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/RingTheory/IsTensorProduct.lean
|
IsTensorProduct.equiv_toLinearMap
|
[] |
[
89,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.trivial_covering
|
[] |
[
255,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/BigOperators/Intervals.lean
|
Finset.prod_Ico_succ_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na✝ : α\ng f✝ : α → β\ninst✝ : CommMonoid β\na b : ℕ\nhab : a ≤ b\nf : ℕ → β\n⊢ ∏ k in Ico a (b + 1), f k = (∏ k in Ico a b, f k) * f b",
"tactic": "rw [Nat.Ico_succ_right_eq_insert_Ico hab, prod_insert right_not_mem_Ico, mul_comm]"
}
] |
[
55,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.cast_pow'
|
[] |
[
394,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
IsSMulRegular.of_mul_eq_one
|
[
{
"state_after": "R : Type u_1\nS : Type ?u.9429\nM : Type u_2\na b : R\ns : S\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\nh : a * b = 1\n⊢ IsSMulRegular M 1",
"state_before": "R : Type u_1\nS : Type ?u.9429\nM : Type u_2\na b : R\ns : S\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\nh : a * b = 1\n⊢ IsSMulRegular M (?m.11206 h * b)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.9429\nM : Type u_2\na b : R\ns : S\ninst✝¹ : Monoid R\ninst✝ : MulAction R M\nh : a * b = 1\n⊢ IsSMulRegular M 1",
"tactic": "exact one M"
}
] |
[
151,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Topology/Separation.lean
|
isOpen_singleton_of_finite_mem_nhds
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\n⊢ IsOpen {x}",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\n⊢ IsOpen {x}",
"tactic": "have A : {x} ⊆ s := by simp only [singleton_subset_iff, mem_of_mem_nhds hs]"
},
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\n⊢ IsOpen {x}",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\n⊢ IsOpen {x}",
"tactic": "have B : IsClosed (s \\ {x}) := (hsf.subset (diff_subset _ _)).isClosed"
},
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\n⊢ IsOpen {x}",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\n⊢ IsOpen {x}",
"tactic": "have C : (s \\ {x})ᶜ ∈ 𝓝 x := B.isOpen_compl.mem_nhds fun h => h.2 rfl"
},
{
"state_after": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\nD : {x} ∈ 𝓝 x\n⊢ IsOpen {x}",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\n⊢ IsOpen {x}",
"tactic": "have D : {x} ∈ 𝓝 x := by simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\nD : {x} ∈ 𝓝 x\n⊢ IsOpen {x}",
"tactic": "rwa [← mem_interior_iff_mem_nhds, ← singleton_subset_iff, subset_interior_iff_isOpen] at D"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\n⊢ {x} ⊆ s",
"tactic": "simp only [singleton_subset_iff, mem_of_mem_nhds hs]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\ninst✝² : TopologicalSpace α✝\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\nx : α\ns : Set α\nhs : s ∈ 𝓝 x\nhsf : Set.Finite s\nA : {x} ⊆ s\nB : IsClosed (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\n⊢ {x} ∈ 𝓝 x",
"tactic": "simpa only [← diff_eq, diff_diff_cancel_left A] using inter_mem hs C"
}
] |
[
773,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/Order/Filter/ENNReal.lean
|
ENNReal.limsup_mul_le
|
[
{
"state_after": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu v : α → ℝ≥0∞\n⊢ u * v ≤ᶠ[f] fun x => limsup u f * v x",
"state_before": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu v : α → ℝ≥0∞\n⊢ limsup (u * v) f ≤ limsup (fun x => limsup u f * v x) f",
"tactic": "refine limsup_le_limsup ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : Filter α\ninst✝ : CountableInterFilter f\nu v : α → ℝ≥0∞\n⊢ u * v ≤ᶠ[f] fun x => limsup u f * v x",
"tactic": "filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl"
}
] |
[
81,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/CategoryTheory/Localization/Predicate.lean
|
CategoryTheory.Localization.essSurj
|
[] |
[
210,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Order/Cover.lean
|
toDual_covby_toDual_iff
|
[] |
[
252,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.he''
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nG_obj : D → C\ne : (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y)\nhe : ∀ (X' X : C) (Y : D) (f : X' ⟶ X) (g : F.obj X ⟶ Y), ↑(e X' Y) (F.map f ≫ g) = f ≫ ↑(e X Y) g\nX' X : C\nY : D\nf : X' ⟶ X\ng : X ⟶ G_obj Y\n⊢ F.map f ≫ ↑(e X Y).symm g = ↑(e X' Y).symm (f ≫ g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nG_obj : D → C\ne : (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y)\nhe : ∀ (X' X : C) (Y : D) (f : X' ⟶ X) (g : F.obj X ⟶ Y), ↑(e X' Y) (F.map f ≫ g) = f ≫ ↑(e X Y) g\nX' X : C\nY : D\nf : X' ⟶ X\ng : X ⟶ G_obj Y\n⊢ F.map f ≫ ↑(e X Y).symm g = ↑(e X' Y).symm (f ≫ g)",
"tactic": "intros"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nG_obj : D → C\ne : (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y)\nhe : ∀ (X' X : C) (Y : D) (f : X' ⟶ X) (g : F.obj X ⟶ Y), ↑(e X' Y) (F.map f ≫ g) = f ≫ ↑(e X Y) g\nX' X : C\nY : D\nf : X' ⟶ X\ng : X ⟶ G_obj Y\n⊢ f ≫ ↑(e X Y) (↑(e X Y).symm g) = f ≫ g",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nG_obj : D → C\ne : (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y)\nhe : ∀ (X' X : C) (Y : D) (f : X' ⟶ X) (g : F.obj X ⟶ Y), ↑(e X' Y) (F.map f ≫ g) = f ≫ ↑(e X Y) g\nX' X : C\nY : D\nf : X' ⟶ X\ng : X ⟶ G_obj Y\n⊢ F.map f ≫ ↑(e X Y).symm g = ↑(e X' Y).symm (f ≫ g)",
"tactic": "rw [Equiv.eq_symm_apply, he]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nG_obj : D → C\ne : (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃ (X ⟶ G_obj Y)\nhe : ∀ (X' X : C) (Y : D) (f : X' ⟶ X) (g : F.obj X ⟶ Y), ↑(e X' Y) (F.map f ≫ g) = f ≫ ↑(e X Y) g\nX' X : C\nY : D\nf : X' ⟶ X\ng : X ⟶ G_obj Y\n⊢ f ≫ ↑(e X Y) (↑(e X Y).symm g) = f ≫ g",
"tactic": "simp"
}
] |
[
549,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
9
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_iff_forall_isBigOWith
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.24405\nE : Type u_2\nF : Type u_3\nG : Type ?u.24414\nE' : Type ?u.24417\nF' : Type ?u.24420\nG' : Type ?u.24423\nE'' : Type ?u.24426\nF'' : Type ?u.24429\nG'' : Type ?u.24432\nR : Type ?u.24435\nR' : Type ?u.24438\n𝕜 : Type ?u.24441\n𝕜' : Type ?u.24444\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\n⊢ f =o[l] g ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g",
"tactic": "rw [IsLittleO_def]"
}
] |
[
150,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasDerivAt.hasDerivAtFilter
|
[] |
[
377,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/Finite/Card.lean
|
Finite.card_le_of_surjective
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2785\ninst✝ : Finite α\nf : α → β\nhf : Function.Surjective f\nthis : Fintype α\n⊢ Nat.card β ≤ Nat.card α",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2785\ninst✝ : Finite α\nf : α → β\nhf : Function.Surjective f\n⊢ Nat.card β ≤ Nat.card α",
"tactic": "haveI := Fintype.ofFinite α"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2785\ninst✝ : Finite α\nf : α → β\nhf : Function.Surjective f\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card β ≤ Nat.card α",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2785\ninst✝ : Finite α\nf : α → β\nhf : Function.Surjective f\nthis : Fintype α\n⊢ Nat.card β ≤ Nat.card α",
"tactic": "haveI := Fintype.ofSurjective f hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.2785\ninst✝ : Finite α\nf : α → β\nhf : Function.Surjective f\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card β ≤ Nat.card α",
"tactic": "simpa only [Nat.card_eq_fintype_card, ge_iff_le] using Fintype.card_le_of_surjective f hf"
}
] |
[
116,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Order/Minimal.lean
|
IsAntichain.minimals_upperClosure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\ninst✝ : PartialOrder α\nhs : IsAntichain (fun x x_1 => x ≤ x_1) s\na : α\nx✝ : a ∈ minimals (fun x x_1 => x ≤ x_1) ↑(upperClosure s)\nb : α\nhb : b ∈ s\nhba : b ≤ a\nright✝ : ∀ ⦃b : α⦄, b ∈ ↑(upperClosure s) → (fun x x_1 => x ≤ x_1) b a → (fun x x_1 => x ≤ x_1) a b\n⊢ a ∈ s",
"tactic": "rwa [eq_of_mem_minimals ‹a ∈ _› (subset_upperClosure hb) hba]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr r₁ r₂ : α → α → Prop\ns t : Set α\na✝ b✝ : α\ninst✝ : PartialOrder α\nhs : IsAntichain (fun x x_1 => x ≤ x_1) s\na : α\nha : a ∈ s\nb : α\nx✝ : b ∈ ↑(upperClosure s)\nhba : (fun x x_1 => x ≤ x_1) b a\nc : α\nhc : c ∈ s\nhcb : c ≤ b\n⊢ (fun x x_1 => x ≤ x_1) a b",
"tactic": "rwa [hs.eq' ha hc (hcb.trans hba)]"
}
] |
[
239,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
FiniteDimensional.lt_aleph0_of_linearIndependent
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (#ι) < lift ℵ₀",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ (#ι) < ℵ₀",
"tactic": "apply Cardinal.lift_lt.1"
},
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (#ι) ≤ ?b\n\ncase a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ ?b < lift ℵ₀\n\ncase b\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ Cardinal",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (#ι) < lift ℵ₀",
