file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csInf_le_of_le
|
[] |
[
463,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.denote_toExpr_go
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\ne : Expr\np : Poly\n⊢ Expr.denote ctx (toExpr.go e p) = Expr.denote ctx e + denote ctx p",
"tactic": "induction p generalizing e with\n| nil => simp [toExpr.go, Poly.denote]\n| cons kv p ih => cases kv; simp [toExpr.go, ih, Expr.denote, Poly.denote]"
},
{
"state_after": "no goals",
"state_before": "case nil\nctx : Context\ne : Expr\n⊢ Expr.denote ctx (toExpr.go e []) = Expr.denote ctx e + denote ctx []",
"tactic": "simp [toExpr.go, Poly.denote]"
},
{
"state_after": "case cons.mk\nctx : Context\np : List (Nat × Var)\nih : ∀ (e : Expr), Expr.denote ctx (toExpr.go e p) = Expr.denote ctx e + denote ctx p\ne : Expr\nfst✝ : Nat\nsnd✝ : Var\n⊢ Expr.denote ctx (toExpr.go e ((fst✝, snd✝) :: p)) = Expr.denote ctx e + denote ctx ((fst✝, snd✝) :: p)",
"state_before": "case cons\nctx : Context\nkv : Nat × Var\np : List (Nat × Var)\nih : ∀ (e : Expr), Expr.denote ctx (toExpr.go e p) = Expr.denote ctx e + denote ctx p\ne : Expr\n⊢ Expr.denote ctx (toExpr.go e (kv :: p)) = Expr.denote ctx e + denote ctx (kv :: p)",
"tactic": "cases kv"
},
{
"state_after": "no goals",
"state_before": "case cons.mk\nctx : Context\np : List (Nat × Var)\nih : ∀ (e : Expr), Expr.denote ctx (toExpr.go e p) = Expr.denote ctx e + denote ctx p\ne : Expr\nfst✝ : Nat\nsnd✝ : Var\n⊢ Expr.denote ctx (toExpr.go e ((fst✝, snd✝) :: p)) = Expr.denote ctx e + denote ctx ((fst✝, snd✝) :: p)",
"tactic": "simp [toExpr.go, ih, Expr.denote, Poly.denote]"
}
] |
[
706,
77
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
703,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.relindex_le_of_le_right
|
[] |
[
407,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/CategoryTheory/Limits/FullSubcategory.lean
|
CategoryTheory.Limits.hasLimit_of_closed_under_limits
|
[] |
[
115,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_symmDiff_distrib_right
|
[] |
[
2119,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2118,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.isLeast_lfp_le
|
[] |
[
96,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.Icc_mem_nhds
|
[] |
[
257,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
Convex.affine_image
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.240342\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t : Set E\nf : E →ᵃ[𝕜] F\nhs : Convex 𝕜 s\nx : E\nhx : x ∈ s\n⊢ StarConvex 𝕜 (↑f x) (↑f '' s)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.240342\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t : Set E\nf : E →ᵃ[𝕜] F\nhs : Convex 𝕜 s\n⊢ Convex 𝕜 (↑f '' s)",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nβ : Type ?u.240342\ninst✝⁴ : OrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns t : Set E\nf : E →ᵃ[𝕜] F\nhs : Convex 𝕜 s\nx : E\nhx : x ∈ s\n⊢ StarConvex 𝕜 (↑f x) (↑f '' s)",
"tactic": "exact (hs hx).affine_image _"
}
] |
[
504,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
lowerCentralSeries_antitone
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ lowerCentralSeries G (n + 1)\n⊢ x ∈ lowerCentralSeries G n",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ Antitone (lowerCentralSeries G)",
"tactic": "refine' antitone_nat_of_succ_le fun n x hx => _"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\n⊢ x ∈ lowerCentralSeries G n",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ lowerCentralSeries G (n + 1)\n⊢ x ∈ lowerCentralSeries G n",
"tactic": "simp only [mem_lowerCentralSeries_succ_iff, exists_prop, mem_top, exists_true_left,\n true_and_iff] at hx"
},
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\n⊢ ∀ (x : G), x ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x} → x ∈ lowerCentralSeries G n",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\n⊢ x ∈ lowerCentralSeries G n",
"tactic": "refine'\n closure_induction hx _ (Subgroup.one_mem _) (@Subgroup.mul_mem _ _ _) (@Subgroup.inv_mem _ _ _)"
},
{
"state_after": "case intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\ny z : G\nhz : z ∈ lowerCentralSeries G n\na : G\nha : z * a * z⁻¹ * a⁻¹ = y\n⊢ y ∈ lowerCentralSeries G n",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\n⊢ ∀ (x : G), x ∈ {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x} → x ∈ lowerCentralSeries G n",
"tactic": "rintro y ⟨z, hz, a, ha⟩"
},
{
"state_after": "case intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\ny z : G\nhz : z ∈ lowerCentralSeries G n\na : G\nha : z * a * z⁻¹ * a⁻¹ = y\n⊢ z * (a * z⁻¹ * a⁻¹) ∈ lowerCentralSeries G n",
"state_before": "case intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\ny z : G\nhz : z ∈ lowerCentralSeries G n\na : G\nha : z * a * z⁻¹ * a⁻¹ = y\n⊢ y ∈ lowerCentralSeries G n",
"tactic": "rw [← ha, mul_assoc, mul_assoc, ← mul_assoc a z⁻¹ a⁻¹]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nn : ℕ\nx : G\nhx : x ∈ closure {x | ∃ p, p ∈ lowerCentralSeries G n ∧ ∃ q, p * q * p⁻¹ * q⁻¹ = x}\ny z : G\nhz : z ∈ lowerCentralSeries G n\na : G\nha : z * a * z⁻¹ * a⁻¹ = y\n⊢ z * (a * z⁻¹ * a⁻¹) ∈ lowerCentralSeries G n",
"tactic": "exact mul_mem hz (Normal.conj_mem (lowerCentralSeries_normal n) z⁻¹ (inv_mem hz) a)"
}
] |
[
323,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.mem_def
|
[] |
[
116,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
Convex.combo_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.11520\nR : Type u_1\nk : Type ?u.11526\nS : Type ?u.11529\nM : Type u_2\nM₂ : Type ?u.11535\nM₃ : Type ?u.11538\nι : Type ?u.11541\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr s : R\nx✝ y : M\na b : R\nh : a + b = 1\nx : M\n⊢ a • x + b • x = x",
"tactic": "rw [← add_smul, h, one_smul]"
}
] |
[
99,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/List/Card.lean
|
List.card_insert_of_not_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort ?u.26060\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\na : α\nas : List α\nh : ¬a ∈ as\n⊢ card (List.insert a as) = card as + 1",
"tactic": "simp [h]"
}
] |
[
96,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
9
] |
Mathlib/Data/Finset/Preimage.lean
|
Finset.prod_preimage'
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : CommMonoid β\nf : α → γ\ninst✝ : DecidablePred fun x => x ∈ Set.range f\ns : Finset γ\nhf : InjOn f (f ⁻¹' ↑s)\ng : γ → β\nthis : DecidableEq γ\n⊢ ∏ x in preimage s f hf, g (f x) = ∏ x in filter (fun x => x ∈ Set.range f) s, g x",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : CommMonoid β\nf : α → γ\ninst✝ : DecidablePred fun x => x ∈ Set.range f\ns : Finset γ\nhf : InjOn f (f ⁻¹' ↑s)\ng : γ → β\n⊢ ∏ x in preimage s f hf, g (f x) = ∏ x in filter (fun x => x ∈ Set.range f) s, g x",
"tactic": "haveI := Classical.decEq γ"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : CommMonoid β\nf : α → γ\ninst✝ : DecidablePred fun x => x ∈ Set.range f\ns : Finset γ\nhf : InjOn f (f ⁻¹' ↑s)\ng : γ → β\nthis : DecidableEq γ\n⊢ ∏ x in preimage s f hf, g (f x) = ∏ x in filter (fun x => x ∈ Set.range f) s, g x",
"tactic": "calc\n (∏ x in preimage s f hf, g (f x)) = ∏ x in image f (preimage s f hf), g x :=\n Eq.symm <| prod_image <| by simpa only [mem_preimage, InjOn] using hf\n _ = ∏ x in s.filter fun x => x ∈ Set.range f, g x := by rw [image_preimage]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : CommMonoid β\nf : α → γ\ninst✝ : DecidablePred fun x => x ∈ Set.range f\ns : Finset γ\nhf : InjOn f (f ⁻¹' ↑s)\ng : γ → β\nthis : DecidableEq γ\n⊢ ∀ (x : α), x ∈ preimage s f hf → ∀ (y : α), y ∈ preimage s f hf → f x = f y → x = y",
"tactic": "simpa only [mem_preimage, InjOn] using hf"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : CommMonoid β\nf : α → γ\ninst✝ : DecidablePred fun x => x ∈ Set.range f\ns : Finset γ\nhf : InjOn f (f ⁻¹' ↑s)\ng : γ → β\nthis : DecidableEq γ\n⊢ ∏ x in image f (preimage s f hf), g x = ∏ x in filter (fun x => x ∈ Set.range f) s, g x",
"tactic": "rw [image_preimage]"
}
] |
[
152,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.prod_iUnion
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.227121\nι : Sort u_3\nι' : Sort ?u.227127\nι₂ : Sort ?u.227130\nκ : ι → Sort ?u.227135\nκ₁ : ι → Sort ?u.227140\nκ₂ : ι → Sort ?u.227145\nκ' : ι' → Sort ?u.227150\ns : Set α\nt : ι → Set β\nx✝ : α × β\n⊢ (x✝ ∈ s ×ˢ ⋃ (i : ι), t i) ↔ x✝ ∈ ⋃ (i : ι), s ×ˢ t i",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.227121\nι : Sort u_3\nι' : Sort ?u.227127\nι₂ : Sort ?u.227130\nκ : ι → Sort ?u.227135\nκ₁ : ι → Sort ?u.227140\nκ₂ : ι → Sort ?u.227145\nκ' : ι' → Sort ?u.227150\ns : Set α\nt : ι → Set β\n⊢ (s ×ˢ ⋃ (i : ι), t i) = ⋃ (i : ι), s ×ˢ t i",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.227121\nι : Sort u_3\nι' : Sort ?u.227127\nι₂ : Sort ?u.227130\nκ : ι → Sort ?u.227135\nκ₁ : ι → Sort ?u.227140\nκ₂ : ι → Sort ?u.227145\nκ' : ι' → Sort ?u.227150\ns : Set α\nt : ι → Set β\nx✝ : α × β\n⊢ (x✝ ∈ s ×ˢ ⋃ (i : ι), t i) ↔ x✝ ∈ ⋃ (i : ι), s ×ˢ t i",
