mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Quotient.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
a b : R
f : (functor r).obj X ⟶ (functor r).obj Y
⊢ (a + b) • f = a • f + b • f

Tactic:
obtain ⟨f, rfl⟩ := (functor r).map_surjective f

Goal state after:
R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
a b : R
f : X ⟶ Y
⊢ (a + b) • (functor r).map f = a • (functor r).map f + b • (functor r).map f
Module: Mathlib.CategoryTheory.Quotient.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
f : (functor r).obj X ⟶ (functor r).obj Y
⊢ 0 • f = 0

Tactic:
obtain ⟨f, rfl⟩ := (functor r).map_surjective f

Goal state after:
R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
X Y : C
smul : SMul R ((functor r).obj X ⟶ (functor r).obj Y) := Linear.smul r hr ((functor r).obj X) ((functor r).obj Y)
f : X ⟶ Y
⊢ 0 • (functor r).map f = 0
Module: Mathlib.CategoryTheory.Quotient.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
this : (X Y : Quotient r) → Module R (X ⟶ Y) := Linear.module r hr
X Y Z : C
a : R
f : { as := X } ⟶ { as := Y }
g : { as := Y } ⟶ { as := Z }
⊢ (a • f) ≫ g = a • f ≫ g

Tactic:
obtain ⟨f, rfl⟩ := (functor r).map_surjective f

Goal state after:
R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
this : (X Y : Quotient r) → Module R (X ⟶ Y) := Linear.module r hr
X Y Z : C
a : R
g : { as := Y } ⟶ { as := Z }
f : X ⟶ Y
⊢ (a • (functor r).map f) ≫ g = a • (functor r).map f ≫ g
Module: Mathlib.CategoryTheory.Quotient.Linear

Goal state before:
pp: R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
this : (X Y : Quotient r) → Module R (X ⟶ Y) := Linear.module r hr
X Y Z : C
f : { as := X } ⟶ { as := Y }
a : R
g : { as := Y } ⟶ { as := Z }
⊢ f ≫ (a • g) = a • f ≫ g

Tactic:
obtain ⟨f, rfl⟩ := (functor r).map_surjective f

Goal state after:
R : Type u_1
C : Type u_2
inst✝⁶ : Semiring R
inst✝⁵ : Category.{v_1, u_2} C
inst✝⁴ : Preadditive C
inst✝³ : Linear R C
r : HomRel C
inst✝² : Congruence r
hr : ∀ (a : R) ⦃X Y : C⦄ (f₁ f₂ : X ⟶ Y), r f₁ f₂ → r (a • f₁) (a • f₂)
inst✝¹ : Preadditive (Quotient r)
inst✝ : (functor r).Additive
this : (X Y : Quotient r) → Module R (X ⟶ Y) := Linear.module r hr
X Y Z : C
a : R
g : { as := Y } ⟶ { as := Z }
f : X ⟶ Y
⊢ (functor r).map f ≫ (a • g) = a • (functor r).map f ≫ g
Module: Mathlib.CategoryTheory.Shift.Induced

Goal state before:
pp: C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
n : A
X : C
⊢ eqToHom ⋯ ≫
      (i n).hom.app X ≫
        eqToHom ⋯ ≫
          F.map ((shiftFunctorAdd C n 0).hom.app X) ≫
            (i 0).inv.app ((shiftFunctor C n).obj X) ≫ (s 0).map ((i n).inv.app X) ≫ (s 0).map ((i n).hom.app X) =
    (eqToHom ⋯).app (F.obj X) ≫ (Induced.zero F s i).inv.app ((s n).obj (F.obj X)) ≫ (s 0).map ((i n).hom.app X)

Tactic:
← (s 0).map_comp,

Goal state after:
C : Type ?u.2
D : Type ?u.4
inst✝⁵ : Category.{v_1, ?u.2} C
inst✝⁴ : Category.{v_2, ?u.4} D
F : C ⥤ D
A : Type ?u.15
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
n : A
X : C
⊢ eqToHom ⋯ ≫
      (i n).hom.app X ≫
        eqToHom ⋯ ≫
          F.map ((shiftFunctorAdd C n 0).hom.app X) ≫
            (i 0).inv.app ((shiftFunctor C n).obj X) ≫ (s 0).map ((i n).inv.app X ≫ (i n).hom.app X) =
    (eqToHom ⋯).app (F.obj X) ≫ (Induced.zero F s i).inv.app ((s n).obj (F.obj X)) ≫ (s 0).map ((i n).hom.app X)
Module: Mathlib.CategoryTheory.Shift.CommShift

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
J : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} E
inst✝¹² : Category.{v_4, u_4} J
F₁ F₂ F₃ : C ⥤ D
τ : F₁ ⟶ F₂
τ' : F₂ ⟶ F₃
e : F₁ ≅ F₂
G G' : D ⥤ E
τ'' : G ⟶ G'
H : E ⥤ J
A : Type u_5
inst✝¹¹ : AddMonoid A
inst✝¹⁰ : HasShift C A
inst✝⁹ : HasShift D A
inst✝⁸ : HasShift E A
inst✝⁷ : HasShift J A
inst✝⁶ : F₁.CommShift A
inst✝⁵ : F₂.CommShift A
inst✝⁴ : F₃.CommShift A
inst✝³ : G.CommShift A
inst✝² : G'.CommShift A
inst✝¹ : H.CommShift A
inst✝ : CommShift e.hom A
a : A
X : C
⊢ (Functor.commShiftIso F₂ a).hom.app X ≫ (shiftFunctor D a).map (e.inv.app X) =
    e.inv.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso F₁ a).hom.app X

Tactic:
← cancel_epi (e.hom.app (X⟦a⟧)),

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
J : Type u_4
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : Category.{v_3, u_3} E
inst✝¹² : Category.{v_4, u_4} J
F₁ F₂ F₃ : C ⥤ D
τ : F₁ ⟶ F₂
τ' : F₂ ⟶ F₃
e : F₁ ≅ F₂
G G' : D ⥤ E
τ'' : G ⟶ G'
H : E ⥤ J
A : Type u_5
inst✝¹¹ : AddMonoid A
inst✝¹⁰ : HasShift C A
inst✝⁹ : HasShift D A
inst✝⁸ : HasShift E A
inst✝⁷ : HasShift J A
inst✝⁶ : F₁.CommShift A
inst✝⁵ : F₂.CommShift A
inst✝⁴ : F₃.CommShift A
inst✝³ : G.CommShift A
inst✝² : G'.CommShift A
inst✝¹ : H.CommShift A
inst✝ : CommShift e.hom A
a : A
X : C
⊢ e.hom.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso F₂ a).hom.app X ≫ (shiftFunctor D a).map (e.inv.app X) =
    e.hom.app ((shiftFunctor C a).obj X) ≫ e.inv.app ((shiftFunctor C a).obj X) ≫ (Functor.commShiftIso F₁ a).hom.app X
Module: Mathlib.CategoryTheory.Triangulated.TriangleShift

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
n : ℤ
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ n.negOnePow •
      ((CategoryTheory.shiftFunctor C n).map X✝.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app X✝.obj₁) ≫
        (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map f.hom₁) =
    n.negOnePow • (CategoryTheory.shiftFunctor C n).map (f.hom₃ ≫ Y✝.mor₃) ≫ (shiftFunctorComm C 1 n).hom.app Y✝.obj₁

Tactic:
← f.comm₃,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive
n : ℤ
X✝ Y✝ : Triangle C
f : X✝ ⟶ Y✝
⊢ n.negOnePow •
      ((CategoryTheory.shiftFunctor C n).map X✝.mor₃ ≫ (shiftFunctorComm C 1 n).hom.app X✝.obj₁) ≫
        (CategoryTheory.shiftFunctor C 1).map ((CategoryTheory.shiftFunctor C n).map f.hom₁) =
    n.negOnePow •
      (CategoryTheory.shiftFunctor C n).map (X✝.mor₃ ≫ (CategoryTheory.shiftFunctor C 1).map f.hom₁) ≫
        (shiftFunctorComm C 1 n).hom.app Y✝.obj₁
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
hT : T ∈ distinguishedTriangles
h₁ : IsZero T.obj₁
h₃ : IsZero T.obj₃
⊢ IsZero T.obj₂

Tactic:
by
  rw [T.isZero₂_iff hT]
  exact ⟨h₁.eq_of_src _ _, h₃.eq_of_tgt _ _⟩

Goal state after:
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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₁
⊢ T ∈ distinguishedTriangles ↔ IsIso T.mor₂

Tactic:
rw [rotate_distinguished_triangle,
    distinguished_iff_of_isZero₃ _ (Functor.map_isZero (CategoryTheory.shiftFunctor C 1) h)]

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), (CategoryTheory.shiftFunctor C n).Additive
hC : Pretriangulated C
T : Triangle C
h : IsZero T.obj₁
⊢ IsIso T.rotate.mor₁ ↔ IsIso T.mor₂
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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : ((invRotate C).obj T).mor₁ ≫ ((invRotate C).map φ).hom₂ = ((invRotate C).map φ).hom₁ ≫ ((invRotate C).obj T').mor₁
⊢ (h ≫ inv ((shiftFunctor C (-1)).map φ.hom₃)) ≫ T.invRotate.mor₁ = g

Tactic:
dsimp only [invRotate] at eq

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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
A : C
g : A ⟶ T.obj₁
hf : g ≫ T.mor₁ ≫ φ.hom₂ = 0
h : A ⟶ T'.invRotate.obj₁
hh : g ≫ φ.hom₁ = h ≫ T'.invRotate.mor₁
eq : T.invRotate.mor₁ ≫ φ.hom₁ = (shiftFunctor C (-1)).map φ.hom₃ ≫ T'.invRotate.mor₁
⊢ (h ≫ inv ((shiftFunctor C (-1)).map φ.hom₃)) ≫ T.invRotate.mor₁ = g
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: case refine_2
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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
y₁ : A ⟶ T'.obj₁
hy₁ : y₂ - x₂ ≫ φ.hom₂ = y₁ ≫ T'.mor₁
x₁ : A ⟶ T.obj₁
hx₁ : x₁ ≫ φ.hom₁ = y₁
⊢ x₂ ≫ φ.hom₂ + y₁ ≫ T'.mor₁ = y₂

Tactic:
← hy₁,

Goal state after:
case refine_2
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
T T' : Triangle C
φ : T ⟶ T'
hT : T ∈ distinguishedTriangles
hT' : T' ∈ distinguishedTriangles
h₁ : IsIso φ.hom₁
h₃ : IsIso φ.hom₃
this : Mono φ.hom₂
A : C
y₂ : A ⟶ T'.obj₂
x₃ : A ⟶ T.obj₃
hx₃ : x₃ ≫ φ.hom₃ = y₂ ≫ T'.mor₂
x₂ : A ⟶ T.obj₂
hx₂ : x₃ = x₂ ≫ T.mor₂
y₁ : A ⟶ T'.obj₁
hy₁ : y₂ - x₂ ≫ φ.hom₂ = y₁ ≫ T'.mor₁
x₁ : A ⟶ T.obj₁
hx₁ : x₁ ≫ φ.hom₁ = y₁
⊢ x₂ ≫ φ.hom₂ + (y₂ - x₂ ≫ φ.hom₂) = y₂
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
⊢ IsIso φ'.hom₁

Tactic:
rw [h₁]

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
⊢ IsIso (𝟙 T'.obj₁)
Module: Mathlib.CategoryTheory.Triangulated.Functor

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ Z₁₂ Z₂₃ Z₁₃ : D
u₁₂ : Y₁ ⟶ Y₂
u₂₃ : Y₂ ⟶ Y₃
u₁₃ : Y₁ ⟶ Y₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Y₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Y₃ ⟶ Z₂₃
⊢ ∀ {w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂} (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃}
    {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁} (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles),
    Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)

Tactic:
w₂₃

Goal state after:
C : Type u_1
D : Type u_2
inst✝¹⁵ : Category.{v_1, u_1} C
inst✝¹⁴ : Category.{v_2, u_2} D
inst✝¹³ : HasShift C ℤ
inst✝¹² : HasShift D ℤ
inst✝¹¹ : HasZeroObject C
inst✝¹⁰ : HasZeroObject D
inst✝⁹ : Preadditive C
inst✝⁸ : Preadditive D
inst✝⁷ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor D n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : Pretriangulated D
F : C ⥤ D
inst✝³ : F.CommShift ℤ
inst✝² : F.IsTriangulated
inst✝¹ : (F.mapComposableArrows 2).EssSurj
inst✝ : IsTriangulated C
Y₁ Y₂ Y₃ Z₁₂ Z₂₃ Z₁₃ : D
u₁₂ : Y₁ ⟶ Y₂
u₂₃ : Y₂ ⟶ Y₃
u₁₃ : Y₁ ⟶ Y₃
comm : u₁₂ ≫ u₂₃ = u₁₃
v₁₂ : Y₂ ⟶ Z₁₂
w₁₂ : Z₁₂ ⟶ (shiftFunctor D 1).obj Y₁
h₁₂ : Triangle.mk u₁₂ v₁₂ w₁₂ ∈ distinguishedTriangles
v₂₃ : Y₃ ⟶ Z₂₃
w₂₃ : Z₂₃ ⟶ (shiftFunctor D 1).obj Y₂
⊢ ∀ (h₂₃ : Triangle.mk u₂₃ v₂₃ w₂₃ ∈ distinguishedTriangles) {v₁₃ : Y₃ ⟶ Z₁₃} {w₁₃ : Z₁₃ ⟶ (shiftFunctor D 1).obj Y₁}
    (h₁₃ : Triangle.mk u₁₃ v₁₃ w₁₃ ∈ distinguishedTriangles), Nonempty (Octahedron comm h₁₂ h₂₃ h₁₃)
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + n + a = a''

Tactic:
← ha'',

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁴ : AddMonoid M
inst✝³ : HasShift C M
inst✝² : HasShift D M
G : Type u_5
inst✝¹ : AddGroup G
inst✝ : HasShift C G
n m a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + n + a = m + a'
Module: Mathlib.CategoryTheory.PathCategory.Basic

Goal state before:
pp: V : Type u₁
inst✝ : Quiver V
P : {a b : Paths V} → (a ⟶ b) → Prop
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)
⊢ ∀ {a b : Paths V} (f : a ⟶ b), P f

Tactic:
by
  intro a b
  revert a
  exact induction_fixed_target (P := fun f ↦ P f) id (fun _ _ ↦ comp _ _)

Goal state after:
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + n + a = a''

Tactic:
← ha'',

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ m + n + a = m + a'
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + n + a = a''

Tactic:
← ha'',

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
n m mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
X : C
⊢ m + n + a = m + a'
Module: Mathlib.CategoryTheory.Shift.ShiftSequence

Goal state before:
pp: C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
X Y : C
m : M
f : X ⟶ (shiftFunctor C m).obj Y
n mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ mn + a = a''

Tactic:
← ha'',

Goal state after:
C : Type u_1
D : Type u_2
A : Type u_3
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : Category.{v_3, u_3} A
F : C ⥤ A
π : C ⥤ D
H : D ⥤ A
e : π ⋙ H ≅ F
M : Type u_4
inst✝⁵ : AddMonoid M
inst✝⁴ : HasShift C M
inst✝³ : HasShift D M
G : Type u_5
inst✝² : AddGroup G
inst✝¹ : HasShift C G
inst✝ : F.ShiftSequence M
X Y : C
m : M
f : X ⟶ (shiftFunctor C m).obj Y
n mn : M
hnm : m + n = mn
a a' a'' : M
ha' : n + a = a'
ha'' : m + a' = a''
⊢ mn + a = m + a'
Module: Mathlib.CategoryTheory.Localization.Predicate

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Category.{v_2, u_2} D
L : C ⥤ D
W : MorphismProperty C
h₁ : Localization.StrictUniversalPropertyFixedTarget L W D
h₂ : Localization.StrictUniversalPropertyFixedTarget L W W.Localization
⊢ W.Q ⋙ 𝟭 W.Localization = W.Q ⋙ Localization.Construction.lift L ⋯ ⋙ h₂.lift W.Q ⋯

Tactic:
simp only [← Functor.assoc, Localization.Construction.fac, h₂.fac, Functor.comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Predicate

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
hW : W ≤ MorphismProperty.isomorphisms C
inst✝ : L.IsEquivalence
⊢ L.IsLocalization W

Tactic:
by
  have : (𝟭 C).IsLocalization W := for_id W hW
  exact of_equivalence_target (𝟭 C) W L L.asEquivalence L.leftUnitor

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Construction

Goal state before:
pp: case refine_2
C : Type uC
inst✝¹ : Category.{uC', uC} C
W : MorphismProperty C
D : Type uD
inst✝ : Category.{uD', uD} D
G : C ⥤ D
hG : W.IsInvertedBy G
G₁ G₂ : W.Localization ⥤ D
τ : G₁ ⟶ G₂
⊢ (𝟭 (W.Localization ⥤ D)).map τ = eqToHom ⋯ ≫ (functor W D ⋙ inverse W D).map τ ≫ eqToHom ⋯

Tactic:
· apply natTrans_hcomp_injective
        ext X
        simp

Goal state after:
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.f ≫ t₁) = L.map (-φ.f ≫ t₂)

Tactic:
hft

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 (-φ.f ≫ t₂) = L.map (-φ.f ≫ t₂)
Module: Mathlib.CategoryTheory.ObjectProperty.LimitsClosure

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
Q : ObjectProperty C
h : P ≤ Q
⊢ P ⊔ ⨆ a, P.strictLimitsOfShape (J a) ≤ Q ⊔ ⨆ a, Q.strictLimitsOfShape (J a)

Tactic:
simp only [sup_le_iff, iSup_le_iff]

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
P : ObjectProperty C
α : Type t
J : α → Type u'
inst✝ : (a : α) → Category.{v', u'} (J a)
Q : ObjectProperty C
h : P ≤ Q
⊢ P ≤ Q ⊔ ⨆ a, Q.strictLimitsOfShape (J a) ∧
    ∀ (i : α), P.strictLimitsOfShape (J i) ≤ Q ⊔ ⨆ a, Q.strictLimitsOfShape (J a)