"tactic": "apply lt_of_le_of_lt"
},
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (Module.rank K V) < lift ℵ₀",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (#ι) ≤ ?b\n\ncase a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ ?b < lift ℵ₀\n\ncase b\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ Cardinal",
"tactic": "apply cardinal_lift_le_rank_of_linearIndependent h"
},
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ ↑(finrank K V) < ℵ₀",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ lift (Module.rank K V) < lift ℵ₀",
"tactic": "rw [← finrank_eq_rank, Cardinal.lift_aleph0, Cardinal.lift_natCast]"
},
{
"state_after": "no goals",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\nι : Type w\ninst✝ : FiniteDimensional K V\nv : ι → V\nh : LinearIndependent K v\n⊢ ↑(finrank K V) < ℵ₀",
"tactic": "apply Cardinal.nat_lt_aleph0"
}
] |
[
301,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Order/Atoms.lean
|
IsCoatom.Ici
|
[] |
[
143,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.mul_iSup
|
[
{
"state_after": "case pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ∀ (i : ι), f i = 0\n⊢ a * iSup f = ⨆ (i : ι), a * f i\n\ncase neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ a * iSup f = ⨆ (i : ι), a * f i",
"state_before": "α : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\n⊢ a * iSup f = ⨆ (i : ι), a * f i",
"tactic": "by_cases hf : ∀ i, f i = 0"
},
{
"state_after": "α : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ∀ (i : ι), f i = 0\n⊢ f = fun x => 0\n\ncase pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\na : ℝ≥0∞\nhf : ∀ (i : ι), (fun x => 0) i = 0\n⊢ (a * ⨆ (x : ι), 0) = ⨆ (i : ι), a * (fun x => 0) i",
"state_before": "case pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ∀ (i : ι), f i = 0\n⊢ a * iSup f = ⨆ (i : ι), a * f i",
"tactic": "obtain rfl : f = fun _ => 0"
},
{
"state_after": "case pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\na : ℝ≥0∞\nhf : ∀ (i : ι), (fun x => 0) i = 0\n⊢ (a * ⨆ (x : ι), 0) = ⨆ (i : ι), a * (fun x => 0) i",
"state_before": "α : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ∀ (i : ι), f i = 0\n⊢ f = fun x => 0\n\ncase pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\na : ℝ≥0∞\nhf : ∀ (i : ι), (fun x => 0) i = 0\n⊢ (a * ⨆ (x : ι), 0) = ⨆ (i : ι), a * (fun x => 0) i",
"tactic": "exact funext hf"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\na : ℝ≥0∞\nhf : ∀ (i : ι), (fun x => 0) i = 0\n⊢ (a * ⨆ (x : ι), 0) = ⨆ (i : ι), a * (fun x => 0) i",
"tactic": "simp only [iSup_zero_eq_zero, mul_zero]"
},
{
"state_after": "case neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ ContinuousAt (fun x => a * id x) (⨆ (i : ι), f i)",
"state_before": "case neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ a * iSup f = ⨆ (i : ι), a * f i",
"tactic": "refine' (monotone_id.const_mul' _).map_iSup_of_continuousAt _ (mul_zero a)"
},
{
"state_after": "case neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ id (⨆ (i : ι), f i) ≠ 0",
"state_before": "case neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ ContinuousAt (fun x => a * id x) (⨆ (i : ι), f i)",
"tactic": "refine' ENNReal.Tendsto.const_mul tendsto_id (Or.inl _)"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.153814\nβ : Type ?u.153817\nγ : Type ?u.153820\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\n⊢ id (⨆ (i : ι), f i) ≠ 0",
"tactic": "exact mt iSup_eq_zero.1 hf"
}
] |
[
648,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
641,
1
] |
Mathlib/Data/Multiset/Sort.lean
|
Multiset.sort_zero
|
[] |
[
63,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Sum/Order.lean
|
OrderIso.sumLexDualAntidistrib_inl
|
[] |
[
712,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
710,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.wOppSide_pointReflection
|
[] |
[
844,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/Data/Sigma/Basic.lean
|
Prod.fst_comp_toSigma
|
[] |
[
174,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
Differentiable.smul
|
[] |
[
214,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.minFac_eq
|
[] |
[
375,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Init/CcLemmas.lean
|
not_eq_of_eq_false
|
[] |
[
76,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.toAddMonoidMom_commutes
|
[] |
[
718,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean
|
LinearMap.quotientInfEquivSupQuotient_symm_apply_right
|
[] |
[
148,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.compl_insert
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type ?u.12293\nγ : Type ?u.12296\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ insert a sᶜ ↔ a✝ ∈ erase (sᶜ) a",
"state_before": "α : Type u_1\nβ : Type ?u.12293\nγ : Type ?u.12296\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na : α\n⊢ insert a sᶜ = erase (sᶜ) a",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type ?u.12293\nγ : Type ?u.12296\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na a✝ : α\n⊢ a✝ ∈ insert a sᶜ ↔ a✝ ∈ erase (sᶜ) a",
"tactic": "simp only [not_or, mem_insert, iff_self_iff, mem_compl, mem_erase]"
}
] |
[
230,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Topology/Separation.lean
|
Inducing.injective
|
[] |
[
198,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
11
] |
Mathlib/Data/Multiset/Dedup.lean
|
Multiset.dedup_cons_of_not_mem
|
[] |
[
54,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.add'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.940671\nδ : Type ?u.940674\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf g : α → β\nhf : Integrable f\nhg : Integrable g\na : α\n⊢ ↑‖f a + g a‖₊ ≤ ↑‖f a‖₊ + ↑‖g a‖₊",
"tactic": "exact_mod_cast nnnorm_add_le _ _"
}
] |
[
660,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.biInter_subset_of_mem
|
[] |
[
873,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
GradedRing.proj_recompose
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type ?u.79391\nA : Type u_2\nσ : Type u_3\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : AddMonoid ι\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\na : ⨁ (i : ι), { x // x ∈ 𝒜 i }\ni : ι\n⊢ ↑(proj 𝒜 i) (↑(decompose 𝒜).symm a) = ↑(decompose 𝒜).symm (↑(of (fun i => (fun i => { x // x ∈ 𝒜 i }) i) i) (↑a i))",
"tactic": "rw [GradedRing.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]"
}
] |
[
122,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_iff_not_smul_mem_span
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nha : ∃ y, y ∈ Finsupp.supported R R (univ \\ {i}) ∧ ↑(Finsupp.total ι M R v) y = a • v i\n⊢ a = 0",
"state_before": "ι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nha : a • v i ∈ span R (v '' (univ \\ {i}))\n⊢ a = 0",
"tactic": "rw [Finsupp.span_image_eq_map_total, mem_map] at ha"
},
{
"state_after": "case intro.intro\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nl : ι →₀ R\nhl : l ∈ Finsupp.supported R R (univ \\ {i})\ne : ↑(Finsupp.total ι M R v) l = a • v i\n⊢ a = 0",
"state_before": "ι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nha : ∃ y, y ∈ Finsupp.supported R R (univ \\ {i}) ∧ ↑(Finsupp.total ι M R v) y = a • v i\n⊢ a = 0",
"tactic": "rcases ha with ⟨l, hl, e⟩"
},
{
"state_after": "case intro.intro\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nl : ι →₀ R\nhl : Finsupp.single i a ∈ Finsupp.supported R R (univ \\ {i})\ne : ↑(Finsupp.total ι M R v) l = a • v i\n⊢ a = 0",
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"tactic": "rw [sub_eq_zero.1 (linearIndependent_iff.1 hv (l - Finsupp.single i a) (by simp [e]))] at hl"
},
{
"state_after": "case intro.intro\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na✝ b : R\nx y : M\nhv✝ hv : LinearIndependent R v\ni : ι\na : R\nl : ι →₀ R\nhl : Finsupp.single i a ∈ Finsupp.supported R R (univ \\ {i})\ne : ↑(Finsupp.total ι M R v) l = a • v i\nhn : ¬a = 0\n⊢ False",
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},
{
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"tactic": "exact (not_mem_of_mem_diff (hl <| by simp [hn])) (mem_singleton _)"
},
{
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"tactic": "simp [e]"
},
{
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"tactic": "simp [hn]"
},
{
"state_after": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\n⊢ ↑l i = ↑0 i",
"state_before": "ι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\n⊢ l = 0",
"tactic": "ext i"
},
{
"state_after": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\n⊢ ↑l i = 0",
"state_before": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\n⊢ ↑l i = ↑0 i",
"tactic": "simp only [Finsupp.zero_apply]"
},
{
"state_after": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ False",
"state_before": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\n⊢ ↑l i = 0",
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},