"tactic": "simp"
}
] |
[
1762,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1760,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Embedding.map_adj_iff
|
[] |
[
1778,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1777,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_ciInf_iff
|
[] |
[
519,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
517,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.normSq_eq_zero
|
[] |
[
489,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_max
|
[] |
[
1168,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1167,
1
] |
Mathlib/Algebra/Category/Mon/FilteredColimits.lean
|
MonCat.FilteredColimits.colimit_mul_mk_eq
|
[
{
"state_after": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\ny : (j : J) × ↑(F.obj j)\nk : J\ng : y.fst ⟶ k\nj₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F y =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) y.snd }",
"state_before": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nx y : (j : J) × ↑(F.obj j)\nk : J\nf : x.fst ⟶ k\ng : y.fst ⟶ k\n⊢ M.mk F x * M.mk F y = M.mk F { fst := k, snd := ↑(F.map f) x.snd * ↑(F.map g) y.snd }",
"tactic": "cases' x with j₁ x"
},
{
"state_after": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F { fst := j₂, snd := y } =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }",
"state_before": "case mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\ny : (j : J) × ↑(F.obj j)\nk : J\ng : y.fst ⟶ k\nj₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F y =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) y.snd }",
"tactic": "cases' y with j₂ y"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F { fst := j₂, snd := y } =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }",
"state_before": "case mk.mk\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F { fst := j₂, snd := y } =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }",
"tactic": "obtain ⟨s, α, β, h₁, h₂⟩ := IsFiltered.bowtie (IsFiltered.leftToMax j₁ j₂) f\n (IsFiltered.rightToMax j₁ j₂) g"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ∃ k_1 f_1 g_1,\n ↑(F.map f_1)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map g_1)\n { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }.snd",
"state_before": "case mk.mk.intro.intro.intro.intro\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ M.mk F { fst := j₁, snd := x } * M.mk F { fst := j₂, snd := y } =\n M.mk F { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }",
"tactic": "apply M.mk_eq"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map β) { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }.snd",
"state_before": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ∃ k_1 f_1 g_1,\n ↑(F.map f_1)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map g_1)\n { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }.snd",
"tactic": "use s, α, β"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map β) (↑(F.map f) x * ↑(F.map g) y)",
"state_before": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α)\n { fst := IsFiltered.max { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst,\n snd :=\n ↑(F.map (IsFiltered.leftToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₁, snd := x }.snd *\n ↑(F.map (IsFiltered.rightToMax { fst := j₁, snd := x }.fst { fst := j₂, snd := y }.fst))\n { fst := j₂, snd := y }.snd }.snd =\n ↑(F.map β) { fst := k, snd := ↑(F.map f) { fst := j₁, snd := x }.snd * ↑(F.map g) { fst := j₂, snd := y }.snd }.snd",
"tactic": "dsimp"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map β) (↑(F.map f) x) * ↑(F.map β) (↑(F.map g) y)",
"state_before": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map β) (↑(F.map f) x * ↑(F.map g) y)",
"tactic": "simp_rw [MonoidHom.map_mul]"
},
{
"state_after": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₁ j₂) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂) ≫ F.map α) y =\n ↑(F.map f ≫ F.map β) x * ↑(F.map g ≫ F.map β) y",
"state_before": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map α) (↑(F.map (IsFiltered.leftToMax j₁ j₂)) x) * ↑(F.map α) (↑(F.map (IsFiltered.rightToMax j₁ j₂)) y) =\n ↑(F.map β) (↑(F.map f) x) * ↑(F.map β) (↑(F.map g) y)",
"tactic": "change (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _ =\n (F.map _ ≫ F.map _) _ * (F.map _ ≫ F.map _) _"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro.intro.intro.h\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ MonCat\ninst✝ : IsFiltered J\nk j₁ : J\nx : ↑(F.obj j₁)\nf : { fst := j₁, snd := x }.fst ⟶ k\nj₂ : J\ny : ↑(F.obj j₂)\ng : { fst := j₂, snd := y }.fst ⟶ k\ns : J\nα : IsFiltered.max j₁ j₂ ⟶ s\nβ : k ⟶ s\nh₁ : IsFiltered.leftToMax j₁ j₂ ≫ α = f ≫ β\nh₂ : IsFiltered.rightToMax j₁ j₂ ≫ α = g ≫ β\n⊢ ↑(F.map (IsFiltered.leftToMax j₁ j₂) ≫ F.map α) x * ↑(F.map (IsFiltered.rightToMax j₁ j₂) ≫ F.map α) y =\n ↑(F.map f ≫ F.map β) x * ↑(F.map g ≫ F.map β) y",
"tactic": "simp_rw [← F.map_comp, h₁, h₂]"
}
] |
[
224,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.comp₂
|
[] |
[
859,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm'_eq_zero_iff
|
[] |
[
725,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsMaxOn.isExtr
|
[] |
[
174,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Topology/Connected.lean
|
IsPreconnected.preimage_of_closed_map
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v))",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nhsf : s ⊆ range f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\n⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v))",
"tactic": "replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v))\n\ncase intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\na : α\nhau : a ∈ u\nhas : f a ∈ s\nhav : f a ∈ f '' v\n⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v))",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v))",
"tactic": "obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty"
},
{
"state_after": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ s ⊆ f '' u ∪ f '' v\n\ncase refine_2\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ f '' u)\n\ncase refine_3\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ f '' v)",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ (f '' u ∩ f '' v))",
"tactic": "refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ s ⊆ f '' u ∪ f '' v",
"tactic": "simpa only [hsf, image_union] using image_subset f hsuv"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ f '' u)",
"tactic": "simpa only [image_preimage_inter] using hsu.image f"
},
{
"state_after": "no goals",
"state_before": "case refine_3\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\n⊢ Set.Nonempty (s ∩ f '' v)",
"tactic": "simpa only [image_preimage_inter] using hsv.image f"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.26506\nπ : ι → Type ?u.26511\ninst✝¹ : TopologicalSpace α\ns✝ t u✝ v✝ : Set α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed u\nhv : IsClosed v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : Set.Nonempty (f ⁻¹' s ∩ u)\nhsv : Set.Nonempty (f ⁻¹' s ∩ v)\nhsf : f '' (f ⁻¹' s) = s\na : α\nhau : a ∈ u\nhas : f a ∈ s\nhav : f a ∈ f '' v\n⊢ Set.Nonempty (f ⁻¹' s ∩ (u ∩ v))",
"tactic": "exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩"
}
] |
[
402,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.fromSubtype_toSubtype
|
[
{
"state_after": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nf : J → C\ninst✝¹ : HasBiproduct f\np : J → Prop\ninst✝ : HasBiproduct (Subtype.restrict p f)\nj : Subtype p\n⊢ (fromSubtype f p ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j =\n 𝟙 (⨁ Subtype.restrict p f) ≫ π (Subtype.restrict p f) j",
"state_before": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nf : J → C\ninst✝¹ : HasBiproduct f\np : J → Prop\ninst✝ : HasBiproduct (Subtype.restrict p f)\n⊢ fromSubtype f p ≫ toSubtype f p = 𝟙 (⨁ Subtype.restrict p f)",
"tactic": "refine' biproduct.hom_ext _ _ fun j => _"
},
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\nf : J → C\ninst✝¹ : HasBiproduct f\np : J → Prop\ninst✝ : HasBiproduct (Subtype.restrict p f)\nj : Subtype p\n⊢ (fromSubtype f p ≫ toSubtype f p) ≫ π (Subtype.restrict p f) j =\n 𝟙 (⨁ Subtype.restrict p f) ≫ π (Subtype.restrict p f) j",
"tactic": "rw [Category.assoc, biproduct.toSubtype_π, biproduct.fromSubtype_π_subtype, Category.id_comp]"
}
] |
[
654,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_lt_one
|
[
{
"state_after": "case intro\nz : ℝ\nhz : 0 < z\nx : ℝ≥0\nhx : ↑x < 1\n⊢ ↑x ^ z < 1",
"state_before": "x : ℝ≥0∞\nz : ℝ\nhx : x < 1\nhz : 0 < z\n⊢ x ^ z < 1",
"tactic": "lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx le_top)"
},
{
"state_after": "case intro\nz : ℝ\nhz : 0 < z\nx : ℝ≥0\nhx : x < 1\n⊢ ↑x ^ z < 1",
"state_before": "case intro\nz : ℝ\nhz : 0 < z\nx : ℝ≥0\nhx : ↑x < 1\n⊢ ↑x ^ z < 1",
"tactic": "simp only [coe_lt_one_iff] at hx"
},
{
"state_after": "no goals",
"state_before": "case intro\nz : ℝ\nhz : 0 < z\nx : ℝ≥0\nhx : x < 1\n⊢ ↑x ^ z < 1",
"tactic": "simp [coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.rpow_lt_one hx hz]"
}
] |
[
668,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
665,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.ext
|
[] |
[
93,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
affineSpan_mono
|
[] |
[
1378,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1377,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.exists_edist_lt_of_hausdorffEdist_lt
|
[] |
[
314,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Complex.hasFDerivAt_cpow
|
[] |
[
70,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
LinearMap.baseChange_sub
|
[
{
"state_after": "case a.h.h\nR : Type u_4\nA : Type u_1\nB : Type ?u.473826\nM : Type u_2\nN : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf g : M →ₗ[R] N\nx✝ : M\n⊢ ↑(↑(AlgebraTensorModule.curry (baseChange A (f - g))) 1) x✝ =\n ↑(↑(AlgebraTensorModule.curry (baseChange A f - baseChange A g)) 1) x✝",
"state_before": "R : Type u_4\nA : Type u_1\nB : Type ?u.473826\nM : Type u_2\nN : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf g : M →ₗ[R] N\n⊢ baseChange A (f - g) = baseChange A f - baseChange A g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h.h\nR : Type u_4\nA : Type u_1\nB : Type ?u.473826\nM : Type u_2\nN : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : Ring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R B\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf g : M →ₗ[R] N\nx✝ : M\n⊢ ↑(↑(AlgebraTensorModule.curry (baseChange A (f - g))) 1) x✝ =\n ↑(↑(AlgebraTensorModule.curry (baseChange A f - baseChange A g)) 1) x✝",
"tactic": "simp [baseChange_eq_ltensor, tmul_sub]"
}
] |
[
319,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.iSup_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.31772\nr : α → α → Prop\nι : Type u_1\nf : ι → Cardinal\nc : Cardinal\nhι : (#ι) < cof (ord c)\n⊢ Cardinal.lift (#ι) < cof (ord c)",
"tactic": "rwa [(#ι).lift_id]"
}
] |
[
377,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/Finset/NoncommProd.lean
|
Multiset.noncommFoldr_cons
|
[
{
"state_after": "case h\nF : Type ?u.2553\nι : Type ?u.2556\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2565\nf : α → β → β\nop : α → α → α\na : α\nb : β\na✝ : List α\nh : Set.Pairwise {x | x ∈ a ::ₘ Quotient.mk (List.isSetoid α) a✝} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\nh' : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid α) a✝} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\n⊢ noncommFoldr f (a ::ₘ Quotient.mk (List.isSetoid α) a✝) h b =\n f a (noncommFoldr f (Quotient.mk (List.isSetoid α) a✝) h' b)",
"state_before": "F : Type ?u.2553\nι : Type ?u.2556\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2565\nf : α → β → β\nop : α → α → α\ns : Multiset α\na : α\nh : Set.Pairwise {x | x ∈ a ::ₘ s} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\nh' : Set.Pairwise {x | x ∈ s} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\nb : β\n⊢ noncommFoldr f (a ::ₘ s) h b = f a (noncommFoldr f s h' b)",
"tactic": "induction s using Quotient.inductionOn"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.2553\nι : Type ?u.2556\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.2565\nf : α → β → β\nop : α → α → α\na : α\nb : β\na✝ : List α\nh : Set.Pairwise {x | x ∈ a ::ₘ Quotient.mk (List.isSetoid α) a✝} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\nh' : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid α) a✝} fun x y => ∀ (b : β), f x (f y b) = f y (f x b)\n⊢ noncommFoldr f (a ::ₘ Quotient.mk (List.isSetoid α) a✝) h b =\n f a (noncommFoldr f (Quotient.mk (List.isSetoid α) a✝) h' b)",
"tactic": "simp"
}
] |
[
70,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/PartitionOfUnity.lean
|
BumpCovering.IsSubordinate.toPartitionOfUnity
|
[] |
[
489,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/Combinatorics/Hall/Finite.lean
|
HallMarriageTheorem.hall_cond_of_restrict
|
[
{
"state_after": "no goals",
"state_before": "ι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card s' ≤ card (Finset.biUnion s' fun a' => t ↑a')",
"tactic": "classical\n rw [← card_image_of_injective s' Subtype.coe_injective]\n convert ht (s'.image fun z => z.1) using 1\n apply congr_arg\n ext y\n simp"
},
{
"state_after": "ι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card (image (fun a => ↑a) s') ≤ card (Finset.biUnion s' fun a' => t ↑a')",
"state_before": "ι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card s' ≤ card (Finset.biUnion s' fun a' => t ↑a')",
"tactic": "rw [← card_image_of_injective s' Subtype.coe_injective]"
},
{
"state_after": "case h.e'_4\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card (Finset.biUnion s' fun a' => t ↑a') = card (Finset.biUnion (image (fun z => ↑z) s') t)",
"state_before": "ι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card (image (fun a => ↑a) s') ≤ card (Finset.biUnion s' fun a' => t ↑a')",
"tactic": "convert ht (s'.image fun z => z.1) using 1"
},
{
"state_after": "case h.e'_4.h\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ (Finset.biUnion s' fun a' => t ↑a') = Finset.biUnion (image (fun z => ↑z) s') t",
"state_before": "case h.e'_4\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ card (Finset.biUnion s' fun a' => t ↑a') = card (Finset.biUnion (image (fun z => ↑z) s') t)",
"tactic": "apply congr_arg"
},
{
"state_after": "case h.e'_4.h.a\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\ny : α\n⊢ (y ∈ Finset.biUnion s' fun a' => t ↑a') ↔ y ∈ Finset.biUnion (image (fun z => ↑z) s') t",
"state_before": "case h.e'_4.h\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\n⊢ (Finset.biUnion s' fun a' => t ↑a') = Finset.biUnion (image (fun z => ↑z) s') t",
"tactic": "ext y"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.a\nι✝ : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt✝ : ι✝ → Finset α\ninst✝ : Fintype ι✝\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), card s ≤ card (Finset.biUnion s t)\ns' : Finset ↑↑s\ny : α\n⊢ (y ∈ Finset.biUnion s' fun a' => t ↑a') ↔ y ∈ Finset.biUnion (image (fun z => ↑z) s') t",
"tactic": "simp"
}
] |
[
136,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.coe_pair
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.108687\nγ : Type ?u.108690\ninst✝ : DecidableEq α\ns t u v : Finset α\na✝ b✝ a b x✝ : α\n⊢ x✝ ∈ ↑{a, b} ↔ x✝ ∈ {a, b}",
"state_before": "α : Type u_1\nβ : Type ?u.108687\nγ : Type ?u.108690\ninst✝ : DecidableEq α\ns t u v : Finset α\na✝ b✝ a b : α\n⊢ ↑{a, b} = {a, b}",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.108687\nγ : Type ?u.108690\ninst✝ : DecidableEq α\ns t u v : Finset α\na✝ b✝ a b x✝ : α\n⊢ x✝ ∈ ↑{a, b} ↔ x✝ ∈ {a, b}",
"tactic": "simp"
}
] |
[
1124,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1122,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
StrictConvex.add_right
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.89300\nE : Type u_2\nF : Type ?u.89306\nβ : Type ?u.89309\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 F\ninst✝ : ContinuousAdd E\ns t : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x => x + z) '' s)",
"tactic": "simpa only [add_comm] using hs.add_left z"
}
] |
[
268,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
UniformCauchySeqOn.div
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\nι : Type u_3\nl : Filter ι\nl' : Filter β\nf f' : ι → β → α\ng g' : β → α\ns : Set β\nhf : UniformCauchySeqOn f l s\nhf' : UniformCauchySeqOn f' l s\nu : Set (α × α)\nhu : u ∈ 𝓤 α\n⊢ ∀ᶠ (m : ι × ι) in l ×ˢ l, ∀ (x : β), x ∈ s → ((f / f') m.fst x, (f / f') m.snd x) ∈ u",
"tactic": "simpa using (uniformContinuous_div.comp_uniformCauchySeqOn (hf.prod' hf')) u hu"
}
] |
[
533,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/Data/PNat/Prime.lean
|
PNat.Coprime.factor_eq_gcd_left
|
[
{
"state_after": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ a = gcd (a * b) m",
"state_before": "a b m n : ℕ+\ncop : Coprime m n\nam : a ∣ m\nbn : b ∣ n\n⊢ a = gcd (a * b) m",
"tactic": "rw [gcd_eq_left_iff_dvd] at am"
},
{
"state_after": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ gcd a m = gcd (a * b) m",
"state_before": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ a = gcd (a * b) m",
"tactic": "conv_lhs => rw [← am]"
},
{
"state_after": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ gcd (a * b) m = gcd a m",
"state_before": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ gcd a m = gcd (a * b) m",
"tactic": "rw [eq_comm]"
},
{
"state_after": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ Coprime b m",
"state_before": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ gcd (a * b) m = gcd a m",
"tactic": "apply Coprime.gcd_mul_right_cancel a"
},
{
"state_after": "no goals",
"state_before": "a b m n : ℕ+\ncop : Coprime m n\nam : gcd a m = a\nbn : b ∣ n\n⊢ Coprime b m",
"tactic": "apply Coprime.coprime_dvd_left bn cop.symm"
}
] |
[
270,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_le_iff_degree_le
|
[] |
[
224,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.zero_kroneckerTMul
|
[] |
[
469,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.kroneckerTMul_assoc'
|
[] |
[
526,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Data/List/Basic.lean
|
List.foldl_ext
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.228451\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β → α\na : α\nl : List β\nH : ∀ (a : α) (b : β), b ∈ l → f a b = g a b\n⊢ foldl f a l = foldl g a l",
"tactic": "induction l generalizing a with\n| nil => rfl\n| cons hd tl ih =>\n unfold foldl\n rw [ih _ fun a b bin => H a b <| mem_cons_of_mem _ bin, H a hd (mem_cons_self _ _)]"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.228451\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β → α\na : α\nH : ∀ (a : α) (b : β), b ∈ [] → f a b = g a b\n⊢ foldl f a [] = foldl g a []",
"tactic": "rfl"
},
{
"state_after": "case cons\nι : Type ?u.228451\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β → α\nhd : β\ntl : List β\nih : ∀ (a : α), (∀ (a : α) (b : β), b ∈ tl → f a b = g a b) → foldl f a tl = foldl g a tl\na : α\nH : ∀ (a : α) (b : β), b ∈ hd :: tl → f a b = g a b\n⊢ foldl f (f a hd) tl = foldl g (g a hd) tl",
"state_before": "case cons\nι : Type ?u.228451\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β → α\nhd : β\ntl : List β\nih : ∀ (a : α), (∀ (a : α) (b : β), b ∈ tl → f a b = g a b) → foldl f a tl = foldl g a tl\na : α\nH : ∀ (a : α) (b : β), b ∈ hd :: tl → f a b = g a b\n⊢ foldl f a (hd :: tl) = foldl g a (hd :: tl)",
"tactic": "unfold foldl"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.228451\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf g : α → β → α\nhd : β\ntl : List β\nih : ∀ (a : α), (∀ (a : α) (b : β), b ∈ tl → f a b = g a b) → foldl f a tl = foldl g a tl\na : α\nH : ∀ (a : α) (b : β), b ∈ hd :: tl → f a b = g a b\n⊢ foldl f (f a hd) tl = foldl g (g a hd) tl",
"tactic": "rw [ih _ fun a b bin => H a b <| mem_cons_of_mem _ bin, H a hd (mem_cons_self _ _)]"
}
] |
[
2383,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2377,
1
] |
Mathlib/Data/Finset/Basic.lean
|
List.toFinset_coe
|
[] |
[
3243,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3242,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
IsLinearMap.mk'_apply
|
[] |
[
689,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.IntegrableOn.congr_fun_ae
|
[] |
[
148,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
NNReal.sqrt_lt_sqrt_iff
|
[] |
[
60,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.filter_single_of_pos
|
[] |
[
932,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
931,
1
] |
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lxor'_right_injective
|
[
{
"state_after": "no goals",
"state_before": "n m m' : ℕ\nh : lxor' n m = lxor' n m'\n⊢ m = m'",
"tactic": "rw [← lxor'_cancel_left n m, ← lxor'_cancel_left n m', h]"
}
] |
[
252,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_toLocalEquiv
|
[] |
[
799,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.singleton_disjUnion
|
[] |
[
1023,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1021,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.snd_equivOfRightInverse
|
[] |
[
2415,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2412,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.subsingleton_of_univ_subsingleton
|
[] |
[
2380,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2379,
1
] |
Mathlib/Data/Set/Intervals/Monotone.lean
|
AntitoneOn.Ioc
|
[] |
[
161,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
11
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_iff_left_eq_or_right_mem_image_Ici
|
[
{
"state_after": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ x = y ∨ z ∈ ↑(lineMap x y) '' Set.Ici 1\n\ncase refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : x = y ∨ z ∈ ↑(lineMap x y) '' Set.Ici 1\n⊢ Wbtw R x y z",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\n⊢ Wbtw R x y z ↔ x = y ∨ z ∈ ↑(lineMap x y) '' Set.Ici 1",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\n⊢ x = ↑(lineMap x z) r ∨ z ∈ ↑(lineMap x (↑(lineMap x z) r)) '' Set.Ici 1",
"state_before": "case refine'_1\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ x = y ∨ z ∈ ↑(lineMap x y) '' Set.Ici 1",
"tactic": "rcases h with ⟨r, ⟨hr0, hr1⟩, rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ x = ↑(lineMap x z) r ∨ z ∈ ↑(lineMap x (↑(lineMap x z) r)) '' Set.Ici 1\n\ncase refine'_1.intro.intro.intro.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nhr0 : 0 ≤ 0\nhr1 : 0 ≤ 1\n⊢ x = ↑(lineMap x z) 0 ∨ z ∈ ↑(lineMap x (↑(lineMap x z) 0)) '' Set.Ici 1",
"state_before": "case refine'_1.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\n⊢ x = ↑(lineMap x z) r ∨ z ∈ ↑(lineMap x (↑(lineMap x z) r)) '' Set.Ici 1",
"tactic": "rcases hr0.lt_or_eq with (hr0' | rfl)"
},
{
"state_after": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ x = ↑(lineMap x z) r ∨ ∃ x_1, x_1 ∈ Set.Ici 1 ∧ ↑(lineMap x (↑(lineMap x z) r)) x_1 = z",
"state_before": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ x = ↑(lineMap x z) r ∨ z ∈ ↑(lineMap x (↑(lineMap x z) r)) '' Set.Ici 1",
"tactic": "rw [Set.mem_image]"
},
{
"state_after": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ ↑(lineMap x (↑(lineMap x z) r)) r⁻¹ = z",
"state_before": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ x = ↑(lineMap x z) r ∨ ∃ x_1, x_1 ∈ Set.Ici 1 ∧ ↑(lineMap x (↑(lineMap x z) r)) x_1 = z",
"tactic": "refine' Or.inr ⟨r⁻¹, one_le_inv hr0' hr1, _⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ (r⁻¹ * r) • (z -ᵥ x) +ᵥ x = z",
"state_before": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ ↑(lineMap x (↑(lineMap x z) r)) r⁻¹ = z",
"tactic": "simp only [lineMap_apply, smul_smul, vadd_vsub]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nr : R\nhr0 : 0 ≤ r\nhr1 : r ≤ 1\nhr0' : 0 < r\n⊢ (r⁻¹ * r) • (z -ᵥ x) +ᵥ x = z",
"tactic": "rw [inv_mul_cancel hr0'.ne', one_smul, vsub_vadd]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.inr\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\nhr0 : 0 ≤ 0\nhr1 : 0 ≤ 1\n⊢ x = ↑(lineMap x z) 0 ∨ z ∈ ↑(lineMap x (↑(lineMap x z) 0)) '' Set.Ici 1",
"tactic": "simp"
},
{
"state_after": "case refine'_2.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\n⊢ Wbtw R x x z\n\ncase refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : r ∈ Set.Ici 1\n⊢ Wbtw R x y (↑(lineMap x y) r)",
"state_before": "case refine'_2\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : x = y ∨ z ∈ ↑(lineMap x y) '' Set.Ici 1\n⊢ Wbtw R x y z",
"tactic": "rcases h with (rfl | ⟨r, ⟨hr, rfl⟩⟩)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inl\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx z : P\n⊢ Wbtw R x x z",
"tactic": "exact wbtw_self_left _ _ _"
},
{
"state_after": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ Wbtw R x y (↑(lineMap x y) r)",
"state_before": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : r ∈ Set.Ici 1\n⊢ Wbtw R x y (↑(lineMap x y) r)",
"tactic": "rw [Set.mem_Ici] at hr"
},
{
"state_after": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ ↑(lineMap x (↑(lineMap x y) r)) r⁻¹ = y",
"state_before": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ Wbtw R x y (↑(lineMap x y) r)",
"tactic": "refine' ⟨r⁻¹, ⟨inv_nonneg.2 (zero_le_one.trans hr), inv_le_one hr⟩, _⟩"
},
{
"state_after": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ (r⁻¹ * r) • (y -ᵥ x) +ᵥ x = y",
"state_before": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ ↑(lineMap x (↑(lineMap x y) r)) r⁻¹ = y",
"tactic": "simp only [lineMap_apply, smul_smul, vadd_vsub]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.inr.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.468028\nP : Type u_3\nP' : Type ?u.468034\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ (r⁻¹ * r) • (y -ᵥ x) +ᵥ x = y",
"tactic": "rw [inv_mul_cancel (one_pos.trans_le hr).ne', one_smul, vsub_vadd]"
}
] |
[
692,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.rat_sub
|
[
{
"state_after": "no goals",
"state_before": "q : ℚ\nx y : ℝ\nh : Irrational x\n⊢ Irrational (↑q - x)",
"tactic": "simpa only [sub_eq_add_neg] using h.neg.rat_add q"
}
] |
[
284,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.isSt_inj_real
|
[] |
[
