awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hf : f ≫ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).d i j = 0
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
dsimp : Eq (α := A ⟶ K.X (j + n)) ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) 0
⊢ Eq (α := A ⟶ K.X (j + n)) ((f ≫ (K.shiftFunctorObjXIso n i (i + n) ⋯).hom) ≫ K.d (i + n) (j + n)) 0

Tactic:
dsimp at hf ⊢
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hf : f ≫ HSMul.hSMul (β := K.X (i + n) ⟶ K.X (j + n)) n.negOnePow (K.d (i + n) (j + n)) = 0
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
⊢ HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow (f ≫ K.d (i + n) (j + n)) = 0

Tactic:
simp only [Linear.comp_units_smul] at hf

Premises used: CategoryTheory.Linear.comp_units_smul
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
hf : HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow (f ≫ K.d (i + n) (j + n)) = 0
apply : HSMul.hSMul (α := ℤˣ) (β := A ⟶ K.X (j + n)) 1 ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) = 0
⊢ Eq (α := A ⟶ K.X (j + n)) ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) 0

Tactic:
apply (one_smul (M := ℤˣ) _).symm.trans _

Premises used: Units, Int, one_smul, Eq.symm, Eq.trans
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hj : i + 1 = j
hi' : n + i = i + n
hj' : i + n + 1 = j + n
hf : HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow (f ≫ K.d (i + n) (j + n)) = 0
rw : HSMul.hSMul (β := A ⟶ K.X (j + n)) (n.negOnePow * n.negOnePow) ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) = 0
rw₁ : n.negOnePow • HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) = 0
rw₂ : n.negOnePow • HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow (f ≫ K.d (i + n) (j + n)) = 0
rw₃ : HSMul.hSMul (β := A ⟶ K.X (j + n)) n.negOnePow 0 = 0
rw₄ rw₅ : Eq.{v_1 + 1} (α := A ⟶ K.X (j + n)) 0 0
⊢ HSMul.hSMul (α := ℤˣ) (β := A ⟶ K.X (j + n)) 1 ((f ≫ 𝟙 (K.X (i + n))) ≫ K.d (i + n) (j + n)) = 0

Tactic:
rw [← Int.units_mul_self n.negOnePow, mul_smul, comp_id, hf, smul_zero]

Premises used: smul_zero, CategoryTheory.Category.comp_id, SemigroupAction.mul_smul, Int.negOnePow, Int.units_mul_self
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hj : (up ℤ).next i = j
hf : f ≫ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj' : (up ℤ).next i' = j'
simp :
  Eq (α := A ⟶ homology (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i)
    ((sc (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i).liftCycles f ⋯ ≫
      (sc (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i).homologyπ)
    ((Functor.obj (shiftFunctor C n ⋙ shortComplexFunctor C (up ℤ) i) K).liftCycles f ⋯ ≫
      (Functor.obj (shiftFunctor C n ⋙ shortComplexFunctor C (up ℤ) i) K).homologyπ)
⊢ Eq (α := A ⟶ homology (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i)
    (liftCycles (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) f j hj hf ≫
      homologyπ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i)
    (CategoryStruct.comp (Y := cycles K i') (liftCycles K (f ≫ (K.shiftFunctorObjXIso n i i' ⋯).hom) j' hj' ⋯)
      (homologyπ K i' ≫ NatTrans.app (Iso.inv (Functor.shiftIso (homologyFunctor C (up ℤ) 0) n i i' hi')) K))

Tactic:
simp only [liftCycles, homologyπ,
    shiftFunctorObjXIso, Functor.shiftIso, Functor.ShiftSequence.shiftIso,
    ShiftSequence.shiftIso_inv_app, ShortComplex.homologyπ_naturality,
    ShortComplex.liftCycles_comp_cyclesMap_assoc, shiftShortComplexFunctorIso_inv_app_τ₂,
    assoc, Iso.hom_inv_id, comp_id]

Premises used: CategoryTheory.Category.assoc, HomologicalComplex.homologyπ, CategoryTheory.ShortComplex.liftCycles_comp_cyclesMap_assoc, CategoryTheory.Iso.hom_inv_id, CategoryTheory.ShortComplex.homologyπ_naturality, CochainComplex.shiftFunctorObjXIso, CategoryTheory.Category.comp_id, CochainComplex.shiftShortComplexFunctorIso_inv_app_τ₂, CategoryTheory.Functor.shiftIso, HomologicalComplex.liftCycles, CategoryTheory.Functor.ShiftSequence.shiftIso, CochainComplex.ShiftSequence.shiftIso_inv_app
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K : CochainComplex.{v_1, u_1, 0} C ℤ
A : C
n i : ℤ
f : A ⟶ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).X i
j : ℤ
hj : (up ℤ).next i = j
hf : f ≫ (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K).d i j = 0
i' : ℤ
hi' : n + i = i'
j' : ℤ
hj' : (up ℤ).next i' = j'
⊢ Eq (α := A ⟶ homology (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i)
    ((sc (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i).liftCycles f ⋯ ≫
      (sc (Functor.obj (CategoryTheory.shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) K) i).homologyπ)
    ((Functor.obj (shiftFunctor C n ⋙ shortComplexFunctor C (up ℤ) i) K).liftCycles f ⋯ ≫
      (Functor.obj (shiftFunctor C n ⋙ shortComplexFunctor C (up ℤ) i) K).homologyπ)

Tactic:
rfl
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
n a a' : ℤ
ha' : n + a = a'
K : CochainComplex.{v_1, u_1, 0} C ℤ
⊢ Eq (α :=
    Functor.obj
        (shiftFunctor (HomotopyCategory.{v_1, u_1, 0} C (up ℤ)) n ⋙ Functor.shift (homologyFunctor C (up ℤ) 0) a)
        (Functor.obj (quotient C (up ℤ)) K) ⟶
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) a') (Functor.obj (quotient C (up ℤ)) K))
    (NatTrans.app (Iso.hom (Functor.shiftIso (homologyFunctor C (up ℤ) 0) n a a' ha'))
      (Functor.obj (quotient C (up ℤ)) K))
    (CategoryTheory.Functor.map (homologyFunctor C (up ℤ) a)
        (NatTrans.app (Iso.inv (Functor.commShiftIso (quotient C (up ℤ)) n)) K) ≫
      NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) a))
          (Functor.obj (shiftFunctor (HomologicalComplex.{v_1, u_1, 0} C (up ℤ)) n) K) ≫
        NatTrans.app (Iso.hom (Functor.shiftIso (HomologicalComplex.homologyFunctor C (up ℤ) 0) n a a' ha')) K ≫
          NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) a')) K)

Tactic:
apply Functor.ShiftSequence.induced_shiftIso_hom_app_obj

Premises used: CategoryTheory.Functor.ShiftSequence.induced_shiftIso_hom_app_obj
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preadditive.{v_1, u_1} C
inst_2 : CategoryWithHomology.{v_1, u_1} C
K L : CochainComplex.{v_1, u_1, 0} C ℤ
n : ℤ
f : K ⟶ Functor.obj (shiftFunctor (CochainComplex.{v_1, u_1, 0} C ℤ) n) L
a a' : ℤ
h : n + a = a'
⊢ Eq (α :=
    Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) a) (Functor.obj (quotient C (up ℤ)) K) ⟶
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) a') (Functor.obj (quotient C (up ℤ)) L))
    (Functor.shiftMap (homologyFunctor C (up ℤ) 0) (ShiftedHom.map f (quotient C (up ℤ))) a a' h)
    (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) a)) K ≫
      Functor.shiftMap (HomologicalComplex.homologyFunctor C (up ℤ) 0) f a a' h ≫
        NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) a')) L)

Tactic:
apply Functor.ShiftSequence.induced_shiftMap

Premises used: CategoryTheory.Functor.ShiftSequence.induced_shiftMap
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
T : Triangle (CochainComplex.{v_1, u_1, 0} C ℤ)
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ 1 + n₀ = n₁

Tactic:
lia
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
T : Triangle (CochainComplex.{v_1, u_1, 0} C ℤ)
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
⊢ Eq (α :=
    Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₀)
        (Triangle.obj₃ (Functor.obj (Functor.mapTriangle (quotient C (up ℤ))) T)) ⟶
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₁)
        (Triangle.obj₁ (Functor.obj (Functor.mapTriangle (quotient C (up ℤ))) T)))
    (Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0) (Functor.obj (Functor.mapTriangle (quotient C (up ℤ))) T) n₀
      n₁ h)
    (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₀)) T.obj₃ ≫
      Functor.shiftMap (HomologicalComplex.homologyFunctor C (up ℤ) 0) T.mor₃ n₀ n₁ ⋯ ≫
        NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₁)) T.obj₁)

Tactic:
apply homologyFunctor_shiftMap

Premises used: HomotopyCategory.homologyFunctor_shiftMap
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
⊢ HomComplex.δ (-1) 0 0 = HomComplex.Cochain.ofHom (S.f ≫ S.g)

Tactic:
simp
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
⊢ inr S.f ≫ descShortComplex S = S.g

Tactic:
simp [descShortComplex]

Premises used: CochainComplex.mappingCone.descShortComplex
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
n : ℤ
⊢ HomologicalComplex.Hom.f (inr S.f) n ≫ (descShortComplex S).f n = S.g.f n

Tactic:
simp [descShortComplex]

Premises used: CochainComplex.mappingCone.descShortComplex
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
i j : ℤ
h : i + -1 = j
⊢ HomComplex.Cochain.v (inl S.f) i j h ≫ (descShortComplex S).f j = 0

Tactic:
simp [descShortComplex]

Premises used: CochainComplex.mappingCone.descShortComplex
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S₁ S₂ : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
f : S₁ ⟶ S₂
h :
  ∀ (i : ℤ),
    HomologicalComplex.Hom.f (map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂) i = (descShortComplex S₁ ≫ f.τ₃).f i
⊢ map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂ = descShortComplex S₁ ≫ f.τ₃

Tactic:
ext i
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S₁ S₂ : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
f : S₁ ⟶ S₂
i : ℤ
⊢ HomologicalComplex.Hom.f (map S₁.f S₂.f f.τ₁ f.τ₂ ⋯ ≫ descShortComplex S₂) i = (descShortComplex S₁ ≫ f.τ₃).f i

Tactic:
simpa [mappingCone.ext_from_iff _ _ _ rfl, map] using
    congr_fun (congr_arg HomologicalComplex.Hom.f f.comm₂₃) i

Premises used: HomologicalComplex.Hom.f, CochainComplex.mappingCone.ext_from_iff, CochainComplex.mappingCone.map, congr_arg, CategoryTheory.ShortComplex.Hom.comm₂₃, congr_fun, rfl
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
dsimp :
  Eq (α :=
    Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₀) (Functor.obj (quotient C (up ℤ)) (mappingCone S.f)) ⟶
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₁) (Functor.obj (quotient C (up ℤ)) S.X₁))
    (Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0)
        (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁) (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
          (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
          (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
            NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))
        n₀ n₁ h :
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₀)
          (Triangle.obj₃
            (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
                NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))) ⟶
        Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₁)
          (Triangle.obj₁
            (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
                NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))))
    (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₁)) S.X₁)
⊢ Eq (α :=
    Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₀) (triangleh S.f).obj₃ ⟶
      Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₁) (triangleh S.f).obj₁)
    (Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0) (triangleh S.f) n₀ n₁ h)
    (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₁)) S.X₁)

Tactic:
dsimp
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
a :
  Eq (α :=
    Quiver.Hom (V := CategoryTheory.Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
      (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f)))
      (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))
    (CategoryTheory.Functor.map yoneda
      (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
        Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0)
            (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
                NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))
            n₀ n₁ h ≫
          NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁))
    (CategoryTheory.Functor.map yoneda
      (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ 𝟙 (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))
⊢ Eq (α :=
    Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f) ⟶
      Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)
    (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
      Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0)
          (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁) (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
            (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
            (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
              NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))
          n₀ n₁ h ≫
        NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁)
    (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
      hS.δ n₀ n₁ h ≫ 𝟙 (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁))

Tactic:
apply yoneda.map_injective

Premises used: CategoryTheory.yoneda, CategoryTheory.Functor.map_injective
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
a :
  ∀ (A : C) (x : A ⟶ Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f)),
    Eq (α :=
      Functor.obj (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁))
        (Opposite.op A))
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (CategoryTheory.Functor.map yoneda
              (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
                Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0)
                    (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁)
                      (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
                      (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
                      (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
                        NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))
                    n₀ n₁ h ≫
                  NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁))
            (Opposite.op A)))
        x)
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (CategoryTheory.Functor.map yoneda
              (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
                hS.δ n₀ n₁ h ≫ 𝟙 (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))
            (Opposite.op A)))
        x)
⊢ Eq (α :=
    Quiver.Hom (V := CategoryTheory.Functor.{v_1, v_1, u_1, v_1 + 1} Cᵒᵖ (Type v_1))
      (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f)))
      (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))
    (CategoryTheory.Functor.map yoneda
      (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
        Functor.homologySequenceδ (homologyFunctor C (up ℤ) 0)
            (Triangle.mk (X := Functor.obj (quotient C (up ℤ)) S.X₁)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) S.f)
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (inr S.f))
              (CategoryTheory.Functor.map (quotient C (up ℤ)) (triangle S.f).mor₃ ≫
                NatTrans.app (Iso.hom (Functor.commShiftIso (quotient C (up ℤ)) 1)) S.X₁))
            n₀ n₁ h ≫
          NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁))
    (CategoryTheory.Functor.map yoneda
      (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
        hS.δ n₀ n₁ h ≫ 𝟙 (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))

Tactic:
ext ⟨A⟩ (x : A ⟶ _)
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f)
⊢ (up ℤ).next n₀ = n₁

Tactic:
simpa using h
-- Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Abelian.{v_1, u_1} C
S : ShortComplex (CochainComplex.{v_1, u_1, 0} C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₀) (mappingCone S.f)
A' : C
π : A' ⟶ A
w_1 : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' :
  π ≫ x = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a :
  x ≫
      CategoryStruct.comp (Y :=
        Functor.obj (homologyFunctor C (up ℤ) n₀) (Functor.obj (quotient C (up ℤ)) (mappingCone S.f)))
        (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f))
        (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
          CategoryStruct.comp (Y := Functor.obj (Functor.shift (HomologicalComplex.homologyFunctor C (up ℤ) 0) n₁) S.X₁)
            (Functor.shiftMap (HomologicalComplex.homologyFunctor C (up ℤ) 0) (triangle S.f).mor₃ n₀ n₁ ⋯ :
              Functor.obj (Functor.shift (HomologicalComplex.homologyFunctor C (up ℤ) 0) n₀) (mappingCone S.f) ⟶
                Functor.obj (Functor.shift (HomologicalComplex.homologyFunctor C (up ℤ) 0) n₁) S.X₁)
            (CategoryStruct.comp (Y :=
                Functor.obj (Functor.shift (homologyFunctor C (up ℤ) 0) n₁) (Functor.obj (quotient C (up ℤ)) S.X₁))
                (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₁)) S.X₁)
                (NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁) :
              Functor.obj (Functor.shift (HomologicalComplex.homologyFunctor C (up ℤ) 0) n₁) S.X₁ ⟶
                HomologicalComplex.homology S.X₁ n₁)) =
    x ≫ HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫ 𝟙 (S.X₁.homology n₁)
⊢ Eq (α :=
    Functor.obj (Functor.obj yoneda (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁))
      (Opposite.op A))
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (CategoryTheory.Functor.map yoneda
            (NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₀)) (mappingCone S.f) ≫
              NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₀)) (triangle S.f).obj₃ ≫
                Functor.shiftMap (HomologicalComplex.homologyFunctor C (up ℤ) 0) (triangle S.f).mor₃ n₀ n₁ ⋯ ≫
                  NatTrans.app (Iso.inv (homologyFunctorFactors C (up ℤ) n₁)) (triangle S.f).obj₁ ≫
                    NatTrans.app (Iso.hom (homologyFunctorFactors C (up ℤ) n₁)) S.X₁))
          (Opposite.op A)))
      x)
    (TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (NatTrans.app
          (CategoryTheory.Functor.map yoneda
            (HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫
              hS.δ n₀ n₁ h ≫ 𝟙 (Functor.obj (HomologicalComplex.homologyFunctor C (up ℤ) n₁) S.X₁)))
          (Opposite.op A)))
      x)

Tactic:
dsimp