mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f ≫
      (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr) =
    𝟙 { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.X₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
⊢ t.IsLE X n₀ ↔ IsZero ((t.truncGE n₁).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
inst✝ : t.IsGE X n
⊢ IsZero ((t.truncLT n).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (σ₂.s ≫ σ₁.s) ≫
      { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.g =
    𝟙 { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.X₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
X : C
inst✝ : t.IsLE X n₀
⊢ IsZero ((t.truncGE n₁).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
this : ∀ (f : (t.truncGE n).obj X ⟶ Y), (t.truncGEπ n).app X ≫ f = 0 → f = 0
⊢ (t.truncGEπ n).app X ≫ (f₁ - f₂) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
this : ∀ (f : (t.truncGE n).obj X ⟶ Y), (t.truncGEπ n).app X ≫ f = 0 → f = 0
⊢ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Abelian C
inst✝² : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
inst✝¹ : K.IsStrictlyGE (n + 1)
inst✝ : L.IsStrictlyGE n
⊢ (mappingCone (𝟙 (I K)) ⊞ L).IsStrictlyGE n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ 1 + 1 = 2

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
f : (t.truncGE n).obj X ⟶ Y
hf : (t.truncGEπ n).app X ≫ f = 0
g : (shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁ ⟶ Y
hg : f = ((t.triangleLTGE n).obj X).mor₃ ≫ g
⊢ n - 2 < n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
p q : ℤ
hpq : p + 2 = q
⊢ 1 + 1 = 2

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
p q : ℤ
hpq : p + 2 = q
⊢ p + 1 = q - 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
f : (t.truncGE n).obj X ⟶ Y
hf : (t.truncGEπ n).app X ≫ f = 0
g : (shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁ ⟶ Y
hg : f = ((t.triangleLTGE n).obj X).mor₃ ≫ g
⊢ 1 + (n - 2) = n - 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
I₁ : C
w✝¹ : Injective I₁
f₁ : I₁ ⟶ X
w₁ : f₁ ≫ g₁ = 0
σ₁ : { X₁ := I₁, X₂ := X, X₃ := Y, f := f₁, g := g₁, zero := w₁ }.Splitting
I₂ : C
w✝ : Injective I₂
f₂ : I₂ ⟶ Y
w₂ : f₂ ≫ g₂ = 0
σ₂ : { X₁ := I₂, X₂ := Y, X₃ := Z, f := f₂, g := g₂, zero := w₂ }.Splitting
⊢ (σ₁.r ≫ biprod.inl + g₁ ≫ σ₂.r ≫ biprod.inr) ≫
        { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.f +
      { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.g ≫
        σ₂.s ≫ σ₁.s =
    𝟙 { X₁ := I₁ ⊞ I₂, X₂ := X, X₃ := Z, f := biprod.fst ≫ f₁ + biprod.snd ≫ f₂ ≫ σ₁.s, g := g₁ ≫ g₂, zero := ⋯ }.X₂

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ HomComplex.δ 1 2 (HomComplex.Cochain.mk fun p q x ↦ K.d p q ≫ Injective.ι (K.X q)) = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
f : (t.truncGE n).obj X ⟶ Y
hf : (t.truncGEπ n).app X ≫ f = 0
g : (shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁ ⟶ Y
hg : f = ((t.triangleLTGE n).obj X).mor₃ ≫ g
⊢ t.IsLE ((shiftFunctor C 1).obj ((t.triangleLTGE n).obj X).obj₁) (n - 2)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ HomComplex.δ 0 1 (HomComplex.Cochain.ofHoms fun n ↦ Injective.ι (K.X n)) +
      (↑(HomComplex.Cocycle.mk (HomComplex.Cochain.mk fun p q x ↦ K.d p q ≫ Injective.ι (K.X q)) 2 i._proof_2 ⋯)).comp
        (HomComplex.Cochain.ofHom (𝟙 (I K))) ⋯ =
    0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
n : ℤ
⊢ (i f).f n ≫ biprod.fst.f n ≫ (mappingCone.snd (𝟙 (I K))).v n n ⋯ = Injective.ι (K.X n)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X Y Z : C
g₁ : X ⟶ Y
g₂ : Y ⟶ Z
hg₁ : epiWithInjectiveKernel g₁
hg₂ : epiWithInjectiveKernel g₂
⊢ epiWithInjectiveKernel (g₁ ≫ g₂)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ i f ≫ p K L = f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
X Y : C
f₁ f₂ : (t.truncGE n).obj X ⟶ Y
h : (t.truncGEπ n).app X ≫ f₁ = (t.truncGEπ n).app X ≫ f₂
inst✝ : t.IsGE Y n
⊢ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
this : ∀ (f : Y ⟶ (t.truncLT n).obj X), f ≫ (t.truncLTι n).app X = 0 → f = 0
⊢ (f₁ - f₂) ≫ (t.truncLTι n).app X = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X' Y' X Y : C
f' : X' ⟶ Y'
f : X ⟶ Y
r : RetractArrow f' f
left✝ : Epi f
hf : Injective (kernel f)
this : Epi f'
⊢ kernel.map f' f (Arrow.Hom.left r.i) (Arrow.Hom.right r.i) ⋯ ≫
      kernel.map f f' (Arrow.Hom.left r.r) (Arrow.Hom.right r.r) ⋯ =
    𝟙 (kernel f')