384,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_of_lt_mul_left
|
[] |
[
260,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Topology/Basic.lean
|
interior_empty
|
[] |
[
329,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
QuasiconcaveOn.inf
|
[] |
[
122,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12543\nE : Type u_2\nF : Type u_3\nG : Type ?u.12552\nE' : Type ?u.12555\nF' : Type ?u.12558\nG' : Type ?u.12561\nE'' : Type ?u.12564\nF'' : Type ?u.12567\nG'' : Type ?u.12570\nR : Type ?u.12573\nR' : Type ?u.12576\n𝕜 : Type ?u.12579\n𝕜' : Type ?u.12582\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ f =O[l] g ↔ ∃ c, ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖",
"tactic": "simp only [IsBigO_def, IsBigOWith_def]"
}
] |
[
120,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
HasCompactMulSupport.eq_one_or_finiteDimensional
|
[
{
"state_after": "case pos\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ∀ (x : E), f x = 1\n⊢ f = 1 ∨ FiniteDimensional 𝕜 E\n\ncase neg\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ¬∀ (x : E), f x = 1\n⊢ f = 1 ∨ FiniteDimensional 𝕜 E",
"state_before": "𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\n⊢ f = 1 ∨ FiniteDimensional 𝕜 E",
"tactic": "by_cases h : ∀ x, f x = 1"
},
{
"state_after": "case neg.h\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ¬∀ (x : E), f x = 1\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ¬∀ (x : E), f x = 1\n⊢ f = 1 ∨ FiniteDimensional 𝕜 E",
"tactic": "apply Or.inr"
},
{
"state_after": "case neg.h\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ∃ x, f x ≠ 1\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg.h\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ¬∀ (x : E), f x = 1\n⊢ FiniteDimensional 𝕜 E",
"tactic": "push_neg at h"
},
{
"state_after": "case neg.h.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg.h\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ∃ x, f x ≠ 1\n⊢ FiniteDimensional 𝕜 E",
"tactic": "obtain ⟨x, hx⟩ : ∃ x, f x ≠ 1 := h"
},
{
"state_after": "case neg.h.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis : Function.mulSupport f ∈ 𝓝 x\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg.h.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\n⊢ FiniteDimensional 𝕜 E",
"tactic": "have : Function.mulSupport f ∈ 𝓝 x := h'f.isOpen_mulSupport.mem_nhds hx"
},
{
"state_after": "case neg.h.intro.intro.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis : Function.mulSupport f ∈ 𝓝 x\nr : ℝ\nrpos : 0 < r\nhr : Metric.closedBall x r ⊆ Function.mulSupport f\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg.h.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis : Function.mulSupport f ∈ 𝓝 x\n⊢ FiniteDimensional 𝕜 E",
"tactic": "obtain ⟨r : ℝ, rpos : 0 < r, hr : Metric.closedBall x r ⊆ Function.mulSupport f⟩ :=\n Metric.nhds_basis_closedBall.mem_iff.1 this"
},
{
"state_after": "case neg.h.intro.intro.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis✝ : Function.mulSupport f ∈ 𝓝 x\nr : ℝ\nrpos : 0 < r\nhr : Metric.closedBall x r ⊆ Function.mulSupport f\nthis : IsCompact (Metric.closedBall x r)\n⊢ FiniteDimensional 𝕜 E",
"state_before": "case neg.h.intro.intro.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis : Function.mulSupport f ∈ 𝓝 x\nr : ℝ\nrpos : 0 < r\nhr : Metric.closedBall x r ⊆ Function.mulSupport f\n⊢ FiniteDimensional 𝕜 E",
"tactic": "have : IsCompact (Metric.closedBall x r) :=\n isCompact_of_isClosed_subset hf Metric.isClosed_ball (hr.trans (subset_mulTSupport _))"
},
{
"state_after": "no goals",
"state_before": "case neg.h.intro.intro.intro\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nx : E\nhx : f x ≠ 1\nthis✝ : Function.mulSupport f ∈ 𝓝 x\nr : ℝ\nrpos : 0 < r\nhr : Metric.closedBall x r ⊆ Function.mulSupport f\nthis : IsCompact (Metric.closedBall x r)\n⊢ FiniteDimensional 𝕜 E",
"tactic": "exact finiteDimensional_of_isCompact_closedBall 𝕜 rpos this"
},
{
"state_after": "case pos.h\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ∀ (x : E), f x = 1\n⊢ f = 1",
"state_before": "case pos\n𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : One X\ninst✝ : T2Space X\nf : E → X\nhf : HasCompactMulSupport f\nh'f : Continuous f\nh : ∀ (x : E), f x = 1\n⊢ f = 1 ∨ FiniteDimensional 𝕜 E",
"tactic": "apply Or.inl"
},
{
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"tactic": "ext x"
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"tactic": "exact h x"
}
] |
[
511,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_comp_subtype
|
[
{
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},
{
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"tactic": "intro h l hl₁ hl₂"
},
{
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"tactic": "have := h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl₁).trans hl₂)"
},
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"tactic": "exact (Finsupp.subtypeDomain_eq_zero_iff hl₁).1 this"
},
{
"state_after": "case mpr\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ l = 0",
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"tactic": "intro h l hl"
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"tactic": "refine' Finsupp.embDomain_eq_zero.1 (h (l.embDomain <| Function.Embedding.subtype s) _ _)"
},
{
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"tactic": "suffices ∀ i hi, ¬l ⟨i, hi⟩ = 0 → i ∈ s by simpa"
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{
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{
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},
{
"state_after": "case mpr.refine'_2.h_zero\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ ∀ (b : ι), 0 • v b = 0\n\ncase mpr.refine'_2.h_add\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ ∀ (b : ι) (m₁ m₂ : R), (m₁ + m₂) • v b = m₁ • v b + m₂ • v b",
"state_before": "case mpr.refine'_2\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ (Finsupp.sum (Finsupp.embDomain (Embedding.subtype s) l) fun i a => a • v i) = 0",
"tactic": "rwa [Finsupp.embDomain_eq_mapDomain, Finsupp.sum_mapDomain_index]"
},
{
"state_after": "no goals",
"state_before": "case mpr.refine'_2.h_zero\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ ∀ (b : ι), 0 • v b = 0\n\ncase mpr.refine'_2.h_add\nι : Type u'\nι' : Type ?u.143047\nR : Type u_1\nK : Type ?u.143053\nM : Type u_2\nM' : Type ?u.143059\nM'' : Type ?u.143062\nV : Type u\nV' : Type ?u.143067\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\ns : Set ι\nh : ∀ (l : ι →₀ R), (∀ (x : ι), x ∈ l.support → x ∈ s) → (Finsupp.sum l fun i a => a • v i) = 0 → l = 0\nl : ↑s →₀ R\nhl : (Finsupp.sum l fun i a => a • v ↑i) = 0\n⊢ ∀ (b : ι) (m₁ m₂ : R), (m₁ + m₂) • v b = m₁ • v b + m₂ • v b",
"tactic": "exacts [fun _ => zero_smul _ _, fun _ _ _ => add_smul _ _ _]"
}
] |
[
365,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_im
|
[] |
[
100,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
Filter.Tendsto.atTop_nsmul_neg_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf✝ : α → R\nr : R\ninst✝¹ : LinearOrderedAddCommGroup R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x => f x • r) l atBot",
"tactic": "simpa using hf.atTop_nsmul_const (neg_pos.2 hr)"
}
] |
[
228,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.toSubgroup_comap
|
[] |
[
286,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.CosimplicialObject.augment_hom_zero
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X.obj [0]\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\n⊢ f ≫ X.map (SimplexCategory.const [0] 0) = f",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X.obj [0]\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\n⊢ (augment X X₀ f w).hom.app [0] = f",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX✝ X : CosimplicialObject C\nX₀ : C\nf : X₀ ⟶ X.obj [0]\nw : ∀ (i : SimplexCategory) (g₁ g₂ : [0] ⟶ i), f ≫ X.map g₁ = f ≫ X.map g₂\n⊢ f ≫ X.map (SimplexCategory.const [0] 0) = f",
"tactic": "rw [SimplexCategory.hom_zero_zero ([0].const 0), X.map_id, Category.comp_id]"
}
] |
[
768,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
765,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
one_le_of_le_mul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.16522\ninst✝² : MulOneClass α\ninst✝¹ : LE α\ninst✝ : ContravariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b : α\nh : b ≤ a * b\n⊢ 1 * ?m.16901 h ≤ a * ?m.16901 h",
"tactic": "simpa only [one_mul]"
}
] |
[
386,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/RingTheory/Localization/Submodule.lean
|
IsLocalization.coeSubmodule_strictMono
|
[] |
[
120,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
MultilinearMap.domCoprod_alternization_eq
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1240826\ninst✝²¹ : Semiring R\nM : Type ?u.1240832\ninst✝²⁰ : AddCommMonoid M\ninst✝¹⁹ : Module R M\nN : Type ?u.1240864\ninst✝¹⁸ : AddCommMonoid N\ninst✝¹⁷ : Module R N\nP : Type ?u.1240894\ninst✝¹⁶ : AddCommMonoid P\ninst✝¹⁵ : Module R P\nM' : Type ?u.1240924\ninst✝¹⁴ : AddCommGroup M'\ninst✝¹³ : Module R M'\nN' : Type ?u.1241312\ninst✝¹² : AddCommGroup N'\ninst✝¹¹ : Module R N'\nι : Type ?u.1241700\nι' : Type ?u.1241703\nι'' : Type ?u.1241706\nιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' Mᵢ\ninst✝¹ : DecidableEq ιa\ninst✝ : DecidableEq ιb\na : AlternatingMap R' Mᵢ N₁ ιa\nb : AlternatingMap R' Mᵢ N₂ ιb\n⊢ ↑alternatization (domCoprod ↑a ↑b) =\n (Nat.factorial (Fintype.card ιa) * Nat.factorial (Fintype.card ιb)) • AlternatingMap.domCoprod a b",
"tactic": "rw [MultilinearMap.domCoprod_alternization, coe_alternatization, coe_alternatization, mul_smul,\n ← AlternatingMap.domCoprod'_apply, ← AlternatingMap.domCoprod'_apply,\n ← TensorProduct.smul_tmul', TensorProduct.tmul_smul,\n LinearMap.map_smul_of_tower AlternatingMap.domCoprod',\n LinearMap.map_smul_of_tower AlternatingMap.domCoprod']"
}
] |
[
1196,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1187,
1
] |
Mathlib/CategoryTheory/EssentiallySmall.lean
|
CategoryTheory.essentiallySmall_iff
|
[
{
"state_after": "case mp\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\n⊢ EssentiallySmall C → Small (Skeleton C) ∧ LocallySmall C\n\ncase mpr\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\n⊢ Small (Skeleton C) ∧ LocallySmall C → EssentiallySmall C",
"state_before": "C✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\n⊢ EssentiallySmall C ↔ Small (Skeleton C) ∧ LocallySmall C",
"tactic": "fconstructor"
},
{
"state_after": "case mp\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ Small (Skeleton C) ∧ LocallySmall C",
"state_before": "case mp\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\n⊢ EssentiallySmall C → Small (Skeleton C) ∧ LocallySmall C",
"tactic": "intro h"
},
{
"state_after": "case mp.left\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ Small (Skeleton C)\n\ncase mp.right\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ LocallySmall C",
"state_before": "case mp\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ Small (Skeleton C) ∧ LocallySmall C",
"tactic": "fconstructor"
},
{
"state_after": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Small (Skeleton C)",
"state_before": "case mp.left\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ Small (Skeleton C)",
"tactic": "rcases h with ⟨S, 𝒮, ⟨e⟩⟩"
},
{
"state_after": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Small (Skeleton C)",
"state_before": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Small (Skeleton C)",
"tactic": "skip"
},
{
"state_after": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Skeleton C ≃ Skeleton S",
"state_before": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Small (Skeleton C)",
"tactic": "refine' ⟨⟨Skeleton S, ⟨_⟩⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.left.mk.intro.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nS : Type w\n𝒮 : SmallCategory S\ne : C ≌ S\n⊢ Skeleton C ≃ Skeleton S",
"tactic": "exact e.skeletonEquiv"
},
{
"state_after": "case mp.right\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ LocallySmall C",
"state_before": "case mp.right\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ LocallySmall C",
"tactic": "skip"
},
{
"state_after": "no goals",
"state_before": "case mp.right\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nh : EssentiallySmall C\n⊢ LocallySmall C",
"tactic": "infer_instance"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\n⊢ EssentiallySmall C",
"state_before": "case mpr\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\n⊢ Small (Skeleton C) ∧ LocallySmall C → EssentiallySmall C",
"tactic": "rintro ⟨⟨S, ⟨e⟩⟩, L⟩"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\n⊢ EssentiallySmall C",
"state_before": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\n⊢ EssentiallySmall C",
"tactic": "skip"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\n⊢ EssentiallySmall C",
"state_before": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\n⊢ EssentiallySmall C",
"tactic": "let e' := (ShrinkHoms.equivalence C).skeletonEquiv.symm"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\nthis : Category S := InducedCategory.category ↑(e'.trans e).symm\n⊢ EssentiallySmall C",
"state_before": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\n⊢ EssentiallySmall C",
"tactic": "letI : Category S := InducedCategory.category (e'.trans e).symm"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\nthis : Category S := InducedCategory.category ↑(e'.trans e).symm\n⊢ C ≌ S",
"state_before": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\nthis : Category S := InducedCategory.category ↑(e'.trans e).symm\n⊢ EssentiallySmall C",
"tactic": "refine' ⟨⟨S, this, ⟨_⟩⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.mk.intro.intro\nC✝ : Type u\ninst✝¹ : Category C✝\nC : Type u\ninst✝ : Category C\nL : LocallySmall C\nS : Type w\ne : Skeleton C ≃ S\ne' : Skeleton (ShrinkHoms C) ≃ Skeleton C := (Equivalence.skeletonEquiv (ShrinkHoms.equivalence C)).symm\nthis : Category S := InducedCategory.category ↑(e'.trans e).symm\n⊢ C ≌ S",
"tactic": "refine' (ShrinkHoms.equivalence C).trans <|\n (skeletonEquivalence (ShrinkHoms C)).symm.trans\n ((inducedFunctor (e'.trans e).symm).asEquivalence.symm)"
}
] |
[
224,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Topology/Maps.lean
|
IsClosedMap.closure_image_subset
|
[] |
[
495,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.count_cons_of_ne
|
[] |
[
2359,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2358,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_mono
|
[] |
[
841,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
840,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.CompactOpens.coe_mk
|
[] |
[
498,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Combinatorics/SimpleGraph/Trails.lean
|
SimpleGraph.Walk.IsEulerian.card_filter_odd_degree
|
[
{
"state_after": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 0 ∨\n Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 2",
"state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\ns : Finset V\nh : s = Finset.filter (fun v => Odd (degree G v)) Finset.univ\n⊢ Finset.card s = 0 ∨ Finset.card s = 2",
"tactic": "subst s"
},
{
"state_after": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬Even (degree G v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v => ¬Even (degree G v)) Finset.univ) = 2",
"state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 0 ∨\n Finset.card (Finset.filter (fun v => Odd (degree G v)) Finset.univ) = 2",
"tactic": "simp only [Nat.odd_iff_not_even, Finset.card_eq_zero]"
},
{
"state_after": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬Even (degree G v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v => ¬Even (degree G v)) Finset.univ) = 2",
"tactic": "simp only [ht.even_degree_iff, Ne.def, not_forall, not_and, Classical.not_not, exists_prop]"
},
{
"state_after": "case inl\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ) = 2\n\ncase inr\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"state_before": "V : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"tactic": "obtain rfl | hn := eq_or_ne u v"
},
{
"state_after": "case inl.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ = ∅",
"state_before": "case inl\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ) = 2",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu : V\np : Walk G u u\nht : IsEulerian p\n⊢ Finset.filter (fun v => ¬u = u ∧ (¬v = u → v = u)) Finset.univ = ∅",
"tactic": "simp"
},
{
"state_after": "case inr.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"state_before": "case inr\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = ∅ ∨\n Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"tactic": "right"
},
{
"state_after": "case h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ 2 = Finset.card {u, v}\n\ncase inr.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = Finset.card {u, v}",
"state_before": "case inr.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = 2",
"tactic": "convert_to _ = ({u, v} : Finset V).card"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ 2 = Finset.card {u, v}",
"tactic": "simp [hn]"
},
{
"state_after": "case inr.h.e_s\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = {u, v}",
"state_before": "case inr.h\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.card (Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ) = Finset.card {u, v}",
"tactic": "congr"
},
{
"state_after": "case inr.h.e_s.a\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\nx : V\n⊢ x ∈ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ ↔ x ∈ {u, v}",
"state_before": "case inr.h.e_s\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\n⊢ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ = {u, v}",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case inr.h.e_s.a\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nu v : V\np : Walk G u v\nht : IsEulerian p\nhn : u ≠ v\nx : V\n⊢ x ∈ Finset.filter (fun v_1 => ¬u = v ∧ (¬v_1 = u → v_1 = v)) Finset.univ ↔ x ∈ {u, v}",
"tactic": "simp [hn, imp_iff_not_or]"
}
] |
[
163,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lt_mul_div_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.230000\nβ : Type ?u.230003\nγ : Type ?u.230006\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal\nh : b ≠ 0\n⊢ a < b * (a / b) + b",
"tactic": "simpa only [mul_succ] using lt_mul_succ_div a h"
}
] |
[
893,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
1
] |
src/lean/Init/Data/Nat/Power2.lean
|
Nat.mul2_isPowerOfTwo_of_isPowerOfTwo
|
[
{
"state_after": "no goals",
"state_before": "n : Nat\nh✝ : isPowerOfTwo n\nk : Nat\nh : n = 2 ^ k\n⊢ n * 2 = 2 ^ (k + 1)",
"tactic": "simp [h, Nat.pow_succ]"
}
] |
[
34,
35
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
32,
1
] |
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.Pivot.exists_list_transvec_mul_diagonal_mul_list_transvec
|
[
{
"state_after": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ ∃ L L' D, M = List.prod (List.map toMatrix L) ⬝ diagonal D ⬝ List.prod (List.map toMatrix L')",
"state_before": "n : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\n⊢ ∃ L L' D, M = List.prod (List.map toMatrix L) ⬝ diagonal D ⬝ List.prod (List.map toMatrix L')",
"tactic": "rcases exists_list_transvec_mul_mul_list_transvec_eq_diagonal M with ⟨L, L', D, h⟩"
},
{
"state_after": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ M =\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L))) ⬝ diagonal D ⬝\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L')))",
"state_before": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ ∃ L L' D, M = List.prod (List.map toMatrix L) ⬝ diagonal D ⬝ List.prod (List.map toMatrix L')",
"tactic": "refine' ⟨L.reverse.map TransvectionStruct.inv, L'.reverse.map TransvectionStruct.inv, D, _⟩"
},
{
"state_after": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ M =\n List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) ⬝ M ⬝\n (List.prod (List.map toMatrix L') ⬝ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L')))",
"state_before": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ M =\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L))) ⬝ diagonal D ⬝\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L')))",
"tactic": "suffices\n M =\n (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod ⬝ (L.map toMatrix).prod ⬝ M ⬝\n ((L'.map toMatrix).prod ⬝ (L'.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod)\n by simpa [← h, Matrix.mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nn : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\n⊢ M =\n List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) ⬝ M ⬝\n (List.prod (List.map toMatrix L') ⬝ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L')))",
"tactic": "rw [reverse_inv_prod_mul_prod, prod_mul_reverse_inv_prod, Matrix.one_mul, Matrix.mul_one]"
},
{
"state_after": "no goals",
"state_before": "n : Type u_1\np : Type ?u.321806\nR : Type u₂\n𝕜 : Type u_2\ninst✝⁵ : Field 𝕜\ninst✝⁴ : DecidableEq n\ninst✝³ : DecidableEq p\ninst✝² : CommRing R\nr : ℕ\nM✝ : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\ninst✝¹ : Fintype n\ninst✝ : Fintype p\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : List.prod (List.map toMatrix L) ⬝ M ⬝ List.prod (List.map toMatrix L') = diagonal D\nthis :\n M =\n List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L)) ⬝ List.prod (List.map toMatrix L) ⬝ M ⬝\n (List.prod (List.map toMatrix L') ⬝ List.prod (List.map (toMatrix ∘ TransvectionStruct.inv) (List.reverse L')))\n⊢ M =\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L))) ⬝ diagonal D ⬝\n List.prod (List.map toMatrix (List.map TransvectionStruct.inv (List.reverse L')))",
"tactic": "simpa [← h, Matrix.mul_assoc]"
}
] |
[
692,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comap_injective_of_surjective
|
[] |
[
906,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
concaveOn_of_deriv2_nonpos'
|
[] |
[
1229,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1225,
1
] |
Mathlib/Data/List/Basic.lean
|
List.takeI_length
|
[] |
[
2326,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2324,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.LeftInvOn.eqOn
|
[] |
[
1026,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1025,
1
] |
Mathlib/Data/Fintype/Order.lean
|
Fintype.bddAbove_range
|
[
{
"state_after": "case intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\n⊢ BddAbove (Set.range f)",
"state_before": "α : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\n⊢ BddAbove (Set.range f)",
"tactic": "obtain ⟨M, hM⟩ := Fintype.exists_le f"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\na : α\nha : a ∈ Set.range f\n⊢ a ≤ M",
"state_before": "case intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\n⊢ BddAbove (Set.range f)",
"tactic": "refine' ⟨M, fun a ha => _⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\nb : β\n⊢ f b ≤ M",
"state_before": "case intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\na : α\nha : a ∈ Set.range f\n⊢ a ≤ M",
"tactic": "obtain ⟨b, rfl⟩ := ha"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : Nonempty α\ninst✝² : Preorder α\ninst✝¹ : IsDirected α fun x x_1 => x ≤ x_1\nβ : Type u_2\ninst✝ : Fintype β\nf : β → α\nM : α\nhM : ∀ (i : β), f i ≤ M\nb : β\n⊢ f b ≤ M",
"tactic": "exact hM b"
}
] |
[
216,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
MulMemClass.mul_mem_add_closure
|
[
{
"state_after": "M : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\nb : R\nhb : b ∈ AddSubmonoid.closure ↑S\n⊢ ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * b ∈ AddSubmonoid.closure ↑S",
"state_before": "M : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na b : R\nha : a ∈ AddSubmonoid.closure ↑S\nhb : b ∈ AddSubmonoid.closure ↑S\n⊢ a * b ∈ AddSubmonoid.closure ↑S",
"tactic": "revert a"
},
{
"state_after": "case refine'_1\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x : R), x ∈ ↑S → (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) x\n\ncase refine'_2\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) 0\n\ncase refine'_3\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x y : R),\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) x →\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) y →\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) (x + y)",
"state_before": "M : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\nb : R\nhb : b ∈ AddSubmonoid.closure ↑S\n⊢ ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * b ∈ AddSubmonoid.closure ↑S",
"tactic": "refine' @AddSubmonoid.closure_induction _ _ _\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S)\n _ hb _ _ _ <;> clear hb b"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x : R), x ∈ ↑S → (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) x",
"tactic": "exact fun r hr b hb => MulMemClass.mul_right_mem_add_closure hb hr"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) 0",
"tactic": "exact fun _ => by simp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\na✝ : R\nx✝ : a✝ ∈ AddSubmonoid.closure ↑S\n⊢ a✝ * 0 ∈ AddSubmonoid.closure ↑S",
"tactic": "simp only [mul_zero, (AddSubmonoid.closure (S : Set R)).zero_mem]"
},
{
"state_after": "case refine'_3\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x y : R),\n (∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * x ∈ AddSubmonoid.closure ↑S) →\n (∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * y ∈ AddSubmonoid.closure ↑S) →\n ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * x + a * y ∈ AddSubmonoid.closure ↑S",
"state_before": "case refine'_3\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x y : R),\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) x →\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) y →\n (fun z => ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * z ∈ AddSubmonoid.closure ↑S) (x + y)",
"tactic": "simp_rw [mul_add]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nM : Type u_2\nA : Type ?u.259525\nB : Type ?u.259528\nR : Type u_1\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : SetLike M R\ninst✝ : MulMemClass M R\nS : M\n⊢ ∀ (x y : R),\n (∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * x ∈ AddSubmonoid.closure ↑S) →\n (∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * y ∈ AddSubmonoid.closure ↑S) →\n ∀ {a : R}, a ∈ AddSubmonoid.closure ↑S → a * x + a * y ∈ AddSubmonoid.closure ↑S",
"tactic": "exact fun r s hr hs b hb => (AddSubmonoid.closure (S : Set R)).add_mem (hr hb) (hs hb)"
}
] |
[
667,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Topology/Basic.lean
|
IsOpen.subset_interior_iff
|
[] |
[
314,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Std/Data/String/Lemmas.lean
|
Substring.Valid.takeWhile
|
[] |
[
1069,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1068,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
left_distrib
|
[] |
[
83,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
mem_powers_iff_mem_range_order_of'
|
[] |
[
269,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Combinatorics/Hindman.lean
|
Hindman.FP_partition_regular
|
[] |
[
224,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.subtype_eq_val
|
[] |
[
203,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Order/BoundedOrder.lean
|
Subtype.mk_eq_top_iff
|
[] |
[
800,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.separatingLeft_toLinearMap₂'_iff_det_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.2867092\nR₁ : Type u_2\nR₂ : Type ?u.2867098\nM✝ : Type ?u.2867101\nM₁ : Type ?u.2867104\nM₂ : Type ?u.2867107\nM₁' : Type ?u.2867110\nM₂' : Type ?u.2867113\nn : Type ?u.2867116\nm : Type ?u.2867119\nn' : Type ?u.2867122\nm' : Type ?u.2867125\nι : Type u_1\ninst✝⁵ : CommRing R₁\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : Module R₁ M₁\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\nB : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁\ninst✝ : IsDomain R₁\nM : Matrix ι ι R₁\n⊢ SeparatingLeft (↑toLinearMap₂' M) ↔ det M ≠ 0",
"tactic": "rw [Matrix.separatingLeft_toLinearMap₂'_iff, Matrix.nondegenerate_iff_det_ne_zero]"
}
] |
[
745,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
HasDerivAt.lhopital_zero_left_on_Ioc
|
[
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x) (𝓝[Iio b] b) (𝓝 0)\n\ncase refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => g x) (𝓝[Iio b] b) (𝓝 0)",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Iio b] b) l",
"tactic": "refine' lhopital_zero_left_on_Ioo hab hff' hgg' hg' _ _ hdiv"
},
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] b) (𝓝 (f b))",
"state_before": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x) (𝓝[Iio b] b) (𝓝 0)",
"tactic": "rw [← hfb, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] b) (𝓝 (f b))",
"tactic": "exact ((hcf b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto"
},
{
"state_after": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] b) (𝓝 (g b))",
"state_before": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => g x) (𝓝[Iio b] b) (𝓝 0)",
"tactic": "rw [← hgb, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhcf : ContinuousOn f (Ioc a b)\nhcg : ContinuousOn g (Ioc a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → g' x ≠ 0\nhfb : f b = 0\nhgb : g b = 0\nhdiv : Tendsto (fun x => f' x / g' x) (𝓝[Iio b] b) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] b) (𝓝 (g b))",
"tactic": "exact ((hcg b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).tendsto"
}
] |
[
145,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Control/Fold.lean
|
Traversable.toList_eq_self
|
[
{
"state_after": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nxs : List α\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) xs) = xs",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nxs : List α\n⊢ toList xs = xs",
"tactic": "simp only [toList_spec, foldMap, traverse]"
},
{
"state_after": "case nil\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) []) = []\n\ncase cons\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) (head✝ :: tail✝)) = head✝ :: tail✝",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nxs : List α\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) xs) = xs",
"tactic": "induction xs"
},
{
"state_after": "case cons\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) (head✝ :: tail✝)) = head✝ :: tail✝",
"state_before": "case nil\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) []) = []\n\ncase cons\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) (head✝ :: tail✝)) = head✝ :: tail✝",
"tactic": "case nil => rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) (head✝ :: tail✝)) = head✝ :: tail✝",
"tactic": "case cons _ _ ih => conv_rhs => rw [← ih]; rfl"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) []) = []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nhead✝ : α\ntail✝ : List α\nih : ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) tail✝) = tail✝\n⊢ ↑FreeMonoid.toList (List.traverse (Const.mk' ∘ FreeMonoid.of) (head✝ :: tail✝)) = head✝ :: tail✝",
"tactic": "conv_rhs => rw [← ih]; rfl"
}
] |
[
391,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_ite_of_false
|
[
{
"state_after": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ (∏ x in ∅, f x) * ∏ x in s, g x = ∏ x in s, g x\n\nι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ ∀ (x : α), x ∈ s → ¬p x\n\nι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ ∀ (x : α), x ∈ s → ¬p x",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ (∏ x in s, if p x then f x else g x) = ∏ x in s, g x",
"tactic": "rw [prod_ite, filter_false_of_mem, filter_true_of_mem]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ ∀ (x : α), x ∈ s → ¬p x\n\nι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ ∀ (x : α), x ∈ s → ¬p x",
"tactic": "all_goals intros; simp; apply h; assumption"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ (∏ x in ∅, f x) * ∏ x in s, g x = ∏ x in s, g x",
"tactic": "simp only [prod_empty, one_mul]"
},
{
"state_after": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ ¬p x✝",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\n⊢ ∀ (x : α), x ∈ s → ¬p x",
"tactic": "intros"
},
{
"state_after": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ ¬p x✝",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ ¬p x✝",
"tactic": "simp"
},
{
"state_after": "case a\nι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ x✝ ∈ s",
"state_before": "ι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ ¬p x✝",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.403432\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝ : CommMonoid β\np : α → Prop\nhp : DecidablePred p\nf g : α → β\nh : ∀ (x : α), x ∈ s → ¬p x\nx✝ : α\na✝ : x✝ ∈ s\n⊢ x✝ ∈ s",
"tactic": "assumption"
}
] |
[
1005,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1001,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean
|
dist_right_lineMap
|
[] |
[
133,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Std/Data/List/Lemmas.lean
|
List.mem_insert_of_mem
|
[] |
[
911,
30
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
910,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
MeasurableEmbedding.essSup_map_measure
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\n⊢ essSup g (Measure.map f μ) ≤ essSup (g ∘ f) μ",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\n⊢ essSup g (Measure.map f μ) = essSup (g ∘ f) μ",
"tactic": "refine' le_antisymm _ (essSup_comp_le_essSup_map_measure hf.measurable.aemeasurable)"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\nc : β\nh_le : ∀ᶠ (n : β) in map (g ∘ f) (Measure.ae μ), n ≤ c\n⊢ ∀ᶠ (n : β) in map g (Measure.ae (Measure.map f μ)), n ≤ c",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\n⊢ essSup g (Measure.map f μ) ≤ essSup (g ∘ f) μ",
"tactic": "refine' limsSup_le_limsSup (by isBoundedDefault) (by isBoundedDefault) (fun c h_le => _)"
},
{
"state_after": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\nc : β\nh_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c\n⊢ ∀ᵐ (a : γ) ∂Measure.map f μ, g a ≤ c",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\nc : β\nh_le : ∀ᶠ (n : β) in map (g ∘ f) (Measure.ae μ), n ≤ c\n⊢ ∀ᶠ (n : β) in map g (Measure.ae (Measure.map f μ)), n ≤ c",
"tactic": "rw [eventually_map] at h_le ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\nc : β\nh_le : ∀ᵐ (a : α) ∂μ, (g ∘ f) a ≤ c\n⊢ ∀ᵐ (a : γ) ∂Measure.map f μ, g a ≤ c",
"tactic": "exact hf.ae_map_iff.mpr h_le"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\n⊢ IsCobounded (fun x x_1 => x ≤ x_1) (map g (Measure.ae (Measure.map f μ)))",
"tactic": "isBoundedDefault"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝ : CompleteLattice β\nγ : Type u_2\nmγ : MeasurableSpace γ\nf : α → γ\ng : γ → β\nhf : MeasurableEmbedding f\n⊢ IsBounded (fun x x_1 => x ≤ x_1) (map (g ∘ f) (Measure.ae μ))",
"tactic": "isBoundedDefault"
}
] |
[
242,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
Monoid.coe_mul
|
[] |
[
1337,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1337,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.succ_le_iff
|
[] |
[
212,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
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