mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
ha'' : ∀ (j : J), ∃ g, (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = g ≫ (T j).mor₂
b : (j : J) → A ⟶ (T j).obj₂ := fun j ↦ ⋯.choose
hb : ∀ (j : J), (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = b j ≫ (T j).mor₂
j : J
⊢ b j ≫ (T j).mor₂ = (Pi.lift b ≫ (productTriangle T).mor₂) ≫ Pi.π (fun j ↦ (T j).obj₃) j

Tactic:
;

Goal state after:
C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A : C
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
ha'' : ∀ (j : J), ∃ g, (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = g ≫ (T j).mor₂
b : (j : J) → A ⟶ (T j).obj₂ := fun j ↦ ⋯.choose
hb : ∀ (j : J), (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun j ↦ (T j).obj₃) j = b j ≫ (T j).mor₂
j : J
⊢ b j ≫ (T j).mor₂ = (Pi.lift b ≫ (productTriangle T).mor₂) ≫ Pi.π (fun j ↦ (T j).obj₃) j
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₁))

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₁))
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₃))

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝²⁰ : Category.{v_1, u_1} C
inst✝¹⁹ : Category.{v_2, u_2} D
inst✝¹⁸ : Category.{v_3, u_3} E
inst✝¹⁷ : HasShift C ℤ
inst✝¹⁶ : HasShift D ℤ
inst✝¹⁵ : HasShift E ℤ
F : C ⥤ D
inst✝¹⁴ : F.CommShift ℤ
G : D ⥤ E
inst✝¹³ : G.CommShift ℤ
inst✝¹² : HasZeroObject C
inst✝¹¹ : HasZeroObject D
inst✝¹⁰ : HasZeroObject E
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : Preadditive E
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor E n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
inst✝¹ : Pretriangulated E
inst✝ : F.IsTriangulated
X₁ X₃ : C
φ : F.mapTriangle.obj (binaryProductTriangle X₁ X₃) ⟶ binaryProductTriangle (F.obj X₁) (F.obj X₃) :=
  { hom₁ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₁, hom₂ := prodComparison F X₁ X₃,
    hom₃ := 𝟙 (F.mapTriangle.obj (binaryProductTriangle X₁ X₃)).obj₃, comm₁ := ⋯, comm₂ := ⋯, comm₃ := ⋯ }
⊢ IsIso (𝟙 (F.obj X₃))
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
F : C ⥤ D
inst✝¹ : F.CommShift ℤ
inst✝ : F.IsTriangulated
⊢ F.map u₁₂ ≫ F.map u₂₃ = F.map u₁₃

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : HasShift C ℤ
inst✝¹⁰ : HasShift D ℤ
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasZeroObject D
inst✝⁷ : Preadditive C
inst✝⁶ : Preadditive D
inst✝⁵ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝³ : Pretriangulated C
inst✝² : Pretriangulated D
X₁ X₂ X₃ Z₁₂ Z₂₃ Z₁₃ : C
u₁₂ : X₁ ⟶ X₂
u₂₃ : X₂ ⟶ X₃
u₁₃ : X₁ ⟶ X₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : X₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor C 1).obj X₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : X₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor C 1).obj X₂
h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles
v₁₃ : X₃ ⟶ Z₁₃
w₁₃ : Z₁₃ ⟶ (shiftFunctor C 1).obj X₁
h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles
h : Octahedron comm h₁₂ h₂₃ h₁₃
F : C ⥤ D
inst✝¹ : F.CommShift ℤ
inst✝ : F.IsTriangulated
⊢ F.map u₁₂ ≫ F.map u₂₃ = F.map u₁₃
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
⊢ IsIso (Arrow.Hom.left e.hom)

Tactic:
;

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
⊢ IsIso (Arrow.Hom.left e.hom)
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
this : IsIso φ.hom₁
⊢ IsIso (Arrow.Hom.right e.hom)

Tactic:
;

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : Preadditive C
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
T₁ T₂ : Triangle C
hT₁ : T₁ ∈ distinguishedTriangles
hT₂ : T₂ ∈ distinguishedTriangles
e : Arrow.mk T₁.mor₁ ≅ Arrow.mk T₂.mor₁
φ : T₁ ⟶ T₂ := completeDistinguishedTriangleMorphism T₁ T₂ hT₁ hT₂ (Arrow.Hom.left e.hom) (Arrow.Hom.right e.hom) ⋯
this : IsIso φ.hom₁
⊢ IsIso (Arrow.Hom.right e.hom)
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ homOfDegreewiseSplit S σ ≫ 𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) =
    𝟙 S.X₃ ≫ homOfDegreewiseSplit S σ

Tactic:
;

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ homOfDegreewiseSplit S σ ≫ 𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) =
    𝟙 S.X₃ ≫ homOfDegreewiseSplit S σ
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.f) ≫ (mappingConeHomOfDegreewiseSplitIso S σ).inv =
    𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) ≫ mappingCone.inr (homOfDegreewiseSplit S σ)

Tactic:
;

