-- Mathlib/Algebra/Homology/HomotopyCofiber.lean
lemma inrX_nullHomotopy_f (j : ι) :
    inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 _).f j

Proof state:
case pos.intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ (K.biprodXIso K j).inv ≫
      biprod.snd.f j ≫ (biprod.lift (𝟙 K) (-𝟙 K)).f j ≫ homotopyCofiber.inrX (biprod.lift (𝟙 K) (-𝟙 K)) j =
    (K.biprodXIso K j).inv ≫ ((biprod.desc (𝟙 K) (𝟙 K)).f j ≫ (ι₀ K).f j - inrX K j)

Tactic:
ext

Result:
case pos.intro.h₀
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ biprod.inl ≫
      (K.biprodXIso K j).inv ≫
        biprod.snd.f j ≫ (biprod.lift (𝟙 K) (-𝟙 K)).f j ≫ homotopyCofiber.inrX (biprod.lift (𝟙 K) (-𝟙 K)) j =
    biprod.inl ≫ (K.biprodXIso K j).inv ≫ ((biprod.desc (𝟙 K) (𝟙 K)).f j ≫ (ι₀ K).f j - inrX K j)

case pos.intro.h₁
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ biprod.inr ≫
      (K.biprodXIso K j).inv ≫
        biprod.snd.f j ≫ (biprod.lift (𝟙 K) (-𝟙 K)).f j ≫ homotopyCofiber.inrX (biprod.lift (𝟙 K) (-𝟙 K)) j =
    biprod.inr ≫ (K.biprodXIso K j).inv ≫ ((biprod.desc (𝟙 K) (𝟙 K)).f j ≫ (ι₀ K).f j - inrX K j)

Proof state:
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
⊢ inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 K.cylinder).f j

Tactic:
obtain ⟨i, hij⟩ := hc j

Result:
case intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
⊢ inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 K.cylinder).f j

Proof state:
case intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
⊢ inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 K.cylinder).f j

Tactic:
dsimp [nullHomotopicMap]

Result:
case intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
⊢ inrX K j ≫
      (Homotopy.nullHomotopicMap' fun i j hij =>
            homotopyCofiber.sndX (biprod.lift (𝟙 K) (-𝟙 K)) i ≫ biprod.snd.f i ≫ inlX K i j hij).f
        j =
    inrX K j ≫ ((π K).f j ≫ (ι₀ K).f j - 𝟙 (homotopyCofiber.X (biprod.lift (𝟙 K) (-𝟙 K)) j))

Proof state:
case pos.intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ inrX K j ≫
      (Homotopy.nullHomotopicMap' fun i j hij =>
            homotopyCofiber.sndX (biprod.lift (𝟙 K) (-𝟙 K)) i ≫ biprod.snd.f i ≫ inlX K i j hij).f
        j =
    inrX K j ≫ ((π K).f j ≫ (ι₀ K).f j - 𝟙 (homotopyCofiber.X (biprod.lift (𝟙 K) (-𝟙 K)) j))

Tactic:
simp only [Homotopy.nullHomotopicMap'_f hij hjk,
  homotopyCofiber_X, homotopyCofiber_d, assoc, comp_add,
  homotopyCofiber.inrX_d_assoc, homotopyCofiber.inrX_sndX_assoc, comp_sub,
  inrX_π_assoc, comp_id, ← Hom.comm_assoc, homotopyCofiber.inlX_d _ _ _ _ _ hjk,
  comp_neg, add_neg_cancel_left]

Result:
case pos.intro
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ biprod.snd.f j ≫ (biprod.lift (𝟙 K) (-𝟙 K)).f j ≫ homotopyCofiber.inrX (biprod.lift (𝟙 K) (-𝟙 K)) j =
    (biprod.desc (𝟙 K) (𝟙 K)).f j ≫ (ι₀ K).f j - inrX K j

Proof state:
case pos.intro.h₁
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
i : ι
hij : c.Rel i j
k : ι
hjk : c.Rel j k
⊢ biprod.inr ≫
      (K.biprodXIso K j).inv ≫
        biprod.snd.f j ≫ (biprod.lift (𝟙 K) (-𝟙 K)).f j ≫ homotopyCofiber.inrX (biprod.lift (𝟙 K) (-𝟙 K)) j =
    biprod.inr ≫ (K.biprodXIso K j).inv ≫ ((biprod.desc (𝟙 K) (𝟙 K)).f j ≫ (ι₀ K).f j - inrX K j)

Tactic:
simp only [inr_biprodXIso_inv_assoc, biprod_inr_snd_f_assoc, comp_sub,
  biprod_inr_desc_f_assoc, id_f, id_comp, ι₀, comp_f, this,
  sub_f_apply, sub_comp, homotopyCofiber_X, homotopyCofiber.inr_f]

Result:
no goals

Proof state:
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
⊢ inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 K.cylinder).f j

Tactic:
have : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr :=
  biprod.hom_ext _ _ (by simp) (by simp)

Result:
C : Type u_1
inst✝⁴ : Category.{u_3, u_1} C
inst✝³ : Preadditive C
ι : Type u_2
c : ComplexShape ι
K : HomologicalComplex C c
inst✝² : DecidableRel c.Rel
inst✝¹ : ∀ (i : ι), HasBinaryBiproduct (K.X i) (K.X i)
inst✝ : HasHomotopyCofiber (biprod.lift (𝟙 K) (-𝟙 K))
hc : ∀ (j : ι), ∃ i, c.Rel i j
j : ι
this : biprod.lift (𝟙 K) (-𝟙 K) = biprod.inl - biprod.inr
⊢ inrX K j ≫ (nullHomotopicMap K).f j = inrX K j ≫ (π K ≫ ι₀ K - 𝟙 K.cylinder).f j