{
"state_after": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ↑l i • v i ∈ span R (v '' (univ \\ {i}))",
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"tactic": "refine' hn (H i _ _)"
},
{
"state_after": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ Finsupp.single i (↑l i) - l ∈ Finsupp.supported R R (univ \\ {i})\n\ncase h.refine'_2\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.single i (↑l i) - l) = ↑l i • v i",
"state_before": "case h\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ↑l i • v i ∈ span R (v '' (univ \\ {i}))",
"tactic": "refine' (Finsupp.mem_span_image_iff_total R).2 ⟨Finsupp.single i (l i) - l, _, _⟩"
},
{
"state_after": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ∀ (x : ι), ¬x ∈ univ \\ {i} → ↑(Finsupp.single i (↑l i) - l) x = 0",
"state_before": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ Finsupp.single i (↑l i) - l ∈ Finsupp.supported R R (univ \\ {i})",
"tactic": "rw [Finsupp.mem_supported']"
},
{
"state_after": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\nj : ι\nhj : ¬j ∈ univ \\ {i}\n⊢ ↑(Finsupp.single i (↑l i) - l) j = 0",
"state_before": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ∀ (x : ι), ¬x ∈ univ \\ {i} → ↑(Finsupp.single i (↑l i) - l) x = 0",
"tactic": "intro j hj"
},
{
"state_after": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\nj : ι\nhj : ¬j ∈ univ \\ {i}\nhij : j = i\n⊢ ↑(Finsupp.single i (↑l i) - l) j = 0",
"state_before": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\nj : ι\nhj : ¬j ∈ univ \\ {i}\n⊢ ↑(Finsupp.single i (↑l i) - l) j = 0",
"tactic": "have hij : j = i :=\n Classical.not_not.1 fun hij : j ≠ i =>\n hj ((mem_diff _).2 ⟨mem_univ _, fun h => hij (eq_of_mem_singleton h)⟩)"
},
{
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"state_before": "case h.refine'_1\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\nj : ι\nhj : ¬j ∈ univ \\ {i}\nhij : j = i\n⊢ ↑(Finsupp.single i (↑l i) - l) j = 0",
"tactic": "simp [hij]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nι : Type u'\nι' : Type ?u.472009\nR : Type u_1\nK : Type ?u.472015\nM : Type u_2\nM' : Type ?u.472021\nM'' : Type ?u.472024\nV : Type u\nV' : Type ?u.472029\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\nH : ∀ (i : ι) (a : R), a • v i ∈ span R (v '' (univ \\ {i})) → a = 0\nl : ι →₀ R\nhl : ↑(Finsupp.total ι M R v) l = 0\ni : ι\nhn : ¬↑l i = 0\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.single i (↑l i) - l) = ↑l i • v i",
"tactic": "simp [hl]"
}
] |
[
851,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
Real.toNNReal_mul_nnnorm
|
[
{
"state_after": "case a\nα : Type ?u.381131\nβ : Type ?u.381134\nγ : Type ?u.381137\nι : Type ?u.381140\nx y : ℝ\nhx : 0 ≤ x\n⊢ ↑(toNNReal x * ‖y‖₊) = ↑‖x * y‖₊",
"state_before": "α : Type ?u.381131\nβ : Type ?u.381134\nγ : Type ?u.381137\nι : Type ?u.381140\nx y : ℝ\nhx : 0 ≤ x\n⊢ toNNReal x * ‖y‖₊ = ‖x * y‖₊",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.381131\nβ : Type ?u.381134\nγ : Type ?u.381137\nι : Type ?u.381140\nx y : ℝ\nhx : 0 ≤ x\n⊢ ↑(toNNReal x * ‖y‖₊) = ↑‖x * y‖₊",
"tactic": "simp only [NNReal.coe_mul, nnnorm_mul, coe_nnnorm, Real.toNNReal_of_nonneg, norm_of_nonneg, hx,\n coe_mk]"
}
] |
[
815,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
MonovaryOn.comp_antitoneOn_right
|
[] |
[
369,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/RepresentationTheory/Action.lean
|
Action.functorCategoryMonoidalEquivalence.μIso_inv_app
|
[
{
"state_after": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nA B : Action V G\n⊢ inv ((MonoidalFunctor.μIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom =\n 𝟙 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A ⊗ B)).obj PUnit.unit)",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nA B : Action V G\n⊢ (MonoidalFunctor.μIso (functorCategoryMonoidalEquivalence V G) A B).inv.app PUnit.unit =\n 𝟙 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A ⊗ B)).obj PUnit.unit)",
"tactic": "rw [← NatIso.app_inv, ← IsIso.Iso.inv_hom]"
},
{
"state_after": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nA B : Action V G\n⊢ ((MonoidalFunctor.μIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom ≫\n 𝟙 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A ⊗ B)).obj PUnit.unit) =\n 𝟙\n (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj A ⊗\n (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj B).obj\n PUnit.unit)",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nA B : Action V G\n⊢ inv ((MonoidalFunctor.μIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom =\n 𝟙 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A ⊗ B)).obj PUnit.unit)",
"tactic": "refine' IsIso.inv_eq_of_hom_inv_id _"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝¹ : LargeCategory V\nG : MonCat\ninst✝ : MonoidalCategory V\nA B : Action V G\n⊢ ((MonoidalFunctor.μIso (functorCategoryMonoidalEquivalence V G) A B).app PUnit.unit).hom ≫\n 𝟙 (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj (A ⊗ B)).obj PUnit.unit) =\n 𝟙\n (((functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj A ⊗\n (functorCategoryMonoidalEquivalence V G).toLaxMonoidalFunctor.toFunctor.obj B).obj\n PUnit.unit)",
"tactic": "rw [Category.comp_id, NatIso.app_hom, MonoidalFunctor.μIso_hom,\n functorCategoryMonoidalEquivalence.μ_app]"
}
] |
[
653,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
648,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.index_eq_zero_of_relindex_eq_zero
|
[] |
[
393,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
affineIndependent_of_ne_of_mem_of_not_mem_of_mem
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\n⊢ AffineIndependent k (![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2))",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\n⊢ AffineIndependent k ![p₁, p₂, p₃]",
"tactic": "rw [← affineIndependent_equiv (Equiv.swap (1 : Fin 3) 2)]"
},
{
"state_after": "case h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\n⊢ ![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2) = ![p₁, p₃, p₂]",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\n⊢ AffineIndependent k (![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2))",
"tactic": "convert affineIndependent_of_ne_of_mem_of_mem_of_not_mem hp₁p₃ hp₁ hp₃ hp₂ using 1"
},
{
"state_after": "case h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\nx : Fin 3\n⊢ (![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2)) x = Matrix.vecCons p₁ ![p₃, p₂] x",
"state_before": "case h.e'_9\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\n⊢ ![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2) = ![p₁, p₃, p₂]",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_9.h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.460888\ns : AffineSubspace k P\np₁ p₂ p₃ : P\nhp₁p₃ : p₁ ≠ p₃\nhp₁ : p₁ ∈ s\nhp₂ : ¬p₂ ∈ s\nhp₃ : p₃ ∈ s\nx : Fin 3\n⊢ (![p₁, p₂, p₃] ∘ ↑(Equiv.swap 1 2)) x = Matrix.vecCons p₁ ![p₃, p₂] x",
"tactic": "fin_cases x <;> rfl"
}
] |
[
704,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
698,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Finset.exists_card_fiber_lt_of_card_lt_nsmul
|
[
{
"state_after": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nht : ∑ a in s, 1 < card t • b\n⊢ ∃ y, y ∈ t ∧ ∑ a in filter (fun x => f x = y) s, 1 < b",
"state_before": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nht : ↑(card s) < card t • b\n⊢ ∃ y, y ∈ t ∧ ↑(card (filter (fun x => f x = y) s)) < b",
"tactic": "simp_rw [cast_card] at ht⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCommSemiring M\nht : ∑ a in s, 1 < card t • b\n⊢ ∃ y, y ∈ t ∧ ∑ a in filter (fun x => f x = y) s, 1 < b",
"tactic": "exact\n exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul\n (fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht"
}
] |
[
258,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_self_add
|
[] |
[
349,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
ContDiffOn.arsinh
|
[] |
[
290,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
mem_generatePiSystem_iUnion_elim
|
[
{
"state_after": "case base\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nh_s : s ∈ ⋃ (b : β), g b\n⊢ ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\n\ncase inter\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s t' : Set α\nh_gen_s : generatePiSystem (⋃ (b : β), g b) s\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) t'\nh_nonempty : Set.Nonempty (s ∩ t')\nh_s : ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\nh_t' : ∃ T f, (t' = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\n⊢ ∃ T f, (s ∩ t' = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"state_before": "α : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nh_t : t ∈ generatePiSystem (⋃ (b : β), g b)\n⊢ ∃ T f, (t = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "induction' h_t with s h_s s t' h_gen_s h_gen_t' h_nonempty h_s h_t'"
},
{
"state_after": "case base.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b => g b) b\n⊢ ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"state_before": "case base\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nh_s : s ∈ ⋃ (b : β), g b\n⊢ ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "rcases h_s with ⟨t', ⟨⟨b, rfl⟩, h_s_in_t'⟩⟩"
},
{
"state_after": "case base.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b => g b) b\n⊢ (s = ⋂ (b_1 : β) (_ : b_1 ∈ {b}), (fun x => s) b_1) ∧ ∀ (b_1 : β), b_1 ∈ {b} → (fun x => s) b_1 ∈ g b_1",
"state_before": "case base.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b => g b) b\n⊢ ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "refine' ⟨{b}, fun _ => s, _⟩"
},
{
"state_after": "no goals",
"state_before": "case base.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b => g b) b\n⊢ (s = ⋂ (b_1 : β) (_ : b_1 ∈ {b}), (fun x => s) b_1) ∧ ∀ (b_1 : β), b_1 ∈ {b} → (fun x => s) b_1 ∈ g b_1",
"tactic": "simpa using h_s_in_t'"
},
{
"state_after": "case inter.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nh_gen_s : generatePiSystem (⋃ (b : β), g b) s\nh_s : ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nh_nonempty : Set.Nonempty (s ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∃ T f, ((s ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"state_before": "case inter\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s t' : Set α\nh_gen_s : generatePiSystem (⋃ (b : β), g b) s\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) t'\nh_nonempty : Set.Nonempty (s ∩ t')\nh_s : ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\nh_t' : ∃ T f, (t' = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\n⊢ ∃ T f, (s ∩ t' = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "rcases h_t' with ⟨T_t', ⟨f_t', ⟨rfl, h_t'⟩⟩⟩"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∃ T f,\n (((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) = ⋂ (b : β) (_ : b ∈ T), f b) ∧\n ∀ (b : β), b ∈ T → f b ∈ g b",
"state_before": "case inter.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nh_gen_s : generatePiSystem (⋃ (b : β), g b) s\nh_s : ∃ T f, (s = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nh_nonempty : Set.Nonempty (s ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∃ T f, ((s ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) = ⋂ (b : β) (_ : b ∈ T), f b) ∧ ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "rcases h_s with ⟨T_s, ⟨f_s, ⟨rfl, h_s⟩⟩⟩"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ (((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) =\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b) ∧\n ∀ (b : β),\n b ∈ T_s ∪ T_t' →\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈\n g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∃ T f,\n (((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) = ⋂ (b : β) (_ : b ∈ T), f b) ∧\n ∀ (b : β), b ∈ T → f b ∈ g b",
"tactic": "use T_s ∪ T_t', fun b : β =>\n if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b\n else if b ∈ T_t' then f_t' b else (∅ : Set α)"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.left\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) =\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b\n\ncase inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∀ (b : β),\n b ∈ T_s ∪ T_t' →\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈ g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ (((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) =\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b) ∧\n ∀ (b : β),\n b ∈ T_s ∪ T_t' →\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈\n g b",
"tactic": "constructor"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\n⊢ (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈ g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ∀ (b : β),\n b ∈ T_s ∪ T_t' →\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈ g b",
"tactic": "intro b h_b"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\n⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\n⊢ (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b ∈ g b",
"tactic": "simp only []"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.right.inl.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\n⊢ f_s b ∩ f_t' b ∈ g b\n\ncase inter.intro.intro.intro.intro.intro.intro.right.inl.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ f_s b ∈ g b\n\ncase inter.intro.intro.intro.intro.intro.intro.right.inr.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\n⊢ f_t' b ∈ g b\n\ncase inter.intro.intro.intro.intro.intro.intro.right.inr.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : ¬b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ ∅ ∈ g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\n⊢ (if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) ∈ g b",
"tactic": "split_ifs with hbs hbt hbt"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ (a ∈ (⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) ↔\n a ∈\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.left\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\n⊢ ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) =\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b",
"tactic": "ext a"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ ((∀ (i : β), i ∈ T_s → a ∈ f_s i) ∧ ∀ (i : β), i ∈ T_t' → a ∈ f_t' i) ↔\n ∀ (i : β),\n (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧\n (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ (a ∈ (⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) ↔\n a ∈\n ⋂ (b : β) (_ : b ∈ T_s ∪ T_t'),\n (fun b => if b ∈ T_s then if b ∈ T_t' then f_s b ∩ f_t' b else f_s b else if b ∈ T_t' then f_t' b else ∅) b",
"tactic": "simp_rw [Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union, or_imp]"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ (∀ (x : β), (x ∈ T_s → a ∈ f_s x) ∧ (x ∈ T_t' → a ∈ f_t' x)) ↔\n ∀ (i : β),\n (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧\n (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ ((∀ (i : β), i ∈ T_s → a ∈ f_s i) ∧ ∀ (i : β), i ∈ T_t' → a ∈ f_t' i) ↔\n ∀ (i : β),\n (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧\n (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)",
"tactic": "rw [← forall_and]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_s b ∧ a ∈ f_t' b\n⊢ a ∈ f_s b ∩ f_t' b\n\ncase neg\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\nh1 : a ∈ f_s b\n⊢ a ∈ f_s b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_t' b\n⊢ a ∈ f_t' b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_s b ∩ f_t' b\n⊢ a ∈ f_s b ∧ a ∈ f_t' b\n\ncase neg\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\nh1 : a ∈ f_s b\n⊢ a ∈ f_s b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_t' b\n⊢ a ∈ f_t' b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.left.h\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\n⊢ (∀ (x : β), (x ∈ T_s → a ∈ f_s x) ∧ (x ∈ T_t' → a ∈ f_t' x)) ↔\n ∀ (i : β),\n (i ∈ T_s → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅) ∧\n (i ∈ T_t' → a ∈ if i ∈ T_s then if i ∈ T_t' then f_s i ∩ f_t' i else f_s i else if i ∈ T_t' then f_t' i else ∅)",
"tactic": "constructor <;> intro h1 b <;> by_cases hbs : b ∈ T_s <;> by_cases hbt : b ∈ T_t' <;>\n specialize h1 b <;>\n simp only [hbs, hbt, if_true, if_false, true_imp_iff, and_self_iff, false_imp_iff,\n and_true_iff, true_and_iff] at h1⊢"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_s b ∧ a ∈ f_t' b\n⊢ a ∈ f_s b ∩ f_t' b\n\ncase neg\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\nh1 : a ∈ f_s b\n⊢ a ∈ f_s b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_t' b\n⊢ a ∈ f_t' b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_s b ∩ f_t' b\n⊢ a ∈ f_s b ∧ a ∈ f_t' b\n\ncase neg\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\nh1 : a ∈ f_s b\n⊢ a ∈ f_s b\n\ncase pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_t' b\n⊢ a ∈ f_t' b",
"tactic": "all_goals exact h1"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\na : α\nb : β\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\nh1 : a ∈ f_t' b\n⊢ a ∈ f_t' b",
"tactic": "exact h1"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.right.inl.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\n⊢ ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) ⊆ f_s b ∩ f_t' b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inl.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\n⊢ f_s b ∩ f_t' b ∈ g b",
"tactic": "refine' h_pi b (f_s b) (h_s b hbs) (f_t' b) (h_t' b hbt) (Set.Nonempty.mono _ h_nonempty)"
},
{
"state_after": "no goals",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inl.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : b ∈ T_t'\n⊢ ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b) ⊆ f_s b ∩ f_t' b",
"tactic": "exact Set.inter_subset_inter (Set.biInter_subset_of_mem hbs) (Set.biInter_subset_of_mem hbt)"
},
{
"state_after": "no goals",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inl.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ f_s b ∈ g b",
"tactic": "exact h_s b hbs"
},
{
"state_after": "no goals",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inr.inl\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : ¬b ∈ T_s\nhbt : b ∈ T_t'\n⊢ f_t' b ∈ g b",
"tactic": "exact h_t' b hbt"
},
{
"state_after": "case inter.intro.intro.intro.intro.intro.intro.right.inr.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∨ b ∈ T_t'\nhbs : ¬b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ ∅ ∈ g b",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inr.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∪ T_t'\nhbs : ¬b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ ∅ ∈ g b",
"tactic": "rw [Finset.mem_union] at h_b"
},
{
"state_after": "no goals",
"state_before": "case inter.intro.intro.intro.intro.intro.intro.right.inr.inr\nα : Type u_1\nβ : Type u_2\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt : Set α\nT_t' : Finset β\nf_t' : β → Set α\nh_t' : ∀ (b : β), b ∈ T_t' → f_t' b ∈ g b\nh_gen_t' : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nT_s : Finset β\nf_s : β → Set α\nh_s : ∀ (b : β), b ∈ T_s → f_s b ∈ g b\nh_gen_s : generatePiSystem (⋃ (b : β), g b) (⋂ (b : β) (_ : b ∈ T_s), f_s b)\nh_nonempty : Set.Nonempty ((⋂ (b : β) (_ : b ∈ T_s), f_s b) ∩ ⋂ (b : β) (_ : b ∈ T_t'), f_t' b)\nb : β\nh_b : b ∈ T_s ∨ b ∈ T_t'\nhbs : ¬b ∈ T_s\nhbt : ¬b ∈ T_t'\n⊢ ∅ ∈ g b",
"tactic": "apply False.elim (h_b.elim hbs hbt)"
}
] |
[
311,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegralClosure.algebraMap_injective
|
[] |
[
834,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.norm_incl
|
[] |
[
729,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal_one
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.138725\nm : Type u_1\nn : Type ?u.138731\no : Type u_3\np : Type ?u.138737\nq : Type ?u.138740\nm' : o → Type ?u.138745\nn' : o → Type ?u.138750\np' : o → Type ?u.138755\nR : Type ?u.138758\nS : Type ?u.138761\nα : Type u_2\nβ : Type ?u.138767\ninst✝⁴ : DecidableEq o\ninst✝³ : Zero α\ninst✝² : Zero β\ninst✝¹ : DecidableEq m\ninst✝ : One α\n⊢ (blockDiagonal fun x => diagonal fun x => 1) = diagonal fun x => 1",
"tactic": "rw [blockDiagonal_diagonal]"
}
] |
[
416,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.drop_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ni : Nat\n⊢ drop i nil = nil",
"tactic": "cases i <;> rfl"
}
] |
[
481,
18
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
480,
9
] |
Mathlib/Data/Set/Finite.lean
|
Set.infinite_of_not_bddBelow
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\ns : Set α\n⊢ ¬Set.Infinite s → BddBelow s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\ns : Set α\n⊢ ¬BddBelow s → Set.Infinite s",
"tactic": "contrapose!"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\ns : Set α\n⊢ Set.Finite s → BddBelow s",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\ns : Set α\n⊢ ¬Set.Infinite s → BddBelow s",
"tactic": "rw [not_infinite]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : SemilatticeInf α\ninst✝ : Nonempty α\ns : Set α\n⊢ Set.Finite s → BddBelow s",
"tactic": "exact Finite.bddBelow"
}
] |
[
1630,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1627,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.bit0_im
|
[] |
[
190,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Algebra/Regular/Pow.lean
|
IsRightRegular.pow_iff
|
[
{
"state_after": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsRightRegular (a ^ n) → IsRightRegular a",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsRightRegular (a ^ n) ↔ IsRightRegular a",
"tactic": "refine' ⟨_, IsRightRegular.pow n⟩"
},
{
"state_after": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsRightRegular (a * a ^ Nat.pred n) → IsRightRegular a",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsRightRegular (a ^ n) → IsRightRegular a",
"tactic": "rw [← Nat.succ_pred_eq_of_pos n0, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\na b : R\ninst✝ : Monoid R\nn : ℕ\nn0 : 0 < n\n⊢ IsRightRegular (a * a ^ Nat.pred n) → IsRightRegular a",
"tactic": "exact IsRightRegular.of_mul"
}
] |
[
60,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_eq_zero_iff
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nh : lcm a b = 0\nthis : Associated (a * b) 0\n⊢ a = 0 ∨ b = 0",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nh : lcm a b = 0\n⊢ a = 0 ∨ b = 0",
"tactic": "have : Associated (a * b) 0 := (gcd_mul_lcm a b).symm.trans <| by rw [h, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nh : lcm a b = 0\nthis : Associated (a * b) 0\n⊢ a = 0 ∨ b = 0",
"tactic": "rwa [← mul_eq_zero, ← associated_zero_iff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nh : lcm a b = 0\n⊢ Associated (gcd a b * lcm a b) 0",
"tactic": "rw [h, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\n⊢ a = 0 ∨ b = 0 → lcm a b = 0",
"tactic": "rintro (rfl | rfl) <;> [apply lcm_zero_left; apply lcm_zero_right]"
}
] |
[
733,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_toMeasurable
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\n⊢ ↑↑μ\n (if h : ∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s then Exists.choose h\n else\n if h' : ∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) then\n Exists.choose h'\n else Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) =\n ↑↑μ s",
"state_before": "α : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\n⊢ ↑↑μ (toMeasurable μ s) = ↑↑μ s",
"tactic": "rw [toMeasurable_def]"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\n⊢ ↑↑μ (Exists.choose hs) = ↑↑μ s\n\ncase inr.inl\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ¬∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\nh's : ∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u)\n⊢ ↑↑μ (Exists.choose h's) = ↑↑μ s\n\ncase inr.inr\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ¬∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\nh's : ¬∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u)\n⊢ ↑↑μ (Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) = ↑↑μ s",
"state_before": "α : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\n⊢ ↑↑μ\n (if h : ∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s then Exists.choose h\n else\n if h' : ∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u) then\n Exists.choose h'\n else Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) =\n ↑↑μ s",
"tactic": "split_ifs with hs h's"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\n⊢ ↑↑μ (Exists.choose hs) = ↑↑μ s",
"tactic": "exact measure_congr hs.choose_spec.snd.2"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ¬∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\nh's : ∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u)\n⊢ ↑↑μ (Exists.choose h's) = ↑↑μ s",
"tactic": "simpa only [inter_univ] using h's.choose_spec.snd.2 univ MeasurableSet.univ"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type ?u.144519\nγ : Type ?u.144522\nδ : Type ?u.144525\nι : Type ?u.144528\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t s : Set α\nhs : ¬∃ t x, MeasurableSet t ∧ t =ᵐ[μ] s\nh's : ¬∃ t x, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → ↑↑μ (t ∩ u) = ↑↑μ (s ∩ u)\n⊢ ↑↑μ (Exists.choose (_ : ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑↑μ t = ↑↑μ s)) = ↑↑μ s",
"tactic": "exact (exists_measurable_superset μ s).choose_spec.2.2"
}
] |
[
623,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/Algebra/Group/Pi.lean
|
Pi.update_eq_div_mul_mulSingle
|
[
{
"state_after": "case h\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\nj : I\n⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j",
"state_before": "ι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\n⊢ Function.update g i x = g / mulSingle i (g i) * mulSingle i x",
"tactic": "ext j"
},
{
"state_after": "case h.inl\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\n⊢ Function.update g i x i = (g / mulSingle i (g i) * mulSingle i x) i\n\ncase h.inr\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\nj : I\nh : i ≠ j\n⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j",
"state_before": "case h\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\nj : I\n⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j",
"tactic": "rcases eq_or_ne i j with (rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\n⊢ Function.update g i x i = (g / mulSingle i (g i) * mulSingle i x) i",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nι : Type ?u.76203\nα : Type ?u.76206\nI : Type u\nf : I → Type v\nx✝ y : (i : I) → f i\ni j✝ : I\ninst✝¹ : DecidableEq I\ninst✝ : (i : I) → Group (f i)\ng : (i : I) → f i\nx : f i\nj : I\nh : i ≠ j\n⊢ Function.update g i x j = (g / mulSingle i (g i) * mulSingle i x) j",
"tactic": "simp [Function.update_noteq h.symm, h]"
}
] |
[
