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 |
|---|---|---|---|---|---|---|
src/lean/Init/Core.lean
|
Nat.add_zero
|
[] |
[
457,
68
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
457,
19
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.succ_le_iff_of_not_isMax
|
[] |
[
258,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.eta_fin_three
|
[
{
"state_after": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix (Fin 3) (Fin 3) α\ni j : Fin 3\n⊢ A i j = ↑of ![![A 0 0, A 0 1, A 0 2], ![A 1 0, A 1 1, A 1 2], ![A 2 0, A 2 1, A 2 2]] i j",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix (Fin 3) (Fin 3) α\n⊢ A = ↑of ![![A 0 0, A 0 1, A 0 2], ![A 1 0, A 1 1, A 1 2], ![A 2 0, A 2 1, A 2 2]]",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\nA : Matrix (Fin 3) (Fin 3) α\ni j : Fin 3\n⊢ A i j = ↑of ![![A 0 0, A 0 1, A 0 2], ![A 1 0, A 1 1, A 1 2], ![A 2 0, A 2 1, A 2 2]] i j",
"tactic": "fin_cases i <;> fin_cases j <;> rfl"
}
] |
[
433,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Order/Heyting/Hom.lean
|
HeytingHom.cancel_right
|
[] |
[
359,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Types.lean
|
CategoryTheory.Limits.Types.binaryCoproductIso_inr_comp_hom
|
[] |
[
242,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
affineSegment_same
|
[
{
"state_after": "R : Type u_2\nV : Type u_3\nV' : Type ?u.18554\nP : Type u_1\nP' : Type ?u.18560\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx : P\n⊢ ↑(lineMap x x) '' Set.Icc 0 1 = {x}",
"state_before": "R : Type u_2\nV : Type u_3\nV' : Type ?u.18554\nP : Type u_1\nP' : Type ?u.18560\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx : P\n⊢ affineSegment R x x = {x}",
"tactic": "rw [affineSegment]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nV : Type u_3\nV' : Type ?u.18554\nP : Type u_1\nP' : Type ?u.18560\ninst✝⁶ : OrderedRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx : P\n⊢ ↑(lineMap x x) '' Set.Icc 0 1 = {x}",
"tactic": "simp_rw [lineMap_same, AffineMap.coe_const _ _, Function.const,\n (Set.nonempty_Icc.mpr zero_le_one).image_const]"
}
] |
[
78,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isPreirreducible_iff_closure
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen (u ∩ v)\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen v\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen u",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ Set.Nonempty (closure s ∩ u) → Set.Nonempty (closure s ∩ v) → Set.Nonempty (closure s ∩ (u ∩ v)) ↔\n Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))",
"tactic": "iterate 3 rw [closure_inter_open_nonempty_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen (u ∩ v)\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen v\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen u",
"tactic": "exacts [hu.inter hv, hv, hu]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen (u ∩ v)\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen v\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen u",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (closure s ∩ (u ∩ v)) ↔\n Set.Nonempty (s ∩ u) → Set.Nonempty (s ∩ v) → Set.Nonempty (s ∩ (u ∩ v))\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen v\n\nα : Type u\nβ : Type v\nι : Type ?u.181256\nπ : ι → Type ?u.181261\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns✝ t s u v : Set α\nhu : IsOpen u\nhv : IsOpen v\n⊢ IsOpen u",
"tactic": "rw [closure_inter_open_nonempty_iff]"
}
] |
[
1722,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1718,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_iff_measure_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.647088\nγ : Type ?u.647091\nδ : Type ?u.647094\nι : Type ?u.647097\nR : Type ?u.647100\nR' : Type ?u.647103\nm0 : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\ninst✝ : IsFiniteMeasure μ\np : α → Prop\nhp : NullMeasurableSet {a | p a}\n⊢ (∀ᵐ (a : α) ∂μ, p a) ↔ ↑↑μ {a | p a} = ↑↑μ univ",
"tactic": "rw [← ae_eq_univ_iff_measure_eq hp, eventuallyEq_univ, eventually_iff]"
}
] |
[
3173,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3171,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.StableUnderCobaseChange.mk
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.40576\ninst✝¹ : Category D\nP : MorphismProperty C\ninst✝ : HasPushouts C\nhP₁ : RespectsIso P\nhP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P pushout.inr\nA A' B B' : C\nf : A ⟶ A'\ng : A ⟶ B\nf' : B ⟶ B'\ng' : A' ⟶ B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' ≅ pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n⊢ P f'",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.40576\ninst✝¹ : Category D\nP : MorphismProperty C\ninst✝ : HasPushouts C\nhP₁ : RespectsIso P\nhP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P pushout.inr\nA A' B B' : C\nf : A ⟶ A'\ng : A ⟶ B\nf' : B ⟶ B'\ng' : A' ⟶ B'\nsq : IsPushout g f f' g'\nhf : P f\n⊢ P f'",
"tactic": "let e := sq.flip.isoPushout"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.40576\ninst✝¹ : Category D\nP : MorphismProperty C\ninst✝ : HasPushouts C\nhP₁ : RespectsIso P\nhP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P pushout.inr\nA A' B B' : C\nf : A ⟶ A'\ng : A ⟶ B\nf' : B ⟶ B'\ng' : A' ⟶ B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' ≅ pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n⊢ P pushout.inr",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.40576\ninst✝¹ : Category D\nP : MorphismProperty C\ninst✝ : HasPushouts C\nhP₁ : RespectsIso P\nhP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P pushout.inr\nA A' B B' : C\nf : A ⟶ A'\ng : A ⟶ B\nf' : B ⟶ B'\ng' : A' ⟶ B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' ≅ pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n⊢ P f'",
"tactic": "rw [← hP₁.cancel_right_isIso _ e.hom, sq.flip.inr_isoPushout_hom]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.40576\ninst✝¹ : Category D\nP : MorphismProperty C\ninst✝ : HasPushouts C\nhP₁ : RespectsIso P\nhP₂ : ∀ (A B A' : C) (f : A ⟶ A') (g : A ⟶ B), P f → P pushout.inr\nA A' B B' : C\nf : A ⟶ A'\ng : A ⟶ B\nf' : B ⟶ B'\ng' : A' ⟶ B'\nsq : IsPushout g f f' g'\nhf : P f\ne : B' ≅ pushout f g := IsPushout.isoPushout (_ : IsPushout f g g' f')\n⊢ P pushout.inr",
"tactic": "exact hP₂ _ _ _ f g hf"
}
] |
[
257,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.zero_lintegral
|
[] |
[
1042,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.Subsingleton.wellFoundedOn
|
[] |
[
503,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
11
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.bijOn
|
[] |
[
193,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
11
] |
Mathlib/Algebra/Star/Module.lean
|
skewAdjointPart_comp_subtype_skewAdjoint
|
[
