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/Filter/Prod.lean
|
Filter.mem_prod_top
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3562\nδ : Type ?u.3565\nι : Sort ?u.3568\ns✝ : Set α\nt : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\n⊢ {a | ∀ (b : β), b ∈ univ → (a, b) ∈ s} ∈ f ↔ {a | ∀ (b : β), (a, b) ∈ s} ∈ f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3562\nδ : Type ?u.3565\nι : Sort ?u.3568\ns✝ : Set α\nt : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\n⊢ s ∈ f ×ˢ ⊤ ↔ {a | ∀ (b : β), (a, b) ∈ s} ∈ f",
"tactic": "rw [← principal_univ, mem_prod_principal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3562\nδ : Type ?u.3565\nι : Sort ?u.3568\ns✝ : Set α\nt : Set β\nf✝ : Filter α\ng : Filter β\nf : Filter α\ns : Set (α × β)\n⊢ {a | ∀ (b : β), b ∈ univ → (a, b) ∈ s} ∈ f ↔ {a | ∀ (b : β), (a, b) ∈ s} ∈ f",
"tactic": "simp only [mem_univ, forall_true_left]"
}
] |
[
103,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Real/CauSeqCompletion.lean
|
CauSeq.le_lim
|
[] |
[
454,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
453,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.symm_trans_apply
|
[] |
[
466,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Biproduct.column_nonzero_of_iso'
|
[
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\n⊢ (∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\n⊢ (∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0",
"tactic": "cases nonempty_fintype τ"
},
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\n⊢ 𝟙 (S s) = 0",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\n⊢ (∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0) → 𝟙 (S s) = 0",
"tactic": "intro z"
},
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\n⊢ 𝟙 (S s) = 0",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\n⊢ 𝟙 (S s) = 0",
"tactic": "have reassoced {t : τ} {W : C} (h : _ ⟶ W) :\n biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h := by\n simp only [← Category.assoc]\n apply eq_whisker\n simp only [Category.assoc]\n apply z"
},
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\n⊢ 𝟙 (S s) = 0",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\n⊢ 𝟙 (S s) = 0",
"tactic": "set x := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s"
},
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ 𝟙 (S s) = 0",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\n⊢ 𝟙 (S s) = 0",
"tactic": "have h₁ : x = 𝟙 (S s) := by simp"
},
{
"state_after": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\nh₀ : x = 0\n⊢ 𝟙 (S s) = 0",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ 𝟙 (S s) = 0",
"tactic": "have h₀ : x = 0 := by\n dsimp\n rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]\n simp only [comp_sum_assoc]\n conv_lhs =>\n congr\n congr\n next => skip\n intro j; simp only [reassoced]\n simp"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\nh₀ : x = 0\n⊢ 𝟙 (S s) = 0",
"tactic": "exact h₁.symm.trans h₀"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ ((biproduct.ι S s ≫ f) ≫ biproduct.π T t) ≫ h = 0 ≫ h",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h",
"tactic": "simp only [← Category.assoc]"
},
{
"state_after": "case w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ (biproduct.ι S s ≫ f) ≫ biproduct.π T t = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ ((biproduct.ι S s ≫ f) ≫ biproduct.π T t) ≫ h = 0 ≫ h",
"tactic": "apply eq_whisker"
},
{
"state_after": "case w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ biproduct.ι S s ≫ f ≫ biproduct.π T t = 0",
"state_before": "case w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ (biproduct.ι S s ≫ f) ≫ biproduct.π T t = 0",
"tactic": "simp only [Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nt : τ\nW : C\nh : T t ⟶ W\n⊢ biproduct.ι S s ≫ f ≫ biproduct.π T t = 0",
"tactic": "apply z"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\n⊢ x = 𝟙 (S s)",
"tactic": "simp"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ x = 0",
"tactic": "dsimp"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ biproduct.ι S s ≫ f ≫ (∑ j : τ, biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s = 0",
"tactic": "rw [← Category.id_comp (inv f), Category.assoc, ← biproduct.total]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ (∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ biproduct.ι S s ≫ f ≫ (∑ j : τ, biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"tactic": "simp only [comp_sum_assoc]"
},
{
"state_after": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ (∑ j : τ, 0 ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ (∑ j : τ, biproduct.ι S s ≫ f ≫ biproduct.π T j ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"tactic": "conv_lhs =>\n congr\n congr\n next => skip\n intro j; simp only [reassoced]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nσ τ : Type\ninst✝³ : Finite τ\nS : σ → C\ninst✝² : HasBiproduct S\nT : τ → C\ninst✝¹ : HasBiproduct T\ns : σ\nf : ⨁ S ⟶ ⨁ T\ninst✝ : IsIso f\nval✝ : Fintype τ\nz : ∀ (t : τ), biproduct.ι S s ≫ f ≫ biproduct.π T t = 0\nreassoced : ∀ {t : τ} {W : C} (h : T t ⟶ W), biproduct.ι S s ≫ f ≫ biproduct.π T t ≫ h = 0 ≫ h\nx : S s ⟶ S s := biproduct.ι S s ≫ f ≫ inv f ≫ biproduct.π S s\nh₁ : x = 𝟙 (S s)\n⊢ (∑ j : τ, 0 ≫ biproduct.ι T j) ≫ inv f ≫ biproduct.π S s = 0",
"tactic": "simp"
}
] |
[
860,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/Combinatorics/SimpleGraph/Acyclic.lean
|
SimpleGraph.isAcyclic_iff_forall_adj_isBridge
|
[
{
"state_after": "V : Type u\nG : SimpleGraph V\n⊢ IsAcyclic G ↔\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "V : Type u\nG : SimpleGraph V\n⊢ IsAcyclic G ↔ ∀ ⦃v w : V⦄, Adj G v w → IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (v, w))",
"tactic": "simp_rw [isBridge_iff_adj_and_forall_cycle_not_mem]"
},
{
"state_after": "case mp\nV : Type u\nG : SimpleGraph V\n⊢ IsAcyclic G →\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\n\ncase mpr\nV : Type u\nG : SimpleGraph V\n⊢ (∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧\n ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n IsAcyclic G",
"state_before": "V : Type u\nG : SimpleGraph V\n⊢ IsAcyclic G ↔\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "constructor"
},
{
"state_after": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\n⊢ Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mp\nV : Type u\nG : SimpleGraph V\n⊢ IsAcyclic G →\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "intro ha v w hvw"
},
{
"state_after": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\n⊢ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\n⊢ Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "apply And.intro hvw"
},
{
"state_after": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\nu : V\np : Walk G u u\nhp : Walk.IsCycle p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"state_before": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\n⊢ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "intro u p hp"
},
{
"state_after": "no goals",
"state_before": "case mp\nV : Type u\nG : SimpleGraph V\nha : IsAcyclic G\nv w : V\nhvw : Adj G v w\nu : V\np : Walk G u u\nhp : Walk.IsCycle p\n⊢ ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p",
"tactic": "cases ha p hp"
},
{
"state_after": "case mpr.nil\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv : V\nhp : Walk.IsCycle Walk.nil\n⊢ False\n\ncase mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ False",
"state_before": "case mpr\nV : Type u\nG : SimpleGraph V\n⊢ (∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧\n ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p) →\n IsAcyclic G",
"tactic": "rintro hb v (_ | ⟨ha, p⟩) hp"
},
{
"state_after": "no goals",
"state_before": "case mpr.nil\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv : V\nhp : Walk.IsCycle Walk.nil\n⊢ False",
"tactic": "exact hp.not_of_nil"
},
{
"state_after": "case mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) ∈ Walk.edges (Walk.cons ha p)",
"state_before": "case mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ False",
"tactic": "apply (hb ha).2 _ hp"
},
{
"state_after": "case mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) ∈ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) :: Walk.edges p",
"state_before": "case mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) ∈ Walk.edges (Walk.cons ha p)",
"tactic": "rw [Walk.edges_cons]"
},
{
"state_after": "no goals",
"state_before": "case mpr.cons\nV : Type u\nG : SimpleGraph V\nhb :\n ∀ ⦃v w : V⦄,\n Adj G v w →\n Adj G v w ∧ ∀ ⦃u : V⦄ (p : Walk G u u), Walk.IsCycle p → ¬Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nv v✝ : V\nha : Adj G v v✝\np : Walk G v✝ v\nhp : Walk.IsCycle (Walk.cons ha p)\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) ∈ Quotient.mk (Sym2.Rel.setoid V) (v, v✝) :: Walk.edges p",
"tactic": "apply List.mem_cons_self"
}
] |
[
78,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.symm_comp_eq
|
[] |
[
427,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.linearIndependent_of_iIsOrtho
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\n⊢ LinearIndependent K v",
"tactic": "classical\n rw [linearIndependent_iff']\n intro s w hs i hi\n have : B (s.sum fun i : n => w i • v i) (v i) = 0 := by rw [hs, zero_left]\n have hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) := by\n apply Finset.sum_eq_single_of_mem i hi\n intro j _ hij\n rw [iIsOrtho_def.1 hv₁ _ _ hij, MulZeroClass.mul_zero]\n simp_rw [sum_left, smul_left, hsum] at this\n exact eq_zero_of_ne_zero_of_mul_right_eq_zero (hv₂ i) this"
},
{
"state_after": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\n⊢ ∀ (s : Finset n) (g : n → K), ∑ i in s, g i • v i = 0 → ∀ (i : n), i ∈ s → g i = 0",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\n⊢ LinearIndependent K v",
"tactic": "rw [linearIndependent_iff']"
},
{
"state_after": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\n⊢ w i = 0",
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"tactic": "intro s w hs i hi"
},
{
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"tactic": "have : B (s.sum fun i : n => w i • v i) (v i) = 0 := by rw [hs, zero_left]"
},
{
"state_after": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\nhsum : ∑ j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)\n⊢ w i = 0",
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"tactic": "have hsum : (s.sum fun j : n => w j * B (v j) (v i)) = w i * B (v i) (v i) := by\n apply Finset.sum_eq_single_of_mem i hi\n intro j _ hij\n rw [iIsOrtho_def.1 hv₁ _ _ hij, MulZeroClass.mul_zero]"
},
{
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"tactic": "simp_rw [sum_left, smul_left, hsum] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nhsum : ∑ j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)\nthis : w i * bilin B (v i) (v i) = 0\n⊢ w i = 0",
"tactic": "exact eq_zero_of_ne_zero_of_mul_right_eq_zero (hv₂ i) this"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\n⊢ bilin B (∑ i in s, w i • v i) (v i) = 0",
"tactic": "rw [hs, zero_left]"
},
{
"state_after": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\n⊢ ∀ (b : n), b ∈ s → b ≠ i → w b * bilin B (v b) (v i) = 0",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\n⊢ ∑ j in s, w j * bilin B (v j) (v i) = w i * bilin B (v i) (v i)",
"tactic": "apply Finset.sum_eq_single_of_mem i hi"
},
{
"state_after": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\nj : n\na✝ : j ∈ s\nhij : j ≠ i\n⊢ w j * bilin B (v j) (v i) = 0",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\n⊢ ∀ (b : n), b ∈ s → b ≠ i → w b * bilin B (v b) (v i) = 0",
"tactic": "intro j _ hij"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.787806\nM : Type ?u.787809\ninst✝²² : Semiring R\ninst✝²¹ : AddCommMonoid M\ninst✝²⁰ : Module R M\nR₁ : Type ?u.787845\nM₁ : Type ?u.787848\ninst✝¹⁹ : Ring R₁\ninst✝¹⁸ : AddCommGroup M₁\ninst✝¹⁷ : Module R₁ M₁\nR₂ : Type ?u.788457\nM₂ : Type ?u.788460\ninst✝¹⁶ : CommSemiring R₂\ninst✝¹⁵ : AddCommMonoid M₂\ninst✝¹⁴ : Module R₂ M₂\nR₃ : Type ?u.788647\nM₃ : Type ?u.788650\ninst✝¹³ : CommRing R₃\ninst✝¹² : AddCommGroup M₃\ninst✝¹¹ : Module R₃ M₃\nV : Type u_2\nK : Type u_1\ninst✝¹⁰ : Field K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.790455\nM₂'' : Type ?u.790458\ninst✝⁷ : AddCommMonoid M₂'\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : Module R₂ M₂'\ninst✝⁴ : Module R₂ M₂''\nR₄ : Type ?u.790741\nM₄ : Type ?u.790744\ninst✝³ : Ring R₄\ninst✝² : IsDomain R₄\ninst✝¹ : AddCommGroup M₄\ninst✝ : Module R₄ M₄\nG : BilinForm R₄ M₄\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : iIsOrtho B v\nhv₂ : ∀ (i : n), ¬IsOrtho B (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i in s, w i • v i = 0\ni : n\nhi : i ∈ s\nthis : bilin B (∑ i in s, w i • v i) (v i) = 0\nj : n\na✝ : j ∈ s\nhij : j ≠ i\n⊢ w j * bilin B (v j) (v i) = 0",