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
this : ∀ (f : Y ⟶ (t.truncLT n).obj X), f ≫ (t.truncLTι n).app X = 0 → f = 0
⊢ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
⊢ ∀ {X Y Z W : C} {f : X ⟶ Y} {g : Z ⟶ W} (h : RetractArrow f g), epiWithInjectiveKernel g → epiWithInjectiveKernel f

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
p : X ⟶ Y
A B : C
i : A ⟶ B
inst✝ : Mono i
I : C
w✝ : Injective I
s : I ⟶ X
hs : s ≫ p = 0
σ : { X₁ := I, X₂ := X, X₃ := Y, f := s, g := p, zero := hs }.Splitting
hI : (MorphismProperty.monomorphisms C).rlp 0
⊢ CommSq σ.r p 0 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
p : X ⟶ Y
A B : C
i : A ⟶ B
inst✝ : Mono i
I : C
w✝ : Injective I
s : I ⟶ X
hs : s ≫ p = 0
σ : { X₁ := I, X₂ := X, X₃ := Y, f := s, g := p, zero := hs }.Splitting
hI : (MorphismProperty.monomorphisms C).rlp 0
t : PullbackCone 0 0
⊢ (t.fst ≫ s + t.snd ≫ σ.s) ≫ σ.r = t.fst

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
f : Y ⟶ (t.truncLT n).obj X
hf : f ≫ (t.truncLTι n).app X = 0
g : Y ⟶ ((t.triangleLTGE n).obj X).invRotate.obj₁
hg : f = g ≫ ((t.triangleLTGE n).obj X).invRotate.mor₁
⊢ n - 1 < n + 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
p : X ⟶ Y
A B : C
i : A ⟶ B
inst✝ : Mono i
I : C
w✝ : Injective I
s : I ⟶ X
hs : s ≫ p = 0
σ : { X₁ := I, X₂ := X, X₃ := Y, f := s, g := p, zero := hs }.Splitting
hI : (MorphismProperty.monomorphisms C).rlp 0
t : PullbackCone 0 0
⊢ (t.fst ≫ s + t.snd ≫ σ.s) ≫ p = t.snd

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.EpiWithInjectiveKernel

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
X Y : C
p : X ⟶ Y
hp : epiWithInjectiveKernel p
A B : C
i : A ⟶ B
inst✝ : Mono i
⊢ HasLiftingProperty i p

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
f : Y ⟶ (t.truncLT n).obj X
hf : f ≫ (t.truncLTι n).app X = 0
g : Y ⟶ ((t.triangleLTGE n).obj X).invRotate.obj₁
hg : f = g ≫ ((t.triangleLTGE n).obj X).invRotate.mor₁
⊢ -1 + (n + 1) = n

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
f : Y ⟶ (t.truncLT n).obj X
hf : f ≫ (t.truncLTι n).app X = 0
g : Y ⟶ ((t.triangleLTGE n).obj X).invRotate.obj₁
hg : f = g ≫ ((t.triangleLTGE n).obj X).invRotate.mor₁
⊢ t.IsGE ((t.triangleLTGE n).obj X).invRotate.obj₁ (n + 1)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomologicalComplexLimitsEventuallyConstant

Goal state before:
pp: C : Type u_1
J : Type u_2
ι : Type u_3
inst✝⁵ : Category.{u_4, u_1} C
inst✝⁴ : Category.{u_5, u_2} J
c : ComplexShape ι
inst✝³ : IsCofiltered J
inst✝² : HasZeroMorphisms C
F : J ⥤ HomologicalComplex C c
inst✝¹ : ∀ (j : ι), HasLimit (F ⋙ eval C c j)
cF : Cone F
hcF : IsLimit cF
inst✝ : CategoryWithHomology C
q₀ q₁ q₂ : ι
h₀ : c.prev q₁ = q₀
h₂ : c.next q₁ = q₂
j : J
hq₀ : (F ⋙ eval C c q₀).IsEventuallyConstantTo j
hq₁ : (F ⋙ eval C c q₁).IsEventuallyConstantTo j
hq₂ : (F ⋙ eval C c q₂).IsEventuallyConstantTo j
⊢ QuasiIsoAt (cF.π.app j) q₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Category.Factorisation

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ : X ⟶ Y
d : Factorisation f✝
f : Factorisation.initial ⟶ d
⊢ f = d.initialHom

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Category.Factorisation

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X Y : C
f✝ : X ⟶ Y
d : Factorisation f✝
f : d ⟶ Factorisation.terminal
⊢ f = d.terminalHom

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
⊢ degreewiseEpiWithInjectiveKernel (p K L)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ X i = X j

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ (HomComplex.Cochain.ofHom (𝟙 (mappingCone (𝟙 (I K))))).comp ↑(mappingCone.fst (𝟙 (I K))) ⋯ =
    -HomComplex.δ 0 1 (mappingCone.snd (𝟙 (I K))) + (HomComplex.Cochain.ofHom 0).comp ↑(mappingCone.fst (𝟙 (I K))) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ X (j + 1) = X (i + 1)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i j : ℕ
h : i = j
⊢ f i = eqToHom ⋯ ≫ f j ≫ eqToHom ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
n : ℤ
Y X : C
f₁ f₂ : Y ⟶ (t.truncLT n).obj X
h : f₁ ≫ (t.truncLTι n).app X = f₂ ≫ (t.truncLTι n).app X
inst✝ : t.IsLE Y (n - 1)
⊢ f₁ = f₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f✝ : (n : ℕ) → X n ⟶ X (n + 1)
x✝¹ : ℕ → C
f : (n : ℕ) → x✝¹ n ⟶ x✝¹ (n + 1)
l : ℕ
x✝ : 0 ≤ l + 1
⊢ 0 ≤ l

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
⊢ (HomComplex.Cochain.ofHom (𝟙 (mappingCone (𝟙 (I K))))).comp (mappingCone.snd (𝟙 (I K))) ⋯ =
    HomComplex.δ (-1) 0 0 + (mappingCone.snd (𝟙 (I K))).comp (HomComplex.Cochain.ofHom (𝟙 (I K))) ⋯ +
      (HomComplex.Cochain.ofHom 0).comp (mappingCone.snd (𝟙 (I K))) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f✝ : (n : ℕ) → X n ⟶ X (n + 1)
x✝¹ : ℕ → C
f : (n : ℕ) → x✝¹ n ⟶ x✝¹ (n + 1)
k l : ℕ
x✝ : k + 1 ≤ l + 1
⊢ k ≤ l

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
⊢ (t.truncLT n).map ((t.truncLTι n).app X) ≫ (t.truncLTι n).app ((𝟭 C).obj X) =
    (t.truncLTι n).app ((t.truncLT n).obj X) ≫ (t.truncLTι n).app ((𝟭 C).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ i ≤ i

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
n : ℤ
X : C
⊢ (t.truncGEπ n).app ((𝟭 C).obj X) ≫ (t.truncGE n).map ((t.truncGEπ n).app X) =
    (t.truncGEπ n).app ((𝟭 C).obj X) ≫ (t.truncGEπ n).app ((t.truncGE n).obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : t.IsLE X n₀
⊢ n₀ < n₁

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: case succ
C : Type u_1
inst✝ : Category.{v_1, u_1} C
n✝ : ℕ
hi : ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f n✝ n✝ ⋯ = 𝟙 (X n✝)
⊢ ∀ {X : ℕ → C} (f : (n : ℕ) → X n ⟶ X (n + 1)), map f (n✝ + 1) (n✝ + 1) ⋯ = 𝟙 (X (n✝ + 1))

Tactic:
intro X f
      apply hi

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.LocalizingSubcategory

Goal state before:
pp: C : Type u_1
D : Type u_2
D₁ : Type u_3
D₂ : Type u_4
inst✝¹³ : Category.{v_1, u_1} C
inst✝¹² : Category.{v_2, u_2} D
inst✝¹¹ : Category.{v_3, u_3} D₁
inst✝¹⁰ : Category.{v_4, u_4} D₂
A B : ObjectProperty C
inst✝⁹ : HasZeroObject C
inst✝⁸ : HasShift C ℤ
inst✝⁷ : Preadditive C
inst✝⁶ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝⁵ : Pretriangulated C
inst✝⁴ : IsTriangulated C
inst✝³ : A.IsTriangulated
inst✝² : B.IsTriangulated
inst✝¹ : B.IsClosedUnderIsomorphisms
inst✝ : A.IsVerdierLeftLocalizing B
L₁ : A.FullSubcategory ⥤ (B.inverseImage A.ι).trW.Localization := (B.inverseImage A.ι).trW.Q
L₂ : C ⥤ B.trW.Localization := B.trW.Q
F : (B.inverseImage A.ι).trW.Localization ⥤ B.trW.Localization :=
  (A.triangulatedLocalizerMorphism B).localizedFunctor L₁ L₂
this✝ : CatCommSq (A.op.triangulatedLocalizerMorphism B.op).functor (A.opEquivalence.functor ⋙ L₁.op) L₂.op F.op :=
  {
    iso :=
      A.opEquivalence.functor.isoWhiskerLeft
        (NatIso.op (CatCommSq.iso (A.triangulatedLocalizerMorphism B).functor L₁ L₂ F).symm) }
this : L₂.op.IsLocalization B.op.trW
⊢ (A.opEquivalence.functor ⋙ L₁.op).IsLocalization (B.op.inverseImage A.op.ι).trW

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ map f i i ⋯ = 𝟙 (X i)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.Factorizations.CM5b

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Abelian C
inst✝ : EnoughInjectives C
K L : CochainComplex C ℤ
f : K ⟶ L
h₀ : Homotopy (𝟙 (mappingCone (𝟙 (I K)))) 0 :=
  mappingCone.liftHomotopy (𝟙 (I K)) (𝟙 (mappingCone (𝟙 (I K)))) 0 (mappingCone.snd (𝟙 (I K))) 0 ⋯ ⋯
h₁ : Homotopy (biprod.fst ≫ 𝟙 (mappingCone (𝟙 (I K))) ≫ biprod.inl) (biprod.fst ≫ 0 ≫ biprod.inl) :=
  (h₀.compRight biprod.inl).compLeft biprod.fst
h₂ : Homotopy (biprod.fst ≫ 𝟙 (mappingCone (𝟙 (I K))) ≫ biprod.inl + biprod.snd ≫ biprod.inr)
  (biprod.fst ≫ 0 ≫ biprod.inl + biprod.snd ≫ biprod.inr) :=
  h₁.add (Homotopy.refl (biprod.snd ≫ biprod.inr))
⊢ p K L ≫ biprod.inr = biprod.fst ≫ 0 ≫ biprod.inl + biprod.snd ≫ biprod.inr

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Functor.OfSequence

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : ℕ → C
f : (n : ℕ) → X n ⟶ X (n + 1)
i : ℕ
⊢ i ≤ i + 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.TruncLTGE

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasZeroObject C
inst✝³ : HasShift C ℤ
inst✝² : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝¹ : Pretriangulated C
t : TStructure C
X Y : C
f : X ⟶ Y
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
inst✝ : t.IsLE X n₀
⊢ t.IsGE ((t.triangleLTGE n₁).obj Y).obj₃ n₁

Tactic:
by

Goal state after: