mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ z₁' : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ (z₁ - z₁').comp z₂ h = z₁.comp z₂ h - z₁'.comp z₂ h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ z₁' : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ ((z₁ - z₁').comp z₂ h).v p q hpq = (z₁.comp z₂ h - z₁'.comp z₂ h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ (-z₁).comp z₂ h = -z₁.comp z₂ h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ ((-z₁).comp z₂ h).v p q hpq = (-z₁.comp z₂ h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
k : R
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ (k • z₁).comp z₂ h = k • z₁.comp z₂ h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
k : R
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ ((k • z₁).comp z₂ h).v p q hpq = (k • z₁.comp z₂ h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
z₂ : Cochain F G n
⊢ (ofHom (𝟙 F)).comp z₂ ⋯ = z₂

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
z₂ : Cochain F G n
p q : ℤ
hpq : p + n = q
⊢ ((ofHom (𝟙 F)).comp z₂ ⋯).v p q hpq = z₂.v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
h : n₁ + n₂ = n₁₂
⊢ z₁.comp 0 h = 0

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp 0 h).v p q hpq = v 0 p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ z₂' : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ z₁.comp (z₂ + z₂') h = z₁.comp z₂ h + z₁.comp z₂' h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ z₂' : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (z₂ + z₂') h).v p q hpq = (z₁.comp z₂ h + z₁.comp z₂' h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ z₂' : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ z₁.comp (z₂ - z₂') h = z₁.comp z₂ h - z₁.comp z₂' h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ z₂' : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (z₂ - z₂') h).v p q hpq = (z₁.comp z₂ h - z₁.comp z₂' h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ z₁.comp (-z₂) h = -z₁.comp z₂ h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (-z₂) h).v p q hpq = (-z₁.comp z₂ h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
k : R
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
⊢ z₁.comp (k • z₂) h = k • z₁.comp z₂ h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
k : R
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ (z₁.comp (k • z₂) h).v p q hpq = (k • z₁.comp z₂ h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
z₁ : Cochain F G n
⊢ z₁.comp (ofHom (𝟙 G)) ⋯ = z₁

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n : ℤ
z₁ : Cochain F G n
p q : ℤ
hpq : p + n = q
⊢ (z₁.comp (ofHom (𝟙 G)) ⋯).v p q hpq = z₁.v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : ¬n + 1 = m
z : Cochain F G n
⊢ δ n m z = 0

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n m : ℤ
hnm : ¬n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
⊢ (δ n m z).v p q hpq = Cochain.v 0 p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
α β : Cochain F G n
h : n + 1 = m
⊢ δ n m (α + β) = δ n m α + δ n m β

Tactic:
ext p q hpq

Goal state after:
case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
α β : Cochain F G n
h : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ (δ n m (α + β)).v p q hpq = (δ n m α + δ n m β).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
r : R
a : Cochain F G n
h : n + 1 = m
⊢ δ n m (r • a) = (RingHom.id R) r • δ n m a

Tactic:
ext p q hpq

Goal state after:
case pos
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
r : R
a : Cochain F G n
h : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ (δ n m (r • a)).v p q hpq = ((RingHom.id R) r • δ n m a).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
⊢ δ n₁ n₂ (δ n₀ n₁ z) = 0

Tactic:
ext p q hpq

Goal state after:
case pos
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ (δ n₁ n₂ (δ n₀ n₁ z)).v p q hpq = Cochain.v 0 p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
⊢ δ (n₁ + n₂) (n₁ + n₂ + 1) (z₁.comp z₂ ⋯) = z₁.comp (δ n₂ (n₂ + 1) z₂) ⋯ + n₂.negOnePow • (δ n₁ (n₁ + 1) z₁).comp z₂ ⋯

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ (δ (n₁ + n₂) (n₁ + n₂ + 1) (z₁.comp z₂ ⋯)).v p q hpq =
    (z₁.comp (δ n₂ (n₂ + 1) z₂) ⋯ + n₂.negOnePow • (δ n₁ (n₁ + 1) z₁).comp z₂ ⋯).v p q hpq
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst

Tactic:
rcases i with (_ | _ | i)

