mathlib-initiative/mathlib-tactics huggingface dataset:
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 }
p : ℤ
eq : (ofHom φ₁).v p p ⋯ = (δ (-1) 0 ↑z + ofHom φ₂).v p p ⋯
⊢ (ComplexShape.up ℤ).Rel (p - 1) 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
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
⊢ m + -1 = n

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''
⊢ mapArrowHom φ φ'' H (α ≫ β) = mapArrowHom φ φ' H α ≫ mapArrowHom φ' φ'' H 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
n m : ℤ
α : Cochain K F m
β : Cochain K G n
h : n + 1 = m
⊢ (liftCochain φ α β h).comp (↑(fst φ)) h = α

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 }
p : ℤ
eq : (ofHom φ₁).v p p ⋯ = (δ (-1) 0 ↑z + ofHom φ₂).v p p ⋯
h₁ : (ComplexShape.up ℤ).Rel (p - 1) p
⊢ (ComplexShape.up ℤ).Rel p (p + 1)

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 K F m
β : Cochain K G n
h : n + 1 = m
⊢ (liftCochain φ α β h).comp (snd φ) ⋯ = 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 K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ p₁ + m = p₃

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 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'
⊢ mapArrowHom φ φ' H α.hom ≫ mapArrowHom φ' φ H α.inv = 𝟙 (homotopyCofiber 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 K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ p₁ + m = p₃

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 K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + 1 = p₃
⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (↑(fst φ)).v p₂ p₃ h₂₃ = α.v p₁ p₃ ⋯

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 K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ p₁ + n = p₂

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 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'
⊢ mapArrowHom φ' φ H α.inv ≫ mapArrowHom φ φ' H α.hom = 𝟙 (homotopyCofiber 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 K F m
β : Cochain K G n
h : n + 1 = m
p₁ p₂ : ℤ
h₁₂ : p₁ + n = p₂
⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (snd φ).v p₂ p₂ ⋯ = β.v p₁ p₂ h₁₂

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 K F m
β : Cochain K G n
h : n + 1 = m
m' : ℤ
hm' : m + 1 = m'
⊢ m' + -1 = m

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i : ι
hi : c.Rel i (c.next i)
⊢ H.obj ((homotopyCofiber φ).X i) ≅ (homotopyCofiber ((H.mapHomologicalComplex c).map φ)).X 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 K F m
β : Cochain K G n
h : n + 1 = m
m' : ℤ
hm' : m + 1 = m'
⊢ δ n m (liftCochain φ α β h) =
    -(δ m m' α).comp (inl φ) ⋯ + (δ n m β + α.comp (Cochain.ofHom φ) ⋯).comp (Cochain.ofHom (inr φ)) ⋯

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
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 j : ℕ
h : i + 1 = j
⊢ (xNextIso P h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ i).snd.fst =
    (mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).fst ≫ (xPrevIso Q h).hom

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i j : ι
hij : c.Rel j i
⊢ inlX ((H.mapHomologicalComplex c).map φ) i j hij ≫ (mapHomologicalComplexObjXIso φ H j).inv = H.map (inlX φ i j hij)

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 : Cocycle K F m
β : Cochain K G n
h : n + 1 = m
eq : δ n m β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
⊢ δ n m (liftCochain φ (↑α) β h) = 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
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
⊢ Cochain.ofHom (lift φ α β eq) = liftCochain φ (↑α) β ⋯

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 }
p : ℤ
⊢ φ₁.f p =
    (((dNext p) fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij else 0) +
        (prevD p) fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij else 0) +
      φ₂.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 : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 1 = q
⊢ (lift φ α β eq).f p ≫ (↑(fst φ)).v p q hpq = (↑α).v p q hpq

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 : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
⊢ (Cochain.ofHom (lift φ α β eq)).comp ↑(fst φ) ⋯ = ↑α

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 : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 0 = q
⊢ q = p

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
i j : ℤ
h : ¬i + -1 = j
h' : (ComplexShape.up ℤ).Rel j i
⊢ i + -1 = j

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 : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 0 = q
⊢ (lift φ α β eq).f p ≫ (snd φ).v p q hpq = β.v p q hpq

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i : ι
⊢ inrX ((H.mapHomologicalComplex c).map φ) i ≫ (mapHomologicalComplexObjXIso φ H i).inv = H.map (inrX 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
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
⊢ (Cochain.ofHom (lift φ α β eq)).comp (snd φ) ⋯ = β

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
⊢ (fun z ↦ { hom := fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij else 0, zero := ⋯, comm := ⋯ })
      ((fun ho ↦ ⟨ofHomotopy ho, ⋯⟩) ho) =
    ho

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''
i j : ℕ
w : ¬(ComplexShape.up ℕ).Rel j i
⊢ (if h : j + 1 = i then (xNextIso P h).inv ≫ (mkCoinductiveAux₂ e zero comm_zero one comm_one succ j).snd.fst else 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
α : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 1 = q
⊢ q + -1 = 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 : Cocycle K F 1
β : Cochain K G 0
eq : δ 0 1 β + (↑α).comp (Cochain.ofHom φ) ⋯ = 0
p q : ℤ
hpq : p + 1 = q
⊢ (lift φ α β eq).f p = (↑α).v p q hpq ≫ (inl φ).v q p ⋯ + β.v p p ⋯ ≫ (inr φ).f p

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i : ι
⊢ H.map (inrX φ i) ≫ (mapHomologicalComplexObjXIso φ H i).hom = inrX ((H.mapHomologicalComplex c).map 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 : 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:
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
f₁ f₂ : F ⟶ G
h : ofHom f₁ = ofHom f₂
⊢ (Cocycle.equivHom F G) f₁ = (Cocycle.equivHom F G) f₂

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 : K ⟶ mappingCone φ
α : Cochain K F 0
β : Cochain K G (-1)
h₁ : (Cochain.ofHom f₁).comp ↑(fst φ) ⋯ = -δ 0 1 α + (Cochain.ofHom f₂).comp ↑(fst φ) ⋯
h₂ : (Cochain.ofHom f₁).comp (snd φ) ⋯ = δ (-1) 0 β + α.comp (Cochain.ofHom φ) ⋯ + (Cochain.ofHom f₂).comp (snd φ) ⋯
⊢ Cochain.ofHom f₁ = δ (-1) 0 (liftCochain φ α β ⋯) + Cochain.ofHom f₂

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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ m + m' = 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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ n + n' = 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
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n : ℤ
hpq : p + n = q
⊢ (single f n).v p q hpq = 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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ m + -1 + n' = ?m.130

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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
⊢ (liftCochain φ α β h).comp (descCochain φ α' β' h') hp = α.comp α' ⋯ + β.comp β' ⋯

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
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hp' : p' ≠ p
h : p = p' ∧ q = q'
⊢ p' = 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
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hp' : p' ≠ p
⊢ (single f n).v p' q' hpq' = 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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + n' = p₃
q : ℤ
hq : p₁ + m = q
⊢ q + m' = 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
K L : CochainComplex C ℤ
p q : ℤ
f : K.X p ⟶ L.X q
n p' q' : ℤ
hpq' : p' + n = q'
hq' : q' ≠ q
⊢ (single f n).v p' q' hpq' = 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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + n' = p₃
q : ℤ
hq : p₁ + m = q
⊢ p₁ + p = p₃

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G K : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
i j : ι
hij : c.Rel i j
⊢ (mapHomologicalComplexObjXIso φ H i).symm.hom ≫ ((H.mapHomologicalComplex c).obj (homotopyCofiber φ)).d i j =
    (homotopyCofiber ((H.mapHomologicalComplex c).map φ)).d i j ≫ (mapHomologicalComplexObjXIso φ H j).symm.hom

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 L : CochainComplex C ℤ
n m : ℤ
α : Cochain K F m
β : Cochain K G n
n' m' : ℤ
α' : Cochain F L m'
β' : Cochain G L n'
h : n + 1 = m
h' : m' + 1 = n'
p : ℤ
hp : n + n' = p
p₁ p₂ p₃ : ℤ
h₁₂ : p₁ + n = p₂
h₂₃ : p₂ + n' = p₃
q : ℤ
hq : p₁ + m = q
⊢ (liftCochain φ α β h).v p₁ p₂ h₁₂ ≫ (descCochain φ α' β' h').v p₂ p₃ h₂₃ =
    α.v p₁ q hq ≫ α'.v q p₃ ⋯ + β.v p₁ p₂ h₁₂ ≫ β'.v p₂ p₃ h₂₃

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
K L : CochainComplex C ℤ
p q n : ℤ
⊢ single 0 n = 0

Tactic:
by

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

Goal state before:
pp: C : Type u_1
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Preadditive C
ι : Type u_2
c : ComplexShape ι
F G : HomologicalComplex C c
φ : F ⟶ G
inst✝⁵ : HasHomotopyCofiber φ
inst✝⁴ : DecidableRel c.Rel
D : Type u_3
inst✝³ : Category.{v_2, u_3} D
inst✝² : Preadditive D
H : C ⥤ D
inst✝¹ : H.Additive
inst✝ : HasHomotopyCofiber ((H.mapHomologicalComplex c).map φ)
⊢ (H.mapHomologicalComplex c).map (inr φ) ≫ (mapHomologicalComplexObjIso φ H).hom =
    inr ((H.mapHomologicalComplex c).map φ)

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 K : HomologicalComplex C c
φ : F ⟶ G
inst✝³ : HasHomotopyCofiber φ
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : K.HasCylinder
φ₀ φ₁ : K ⟶ F
h : Homotopy φ₀ φ₁
⊢ biprod.lift (𝟙 K) (-𝟙 K) ≫ biprod.desc φ₀ φ₁ = φ₀ - φ₁

Tactic:
by

Goal state after: