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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)
i : ι
⊢ biprod.inl.f i ≫ biprod.snd.f i = 0

Tactic:
by
  rw [← comp_f, biprod.inl_snd, zero_f]

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

Goal state before:
pp: C : Type u_1
ι : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
c : ComplexShape ι
K L : HomologicalComplex C c
inst✝ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (L.X i)
M : HomologicalComplex C c
α : M ⟶ K
β : M ⟶ L
i : ι
⊢ (biprod.lift α β).f i ≫ biprod.snd.f i = β.f i

Tactic:
by
  rw [← comp_f, biprod.lift_snd]

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 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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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 n₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
h₁₂ : n₁ + n₂ = n₁₂
h₂₃ : n₂ + n₃ = n₂₃
h₁₂₃ : n₁ + n₂ + n₃ = n₁₂₃
⊢ n₁ + (n₂ + n₃) = n₁₂₃

Tactic:
← h₁₂₃,

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 n₁ n₂ n₃ n₁₂ n₂₃ n₁₂₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
h₁₂ : n₁ + n₂ = n₁₂
h₂₃ : n₂ + n₃ = n₂₃
h₁₂₃ : n₁ + n₂ + n₃ = n₁₂₃
⊢ n₁ + (n₂ + n₃) = n₁ + n₂ + n₃
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 L : CochainComplex C ℤ
n₁ n₂ n₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
p q : ℤ
hpq : p + (n₁ + n₂ + n₃) = q
⊢ p + n₁ + n₂ + n₃ = q

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₂ n₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
p q : ℤ
hpq : p + (n₁ + n₂ + n₃) = q
⊢ p + n₁ = p + n₁

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₂ n₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
p q : ℤ
hpq : p + (n₁ + n₂ + n₃) = q
⊢ p + n₁ + n₂ = p + n₁ + n₂

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₂ n₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
p q : ℤ
hpq : p + (n₁ + n₂ + n₃) = q
⊢ p + n₁ + (n₂ + n₃) = q

Tactic:
lia

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 L : CochainComplex C ℤ
n₂ n₃ n₂₃ : ℤ
z₁ : Cochain F G 0
z₂ : Cochain G K n₂
z₃ : Cochain K L n₃
h₂₃ : n₂ + n₃ = n₂₃
⊢ 0 + n₂ + n₃ = n₂₃

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₃ n₁₃ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K 0
z₃ : Cochain K L n₃
h₁₃ : n₁ + n₃ = n₁₃
⊢ n₁ + 0 + n₃ = n₁₃

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
z₃ : Cochain K L 0
h₁₂ : n₁ + n₂ = n₁₂
⊢ n₁ + n₂ + 0 = n₁₂

Tactic:
lia

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 L : CochainComplex C ℤ
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K (-n₂)
z₃ : Cochain K L n₂
h₁₂ : n₁ + -n₂ = n₁₂
⊢ n₁ + -n₂ + n₂ = n₁

Tactic:
lia

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
n₁ n₂ n₁₂ : ℤ
z₂ : Cochain G K n₂
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ p + n₁ + n₂ = q

Tactic:
lia

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
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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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
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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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 : 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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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
n₁ n₂ n₁₂ : ℤ
z₁ : Cochain F G n₁
h : n₁ + n₂ = n₁₂
p q : ℤ
hpq : p + n₁₂ = q
⊢ p + n₁ + n₂ = q

Tactic:
lia

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
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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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 : 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
⊢ p + n₁ + n₂ = q

Tactic:
lia

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
p q : ℤ
hpq : p + -1 = q
⊢ q + 1 = p

Tactic:
lia

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
⊢ 1 + 1 = 2

Tactic:
lia

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
p : ℤ
⊢ 1 + 1 = 2

Tactic:
lia

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
p : ℤ
⊢ p + 2 = p + 2

Tactic:
lia

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
p : ℤ
⊢ p + 1 = p + 2 - 1

Tactic:
lia

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
p : ℤ
⊢ p - 1 + 1 = p

Tactic:
lia

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
⊢ (inl φ).comp (snd φ) ⋯ = 0

Tactic:
ext
  simp

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
⊢ (Cochain.ofHom (inr φ)).comp ↑(fst φ) ⋯ = 0

Tactic:
ext
  simp

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
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₁ q₂ : ℤ
hq₁ : q₁ = q - 1
hq₂ : p + 1 = q₂
⊢ q₁ = p + n

Tactic:
lia

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
d e : ℤ
γ : Cochain F K d
he : 1 + d = e
⊢ 1 + d = e

Tactic:
lia

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
d e : ℤ
γ : Cochain F K d
he : 1 + d = e
⊢ -1 + 1 + d = d

Tactic:
lia

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
n m : ℤ
hnm : n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
q₂ : ℤ
hq₂ : p + 1 = q₂
hq₁ : p + n = q - 1
⊢ q₂ = p + m - n

Tactic:
lia

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
d e f : ℤ
γ : Cochain G K d
he : 0 + d = e
hf : -1 + e = f
⊢ e = d

Tactic:
lia

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
d e f : ℤ
γ : Cochain F K d
he : 1 + d = e
hf : 0 + e = f
⊢ e = f

Tactic:
lia

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
d e : ℤ
γ : Cochain G K d
he : 0 + d = e
⊢ d = e

Tactic:
lia

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
n m : ℤ
hnm : ¬n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
x✝ : p + n + 1 = q
⊢ n + 1 = m

Tactic:
lia

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
n m : ℤ
hnm : ¬n + 1 = m
z : Cochain F G n
p q : ℤ
hpq : p + m = q
x✝ : p + 1 = p + m - n
⊢ n + 1 = m