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝⁷ : Semiring R\ninst✝⁶ : StarSemigroup R\ninst✝⁵ : TrivialStar R\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module R A\ninst✝² : StarAddMonoid A\ninst✝¹ : StarModule R A\ninst✝ : Invertible 2\nx✝ : { x // x ∈ skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n⊢ x = ↑(↑LinearMap.id { val := x, property := hx })",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝⁷ : Semiring R\ninst✝⁶ : StarSemigroup R\ninst✝⁵ : TrivialStar R\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module R A\ninst✝² : StarAddMonoid A\ninst✝¹ : StarModule R A\ninst✝ : Invertible 2\nx✝ : { x // x ∈ skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n⊢ ↑(↑(LinearMap.comp (skewAdjointPart R) (Submodule.subtype (skewAdjoint.submodule R A)))\n { val := x, property := hx }) =\n ↑(↑LinearMap.id { val := x, property := hx })",
"tactic": "simp only [LinearMap.comp_apply, Submodule.subtype_apply, skewAdjointPart_apply_coe, hx,\n sub_neg_eq_add, smul_add, invOf_two_smul_add_invOf_two_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝⁷ : Semiring R\ninst✝⁶ : StarSemigroup R\ninst✝⁵ : TrivialStar R\ninst✝⁴ : AddCommGroup A\ninst✝³ : Module R A\ninst✝² : StarAddMonoid A\ninst✝¹ : StarModule R A\ninst✝ : Invertible 2\nx✝ : { x // x ∈ skewAdjoint.submodule R A }\nx : A\nhx : star x = -x\n⊢ x = ↑(↑LinearMap.id { val := x, property := hx })",
"tactic": "rfl"
}
] |
[
173,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_Ico
|
[] |
[
755,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Algebra/Parity.lean
|
Even.neg_one_zpow
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.44652\nα : Type u_1\nβ : Type ?u.44658\nR : Type ?u.44661\ninst✝¹ : DivisionMonoid α\na : α\ninst✝ : HasDistribNeg α\nn : ℤ\nh : Even n\n⊢ (-1) ^ n = 1",
"tactic": "rw [h.neg_zpow, one_zpow]"
}
] |
[
194,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Geometry/Manifold/LocalInvariantProperties.lean
|
LocalHomeomorph.isLocalStructomorphWithinAt_iff
|
[
{
"state_after": "case mp\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\n⊢ IsLocalStructomorphWithinAt G (↑f) s x →\n x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source\n\ncase mpr\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\n⊢ (x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source) →\n IsLocalStructomorphWithinAt G (↑f) s x",
"state_before": "H : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\n⊢ IsLocalStructomorphWithinAt G (↑f) s x ↔\n x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source",
"tactic": "constructor"
},
{
"state_after": "case mp\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\n⊢ ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source",
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"tactic": "intro hf h2x"
},
{
"state_after": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source",
"state_before": "case mp\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\n⊢ ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source",
"tactic": "obtain ⟨e, he, hfe, hxe⟩ := hf h2x"
},
{
"state_after": "case mp.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ (LocalHomeomorph.restr e f.source).toLocalEquiv.source ⊆ f.source\n\ncase mp.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ EqOn (↑f) (↑(LocalHomeomorph.restr e f.source)) (s ∩ (LocalHomeomorph.restr e f.source).toLocalEquiv.source)\n\ncase mp.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ x ∈ interior f.source",
"state_before": "case mp.intro.intro.intro\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source",
"tactic": "refine' ⟨e.restr f.source, closedUnderRestriction' he f.open_source, _, _, hxe, _⟩"
},
{
"state_after": "case mp.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ e.source ∩ interior f.source ⊆ f.source",
"state_before": "case mp.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ (LocalHomeomorph.restr e f.source).toLocalEquiv.source ⊆ f.source",
"tactic": "simp_rw [LocalHomeomorph.restr_source]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.refine'_1\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ e.source ∩ interior f.source ⊆ f.source",
"tactic": "refine' (inter_subset_right _ _).trans interior_subset"
},
{
"state_after": "case mp.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\nx' : H\nhx' : x' ∈ s ∩ (LocalHomeomorph.restr e f.source).toLocalEquiv.source\n⊢ ↑f x' = ↑(LocalHomeomorph.restr e f.source) x'",
"state_before": "case mp.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ EqOn (↑f) (↑(LocalHomeomorph.restr e f.source)) (s ∩ (LocalHomeomorph.restr e f.source).toLocalEquiv.source)",
"tactic": "intro x' hx'"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.refine'_2\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\nx' : H\nhx' : x' ∈ s ∩ (LocalHomeomorph.restr e f.source).toLocalEquiv.source\n⊢ ↑f x' = ↑(LocalHomeomorph.restr e f.source) x'",
"tactic": "exact hfe ⟨hx'.1, hx'.2.1⟩"
},
{
"state_after": "case mp.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ x ∈ f.source",
"state_before": "case mp.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ x ∈ interior f.source",
"tactic": "rw [f.open_source.interior_eq]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.refine'_3\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\nhf : IsLocalStructomorphWithinAt G (↑f) s x\nh2x : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nhfe : EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ x ∈ f.source",
"tactic": "exact Or.resolve_right hx (not_not.mpr h2x)"
},
{
"state_after": "case mpr\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx✝ : x ∈ f.source ∪ sᶜ\nhf : x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source\nhx : x ∈ s\n⊢ ∃ e, e ∈ G ∧ EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source) ∧ x ∈ e.source",
"state_before": "case mpr\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\n⊢ (x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source) →\n IsLocalStructomorphWithinAt G (↑f) s x",
"tactic": "intro hf hx"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx✝ : x ∈ f.source ∪ sᶜ\nhf : x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source\nhx : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nleft✝ : e.source ⊆ f.source\nhfe : EqOn (↑f) (↑e) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ ∃ e, e ∈ G ∧ EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source) ∧ x ∈ e.source",
"state_before": "case mpr\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx✝ : x ∈ f.source ∪ sᶜ\nhf : x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source\nhx : x ∈ s\n⊢ ∃ e, e ∈ G ∧ EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source) ∧ x ∈ e.source",
"tactic": "obtain ⟨e, he, _, hfe, hxe⟩ := hf hx"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nH : Type u_1\nM : Type ?u.69060\nH' : Type ?u.69063\nM' : Type ?u.69066\nX : Type ?u.69069\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\ninst✝¹ : TopologicalSpace X\nG✝ : StructureGroupoid H\nG' : StructureGroupoid H'\ne✝ e' : LocalHomeomorph M H\nf✝ f' : LocalHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\ng g' : M → M'\ns✝ t : Set M\nx✝ : M\nQ : (H → H) → Set H → H → Prop\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : LocalHomeomorph H H\ns : Set H\nx : H\nhx✝ : x ∈ f.source ∪ sᶜ\nhf : x ∈ s → ∃ e, e ∈ G ∧ e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source\nhx : x ∈ s\ne : LocalHomeomorph H H\nhe : e ∈ G\nleft✝ : e.source ⊆ f.source\nhfe : EqOn (↑f) (↑e) (s ∩ e.source)\nhxe : x ∈ e.source\n⊢ ∃ e, e ∈ G ∧ EqOn (↑f) (↑e.toLocalEquiv) (s ∩ e.source) ∧ x ∈ e.source",
"tactic": "exact ⟨e, he, hfe, hxe⟩"
}
] |
[
626,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
1
] |
Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
|
AddMonoidAlgebra.toDirectSum_zero
|
[] |
[
125,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.prod_subset_preimage_fst
|
[] |
[
349,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.clog_zero_right
|
[] |
[
265,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.zero_ndinter
|
[] |
[
229,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
LowerSemicontinuous.isOpen_preimage
|
[] |
[
265,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Init/Function.lean
|
Function.Surjective.comp
|
[] |
[
76,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Control/Applicative.lean
|
Functor.Comp.applicative_id_comp
|
[] |
[
127,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_self_ne_zero
|
[] |
[
618,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/Topology/Algebra/UniformMulAction.lean
|
UniformSpace.Completion.smul_def
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.IsZero.eq_zero_of_tgt
|
[] |
[
186,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_sub
|
[] |
[
1231,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1230,
11
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasStrictFDerivAt.const_add
|
[] |
[
254,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrtAux_nonneg
|
[
{
"state_after": "f : CauSeq ℚ abs\n⊢ 0 ≤ ↑↑(Nat.sqrt (Int.toNat (↑f 0).num)) / ↑↑(Nat.sqrt (↑f 0).den)",
"state_before": "f : CauSeq ℚ abs\n⊢ 0 ≤ sqrtAux f 0",
"tactic": "rw [sqrtAux, Rat.mkRat_eq, Rat.divInt_eq_div]"
},
{
"state_after": "no goals",
"state_before": "f : CauSeq ℚ abs\n⊢ 0 ≤ ↑↑(Nat.sqrt (Int.toNat (↑f 0).num)) / ↑↑(Nat.sqrt (↑f 0).den)",
"tactic": "apply div_nonneg <;>\nexact Int.cast_nonneg.2 (Int.ofNat_nonneg _)"
}
] |
[
149,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Computability/Reduce.lean
|
ManyOneDegree.le_add_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝⁵ : Primcodable α\ninst✝⁴ : Inhabited α\nβ : Type v\ninst✝³ : Primcodable β\ninst✝² : Inhabited β\nγ : Type w\ninst✝¹ : Primcodable γ\ninst✝ : Inhabited γ\nd₁ d₂ : ManyOneDegree\n⊢ ?m.115538 d₁ d₂ + d₂ ≤ d₁ + d₂",
"tactic": "rfl"
}
] |
[
505,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
504,
11
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
sub_self_div_two
|
[
{
"state_after": "ι : Type ?u.183426\nα : Type u_1\nβ : Type ?u.183432\ninst✝ : LinearOrderedField α\na✝ b c d : α\nn : ℤ\na : α\n⊢ a / 2 + a / 2 - a / 2 = a / 2",
"state_before": "ι : Type ?u.183426\nα : Type u_1\nβ : Type ?u.183432\ninst✝ : LinearOrderedField α\na✝ b c d : α\nn : ℤ\na : α\n⊢ a - a / 2 = a / 2",
"tactic": "suffices a / 2 + a / 2 - a / 2 = a / 2 by rwa [add_halves] at this"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.183426\nα : Type u_1\nβ : Type ?u.183432\ninst✝ : LinearOrderedField α\na✝ b c d : α\nn : ℤ\na : α\n⊢ a / 2 + a / 2 - a / 2 = a / 2",
"tactic": "rw [add_sub_cancel]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.183426\nα : Type u_1\nβ : Type ?u.183432\ninst✝ : LinearOrderedField α\na✝ b c d : α\nn : ℤ\na : α\nthis : a / 2 + a / 2 - a / 2 = a / 2\n⊢ a - a / 2 = a / 2",
"tactic": "rwa [add_halves] at this"
}
] |
[
909,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
907,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
lie_sub
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\nN : Type w₁\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : LieRingModule L N\ninst✝ : LieModule R L N\nt : R\nx y z : L\nm n : M\n⊢ ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
188,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Algebra/Associated.lean
|
irreducible_units_mul
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\n⊢ ¬IsUnit b →\n ((∀ (a_2 b_1 : α), ↑a * b = a_2 * b_1 → IsUnit a_2 ∨ IsUnit b_1) ↔\n ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1)",
"state_before": "α : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\n⊢ Irreducible (↑a * b) ↔ Irreducible b",
"tactic": "simp only [irreducible_iff, Units.isUnit_units_mul, and_congr_right_iff]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ IsUnit A ∨ IsUnit B\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ IsUnit A ∨ IsUnit B",
"state_before": "α : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\n⊢ ¬IsUnit b →\n ((∀ (a_2 b_1 : α), ↑a * b = a_2 * b_1 → IsUnit a_2 ∨ IsUnit b_1) ↔\n ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1)",
"tactic": "refine' fun _ => ⟨fun h A B HAB => _, fun h A B HAB => _⟩"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ IsUnit (↑a * A) ∨ IsUnit B",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ IsUnit A ∨ IsUnit B",
"tactic": "rw [← a.isUnit_units_mul]"
},
{
"state_after": "case refine'_1.a\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ ↑a * b = ↑a * A * B",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ IsUnit (↑a * A) ∨ IsUnit B",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.a\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a_1 b_1 : α), ↑a * b = a_1 * b_1 → IsUnit a_1 ∨ IsUnit b_1\nA B : α\nHAB : b = A * B\n⊢ ↑a * b = ↑a * A * B",
"tactic": "rw [mul_assoc, ← HAB]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ IsUnit (↑a⁻¹ * A) ∨ IsUnit B",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ IsUnit A ∨ IsUnit B",
"tactic": "rw [← a⁻¹.isUnit_units_mul]"
},
{
"state_after": "case refine'_2.a\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ b = ↑a⁻¹ * A * B",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ IsUnit (↑a⁻¹ * A) ∨ IsUnit B",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.a\nα : Type u_1\nβ : Type ?u.87841\nγ : Type ?u.87844\nδ : Type ?u.87847\ninst✝ : Monoid α\na : αˣ\nb : α\nx✝ : ¬IsUnit b\nh : ∀ (a b_1 : α), b = a * b_1 → IsUnit a ∨ IsUnit b_1\nA B : α\nHAB : ↑a * b = A * B\n⊢ b = ↑a⁻¹ * A * B",
"tactic": "rw [mul_assoc, ← HAB, Units.inv_mul_cancel_left]"
}
] |
[
258,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_range_eq
|
[] |
[
2017,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2016,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.zero_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\n⊢ 0 = m * n ↔ m = 0 ∨ n = 0",
"tactic": "rw [eq_comm, Nat.mul_eq_zero]"
}
] |
[
102,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
11
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
LinearMap.proj_stdBasis_ne
|
[] |
[
98,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Topology/Constructions.lean
|
Dense.quotient
|
[] |
[
194,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
commutatorElement_self
|
[] |
[
58,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_lt_mul
|
[
{
"state_after": "case intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nac : a < c\nbd : b < d\na' : ℝ≥0\naa' : a < ↑a'\na'c : ↑a' < c\n⊢ a * b < c * d",
"state_before": "α : Type ?u.152394\nβ : Type ?u.152397\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nac : a < c\nbd : b < d\n⊢ a * b < c * d",
"tactic": "rcases lt_iff_exists_nnreal_btwn.1 ac with ⟨a', aa', a'c⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\n⊢ ↑a * b < c * d",
"state_before": "case intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nac : a < c\nbd : b < d\na' : ℝ≥0\naa' : a < ↑a'\na'c : ↑a' < c\n⊢ a * b < c * d",
"tactic": "lift a to ℝ≥0 using ne_top_of_lt aa'"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\nb' : ℝ≥0\nbb' : b < ↑b'\nb'd : ↑b' < d\n⊢ ↑a * b < c * d",
"state_before": "case intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\n⊢ ↑a * b < c * d",
"tactic": "rcases lt_iff_exists_nnreal_btwn.1 bd with ⟨b', bb', b'd⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nc d : ℝ≥0∞\nr p q a' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\nb' : ℝ≥0\nb'd : ↑b' < d\nb : ℝ≥0\nbd : ↑b < d\nbb' : ↑b < ↑b'\n⊢ ↑a * ↑b < c * d",
"state_before": "case intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nb c d : ℝ≥0∞\nr p q : ℝ≥0\nbd : b < d\na' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\nb' : ℝ≥0\nbb' : b < ↑b'\nb'd : ↑b' < d\n⊢ ↑a * b < c * d",
"tactic": "lift b to ℝ≥0 using ne_top_of_lt bb'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nc d : ℝ≥0∞\nr p q a' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\nb' : ℝ≥0\nb'd : ↑b' < d\nb : ℝ≥0\nbd : ↑b < d\naa' : a < a'\nbb' : b < b'\n⊢ ↑(a * b) < c * d",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nc d : ℝ≥0∞\nr p q a' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\naa' : ↑a < ↑a'\nb' : ℝ≥0\nb'd : ↑b' < d\nb : ℝ≥0\nbd : ↑b < d\nbb' : ↑b < ↑b'\n⊢ ↑a * ↑b < c * d",
"tactic": "norm_cast at *"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ?u.152394\nβ : Type ?u.152397\nc d : ℝ≥0∞\nr p q a' : ℝ≥0\na'c : ↑a' < c\na : ℝ≥0\nac : ↑a < c\nb' : ℝ≥0\nb'd : ↑b' < d\nb : ℝ≥0\nbd : ↑b < d\naa' : a < a'\nbb' : b < b'\n⊢ ↑(a * b) < c * d",
"tactic": "calc\n ↑(a * b) < ↑(a' * b') := coe_lt_coe.2 (mul_lt_mul₀ aa' bb')\n _ = ↑a' * ↑b' := coe_mul\n _ ≤ c * d := mul_le_mul' a'c.le b'd.le"
}
] |
[
971,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.coe_ringHom_apply
|
[] |
[
236,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equivalence.lean
|
CategoryTheory.Equivalence.hasInitial_iff
|
[] |
[
36,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
Measurable.dite
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.74519\nδ : Type ?u.74522\nδ' : Type ?u.74525\nι : Sort uι\ns t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : Measurable f\ng : ↑(sᶜ) → β\nhg : Measurable g\nhs : MeasurableSet s\n⊢ Measurable (restrict s fun x => if hx : x ∈ s then f { val := x, property := hx } else g { val := x, property := hx })",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.74519\nδ : Type ?u.74522\nδ' : Type ?u.74525\nι : Sort uι\ns t u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : (x : α) → Decidable (x ∈ s)\nf : ↑s → β\nhf : Measurable f\ng : ↑(sᶜ) → β\nhg : Measurable g\nhs : MeasurableSet s\n⊢ Measurable\n (restrict (sᶜ) fun x => if hx : x ∈ s then f { val := x, property := hx } else g { val := x, property := hx })",
"tactic": "simpa"
}
] |
[
612,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
|
Gamma_zero_bot
|
[
{
"state_after": "case h\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ x✝ ∈ Gamma 0 ↔ x✝ ∈ ⊥",
"state_before": "N : ℕ\n⊢ Gamma 0 = ⊥",
"tactic": "ext"
},
{
"state_after": "case h\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1 ↔ x✝ = 1",
"state_before": "case h\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ x✝ ∈ Gamma 0 ↔ x✝ ∈ ⊥",
"tactic": "simp only [Gamma_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply, Int.cast_id,\n Subgroup.mem_bot]"
},
{
"state_after": "case h.mp\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1 → x✝ = 1\n\ncase h.mpr\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ x✝ = 1 → ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1",
"state_before": "case h\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1 ↔ x✝ = 1",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ x✝ = 1",
"state_before": "case h.mp\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1 → x✝ = 1",
"tactic": "intro h"
},
{
"state_after": "case h.mp.a\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\ni : Fin 2\n⊢ ∀ (j : Fin 2), ↑x✝ i j = ↑1 i j",
"state_before": "case h.mp\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ x✝ = 1",
"tactic": "ext i"
},
{
"state_after": "case h.mp.a\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\ni j : Fin 2\n⊢ ↑x✝ i j = ↑1 i j",
"state_before": "case h.mp.a\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\ni : Fin 2\n⊢ ∀ (j : Fin 2), ↑x✝ i j = ↑1 i j",
"tactic": "intro j"
},
{
"state_after": "case h.mp.a.head.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) } =\n ↑1 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n\ncase h.mp.a.head.tail.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n ↑1 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\ncase h.mp.a.tail.head.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) } =\n ↑1 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\n\ncase h.mp.a.tail.head.tail.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n ↑1 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }",
"state_before": "case h.mp.a\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\ni j : Fin 2\n⊢ ↑x✝ i j = ↑1 i j",
"tactic": "fin_cases i <;> fin_cases j <;> simp only [h]"
},
{
"state_after": "no goals",
"state_before": "case h.mp.a.head.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) } =\n ↑1 { val := 0, isLt := (_ : 0 < 2) } { val := 0, isLt := (_ : 0 < 2) }\n\ncase h.mp.a.head.tail.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n ↑1 { val := 0, isLt := (_ : 0 < 2) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }\n\ncase h.mp.a.tail.head.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) } =\n ↑1 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 0, isLt := (_ : 0 < 2) }\n\ncase h.mp.a.tail.head.tail.head\nN : ℕ\nx✝ : SL(2, ℤ)\nh : ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1\n⊢ ↑x✝ { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) } =\n ↑1 { val := 1, isLt := (_ : (fun a => a < 2) 1) } { val := 1, isLt := (_ : (fun a => a < 2) 1) }",
"tactic": "exacts [h.1, h.2.1, h.2.2.1, h.2.2.2]"
},
{
"state_after": "case h.mpr\nN : ℕ\nx✝ : SL(2, ℤ)\nh : x✝ = 1\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1",
"state_before": "case h.mpr\nN : ℕ\nx✝ : SL(2, ℤ)\n⊢ x✝ = 1 → ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nN : ℕ\nx✝ : SL(2, ℤ)\nh : x✝ = 1\n⊢ ↑(↑x✝ 0 0) = 1 ∧ ↑(↑x✝ 0 1) = 0 ∧ ↑(↑x✝ 1 0) = 0 ∧ ↑(↑x✝ 1 1) = 1",
"tactic": "simp [h]"
}
] |
[
91,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean
|
HasDerivAtFilter.scomp
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\nhg : HasDerivAtFilter g₁ g₁' (h x) L'\nhh : HasDerivAtFilter h h' x L\nhL : Tendsto h L L'\n⊢ HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L",
"tactic": "simpa using ((hg.restrictScalars 𝕜).comp x hh hL).hasDerivAtFilter"
}
] |
[
82,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.eq_of_mem_singleton
|
[] |
[
1277,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1276,
1
] |
Mathlib/Data/Finset/Prod.lean
|
Finset.offDiag_inter
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.122500\nγ : Type ?u.122503\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ Set.offDiag (↑s ∩ ↑t) = Set.offDiag ↑s ∩ Set.offDiag ↑t",
"state_before": "α : Type u_1\nβ : Type ?u.122500\nγ : Type ?u.122503\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ ↑(offDiag (s ∩ t)) = ↑(offDiag s ∩ offDiag t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.122500\nγ : Type ?u.122503\ninst✝ : DecidableEq α\ns t : Finset α\nx : α × α\n⊢ Set.offDiag (↑s ∩ ↑t) = Set.offDiag ↑s ∩ Set.offDiag ↑t",
"tactic": "exact Set.offDiag_inter _ _"
}
] |
[
389,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.mem_iInf
|
[
{
"state_after": "R : Type u_2\nS : Type ?u.144585\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\nx : M\n⊢ (∀ (i : ι), x ∈ ↑(p i)) ↔ ∀ (i : ι), x ∈ p i",
"state_before": "R : Type u_2\nS : Type ?u.144585\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\nx : M\n⊢ (x ∈ ⨅ (i : ι), p i) ↔ ∀ (i : ι), x ∈ p i",
"tactic": "rw [← SetLike.mem_coe, iInf_coe, Set.mem_iInter]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type ?u.144585\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Sort u_1\np : ι → Submodule R M\nx : M\n⊢ (∀ (i : ι), x ∈ ↑(p i)) ↔ ∀ (i : ι), x ∈ p i",
"tactic": "rfl"
}
] |
[
269,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.splits_iff_roots_eq_three
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.714741\nS : Type ?u.714744\nF : Type u_1\nK : Type u_2\nP : Cubic F\ninst✝¹ : Field F\ninst✝ : Field K\nφ : F →+* K\nx y z : K\nha : P.a ≠ 0\n⊢ Splits φ (toPoly P) ↔ ∃ x y z, roots (map φ P) = {x, y, z}",
"tactic": "rw [splits_iff_card_roots ha, card_eq_three]"
}
] |
[
521,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Topology/Algebra/Order/Group.lean
|
Continuous.abs
|
[] |
[
83,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
11
] |
Mathlib/Deprecated/Group.lean
|
RingHom.to_isAddMonoidHom
|
[] |
[
381,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat.pow
|
[
{
"state_after": "no goals",
"state_before": "p a b : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nha : a ≠ 0\n⊢ padicValNat p (a ^ n) = n * padicValNat p a",
"tactic": "simpa only [← @Nat.cast_inj ℤ, push_cast] using padicValRat.pow (Nat.cast_ne_zero.mpr ha)"
}
] |
[
427,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
11
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.setToL1_indicatorConstLp
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1189632\nG : Type ?u.1189635\n𝕜 : Type ?u.1189638\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ ↑(setToL1 hT) ↑(indicatorConst 1 hs hμs x) = ↑(T s) x",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1189632\nG : Type ?u.1189635\n𝕜 : Type ?u.1189638\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ ↑(setToL1 hT) (indicatorConstLp 1 hs hμs x) = ↑(T s) x",
"tactic": "rw [← Lp.simpleFunc.coe_indicatorConst hs hμs x]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1189632\nG : Type ?u.1189635\n𝕜 : Type ?u.1189638\np : ℝ≥0∞\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedAddCommGroup F'\ninst✝⁵ : NormedSpace ℝ F'\ninst✝⁴ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s ≠ ⊤\nx : E\n⊢ ↑(setToL1 hT) ↑(indicatorConst 1 hs hμs x) = ↑(T s) x",
"tactic": "exact setToL1_simpleFunc_indicatorConst hT hs hμs.lt_top x"
}
] |
[
1153,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1149,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.abs_mul_exp_arg_mul_I
|
[
{
"state_after": "case inl\n\n⊢ ↑(↑abs 0) * exp (↑(arg 0) * I) = 0\n\ncase inr\nx : ℂ\nhx : x ≠ 0\n⊢ ↑(↑abs x) * exp (↑(arg x) * I) = x",
"state_before": "x : ℂ\n⊢ ↑(↑abs x) * exp (↑(arg x) * I) = x",
"tactic": "rcases eq_or_ne x 0 with (rfl | hx)"
},
{
"state_after": "no goals",
"state_before": "case inl\n\n⊢ ↑(↑abs 0) * exp (↑(arg 0) * I) = 0",
"tactic": "simp"
},
{
"state_after": "case inr\nx : ℂ\nhx : x ≠ 0\nthis : ↑abs x ≠ 0\n⊢ ↑(↑abs x) * exp (↑(arg x) * I) = x",
"state_before": "case inr\nx : ℂ\nhx : x ≠ 0\n⊢ ↑(↑abs x) * exp (↑(arg x) * I) = x",
"tactic": "have : abs x ≠ 0 := abs.ne_zero hx"
},
{
"state_after": "no goals",
"state_before": "case inr\nx : ℂ\nhx : x ≠ 0\nthis : ↑abs x ≠ 0\n⊢ ↑(↑abs x) * exp (↑(arg x) * I) = x",
"tactic": "ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)]"
}
] |
[
64,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.cardDistinctFactors_apply_prime
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.570430\np : ℕ\nhp : Prime p\n⊢ ↑ω p = 1",
"tactic": "rw [← pow_one p, cardDistinctFactors_apply_prime_pow hp one_ne_zero]"
}
] |
[
939,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_norm_eq_lintegral_nnnorm
|
[
{
"state_after": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ) = ENNReal.toReal (∫⁻ (x : α), ↑‖f x‖₊ ∂μ)\n\nα : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ 0 ≤ᵐ[μ] fun x => ‖f x‖",
"state_before": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ (∫ (x : α), ‖f x‖ ∂μ) = ENNReal.toReal (∫⁻ (x : α), ↑‖f x‖₊ ∂μ)",
"tactic": "rw [integral_eq_lintegral_of_nonneg_ae _ hf.norm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ) = ENNReal.toReal (∫⁻ (x : α), ↑‖f x‖₊ ∂μ)",
"tactic": "simp_rw [ofReal_norm_eq_coe_nnnorm]"
},
{
"state_after": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ ∀ (a : α), OfNat.ofNat 0 a ≤ (fun x => ‖f x‖) a",
"state_before": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ 0 ≤ᵐ[μ] fun x => ‖f x‖",
"tactic": "refine' ae_of_all _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type ?u.1049026\nF : Type ?u.1049029\n𝕜 : Type ?u.1049032\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℝ 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1051723\ninst✝² : TopologicalSpace X\ninst✝¹ : FirstCountableTopology X\nG : Type u_1\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : AEStronglyMeasurable f μ\n⊢ ∀ (a : α), OfNat.ofNat 0 a ≤ (fun x => ‖f x‖) a",
"tactic": "simp_rw [Pi.zero_apply, norm_nonneg, imp_true_iff]"
}
] |
[
1143,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1139,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.nullHomotopicMap_comp
|
[
{
"state_after": "case h\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nhom : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nn : ι\n⊢ Hom.f (nullHomotopicMap hom ≫ g) n = Hom.f (nullHomotopicMap fun i j => hom i j ≫ Hom.f g j) n",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nhom : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\n⊢ nullHomotopicMap hom ≫ g = nullHomotopicMap fun i j => hom i j ≫ Hom.f g j",
"tactic": "ext n"
},
{
"state_after": "case h\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nhom : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nn : ι\n⊢ (dFrom C n ≫ hom (ComplexShape.next c n) n + hom n (ComplexShape.prev c n) ≫ dTo D n) ≫ Hom.f g n =\n dFrom C n ≫ hom (ComplexShape.next c n) n ≫ Hom.f g n +\n (hom n (ComplexShape.prev c n) ≫ Hom.f g (ComplexShape.prev c n)) ≫ dTo E n",
"state_before": "case h\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nhom : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nn : ι\n⊢ Hom.f (nullHomotopicMap hom ≫ g) n = Hom.f (nullHomotopicMap fun i j => hom i j ≫ Hom.f g j) n",
"tactic": "dsimp [nullHomotopicMap, fromNext, toPrev, AddMonoidHom.mk'_apply]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nhom : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nn : ι\n⊢ (dFrom C n ≫ hom (ComplexShape.next c n) n + hom n (ComplexShape.prev c n) ≫ dTo D n) ≫ Hom.f g n =\n dFrom C n ≫ hom (ComplexShape.next c n) n ≫ Hom.f g n +\n (hom n (ComplexShape.prev c n) ≫ Hom.f g (ComplexShape.prev c n)) ≫ dTo E n",
"tactic": "simp only [Preadditive.add_comp, Category.assoc, g.comm]"
}
] |
[
275,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_mono_left
|
[] |
[
1481,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1478,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.sup'
|
[] |
[
639,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.epi_of_cokernel_iso_zero
|
[] |
[
290,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.prod_apply_inr_snd
|
[
{
"state_after": "case h\nι : Type u_3\nι' : Type u_4\nR : Type u_5\nR₂ : Type ?u.583731\nK : Type ?u.583734\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.583743\nV : Type u\nV' : Type ?u.583748\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni✝ i : ι'\n⊢ ↑(↑b'.repr (↑(Basis.prod b b') (Sum.inr i✝)).snd) i = ↑(↑b'.repr (↑b' i✝)) i",
"state_before": "ι : Type u_3\nι' : Type u_4\nR : Type u_5\nR₂ : Type ?u.583731\nK : Type ?u.583734\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.583743\nV : Type u\nV' : Type ?u.583748\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni : ι'\n⊢ ↑b'.repr (↑(Basis.prod b b') (Sum.inr i)).snd = ↑b'.repr (↑b' i)",
"tactic": "ext i"
},
{
"state_after": "case h\nι : Type u_3\nι' : Type u_4\nR : Type u_5\nR₂ : Type ?u.583731\nK : Type ?u.583734\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.583743\nV : Type u\nV' : Type ?u.583748\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni✝ i : ι'\n⊢ ↑(Finsupp.single (Sum.inr i✝) 1) (Sum.inr i) = ↑(Finsupp.single i✝ 1) i",
"state_before": "case h\nι : Type u_3\nι' : Type u_4\nR : Type u_5\nR₂ : Type ?u.583731\nK : Type ?u.583734\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.583743\nV : Type u\nV' : Type ?u.583748\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni✝ i : ι'\n⊢ ↑(↑b'.repr (↑(Basis.prod b b') (Sum.inr i✝)).snd) i = ↑(↑b'.repr (↑b' i✝)) i",
"tactic": "simp only [Basis.prod, Basis.coe_ofRepr, LinearEquiv.symm_trans_apply, LinearEquiv.prod_symm,\n LinearEquiv.prod_apply, b'.repr.apply_symm_apply, LinearEquiv.symm_symm, repr_self,\n Equiv.toFun_as_coe, Finsupp.snd_sumFinsuppLEquivProdFinsupp]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_3\nι' : Type u_4\nR : Type u_5\nR₂ : Type ?u.583731\nK : Type ?u.583734\nM : Type u_2\nM' : Type u_1\nM'' : Type ?u.583743\nV : Type u\nV' : Type ?u.583748\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni✝¹ : ι\nc : R\nx : M\nb' : Basis ι' R M'\ni✝ i : ι'\n⊢ ↑(Finsupp.single (Sum.inr i✝) 1) (Sum.inr i) = ↑(Finsupp.single i✝ 1) i",
"tactic": "apply Finsupp.single_apply_left Sum.inr_injective"
}
] |
[
761,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.list_cycles_perm_list_cycles
|
[
{
"state_after": "ι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"state_before": "ι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\n⊢ l₁ ~ l₂",
"tactic": "refine'\n (List.perm_ext (nodup_of_pairwise_disjoint_cycles h₁l₁ h₂l₁)\n (nodup_of_pairwise_disjoint_cycles h₁l₂ h₂l₂)).mpr\n fun σ => _"
},
{
"state_after": "case pos\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : IsCycle σ\n⊢ σ ∈ l₁ ↔ σ ∈ l₂\n\ncase neg\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : ¬IsCycle σ\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"state_before": "ι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"tactic": "by_cases hσ : σ.IsCycle"
},
{
"state_after": "case pos\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : IsCycle σ\nx✝ : ∃ x, ¬↑σ x = ↑1 x\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"state_before": "case pos\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : IsCycle σ\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"tactic": "obtain _ := not_forall.mp (mt ext hσ.ne_one)"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : IsCycle σ\nx✝ : ∃ x, ¬↑σ x = ↑1 x\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"tactic": "rw [mem_list_cycles_iff h₁l₁ h₂l₁, mem_list_cycles_iff h₁l₂ h₂l₂, h₀]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.2673155\nα✝ : Type ?u.2673158\nβ : Type ?u.2673161\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl₁ l₂ : List (Perm α)\nh₀ : List.prod l₁ = List.prod l₂\nh₁l₁ : ∀ (σ : Perm α), σ ∈ l₁ → IsCycle σ\nh₁l₂ : ∀ (σ : Perm α), σ ∈ l₂ → IsCycle σ\nh₂l₁ : List.Pairwise Disjoint l₁\nh₂l₂ : List.Pairwise Disjoint l₂\nσ : Perm α\nhσ : ¬IsCycle σ\n⊢ σ ∈ l₁ ↔ σ ∈ l₂",
"tactic": "exact iff_of_false (mt (h₁l₁ σ) hσ) (mt (h₁l₂ σ) hσ)"
}
] |
[
1325,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1313,
1
] |
Mathlib/Init/Data/Int/Basic.lean
|
Int.ofNat_add_out
|
[] |
[
48,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
11
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.prod
|
[] |
[
103,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
11
] |
Mathlib/Topology/Inseparable.lean
|
IsClosed.not_specializes
|
[] |
[
136,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.lintegral_norm_eq_lintegral_edist
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.688837\nδ : Type ?u.688840\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝² : MeasurableSpace δ\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedAddCommGroup γ\nf : α → β\n⊢ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ) = ∫⁻ (a : α), edist (f a) 0 ∂μ",
"tactic": "simp only [ofReal_norm_eq_coe_nnnorm, edist_eq_coe_nnnorm]"
}
] |
[
76,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
Subgroup.commutator_bot_left
|
[] |
[
160,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
exists_finite_card_le_of_finite_of_linearIndependent_of_span
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\n⊢ s ⊆ ↑(span K t)",
"state_before": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\n⊢ s ⊆ ↑(span K ↑(Finite.toFinset ht))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\n⊢ s ⊆ ↑(span K t)",
"tactic": "assumption"
},
{
"state_after": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\nthis✝ : s ⊆ ↑(span K ↑(Finite.toFinset ht))\nu : Finset V\n_hust : ↑u ⊆ s ∪ ↑(Finite.toFinset ht)\nhsu : s ⊆ ↑u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n⊢ Finset.card (Finite.toFinset this) ≤ Finset.card u",
"state_before": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\nthis✝ : s ⊆ ↑(span K ↑(Finite.toFinset ht))\nu : Finset V\n_hust : ↑u ⊆ s ∪ ↑(Finite.toFinset ht)\nhsu : s ⊆ ↑u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n⊢ Finset.card (Finite.toFinset this) ≤ Finset.card (Finite.toFinset ht)",
"tactic": "rw [← Eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\nthis✝ : s ⊆ ↑(span K ↑(Finite.toFinset ht))\nu : Finset V\n_hust : ↑u ⊆ s ∪ ↑(Finite.toFinset ht)\nhsu : s ⊆ ↑u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n⊢ Finset.card (Finite.toFinset this) ≤ Finset.card u",
"tactic": "exact Finset.card_le_of_subset <| Finset.coe_subset.mp <| by simp [hsu]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.1312198\nR : Type ?u.1312201\nK : Type u_1\nM : Type ?u.1312207\nM' : Type ?u.1312210\nM'' : Type ?u.1312213\nV : Type u\nV' : Type ?u.1312218\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V\ninst✝² : AddCommGroup V'\ninst✝¹ : Module K V\ninst✝ : Module K V'\nv : ι → V\ns t : Set V\nx y z : V\nht : Set.Finite t\nhs : LinearIndependent K fun x => ↑x\nhst : s ⊆ ↑(span K t)\nthis✝ : s ⊆ ↑(span K ↑(Finite.toFinset ht))\nu : Finset V\n_hust : ↑u ⊆ s ∪ ↑(Finite.toFinset ht)\nhsu : s ⊆ ↑u\nEq : Finset.card u = Finset.card (Finite.toFinset ht)\nthis : Set.Finite s\n⊢ ↑(Finite.toFinset this) ⊆ ↑u",
"tactic": "simp [hsu]"
}
] |
[
1405,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1399,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.coe_copy
|
[] |
[
510,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Order/SymmDiff.lean
|
bot_bihimp
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.57742\nα : Type u_1\nβ : Type ?u.57748\nπ : ι → Type ?u.57753\ninst✝ : HeytingAlgebra α\na : α\n⊢ ⊥ ⇔ a = aᶜ",
"tactic": "simp [bihimp]"
}
] |
[
373,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
mul_lt_iff_lt_one_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c d : α\ninst✝⁴ : MulOneClass α\ninst✝³ : Zero α\ninst✝² : Preorder α\ninst✝¹ : MulPosStrictMono α\ninst✝ : MulPosReflectLT α\nb0 : 0 < b\n⊢ a * b < b ↔ a * b < 1 * b",
"tactic": "rw [one_mul]"
}
] |
[
660,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Set.pairwise_sUnion
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.513\nγ : Type ?u.516\nι : Type ?u.519\nι' : Type ?u.522\nr✝ p q : α → α → Prop\nf g : ι → α\ns✝ t u : Set α\na b : α\nr : α → α → Prop\ns : Set (Set α)\nh : DirectedOn (fun x x_1 => x ⊆ x_1) s\n⊢ Set.Pairwise (⋃₀ s) r ↔ ∀ (a : Set α), a ∈ s → Set.Pairwise a r",
"tactic": "rw [sUnion_eq_iUnion, pairwise_iUnion h.directed_val, SetCoe.forall]"
}
] |
[
45,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Algebra/Divisibility/Units.lean
|
isUnit_iff_forall_dvd
|
[] |
[
132,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineMap.lineMap_vsub
|
[] |
[
640,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_subset_Ioc_self
|
[] |
[
511,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/RingTheory/DedekindDomain/Factorization.lean
|
Associates.finprod_ne_zero
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\n⊢ (if h : Set.Finite (mulSupport fun v => maxPowDividing v I) then ∏ i in Finite.toFinset h, maxPowDividing i I\n else 1) ≠\n 0",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\n⊢ Associates.mk (∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I) ≠ 0",
"tactic": "rw [Associates.mk_ne_zero, finprod_def]"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ∏ v in Finite.toFinset h✝, maxPowDividing v I ≠ 0\n\ncase inr\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : ¬Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ 1 ≠ 0",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\n⊢ (if h : Set.Finite (mulSupport fun v => maxPowDividing v I) then ∏ i in Finite.toFinset h, maxPowDividing i I\n else 1) ≠\n 0",
"tactic": "split_ifs"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ∀ (a : HeightOneSpectrum R), a ∈ Finite.toFinset h✝ → maxPowDividing a I ≠ 0",
"state_before": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ∏ v in Finite.toFinset h✝, maxPowDividing v I ≠ 0",
"tactic": "rw [Finset.prod_ne_zero_iff]"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\nv : HeightOneSpectrum R\na✝ : v ∈ Finite.toFinset h✝\n⊢ maxPowDividing v I ≠ 0",
"state_before": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ∀ (a : HeightOneSpectrum R), a ∈ Finite.toFinset h✝ → maxPowDividing a I ≠ 0",
"tactic": "intro v _"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\nI : Ideal R\nh✝ : Set.Finite (mulSupport fun v => maxPowDividing v I)\nv : HeightOneSpectrum R\na✝ : v ∈ Finite.toFinset h✝\n⊢ maxPowDividing v I ≠ 0",
"tactic": "apply pow_ne_zero _ v.ne_bot"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.240355\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nh✝ : ¬Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ 1 ≠ 0",
"tactic": "exact one_ne_zero"
}
] |
[
135,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Order/PartialSups.lean
|
partialSups_eq_sup_range
|
[
{
"state_after": "case zero\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\n⊢ ↑(partialSups f) Nat.zero = Finset.sup (Finset.range (Nat.zero + 1)) f\n\ncase succ\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\nn : ℕ\nih : ↑(partialSups f) n = Finset.sup (Finset.range (n + 1)) f\n⊢ ↑(partialSups f) (Nat.succ n) = Finset.sup (Finset.range (Nat.succ n + 1)) f",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\nn : ℕ\n⊢ ↑(partialSups f) n = Finset.sup (Finset.range (n + 1)) f",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\n⊢ ↑(partialSups f) Nat.zero = Finset.sup (Finset.range (Nat.zero + 1)) f",
"tactic": "simp"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\nn : ℕ\nih : Nat.rec (f 0) (fun n a => a ⊔ f (n + 1)) n = Finset.sup (Finset.range (n + 1)) f\n⊢ Nat.rec (f 0) (fun n a => a ⊔ f (n + 1)) n ⊔ f (n + 1) = Finset.sup (Finset.range (Nat.succ n + 1)) f",
"state_before": "case succ\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\nn : ℕ\nih : ↑(partialSups f) n = Finset.sup (Finset.range (n + 1)) f\n⊢ ↑(partialSups f) (Nat.succ n) = Finset.sup (Finset.range (Nat.succ n + 1)) f",
"tactic": "dsimp [partialSups] at ih⊢"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\nf : ℕ → α\nn : ℕ\nih : Nat.rec (f 0) (fun n a => a ⊔ f (n + 1)) n = Finset.sup (Finset.range (n + 1)) f\n⊢ Nat.rec (f 0) (fun n a => a ⊔ f (n + 1)) n ⊔ f (n + 1) = Finset.sup (Finset.range (Nat.succ n + 1)) f",
"tactic": "rw [Finset.range_succ, Finset.sup_insert, sup_comm, ih]"
}
] |
[
138,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.count_mul
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : Associates α\nha : a ≠ 0\nb : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : Associates α\nha : a ≠ 0\nb : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\n⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)",
"tactic": "obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : Associates α\nha : a ≠ 0\nb : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\nb0 : α\nnzb : b0 ≠ 0\nhb' : Associates.mk b0 = b\n⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : Associates α\nha : a ≠ 0\nb : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)",
"tactic": "obtain ⟨b0, nzb, hb'⟩ := exists_non_zero_rep hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : Associates α\nha : a ≠ 0\nb : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\nb0 : α\nnzb : b0 ≠ 0\nhb' : Associates.mk b0 = b\n⊢ count p (factors (a * b)) = count p (factors a) + count p (factors b)",
"tactic": "rw [factors_mul, ← ha', ← hb', factors_mk a0 nza, factors_mk b0 nzb, ← FactorSet.coe_add, ←\n WithTop.some_eq_coe, ← WithTop.some_eq_coe, ← WithTop.some_eq_coe, count_some hp,\n Multiset.count_add, count_some hp, count_some hp]"
}
] |
[
1738,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1732,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.submatrix_diagonal_embedding
|
[] |
[
2478,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2476,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_iInter_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.210965\nι : Sort u_3\nι' : Sort ?u.210971\nι₂ : Sort ?u.210974\nκ : ι → Sort ?u.210979\nκ₁ : ι → Sort ?u.210984\nκ₂ : ι → Sort ?u.210989\nκ' : ι' → Sort ?u.210994\nf : ι → α\ng : α → Set β\n⊢ (⋂ (x : α) (y : ι) (_ : f y = x), g x) = ⋂ (y : ι), g (f y)",
"tactic": "simpa using biInter_range"
}
] |
[
1694,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1693,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjoint_erase_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.245796\nγ : Type ?u.245799\ninst✝ : DecidableEq α\ns t u v : Finset α\na b : α\n⊢ _root_.Disjoint (erase s a) t ↔ _root_.Disjoint s (erase t a)",
"tactic": "simp_rw [erase_eq, disjoint_sdiff_comm]"
}
] |
[
2231,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2230,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.Germ.coe_inv
|
[] |
[
454,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_neg
|
[] |
[
835,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
834,
11
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
lt_one_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.81079\nα : Type u_1\nβ : Type ?u.81085\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nhb : 0 < b\n⊢ a < 1 / b ↔ b < 1 / a",
"tactic": "simpa using lt_inv ha hb"
}
] |
[
445,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
445,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieAlgebra.nilpotent_ad_of_nilpotent_algebra
|
[] |
[
536,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean
|
FreeAbelianGroup.support_neg
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\na : FreeAbelianGroup X\n⊢ support (-a) = support a",
"tactic": "simp only [support, AddMonoidHom.map_neg, Finsupp.support_neg]"
}
] |
[
178,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
177,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.of_hasBinaryBiproduct
|
[] |
[
836,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
segment_translate_image
|
[] |
[
284,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/Nat/Log.lean
|
Nat.lt_pow_of_log_lt
|
[] |
[
131,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Multiset.toFinset_cons
|
[] |
[
3148,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3147,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.quotientEquivSigmaZMod_symm_apply
|
[] |
[
586,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Data/Finset/Sum.lean
|
Finset.disj_sum_strictMono_right
|
[] |
[
113,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Polynomial/Induction.lean
|
Polynomial.exists_C_coeff_not_mem
|
[] |
[
102,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infinitesimal_of_tendsto_zero
|
[] |
[
715,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.one_lt_top
|
[] |
[
695,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
9
] |
Std/Data/Nat/Lemmas.lean
|
Nat.dvd_antisymm
|
[] |
[
710,
96
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
707,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
LowerSemicontinuousAt.add
|
[] |
[
495,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.closure_univ
|
[] |
[
994,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
993,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.coe_image₂
|
[] |
[
54,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sin_sub_sin
|
[
{
"state_after": "x y : ℂ\ns1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"state_before": "x y : ℂ\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "have s1 := sin_add ((x + y) / 2) ((x - y) / 2)"
},
{
"state_after": "x y : ℂ\ns1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"state_before": "x y : ℂ\ns1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "have s2 := sin_sub ((x + y) / 2) ((x - y) / 2)"
},
{
"state_after": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"state_before": "x y : ℂ\ns1 : sin ((x + y) / 2 + (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "rw [div_add_div_same, add_sub, add_right_comm, add_sub_cancel, half_add_self] at s1"
},
{
"state_after": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"state_before": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin ((x + y) / 2 - (x - y) / 2) = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "rw [div_sub_div_same, ← sub_add, add_sub_cancel', half_add_self] at s2"
},
{
"state_after": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2) -\n (sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)) =\n 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"state_before": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin x - sin y = 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "rw [s1, s2]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\ns1 : sin x = sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2)\ns2 : sin y = sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)\n⊢ sin ((x + y) / 2) * cos ((x - y) / 2) + cos ((x + y) / 2) * sin ((x - y) / 2) -\n (sin ((x + y) / 2) * cos ((x - y) / 2) - cos ((x + y) / 2) * sin ((x - y) / 2)) =\n 2 * sin ((x - y) / 2) * cos ((x + y) / 2)",
"tactic": "ring"
}
] |
[
900,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Finset/Card.lean
|
Finset.card_sdiff_add_card
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41772\ns t : Finset α\nf : α → β\nn : ℕ\ninst✝ : DecidableEq α\n⊢ card (s \\ t) + card t = card (s ∪ t)",
"tactic": "rw [← card_disjoint_union sdiff_disjoint, sdiff_union_self_eq_union]"
}
] |
[
453,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
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