Goal state after:
case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst

case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst

case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : ChainComplex V ℕ
e : P ⟶ Q
zero : P.X 0 ⟶ Q.X 1
comm_zero : e.f 0 = zero ≫ Q.d 1 0
one : P.X 1 ⟶ Q.X 2
comm_one : e.f 1 = P.d 1 0 ≫ zero + one ≫ Q.d 2 1
succ :
  (n : ℕ) →
    (p :
        (f : P.X n ⟶ Q.X (n + 1)) ×'
          (f' : P.X (n + 1) ⟶ Q.X (n + 2)) ×' e.f (n + 1) = P.d (n + 1) n ≫ f + f' ≫ Q.d (n + 2) (n + 1)) →
      (f'' : P.X (n + 2) ⟶ Q.X (n + 3)) ×' e.f (n + 2) = P.d (n + 2) (n + 1) ≫ p.snd.fst + f'' ≫ Q.d (n + 3) (n + 2)
i : ℕ
⊢ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)).snd.fst ≫ (xPrevIso Q ⋯).hom =
    (xNextIso P ⋯).inv ≫ (mkInductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1)).fst
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: case mpr
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain K (mappingCone φ) i
h₁ : γ₁.comp (↑(fst φ)) hij = γ₂.comp (↑(fst φ)) hij
h₂ : γ₁.comp (snd φ) ⋯ = γ₂.comp (snd φ) ⋯
⊢ γ₁ = γ₂

Tactic:
ext p q hpq

Goal state after:
case mpr
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain K (mappingCone φ) i
h₁ : γ₁.comp (↑(fst φ)) hij = γ₂.comp (↑(fst φ)) hij
h₂ : γ₁.comp (snd φ) ⋯ = γ₂.comp (snd φ) ⋯
p q : ℤ
hpq : p + i = q
⊢ γ₁.v p q hpq = γ₂.v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G K : CochainComplex C ℤ
n₁ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K 0
m₁ : ℤ
h₁ : n₁ + 1 = m₁
⊢ δ n₁ m₁ (z₁.comp z₂ ⋯) = z₁.comp (δ 0 1 z₂) h₁ + (δ n₁ m₁ z₁).comp z₂ ⋯

Tactic:
by
  simp only [δ_comp z₁ z₂ (add_zero n₁) m₁ 1 m₁ h₁ h₁ (zero_add 1), one_smul,
    Int.negOnePow_zero]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: case mpr
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
h₁ : (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯
h₂ : (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯
⊢ γ₁ = γ₂

Tactic:
ext p q hpq

Goal state after:
case mpr
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
h₁ : (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯
h₂ : (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯
p q : ℤ
hpq : p + j = q
⊢ γ₁.v p q hpq = γ₂.v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCofiber

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
K : HomologicalComplex C c
y : (α : G ⟶ K) × Homotopy (φ ≫ α) 0
x₁ : G ⟶ K
x₂ : Homotopy (φ ≫ x₁) 0
⊢ (⟨x₁, x₂⟩.fst = y.fst ∧ ∀ (i j : ι), c.Rel j i → ⟨x₁, x₂⟩.snd.hom i j = y.snd.hom i j) → ⟨x₁, x₂⟩ = y

Tactic:
obtain ⟨y₁, y₂⟩ := y

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
K : HomologicalComplex C c
x₁ : G ⟶ K
x₂ : Homotopy (φ ≫ x₁) 0
y₁ : G ⟶ K
y₂ : Homotopy (φ ≫ y₁) 0
⊢ (⟨x₁, x₂⟩.fst = ⟨y₁, y₂⟩.fst ∧ ∀ (i j : ι), c.Rel j i → ⟨x₁, x₂⟩.snd.hom i j = ⟨y₁, y₂⟩.snd.hom i j) →
    ⟨x₁, x₂⟩ = ⟨y₁, y₂⟩
Module: Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
⊢ δ 0 1 (snd φ) = -(↑(fst φ)).comp (Cochain.ofHom φ) ⋯

Tactic:
ext p q hpq

Goal state after:
C : Type u_1
inst✝² : Category.{v, u_1} C
inst✝¹ : Preadditive C
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
p q : ℤ
hpq : p + 1 = q
⊢ (δ 0 1 (snd φ)).v p q hpq = (-(↑(fst φ)).comp (Cochain.ofHom φ) ⋯).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
⊢ δ 1 2 (Cochain.diff K) = 0

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m p q : ℤ
hpq : p + 2 = q
⊢ (δ 1 2 (Cochain.diff K)).v p q hpq = Cochain.v 0 p q hpq
Module: Mathlib.Algebra.Homology.Homotopy

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
i : ℕ
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (i + 1)).fst ≫ (xPrevIso Q ⋯).hom

Tactic:
rcases i with (_ | _ | i)

Goal state after:
case zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ 0).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).fst ≫ (xPrevIso Q ⋯).hom

case succ.zero
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1)).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (0 + 1 + 1)).fst ≫ (xPrevIso Q ⋯).hom

case succ.succ
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
P Q : CochainComplex V ℕ
e : P ⟶ Q
zero : P.X 1 ⟶ Q.X 0
comm_zero : e.f 0 = P.d 0 1 ≫ zero
one : P.X 2 ⟶ Q.X 1
comm_one : e.f 1 = zero ≫ Q.d 0 1 + P.d 1 2 ≫ one
succ :
  (n : ℕ) →
    (p :
        (f : P.X (n + 1) ⟶ Q.X n) ×'
          (f' : P.X (n + 2) ⟶ Q.X (n + 1)) ×' e.f (n + 1) = f ≫ Q.d n (n + 1) + P.d (n + 1) (n + 2) ≫ f') →
      (f'' : P.X (n + 3) ⟶ Q.X (n + 2)) ×' e.f (n + 2) = p.snd.fst ≫ Q.d (n + 1) (n + 2) + P.d (n + 2) (n + 3) ≫ f''
i : ℕ
⊢ (xNextIso P ⋯).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1)).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ (i + 1 + 1 + 1)).fst ≫ (xPrevIso Q ⋯).hom
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
⊢ (fun ho ↦ ⟨ofHomotopy ho, ⋯⟩)
      ((fun z ↦ { hom := fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij else 0, zero := ⋯, comm := ⋯ }) z) =
    z

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preadditive C
R : Type u_1
inst✝¹ : Ring R
inst✝ : Linear R C
F G K L : CochainComplex C ℤ
n m : ℤ
φ₁ φ₂ : F ⟶ G
z : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }
p q : ℤ
hpq : p + -1 = q
⊢ (↑((fun ho ↦ ⟨ofHomotopy ho, ⋯⟩)
            ((fun z ↦ { hom := fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij else 0, zero := ⋯, comm := ⋯ }) z))).v
      p q hpq =
    (↑z).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
F G K : CochainComplex C ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
Φ : C ⥤ D
inst✝ : Φ.Additive
⊢ (z₁.comp z₂ h).map Φ = (z₁.map Φ).comp (z₂.map Φ) h

Tactic:
ext p q hpq

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
F G K : CochainComplex C ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
Φ : C ⥤ D
inst✝ : Φ.Additive
p q : ℤ
hpq : p + n₁₂ = q
⊢ ((z₁.comp z₂ h).map Φ).v p q hpq = ((z₁.map Φ).comp (z₂.map Φ) h).v p q hpq
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplex

Goal state before:
pp: case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : n + 1 = m
⊢ δ n m (z.map Φ) = (δ n m z).map Φ

Tactic:
ext p q hpq

Goal state after:
case pos
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Preadditive C
K L : CochainComplex C ℤ
n m : ℤ
D : Type u_2
inst✝² : Category.{v_1, u_2} D
inst✝¹ : Preadditive D
z : Cochain K L n
Φ : C ⥤ D
inst✝ : Φ.Additive
hnm : n + 1 = m
p q : ℤ
hpq : p + m = q
⊢ (δ n m (z.map Φ)).v p q hpq = ((δ n m z).map Φ).v p q hpq
Module: Mathlib.CategoryTheory.Shift.Quotient

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X : C
⊢ (lift r F hF).map ((shiftFunctorZero (Quotient r) A).hom.app ((functor r).obj X)) ≫
      F.map (𝟙 ((𝟭 C).obj X)) ≫ (shiftFunctorZero D A).inv.app (F.obj X) =
    (lift r F hF).map ((shiftFunctorZero (Quotient r) A).hom.app ((functor r).obj X)) ≫
      (shiftFunctorZero D A).inv.app ((lift r F hF).obj ((functor r).obj X))

Tactic:
dsimp [lift_obj_functor_obj]

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v', u'} D
F : C ⥤ D
r : HomRel C
A : Type w
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : r.IsCompatibleWithShift A
inst✝ : F.CommShift A
hF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ → F.map f₁ = F.map f₂
X : C
⊢ (lift r F hF).map ((shiftFunctorZero (Quotient r) A).hom.app ((functor r).obj X)) ≫
      F.map (𝟙 X) ≫ (shiftFunctorZero D A).inv.app (F.obj X) =
    (lift r F hF).map ((shiftFunctorZero (Quotient r) A).hom.app ((functor r).obj X)) ≫
      (shiftFunctorZero D A).inv.app (F.obj X)