Tactic:
lia

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
i j : ℤ
hij : j + 1 = i
A : C
f g : (mappingCone φ).X j ⟶ A
⊢ i + -1 = j

Tactic:
lia

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
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ p + n₁ = q - 1

Tactic:
lia

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
i : ℤ
A : C
f : A ⟶ (mappingCone φ).X i
j : ℤ
hij : i + 1 = j
a : A ⟶ F.X j
b : A ⟶ G.X i
⊢ j + -1 = i

Tactic:
lia

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
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ q - 2 = q - 1 - 1

Tactic:
lia

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
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ p + 1 + n₁ = q

Tactic:
lia

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
n₀ n₁ n₂ : ℤ
z : Cochain F G n₀
h₁₂ : n₁ + 1 = n₂
h₀₁ : n₀ + 1 = n₁
p q : ℤ
hpq : p + n₂ = q
⊢ p + 1 + 1 = p + 2

Tactic:
lia

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
j : ℤ
A : C
f : (mappingCone φ).X j ⟶ A
i : ℤ
hij : j + 1 = i
⊢ i + -1 = j

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + n₁ + (n₂ + 1) = q

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ n₁ + 1 + n₂ = n₁ + n₂ + 1

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + (n₁ + 1) = p + n₁ + 1

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + n₁ + 1 + n₂ = q

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + n₁ + n₂ = q - 1

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + 1 + n₁ = p + n₁ + 1

Tactic:
lia

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
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
⊢ p + j = q + 1

Tactic:
lia

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
n₁ n₂ : ℤ
z₁ : Cochain F G n₁
z₂ : Cochain G K n₂
p q : ℤ
hpq : p + (n₁ + n₂ + 1) = q
⊢ p + n₁ = p + n₁ + 1 - 1

Tactic:
lia

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
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
⊢ -1 + j = i

Tactic:
lia

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 : Cochain F G 0
p q : ℤ
hpq : p + 1 = q
⊢ p = q - 1

Tactic:
lia

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

Goal state before:
pp: case neg.hnm
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
p : ℤ
φ : F ⟶ G
h : ¬p = 1
⊢ ¬0 + 1 = p

Tactic:
lia

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
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
p : ℤ
eq : φ₁.f p = F.d p (p + 1) ≫ h.hom (p + 1) p + h.hom p (p - 1) ≫ G.d (p - 1) p + φ₂.f p
⊢ (δ (-1) 0 (Cochain.ofHomotopy h)).v p p ⋯ = (Cochain.ofHom φ₁ - Cochain.ofHom φ₂).v p p ⋯

Tactic:
Cochain.ofHomotopy,

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
F G : CochainComplex C ℤ
φ₁ φ₂ : F ⟶ G
h : Homotopy φ₁ φ₂
p : ℤ
eq : φ₁.f p = F.d p (p + 1) ≫ h.hom (p + 1) p + h.hom p (p - 1) ≫ G.d (p - 1) p + φ₂.f p
⊢ (δ (-1) 0 (Cochain.mk fun p q x ↦ h.hom p q)).v p p ⋯ = (Cochain.ofHom φ₁ - Cochain.ofHom φ₂).v p p ⋯
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
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
⊢ p + 1 + i = q

Tactic:
lia

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
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
h₂ : (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯
p q : ℤ
hpq : p + j = q
h₁ : ((inl φ).comp γ₁ ⋯).v (p + 1) q ⋯ = ((inl φ).comp γ₂ ⋯).v (p + 1) q ⋯
⊢ p + j = q

Tactic:
lia

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
i j : ℤ
hij : i + 1 = j
K : CochainComplex C ℤ
γ₁ γ₂ : Cochain (mappingCone φ) K j
p q : ℤ
hpq : p + j = q
h₁ : ((inl φ).comp γ₁ ⋯).v (p + 1) q ⋯ = ((inl φ).comp γ₂ ⋯).v (p + 1) q ⋯
h₂ : ((Cochain.ofHom (inr φ)).comp γ₁ ⋯).v p q ⋯ = ((Cochain.ofHom (inr φ)).comp γ₂ ⋯).v p q ⋯
⊢ p + 1 + -1 = p

Tactic:
lia

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
p q : ℤ
hpq : p + 1 = q
⊢ q + -1 = p

Tactic:
lia

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
p q : ℤ
hpq : p + 1 = q
⊢ q + -1 = p

Tactic:
lia

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
i j k : ℤ
hij : i + -1 = j
hik : k + -1 = i
⊢ (inl φ).v i j hij ≫ (mappingCone φ).d j i = φ.f i ≫ (inr φ).f i - F.d i k ≫ (inl φ).v k i hik

Tactic:
dsimp [mappingCone, inl, inr]

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
i j k : ℤ
hij : i + -1 = j
hik : k + -1 = i
⊢ homotopyCofiber.inlX φ i j ⋯ ≫ homotopyCofiber.d φ j i =
    φ.f i ≫ homotopyCofiber.inrX φ i - F.d i k ≫ homotopyCofiber.inlX φ k i ⋯
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
i j k : ℤ
hij : i + -1 = j
hik : k + -1 = i
⊢ j + 1 = i

Tactic:
lia

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
i j k : ℤ
hij : i + -1 = j
hik : k + -1 = i
⊢ i + 1 = k

Tactic:
lia

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
i j : ℤ
hij : i + 1 = j
⊢ i - 1 + 1 = i

Tactic:
lia

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
i j : ℤ
hij : i + 1 = j
⊢ i - 1 + 1 = i

Tactic:
lia

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
n : ℤ
⊢ n - 1 + 1 = n

Tactic:
lia

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
p : ℤ
⊢ p + -1 = p - 1

Tactic:
lia

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
p : ℤ
⊢ p + 1 + -1 = p

Tactic:
lia

Goal state after: