_of_subset <| image_subset_iff.2 hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns t✝ : Finset α\nf✝ : α → β\nn : ℕ\nt : Finset β\nf : α → β\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nf_inj : ∀ (a₁ : α), a₁ ∈ s → ∀ (a₂ : α), a₂ ∈ s → f a₁ = f a₂ → a₁ = a₂\n⊢ card s ≤ card t",
"tactic": "calc\ns.card = (s.image f).card := (card_image_of_injOn f_inj).symm\n_ ≤ t.card := card_le_of_subset <| image_subset_iff.2 hf"
}
] |
[
334,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
Algebra.FinitePresentation.of_finiteType
|
[
{
"state_after": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\n⊢ FinitePresentation R A",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\n⊢ FiniteType R A ↔ FinitePresentation R A",
"tactic": "refine' ⟨fun h => _, Algebra.FiniteType.of_finitePresentation⟩"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\n⊢ FinitePresentation R A",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\n⊢ FinitePresentation R A",
"tactic": "obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.1 h"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\n⊢ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\n⊢ FinitePresentation R A",
"tactic": "refine' ⟨n, f, hf, _⟩"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherianRing (MvPolynomial (Fin n) R)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\n⊢ Ideal.FG (RingHom.ker ↑f)",
"tactic": "have hnoet : IsNoetherianRing (MvPolynomial (Fin n) R) := by infer_instance"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherianRing (MvPolynomial (Fin n) R)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"tactic": "rw [isNoetherianRing_iff] at hnoet"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\nthis : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule\n⊢ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"tactic": "letI : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule"
},
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\nthis✝ : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule\nthis : Submodule.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\nthis : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule\n⊢ Ideal.FG (RingHom.ker ↑f)",
"tactic": "have := hnoet.noetherian (RingHom.ker f.toRingHom)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\nhnoet : IsNoetherian (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R)\nthis✝ : Module (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) R) := Semiring.toModule\nthis : Submodule.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑f)",
"tactic": "convert this"
},
{
"state_after": "no goals",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsNoetherianRing R\nh : FiniteType R A\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f\n⊢ IsNoetherianRing (MvPolynomial (Fin n) R)",
"tactic": "infer_instance"
}
] |
[
80,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMin.add
|
[] |
[
369,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
8
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.range_coe_union_infty
|
[] |
[
122,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Geometry/Euclidean/Inversion.lean
|
EuclideanGeometry.dist_inversion_center
|
[
{
"state_after": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d x✝ y z : P\nR✝ : ℝ\nx : P\nR : ℝ\n⊢ dist (inversion x R x) x = R ^ 2 / dist x x\n\ncase inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR✝ : ℝ\nc x : P\nR : ℝ\nhx : x ≠ c\n⊢ dist (inversion c R x) c = R ^ 2 / dist x c",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR✝ : ℝ\nc x : P\nR : ℝ\n⊢ dist (inversion c R x) c = R ^ 2 / dist x c",
"tactic": "rcases eq_or_ne x c with (rfl | hx)"
},
{
"state_after": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR✝ : ℝ\nc x : P\nR : ℝ\nhx : x ≠ c\nthis : dist x c ≠ 0\n⊢ dist (inversion c R x) c = R ^ 2 / dist x c",
"state_before": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR✝ : ℝ\nc x : P\nR : ℝ\nhx : x ≠ c\n⊢ dist (inversion c R x) c = R ^ 2 / dist x c",
"tactic": "have : dist x c ≠ 0 := dist_ne_zero.2 hx"
},
{
"state_after": "no goals",
"state_before": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR✝ : ℝ\nc x : P\nR : ℝ\nhx : x ≠ c\nthis : dist x c ≠ 0\n⊢ dist (inversion c R x) c = R ^ 2 / dist x c",
"tactic": "field_simp [inversion, norm_smul, abs_div, ← dist_eq_norm_vsub, sq, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d x✝ y z : P\nR✝ : ℝ\nx : P\nR : ℝ\n⊢ dist (inversion x R x) x = R ^ 2 / dist x x",
"tactic": "simp"
}
] |
[
72,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.div
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.98980\nP : Type ?u.98983\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nw x y z : M\nh1 : ↑c w x\nh2 : ↑c y z\n⊢ ↑c (w / y) (x / z)",
"tactic": "simpa only [div_eq_mul_inv] using c.mul h1 (c.inv h2)"
}
] |
[
1232,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
11
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_conjTranspose
|
[
{
"state_after": "l : Type ?u.201579\nm : Type ?u.201582\nn : Type ?u.201585\no : Type u_4\np : Type ?u.201591\nq : Type ?u.201594\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.201609\nR : Type ?u.201612\nS : Type ?u.201615\nα✝ : Type ?u.201618\nβ : Type ?u.201621\ninst✝⁴ : DecidableEq o\ninst✝³ : Zero α✝\ninst✝² : Zero β\nα : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\nM : (i : o) → Matrix (m' i) (n' i) α\n⊢ map (blockDiagonal' fun k => (M k)ᵀ) star = blockDiagonal' fun k => map (M k)ᵀ star",
"state_before": "l : Type ?u.201579\nm : Type ?u.201582\nn : Type ?u.201585\no : Type u_4\np : Type ?u.201591\nq : Type ?u.201594\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.201609\nR : Type ?u.201612\nS : Type ?u.201615\nα✝ : Type ?u.201618\nβ : Type ?u.201621\ninst✝⁴ : DecidableEq o\ninst✝³ : Zero α✝\ninst✝² : Zero β\nα : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\nM : (i : o) → Matrix (m' i) (n' i) α\n⊢ (blockDiagonal' M)ᴴ = blockDiagonal' fun k => (M k)ᴴ",
"tactic": "simp only [conjTranspose, blockDiagonal'_transpose]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.201579\nm : Type ?u.201582\nn : Type ?u.201585\no : Type u_4\np : Type ?u.201591\nq : Type ?u.201594\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.201609\nR : Type ?u.201612\nS : Type ?u.201615\nα✝ : Type ?u.201618\nβ : Type ?u.201621\ninst✝⁴ : DecidableEq o\ninst✝³ : Zero α✝\ninst✝² : Zero β\nα : Type u_1\ninst✝¹ : AddMonoid α\ninst✝ : StarAddMonoid α\nM : (i : o) → Matrix (m' i) (n' i) α\n⊢ map (blockDiagonal' fun k => (M k)ᵀ) star = blockDiagonal' fun k => map (M k)ᵀ star",
"tactic": "exact blockDiagonal'_map _ star (star_zero α)"
}
] |
[
691,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.bot_coe
|
[] |
[
437,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.map_add_right
|
[
{
"state_after": "case H.h.h\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.989827\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.989833\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.989851\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.990978\nQ' : Type ?u.990981\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] P\ng₁ g₂ : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (mk R M N) (map f (g₁ + g₂))) x✝¹) x✝ = ↑(↑(compr₂ (mk R M N) (map f g₁ + map f g₂)) x✝¹) x✝",
"state_before": "R : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.989827\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.989833\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.989851\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.990978\nQ' : Type ?u.990981\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] P\ng₁ g₂ : N →ₗ[R] Q\n⊢ map f (g₁ + g₂) = map f g₁ + map f g₂",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.h.h\nR : Type u_1\ninst✝¹⁸ : CommSemiring R\nR' : Type ?u.989827\ninst✝¹⁷ : Monoid R'\nR'' : Type ?u.989833\ninst✝¹⁶ : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type ?u.989851\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid N\ninst✝¹³ : AddCommMonoid P\ninst✝¹² : AddCommMonoid Q\ninst✝¹¹ : AddCommMonoid S\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R N\ninst✝⁸ : Module R P\ninst✝⁷ : Module R Q\ninst✝⁶ : Module R S\ninst✝⁵ : DistribMulAction R' M\ninst✝⁴ : Module R'' M\nP' : Type ?u.990978\nQ' : Type ?u.990981\ninst✝³ : AddCommMonoid P'\ninst✝² : Module R P'\ninst✝¹ : AddCommMonoid Q'\ninst✝ : Module R Q'\nf : M →ₗ[R] P\ng₁ g₂ : N →ₗ[R] Q\nx✝¹ : M\nx✝ : N\n⊢ ↑(↑(compr₂ (mk R M N) (map f (g₁ + g₂))) x✝¹) x✝ = ↑(↑(compr₂ (mk R M N) (map f g₁ + map f g₂)) x✝¹) x✝",
"tactic": "simp only [tmul_add, compr₂_apply, mk_apply, map_tmul, add_apply]"
}
] |
[
800,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
797,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.reindexRange_repr'
|
[
{
"state_after": "ι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx bi : M\ni : ι\nh : ↑b i = bi\n✝ : Nontrivial ((fun x => R) { val := bi, property := (_ : ∃ y, ↑b y = bi) })\n⊢ ↑(↑(reindexRange b).repr x) { val := bi, property := (_ : ∃ y, ↑b y = bi) } = ↑(↑b.repr x) i",
"state_before": "ι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx bi : M\ni : ι\nh : ↑b i = bi\n⊢ ↑(↑(reindexRange b).repr x) { val := bi, property := (_ : ∃ y, ↑b y = bi) } = ↑(↑b.repr x) i",
"tactic": "nontriviality"
},
{
"state_after": "ι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\ni : ι\n✝ : Nontrivial ((fun x => R) { val := ↑b i, property := (_ : ∃ y, ↑b y = ↑b i) })\n⊢ ↑(↑(reindexRange b).repr x) { val := ↑b i, property := (_ : ∃ y, ↑b y = ↑b i) } = ↑(↑b.repr x) i",
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"tactic": "simp only [reindexRange_repr_self]"
},
{
"state_after": "case h.hf\nι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\ni✝ : ι\n✝ : Nontrivial ((fun x => R) { val := ↑b i✝, property := (_ : ∃ y, ↑b y = ↑b i✝) })\ni j : ι\n⊢ Injective fun i => { val := ↑b i, property := (_ : ↑b i ∈ range ↑b) }",
"state_before": "case h\nι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\ni✝ : ι\n✝ : Nontrivial ((fun x => R) { val := ↑b i✝, property := (_ : ∃ y, ↑b y = ↑b i✝) })\ni j : ι\n⊢ ↑(Finsupp.single { val := ↑b i, property := (_ : ↑b i ∈ range ↑b) } 1)\n { val := ↑b j, property := (_ : ∃ y, ↑b y = ↑b j) } =\n ↑(Finsupp.single i 1) j",
"tactic": "apply Finsupp.single_apply_left (f := fun i => (⟨b i, _⟩ : Set.range b))"
},
{
"state_after": "no goals",
"state_before": "case h.hf\nι : Type u_2\nι' : Type ?u.390992\nR : Type u_3\nR₂ : Type ?u.390998\nK : Type ?u.391001\nM : Type u_1\nM' : Type ?u.391007\nM'' : Type ?u.391010\nV : Type u\nV' : Type ?u.391015\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\ni✝ : ι\n✝ : Nontrivial ((fun x => R) { val := ↑b i✝, property := (_ : ∃ y, ↑b y = ↑b i✝) })\ni j : ι\n⊢ Injective fun i => { val := ↑b i, property := (_ : ↑b i ∈ range ↑b) }",
"tactic": "exact fun i j h => b.injective (Subtype.mk.inj h)"
}
] |
[
507,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_add
|
[] |
[
499,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/Algebra/Lie/Normalizer.lean
|
LieSubalgebra.normalizer_eq_self_iff
|
[
{
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"tactic": "rw [LieSubmodule.eq_bot_iff]"
},
{
"state_after": "case refine'_1\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\n⊢ ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\n\ncase refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\n⊢ normalizer H ≤ H",
"state_before": "R : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\n⊢ normalizer H = H ↔\n ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0",
"tactic": "refine' ⟨fun h => _, fun h => le_antisymm _ H.le_normalizer⟩"
},
{
"state_after": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ Quot.mk Setoid.r x = 0",
"state_before": "case refine'_1\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\n⊢ ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0",
"tactic": "rintro ⟨x⟩ hx"
},
{
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"state_before": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ Quot.mk Setoid.r x = 0",
"tactic": "suffices x ∈ H by rwa [Submodule.Quotient.quot_mk_eq_mk, Submodule.Quotient.mk_eq_zero,\n coe_toLieSubmodule, mem_coe_submodule]"
},
{
"state_after": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ ∀ (y : L), y ∈ H → ⁅y, x⁆ ∈ H",
"state_before": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ x ∈ H",
"tactic": "rw [← h, H.mem_normalizer_iff']"
},
{
"state_after": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\ny : L\nhy : y ∈ H\n⊢ ⁅y, x⁆ ∈ H",
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"tactic": "intro y hy"
},
{
"state_after": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx y : L\nhy : y ∈ H\nhx : ⁅{ val := y, property := hy }, ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x⁆ = 0\n⊢ ⁅y, x⁆ ∈ H",
"state_before": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\ny : L\nhy : y ∈ H\n⊢ ⁅y, x⁆ ∈ H",
"tactic": "replace hx : ⁅_, LieSubmodule.Quotient.mk' _ x⁆ = 0 := hx ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx y : L\nhy : y ∈ H\nhx : ⁅{ val := y, property := hy }, ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x⁆ = 0\n⊢ ⁅y, x⁆ ∈ H",
"tactic": "rwa [← LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero] at hx"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : normalizer H = H\nm✝ : L ⧸ toLieSubmodule H\nx : L\nhx : Quot.mk Setoid.r x ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\nthis : x ∈ H\n⊢ Quot.mk Setoid.r x = 0",
"tactic": "rwa [Submodule.Quotient.quot_mk_eq_mk, Submodule.Quotient.mk_eq_zero,\ncoe_toLieSubmodule, mem_coe_submodule]"
},
{
"state_after": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\n⊢ x ∈ H",
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"tactic": "intro x hx"
},
{
"state_after": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\n⊢ x ∈ H",
"state_before": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\n⊢ x ∈ H",
"tactic": "let y := LieSubmodule.Quotient.mk' H.toLieSubmodule x"
},
{
"state_after": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nhy : y ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ x ∈ H",
"state_before": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\n⊢ x ∈ H",
"tactic": "have hy : y ∈ LieModule.maxTrivSubmodule R H (L ⧸ H.toLieSubmodule) := by\n rintro ⟨z, hz⟩\n rw [← LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero, coe_bracket_of_module,\n Submodule.coe_mk, mem_toLieSubmodule]\n exact (H.mem_normalizer_iff' x).mp hx z hz"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nhy : y ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)\n⊢ x ∈ H",
"tactic": "simpa using h y hy"
},
{
"state_after": "case mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z ∈ H\n⊢ ⁅{ val := z, property := hz }, y⁆ = 0",
"state_before": "R : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\n⊢ y ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H)",
"tactic": "rintro ⟨z, hz⟩"
},
{
"state_after": "case mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z ∈ H\n⊢ ⁅z, x⁆ ∈ H",
"state_before": "case mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z ∈ H\n⊢ ⁅{ val := z, property := hz }, y⁆ = 0",
"tactic": "rw [← LieModuleHom.map_lie, LieSubmodule.Quotient.mk_eq_zero, coe_bracket_of_module,\n Submodule.coe_mk, mem_toLieSubmodule]"
},
{
"state_after": "no goals",
"state_before": "case mk\nR : Type u_2\nL : Type u_1\nM : Type ?u.62528\nM' : Type ?u.62531\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieModule R L M'\nH : LieSubalgebra R L\nh : ∀ (m : L ⧸ toLieSubmodule H), m ∈ LieModule.maxTrivSubmodule R { x // x ∈ H } (L ⧸ toLieSubmodule H) → m = 0\nx : L\nhx : x ∈ normalizer H\ny : (fun x => L ⧸ toLieSubmodule H) x := ↑(LieSubmodule.Quotient.mk' (toLieSubmodule H)) x\nz : L\nhz : z ∈ H\n⊢ ⁅z, x⁆ ∈ H",
"tactic": "exact (H.mem_normalizer_iff' x).mp hx z hz"
}
] |
[
189,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/Connected.lean
|
isPreconnected_univ_pi
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u v : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\n⊢ IsPreconnected (pi univ s)",
"tactic": "rintro u v uo vo hsuv ⟨f, hfs, hfu⟩ ⟨g, hgs, hgv⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI : Finset.piecewise I f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "rcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with ⟨I, hI⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.empty\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI✝ : Finset.piecewise I f g ∈ u\nhI : Finset.piecewise ∅ f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))\n\ncase intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : Finset.piecewise (insert i I) f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI : Finset.piecewise I f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "induction' I using Finset.induction_on with i I _ ihI"
},
{
"state_after": "case intro.intro.intro.intro.intro.empty\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI✝ : Finset.piecewise I f g ∈ u\nhI : Finset.piecewise ∅ f g ∈ u\n⊢ g ∈ u",
"state_before": "case intro.intro.intro.intro.intro.empty\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI✝ : Finset.piecewise I f g ∈ u\nhI : Finset.piecewise ∅ f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "refine' ⟨g, hgs, ⟨_, hgv⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.empty\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI : Finset ι\nhI✝ : Finset.piecewise I f g ∈ u\nhI : Finset.piecewise ∅ f g ∈ u\n⊢ g ∈ u",
"tactic": "simpa using hI"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : Finset.piecewise (insert i I) f g ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "rw [Finset.piecewise_insert] at hI"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "have := I.piecewise_mem_set_pi hfs hgs"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "refine' (hsuv this).elim ihI fun h => _"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "set S := update (I.piecewise f g) i '' s i"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "have hsub : S ⊆ pi univ s := by\n refine' image_subset_iff.2 fun z hz => _\n rwa [update_preimage_univ_pi]\n exact fun j _ => this j trivial"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "have hconn : IsPreconnected S :=\n (hs i).image _ (continuous_const.update i continuous_id).continuousOn"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "have hSu : (S ∩ u).Nonempty := ⟨_, mem_image_of_mem _ (hfs _ trivial), hI⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\nhSv : Set.Nonempty (S ∩ v)\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "have hSv : (S ∩ v).Nonempty := ⟨_, ⟨_, this _ trivial, update_eq_self _ _⟩, h⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\nhSv : Set.Nonempty (S ∩ v)\n⊢ S ∩ (u ∩ v) ⊆ pi univ s ∩ (u ∩ v)",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\nhSv : Set.Nonempty (S ∩ v)\n⊢ Set.Nonempty (pi univ s ∩ (u ∩ v))",
"tactic": "refine' (hconn u v uo vo (hsub.trans hsuv) hSu hSv).mono _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.insert\nα : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nhsub : S ⊆ pi univ s\nhconn : IsPreconnected S\nhSu : Set.Nonempty (S ∩ u)\nhSv : Set.Nonempty (S ∩ v)\n⊢ S ∩ (u ∩ v) ⊆ pi univ s ∩ (u ∩ v)",
"tactic": "exact inter_subset_inter_left _ hsub"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nz : π i\nhz : z ∈ s i\n⊢ z ∈ update (Finset.piecewise I f g) i ⁻¹' pi univ s",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\n⊢ S ⊆ pi univ s",
"tactic": "refine' image_subset_iff.2 fun z hz => _"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nz : π i\nhz : z ∈ s i\n⊢ ∀ (j : ι), j ≠ i → Finset.piecewise I f g j ∈ s j",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nz : π i\nhz : z ∈ s i\n⊢ z ∈ update (Finset.piecewise I f g) i ⁻¹' pi univ s",
"tactic": "rwa [update_preimage_univ_pi]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type u_2\nπ : ι → Type u_1\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : (i : ι) → TopologicalSpace (π i)\ns : (i : ι) → Set (π i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → π i)\nuo : IsOpen u\nvo : IsOpen v\nhsuv : pi univ s ⊆ u ∪ v\nf : (i : ι) → π i\nhfs : f ∈ pi univ s\nhfu : f ∈ u\ng : (i : ι) → π i\nhgs : g ∈ pi univ s\nhgv : g ∈ v\nI✝ : Finset ι\nhI✝ : Finset.piecewise I✝ f g ∈ u\ni : ι\nI : Finset ι\na✝ : ¬i ∈ I\nihI : Finset.piecewise I f g ∈ u → Set.Nonempty (pi univ s ∩ (u ∩ v))\nhI : update (Finset.piecewise I f g) i (f i) ∈ u\nthis : Finset.piecewise I f g ∈ pi univ s\nh : Finset.piecewise I f g ∈ v\nS : Set ((a : ι) → π a) := update (Finset.piecewise I f g) i '' s i\nz : π i\nhz : z ∈ s i\n⊢ ∀ (j : ι), j ≠ i → Finset.piecewise I f g j ∈ s j",
"tactic": "exact fun j _ => this j trivial"
}
] |
[
503,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.cos_two_mul'
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ cos (2 * x) = cos x ^ 2 - sin x ^ 2",
"tactic": "rw [two_mul, cos_add, ← sq, ← sq]"
}
] |
[
1029,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1029,
1
] |
Mathlib/Computability/EpsilonNFA.lean
|
εNFA.evalFrom_append_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nσ σ' : Type v\nM : εNFA α σ\nS✝ : Set σ\nx✝ : List α\ns : σ\na✝ : α\nS : Set σ\nx : List α\na : α\n⊢ evalFrom M S (x ++ [a]) = stepSet M (evalFrom M S x) a",
"tactic": "rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]"
}
] |
[
110,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorPrebundle.to_vectorBundle
|
[
{
"state_after": "case mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\n⊢ Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)",
"state_before": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n⊢ ∀ (e : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e], Trivialization.IsLinear R e",
"tactic": "rintro _ ⟨e, he, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\n⊢ Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)",
"tactic": "apply linear_trivializationOfMemPretrivializationAtlas"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\n⊢ ContinuousOn\n (fun b =>\n ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b))\n ((FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet)",
"state_before": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\n⊢ ∀ (e e' : Trivialization F TotalSpace.proj) [inst : MemTrivializationAtlas e] [inst_1 : MemTrivializationAtlas e'],\n ContinuousOn (fun b => ↑(Trivialization.coordChangeL R e e' b)) (e.baseSet ∩ e'.baseSet)",
"tactic": "rintro _ _ ⟨e, he, rfl⟩ ⟨e', he', rfl⟩"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\n⊢ ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) =\n coordChange a he he' b",
"state_before": "case mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\n⊢ ContinuousOn\n (fun b =>\n ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b))\n ((FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet)",
"tactic": "refine (a.continuousOn_coordChange he he').congr fun b hb ↦ ?_"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"state_before": "case mk.intro.intro.mk.intro.intro\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\n⊢ ↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b) =\n coordChange a he he' b",
"tactic": "ext v"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"state_before": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"tactic": "haveI h₁ := a.linear_trivializationOfMemPretrivializationAtlas he"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\nh₂ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he')\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"state_before": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"tactic": "haveI h₂ := a.linear_trivializationOfMemPretrivializationAtlas he'"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"state_before": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he)\nh₂ : Trivialization.IsLinear R (trivializationOfMemPretrivializationAtlas a he')\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"tactic": "rw [trivializationOfMemPretrivializationAtlas] at h₁ h₂"
},
{
"state_after": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ (↑(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n (totalSpaceMk b\n (Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b\n v))).snd =\n (↑e' (totalSpaceMk b (Pretrivialization.symm e b v))).snd\n\ncase mk.intro.intro.mk.intro.intro.h.hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ b ∈\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet",
"state_before": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ ↑↑(Trivialization.coordChangeL R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he') b)\n v =\n ↑(coordChange a he he' b) v",