"tactic": "rw [iIsOrtho_def.1 hv₁ _ _ hij, MulZeroClass.mul_zero]"
}
] |
[
823,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_of_neg
|
[] |
[
123,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.const_equiv
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nq r : ℚ\nheq : q = r\n⊢ const (padicNorm p) q ≈ const (padicNorm p) r",
"tactic": "rw [heq]"
}
] |
[
525,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Dense.exists_dist_lt
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.155993\nι : Type ?u.155996\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ s : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\nthis : Set.Nonempty (ball x ε)\n⊢ ∃ y, y ∈ s ∧ dist x y < ε",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.155993\nι : Type ?u.155996\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ s : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\n⊢ ∃ y, y ∈ s ∧ dist x y < ε",
"tactic": "have : (ball x ε).Nonempty := by simp [hε]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.155993\nι : Type ?u.155996\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ s : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\nthis : Set.Nonempty (ball x ε)\n⊢ ∃ y, y ∈ s ∧ dist x y < ε",
"tactic": "simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.155993\nι : Type ?u.155996\ninst✝ : PseudoMetricSpace α\nx✝ y z : α\nδ ε✝ ε₁ ε₂ : ℝ\ns✝ s : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\n⊢ Set.Nonempty (ball x ε)",
"tactic": "simp [hε]"
}
] |
[
1135,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1132,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Cocone.ofCotrident_ι
|
[
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝ : Category C\nX Y : C\nf : J → (X ⟶ Y)\nF : WalkingParallelFamily J ⥤ C\nt : Cotrident fun j => F.map (line j)\nj : WalkingParallelFamily J\n⊢ F.obj j = (parallelFamily fun j => F.map (line j)).obj j",
"tactic": "cases j <;> aesop_cat"
}
] |
[
475,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
IsAbsoluteValue.abv_inv
|
[] |
[
474,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.pos_of_mem_properDivisors
|
[] |
[
194,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
GCongr.prod_lt_prod_of_nonempty'
|
[] |
[
469,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_const
|
[] |
[
1441,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1440,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
HasFDerivWithinAt.hasDerivWithinAt
|
[] |
[
185,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Algebra/Associated.lean
|
MulEquiv.prime_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.25765\nδ : Type ?u.25768\ninst✝³ : CommMonoidWithZero α\ninst✝² : CommMonoidWithZero β\nF : Type ?u.25777\nG : Type ?u.25780\ninst✝¹ : MonoidWithZeroHomClass F α β\ninst✝ : MulHomClass G β α\nf : F\ng : G\np : α\ne : α ≃* β\nh : Prime p\na : β\n⊢ ↑e (↑(symm e) a) = a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.25765\nδ : Type ?u.25768\ninst✝³ : CommMonoidWithZero α\ninst✝² : CommMonoidWithZero β\nF : Type ?u.25777\nG : Type ?u.25780\ninst✝¹ : MonoidWithZeroHomClass F α β\ninst✝ : MulHomClass G β α\nf : F\ng : G\np : α\ne : α ≃* β\na : α\n⊢ ↑(symm e) (↑e a) = a",
"tactic": "simp"
}
] |
[
94,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasStrictDerivAt.add_const
|
[] |
[
89,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
zero_pow'
|
[
{
"state_after": "R : Type ?u.3804\nS : Type ?u.3807\nM : Type u_1\ninst✝ : MonoidWithZero M\nk : ℕ\nx✝ : k + 1 ≠ 0\n⊢ 0 * 0 ^ k = 0",
"state_before": "R : Type ?u.3804\nS : Type ?u.3807\nM : Type u_1\ninst✝ : MonoidWithZero M\nk : ℕ\nx✝ : k + 1 ≠ 0\n⊢ 0 ^ (k + 1) = 0",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.3804\nS : Type ?u.3807\nM : Type u_1\ninst✝ : MonoidWithZero M\nk : ℕ\nx✝ : k + 1 ≠ 0\n⊢ 0 * 0 ^ k = 0",
"tactic": "exact zero_mul _"
}
] |
[
44,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean
|
Matrix.circulant_single_one
|
[
{
"state_after": "case a.h\nα✝ : Type ?u.41028\nβ : Type ?u.41031\nm : Type ?u.41034\nn✝ : Type ?u.41037\nR : Type ?u.41040\nα : Type u_1\nn : Type u_2\ninst✝³ : Zero α\ninst✝² : One α\ninst✝¹ : DecidableEq n\ninst✝ : AddGroup n\ni j : n\n⊢ circulant (Pi.single 0 1) i j = OfNat.ofNat 1 i j",
"state_before": "α✝ : Type ?u.41028\nβ : Type ?u.41031\nm : Type ?u.41034\nn✝ : Type ?u.41037\nR : Type ?u.41040\nα : Type u_1\nn : Type u_2\ninst✝³ : Zero α\ninst✝² : One α\ninst✝¹ : DecidableEq n\ninst✝ : AddGroup n\n⊢ circulant (Pi.single 0 1) = 1",
"tactic": "ext (i j)"
},
{
"state_after": "no goals",
"state_before": "case a.h\nα✝ : Type ?u.41028\nβ : Type ?u.41031\nm : Type ?u.41034\nn✝ : Type ?u.41037\nR : Type ?u.41040\nα : Type u_1\nn : Type u_2\ninst✝³ : Zero α\ninst✝² : One α\ninst✝¹ : DecidableEq n\ninst✝ : AddGroup n\ni j : n\n⊢ circulant (Pi.single 0 1) i j = OfNat.ofNat 1 i j",
"tactic": "simp [one_apply, Pi.single_apply, sub_eq_zero]"
}
] |
[
170,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.mem_append_of_mem_right
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nb : α\nbs as : List α\nh : b ∈ bs\n⊢ b ∈ as ++ bs",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nb : α\nbs as : List α\n⊢ b ∈ bs → b ∈ as ++ bs",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nb : α\nbs as : List α\nh : b ∈ bs\n⊢ b ∈ as ++ bs",
"tactic": "induction as with\n| nil => simp [h]\n| cons => apply Mem.tail; assumption"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\nh : b ∈ bs\n⊢ b ∈ nil ++ bs",
"tactic": "simp [h]"
},
{
"state_after": "case cons.a\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\nh : b ∈ bs\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : b ∈ tail✝ ++ bs\n⊢ Mem b (List.append tail✝ bs)",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\nh : b ∈ bs\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : b ∈ tail✝ ++ bs\n⊢ b ∈ head✝ :: tail✝ ++ bs",
"tactic": "apply Mem.tail"
},
{
"state_after": "no goals",
"state_before": "case cons.a\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\nh : b ∈ bs\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : b ∈ tail✝ ++ bs\n⊢ Mem b (List.append tail✝ bs)",
"tactic": "assumption"
}
] |
[
386,
39
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
382,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.moveLeft_neg_symm
|
[
{
"state_after": "no goals",
"state_before": "x : PGame\ni : RightMoves (-x)\n⊢ moveLeft x (↑toRightMovesNeg.symm i) = -moveRight (-x) i",
"tactic": "simp"
}
] |
[
1289,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1288,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_tsum_eq_single
|
[] |
[
531,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_updateRow_smul'
|
[] |
[
415,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.eval_image_pi
|
[] |
[
814,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
813,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
deriv.lhopital_zero_right_on_Ico
|
[
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioi a] a) (𝓝 0)\n\ncase refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioi a] a) (𝓝 0)",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x / g x) (𝓝[Ioi a] a) l",
"tactic": "refine' lhopital_zero_right_on_Ioo hab hdf hg' _ _ hdiv"
},
{
"state_after": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] a) (𝓝 (f a))",
"state_before": "case refine'_1\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioi a] a) (𝓝 0)",
"tactic": "rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi 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\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => f x) (𝓝[Ioo a b] a) (𝓝 (f a))",
"tactic": "exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto"
},
{
"state_after": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] a) (𝓝 (g a))",
"state_before": "case refine'_2\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioi a] a) (𝓝 0)",
"tactic": "rw [← hga, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi 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\nhdf : DifferentiableOn ℝ f (Ioo a b)\nhcf : ContinuousOn f (Ico a b)\nhcg : ContinuousOn g (Ico a b)\nhg' : ∀ (x : ℝ), x ∈ Ioo a b → deriv g x ≠ 0\nhfa : f a = 0\nhga : g a = 0\nhdiv : Tendsto (fun x => deriv f x / deriv g x) (𝓝[Ioi a] a) l\n⊢ Tendsto (fun x => g x) (𝓝[Ioo a b] a) (𝓝 (g a))",
"tactic": "exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto"
}
] |
[
232,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.comap_hasBasis
|
[] |
[
805,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/Data/Set/Image.lean
|
Function.Injective.preimage_image
|
[] |
[
1287,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1286,
1
] |
Mathlib/Algebra/Category/ModuleCat/Basic.lean
|
ModuleCat.coe_of
|
[] |
[
201,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.cauchy_natCast
|
[] |
[
193,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.Flat.pointed
|
[
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.77348\nG : Type ?u.77351\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\nx : E\nhx : x ∈ S\nleft✝ : x ≠ 0\nhxneg : -x ∈ S\n⊢ Pointed S",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.77348\nG : Type ?u.77351\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\nhS : Flat S\n⊢ Pointed S",
"tactic": "obtain ⟨x, hx, _, hxneg⟩ := hS"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.77348\nG : Type ?u.77351\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\nx : E\nhx : x ∈ S\nleft✝ : x ≠ 0\nhxneg : -x ∈ S\n⊢ x + -x ∈ S",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.77348\nG : Type ?u.77351\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\nx : E\nhx : x ∈ S\nleft✝ : x ≠ 0\nhxneg : -x ∈ S\n⊢ Pointed S",
"tactic": "rw [Pointed, ← add_neg_self x]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.77348\nG : Type ?u.77351\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\nx : E\nhx : x ∈ S\nleft✝ : x ≠ 0\nhxneg : -x ∈ S\n⊢ x + -x ∈ S",
"tactic": "exact add_mem S hx hxneg"
}
] |
[
403,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.dist_lt_iff_of_compact
|
[
{
"state_after": "case mp\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\n⊢ dist f g < C → ∀ (x : α), dist (↑f x) (↑g x) < C\n\ncase mpr\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C",
"state_before": "F : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\n⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C",
"tactic": "fconstructor"
},
{
"state_after": "case mp\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx✝ : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nw : dist f g < C\nx : α\n⊢ dist (↑f x) (↑g x) < C",
"state_before": "case mp\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\n⊢ dist f g < C → ∀ (x : α), dist (↑f x) (↑g x) < C",
"tactic": "intro w x"
},
{
"state_after": "no goals",
"state_before": "case mp\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx✝ : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nw : dist f g < C\nx : α\n⊢ dist (↑f x) (↑g x) < C",
"tactic": "exact lt_of_le_of_lt (dist_coe_le_dist x) w"
},
{
"state_after": "case pos\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : Nonempty α\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C\n\ncase neg\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C",
"state_before": "case mpr\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C",
"tactic": "by_cases h : Nonempty α"
},
{
"state_after": "no goals",
"state_before": "case pos\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : Nonempty α\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C",
"tactic": "exact dist_lt_of_nonempty_compact"
},
{
"state_after": "case neg\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g < C",
"state_before": "case neg\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ (∀ (x : α), dist (↑f x) (↑g x) < C) → dist f g < C",
"tactic": "rintro -"
},
{
"state_after": "case h.e'_3\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g = 0",
"state_before": "case neg\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g < C",
"tactic": "convert C0"
},
{
"state_after": "F : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g ≤ 0",
"state_before": "case h.e'_3\nF : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g = 0",
"tactic": "apply le_antisymm _ dist_nonneg'"
},
{
"state_after": "F : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} ≤ 0",
