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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝³ : HasHomotopyCofiber φ
inst✝² : DecidableRel c.Rel
F' F'' G' G'' : HomologicalComplex C c
φ' : F' ⟶ G'
φ'' : F'' ⟶ G''
inst✝¹ : HasHomotopyCofiber φ'
inst✝ : HasHomotopyCofiber φ''
H : ∀ (j : ι), ∃ i, c.Rel i j
α : Arrow.mk φ ⟶ Arrow.mk φ'
⊢ Arrow.Hom.left α ≫ 0 = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ 1 + m = n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ -1 + n = m

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ (inl φ).comp (descCochain φ α β h) ⋯ = α

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i✝ : ι
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
⊢ e.f i =
    (((dNext i) fun i j ↦
          if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom
          else 0) +
        (prevD i) fun i j ↦
          if h : i + 1 = j then (mkInductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst ≫ (xPrevIso Q h).hom
          else 0) +
      Hom.f 0 i

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
⊢ (Cochain.ofHom (inr φ)).comp (descCochain φ α β h) ⋯ = β

Tactic:
by

Goal state after:
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
f : (i j : ℕ) → P.X i ⟶ Q.X j
j : ℕ
⊢ (dNext j) f = P.d j (j + 1) ≫ f (j + 1) j

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₁ + m = p₃

Tactic:
by

Goal state after:
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
z : Cocycle F G 0
⊢ ∀ (i j : ℤ), (ComplexShape.up ℤ).Rel i j → (↑z).v i i ⋯ ≫ G.d i j = F.d i j ≫ (↑z).v j j ⋯

Tactic:
by

Goal state after:
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
φ : F ⟶ G
⊢ (ofHom φ).homOf = φ

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₁ + -1 = p₂

Tactic:
by

Goal state after:
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
z : Cocycle F G 0
⊢ ofHom z.homOf = z

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₂ + n = p₃

Tactic:
by

Goal state after:
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
z : Cocycle F G 0
⊢ Cochain.ofHom z.homOf = ↑z

Tactic:
by

Goal state after:
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
f : (i j : ℕ) → P.X i ⟶ Q.X j
i : ℕ
⊢ (prevD (i + 1)) f = f (i + 1) i ≫ Q.d i (i + 1)

Tactic:
by

Goal state after:
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
⊢ ∀ (x y : F ⟶ G), ofHom (x + y) = ofHom x + ofHom y

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
H : ∀ (j : ι), ∃ i, c.Rel i j
i : ι
hi : ¬c.Rel i (c.next i)
⊢ (mapArrowHom φ φ H (𝟙 (Arrow.mk φ))).f i ≫ sndX φ i = 𝟙 (X φ i) ≫ sndX φ i

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ p₁ + m = p₃

Tactic:
by

Goal state after:
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:
by

Goal state after:
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
⊢ ↑0 = 0

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + -1 = p₂
h₂₃ : p₂ + n = p₃
⊢ (inl φ).v p₁ p₂ h₁₂ ≫ (descCochain φ α β h).v p₂ p₃ h₂₃ = α.v p₁ p₃ ⋯

Tactic:
by

Goal state after:
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
⊢ ∀ (x y : Cocycle K L n), ↑(x + y) = ↑x + ↑y

Tactic:
by

Goal state after:
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
hm : n + 1 = m
⊢ AddCommGrpCat.ofHom (toCochainAddMonoidHom K L n) ≫ (K.HomComplex L).d n m = 0

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ p₁ + n = p₂

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ (inr φ).f p₁ ≫ (descCochain φ α β h).v p₁ p₂ h₁₂ = β.v p₁ p₂ h₁₂

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝¹ : HasHomotopyCofiber φ
inst✝ : DecidableRel c.Rel
H : ∀ (j : ι), ∃ i, c.Rel i j
⊢ mapArrowHom φ φ H (𝟙 (Arrow.mk φ)) = 𝟙 (homotopyCofiber φ)

Tactic:
by

Goal state after:
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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
x : ↑s.1
⊢ (ConcreteCategory.hom s.ι) x ∈ cocycle K L n

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ 1 + n = n'

Tactic:
by

Goal state after:
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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
⊢ ⟨(ConcreteCategory.hom s.ι) 0, ⋯⟩ = 0

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ 1 + 1 = 2

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ m + 1 = n

Tactic:
by

Goal state after:
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
f : (i j : ℕ) → P.X i ⟶ Q.X j
⊢ (prevD 0) f = 0

Tactic:
by

Goal state after:
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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
x✝¹ x✝ : ↑s.1
⊢ ⟨(ConcreteCategory.hom s.ι) (x✝¹ + x✝), ⋯⟩ = ⟨(ConcreteCategory.hom s.ι) x✝¹, ⋯⟩ + ⟨(ConcreteCategory.hom s.ι) x✝, ⋯⟩

Tactic:
by

Goal state after:
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
hm : n + 1 = m
⊢ ∀ (s : Fork ((K.HomComplex L).d n m) 0),
    AddCommGrpCat.ofHom { toFun := fun x ↦ ⟨(ConcreteCategory.hom s.ι) x, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ } ≫
        Fork.ι (KernelFork.ofι (AddCommGrpCat.ofHom (toCochainAddMonoidHom K L n)) ⋯) =
      s.ι

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cochain G K n
h : m + 1 = n
n' : ℤ
hn' : n + 1 = n'
⊢ δ n n' (descCochain φ α β h) =
    (↑(fst φ)).comp (δ m n α + n'.negOnePow • (Cochain.ofHom φ).comp β ⋯) ⋯ + (snd φ).comp (δ n n' β) ⋯

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
n m : ℤ
α : Cochain F K m
β : Cocycle G K n
h : m + 1 = n
eq : δ m n α = n.negOnePow • (Cochain.ofHom φ).comp ↑β ⋯
⊢ δ n (n + 1) (descCochain φ α (↑β) h) = 0

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i : ι
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''
⊢ e.f 0 = 0 ≫ dTo Q 0 + dFrom P 0 ≫ (xNextIso P ⋯).hom ≫ zero

Tactic:
by

Goal state after:
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
hm : n + 1 = m
s : Fork ((K.HomComplex L).d n m) 0
l : s.pt ⟶ (KernelFork.ofι (AddCommGrpCat.ofHom (toCochainAddMonoidHom K L n)) ⋯).pt
hl : l ≫ Fork.ι (KernelFork.ofι (AddCommGrpCat.ofHom (toCochainAddMonoidHom K L n)) ⋯) = s.ι
⊢ l = AddCommGrpCat.ofHom { toFun := fun x ↦ ⟨(ConcreteCategory.hom s.ι) x, ⋯⟩, map_zero' := ⋯, map_add' := ⋯ }

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
⊢ δ (-1) 0 α = Int.negOnePow 0 • (Cochain.ofHom φ).comp ↑(Cocycle.ofHom β) ⋯

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
⊢ Cochain.ofHom (desc φ α β eq) = descCochain φ α (Cochain.ofHom β) ⋯

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
p q : ℤ
h : p + -1 = q
⊢ (inl φ).v p q h ≫ (desc φ α β eq).f q = α.v p q h

Tactic:
by

Goal state after:
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
n : ℤ
z₁ : Cochain F G n
z₂ : Cocycle G K 0
m : ℤ
⊢ δ n m (z₁.comp ↑z₂ ⋯) = (δ n m z₁).comp ↑z₂ ⋯

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
⊢ (inl φ).comp (Cochain.ofHom (desc φ α β eq)) ⋯ = α

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : Preadditive V
c : ComplexShape ι
C D E : HomologicalComplex V c
f g : C ⟶ D
h k : D ⟶ E
i : ι
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''
n : ℕ
I : (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' :=
  mkCoinductiveAux₁ e zero one comm_one succ n
⊢ e.f (n + 1) = (I.fst ≫ (xPrevIso Q ⋯).inv) ≫ dTo Q (n + 1) + dFrom P (n + 1) ≫ (xNextIso P ⋯).hom ≫ I.snd.fst

Tactic:
by

Goal state after:
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
n : ℤ
z₁ : Cochain F G n
f : G ⟶ K
m : ℤ
⊢ δ n m (z₁.comp (Cochain.ofHom f) ⋯) = (δ n m z₁).comp (Cochain.ofHom f) ⋯

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
p : ℤ
⊢ (inr φ).f p ≫ (desc φ α β eq).f p = β.f p

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
⊢ inr φ ≫ desc φ α β eq = β

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
p q : ℤ
hpq : p + 1 = q
⊢ q + -1 = p