"tactic": "rw [a.coordChange_apply he he' hb v, ContinuousLinearEquiv.coe_coe,\n Trivialization.coordChangeL_apply]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro.mk.intro.intro.h\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ (↑(FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n (totalSpaceMk b\n (Trivialization.symm (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he) b\n v))).snd =\n (↑e' (totalSpaceMk b (Pretrivialization.symm e b v))).snd\n\ncase mk.intro.intro.mk.intro.intro.h.hb\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebundle R F E\nthis✝ : TopologicalSpace (TotalSpace E) := totalSpaceTopology a\nthis : FiberBundle F E := toFiberBundle a\ne : Pretrivialization F TotalSpace.proj\nhe : e ∈ (toFiberPrebundle a).pretrivializationAtlas\ne' : Pretrivialization F TotalSpace.proj\nhe' : e' ∈ (toFiberPrebundle a).pretrivializationAtlas\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\nv : F\nh₁ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he)\nh₂ : Trivialization.IsLinear R (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he')\n⊢ b ∈\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he).baseSet ∩\n (FiberPrebundle.trivializationOfMemPretrivializationAtlas (toFiberPrebundle a) he').baseSet",
"tactic": "exacts [rfl, hb]"
}
] |
[
992,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_restrict_uIoc_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.565282\nγ : Type ?u.565285\nδ : Type ?u.565288\nι : Type ?u.565291\nR : Type ?u.565294\nR' : Type ?u.565297\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : LinearOrder α\na b : α\n⊢ ae (Measure.restrict μ (Ι a b)) = ae (Measure.restrict μ (Ioc a b)) ⊔ ae (Measure.restrict μ (Ioc b a))",
"tactic": "simp only [uIoc_eq_union, ae_restrict_union_eq]"
}
] |
[
2787,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2785,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.coe_alternatization
|
[
{
"state_after": "case a\nR : Type u_2\ninst✝¹² : Semiring R\nM : Type u_3\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type ?u.719894\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nP : Type ?u.719924\ninst✝⁷ : AddCommMonoid P\ninst✝⁶ : Module R P\nM' : Type ?u.719954\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nN' : Type u_4\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nι : Type u_1\nι' : Type ?u.720733\nι'' : Type ?u.720736\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\na : AlternatingMap R M N' ι\n⊢ ↑(↑MultilinearMap.alternatization ↑a) = ↑(Nat.factorial (Fintype.card ι) • a)",
"state_before": "R : Type u_2\ninst✝¹² : Semiring R\nM : Type u_3\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type ?u.719894\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nP : Type ?u.719924\ninst✝⁷ : AddCommMonoid P\ninst✝⁶ : Module R P\nM' : Type ?u.719954\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nN' : Type u_4\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nι : Type u_1\nι' : Type ?u.720733\nι'' : Type ?u.720736\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\na : AlternatingMap R M N' ι\n⊢ ↑MultilinearMap.alternatization ↑a = Nat.factorial (Fintype.card ι) • a",
"tactic": "apply AlternatingMap.coe_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_2\ninst✝¹² : Semiring R\nM : Type u_3\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\nN : Type ?u.719894\ninst✝⁹ : AddCommMonoid N\ninst✝⁸ : Module R N\nP : Type ?u.719924\ninst✝⁷ : AddCommMonoid P\ninst✝⁶ : Module R P\nM' : Type ?u.719954\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nN' : Type u_4\ninst✝³ : AddCommGroup N'\ninst✝² : Module R N'\nι : Type u_1\nι' : Type ?u.720733\nι'' : Type ?u.720736\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\na : AlternatingMap R M N' ι\n⊢ ↑(↑MultilinearMap.alternatization ↑a) = ↑(Nat.factorial (Fintype.card ι) • a)",
"tactic": "simp_rw [MultilinearMap.alternatization_def, ← coe_domDomCongr, domDomCongr_perm, coe_smul,\n smul_smul, Int.units_mul_self, one_smul, Finset.sum_const, Finset.card_univ, Fintype.card_perm,\n ← coe_multilinearMap, coe_smul]"
}
] |
[
906,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
900,
1
] |
Mathlib/Data/Bool/Count.lean
|
List.count_true_add_count_false
|
[] |
[
47,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Topology/Algebra/Order/Compact.lean
|
IsCompact.isGreatest_sSup
|
[] |
[
184,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_self_const_mul'
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.325839\nE : Type ?u.325842\nF : Type ?u.325845\nG : Type ?u.325848\nE' : Type ?u.325851\nF' : Type ?u.325854\nG' : Type ?u.325857\nE'' : Type ?u.325860\nF'' : Type ?u.325863\nG'' : Type ?u.325866\nR : Type u_1\nR' : Type ?u.325872\n𝕜 : Type ?u.325875\n𝕜' : Type ?u.325878\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nu : Rˣ\nf : α → R\nl : Filter α\n⊢ ∀ (x : α), ↑u⁻¹ * (↑u * f x) = f x",
"state_before": "α : Type u_2\nβ : Type ?u.325839\nE : Type ?u.325842\nF : Type ?u.325845\nG : Type ?u.325848\nE' : Type ?u.325851\nF' : Type ?u.325854\nG' : Type ?u.325857\nE'' : Type ?u.325860\nF'' : Type ?u.325863\nG'' : Type ?u.325866\nR : Type u_1\nR' : Type ?u.325872\n𝕜 : Type ?u.325875\n𝕜' : Type ?u.325878\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nu : Rˣ\nf : α → R\nl : Filter α\n⊢ IsBigOWith ‖↑u⁻¹‖ l f fun x => ↑u * f x",
"tactic": "refine' (isBigOWith_const_mul_self ↑u⁻¹ _ l).congr_left _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.325839\nE : Type ?u.325842\nF : Type ?u.325845\nG : Type ?u.325848\nE' : Type ?u.325851\nF' : Type ?u.325854\nG' : Type ?u.325857\nE'' : Type ?u.325860\nF'' : Type ?u.325863\nG'' : Type ?u.325866\nR : Type u_1\nR' : Type ?u.325872\n𝕜 : Type ?u.325875\n𝕜' : Type ?u.325878\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nu : Rˣ\nf : α → R\nl : Filter α\n⊢ ∀ (x : α), ↑u⁻¹ * (↑u * f x) = f x",
"tactic": "exact fun x => u.inv_mul_cancel_left (f x)"
}
] |
[
1462,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1459,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Frequently.exists
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ¬∃ x, p x\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\n⊢ ∃ x, p x",
"tactic": "by_contra H"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ¬∃ x, p x\n⊢ ∀ᶠ (x : α) in f, ¬p x\n\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ∀ᶠ (x : α) in f, ¬p x\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ¬∃ x, p x\n⊢ False",
"tactic": "replace H : ∀ᶠ x in f, ¬p x"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ∀ᶠ (x : α) in f, ¬p x\n⊢ False",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ¬∃ x, p x\n⊢ ∀ᶠ (x : α) in f, ¬p x\n\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ∀ᶠ (x : α) in f, ¬p x\n⊢ False",
"tactic": "exact eventually_of_forall (not_exists.1 H)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.160311\nι : Sort x\np : α → Prop\nf : Filter α\nhp : ∃ᶠ (x : α) in f, p x\nH : ∀ᶠ (x : α) in f, ¬p x\n⊢ False",
"tactic": "exact hp H"
}
] |
[
1295,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1292,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.coe_fst
|
[] |
[
118,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
LocallyConstant.toFun_eq_coe
|
[] |
[
265,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean
|
CategoryTheory.Monoidal.tensorObj_obj
|
[] |
[
93,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/Nat/Cast/Defs.lean
|
one_add_one_eq_two
|
[
{
"state_after": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ ↑(1 + 1) = 2",
"state_before": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 1 + 1 = 2",
"tactic": "rw [←Nat.cast_one, ←Nat.cast_add]"
},
{
"state_after": "case h\nα : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 1 + 1 = 2",
"state_before": "α : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ ↑(1 + 1) = 2",
"tactic": "apply congrArg"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : AddMonoidWithOne α\n⊢ 1 + 1 = 2",
"tactic": "decide"
}
] |
[
230,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
CategoryTheory.Limits.initialMul_inv
|
[] |
[
157,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_subset_Ioo_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.25908\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\nh₁ : a₁ ≤ b₁\n⊢ Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂",
"tactic": "rw [← coe_subset, coe_Icc, coe_Ioo, Set.Icc_subset_Ioo_iff h₁]"
}
] |
[
254,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Order/GameAdd.lean
|
Sym2.GameAdd.fst
|
[] |
[
183,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.cpow_add
|
[
{
"state_after": "x y z : ℂ\nhx : x ≠ 0\n⊢ (if x = 0 then if y + z = 0 then 1 else 0 else exp (log x * (y + z))) =\n if x = 0 then\n if z = 0 then if x = 0 then if y = 0 then 1 * 1 else 0 * 1 else exp (log x * y) * 1\n else if x = 0 then if y = 0 then 1 * 0 else 0 * 0 else exp (log x * y) * 0\n else if x = 0 then if y = 0 then 1 * exp (log x * z) else 0 * exp (log x * z) else exp (log x * y) * exp (log x * z)",
"state_before": "x y z : ℂ\nhx : x ≠ 0\n⊢ x ^ (y + z) = x ^ y * x ^ z",
"tactic": "simp only [cpow_def, ite_mul, boole_mul, mul_ite, mul_boole]"
},
{
"state_after": "no goals",
"state_before": "x y z : ℂ\nhx : x ≠ 0\n⊢ (if x = 0 then if y + z = 0 then 1 else 0 else exp (log x * (y + z))) =\n if x = 0 then\n if z = 0 then if x = 0 then if y = 0 then 1 * 1 else 0 * 1 else exp (log x * y) * 1\n else if x = 0 then if y = 0 then 1 * 0 else 0 * 0 else exp (log x * y) * 0\n else if x = 0 then if y = 0 then 1 * exp (log x * z) else 0 * exp (log x * z) else exp (log x * y) * exp (log x * z)",
"tactic": "simp_all [exp_add, mul_add]"
}
] |
[
96,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Algebra/Order/Interval.lean
|
Interval.bot_ne_one
|
[] |
[
103,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/MeasureTheory/Function/Floor.lean
|
MeasurableSet.image_fract
|
[
{
"state_after": "α : Type ?u.4493\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedRing R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n⊢ MeasurableSet (⋃ (m : ℤ), (fun x => x + ↑m) ⁻¹' s ∩ Ico 0 1)",
"state_before": "α : Type ?u.4493\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedRing R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n⊢ MeasurableSet (Int.fract '' s)",
"tactic": "simp only [Int.image_fract, sub_eq_add_neg, image_add_right']"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.4493\nR : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : LinearOrderedRing R\ninst✝⁴ : FloorRing R\ninst✝³ : TopologicalSpace R\ninst✝² : OrderTopology R\ninst✝¹ : MeasurableSpace R\ninst✝ : BorelSpace R\ns : Set R\nhs : MeasurableSet s\n⊢ MeasurableSet (⋃ (m : ℤ), (fun x => x + ↑m) ⁻¹' s ∩ Ico 0 1)",
"tactic": "exact MeasurableSet.iUnion fun m => (measurable_add_const _ hs).inter measurableSet_Ico"
}
] |
[
65,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
compl_sup
|
[] |
[
795,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
794,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj.eq
|
[] |
[
48,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
11
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.FinStronglyMeasurable.add
|
[] |
[
1076,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1070,
11
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.singleton_disjoint
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.482994\nγ : Type ?u.482997\nl : Multiset α\na : α\n⊢ Disjoint {a} l ↔ ¬a ∈ l",
"tactic": "simp [Disjoint]"
}
] |
[
2939,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2938,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
min_mul_of_nonneg
|
[] |
[
981,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
980,
1
] |
Mathlib/GroupTheory/Subgroup/Finite.lean
|
Subgroup.prod_mem
|
[] |
[
80,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
11
] |
Mathlib/Analysis/Complex/OperatorNorm.lean
|
Complex.imClm_nnnorm
|
[] |
[
63,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Fintype.finsetEquivSet_apply
|
[] |
[
1049,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1048,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
deriv.lhopital_zero_nhds'
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, DifferentiableAt ℝ f x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, DifferentiableAt ℝ f x\nhg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, deriv g x ≠ 0) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : ∀ᶠ (x : ℝ) in 𝓝[{a}ᶜ] a, DifferentiableAt ℝ f x\nhg' : ∀ᶠ (x : ℝ) in 𝓝[{a}ᶜ] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[{a}ᶜ] a) (𝓝 0)\nhga : Tendsto g (𝓝[{a}ᶜ] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[{a}ᶜ] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[{a}ᶜ] a) l",
"tactic": "simp only [← Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, DifferentiableAt ℝ f x) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, DifferentiableAt ℝ f x\nhg' : (∀ᶠ (x : ℝ) in 𝓝[Iio a] a, deriv g x ≠ 0) ∧ ∀ᶠ (x : ℝ) in 𝓝[Ioi a] a, deriv g x ≠ 0\nhfa : Tendsto f (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto f (𝓝[Ioi a] a) (𝓝 0)\nhga : Tendsto g (𝓝[Iio a] a) (𝓝 0) ∧ Tendsto g (𝓝[Ioi a] a) (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio a] a) l ∧ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "exact ⟨lhopital_zero_nhds_left hdf.1 hg'.1 hfa.1 hga.1 hdiv.1,\n lhopital_zero_nhds_right hdf.2 hg'.2 hfa.2 hga.2 hdiv.2⟩"
}
] |
[
423,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.monic_monomial_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m✝ : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np q : MvPolynomial σ R\nm : σ →₀ ℕ\n⊢ ↑(monomial m) 1 = Finsupp.prod m fun n e => X n ^ e",
"tactic": "simp [monomial_eq]"
}
] |
[
645,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigO.eq_zero_of_norm_pow
|
[
{
"state_after": "α : Type ?u.656218\nβ : Type ?u.656221\nE : Type ?u.656224\nF : Type ?u.656227\nG : Type ?u.656230\nE' : Type ?u.656233\nF' : Type ?u.656236\nG' : Type ?u.656239\nE'' : Type u_1\nF'' : Type u_2\nG'' : Type ?u.656248\nR : Type ?u.656251\nR' : Type ?u.656254\n𝕜 : Type ?u.656257\n𝕜' : Type ?u.656260\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : E'' → F''\nx₀ : E''\nn : ℕ\nh : f =O[𝓝[univ] x₀] fun x => ‖x - x₀‖ ^ n\nhn : 0 < n\n⊢ f x₀ = 0",
"state_before": "α : Type ?u.656218\nβ : Type ?u.656221\nE : Type ?u.656224\nF : Type ?u.656227\nG : Type ?u.656230\nE' : Type ?u.656233\nF' : Type ?u.656236\nG' : Type ?u.656239\nE'' : Type u_1\nF'' : Type u_2\nG'' : Type ?u.656248\nR : Type ?u.656251\nR' : Type ?u.656254\n𝕜 : Type ?u.656257\n𝕜' : Type ?u.656260\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : E'' → F''\nx₀ : E''\nn : ℕ\nh : f =O[𝓝 x₀] fun x => ‖x - x₀‖ ^ n\nhn : 0 < n\n⊢ f x₀ = 0",
"tactic": "rw [← nhdsWithin_univ] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.656218\nβ : Type ?u.656221\nE : Type ?u.656224\nF : Type ?u.656227\nG : Type ?u.656230\nE' : Type ?u.656233\nF' : Type ?u.656236\nG' : Type ?u.656239\nE'' : Type u_1\nF'' : Type u_2\nG'' : Type ?u.656248\nR : Type ?u.656251\nR' : Type ?u.656254\n𝕜 : Type ?u.656257\n𝕜' : Type ?u.656260\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf✝ : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf : E'' → F''\nx₀ : E''\nn : ℕ\nh : f =O[𝓝[univ] x₀] fun x => ‖x - x₀‖ ^ n\nhn : 0 < n\n⊢ f x₀ = 0",
"tactic": "exact h.eq_zero_of_norm_pow_within (mem_univ _) hn"
}
] |
[
2054,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2051,
1
] |
Mathlib/RingTheory/Polynomial/Quotient.lean
|
Ideal.eq_zero_of_polynomial_mem_map_range
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\n⊢ x = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ x = 0",
"tactic": "let i := ((Quotient.mk I).comp C).rangeRestrict"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\n⊢ x = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\n⊢ x = 0",
"tactic": "have hi' : RingHom.ker (Polynomial.mapRingHom i) ≤ I := by\n refine' fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => _\n rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← RingHom.comp_apply]\n rw [RingHom.mem_ker, coe_mapRingHom] at hf\n replace hf := congr_arg (fun f : Polynomial _ => f.coeff n) hf\n simp only [coeff_map, coeff_zero] at hf\n rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf"
},
{
"state_after": "case mk\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\nx : R[X] ⧸ I\nhx' : x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx : ↑C { val := x, property := hx' } ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ { val := x, property := hx' } = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\n⊢ x = 0",
"tactic": "obtain ⟨x, hx'⟩ := x"
},
{
"state_after": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } = 0",
"state_before": "case mk\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\nx : R[X] ⧸ I\nhx' : x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx : ↑C { val := x, property := hx' } ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ { val := x, property := hx' } = 0",
"tactic": "obtain ⟨y, rfl⟩ := RingHom.mem_range.1 hx'"
},
{
"state_after": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑{ val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } = ↑0",
"state_before": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } = 0",
"tactic": "refine' Subtype.eq _"
},
{
"state_after": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑C y ∈ I",
"state_before": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑{ val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } = ↑0",
"tactic": "simp only [RingHom.comp_apply, Quotient.eq_zero_iff_mem, ZeroMemClass.coe_zero]"
},
{
"state_after": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑C (↑i y) ∈ map (mapRingHom i) I",
"state_before": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑C y ∈ I",
"tactic": "suffices C (i y) ∈ I.map (Polynomial.mapRingHom i) by\n obtain ⟨f, hf⟩ := mem_image_of_mem_map_of_surjective (Polynomial.mapRingHom i)\n (Polynomial.map_surjective _ (RingHom.rangeRestrict_surjective ((Quotient.mk I).comp C))) this\n refine' sub_add_cancel (C y) f ▸ I.add_mem (hi' _ : C y - f ∈ I) hf.1\n rw [RingHom.mem_ker, RingHom.map_sub, hf.2, sub_eq_zero, coe_mapRingHom, map_C]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\n⊢ ↑C (↑i y) ∈ map (mapRingHom i) I",
"tactic": "exact hx"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : f ∈ RingHom.ker (mapRingHom i)\nn : ℕ\n⊢ coeff f n ∈ comap C I",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\n⊢ RingHom.ker (mapRingHom i) ≤ I",
"tactic": "refine' fun f hf => polynomial_mem_ideal_of_coeff_mem_ideal I f fun n => _"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : f ∈ RingHom.ker (mapRingHom i)\nn : ℕ\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : f ∈ RingHom.ker (mapRingHom i)\nn : ℕ\n⊢ coeff f n ∈ comap C I",
"tactic": "rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← RingHom.comp_apply]"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : Polynomial.map i f = 0\nn : ℕ\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : f ∈ RingHom.ker (mapRingHom i)\nn : ℕ\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"tactic": "rw [RingHom.mem_ker, coe_mapRingHom] at hf"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nn : ℕ\nhf : (fun f => coeff f n) (Polynomial.map i f) = (fun f => coeff f n) 0\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nhf : Polynomial.map i f = 0\nn : ℕ\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"tactic": "replace hf := congr_arg (fun f : Polynomial _ => f.coeff n) hf"
},
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nn : ℕ\nhf : ↑(RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)) (coeff f n) = 0\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nn : ℕ\nhf : (fun f => coeff f n) (Polynomial.map i f) = (fun f => coeff f n) 0\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"tactic": "simp only [coeff_map, coeff_zero] at hf"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\nx : { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) }\nhx : ↑C x ∈ map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nf : R[X]\nn : ℕ\nhf : ↑(RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)) (coeff f n) = 0\n⊢ ↑(RingHom.comp (Quotient.mk I) C) (coeff f n) = 0",
"tactic": "rwa [Subtype.ext_iff, RingHom.coe_rangeRestrict] at hf"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\nthis : ↑C (↑i y) ∈ map (mapRingHom i) I\nf : R[X]\nhf : f ∈ ↑I ∧ ↑(mapRingHom i) f = ↑C (↑i y)\n⊢ ↑C y ∈ I",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\nthis : ↑C (↑i y) ∈ map (mapRingHom i) I\n⊢ ↑C y ∈ I",
"tactic": "obtain ⟨f, hf⟩ := mem_image_of_mem_map_of_surjective (Polynomial.mapRingHom i)\n (Polynomial.map_surjective _ (RingHom.rangeRestrict_surjective ((Quotient.mk I).comp C))) this"
},
{
"state_after": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\nthis : ↑C (↑i y) ∈ map (mapRingHom i) I\nf : R[X]\nhf : f ∈ ↑I ∧ ↑(mapRingHom i) f = ↑C (↑i y)\n⊢ ↑C y - f ∈ RingHom.ker (mapRingHom i)",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\nthis : ↑C (↑i y) ∈ map (mapRingHom i) I\nf : R[X]\nhf : f ∈ ↑I ∧ ↑(mapRingHom i) f = ↑C (↑i y)\n⊢ ↑C y ∈ I",
"tactic": "refine' sub_add_cancel (C y) f ▸ I.add_mem (hi' _ : C y - f ∈ I) hf.1"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_1\ninst✝ : CommRing R\nI : Ideal R[X]\ni : R →+* { x // x ∈ RingHom.range (RingHom.comp (Quotient.mk I) C) } :=\n RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C)\nhi' : RingHom.ker (mapRingHom i) ≤ I\ny : R\nhx' : ↑(RingHom.comp (Quotient.mk I) C) y ∈ RingHom.range (RingHom.comp (Quotient.mk I) C)\nhx :\n ↑C { val := ↑(RingHom.comp (Quotient.mk I) C) y, property := hx' } ∈\n map (mapRingHom (RingHom.rangeRestrict (RingHom.comp (Quotient.mk I) C))) I\nthis : ↑C (↑i y) ∈ map (mapRingHom i) I\nf : R[X]\nhf : f ∈ ↑I ∧ ↑(mapRingHom i) f = ↑C (↑i y)\n⊢ ↑C y - f ∈ RingHom.ker (mapRingHom i)",
"tactic": "rw [RingHom.mem_ker, RingHom.map_sub, hf.2, sub_eq_zero, coe_mapRingHom, map_C]"
}
] |
[
186,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
CategoryTheory.Limits.Types.equalizerIso_hom_comp_subtype
|
[
{
"state_after": "no goals",
"state_before": "X Y Z : Type u\nf : X ⟶ Y\ng h : Y ⟶ Z\nw : f ≫ g = f ≫ h\n⊢ (equalizerIso g h).hom ≫ Subtype.val = equalizer.ι g h",
"tactic": "rfl"
}
] |
[
441,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.normalizer_sup_eq_top
|
[
{
"state_after": "p✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\n⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N",
"state_before": "p✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\n⊢ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N = ⊤",
"tactic": "refine' top_le_iff.mp fun g _ => _"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N",
"state_before": "p✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\n⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N",
"tactic": "obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ (↑n)⁻¹ * (↑n * g) ∈ N ⊔ normalizer (map (Subgroup.subtype N) ↑P)",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ g ∈ normalizer (map (Subgroup.subtype N) ↑P) ⊔ N",
"tactic": "rw [← inv_mul_cancel_left (↑n) g, sup_comm]"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P)",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ (↑n)⁻¹ * (↑n * g) ∈ N ⊔ normalizer (map (Subgroup.subtype N) ↑P)",
"tactic": "apply mul_mem_sup (N.inv_mem n.2)"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\n⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P)",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn : n • ↑MulAut.conjNormal g • P = P\n⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P)",
"tactic": "rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,\n Sylow.ext_iff, Sylow.pointwise_smul_def, Subgroup.pointwise_smul_def] at hn"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\nx : G\n⊢ ↑(MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) x ∈\n map (MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) (map (Subgroup.subtype N) ↑P) ↔\n ↑n * g * x * (↑n * g)⁻¹ ∈ map (Subgroup.subtype N) ↑P",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\n⊢ ↑n * g ∈ normalizer (map (Subgroup.subtype N) ↑P)",
"tactic": "refine' fun x =>\n (mem_map_iff_mem\n (show Function.Injective (MulAut.conj (↑n * g)).toMonoidHom from\n (MulAut.conj (↑n * g)).injective)).symm.trans\n _"
},
{
"state_after": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\nx : G\n⊢ ↑(MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) x ∈\n map (MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) (Subgroup.subtype N)) ↑P ↔\n ↑n * g * x * (↑n * g)⁻¹ ∈\n map\n (MonoidHom.comp (Subgroup.subtype N)\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g))))\n ↑P",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\nx : G\n⊢ ↑(MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) x ∈\n map (MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) (map (Subgroup.subtype N) ↑P) ↔\n ↑n * g * x * (↑n * g)⁻¹ ∈ map (Subgroup.subtype N) ↑P",
"tactic": "rw [map_map, ← congr_arg (map N.subtype) hn, map_map]"
},
{
"state_after": "no goals",
"state_before": "case intro\np✝ : ℕ\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\nN : Subgroup G\ninst✝¹ : Normal N\ninst✝ : Finite (Sylow p { x // x ∈ N })\nP : Sylow p { x // x ∈ N }\ng : G\nx✝ : g ∈ ⊤\nn : { x // x ∈ N }\nhn✝ : (↑MulAut.conj n * ↑MulAut.conjNormal g) • P = P\nhn :\n map\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g)))\n ↑P =\n ↑P\nx : G\n⊢ ↑(MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) x ∈\n map (MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑n * g))) (Subgroup.subtype N)) ↑P ↔\n ↑n * g * x * (↑n * g)⁻¹ ∈\n map\n (MonoidHom.comp (Subgroup.subtype N)\n (↑(MulDistribMulAction.toMonoidEnd ((fun x => MulAut { x // x ∈ N }) n) { x // x ∈ N })\n (↑MulAut.conjNormal (↑n * g))))\n ↑P",
"tactic": "rfl"
}
] |
[
476,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.toOneHom_injective
|
[] |
[
1225,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1223,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
compl_inf_eq_bot
|
[] |
[
886,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
885,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Ici_eq''
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) ⊓ 𝓟 (Ici a) =\n (⨅ (u : α) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[Ici a] a = (⨅ (u : α) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a)",
"tactic": "rw [nhdsWithin, nhds_eq_order]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ (⨅ (u : α) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) ≤ ⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) ⊓ 𝓟 (Ici a) =\n (⨅ (u : α) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a)",
"tactic": "refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ (⨅ (u : α) (_ : a < u), 𝓟 (Iio u)) ⊓ 𝓟 (Ici a) ≤ ⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)",
"tactic": "exact inf_le_right.trans (le_iInf₂ fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl)"
}
] |
[
1087,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1083,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Hom.map_constants
|
[] |
[
541,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetLen_empty
|
[] |
[
287,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_lt_zero
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.61787\nι : Type ?u.61790\nα : Type ?u.61793\nβ : Type ?u.61796\nK : Type u_1\ninst✝ : LinearOrderedField K\nn : ℚ\n⊢ ↑n < 0 ↔ n < 0",
"tactic": "norm_cast"
}
] |
[
348,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_mono
|
[] |
[
73,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
inv_mul_lt_iff_lt_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c : α\n⊢ b⁻¹ * a < c ↔ a < b * c",
"tactic": "rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]"
}
] |
[
182,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/GroupTheory/NielsenSchreier.lean
|
IsFreeGroupoid.SpanningTree.treeHom_root
|
[] |
[
189,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
mul_right_inj
|
[] |
[
197,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.isTheta_const_const_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.49940\nE : Type ?u.49943\nF : Type ?u.49946\nG : Type ?u.49949\nE' : Type ?u.49952\nF' : Type ?u.49955\nG' : Type ?u.49958\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.49967\nR : Type ?u.49970\nR' : Type ?u.49973\n𝕜 : Type ?u.49976\n𝕜' : Type ?u.49979\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc c' c₁✝ c₂✝ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nl l' : Filter α\ninst✝ : NeBot l\nc₁ : E''\nc₂ : F''\n⊢ ((fun x => c₁) =Θ[l] fun x => c₂) ↔ (c₁ = 0 ↔ c₂ = 0)",
"tactic": "simpa only [IsTheta, isBigO_const_const_iff, ← iff_def] using Iff.comm"
}
] |
[
268,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/RepresentationTheory/Maschke.lean
|
LinearMap.conjugate_i
|
[
{
"state_after": "no goals",
"state_before": "k : Type u\ninst✝⁹ : CommRing k\nG : Type u\ninst✝⁸ : Group G\nV : Type v\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : Module (MonoidAlgebra k G) V\ninst✝⁴ : IsScalarTower k (MonoidAlgebra k G) V\nW : Type w\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : Module (MonoidAlgebra k G) W\ninst✝ : IsScalarTower k (MonoidAlgebra k G) W\nπ : W →ₗ[k] V\ni : V →ₗ[MonoidAlgebra k G] W\nh : ∀ (v : V), ↑π (↑i v) = v\ng : G\nv : V\n⊢ ↑(conjugate π g) (↑i v) = v",
"tactic": "rw [conjugate_apply, ← i.map_smul, h, ← mul_smul, single_mul_single, mul_one, mul_left_inv,\n ← one_def, one_smul]"
}
] |
[
87,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.Tendsto.piecewise
|
[] |
[
3107,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3104,
11
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
Left.self_lt_inv
|
[] |
[
474,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.generate_of_singleton_isSplitEpi
|
[] |
[
445,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.forall_ennreal
|
[] |
[
253,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.trans_eventually_abs_le
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\n⊢ ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * g a)) l (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : SuperpolynomialDecay l k f\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\n⊢ SuperpolynomialDecay l k g",
"tactic": "rw [superpolynomialDecay_iff_abs_tendsto_zero] at hf⊢"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ g) x ≤ (abs ∘ f) x\n⊢ abs (k x ^ z * g x) ≤ abs (k x ^ z * f x)",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\n⊢ ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * g a)) l (𝓝 0)",
"tactic": "refine' fun z =>\n tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds (hf z)\n (eventually_of_forall fun x => abs_nonneg _) (hfg.mono fun x hx => _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ g) x ≤ (abs ∘ f) x\n⊢ abs (k x ^ z * g x) ≤ abs (k x ^ z * f x)",
"tactic": "calc\n |k x ^ z * g x| = |k x ^ z| * |g x| := abs_mul (k x ^ z) (g x)\n _ ≤ |k x ^ z| * |f x| := by gcongr; exact hx\n _ = |k x ^ z * f x| := (abs_mul (k x ^ z) (f x)).symm"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ g) x ≤ (abs ∘ f) x\n⊢ abs (g x) ≤ abs (f x)",
"state_before": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ g) x ≤ (abs ∘ f) x\n⊢ abs (k x ^ z) * abs (g x) ≤ abs (k x ^ z) * abs (f x)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nl : Filter α\nk f g g' : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : LinearOrderedCommRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a => abs (k a ^ n * f a)) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ g) x ≤ (abs ∘ f) x\n⊢ abs (g x) ≤ abs (f x)",
"tactic": "exact hx"
}
] |
[
189,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
|
AlgebraicTopology.DoldKan.PInfty_comp_QInfty
|
[
{
"state_after": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (PInfty ≫ QInfty) n = HomologicalComplex.Hom.f 0 n",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\n⊢ PInfty ≫ QInfty = 0",
"tactic": "ext n"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f (PInfty ≫ QInfty) n = HomologicalComplex.Hom.f 0 n",
"tactic": "apply PInfty_f_comp_QInfty_f"
}
] |
[
142,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Data/Subtype.lean
|
exists_subtype_mk_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\nβ : Sort ?u.5647\nγ : Sort ?u.5650\np q : α → Prop\na : Subtype p\nb : α\n⊢ (∃ h, { val := b, property := h } = a) ↔ b = ↑a",
"tactic": "simp only [@eq_comm _ b, exists_eq_subtype_mk_iff, @eq_comm _ _ a]"
}
] |
[
150,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.extend_comp_eq
|
[] |
[
395,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Analysis/Analytic/Linear.lean
|
ContinuousLinearMap.analyticAt
|
[] |
[
66,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
11
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometry.ediam_range
|
[] |
[
329,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Tactic/Abel.lean
|
Mathlib.Tactic.Abel.term_neg
|
[
{
"state_after": "α : Type u_1\ninst✝ : AddCommGroup α\nn : ℤ\nx a : α\nn' : ℤ\na' : α\nh₁ : -n = n'\nh₂ : -a = a'\n⊢ -a + -(n • x) = -(n • x) + -a",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\nn : ℤ\nx a : α\nn' : ℤ\na' : α\nh₁ : -n = n'\nh₂ : -a = a'\n⊢ -termg n x a = termg n' x a'",
"tactic": "simp [h₂.symm, h₁.symm, termg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddCommGroup α\nn : ℤ\nx a : α\nn' : ℤ\na' : α\nh₁ : -n = n'\nh₂ : -a = a'\n⊢ -a + -(n • x) = -(n • x) + -a",
"tactic": "exact add_comm _ _"
}
] |
[
188,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Setoid.eq_of_mem_eqv_class
|
[] |
[
52,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean
|
ContinuousMap.linearIsometryBoundedOfCompact_apply_apply
|
[] |
[
325,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
Commute.smul_left
|
[] |
[
444,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Topology/Algebra/Module/LocallyConvex.lean
|
locallyConvexSpace_induced
|
[
{
"state_after": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\n⊢ LocallyConvexSpace 𝕜 E",
"state_before": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\n⊢ LocallyConvexSpace 𝕜 E",
"tactic": "letI : TopologicalSpace E := t.induced f"
},
{
"state_after": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\nx : E\n⊢ HasBasis (𝓝 x) ((fun x s => s ∈ 𝓝 (↑f x) ∧ Convex 𝕜 s) x) ((fun x => preimage ↑f) x)",
"state_before": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\n⊢ LocallyConvexSpace 𝕜 E",
"tactic": "refine' LocallyConvexSpace.ofBases 𝕜 E (fun _ => preimage f)\n (fun x => fun s : Set F => s ∈ 𝓝 (f x) ∧ Convex 𝕜 s) (fun x => _) fun x s ⟨_, hs⟩ =>\n hs.linear_preimage f"
},
{
"state_after": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\nx : E\n⊢ HasBasis (comap (↑f) (𝓝 (↑f x))) ((fun x s => s ∈ 𝓝 (↑f x) ∧ Convex 𝕜 s) x) ((fun x => preimage ↑f) x)",
"state_before": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\nx : E\n⊢ HasBasis (𝓝 x) ((fun x s => s ∈ 𝓝 (↑f x) ∧ Convex 𝕜 s) x) ((fun x => preimage ↑f) x)",
"tactic": "rw [nhds_induced]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.45774\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module 𝕜 E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 F\nt : TopologicalSpace F\ninst✝ : LocallyConvexSpace 𝕜 F\nf : E →ₗ[𝕜] F\nthis : TopologicalSpace E := induced (↑f) t\nx : E\n⊢ HasBasis (comap (↑f) (𝓝 (↑f x))) ((fun x s => s ∈ 𝓝 (↑f x) ∧ Convex 𝕜 s) x) ((fun x => preimage ↑f) x)",
"tactic": "exact (LocallyConvexSpace.convex_basis <| f x).comap f"
}
] |
[
188,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Analysis/Convex/Jensen.lean
|
ConcaveOn.le_map_sum
|
[] |
[
76,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Std/Data/List/Lemmas.lean
|
List.leftpad_length
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nn : Nat\na : α\nl : List α\n⊢ length (leftpad n a l) = max n (length l)",
"tactic": "simp only [leftpad, length_append, length_replicate, Nat.sub_add_eq_max]"
}
] |
[
1461,
75
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1459,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.IsSymm.neg
|
[] |
[
103,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean
|
CompHaus.effectiveEpiFamily_of_jointly_surjective
|
[] |
[
189,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/GroupTheory/Schreier.lean
|
Subgroup.rank_le_index_mul_rank
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\nthis : Group.FG { x // x ∈ H }\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"tactic": "haveI := H.fG_of_index_ne_zero"
},
{
"state_after": "case intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S✝ : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\nthis : Group.FG { x // x ∈ H }\nS : Finset G\nhS₀ : Finset.card S = Group.rank G\nhS : closure ↑S = ⊤\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\nthis : Group.FG { x // x ∈ H }\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"tactic": "obtain ⟨S, hS₀, hS⟩ := Group.rank_spec G"
},
{
"state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S✝ : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\nthis : Group.FG { x // x ∈ H }\nS : Finset G\nhS₀ : Finset.card S = Group.rank G\nhS : closure ↑S = ⊤\nT : Finset { x // x ∈ H }\nhT₀ : Finset.card T ≤ index H * Finset.card S\nhT : closure ↑T = ⊤\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"state_before": "case intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S✝ : Set G\nhG : Group.FG G\ninst✝ : FiniteIndex H\nthis : Group.FG { x // x ∈ H }\nS : Finset G\nhS₀ : Finset.card S = Group.rank G\nhS : closure ↑S = ⊤\n⊢ Group.rank { x // x ∈ H } ≤ index H * Group.rank G",
"tactic": "obtain ⟨T, hT₀, hT⟩ := exists_finset_card_le_mul H hS"
}
] |
[
146,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.one_lt_ncard_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.156869\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ (∃ a, a ∈ s ∧ ∃ b, b ∈ s ∧ a ≠ b) ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b",
"state_before": "α : Type u_1\nβ : Type ?u.156869\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ 1 < ncard s ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b",
"tactic": "rw [one_lt_ncard hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.156869\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\n⊢ (∃ a, a ∈ s ∧ ∃ b, b ∈ s ∧ a ≠ b) ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b",
"tactic": "simp only [exists_prop, exists_and_left]"
}
] |
[
705,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
702,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.foldlM_eq_foldlM_data
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nβ : Type u_1\nα : Type u_3\ninst✝ : Monad m\nf : β → α → m β\ninit : β\narr : Array α\n⊢ foldlM f init arr 0 (size arr) = List.foldlM f init arr.data",
"tactic": "simp [foldlM, foldlM_eq_foldlM_data.aux]"
}
] |
[
45,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
42,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.max_def
|
[] |
[
517,
80
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
517,
11
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
BilinForm.toQuadraticForm_eq_zero
|
[] |
[
713,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/Data/Finset/Fold.lean
|
Finset.fold_disjiUnion
|
[] |
[
112,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.ofFractionRing_neg
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : CommRing K\np : FractionRing K[X]\n⊢ { toFractionRing := -p } = -{ toFractionRing := p }",
"tactic": "simp only [Neg.neg, RatFunc.neg]"
}
] |
[
356,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
355,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
Subsingleton.aemeasurable
|
[] |
[
33,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
32,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.ValidFor.get
|
[
{
"state_after": "no goals",
"state_before": "l m₁ : List Char\nc : Char\nm₂ r : List Char\n⊢ Substring.get\n { str := { data := l ++ (m₁ ++ c :: m₂) ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len (m₁ ++ c :: m₂) } }\n { byteIdx := utf8Len m₁ } =\n c",
"tactic": "simpa using get_of_valid (l ++ m₁) (c :: m₂ ++ r)"
}
] |
[
827,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
826,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.finite_singleton
|
[] |
[
838,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.iUnion_div_left_image
|
[] |
[
744,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/Init/Data/List/Lemmas.lean
|
List.length_mapAccumr
|
[] |
[
48,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Nat/Choose/Cast.lean
|
Nat.cast_add_choose
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(choose (a + b) a) = ↑(a + b)! / (↑a ! * ↑b !)",
"tactic": "rw [cast_choose K (_root_.le_add_right le_rfl), add_tsub_cancel_left]"
}
] |
[
35,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
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