Theorem: CochainComplex.mappingCone.ext_cochain_from_iff
File: Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{?u.71645, 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
⊢ -1 + j = i
state_after: no goals

tactic: constructor
state_before: C : Type u_1
inst✝² : Category.{u_2, 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
⊢ γ₁ = γ₂ ↔
    (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯
state_after: case mp
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ γ₁ = γ₂ →
    (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯

case mpr
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯ →
    γ₁ = γ₂

tactic: rintro rfl
state_before: case mp
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ γ₁ = γ₂ →
    (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯
state_after: case mp
C : Type u_1
inst✝² : Category.{u_2, 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 : Cochain (mappingCone φ) K j
⊢ (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₁ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₁ ⋯

tactic: tauto
state_before: case mp
C : Type u_1
inst✝² : Category.{u_2, 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 : Cochain (mappingCone φ) K j
⊢ (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₁ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₁ ⋯
state_after: no goals

tactic: rintro ⟨h₁, h₂⟩
state_before: case mpr
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ (inl φ).comp γ₁ ⋯ = (inl φ).comp γ₂ ⋯ ∧ (Cochain.ofHom (inr φ)).comp γ₁ ⋯ = (Cochain.ofHom (inr φ)).comp γ₂ ⋯ →
    γ₁ = γ₂
state_after: case mpr.intro
C : Type u_1
inst✝² : Category.{u_2, 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
state_before: case mpr.intro
C : Type u_1
inst✝² : Category.{u_2, 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
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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

tactic: rw [ext_from_iff φ (p + 1) p rfl]
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq

tactic: replace h₁ := Cochain.congr_v h₁ (p + 1) q (by omega)
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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 ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq

tactic: replace h₂ := Cochain.congr_v h₂ p q (by omega)
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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 ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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 ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq

tactic: simp only [Cochain.comp_v (inl φ) _ _ (p + 1) p q (by omega) hpq] at h₁
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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 ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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
h₂ : ((Cochain.ofHom (inr φ)).comp γ₁ ⋯).v p q ⋯ = ((Cochain.ofHom (inr φ)).comp γ₂ ⋯).v p q ⋯
h₁ : (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq

tactic: simp only [Cochain.zero_cochain_comp_v, Cochain.ofHom_v] at h₂
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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
h₂ : ((Cochain.ofHom (inr φ)).comp γ₁ ⋯).v p q ⋯ = ((Cochain.ofHom (inr φ)).comp γ₂ ⋯).v p q ⋯
h₁ : (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
state_after: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq
h₂ : (inr φ).f p ≫ γ₁.v p q ⋯ = (inr φ).f p ≫ γ₂.v p q ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq

tactic: exact ⟨h₁, h₂⟩
state_before: case mpr.intro.h
C : Type u_1
inst✝² : Category.{u_2, 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).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq
h₂ : (inr φ).f p ≫ γ₁.v p q ⋯ = (inr φ).f p ≫ γ₂.v p q ⋯
⊢ (inl φ).v (p + 1) p ⋯ ≫ γ₁.v p q hpq = (inl φ).v (p + 1) p ⋯ ≫ γ₂.v p q hpq ∧
    (inr φ).f p ≫ γ₁.v p q hpq = (inr φ).f p ≫ γ₂.v p q hpq
state_after: no goals

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, 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
state_after: no goals

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, 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
state_after: no goals

tactic: omega
state_before: C : Type u_1
inst✝² : Category.{u_2, 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
state_after: no goals