"state_before": "F : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g ≤ 0",
"tactic": "rw [dist_eq]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.285808\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : PseudoMetricSpace γ\nf g : α →ᵇ β\nx : α\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (↑f x) (↑g x) ≤ C} ≤ 0",
"tactic": "exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩"
}
] |
[
211,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.disjoint_span_singleton
|
[
{
"state_after": "R : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\n⊢ (x ∈ s → x = 0) → ∀ (x_1 : E), x_1 ∈ s → x_1 ∈ span K {x} → x_1 = 0",
"state_before": "R : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\n⊢ Disjoint s (span K {x}) ↔ x ∈ s → x = 0",
"tactic": "refine' disjoint_def.trans ⟨fun H hx => H x hx <| subset_span <| mem_singleton x, _⟩"
},
{
"state_after": "R : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\ny : E\nhy : y ∈ s\nhyx : y ∈ span K {x}\n⊢ y = 0",
"state_before": "R : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\n⊢ (x ∈ s → x = 0) → ∀ (x_1 : E), x_1 ∈ s → x_1 ∈ span K {x} → x_1 = 0",
"tactic": "intro H y hy hyx"
},
{
"state_after": "case intro\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\n⊢ c • x = 0",
"state_before": "R : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\ny : E\nhy : y ∈ s\nhyx : y ∈ span K {x}\n⊢ y = 0",
"tactic": "obtain ⟨c, rfl⟩ := mem_span_singleton.1 hyx"
},
{
"state_after": "case pos\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\nhc : c = 0\n⊢ c • x = 0\n\ncase neg\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\nhc : ¬c = 0\n⊢ c • x = 0",
"state_before": "case intro\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\n⊢ c • x = 0",
"tactic": "by_cases hc : c = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\nhc : c = 0\n⊢ c • x = 0",
"tactic": "rw [hc, zero_smul]"
},
{
"state_after": "case neg\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : x ∈ s\nhyx : c • x ∈ span K {x}\nhc : ¬c = 0\n⊢ c • x = 0",
"state_before": "case neg\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : c • x ∈ s\nhyx : c • x ∈ span K {x}\nhc : ¬c = 0\n⊢ c • x = 0",
"tactic": "rw [s.smul_mem_iff hc] at hy"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type ?u.153486\nR₂ : Type ?u.153489\nK✝ : Type ?u.153492\nM : Type ?u.153495\nM₂ : Type ?u.153498\nV : Type ?u.153501\nS : Type ?u.153504\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁵ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : Module R₂ M₂\ns✝ t : Set M\nK : Type u_1\nE : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup E\ninst✝ : Module K E\ns : Submodule K E\nx : E\nH : x ∈ s → x = 0\nc : K\nhy : x ∈ s\nhyx : c • x ∈ span K {x}\nhc : ¬c = 0\n⊢ c • x = 0",
"tactic": "rw [H hy, smul_zero]"
}
] |
[
464,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.sInter_insert
|
[] |
[
1145,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1144,
1
] |
Mathlib/Data/Sum/Order.lean
|
Sum.Lex.toLex_strictMono
|
[] |
[
402,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.finite
|
[] |
[
408,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
11
] |
Mathlib/Probability/Independence/ZeroOne.lean
|
ProbabilityTheory.indep_limsup_atBot_self
|
[
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\n⊢ Indep (limsup s atBot) (limsup s atBot)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\n⊢ Indep (limsup s atBot) (limsup s atBot)",
"tactic": "let ns : ι → Set ι := Set.Ici"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ Indep (limsup s atBot) (limsup s atBot)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\n⊢ Indep (limsup s atBot) (limsup s atBot)",
"tactic": "have hnsp : ∀ i, BddBelow (ns i) := fun i => bddBelow_Ici"
},
{
"state_after": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (t : Set ι), BddBelow t → tᶜ ∈ atBot\n\ncase refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ns a\n\ncase refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (n : ι), ∃ a, n ∈ ns a",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ Indep (limsup s atBot) (limsup s atBot)",
"tactic": "refine' indep_limsup_self h_le h_indep _ _ hnsp _"
},
{
"state_after": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (t : Set ι), (∃ x, x ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}) → ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"state_before": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (t : Set ι), BddBelow t → tᶜ ∈ atBot",
"tactic": "simp only [mem_atBot_sets, ge_iff_le, Set.mem_compl_iff, BddBelow, lowerBounds, Set.Nonempty]"
},
{
"state_after": "case refine'_1.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\n⊢ ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"state_before": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (t : Set ι), (∃ x, x ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}) → ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"tactic": "rintro t ⟨a, ha⟩"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\n⊢ ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"state_before": "case refine'_1.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\n⊢ ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"tactic": "obtain ⟨b, hb⟩ : ∃ b, b < a := exists_lt a"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ False",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\n⊢ ∃ a, ∀ (b : ι), b ≤ a → ¬b ∈ t",
"tactic": "refine' ⟨b, fun c hc hct => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\nthis : ∀ (i : ι), i ∈ t → c < i\n⊢ False\n\ncase this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → c < i",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ False",
"tactic": "suffices : ∀ i ∈ t, c < i"
},
{
"state_after": "case this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → c < i",
"state_before": "case refine'_1.intro.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\nthis : ∀ (i : ι), i ∈ t → c < i\n⊢ False\n\ncase this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → c < i",
"tactic": "exact lt_irrefl c (this c hct)"
},
{
"state_after": "no goals",
"state_before": "case this\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\nt : Set ι\na : ι\nha : a ∈ {x | ∀ ⦃a : ι⦄, a ∈ t → x ≤ a}\nb : ι\nhb : b < a\nc : ι\nhc : c ≤ b\nhct : c ∈ t\n⊢ ∀ (i : ι), i ∈ t → c < i",
"tactic": "exact fun i hi => hc.trans_lt (hb.trans_le (ha hi))"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ns a",
"tactic": "exact directed_of_inf fun i j hij k hki => hij.trans hki"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝³ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\ninst✝² : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nns : ι → Set ι := Set.Ici\nhnsp : ∀ (i : ι), BddBelow (ns i)\n⊢ ∀ (n : ι), ∃ a, n ∈ ns a",
"tactic": "exact fun n => ⟨n, le_rfl⟩"
}
] |
[
176,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/CategoryTheory/Adjunction/Opposites.lean
|
CategoryTheory.Adjunction.rightAdjointUniq_hom_app_counit
|
[
{
"state_after": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op = (adj1.counit.app x).op",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\n⊢ F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x = adj1.counit.app x",
"tactic": "apply Quiver.Hom.op_inj"
},
{
"state_after": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op =\n (opAdjointOpOfAdjoint G' F adj2).unit.app x.op ≫\n (Functor.op F).map\n ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op)\n\ncase h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op = (opAdjointOpOfAdjoint G F adj1).unit.app x.op",
"state_before": "case a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op = (adj1.counit.app x).op",
"tactic": "convert\n unit_leftAdjointUniq_hom_app (opAdjointOpOfAdjoint _ _ adj2)\n (opAdjointOpOfAdjoint _ _ adj1) (Opposite.op x) using 1"
},
{
"state_after": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj2.counit.app x).op ≫ (F.map ((rightAdjointUniq adj1 adj2).hom.app x)).op =\n ↑(opEquiv x.op (F.toPrefunctor.1 (G'.obj x)).op).symm\n (F.map (↑(opEquiv (G'.obj x).op (G'.obj x).op) (𝟙 (G'.obj x).op)) ≫ adj2.counit.app x) ≫\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop).op",
"state_before": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op =\n (opAdjointOpOfAdjoint G' F adj2).unit.app x.op ≫\n (Functor.op F).map\n ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op)",
"tactic": "simp only [Functor.id_obj, op_comp, Functor.comp_obj, Functor.op_obj, Opposite.unop_op,\n opAdjointOpOfAdjoint_unit_app, Functor.op_map]"
},
{
"state_after": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj2.counit.app x).op ≫ (F.map ((rightAdjointUniq adj1 adj2).hom.app x)).op =\n ((adj2.counit.app x).op ≫ (F.map (𝟙 (G'.obj x))).op) ≫\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop).op",
"state_before": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj2.counit.app x).op ≫ (F.map ((rightAdjointUniq adj1 adj2).hom.app x)).op =\n ↑(opEquiv x.op (F.toPrefunctor.1 (G'.obj x)).op).symm\n (F.map (↑(opEquiv (G'.obj x).op (G'.obj x).op) (𝟙 (G'.obj x).op)) ≫ adj2.counit.app x) ≫\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop).op",
"tactic": "dsimp [opEquiv]"
},
{
"state_after": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op =\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop ≫\n F.map (𝟙 (G'.obj x)) ≫ adj2.counit.app x).op",
"state_before": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj2.counit.app x).op ≫ (F.map ((rightAdjointUniq adj1 adj2).hom.app x)).op =\n ((adj2.counit.app x).op ≫ (F.map (𝟙 (G'.obj x))).op) ≫\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop).op",
"tactic": "simp only [← op_comp]"
},
{
"state_after": "case h.e'_2.h.e_f.e_a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ adj2.counit.app x = F.map (𝟙 (G'.obj x)) ≫ adj2.counit.app x",
"state_before": "case h.e'_2.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (F.map ((rightAdjointUniq adj1 adj2).hom.app x) ≫ adj2.counit.app x).op =\n (F.map ((leftAdjointUniq (opAdjointOpOfAdjoint G' F adj2) (opAdjointOpOfAdjoint G F adj1)).hom.app x.op).unop ≫\n F.map (𝟙 (G'.obj x)) ≫ adj2.counit.app x).op",
"tactic": "congr 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e_f.e_a\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ adj2.counit.app x = F.map (𝟙 (G'.obj x)) ≫ adj2.counit.app x",
"tactic": "simp"
},
{
"state_after": "case h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op =\n ↑(opEquiv x.op (F.toPrefunctor.1 (G.obj x)).op).symm\n (F.map (↑(opEquiv (G.obj x).op (G.obj x).op) (𝟙 (G.obj x).op)) ≫ adj1.counit.app x)",
"state_before": "case h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op = (opAdjointOpOfAdjoint G F adj1).unit.app x.op",
"tactic": "simp only [Functor.id_obj, opAdjointOpOfAdjoint_unit_app, Opposite.unop_op]"
},
{
"state_after": "case h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op = ↑(opEquiv x.op (F.toPrefunctor.1 (G.obj x)).op).symm (adj1.counit.app x)",
"state_before": "case h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op =\n ↑(opEquiv x.op (F.toPrefunctor.1 (G.obj x)).op).symm\n (F.map (↑(opEquiv (G.obj x).op (G.obj x).op) (𝟙 (G.obj x).op)) ≫ adj1.counit.app x)",
"tactic": "erw [Functor.map_id, Category.id_comp]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG G' : D ⥤ C\nadj1 : F ⊣ G\nadj2 : F ⊣ G'\nx : D\ne_1✝ : (((𝟭 D).obj x).op ⟶ (F.obj (G.obj x)).op) = ((𝟭 Dᵒᵖ).obj x.op ⟶ (Functor.op F).obj ((Functor.op G).obj x.op))\n⊢ (adj1.counit.app x).op = ↑(opEquiv x.op (F.toPrefunctor.1 (G.obj x)).op).symm (adj1.counit.app x)",
"tactic": "rfl"
}
] |
[
278,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
Measurable.mul_const
|
[] |
[
125,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.Iso.card_eq_of_iso
|
[
{
"state_after": "ι : Sort ?u.637529\n𝕜 : Type ?u.637532\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf✝ : G ≃g G'\ninst✝¹ : Fintype V\ninst✝ : Fintype W\nf : G ≃g G'\n⊢ Fintype.card W = Fintype.card W",
"state_before": "ι : Sort ?u.637529\n𝕜 : Type ?u.637532\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf✝ : G ≃g G'\ninst✝¹ : Fintype V\ninst✝ : Fintype W\nf : G ≃g G'\n⊢ Fintype.card V = Fintype.card W",
"tactic": "rw [← Fintype.ofEquiv_card f.toEquiv]"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.637529\n𝕜 : Type ?u.637532\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\nf✝ : G ≃g G'\ninst✝¹ : Fintype V\ninst✝ : Fintype W\nf : G ≃g G'\n⊢ Fintype.card W = Fintype.card W",
"tactic": "apply @Fintype.card_congr' _ _ (_) (_) rfl"
}
] |
[
1965,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1962,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.sum_int_mod
|
[
{
"state_after": "ι : Type ?u.877325\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ns : Finset α\nn : ℤ\nf : α → ℤ\n⊢ Multiset.sum (Multiset.map ((fun x => x % n) ∘ fun i => f i) s.val) % n =\n Multiset.sum (Multiset.map (fun i => f i % n) s.val) % n",
"state_before": "ι : Type ?u.877325\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ns : Finset α\nn : ℤ\nf : α → ℤ\n⊢ Multiset.sum (Multiset.map (fun x => x % n) (Multiset.map (fun i => f i) s.val)) % n = (∑ i in s, f i % n) % n",
"tactic": "rw [Finset.sum, Multiset.map_map]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.877325\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ns : Finset α\nn : ℤ\nf : α → ℤ\n⊢ Multiset.sum (Multiset.map ((fun x => x % n) ∘ fun i => f i) s.val) % n =\n Multiset.sum (Multiset.map (fun i => f i % n) s.val) % n",
"tactic": "rfl"
}
] |
[
1957,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1955,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.lcm_self
|
[
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ lcm m m = m",
"tactic": "match eq_zero_or_pos m with\n| .inl h => rw [h, lcm_zero_left]\n| .inr h => simp [lcm, Nat.mul_div_cancel _ h]"
},
{
"state_after": "no goals",
"state_before": "m : Nat\nh : m = 0\n⊢ lcm m m = m",
"tactic": "rw [h, lcm_zero_left]"
},
{
"state_after": "no goals",
"state_before": "m : Nat\nh : m > 0\n⊢ lcm m m = m",
"tactic": "simp [lcm, Nat.mul_div_cancel _ h]"
}
] |
[
196,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
193,
9
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
invOf_exp
|
[] |
[
480,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.integrable_add_measure
|
[] |
[
549,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.nontrivial_mono
|
[] |
[
2601,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2599,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.bind_dirac
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → Measure β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(bind (dirac a) f) s = ↑↑(f a) s",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → Measure β\nhf : Measurable f\na : α\n⊢ bind (dirac a) f = f a",
"tactic": "ext1 s hs"
},
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → Measure β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ((fun μ => ↑↑μ s) ∘ f) a = ↑↑(f a) s",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → Measure β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ↑↑(bind (dirac a) f) s = ↑↑(f a) s",
"tactic": "erw [bind_apply hs hf, lintegral_dirac' a ((measurable_coe hs).comp hf)]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → Measure β\nhf : Measurable f\na : α\ns : Set β\nhs : MeasurableSet s\n⊢ ((fun μ => ↑↑μ s) ∘ f) a = ↑↑(f a) s",
"tactic": "rfl"
}
] |
[
202,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
hasSum_ite_eq
|
[
{
"state_after": "case h.e'_6\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41040\nδ : Type ?u.41043\ninst✝² : AddCommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na✝ b✝ : α\ns : Finset β\nb : β\ninst✝ : DecidablePred fun x => x = b\na : α\n⊢ a = if b = b then a else 0",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.41040\nδ : Type ?u.41043\ninst✝² : AddCommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na✝ b✝ : α\ns : Finset β\nb : β\ninst✝ : DecidablePred fun x => x = b\na : α\n⊢ HasSum (fun b' => if b' = b then a else 0) a",
"tactic": "convert @hasSum_single _ _ _ _ (fun b' => if b' = b then a else 0) b (fun b' hb' => if_neg hb')"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.41040\nδ : Type ?u.41043\ninst✝² : AddCommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na✝ b✝ : α\ns : Finset β\nb : β\ninst✝ : DecidablePred fun x => x = b\na : α\n⊢ a = if b = b then a else 0",
"tactic": "exact (if_pos rfl).symm"
}
] |
[
211,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
UniformCauchySeqOn.comp
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ✝ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nγ : Type u_1\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\ng : γ → α\n⊢ UniformCauchySeqOnFilter (fun n => F n ∘ g) p (𝓟 (g ⁻¹' s))",
"state_before": "α : Type u\nβ : Type v\nγ✝ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nγ : Type u_1\nhf : UniformCauchySeqOn F p s\ng : γ → α\n⊢ UniformCauchySeqOn (fun n => F n ∘ g) p (g ⁻¹' s)",
"tactic": "rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ✝ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng✝ : ι → α\nγ : Type u_1\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\ng : γ → α\n⊢ UniformCauchySeqOnFilter (fun n => F n ∘ g) p (𝓟 (g ⁻¹' s))",
"tactic": "simpa only [UniformCauchySeqOn, comap_principal] using hf.comp g"
}
] |
[
509,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.list_head?
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.248636\nγ : Type ?u.248639\nσ : Type ?u.248642\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nl : List α\n⊢ (List.casesOn (id l) none fun b l_1 => some (l, b, l_1).snd.fst) = List.head? l",
"tactic": "cases l <;> rfl"
}
] |
[
1043,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_univ
|
[] |
[
665,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
664,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.PolyCnstr.denote_combine
|
[
{
"state_after": "case mk\nctx : Context\nc₂ : PolyCnstr\nh₂ : denote ctx c₂\neq✝ : Bool\nlhs✝ rhs✝ : Poly\nh₁ : denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }\n⊢ denote ctx (combine { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } c₂)",
"state_before": "ctx : Context\nc₁ c₂ : PolyCnstr\nh₁ : denote ctx c₁\nh₂ : denote ctx c₂\n⊢ denote ctx (combine c₁ c₂)",
"tactic": "cases c₁"
},
{
"state_after": "case mk.mk\nctx : Context\neq✝¹ : Bool\nlhs✝¹ rhs✝¹ : Poly\nh₁ : denote ctx { eq := eq✝¹, lhs := lhs✝¹, rhs := rhs✝¹ }\neq✝ : Bool\nlhs✝ rhs✝ : Poly\nh₂ : denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }\n⊢ denote ctx (combine { eq := eq✝¹, lhs := lhs✝¹, rhs := rhs✝¹ } { eq := eq✝, lhs := lhs✝, rhs := rhs✝ })",
"state_before": "case mk\nctx : Context\nc₂ : PolyCnstr\nh₂ : denote ctx c₂\neq✝ : Bool\nlhs✝ rhs✝ : Poly\nh₁ : denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }\n⊢ denote ctx (combine { eq := eq✝, lhs := lhs✝, rhs := rhs✝ } c₂)",
"tactic": "cases c₂"
},
{
"state_after": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : denote ctx { eq := eq₁, lhs := lhs₁, rhs := rhs₁ }\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : denote ctx { eq := eq₂, lhs := lhs₂, rhs := rhs₂ }\n⊢ denote ctx (combine { eq := eq₁, lhs := lhs₁, rhs := rhs₁ } { eq := eq₂, lhs := lhs₂, rhs := rhs₂ })",
"state_before": "case mk.mk\nctx : Context\neq✝¹ : Bool\nlhs✝¹ rhs✝¹ : Poly\nh₁ : denote ctx { eq := eq✝¹, lhs := lhs✝¹, rhs := rhs✝¹ }\neq✝ : Bool\nlhs✝ rhs✝ : Poly\nh₂ : denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }\n⊢ denote ctx (combine { eq := eq✝¹, lhs := lhs✝¹, rhs := rhs✝¹ } { eq := eq✝, lhs := lhs✝, rhs := rhs✝ })",
"tactic": "rename_i eq₁ lhs₁ rhs₁ eq₂ lhs₂ rhs₂"
},
{
"state_after": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : bif eq₁ then Poly.denote_eq ctx (lhs₁, rhs₁) else Poly.denote_le ctx (lhs₁, rhs₁)\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : bif eq₂ then Poly.denote_eq ctx (lhs₂, rhs₂) else Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ denote ctx (combine { eq := eq₁, lhs := lhs₁, rhs := rhs₁ } { eq := eq₂, lhs := lhs₂, rhs := rhs₂ })",
"state_before": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : denote ctx { eq := eq₁, lhs := lhs₁, rhs := rhs₁ }\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : denote ctx { eq := eq₂, lhs := lhs₂, rhs := rhs₂ }\n⊢ denote ctx (combine { eq := eq₁, lhs := lhs₁, rhs := rhs₁ } { eq := eq₂, lhs := lhs₂, rhs := rhs₂ })",
"tactic": "simp [denote] at h₁ h₂"
},
{
"state_after": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : bif eq₁ then Poly.denote_eq ctx (lhs₁, rhs₁) else Poly.denote_le ctx (lhs₁, rhs₁)\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : bif eq₂ then Poly.denote_eq ctx (lhs₂, rhs₂) else Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ bif eq₁ && eq₂ then\n Poly.denote_eq ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)\n else\n Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"state_before": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : bif eq₁ then Poly.denote_eq ctx (lhs₁, rhs₁) else Poly.denote_le ctx (lhs₁, rhs₁)\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : bif eq₂ then Poly.denote_eq ctx (lhs₂, rhs₂) else Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ denote ctx (combine { eq := eq₁, lhs := lhs₁, rhs := rhs₁ } { eq := eq₂, lhs := lhs₂, rhs := rhs₂ })",
"tactic": "simp [PolyCnstr.combine, denote]"
},
{
"state_after": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_eq ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)\n\ncase mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)\n\ncase mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)\n\ncase mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"state_before": "case mk.mk\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\nh₁ : bif eq₁ then Poly.denote_eq ctx (lhs₁, rhs₁) else Poly.denote_le ctx (lhs₁, rhs₁)\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nh₂ : bif eq₂ then Poly.denote_eq ctx (lhs₂, rhs₂) else Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ bif eq₁ && eq₂ then\n Poly.denote_eq ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)\n else\n Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"tactic": "by_cases he₁ : eq₁ = true <;> by_cases he₂ : eq₂ = true <;> simp [he₁, he₂] at h₁ h₂ |-"
},
{
"state_after": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_eq ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"state_before": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_eq ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"tactic": "rw [Poly.denote_eq_cancel_eq]"
},
{
"state_after": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ = Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_eq ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"tactic": "simp [Poly.denote_eq] at h₁ h₂ |-"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.inl.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ = Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "simp [h₁, h₂]"
},
{
"state_after": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"state_before": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"tactic": "rw [Poly.denote_le_cancel_eq]"
},
{
"state_after": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_eq ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"tactic": "simp [Poly.denote_eq, Poly.denote_le] at h₁ h₂ |-"
},
{
"state_after": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx rhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "rw [h₁]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.inl.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ = Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx rhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "apply Nat.add_le_add_left h₂"
},
{
"state_after": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"state_before": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"tactic": "rw [Poly.denote_le_cancel_eq]"
},
{
"state_after": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_eq ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"tactic": "simp [Poly.denote_eq, Poly.denote_le] at h₁ h₂ |-"
},
{
"state_after": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx rhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "rw [h₂]"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.inr.inl\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ = Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx rhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "apply Nat.add_le_add_right h₁"
},
{
"state_after": "case mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"state_before": "case mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx\n ((Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).fst,\n (Poly.cancel (Poly.combine lhs₁ lhs₂) (Poly.combine rhs₁ rhs₂)).snd)",
"tactic": "rw [Poly.denote_le_cancel_eq]"
},
{
"state_after": "case mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"state_before": "case mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote_le ctx (lhs₁, rhs₁)\nh₂ : Poly.denote_le ctx (lhs₂, rhs₂)\n⊢ Poly.denote_le ctx (Poly.combine lhs₁ lhs₂, Poly.combine rhs₁ rhs₂)",
"tactic": "simp [Poly.denote_eq, Poly.denote_le] at h₁ h₂ |-"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.inr.inr\nctx : Context\neq₁ : Bool\nlhs₁ rhs₁ : Poly\neq₂ : Bool\nlhs₂ rhs₂ : Poly\nhe₁ : ¬eq₁ = true\nhe₂ : ¬eq₂ = true\nh₁ : Poly.denote ctx lhs₁ ≤ Poly.denote ctx rhs₁\nh₂ : Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₂\n⊢ Poly.denote ctx lhs₁ + Poly.denote ctx lhs₂ ≤ Poly.denote ctx rhs₁ + Poly.denote ctx rhs₂",
"tactic": "apply Nat.add_le_add h₁ h₂"
}
] |
[
613,
113
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
605,
1
] |
Mathlib/Init/Data/Nat/Bitwise.lean
|
Nat.bit_zero
|
[] |
[
181,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.cauchy_schwarz_aux
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ inner (inner x y • x - inner x x • y) (inner x y • x - inner x x • y) =\n ↑(normSq x * (normSq x * normSq y - ‖inner x y‖ ^ 2))",
"state_before": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ normSq (inner x y • x - inner x x • y) = normSq x * (normSq x * normSq y - ‖inner x y‖ ^ 2)",
"tactic": "rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ inner x y * (↑(starRingEnd 𝕜) (inner x y) * ↑(normSq x)) - ↑(normSq x) * (↑(starRingEnd 𝕜) (inner x y) * inner x y) -\n inner x y * (↑(normSq x) * inner y x) +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x * (normSq x * normSq y) - normSq x * ‖inner x y‖ ^ 2)",
"state_before": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ inner (inner x y • x - inner x x • y) (inner x y • x - inner x x • y) =\n ↑(normSq x * (normSq x * normSq y - ‖inner x y‖ ^ 2))",
"tactic": "simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, ←\n ofReal_normSq_eq_inner_self x, ← ofReal_normSq_eq_inner_self y]"
},
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑(‖inner x y‖ ^ 2) * ↑(normSq x) - ↑(normSq x) * ↑(‖inner x y‖ ^ 2) - ↑(normSq x) * ↑(‖inner x y‖ ^ 2) +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x * (normSq x * normSq y) - normSq x * ‖inner x y‖ ^ 2)",
"state_before": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ inner x y * (↑(starRingEnd 𝕜) (inner x y) * ↑(normSq x)) - ↑(normSq x) * (↑(starRingEnd 𝕜) (inner x y) * inner x y) -\n inner x y * (↑(normSq x) * inner y x) +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x * (normSq x * normSq y) - normSq x * ‖inner x y‖ ^ 2)",
"tactic": "rw [← mul_assoc, mul_conj, IsROrC.conj_mul, normSq_eq_def', mul_left_comm, ← inner_conj_symm y,\n mul_conj, normSq_eq_def']"
},
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑‖inner x y‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ↑‖inner x y‖ ^ 2 - ↑(normSq x) * ↑‖inner x y‖ ^ 2 +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) - ↑(normSq x) * ↑‖inner x y‖ ^ 2",
"state_before": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑(‖inner x y‖ ^ 2) * ↑(normSq x) - ↑(normSq x) * ↑(‖inner x y‖ ^ 2) - ↑(normSq x) * ↑(‖inner x y‖ ^ 2) +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x * (normSq x * normSq y) - normSq x * ‖inner x y‖ ^ 2)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.645181\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\n⊢ ↑‖inner x y‖ ^ 2 * ↑(normSq x) - ↑(normSq x) * ↑‖inner x y‖ ^ 2 - ↑(normSq x) * ↑‖inner x y‖ ^ 2 +\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) =\n ↑(normSq x) * (↑(normSq x) * ↑(normSq y)) - ↑(normSq x) * ↑‖inner x y‖ ^ 2",
"tactic": "ring"
}
] |
[
327,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.discr_comp
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.617859\nR : Type ?u.617862\nR₁ : Type u_1\nM : Type ?u.617868\nn : Type w\ninst✝⁵ : Fintype n\ninst✝⁴ : CommRing R₁\ninst✝³ : DecidableEq n\ninst✝² : Invertible 2\nm : Type w\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nQ : QuadraticForm R₁ (n → R₁)\nf : (n → R₁) →ₗ[R₁] n → R₁\n⊢ discr (comp Q f) = det (↑LinearMap.toMatrix' f) * det (↑LinearMap.toMatrix' f) * discr Q",
"tactic": "simp only [Matrix.det_transpose, mul_left_comm, QuadraticForm.toMatrix'_comp, mul_comm,\n Matrix.det_mul, discr]"
}
] |
[
1041,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1038,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.index
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\n⊢ ∃ n, Subgroup.index H = p ^ n",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\n⊢ ∃ n, Subgroup.index H = p ^ n",
"tactic": "haveI := H.normalCore.fintypeQuotientOfFiniteIndex"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\nn : ℕ\nhn : card (G ⧸ Subgroup.normalCore H) = p ^ n\n⊢ ∃ n, Subgroup.index H = p ^ n",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\n⊢ ∃ n, Subgroup.index H = p ^ n",
"tactic": "obtain ⟨n, hn⟩ := iff_card.mp (hG.to_quotient H.normalCore)"
},
{
"state_after": "case intro.intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\nn : ℕ\nhn : card (G ⧸ Subgroup.normalCore H) = p ^ n\nk : ℕ\nleft✝ : k ≤ n\nhk2 : Subgroup.index H = p ^ k\n⊢ ∃ n, Subgroup.index H = p ^ n",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\nn : ℕ\nhn : card (G ⧸ Subgroup.normalCore H) = p ^ n\n⊢ ∃ n, Subgroup.index H = p ^ n",
"tactic": "obtain ⟨k, _, hk2⟩ :=\n (Nat.dvd_prime_pow hp.out).mp\n ((congr_arg _ (H.normalCore.index_eq_card.trans hn)).mp\n (Subgroup.index_dvd_of_le H.normalCore_le))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nH : Subgroup G\ninst✝ : Subgroup.FiniteIndex H\nthis : Fintype (G ⧸ Subgroup.normalCore H)\nn : ℕ\nhn : card (G ⧸ Subgroup.normalCore H) = p ^ n\nk : ℕ\nleft✝ : k ≤ n\nhk2 : Subgroup.index H = p ^ k\n⊢ ∃ n, Subgroup.index H = p ^ n",
"tactic": "exact ⟨k, hk2⟩"
}
] |
[
145,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_zero
|
[] |
[
280,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_subset_iff₀
|
[] |
[
2053,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2052,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.star_imK
|
[] |
[
1102,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1102,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.CokernelCofork.π_ofπ
|
[] |
[
566,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.L1.SimpleFunc.integral_eq_integral
|
[] |
[
525,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.rescale_injective
|
[
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh : ↑(rescale a) p = ↑(rescale a) q\n⊢ p = q",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\n⊢ Function.Injective ↑(rescale a)",
"tactic": "intro p q h"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nh : ∀ (n : ℕ), ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\n⊢ ∀ (n : ℕ), ↑(coeff R n) p = ↑(coeff R n) q",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh : ↑(rescale a) p = ↑(rescale a) q\n⊢ p = q",
"tactic": "rw [PowerSeries.ext_iff] at *"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nh : ∀ (n : ℕ), ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\nn : ℕ\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nh : ∀ (n : ℕ), ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\n⊢ ∀ (n : ℕ), ↑(coeff R n) p = ↑(coeff R n) q",
"tactic": "intro n"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nh : ∀ (n : ℕ), ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\nn : ℕ\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"tactic": "specialize h n"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) (↑(rescale a) p) = ↑(coeff R n) (↑(rescale a) q)\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"tactic": "rw [coeff_rescale, coeff_rescale, mul_eq_mul_left_iff] at h"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\n⊢ ¬a ^ n = 0",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\n⊢ ↑(coeff R n) p = ↑(coeff R n) q",
"tactic": "apply h.resolve_right"
},
{
"state_after": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\nh' : a ^ n = 0\n⊢ False",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\n⊢ ¬a ^ n = 0",
"tactic": "intro h'"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na : R\nha : a ≠ 0\np q : PowerSeries R\nh✝ : ↑(rescale a) p = ↑(rescale a) q\nn : ℕ\nh : ↑(coeff R n) p = ↑(coeff R n) q ∨ a ^ n = 0\nh' : a ^ n = 0\n⊢ False",
"tactic": "exact ha (pow_eq_zero h')"
}
] |
[
2065,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2057,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.monotone_sizeUpTo
|
[] |
[
254,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.destruct_eq_pure
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ (match ↑s 0 with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a →\n s = pure a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ destruct s = Sum.inl a → s = pure a",
"tactic": "dsimp [destruct]"
},
{
"state_after": "case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nf0 : ↑s 0 = none\nh :\n (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ s = pure a\n\ncase some\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ s = pure a",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\n⊢ (match ↑s 0 with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a →\n s = pure a",
"tactic": "induction' f0 : s.1 0 with _ <;> intro h"
},
{
"state_after": "no goals",
"state_before": "case none\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nf0 : ↑s 0 = none\nh :\n (match none with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ s = pure a",
"tactic": "contradiction"
},
{
"state_after": "case some.a\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ ↑s = ↑(pure a)",
"state_before": "case some\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ s = pure a",
"tactic": "apply Subtype.eq"
},
{
"state_after": "case some.a.h\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : ℕ\n⊢ ↑s n = ↑(pure a) n",
"state_before": "case some.a\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ ↑s = ↑(pure a)",
"tactic": "funext n"
},
{
"state_after": "case some.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ ↑s Nat.zero = ↑(pure a) Nat.zero\n\ncase some.a.h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : ℕ\nIH : ↑s n = ↑(pure a) n\n⊢ ↑s (Nat.succ n) = ↑(pure a) (Nat.succ n)",
"state_before": "case some.a.h\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : ℕ\n⊢ ↑s n = ↑(pure a) n",
"tactic": "induction' n with n IH"
},
{
"state_after": "case some.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh' : val✝ = a\n⊢ ↑s Nat.zero = ↑(pure a) Nat.zero",
"state_before": "case some.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\n⊢ ↑s Nat.zero = ↑(pure a) Nat.zero",
"tactic": "injection h with h'"
},
{
"state_after": "no goals",
"state_before": "case some.a.h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh' : val✝ = a\n⊢ ↑s Nat.zero = ↑(pure a) Nat.zero",
"tactic": "rwa [h'] at f0"
},
{
"state_after": "no goals",
"state_before": "case some.a.h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\na : α\nx✝ : Option α\nf0✝ : ↑s 0 = x✝\nval✝ : α\nf0 : ↑s 0 = some val✝\nh :\n (match some val✝ with\n | none => Sum.inr (tail s)\n | some a => Sum.inl a) =\n Sum.inl a\nn : ℕ\nIH : ↑s n = ↑(pure a) n\n⊢ ↑s (Nat.succ n) = ↑(pure a) (Nat.succ n)",
"tactic": "exact s.2 IH"
}
] |
[
128,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.rightUnitor_naturality
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf g : a ⟶ b\nη : f ⟶ g\n⊢ η ▷ 𝟙 b ≫ (ρ_ g).hom = (ρ_ f).hom ≫ η",
"tactic": "simp"
}
] |
[
405,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/RingTheory/Polynomial/Content.lean
|
Polynomial.isUnit_primPart_C
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : r = 0\n⊢ IsUnit (primPart (↑C r))\n\ncase neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ IsUnit (primPart (↑C r))",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\n⊢ IsUnit (primPart (↑C r))",
"tactic": "by_cases h0 : r = 0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ∃ u, ↑u = primPart (↑C r)",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ IsUnit (primPart (↑C r))",
"tactic": "unfold IsUnit"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n primPart (↑C r)",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ∃ u, ↑u = primPart (↑C r)",
"tactic": "refine'\n ⟨⟨C ↑(normUnit r)⁻¹, C ↑(normUnit r), by rw [← RingHom.map_mul, Units.inv_mul, C_1], by\n rw [← RingHom.map_mul, Units.mul_inv, C_1]⟩,\n _⟩"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n primPart (↑C r)",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n primPart (↑C r)",
"tactic": "rw [← normalize_eq_zero, ← C_eq_zero] at h0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C (↑normalize r) *\n ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n ↑C (↑normalize r) * primPart (↑C r)",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n primPart (↑C r)",
"tactic": "apply mul_left_cancel₀ h0"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C (↑normalize r) *\n ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n ↑C r",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C (↑normalize r) *\n ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n ↑C (↑normalize r) * primPart (↑C r)",
"tactic": "conv_rhs => rw [← content_C, ← (C r).eq_C_content_mul_primPart]"
},
{
"state_after": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C r * ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = ↑C r",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C (↑normalize r) *\n ↑{ val := ↑C ↑(normUnit r)⁻¹, inv := ↑C ↑(normUnit r), val_inv := (_ : ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1),\n inv_val := (_ : ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1) } =\n ↑C r",
"tactic": "simp only [Units.val_mk, normalize_apply, RingHom.map_mul]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0✝ : ¬r = 0\nh0 : ¬↑C (↑normalize r) = 0\n⊢ ↑C r * ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = ↑C r",
"tactic": "rw [mul_assoc, ← RingHom.map_mul, Units.mul_inv, C_1, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : r = 0\n⊢ IsUnit (primPart (↑C r))",
"tactic": "simp [h0]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ↑C ↑(normUnit r)⁻¹ * ↑C ↑(normUnit r) = 1",
"tactic": "rw [← RingHom.map_mul, Units.inv_mul, C_1]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NormalizedGCDMonoid R\nr : R\nh0 : ¬r = 0\n⊢ ↑C ↑(normUnit r) * ↑C ↑(normUnit r)⁻¹ = 1",
"tactic": "rw [← RingHom.map_mul, Units.mul_inv, C_1]"
}
] |
[
308,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.le_of_le_le
|
[] |
[
741,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
738,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zipWith_nil_left
|
[] |
[
47,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.foldr_cons
|
[] |
[
1390,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1389,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_add_left'
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1352463\nG : Type ?u.1352466\n𝕜 : Type ?u.1352469\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nhT'' : DominatedFinMeasAdditive μ T'' C''\nh_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1352463\nG : Type ?u.1352466\n𝕜 : Type ?u.1352469\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nhT'' : DominatedFinMeasAdditive μ T'' C''\nh_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1352463\nG : Type ?u.1352466\n𝕜 : Type ?u.1352469\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nhT'' : DominatedFinMeasAdditive μ T'' C''\nh_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s\nf : α → E\n⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1352463\nG : Type ?u.1352466\n𝕜 : Type ?u.1352469\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nhT'' : DominatedFinMeasAdditive μ T'' C''\nh_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f",
"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_add_left' hT hT' hT'' h_add]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1352463\nG : Type ?u.1352466\n𝕜 : Type ?u.1352469\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nhT'' : DominatedFinMeasAdditive μ T'' C''\nh_add : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T'' s = T s + T' s\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T'' hT'' f = setToFun μ T hT f + setToFun μ T' hT' f",
"tactic": "simp_rw [setToFun_undef _ hf, add_zero]"
}
] |
[
1332,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1326,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.foldr_zero
|
[] |
[
1385,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/Order/BoundedOrder.lean
|
StrictAnti.apply_eq_top_iff
|
[] |
[
204,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/MeasureTheory/Covering/Vitali.lean
|
Vitali.exists_disjoint_subfamily_covering_enlargment_closedBall
|
[
{
"state_after": "case inl\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ ∅ → r a ≤ R\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ ∅ → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\n\ncase inr\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases eq_empty_or_nonempty t with (rfl | _)"
},
{
"state_after": "case pos\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∀ (a : ι), a ∈ t → r a < 0\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\n\ncase neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ¬∀ (a : ι), a ∈ t → r a < 0\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case inr\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "by_cases ht : ∀ a ∈ t, r a < 0"
},
{
"state_after": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ¬∀ (a : ι), a ∈ t → r a < 0\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "push_neg at ht"
},
{
"state_after": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "let t' := { a ∈ t | 0 ≤ r a }"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases exists_disjoint_subfamily_covering_enlargment (fun a => closedBall (x a) (r a)) t' r 2\n one_lt_two (fun a ha => ha.2) R (fun a ha => hr a ha.1) fun a ha =>\n ⟨x a, mem_closedBall_self ha.2⟩ with\n ⟨u, ut', u_disj, hu⟩"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "have A : ∀ a ∈ t', ∃ b ∈ u, closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b) := by\n intro a ha\n rcases hu a ha with ⟨b, bu, hb, rb⟩\n refine' ⟨b, bu, _⟩\n have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb\n apply closedBall_subset_closedBall'\n linarith"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "refine' ⟨u, ut'.trans fun a ha => ha.1, u_disj, fun a ha => _⟩"
},
{
"state_after": "case neg.intro.intro.intro.inl\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : 0 ≤ r a\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\n\ncase neg.intro.intro.intro.inr\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases le_or_lt 0 (r a) with (h'a | h'a)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ ∅ → r a ≤ R\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ ∅ → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "exact ⟨∅, Subset.refl _, pairwiseDisjoint_empty, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ ∅ → r a ≤ R\n⊢ ∀ (a : ι), a ∈ ∅ → ∃ b, b ∈ ∅ ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∀ (a : ι), a ∈ t → r a < 0\n⊢ ∃ u x_1,\n (PairwiseDisjoint u fun a => closedBall (x a) (r a)) ∧\n ∀ (a : ι), a ∈ t → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "exact ⟨t, Subset.rfl, fun a ha b _ _ => by\n simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint],\n fun a ha => ⟨a, ha, by simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∀ (a : ι), a ∈ t → r a < 0\na : ι\nha : a ∈ t\nb : ι\nx✝¹ : b ∈ t\nx✝ : a ≠ b\n⊢ (Disjoint on fun a => closedBall (x a) (r a)) a b",
"tactic": "simp only [Function.onFun, closedBall_eq_empty.2 (ht a ha), empty_disjoint]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∀ (a : ι), a ∈ t → r a < 0\na : ι\nha : a ∈ t\n⊢ closedBall (x a) (r a) ⊆ closedBall (x a) (5 * r a)",
"tactic": "simp only [closedBall_eq_empty.2 (ht a ha), empty_subset]"
},
{
"state_after": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\n⊢ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "intro a ha"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases hu a ha with ⟨b, bu, hb, rb⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\n⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "refine' ⟨b, bu, _⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\nthis : dist (x a) (x b) ≤ r a + r b\n⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\n⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "have : dist (x a) (x b) ≤ r a + r b := dist_le_add_of_nonempty_closedBall_inter_closedBall hb"
},
{
"state_after": "case intro.intro.intro.h\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\nthis : dist (x a) (x b) ≤ r a + r b\n⊢ r a + dist (x a) (x b) ≤ 5 * r b",
"state_before": "case intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\nthis : dist (x a) (x b) ≤ r a + r b\n⊢ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "apply closedBall_subset_closedBall'"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.h\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\na : ι\nha : a ∈ t'\nb : ι\nbu : b ∈ u\nhb : Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b))\nrb : r a ≤ 2 * r b\nthis : dist (x a) (x b) ≤ r a + r b\n⊢ r a + dist (x a) (x b) ≤ 5 * r b",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.inl\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : 0 ≤ r a\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "exact A a ⟨ha, h'a⟩"
},
{
"state_after": "case neg.intro.intro.intro.inr.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\nb : ι\nrb : b ∈ t ∧ 0 ≤ r b\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg.intro.intro.intro.inr\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nht : ∃ a, a ∈ t ∧ 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases ht with ⟨b, rb⟩"
},
{
"state_after": "case neg.intro.intro.intro.inr.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\nb : ι\nrb : b ∈ t ∧ 0 ≤ r b\nc : ι\ncu : c ∈ u\nright✝ : closedBall (x b) (r b) ⊆ closedBall (x c) (5 * r c)\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"state_before": "case neg.intro.intro.intro.inr.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\nb : ι\nrb : b ∈ t ∧ 0 ≤ r b\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.inr.intro.intro.intro\nα : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\nb : ι\nrb : b ∈ t ∧ 0 ≤ r b\nc : ι\ncu : c ∈ u\nright✝ : closedBall (x b) (r b) ⊆ closedBall (x c) (5 * r c)\n⊢ ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)",
"tactic": "refine' ⟨c, cu, by simp only [closedBall_eq_empty.2 h'a, empty_subset]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type u_2\ninst✝ : MetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ (a : ι), a ∈ t → r a ≤ R\nh✝ : Set.Nonempty t\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : PairwiseDisjoint u fun a => closedBall (x a) (r a)\nhu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (closedBall (x a) (r a) ∩ closedBall (x b) (r b)) ∧ r a ≤ 2 * r b\nA : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ closedBall (x a) (r a) ⊆ closedBall (x b) (5 * r b)\na : ι\nha : a ∈ t\nh'a : r a < 0\nb : ι\nrb : b ∈ t ∧ 0 ≤ r b\nc : ι\ncu : c ∈ u\nright✝ : closedBall (x b) (r b) ⊆ closedBall (x c) (5 * r c)\n⊢ closedBall (x a) (r a) ⊆ closedBall (x c) (5 * r c)",
"tactic": "simp only [closedBall_eq_empty.2 h'a, empty_subset]"
}
] |
[
194,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_dvd
|
[] |
[
291,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Topology/Maps.lean
|
IsClosedMap.id
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.198171\nγ : Type ?u.198174\nδ : Type ?u.198177\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ns : Set α\nhs : IsClosed s\n⊢ IsClosed (id '' s)",
"tactic": "rwa [image_id]"
}
] |
[
483,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
11
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mul_div_assoc
|
[
{
"state_after": "no goals",
"state_before": "k n m : Nat\nH : k ∣ n\n⊢ m * n / k = m * (n / k)",
"tactic": "match Nat.eq_zero_or_pos k with\n| .inl h0 => rw [h0, Nat.div_zero, Nat.div_zero, Nat.mul_zero]\n| .inr hpos =>\n have h1 : m * n / k = m * (n / k * k) / k := by rw [Nat.div_mul_cancel H]\n rw [h1, ← Nat.mul_assoc, Nat.mul_div_cancel _ hpos]"
},
{
"state_after": "no goals",
"state_before": "k n m : Nat\nH : k ∣ n\nh0 : k = 0\n⊢ m * n / k = m * (n / k)",
"tactic": "rw [h0, Nat.div_zero, Nat.div_zero, Nat.mul_zero]"
},
{
"state_after": "k n m : Nat\nH : k ∣ n\nhpos : k > 0\nh1 : m * n / k = m * (n / k * k) / k\n⊢ m * n / k = m * (n / k)",
"state_before": "k n m : Nat\nH : k ∣ n\nhpos : k > 0\n⊢ m * n / k = m * (n / k)",
"tactic": "have h1 : m * n / k = m * (n / k * k) / k := by rw [Nat.div_mul_cancel H]"
},
{
"state_after": "no goals",
"state_before": "k n m : Nat\nH : k ∣ n\nhpos : k > 0\nh1 : m * n / k = m * (n / k * k) / k\n⊢ m * n / k = m * (n / k)",
"tactic": "rw [h1, ← Nat.mul_assoc, Nat.mul_div_cancel _ hpos]"
},
{
"state_after": "no goals",
"state_before": "k n m : Nat\nH : k ∣ n\nhpos : k > 0\n⊢ m * n / k = m * (n / k * k) / k",
"tactic": "rw [Nat.div_mul_cancel H]"
}
] |
[
744,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
739,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_id'
|
[] |
[
1248,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1247,
1
] |
Mathlib/Deprecated/Group.lean
|
MulEquiv.isMulHom
|
[] |
[
149,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.nat_cast_val
|
[] |
[
260,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.seq_seq
|
[
{
"state_after": "case refine'_1\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nc : γ\n⊢ c ∈ seq s (seq t u) → c ∈ seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u\n\ncase refine'_2\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nc : γ\n⊢ c ∈ seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u → c ∈ seq s (seq t u)",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\n⊢ seq s (seq t u) = seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u",
"tactic": "refine' Set.ext fun c => Iff.intro _ _"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nf : β → γ\nhfs : f ∈ s\ng : α → β\nhg : g ∈ t\na : α\nhau : a ∈ u\n⊢ f (g a) ∈ seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u",
"state_before": "case refine'_1\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nc : γ\n⊢ c ∈ seq s (seq t u) → c ∈ seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u",
"tactic": "rintro ⟨f, hfs, b, ⟨g, hg, a, hau, rfl⟩, rfl⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nf : β → γ\nhfs : f ∈ s\ng : α → β\nhgt : g ∈ t\na : α\nha : a ∈ u\n⊢ (fun x x_1 => x ∘ x_1) f g a ∈ seq s (seq t u)",
"state_before": "case refine'_2\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nc : γ\n⊢ c ∈ seq (seq ((fun x x_1 => x ∘ x_1) '' s) t) u → c ∈ seq s (seq t u)",
"tactic": "rintro ⟨fg, ⟨fc, ⟨f, hfs, rfl⟩, g, hgt, rfl⟩, a, ha, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Sort ?u.273159\nι' : Sort ?u.273162\nι₂ : Sort ?u.273165\nκ : ι → Sort ?u.273170\nκ₁ : ι → Sort ?u.273175\nκ₂ : ι → Sort ?u.273180\nκ' : ι' → Sort ?u.273185\ns : Set (β → γ)\nt : Set (α → β)\nu : Set α\nf : β → γ\nhfs : f ∈ s\ng : α → β\nhgt : g ∈ t\na : α\nha : a ∈ u\n⊢ (fun x x_1 => x ∘ x_1) f g a ∈ seq s (seq t u)",
"tactic": "exact ⟨f, hfs, g a, ⟨g, hgt, a, ha, rfl⟩, rfl⟩"
}
] |
[
1984,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1978,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.edgeFinset_deleteEdges
|
[
{
"state_after": "case a\nι : Sort ?u.165735\n𝕜 : Type ?u.165738\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne✝ : Sym2 V\ninst✝³ : Fintype (Sym2 V)\ninst✝² : DecidableEq V\ninst✝¹ : DecidableRel G.Adj\ns : Finset (Sym2 V)\ninst✝ : DecidableRel (deleteEdges G ↑s).Adj\ne : Sym2 V\n⊢ e ∈ edgeFinset (deleteEdges G ↑s) ↔ e ∈ edgeFinset G \\ s",
"state_before": "ι : Sort ?u.165735\n𝕜 : Type ?u.165738\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝³ : Fintype (Sym2 V)\ninst✝² : DecidableEq V\ninst✝¹ : DecidableRel G.Adj\ns : Finset (Sym2 V)\ninst✝ : DecidableRel (deleteEdges G ↑s).Adj\n⊢ edgeFinset (deleteEdges G ↑s) = edgeFinset G \\ s",
"tactic": "ext e"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Sort ?u.165735\n𝕜 : Type ?u.165738\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne✝ : Sym2 V\ninst✝³ : Fintype (Sym2 V)\ninst✝² : DecidableEq V\ninst✝¹ : DecidableRel G.Adj\ns : Finset (Sym2 V)\ninst✝ : DecidableRel (deleteEdges G ↑s).Adj\ne : Sym2 V\n⊢ e ∈ edgeFinset (deleteEdges G ↑s) ↔ e ∈ edgeFinset G \\ s",
"tactic": "simp [edgeSet_deleteEdges]"
}
] |
[
1185,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1181,
1
] |
Mathlib/Topology/Order/Basic.lean
|
closure_Ioo
|
[
{
"state_after": "case h₁\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab : a ≠ b\n⊢ closure (Ioo a b) ⊆ Icc a b\n\ncase h₂\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab : a ≠ b\n⊢ Icc a b ⊆ closure (Ioo a b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab : a ≠ b\n⊢ closure (Ioo a b) = Icc a b",
"tactic": "apply Subset.antisymm"
},
{
"state_after": "no goals",
"state_before": "case h₁\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab : a ≠ b\n⊢ closure (Ioo a b) ⊆ Icc a b",
"tactic": "exact closure_minimal Ioo_subset_Icc_self isClosed_Icc"
},
{
"state_after": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\n⊢ Icc a b ⊆ closure (Ioo a b)\n\ncase h₂.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : b < a\n⊢ Icc a b ⊆ closure (Ioo a b)",
"state_before": "case h₂\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab : a ≠ b\n⊢ Icc a b ⊆ closure (Ioo a b)",
"tactic": "cases' hab.lt_or_lt with hab hab"
},
{
"state_after": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\n⊢ {a, b} ⊆ closure (Ioo a b)",
"state_before": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\n⊢ Icc a b ⊆ closure (Ioo a b)",
"tactic": "rw [← diff_subset_closure_iff, Icc_diff_Ioo_same hab.le]"
},
{
"state_after": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\nhab' : Set.Nonempty (Ioo a b)\n⊢ {a, b} ⊆ closure (Ioo a b)",
"state_before": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\n⊢ {a, b} ⊆ closure (Ioo a b)",
"tactic": "have hab' : (Ioo a b).Nonempty := nonempty_Ioo.2 hab"
},
{
"state_after": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\nhab' : Set.Nonempty (Ioo a b)\n⊢ a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)",
"state_before": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\nhab' : Set.Nonempty (Ioo a b)\n⊢ {a, b} ⊆ closure (Ioo a b)",
"tactic": "simp only [insert_subset, singleton_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case h₂.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : a < b\nhab' : Set.Nonempty (Ioo a b)\n⊢ a ∈ closure (Ioo a b) ∧ b ∈ closure (Ioo a b)",
"tactic": "exact ⟨(isGLB_Ioo hab).mem_closure hab', (isLUB_Ioo hab).mem_closure hab'⟩"
},
{
"state_after": "case h₂.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : b < a\n⊢ ∅ ⊆ closure (Ioo a b)",
"state_before": "case h₂.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : b < a\n⊢ Icc a b ⊆ closure (Ioo a b)",
"tactic": "rw [Icc_eq_empty_of_lt hab]"
},
{
"state_after": "no goals",
"state_before": "case h₂.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\na b : α\nhab✝ : a ≠ b\nhab : b < a\n⊢ ∅ ⊆ closure (Ioo a b)",
"tactic": "exact empty_subset _"
}
] |
[
2250,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2241,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.tail_iterate
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn : ℕ\n⊢ nth (tail (iterate f a)) n = nth (iterate f (f a)) n",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\n⊢ tail (iterate f a) = iterate f (f a)",
"tactic": "ext n"
},
{
"state_after": "case a\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn : ℕ\n⊢ nth (iterate f a) (n + 1) = nth (iterate f (f a)) n",
"state_before": "case a\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn : ℕ\n⊢ nth (tail (iterate f a)) n = nth (iterate f (f a)) n",
"tactic": "rw [nth_tail]"
},
{
"state_after": "case a.zero\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\n⊢ nth (iterate f a) (zero + 1) = nth (iterate f (f a)) zero\n\ncase a.succ\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn' : ℕ\nih : nth (iterate f a) (n' + 1) = nth (iterate f (f a)) n'\n⊢ nth (iterate f a) (succ n' + 1) = nth (iterate f (f a)) (succ n')",
"state_before": "case a\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn : ℕ\n⊢ nth (iterate f a) (n + 1) = nth (iterate f (f a)) n",
"tactic": "induction' n with n' ih"
},
{
"state_after": "no goals",
"state_before": "case a.zero\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\n⊢ nth (iterate f a) (zero + 1) = nth (iterate f (f a)) zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case a.succ\nα : Type u\nβ : Type v\nδ : Type w\nf : α → α\na : α\nn' : ℕ\nih : nth (iterate f a) (n' + 1) = nth (iterate f (f a)) n'\n⊢ nth (iterate f a) (succ n' + 1) = nth (iterate f (f a)) (succ n')",
"tactic": "rw [nth_succ_iterate', ih, nth_succ_iterate']"
}
] |
[
281,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventuallyEq_bind
|
[] |
[
2725,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2723,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.image.fac
|
[] |
[
340,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.span_pair_add_mul_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : Semiring α\nI : Ideal α\na b : α\nR : Type u\ninst✝ : CommRing R\nx y z : R\n⊢ span {x, y + x * z} = span {x, y}",
"tactic": "rw [span_pair_comm, span_pair_add_mul_left, span_pair_comm]"
}
] |
[
388,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/NumberTheory/Fermat4.lean
|
Fermat42.exists_pos_odd_minimal
|
[
{
"state_after": "case intro.intro.intro.intro\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"state_before": "a b c : ℤ\nh : Fermat42 a b c\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "obtain ⟨a0, b0, c0, hf, hc⟩ := exists_odd_minimal h"
},
{
"state_after": "case intro.intro.intro.intro.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 < c0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0\n\ncase intro.intro.intro.intro.inr.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 = c0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0\n\ncase intro.intro.intro.intro.inr.inr\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : c0 < 0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"state_before": "case intro.intro.intro.intro\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "rcases lt_trichotomy 0 c0 with (h1 | h1 | h1)"
},
{
"state_after": "case intro.intro.intro.intro.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 < c0\n⊢ Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"state_before": "case intro.intro.intro.intro.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 < c0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "use a0, b0, c0"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 < c0\n⊢ Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "tauto"
},
{
"state_after": "case intro.intro.intro.intro.inr.inl.h\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 = c0\n⊢ False",
"state_before": "case intro.intro.intro.intro.inr.inl\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 = c0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "exfalso"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inr.inl.h\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : 0 = c0\n⊢ False",
"tactic": "exact ne_zero hf.1 h1.symm"
},
{
"state_after": "case intro.intro.intro.intro.inr.inr\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : c0 < 0\n⊢ 0 < -c0",
"state_before": "case intro.intro.intro.intro.inr.inr\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : c0 < 0\n⊢ ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0",
"tactic": "use a0, b0, -c0, neg_of_minimal hf, hc"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.inr.inr\na b c : ℤ\nh : Fermat42 a b c\na0 b0 c0 : ℤ\nhf : Minimal a0 b0 c0\nhc : a0 % 2 = 1\nh1 : c0 < 0\n⊢ 0 < -c0",
"tactic": "exact neg_pos.mpr h1"
}
] |
[
154,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.homogeneousCore_eq_sSup
|
[] |
[
509,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.add_lt_add_of_lt_of_le
|
[] |
[
809,
76
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
808,
11
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.FineSubfamilyOn.exists_disjoint_covering_ae
|
[] |
[
117,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Set/Intervals/Monoid.lean
|
Set.image_add_const_Ico
|
[] |
[
97,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.ext_ring_op
|
[
{
"state_after": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑f (MulOpposite.op x • 1) = ↑g (MulOpposite.op x • 1)",
"state_before": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑f x = ↑g x",
"tactic": "rw [← one_mul x, ← op_smul_eq_mul]"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑σ (MulOpposite.op x) • ↑f 1 = ↑g (MulOpposite.op x • 1)",
"state_before": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑f (MulOpposite.op x • 1) = ↑g (MulOpposite.op x • 1)",
"tactic": "refine (f.map_smulₛₗ (MulOpposite.op x) 1).trans ?_"
},
{
"state_after": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑σ (MulOpposite.op x) • ↑g 1 = ↑g (MulOpposite.op x • 1)",
"state_before": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑σ (MulOpposite.op x) • ↑f 1 = ↑g (MulOpposite.op x • 1)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.223404\nR₂ : Type ?u.223407\nR₃ : Type ?u.223410\nk : Type ?u.223413\nS : Type u_2\nS₃ : Type ?u.223419\nT : Type ?u.223422\nM : Type ?u.223425\nM₁ : Type ?u.223428\nM₂ : Type ?u.223431\nM₃ : Type u_3\nN₁ : Type ?u.223437\nN₂ : Type ?u.223440\nN₃ : Type ?u.223443\nι : Type ?u.223446\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring S\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : AddCommMonoid M₃\ninst✝⁵ : AddCommMonoid N₁\ninst✝⁴ : AddCommMonoid N₂\ninst✝³ : AddCommMonoid N₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module S M₃\nσ✝ : R →+* S\nfₗ gₗ : M →ₗ[R] M₂\nf✝ g✝ : M →ₛₗ[σ✝] M₃\nσ : Rᵐᵒᵖ →+* S\nf g : R →ₛₗ[σ] M₃\nh : ↑f 1 = ↑g 1\nx : R\n⊢ ↑σ (MulOpposite.op x) • ↑g 1 = ↑g (MulOpposite.op x • 1)",
"tactic": "exact (g.map_smulₛₗ (MulOpposite.op x) 1).symm"
}
] |
[
505,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_uIoc
|
[] |
[
254,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.product
|
[] |
[
1119,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/MeasureTheory/Measure/Regular.lean
|
MeasurableSet.exists_lt_isClosed_of_ne_top
|
[] |
[
604,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
1
] |
Mathlib/Geometry/Euclidean/Inversion.lean
|
EuclideanGeometry.inversion_dist_center
|
[
{
"state_after": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d x✝ y z : P\nR : ℝ\nx : P\n⊢ inversion x (dist x x) x = x\n\ncase inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR : ℝ\nc x : P\nhne : x ≠ c\n⊢ inversion c (dist x c) x = x",
"state_before": "V : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR : ℝ\nc x : P\n⊢ inversion c (dist x c) x = x",
"tactic": "rcases eq_or_ne x c with (rfl | hne)"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d x✝ y z : P\nR : ℝ\nx : P\n⊢ inversion x (dist x x) x = x",
"tactic": "apply inversion_self"
},
{
"state_after": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR : ℝ\nc x : P\nhne : x ≠ c\n⊢ dist x c ≠ 0",
"state_before": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR : ℝ\nc x : P\nhne : x ≠ c\n⊢ inversion c (dist x c) x = x",
"tactic": "rw [inversion, div_self, one_pow, one_smul, vsub_vadd]"
},
{
"state_after": "no goals",
"state_before": "case inr\nV : Type u_2\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c✝ d x✝ y z : P\nR : ℝ\nc x : P\nhne : x ≠ c\n⊢ dist x c ≠ 0",
"tactic": "rwa [dist_ne_zero]"
}
] |
[
59,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.multiset_noncommProd_mem
|
[
{
"state_after": "case h\nM : Type u_1\nA : Type ?u.28763\nB : Type ?u.28766\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS✝ : B\ns S : Submonoid M\nm : Multiset M\ncomm✝ : Set.Pairwise {x | x ∈ m} Commute\nh✝ : ∀ (x : M), x ∈ m → x ∈ S\nl : List M\ncomm : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid M) l} Commute\nh : ∀ (x : M), x ∈ Quotient.mk (List.isSetoid M) l → x ∈ S\n⊢ Multiset.noncommProd (Quotient.mk (List.isSetoid M) l) comm ∈ S",
"state_before": "M : Type u_1\nA : Type ?u.28763\nB : Type ?u.28766\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS✝ : B\ns S : Submonoid M\nm : Multiset M\ncomm : Set.Pairwise {x | x ∈ m} Commute\nh : ∀ (x : M), x ∈ m → x ∈ S\n⊢ Multiset.noncommProd m comm ∈ S",
"tactic": "induction' m using Quotient.inductionOn with l"
},
{
"state_after": "case h\nM : Type u_1\nA : Type ?u.28763\nB : Type ?u.28766\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS✝ : B\ns S : Submonoid M\nm : Multiset M\ncomm✝ : Set.Pairwise {x | x ∈ m} Commute\nh✝ : ∀ (x : M), x ∈ m → x ∈ S\nl : List M\ncomm : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid M) l} Commute\nh : ∀ (x : M), x ∈ Quotient.mk (List.isSetoid M) l → x ∈ S\n⊢ List.prod l ∈ S",
"state_before": "case h\nM : Type u_1\nA : Type ?u.28763\nB : Type ?u.28766\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS✝ : B\ns S : Submonoid M\nm : Multiset M\ncomm✝ : Set.Pairwise {x | x ∈ m} Commute\nh✝ : ∀ (x : M), x ∈ m → x ∈ S\nl : List M\ncomm : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid M) l} Commute\nh : ∀ (x : M), x ∈ Quotient.mk (List.isSetoid M) l → x ∈ S\n⊢ Multiset.noncommProd (Quotient.mk (List.isSetoid M) l) comm ∈ S",
"tactic": "simp only [Multiset.quot_mk_to_coe, Multiset.noncommProd_coe]"
},
{
"state_after": "no goals",
"state_before": "case h\nM : Type u_1\nA : Type ?u.28763\nB : Type ?u.28766\ninst✝² : Monoid M\ninst✝¹ : SetLike B M\ninst✝ : SubmonoidClass B M\nS✝ : B\ns S : Submonoid M\nm : Multiset M\ncomm✝ : Set.Pairwise {x | x ∈ m} Commute\nh✝ : ∀ (x : M), x ∈ m → x ∈ S\nl : List M\ncomm : Set.Pairwise {x | x ∈ Quotient.mk (List.isSetoid M) l} Commute\nh : ∀ (x : M), x ∈ Quotient.mk (List.isSetoid M) l → x ∈ S\n⊢ List.prod l ∈ S",
"tactic": "exact Submonoid.list_prod_mem _ h"
}
] |
[
158,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Order/SemiconjSup.lean
|
isOrderRightAdjoint_sSup
|
[] |
[
52,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sub_lt_of_le
|
[] |
[
580,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/CategoryTheory/Limits/Presheaf.lean
|
CategoryTheory.ColimitAdj.extendAlongYoneda_map
|
[
{
"state_after": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫ (extendAlongYoneda A).map f =\n colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫\n colimit.pre ((CategoryOfElements.π Y).leftOp ⋙ A) (Functor.op (CategoryOfElements.map f))",
"state_before": "C : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\n⊢ (extendAlongYoneda A).map f =\n colimit.pre ((CategoryOfElements.π Y).leftOp ⋙ A) (Functor.op (CategoryOfElements.map f))",
"tactic": "ext J"
},
{
"state_after": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫ (extendAlongYoneda A).map f =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"state_before": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫ (extendAlongYoneda A).map f =\n colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫\n colimit.pre ((CategoryOfElements.π Y).leftOp ⋙ A) (Functor.op (CategoryOfElements.map f))",
"tactic": "erw [colimit.ι_pre ((CategoryOfElements.π Y).leftOp ⋙ A) (CategoryOfElements.map f).op]"
},
{
"state_after": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫\n (Adjunction.leftAdjointOfEquiv\n (fun P E =>\n ((Iso.mk (fun a => a.down) fun a => { down := a }).symm ≪≫\n (Iso.mk (fun f => (Cocone.extend (colimit.cocone ((CategoryOfElements.π P).leftOp ⋙ A)) f.down).ι)\n fun ι =>\n {\n down :=\n IsColimit.desc (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))\n { pt := E, ι := ι } }) ≪≫\n Iso.mk\n (fun ι =>\n { val := fun j => ι.app j,\n property :=\n (_ :\n ∀ {j j' : (Functor.Elements P)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ (fun j => ι.app j) j' =\n (fun j => ι.app j) j) })\n fun p => NatTrans.mk fun j => ↑p j).trans\n { toFun := fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op,\n invFun := fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) },\n left_inv :=\n (_ :\n ∀\n (x :\n { p //\n ∀ {j j' : (Functor.Elements P)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ p j' = p j }),\n (fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫\n (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) })\n ((fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op) x) =\n x),\n right_inv :=\n (_ :\n ∀ (x : P ⟶ (restrictedYoneda A).obj E),\n (fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op)\n ((fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫\n (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) })\n x) =\n x) })\n (_ :\n ∀ (P : Cᵒᵖ ⥤ Type u₁) (E E' : ℰ) (g : E ⟶ E')\n (k : (colimit.cocone ((CategoryOfElements.π P).leftOp ⋙ A)).pt ⟶ E),\n ↑(restrictYonedaHomEquiv A P E' (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))) (k ≫ g) =\n ↑(restrictYonedaHomEquiv A P E (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))) k ≫\n (restrictedYoneda A).map g)).map\n f =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"state_before": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫ (extendAlongYoneda A).map f =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"tactic": "dsimp only [extendAlongYoneda, restrictYonedaHomEquiv, IsColimit.homIso', IsColimit.homIso,\n uliftTrivial]"
},
{
"state_after": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ (((fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op) ∘\n ((fun ι =>\n { val := fun j => ι.app j,\n property :=\n (_ :\n ∀ {j j' : (Functor.Elements Y)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π Y).leftOp ⋙ A).map f ≫ (fun j => ι.app j) j' =\n (fun j => ι.app j) j) }) ∘\n fun f =>\n (colimit.cocone ((CategoryOfElements.π Y).leftOp ⋙ A)).ι ≫\n (Functor.const (Functor.Elements Y)ᵒᵖ).map f.down) ∘\n fun a => { down := a })\n (𝟙 (colimit.cocone ((CategoryOfElements.π Y).leftOp ⋙ A)).pt)).app\n J.unop.fst (f.app J.unop.fst J.unop.snd) =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"state_before": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π X).leftOp ⋙ A) J ≫\n (Adjunction.leftAdjointOfEquiv\n (fun P E =>\n ((Iso.mk (fun a => a.down) fun a => { down := a }).symm ≪≫\n (Iso.mk (fun f => (Cocone.extend (colimit.cocone ((CategoryOfElements.π P).leftOp ⋙ A)) f.down).ι)\n fun ι =>\n {\n down :=\n IsColimit.desc (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))\n { pt := E, ι := ι } }) ≪≫\n Iso.mk\n (fun ι =>\n { val := fun j => ι.app j,\n property :=\n (_ :\n ∀ {j j' : (Functor.Elements P)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ (fun j => ι.app j) j' =\n (fun j => ι.app j) j) })\n fun p => NatTrans.mk fun j => ↑p j).trans\n { toFun := fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op,\n invFun := fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) },\n left_inv :=\n (_ :\n ∀\n (x :\n { p //\n ∀ {j j' : (Functor.Elements P)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫ p j' = p j }),\n (fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫\n (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) })\n ((fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op) x) =\n x),\n right_inv :=\n (_ :\n ∀ (x : P ⟶ (restrictedYoneda A).obj E),\n (fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op)\n ((fun τ =>\n { val := fun p => τ.app p.unop.fst p.unop.snd,\n property :=\n (_ :\n ∀ (p p' : (Functor.Elements P)ᵒᵖ) (f : p ⟶ p'),\n ((CategoryOfElements.π P).leftOp ⋙ A).map f ≫\n (fun p => τ.app p.unop.fst p.unop.snd) p' =\n (fun p => τ.app p.unop.fst p.unop.snd) p) })\n x) =\n x) })\n (_ :\n ∀ (P : Cᵒᵖ ⥤ Type u₁) (E E' : ℰ) (g : E ⟶ E')\n (k : (colimit.cocone ((CategoryOfElements.π P).leftOp ⋙ A)).pt ⟶ E),\n ↑(restrictYonedaHomEquiv A P E' (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))) (k ≫ g) =\n ↑(restrictYonedaHomEquiv A P E (colimit.isColimit ((CategoryOfElements.π P).leftOp ⋙ A))) k ≫\n (restrictedYoneda A).map g)).map\n f =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"tactic": "simp only [Adjunction.leftAdjointOfEquiv_map, Iso.symm_mk, Iso.toEquiv_comp, Equiv.coe_trans,\n Equiv.coe_fn_mk, Iso.toEquiv_fun, Equiv.symm_trans_apply, Equiv.coe_fn_symm_mk,\n Iso.toEquiv_symm_fun, id.def, colimit.isColimit_desc, colimit.ι_desc, FunctorToTypes.comp,\n Cocone.extend_ι, Cocone.extensions_app, Functor.map_id, Category.comp_id, colimit.cocone_ι]"
},
{
"state_after": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) { fst := J.unop.fst, snd := f.app J.unop.fst J.unop.snd }.op =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((CategoryOfElements.map f).obj J.unop).op",
"state_before": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ (((fun k => NatTrans.mk fun c p => ↑k { fst := c, snd := p }.op) ∘\n ((fun ι =>\n { val := fun j => ι.app j,\n property :=\n (_ :\n ∀ {j j' : (Functor.Elements Y)ᵒᵖ} (f : j ⟶ j'),\n ((CategoryOfElements.π Y).leftOp ⋙ A).map f ≫ (fun j => ι.app j) j' =\n (fun j => ι.app j) j) }) ∘\n fun f =>\n (colimit.cocone ((CategoryOfElements.π Y).leftOp ⋙ A)).ι ≫\n (Functor.const (Functor.Elements Y)ᵒᵖ).map f.down) ∘\n fun a => { down := a })\n (𝟙 (colimit.cocone ((CategoryOfElements.π Y).leftOp ⋙ A)).pt)).app\n J.unop.fst (f.app J.unop.fst J.unop.snd) =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((Functor.op (CategoryOfElements.map f)).obj J)",
"tactic": "simp only [Functor.comp_obj, Functor.leftOp_obj, CategoryOfElements.π_obj, colimit.cocone_x,\n Functor.comp_map, Functor.leftOp_map, CategoryOfElements.π_map, Opposite.unop_op,\n Adjunction.leftAdjointOfEquiv_obj, Function.comp_apply, Functor.map_id, comp_id,\n colimit.cocone_ι, Functor.op_obj]"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝² : SmallCategory C\nℰ : Type u₂\ninst✝¹ : Category ℰ\nA : C ⥤ ℰ\ninst✝ : HasColimits ℰ\nX Y : Cᵒᵖ ⥤ Type u₁\nf : X ⟶ Y\nJ : (Functor.Elements X)ᵒᵖ\n⊢ colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) { fst := J.unop.fst, snd := f.app J.unop.fst J.unop.snd }.op =\n colimit.ι ((CategoryOfElements.π Y).leftOp ⋙ A) ((CategoryOfElements.map f).obj J.unop).op",
"tactic": "rfl"
}
] |
[
175,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Init/Data/Nat/GCD.lean
|
Nat.gcd_def
|
[
{
"state_after": "no goals",
"state_before": "x y : ℕ\n⊢ gcd x y = if x = 0 then y else gcd (y % x) x",
"tactic": "cases x <;> simp [Nat.gcd_succ]"
}
] |
[
38,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
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