awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
        Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
      (S₁.op.iCycles : S₁.op.cycles ⟶ S₁.op.X₂))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₁ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₂ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₃ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₄ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ S₂.op.iCycles ≫ (opMap φ).τ₂)
⊢ Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
      Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ) : Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
have rw₅ : Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
  (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
  (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
Tactic: rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
    cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
        Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
      (S₁.op.iCycles : S₁.op.cycles ⟶ S₁.op.X₂))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₁ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₂ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₃ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₄ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ S₂.op.iCycles ≫ (opMap φ).τ₂)
rw₅ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
⊢ Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
      Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ) : Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
have rw₆ : Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
  (Quiver.Hom.op (CategoryStruct.comp (Y := S₁.opcycles) S₁.pOpcycles (opcyclesMap φ)))
  (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
Tactic: rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
    cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
        Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
      (S₁.op.iCycles : S₁.op.cycles ⟶ S₁.op.X₂))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₁ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₂ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₃ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₄ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ S₂.op.iCycles ≫ (opMap φ).τ₂)
rw₅ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₆ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (Quiver.Hom.op (CategoryStruct.comp (Y := S₁.opcycles) S₁.pOpcycles (opcyclesMap φ)))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
⊢ Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
      Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ) : Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
have rw₇ : Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op (φ.τ₂ ≫ S₂.pOpcycles))
  (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
Tactic: rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
    cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
        Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
      (S₁.op.iCycles : S₁.op.cycles ⟶ S₁.op.X₂))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₁ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₂ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₃ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₄ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ S₂.op.iCycles ≫ (opMap φ).τ₂)
rw₅ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₆ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (Quiver.Hom.op (CategoryStruct.comp (Y := S₁.opcycles) S₁.pOpcycles (opcyclesMap φ)))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₇ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op (φ.τ₂ ≫ S₂.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
⊢ Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
      Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ) : Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
have rw₈ : Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op S₂.pOpcycles ≫ φ.τ₂.op)
  (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
Tactic: rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
    cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
        Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
      (S₁.op.iCycles : S₁.op.cycles ⟶ S₁.op.X₂))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₁ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (CategoryStruct.comp (Y := S₁.op.cycles) (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ)) S₁.op.iCycles)
rw₂ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ))
      (Iso.inv S₁.cyclesOpIso ≫ S₁.op.iCycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₃ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ)) S₁.op.iCycles)
rw₄ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Iso.inv S₂.cyclesOpIso ≫ S₂.op.iCycles ≫ (opMap φ).τ₂)
rw₅ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Quiver.Hom.op S₁.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₆ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂)
    (Quiver.Hom.op (CategoryStruct.comp (Y := S₁.opcycles) S₁.pOpcycles (opcyclesMap φ)))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₇ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op (φ.τ₂ ≫ S₂.pOpcycles))
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
rw₈ :
  Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op S₂.pOpcycles ≫ φ.τ₂.op)
    (Quiver.Hom.op S₂.pOpcycles ≫ (opMap φ).τ₂)
⊢ Eq (α := Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso) :
      Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
    (Iso.inv S₂.cyclesOpIso ≫ cyclesMap (opMap φ) : Opposite.op S₂.opcycles ⟶ S₁.op.cycles)
have rw₉ : Eq.{v_1 + 1} (α := Opposite.op S₂.opcycles ⟶ S₁.op.X₂) (Quiver.Hom.op S₂.pOpcycles ≫ φ.τ₂.op)
  (Quiver.Hom.op S₂.pOpcycles ≫ φ.τ₂.op)
Tactic: rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
    cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
⊢ Eq (α := S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso : S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)) :
      S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
have rw : Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso)
      (Iso.inv S₁.cyclesOpIso) :
    S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
      (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
      (Iso.inv S₁.cyclesOpIso) :
    S₂.op.cycles ⟶ S₁.op.cycles)
Tactic: rw [← cancel_mono (S₁.cyclesOpIso).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
    cyclesOpIso_inv_naturality, Iso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso)
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
        (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
⊢ Eq (α := S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso : S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)) :
      S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
have rw₁ : Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ) : S₂.op.cycles ⟶ S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Iso.hom S₁.cyclesOpIso) (Iso.inv S₁.cyclesOpIso) :
        S₁.op.cycles ⟶ S₁.op.cycles) :
    S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
      (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
      (Iso.inv S₁.cyclesOpIso) :
    S₂.op.cycles ⟶ S₁.op.cycles)
Tactic: rw [← cancel_mono (S₁.cyclesOpIso).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
    cyclesOpIso_inv_naturality, Iso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso)
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
        (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
rw₁ :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ) : S₂.op.cycles ⟶ S₁.op.cycles)
        (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Iso.hom S₁.cyclesOpIso) (Iso.inv S₁.cyclesOpIso) :
          S₁.op.cycles ⟶ S₁.op.cycles) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
        (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
⊢ Eq (α := S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso : S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)) :
      S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
have rw₂ : Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ) : S₂.op.cycles ⟶ S₁.op.cycles)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Iso.hom S₁.cyclesOpIso) (Iso.inv S₁.cyclesOpIso) :
        S₁.op.cycles ⟶ S₁.op.cycles) :
    S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso)) :
    S₂.op.cycles ⟶ S₁.op.cycles)
Tactic: rw [← cancel_mono (S₁.cyclesOpIso).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
    cyclesOpIso_inv_naturality, Iso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.RightHomology (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroMorphisms.{v_1, u_1} C
S₁ S₂ : ShortComplex.{v_1, u_1} C
φ : S₁ ⟶ S₂
inst_2 : S₁.HasRightHomology
inst_3 : S₂.HasRightHomology
rw :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso)
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
        (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
rw₁ :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ) : S₂.op.cycles ⟶ S₁.op.cycles)
        (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Iso.hom S₁.cyclesOpIso) (Iso.inv S₁.cyclesOpIso) :
          S₁.op.cycles ⟶ S₁.op.cycles) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₁.opcycles)
        (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)))
        (Iso.inv S₁.cyclesOpIso) :
      S₂.op.cycles ⟶ S₁.op.cycles)
rw₂ :
  Eq (α := S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := S₁.op.cycles) (cyclesMap (opMap φ) : S₂.op.cycles ⟶ S₁.op.cycles)
        (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Iso.hom S₁.cyclesOpIso) (Iso.inv S₁.cyclesOpIso) :
          S₁.op.cycles ⟶ S₁.op.cycles) :
      S₂.op.cycles ⟶ S₁.op.cycles)
    (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso)
        (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso)) :
      S₂.op.cycles ⟶ S₁.op.cycles)
⊢ Eq (α := S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (cyclesMap (opMap φ) ≫ Iso.hom S₁.cyclesOpIso : S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
    (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso) (Quiver.Hom.op (opcyclesMap φ)) :
      S₂.op.cycles ⟶ Opposite.op S₁.opcycles)
have rw₃ : Eq (α := S₂.op.cycles ⟶ S₁.op.cycles) (cyclesMap (opMap φ) ≫ 𝟙 S₁.op.cycles : S₂.op.cycles ⟶ S₁.op.cycles)
  (CategoryStruct.comp (Y := Opposite.op S₂.opcycles) (Iso.hom S₂.cyclesOpIso)
      (CategoryStruct.comp (Y := Opposite.op S₁.opcycles) (Quiver.Hom.op (opcyclesMap φ)) (Iso.inv S₁.cyclesOpIso)) :
    S₂.op.cycles ⟶ S₁.op.cycles)
Tactic: rw [← cancel_mono (S₁.cyclesOpIso).inv, assoc, assoc, Iso.hom_inv_id, comp_id,
    cyclesOpIso_inv_naturality, Iso.hom_inv_id_assoc]