572,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.toAffineMap_injective
|
[
{
"state_after": "case mk.mk\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : ↑{ toEquiv := e, linear := el, map_vadd' := h } = ↑{ toEquiv := e', linear := el', map_vadd' := h' }\n⊢ { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }",
"state_before": "k : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\n⊢ Injective toAffineMap",
"tactic": "rintro ⟨e, el, h⟩ ⟨e', el', h'⟩ H"
},
{
"state_after": "case mk.mk\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }",
"state_before": "case mk.mk\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : ↑{ toEquiv := e, linear := el, map_vadd' := h } = ↑{ toEquiv := e', linear := el', map_vadd' := h' }\n⊢ { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }",
"tactic": "simp only [(toAffineMap_mk), (AffineMap.mk.injEq), Equiv.coe_inj,\n LinearEquiv.toLinearMap_inj] at H"
},
{
"state_after": "case mk.mk.e_toEquiv\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ e = e'\n\ncase mk.mk.e_linear\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ el = el'",
"state_before": "case mk.mk\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ { toEquiv := e, linear := el, map_vadd' := h } = { toEquiv := e', linear := el', map_vadd' := h' }",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.e_toEquiv\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ e = e'\n\ncase mk.mk.e_linear\nk : Type u_5\nP₁ : Type u_1\nP₂ : Type u_2\nP₃ : Type ?u.29718\nP₄ : Type ?u.29721\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type ?u.29730\nV₄ : Type ?u.29733\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : Module k V₁\ninst✝⁹ : AffineSpace V₁ P₁\ninst✝⁸ : AddCommGroup V₂\ninst✝⁷ : Module k V₂\ninst✝⁶ : AffineSpace V₂ P₂\ninst✝⁵ : AddCommGroup V₃\ninst✝⁴ : Module k V₃\ninst✝³ : AffineSpace V₃ P₃\ninst✝² : AddCommGroup V₄\ninst✝¹ : Module k V₄\ninst✝ : AffineSpace V₄ P₄\ne : P₁ ≃ P₂\nel : V₁ ≃ₗ[k] V₂\nh : ∀ (p : P₁) (v : V₁), ↑e (v +ᵥ p) = ↑el v +ᵥ ↑e p\ne' : P₁ ≃ P₂\nel' : V₁ ≃ₗ[k] V₂\nh' : ∀ (p : P₁) (v : V₁), ↑e' (v +ᵥ p) = ↑el' v +ᵥ ↑e' p\nH : e = e' ∧ el = el'\n⊢ el = el'",
"tactic": "exacts [H.1, H.2]"
}
] |
[
87,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.ne_iff
|
[] |
[
90,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.restr_trans
|
[] |
[
893,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
1
] |
Mathlib/Data/Rat/Lemmas.lean
|
Rat.mul_num_den'
|
[
{
"state_after": "q r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"state_before": "q r : ℚ\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"tactic": "let s := q.num * r.num /. (q.den * r.den : ℤ)"
},
{
"state_after": "q r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"state_before": "q r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"tactic": "have hs : (q.den * r.den : ℤ) ≠ 0 := Int.coe_nat_ne_zero_iff_pos.mpr (mul_pos q.pos r.pos)"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"state_before": "q r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"tactic": "obtain ⟨c, ⟨c_mul_num, c_mul_den⟩⟩ :=\n exists_eq_mul_div_num_and_eq_mul_div_den (q.num * r.num) hs"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ↑q.den * ↑r.den * (q * r).num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ (q * r).num * ↑q.den * ↑r.den = q.num * r.num * ↑(q * r).den",
"tactic": "rw [c_mul_num, mul_assoc, mul_comm]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ↑q.den * ↑r.den * (q * r).num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "nth_rw 1 [c_mul_den]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * (↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num) =\n c * ((↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den)",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "repeat' rw [Int.mul_assoc]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den ∨\n c = 0",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * (↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num) =\n c * ((↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den)",
"tactic": "apply mul_eq_mul_left_iff.2"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ¬c = 0 →\n ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den ∨\n c = 0",
"tactic": "rw [or_iff_not_imp_right]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ ¬c = 0 →\n ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "intro"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q.num /. ↑q.den * (r.num /. ↑r.den) = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "have h : _ = s :=\n @mul_def' q.num q.den r.num r.den (Int.coe_nat_ne_zero_iff_pos.mpr q.pos)\n (Int.coe_nat_ne_zero_iff_pos.mpr r.pos)"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q.num /. ↑q.den * (r.num /. ↑r.den) = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "rw [num_den, num_den] at h"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * s.num = (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑s.den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num =\n (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den",
"tactic": "rw [h]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ s.num * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den = (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑s.den",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * s.num = (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑s.den",
"tactic": "rw [mul_comm]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ s = ↑(q.num * r.num) / ↑(↑q.den * ↑r.den)",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ s.num * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den = (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑s.den",
"tactic": "apply Rat.eq_iff_mul_eq_mul.mp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\na✝ : ¬c = 0\nh : q * r = s\n⊢ s = ↑(q.num * r.num) / ↑(↑q.den * ↑r.den)",
"tactic": "rw [← divInt_eq_div]"
},
{
"state_after": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * (↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num) =\n c * ((↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den)",
"state_before": "case intro.intro\nq r : ℚ\ns : ℚ := q.num * r.num /. (↑q.den * ↑r.den)\nhs : ↑q.den * ↑r.den ≠ 0\nc : ℤ\nc_mul_num : q.num * r.num = c * (↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num\nc_mul_den : ↑q.den * ↑r.den = c * ↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den\n⊢ c * (↑(↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).den * (q * r).num) =\n c * ((↑(q.num * r.num) / ↑(↑q.den * ↑r.den)).num * ↑(q * r).den)",
"tactic": "rw [Int.mul_assoc]"
}
] |
[
145,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
ae_eq_const_or_norm_average_lt_of_norm_le_const
|
[
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C\n\ncase inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "cases' le_or_lt C 0 with hC0 hC0"
},
{
"state_after": "case pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C\n\ncase neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : ¬Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "by_cases hfi : Integrable f μ"
},
{
"state_after": "case neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : ¬Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C\n\ncase pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C\n\ncase neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : ¬Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "swap"
},
{
"state_after": "case pos.inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ = ⊤\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C\n\ncase pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case pos\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "cases' (le_top : μ univ ≤ ∞).eq_or_lt with hμt hμt"
},
{
"state_after": "case pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "haveI : IsFiniteMeasure μ := ⟨hμt⟩"
},
{
"state_after": "case h_le\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closedBall 0 C\n\ncase pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\nh_le : ∀ᵐ (x : α) ∂μ, f x ∈ closedBall 0 C\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "replace h_le : ∀ᵐ x ∂μ, f x ∈ closedBall (0 : E) C"
},
{
"state_after": "no goals",
"state_before": "case pos.inr\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\nh_le : ∀ᵐ (x : α) ∂μ, f x ∈ closedBall 0 C\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "simpa only [interior_closedBall _ hC0.ne', mem_ball_zero_iff] using\n (strictConvex_closedBall ℝ (0 : E) C).ae_eq_const_or_average_mem_interior isClosed_ball h_le\n hfi"
},
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\nthis : f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "have : f =ᵐ[μ] 0 := h_le.mono fun x hx => norm_le_zero_iff.1 (hx.trans hC0)"
},
{
"state_after": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\nthis : f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] const α 0 ∨ ‖0‖ < C",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\nthis : f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "simp only [average_congr this, Pi.zero_apply, average_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : C ≤ 0\nthis : f =ᶠ[ae μ] 0\n⊢ f =ᶠ[ae μ] const α 0 ∨ ‖0‖ < C",
"tactic": "exact Or.inl this"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : ¬Integrable f\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos.inl\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ = ⊤\n⊢ f =ᶠ[ae μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C",
"tactic": "simp [average_eq, hμt, hC0]"
},
{
"state_after": "no goals",
"state_before": "case h_le\nα : Type u_2\nE : Type u_1\nF : Type ?u.2443666\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : StrictConvexSpace ℝ E\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nhC0 : 0 < C\nhfi : Integrable f\nhμt : ↑↑μ univ < ⊤\nthis : IsFiniteMeasure μ\n⊢ ∀ᵐ (x : α) ∂μ, f x ∈ closedBall 0 C",
"tactic": "simpa only [mem_closedBall_zero_iff]"
}
] |
[
343,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.exists_ne_top'
|
[] |
[
261,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.Homotopy.coe_toContinuousMap
|
[] |
[
154,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.iSup_finset_image
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.450915\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nι : Type ?u.450927\nκ : Type ?u.450930\ninst✝¹ : CompleteLattice β\ninst✝ : DecidableEq α\nf : γ → α\ng : α → β\ns : Finset γ\n⊢ (⨆ (x : α) (_ : x ∈ image f s), g x) = ⨆ (y : γ) (_ : y ∈ s), g (f y)",
"tactic": "rw [← iSup_coe, coe_image, iSup_image, iSup_coe]"
}
] |
[
1969,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1968,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
RingHom.ker_coe_equiv
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nF : Type ?u.1772041\ninst✝¹ : Ring R\ninst✝ : Semiring S\nrc : RingHomClass F R S\nf✝ : F\nf : R ≃+* S\n⊢ ker ↑f = ⊥",
"tactic": "simpa only [← injective_iff_ker_eq_bot] using EquivLike.injective f"
}
] |
[
2043,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2042,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.Nonempty.mul_zero
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.106599\nα : Type u_1\nβ : Type ?u.106605\nγ : Type ?u.106608\ninst✝ : MulZeroClass α\ns t : Set α\nhs : Set.Nonempty s\n⊢ 0 ⊆ s * 0",
"tactic": "simpa [mem_mul] using hs"
}
] |
[
1145,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.homOfLe_apply
|
[] |
[
610,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Topology/UrysohnsLemma.lean
|
Urysohns.CU.tendsto_approx_atTop
|
[] |
[
223,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.nontrivial_of_infsep_pos
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.55856\ninst✝ : EDist α\nx y : α\ns : Set α\nhs : ¬Set.Nontrivial s\n⊢ ¬0 < infsep s",
"state_before": "α : Type u_1\nβ : Type ?u.55856\ninst✝ : EDist α\nx y : α\ns : Set α\nhs : 0 < infsep s\n⊢ Set.Nontrivial s",
"tactic": "contrapose hs"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55856\ninst✝ : EDist α\nx y : α\ns : Set α\nhs : Set.Subsingleton s\n⊢ ¬0 < infsep s",
"state_before": "α : Type u_1\nβ : Type ?u.55856\ninst✝ : EDist α\nx y : α\ns : Set α\nhs : ¬Set.Nontrivial s\n⊢ ¬0 < infsep s",
"tactic": "rw [not_nontrivial_iff] at hs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.55856\ninst✝ : EDist α\nx y : α\ns : Set α\nhs : Set.Subsingleton s\n⊢ ¬0 < infsep s",
"tactic": "exact hs.infsep_zero ▸ lt_irrefl _"
}
] |
[
362,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_const_mul_Icc_of_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\na✝ a b c : α\nh : c < 0\n⊢ (fun x x_1 => x * x_1) c ⁻¹' Icc a b = Icc (b / c) (a / c)",
"tactic": "simpa only [mul_comm] using preimage_mul_const_Icc_of_neg a b h"
}
] |
[
685,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Data/PEquiv.lean
|
PEquiv.trans_single_of_eq_none
|
[
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nh : ↑(PEquiv.symm f) b = none\nx✝ : δ\na✝ : γ\n⊢ a✝ ∈ ↑(PEquiv.trans f (single b c)) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nh : ↑(PEquiv.symm f) b = none\n⊢ PEquiv.trans f (single b c) = ⊥",
"tactic": "ext"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ a✝ ∈ ↑(PEquiv.trans f (single b c)) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nh : ↑(PEquiv.symm f) b = none\nx✝ : δ\na✝ : γ\n⊢ a✝ ∈ ↑(PEquiv.trans f (single b c)) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"tactic": "simp only [eq_none_iff_forall_not_mem, Option.mem_def, f.eq_some_iff] at h"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ (a✝ ∈ Option.bind (↑f x✝) fun x => if x = b then some c else none) ↔ a✝ ∈ none",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ a✝ ∈ ↑(PEquiv.trans f (single b c)) x✝ ↔ a✝ ∈ ↑⊥ x✝",
"tactic": "dsimp [PEquiv.trans, single]"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ ∀ (x : β), ↑f x✝ = some x → ¬(if x = b then some c else none) = some a✝",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ (a✝ ∈ Option.bind (↑f x✝) fun x => if x = b then some c else none) ↔ a✝ ∈ none",
"tactic": "simp"
},
{
"state_after": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝¹ : δ\na✝¹ : γ\nh : ∀ (a : δ), ¬↑f a = some b\nx✝ : β\na✝ : ↑f x✝¹ = some x✝\n⊢ ¬(if x✝ = b then some c else none) = some a✝¹",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝ : δ\na✝ : γ\nh : ∀ (a : δ), ¬↑f a = some b\n⊢ ∀ (x : β), ↑f x✝ = some x → ¬(if x = b then some c else none) = some a✝",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case h.a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nb : β\nc : γ\nf : δ ≃. β\nx✝¹ : δ\na✝¹ : γ\nh : ∀ (a : δ), ¬↑f a = some b\nx✝ : β\na✝ : ↑f x✝¹ = some x✝\n⊢ ¬(if x✝ = b then some c else none) = some a✝¹",
"tactic": "split_ifs <;> simp_all"
}
] |
[
400,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Std/Data/List/Basic.lean
|
List.takeDTR_go_eq
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ndflt : α✝\nacc : Array α✝\nx✝ : List α✝\n⊢ takeDTR.go dflt 0 x✝ acc = acc.data ++ takeD 0 x✝ dflt",
"tactic": "simp [takeDTR.go]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ndflt : α✝\nacc : Array α✝\nn✝ : Nat\n⊢ takeDTR.go dflt (n✝ + 1) [] acc = acc.data ++ takeD (n✝ + 1) [] dflt",
"tactic": "simp [takeDTR.go]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u_1\ndflt : α✝\nacc : Array α✝\nn✝ : Nat\nhead✝ : α✝\nl : List α✝\n⊢ takeDTR.go dflt (n✝ + 1) (head✝ :: l) acc = acc.data ++ takeD (n✝ + 1) (head✝ :: l) dflt",
"tactic": "simp [takeDTR.go, takeDTR_go_eq _ l]"
}
] |
[
605,
57
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
602,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.single_blocksFun
|
[
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nc : Composition n✝\nn : ℕ\nh : 0 < n\ni : Fin (length (single n h))\n⊢ blocksFun (single n h) i = n",
"tactic": "simp [blocksFun, single, blocks, i.2]"
}
] |
[
588,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
587,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Ideal.leadingCoeffNth_mono
|
[
{
"state_after": "R : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\nr : R\nhr : r ∈ leadingCoeffNth I m\n⊢ r ∈ leadingCoeffNth I n",
"state_before": "R : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\n⊢ leadingCoeffNth I m ≤ leadingCoeffNth I n",
"tactic": "intro r hr"
},
{
"state_after": "R : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\nr : R\nhr : ∃ p, p ∈ I ∧ degree p ≤ ↑m ∧ Polynomial.leadingCoeff p = r\n⊢ ∃ p, p ∈ I ∧ degree p ≤ ↑n ∧ Polynomial.leadingCoeff p = r",
"state_before": "R : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\nr : R\nhr : r ∈ leadingCoeffNth I m\n⊢ r ∈ leadingCoeffNth I n",
"tactic": "simp only [SetLike.mem_coe, mem_leadingCoeffNth] at hr⊢"
},
{
"state_after": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ ∃ p_1, p_1 ∈ I ∧ degree p_1 ≤ ↑n ∧ Polynomial.leadingCoeff p_1 = Polynomial.leadingCoeff p",
"state_before": "R : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\nr : R\nhr : ∃ p, p ∈ I ∧ degree p ≤ ↑m ∧ Polynomial.leadingCoeff p = r\n⊢ ∃ p, p ∈ I ∧ degree p ≤ ↑n ∧ Polynomial.leadingCoeff p = r",
"tactic": "rcases hr with ⟨p, hpI, hpdeg, rfl⟩"
},
{
"state_after": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ degree (p * X ^ (n - m)) ≤ ↑n",
"state_before": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ ∃ p_1, p_1 ∈ I ∧ degree p_1 ≤ ↑n ∧ Polynomial.leadingCoeff p_1 = Polynomial.leadingCoeff p",
"tactic": "refine' ⟨p * X ^ (n - m), I.mul_mem_right _ hpI, _, leadingCoeff_mul_X_pow⟩"
},
{
"state_after": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ degree p + degree (X ^ (n - m)) ≤ ↑n",
"state_before": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ degree (p * X ^ (n - m)) ≤ ↑n",
"tactic": "refine' le_trans (degree_mul_le _ _) _"
},
{
"state_after": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ ↑m + ↑(n - m) ≤ ↑n",
"state_before": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ degree p + degree (X ^ (n - m)) ≤ ↑n",
"tactic": "refine' le_trans (add_le_add hpdeg (degree_X_pow_le _)) _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u\nS : Type ?u.180424\ninst✝¹ : CommSemiring R\ninst✝ : Semiring S\nI : Ideal R[X]\nm n : ℕ\nH : m ≤ n\np : R[X]\nhpI : p ∈ I\nhpdeg : degree p ≤ ↑m\n⊢ ↑m + ↑(n - m) ≤ ↑n",
"tactic": "rw [Nat.cast_withBot, Nat.cast_withBot, ← WithBot.coe_add, add_tsub_cancel_of_le H,\n Nat.cast_withBot]"
}
] |
[
598,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.const_mul_lintegral
|
[] |
[
1026,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.toMap_injective
|
[] |
[
571,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
570,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.mk'_eq_coe
|
[] |
[
137,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Topology/MetricSpace/Algebra.lean
|
dist_pair_smul
|
[] |
[
149,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.heq_iff_coe_eq
|
[] |
[
79,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.toSubmonoid_strictMono
|
[] |
[
288,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.continuous
|
[] |
[
677,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
11
] |
Mathlib/Algebra/DirectSum/Ring.lean
|
DirectSum.of_zero_mul
|
[] |
[
444,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_bot
|
[] |
[
864,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
863,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Antivary.sum_smul_lt_sum_comp_perm_smul_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrderedRing α\ninst✝³ : LinearOrderedAddCommGroup β\ninst✝² : Module α β\ninst✝¹ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\ninst✝ : Fintype ι\nhfg : Antivary f g\n⊢ ∑ i : ι, f i • g i < ∑ i : ι, f (↑σ i) • g i ↔ ¬Antivary (f ∘ ↑σ) g",
"tactic": "simp [(hfg.antivaryOn _).sum_smul_lt_sum_comp_perm_smul_iff fun _ _ ↦ mem_univ _]"
}
] |
[
334,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
isRightRegular_zero_iff_subsingleton
|
[] |
[
221,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Algebra/Order/Field/Power.lean
|
zpow_le_max_of_min_le
|
[] |
[
109,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.self_trans_symm
|
[] |
[
856,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
855,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.join_pure
|
[] |
[
1995,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1995,
9
] |
Mathlib/RingTheory/Polynomial/Chebyshev.lean
|
Polynomial.Chebyshev.T_one
|
[] |
[
81,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.toMatrix₂_basisFun
|
[
{
"state_after": "case h.a.h\nR : Type u_1\nR₁ : Type ?u.1414675\nR₂ : Type ?u.1414678\nM : Type ?u.1414681\nM₁ : Type ?u.1414684\nM₂ : Type ?u.1414687\nM₁' : Type ?u.1414690\nM₂' : Type ?u.1414693\nn : Type u_2\nm : Type u_3\nn' : Type ?u.1414702\nm' : Type ?u.1414705\nι : Type ?u.1414708\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\nB : (n → R) →ₗ[R] (m → R) →ₗ[R] R\ni✝ : n\nx✝ : m\n⊢ ↑(toMatrix₂ (Pi.basisFun R n) (Pi.basisFun R m)) B i✝ x✝ = ↑toMatrix₂' B i✝ x✝",
"state_before": "R : Type u_1\nR₁ : Type ?u.1414675\nR₂ : Type ?u.1414678\nM : Type ?u.1414681\nM₁ : Type ?u.1414684\nM₂ : Type ?u.1414687\nM₁' : Type ?u.1414690\nM₂' : Type ?u.1414693\nn : Type u_2\nm : Type u_3\nn' : Type ?u.1414702\nm' : Type ?u.1414705\nι : Type ?u.1414708\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\n⊢ toMatrix₂ (Pi.basisFun R n) (Pi.basisFun R m) = toMatrix₂'",
"tactic": "ext B"
},
{
"state_after": "no goals",
"state_before": "case h.a.h\nR : Type u_1\nR₁ : Type ?u.1414675\nR₂ : Type ?u.1414678\nM : Type ?u.1414681\nM₁ : Type ?u.1414684\nM₂ : Type ?u.1414687\nM₁' : Type ?u.1414690\nM₂' : Type ?u.1414693\nn : Type u_2\nm : Type u_3\nn' : Type ?u.1414702\nm' : Type ?u.1414705\nι : Type ?u.1414708\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ : Module R M₁\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\nB : (n → R) →ₗ[R] (m → R) →ₗ[R] R\ni✝ : n\nx✝ : m\n⊢ ↑(toMatrix₂ (Pi.basisFun R n) (Pi.basisFun R m)) B i✝ x✝ = ↑toMatrix₂' B i✝ x✝",
"tactic": "rw [LinearMap.toMatrix₂_apply, LinearMap.toMatrix₂'_apply, Pi.basisFun_apply, Pi.basisFun_apply]"
}
] |
[
417,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
Real.deriv_tan
|
[
{
"state_after": "x : ℝ\nh : cos x = 0\nthis : ¬DifferentiableAt ℝ tan x\n⊢ deriv tan x = ↑1 / cos x ^ 2",
"state_before": "x : ℝ\nh : cos x = 0\n⊢ deriv tan x = ↑1 / cos x ^ 2",
"tactic": "have : ¬DifferentiableAt ℝ tan x := mt differentiableAt_tan.1 (Classical.not_not.2 h)"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : cos x = 0\nthis : ¬DifferentiableAt ℝ tan x\n⊢ deriv tan x = ↑1 / cos x ^ 2",
"tactic": "simp [deriv_zero_of_not_differentiableAt this, h, sq]"
}
] |
[
68,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/List/Basic.lean
|
List.choose_property
|
[] |
[
3983,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3982,
1
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_add_nat_iff
|
[] |
[
558,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
normalize_eq_one
|
[] |
[
160,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.coe_edgeFinset
|
[] |
[
892,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasFDerivAt.sinh
|
[] |
[
1130,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1128,
1
] |
Mathlib/Algebra/Module/Torsion.lean
|
Submodule.torsion_gc
|
[] |
[
382,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.map_smul
|
[] |
[
971,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
970,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Iic_prod_eq
|
[] |
[
1896,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1895,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
NonUnitalRingHom.copy_eq
|
[] |
[
169,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Basic.lean
|
Units.inv_mul'
|
[] |
[
212,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.mem_carrier
|
[] |
[
66,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
balancedCoreAux_balanced
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\n⊢ (fun x => a • x) y ∈ balancedCoreAux 𝕜 s",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\n⊢ Balanced 𝕜 (balancedCoreAux 𝕜 s)",
"tactic": "rintro a ha x ⟨y, hy, rfl⟩"
},
{
"state_after": "case intro.intro.inl\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\nha : ‖0‖ ≤ 1\n⊢ (fun x => 0 • x) y ∈ balancedCoreAux 𝕜 s\n\ncase intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\nh : a ≠ 0\n⊢ (fun x => a • x) y ∈ balancedCoreAux 𝕜 s",
"state_before": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\n⊢ (fun x => a • x) y ∈ balancedCoreAux 𝕜 s",
"tactic": "obtain rfl | h := eq_or_ne a 0"
},
{
"state_after": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\n⊢ ∀ (r : 𝕜), 1 ≤ ‖r‖ → (fun x => a • x) y ∈ r • s",
"state_before": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\nh : a ≠ 0\n⊢ (fun x => a • x) y ∈ balancedCoreAux 𝕜 s",
"tactic": "rw [mem_balancedCoreAux_iff] at hy⊢"
},
{
"state_after": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\n⊢ (fun x => a • x) y ∈ r • s",
"state_before": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\n⊢ ∀ (r : 𝕜), 1 ≤ ‖r‖ → (fun x => a • x) y ∈ r • s",
"tactic": "intro r hr"
},
{
"state_after": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\n⊢ (fun x => a • x) y ∈ r • s",
"state_before": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\n⊢ (fun x => a • x) y ∈ r • s",
"tactic": "have h'' : 1 ≤ ‖a⁻¹ • r‖ := by\n rw [norm_smul, norm_inv]\n exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr"
},
{
"state_after": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\nh' : y ∈ (a⁻¹ • r) • s\n⊢ (fun x => a • x) y ∈ r • s",
"state_before": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\n⊢ (fun x => a • x) y ∈ r • s",
"tactic": "have h' := hy (a⁻¹ • r) h''"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inr\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\nh' : y ∈ (a⁻¹ • r) • s\n⊢ (fun x => a • x) y ∈ r • s",
"tactic": "rwa [smul_assoc, mem_inv_smul_set_iff₀ h] at h'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.inl\n𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\ny : E\nhy : y ∈ balancedCoreAux 𝕜 s\nha : ‖0‖ ≤ 1\n⊢ (fun x => 0 • x) y ∈ balancedCoreAux 𝕜 s",
"tactic": "simp_rw [zero_smul, h0]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\n⊢ 1 ≤ ‖a‖⁻¹ * ‖r‖",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\n⊢ 1 ≤ ‖a⁻¹ • r‖",
"tactic": "rw [norm_smul, norm_inv]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nι : Type ?u.47476\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\n⊢ 1 ≤ ‖a‖⁻¹ * ‖r‖",
"tactic": "exact one_le_mul_of_one_le_of_one_le (one_le_inv (norm_pos_iff.mpr h) ha) hr"
}
] |
[
186,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
deriv_sum
|
[] |
[
194,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
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