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.f) ≫ (mappingConeHomOfDegreewiseSplitIso S σ).inv =
    𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).obj S.X₁) ≫ mappingCone.inr (homOfDegreewiseSplit S σ)
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.g) ≫
      (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (𝟙 S.X₃) =
    (mappingConeHomOfDegreewiseSplitIso S σ).inv ≫ (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₃

Tactic:
;

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
⊢ (-(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map S.g) ≫
      (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (𝟙 S.X₃) =
    (mappingConeHomOfDegreewiseSplitIso S σ).inv ≫ (mappingCone.triangle (homOfDegreewiseSplit S σ)).mor₃
Module: Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
K L : CochainComplex C ℤ
φ : K ⟶ L
i✝ : ℤ
⊢ ((triangle φ).rotate.mor₃ ≫
          (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (Iso.refl (triangle φ).rotate.obj₁).hom).f
      i✝ =
    ((Iso.refl (triangle φ).rotate.obj₃).hom ≫
          (triangleOfDegreewiseSplit (triangleRotateShortComplex φ) (triangleRotateShortComplexSplitting φ)).mor₃).f
      i✝

Tactic:
;

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex (CochainComplex C ℤ)
σ : (n : ℤ) → (S.map (eval C (ComplexShape.up ℤ) n)).Splitting
inst✝ : HasBinaryBiproducts C
K L : CochainComplex C ℤ
φ : K ⟶ L
i✝ : ℤ
⊢ ((triangle φ).rotate.mor₃ ≫
          (CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map (Iso.refl (triangle φ).rotate.obj₁).hom).f
      i✝ =
    ((Iso.refl (triangle φ).rotate.obj₃).hom ≫
          (triangleOfDegreewiseSplit (triangleRotateShortComplex φ) (triangleRotateShortComplexSplitting φ)).mor₃).f
      i✝
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
a : Paths V
P : {b : Paths V} → (a ⟶ b) → Prop
id : P (𝟙 a)
comp : ∀ {u v : V} (p : a ⟶ (of V).obj u) (q : u ⟶ v), P p → P (p ≫ (of V).map q)
⊢ ∀ {b : Paths V} (f : a ⟶ b), P f

Tactic:
intro _ f

Goal state after:
V : Type u₁
inst✝ : Quiver V
a : Paths V
P : {b : Paths V} → (a ⟶ b) → Prop
id : P (𝟙 a)
comp : ∀ {u v : V} (p : a ⟶ (of V).obj u) (q : u ⟶ v), P p → P (p ≫ (of V).map q)
b✝ : Paths V
f : a ⟶ b✝
⊢ P f
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ Epi (S.g.f i)

Tactic:
;

Goal state after:
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Epi S.g
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ Epi (S.g.f i)
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ Mono (S.f.f i)

Tactic:
;

Goal state after:
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
⊢ Mono (S.f.f i)
Module: Mathlib.Algebra.Homology.HomologySequence

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
⊢ { X₁ := S.X₁.cycles i, X₂ := S.X₂.cycles i, X₃ := S.X₃.cycles i, f := cyclesMap S.f i, g := cyclesMap S.g i,
      zero := ⋯ }.Exact

Tactic:
apply ShortComplex.exact_of_f_is_kernel

Goal state after:
C : Type u_1
ι : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.Exact
inst✝³ : Mono S.f
i : ι
inst✝² : S.X₁.HasHomology i
inst✝¹ : S.X₂.HasHomology i
inst✝ : S.X₃.HasHomology i
this : Mono (S.map (eval C c i)).f
hi : IsLimit (KernelFork.ofι (S.map (eval C c i)).f ⋯)
⊢ IsLimit
    (KernelFork.ofι
      { X₁ := S.X₁.cycles i, X₂ := S.X₂.cycles i, X₃ := S.X₃.cycles i, f := cyclesMap S.f i, g := cyclesMap S.g i,
          zero := ⋯ }.f
      ⋯)
Module: Mathlib.CategoryTheory.PathCategory.MorphismProperty

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : (of V).obj u ⟶ (of V).obj v) (q : v ⟶ w), P p → P (p ≫ (of V).map q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝

Tactic:
;

Goal state after:
V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : (of V).obj u ⟶ (of V).obj v) (q : v ⟶ w), P p → P (p ≫ (of V).map q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝
Module: Mathlib.CategoryTheory.PathCategory.MorphismProperty

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝

Tactic:
;

Goal state after:
V : Type u₁
inst✝ : Quiver V
P : MorphismProperty (Paths V)
id : ∀ {v : V}, P (𝟙 ((of V).obj v))
comp : ∀ {u v w : V} (p : u ⟶ v) (q : (of V).obj v ⟶ (of V).obj w), P q → P ((of V).map p ≫ q)
X✝ Y✝ : Paths V
f✝ : X✝ ⟶ Y✝
⊢ P f✝ ↔ ⊤ f✝
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: case mk
C : Type u₁
inst✝ : Category.{v₁, u₁} C
as✝ : Paths C
⊢ (𝟭 (Quotient (pathsHomRel C))).obj { as := as✝ } ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj { as := as✝ }

Tactic:
;

Goal state after:
case mk
C : Type u₁
inst✝ : Category.{v₁, u₁} C
as✝ : Paths C
⊢ (𝟭 (Quotient (pathsHomRel C))).obj { as := as✝ } ≅ (quotientPathsTo C ⋙ toQuotientPaths C).obj { as := as✝ }
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
X Y : C
w : X ⟶ Y
hw : W w
⊢ { obj := Y } ⟶ { obj := X }

Tactic:
;

Goal state after:
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
X Y : C
w : X ⟶ Y
hw : W w
⊢ { obj := Y } ⟶ { obj := X }
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ (wIso f hf).hom = (isoOfHom W.Q W f hf).hom

Tactic:
;

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
W : MorphismProperty C
X Y : C
f : X ⟶ Y
hf : W f
⊢ (wIso f hf).hom = (isoOfHom W.Q W f hf).hom
Module: Mathlib.CategoryTheory.CatCommSq

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ≌ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ≌ C₄
h : CatCommSq T.functor L R B.functor
X : C₁
⊢ (iso T.symm.inverse L R B.symm.inverse).hom.app X = (iso T.symm.inverse L R B.symm.inverse).hom.app X

Tactic:
rw [← cancel_mono (B.functor.map (L.map (T.unitIso.hom.app X)))]

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} C₃
inst✝ : Category.{v_4, u_4} C₄
T : C₁ ≌ C₂
L : C₁ ⥤ C₃
R : C₂ ⥤ C₄
B : C₃ ≌ C₄
h : CatCommSq T.functor L R B.functor
X : C₁
⊢ (iso T.symm.inverse L R B.symm.inverse).hom.app X ≫ B.functor.map (L.map (T.unitIso.hom.app X)) =
    (iso T.symm.inverse L R B.symm.inverse).hom.app X ≫ B.functor.map (L.map (T.unitIso.hom.app X))
Module: Mathlib.CategoryTheory.Localization.Equivalence

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} D
L₁ : C₁ ⥤ D
W₁ : MorphismProperty C₁
L₂ : C₂ ⥤ D
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
inst✝ : L₁.IsLocalization W₁
iso : E.functor ⋙ L₂ ≅ L₁
x✝¹ x✝ : C₁
f : x✝¹ ⟶ x✝
hf : W₁ f
⊢ IsIso ((E.functor ⋙ W₂.Q).map f)

Tactic:
obtain ⟨_, _, f', hf', ⟨e⟩⟩ := hW₁ f hf

Goal state after:
C₁ : Type u_1
C₂ : Type u_2
D : Type u_3
inst✝³ : Category.{v_1, u_1} C₁
inst✝² : Category.{v_2, u_2} C₂
inst✝¹ : Category.{v_3, u_3} D
L₁ : C₁ ⥤ D
W₁ : MorphismProperty C₁
L₂ : C₂ ⥤ D
W₂ : MorphismProperty C₂
E : C₁ ≌ C₂
hW₁ : W₁ ≤ W₂.isoClosure.inverseImage E.functor
hW₂ : W₂.IsInvertedBy L₂
inst✝ : L₁.IsLocalization W₁
iso : E.functor ⋙ L₂ ≅ L₁
x✝¹ x✝ : C₁
f : x✝¹ ⟶ x✝
hf : W₁ f
w✝¹ w✝ : C₂
f' : w✝¹ ⟶ w✝
hf' : W₂ f'
e : Arrow.mk f' ≅ Arrow.mk (E.functor.map f)
⊢ IsIso ((E.functor ⋙ W₂.Q).map f)
Module: Mathlib.CategoryTheory.Localization.Opposite

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X Y : Cᵒᵖ
w : X ⟶ Y
hw : W.op w
⊢ (isoOfHom L.op W.op w hw).unop.hom = (isoOfHom L W w.unop hw).hom

Tactic:
;

Goal state after:
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
inst✝ : L.IsLocalization W
X Y : Cᵒᵖ
w : X ⟶ Y
hw : W.op w
⊢ (isoOfHom L.op W.op w hw).unop.hom = (isoOfHom L W w.unop hw).hom
Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions.Preadditive

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ L.map φ₀.neg.f ≫ L.map t₁ = L.map φ.neg.f ≫ L.map t₂

Tactic:
← L.map_comp,

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
L : C ⥤ D
W : MorphismProperty C
inst✝¹ : L.IsLocalization W
inst✝ : W.HasLeftCalculusOfFractions
X Y : C
φ φ₀ : W.LeftFraction X Y
hφ₀ : neg' W (φ₀.map L ⋯) = φ₀.neg.map L ⋯
Y' : C
t₁ : φ₀.Y' ⟶ Y'
t₂ : φ.Y' ⟶ Y'
hst : φ₀.s ≫ t₁ = φ.s ≫ t₂
hft : φ₀.f ≫ t₁ = φ.f ≫ t₂
ht : W (φ₀.s ≫ t₁)
this : IsIso (L.map (φ₀.s ≫ t₁))
⊢ L.map (φ₀.neg.f ≫ t₁) = L.map φ.neg.f ≫ L.map t₂