awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
simp :
  HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((dNext (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((prevD (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫
        ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯))
      (Iso.hom (K.extendXIso e ⋯) ≫
        ((prevD i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯))
⊢ f.f i =
    HAdd.hAdd (α := K.X i ⟶ L.X i)
        ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
          Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯))
        ((prevD i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
          Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) +
      g.f i

Tactic:
simp only [← cancel_mono (L.extendXIso e rfl).inv,
      ← cancel_epi (K.extendXIso e rfl).hom, this, Preadditive.add_comp,
      Preadditive.comp_add, add_left_inj]

Premises used: add_left_inj, CategoryTheory.cancel_mono, CategoryTheory.Preadditive.add_comp, CategoryTheory.Preadditive.comp_add, CategoryTheory.cancel_epi, CategoryTheory.Iso.hom, HomologicalComplex.extendXIso, CategoryTheory.Iso.inv, rfl
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
pos :
  ∀ (hi : c.Rel i (c.next i)),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((dNext (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫
          ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
              Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
            Iso.inv (L.extendXIso e ⋯) :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
neg :
  ∀ (hi : ¬c.Rel i (c.next i)),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((dNext (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫
          ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
              Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
            Iso.inv (L.extendXIso e ⋯) :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫
        ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯) :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
by_cases hi : c.Rel i (c.next i)

Premises used: ComplexShape.next, ComplexShape.Rel
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi rwa rwa₁ rwa₂ rwa₃ : c.Rel i (c.next i)
⊢ c'.Rel (e.f i) (e.f (c.next i))

Tactic:
rwa [e.rel_iff]

Premises used: ComplexShape.Embedding.rel_iff
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : c.Rel i (c.next i)
⊢ c'.Rel (e.f i) (e.f (c.next i))

Tactic:
have hi' : c'.Rel (e.f i) (e.f (c.next i)) := by rwa [e.rel_iff]
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : c.Rel i (c.next i)
hi' : c'.Rel (e.f i) (e.f (c.next i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫
        ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯) :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
simp [dNext_eq _ hi, dNext_eq _ hi', K.extend_d_eq _ rfl rfl]

Premises used: dNext_eq, HomologicalComplex.extend_d_eq, rfl
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : ¬c.Rel i (c.next i)
neg :
  Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫
        ((dNext i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯) :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
rw [dNext_eq_zero _ _ hi]

Premises used: dNext_eq_zero
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : ¬c.Rel i (c.next i)
pos :
  ∀ (hi' : c'.Rel (e.f i) (c'.next (e.f i))),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((dNext (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
neg :
  ∀ (hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((dNext (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
by_cases hi' : c'.Rel (e.f i) (c'.next (e.f i))

Premises used: ComplexShape.next, ComplexShape.Embedding.f, ComplexShape.Rel
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : ¬c.Rel i (c.next i)
hi' : c'.Rel (e.f i) (c'.next (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
simp [dNext_eq _ hi', extend_d_from_eq_zero _ _ _ _ _ rfl hi]

Premises used: HomologicalComplex.extend_d_from_eq_zero, dNext_eq, rfl
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi : ¬c.Rel i (c.next i)
hi' : ¬c'.Rel (e.f i) (c'.next (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((dNext (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫ 0 ≫ Iso.inv (L.extendXIso e ⋯) : (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
simp [dNext_eq_zero _ _ hi']

Premises used: dNext_eq_zero
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
pos :
  ∀ (hi : c.Rel (c.prev i) i),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((prevD (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫
          ((prevD i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
              Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
            Iso.inv (L.extendXIso e ⋯) :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
neg :
  ∀ (hi : ¬c.Rel (c.prev i) i),
    Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      ((prevD (e.f i) :
            (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
          h.hom :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
      (Iso.hom (K.extendXIso e ⋯) ≫
          ((prevD i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
              Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
            Iso.inv (L.extendXIso e ⋯) :
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
⊢ Eq.{v_1 + 1} (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    ((prevD (e.f i) :
          (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        h.hom :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
    (Iso.hom (K.extendXIso e ⋯) ≫
        ((prevD i : (a : (i j : ι) → K.X i ⟶ L.X j) → K.X i ⟶ L.X i) fun (i j : ι) ↦
            Iso.inv (K.extendXIso e ⋯) ≫ h.hom (e.f i) (e.f j) ≫ Iso.hom (L.extendXIso e ⋯)) ≫
          Iso.inv (L.extendXIso e ⋯) :
      (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))

Tactic:
by_cases hi : c.Rel (c.prev i) i

Premises used: ComplexShape.prev, ComplexShape.Rel
-- Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy
Goal:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst : Category.{v_1, u_3} C
inst_1 : HasZeroObject.{v_1, u_3} C
inst_2 : Preadditive.{v_1, u_3} C
K L : HomologicalComplex.{v_1, u_3, u_1} C c
f g : K ⟶ L
e : c.Embedding c'
inst_3 : e.IsRelIff
h : Homotopy (C := K.extend e) (extendMap f e) (extendMap g e)
i : ι
this :
  Iso.hom (K.extendXIso e ⋯) ≫ f.f i ≫ Iso.inv (L.extendXIso e ⋯) =
    HAdd.hAdd (α := (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) (β :=
        (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((dNext (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
        ((prevD (e.f i) :
              (a : (i j : ι') → (K.extend e).X i ⟶ (L.extend e).X j) → (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i))
            h.hom :
          (K.extend e).X (e.f i) ⟶ (L.extend e).X (e.f i)) +
      Iso.hom (K.extendXIso e ⋯) ≫ g.f i ≫ Iso.inv (L.extendXIso e ⋯)
hi rwa rwa₁ rwa₂ rwa₃ : c.Rel (c.prev i) i
⊢ c'.Rel (e.f (c.prev i)) (e.f i)

Tactic:
rwa [e.rel_iff]

Premises used: ComplexShape.Embedding.rel_iff