Tactic:
by

Goal state after:
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
n : ℤ
z₁ : Cocycle F G 0
z₂ : Cochain G K n
m : ℤ
⊢ δ n m ((↑z₁).comp z₂ ⋯) = (↑z₁).comp (δ n m z₂) ⋯

Tactic:
by

Goal state after:
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
K : CochainComplex C ℤ
α : Cochain F K (-1)
β : G ⟶ K
eq : δ (-1) 0 α = Cochain.ofHom (φ ≫ β)
p q : ℤ
hpq : p + 1 = q
⊢ (desc φ α β eq).f p = (↑(fst φ)).v p q hpq ≫ α.v q p ⋯ + (snd φ).v p p ⋯ ≫ β.f p

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
f₁ f₂ : mappingCone φ ⟶ K
γ₁ : Cochain F K (-2)
γ₂ : Cochain G K (-1)
h₁ :
  (inl φ).comp (Cochain.ofHom f₁) ⋯ = δ (-2) (-1) γ₁ + (Cochain.ofHom φ).comp γ₂ ⋯ + (inl φ).comp (Cochain.ofHom f₂) ⋯
h₂ : Cochain.ofHom (inr φ ≫ f₁) = δ (-1) 0 γ₂ + Cochain.ofHom (inr φ ≫ f₂)
⊢ -2 + 1 = -1

Tactic:
by

Goal state after:
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
n : ℤ
f : F ⟶ G
z : Cochain G K n
m : ℤ
⊢ δ n m ((Cochain.ofHom f).comp z ⋯) = (Cochain.ofHom f).comp (δ n m z) ⋯

Tactic:
by

Goal state after:
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 n : ℤ
z : Cocycle G K n
f : F ⟶ G
⊢ δ n (n + 1) ((Cochain.ofHom f).comp ↑z ⋯) = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝³ : HasHomotopyCofiber φ
inst✝² : DecidableRel c.Rel
F' F'' G' G'' : HomologicalComplex C c
φ' : F' ⟶ G'
φ'' : F'' ⟶ G''
inst✝¹ : HasHomotopyCofiber φ'
inst✝ : HasHomotopyCofiber φ''
H : ∀ (j : ι), ∃ i, c.Rel i j
α : Arrow.mk φ ⟶ Arrow.mk φ'
β : Arrow.mk φ' ⟶ Arrow.mk φ''
i : ι
hi : ¬c.Rel i (c.next i)
⊢ (mapArrowHom φ φ'' H (α ≫ β)).f i ≫ sndX φ'' i =
    ((mapArrowHom φ φ' H α).f i ≫ (mapArrowHom φ' φ'' H β).f i) ≫ sndX φ'' i

Tactic:
by

Goal state after:
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 n : ℤ
z : Cocycle F G n
f : G ⟶ K
⊢ δ n (n + 1) ((↑z).comp (Cochain.ofHom f) ⋯) = 0

Tactic:
by

Goal state after:
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
ho : Homotopy φ₁ φ₂
⊢ ofHom φ₁ = δ (-1) 0 (ofHomotopy ho) + ofHom φ₂

Tactic:
by

Goal state after:
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 }
i j : ℤ
hij : j + 1 ≠ i
x✝ : i + -1 = j
⊢ j + 1 = i

Tactic:
by

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

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v, u_1} C
inst✝³ : Category.{v', u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
F G : CochainComplex C ℤ
φ : F ⟶ G
inst✝ : HasHomotopyCofiber φ
K : CochainComplex C ℤ
f₁ f₂ : mappingCone φ ⟶ K
γ₁ : Cochain F K (-2)
γ₂ : Cochain G K (-1)
h₁ :
  (inl φ).comp (Cochain.ofHom f₁) ⋯ = δ (-2) (-1) γ₁ + (Cochain.ofHom φ).comp γ₂ ⋯ + (inl φ).comp (Cochain.ofHom f₂) ⋯
h₂ : Cochain.ofHom (inr φ ≫ f₁) = δ (-1) 0 γ₂ + Cochain.ofHom (inr φ ≫ f₂)
⊢ Cochain.ofHom f₁ = δ (-1) 0 (descCochain φ γ₁ γ₂ descHomotopy._proof_1) + Cochain.ofHom f₂

Tactic:
by

Goal state after: