awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw₁ : Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
  (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
rw₁ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw₂ : Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ ≫ (F.obj j).g : s.pt.X₂ ⟶ (F.obj j).X₃)
  (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
rw₁ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
rw₂ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ ≫ (F.obj j).g : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw₃ : (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
rw₁ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
rw₂ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ ≫ (F.obj j).g : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
rw₃ : (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw₄ : (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
eq₂ : ∀ (j : J), h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ = (s.π.app j).τ₂
eq₃ : ∀ (j : J), h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ = (s.π.app j).τ₃
eq₂₃ : ∀ (j : J), (c.π.app j).τ₂ ≫ (F.obj j).g = c.pt.g ≫ (c.π.app j).τ₃
eq₂₃' : ∀ (j : J), (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
rw : h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
rw₁ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ c.pt.g ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
rw₂ :
  Eq (α := s.pt.X₂ ⟶ (F.obj j).X₃) (h₂.lift (mapCone π₂ s) ≫ (c.π.app j).τ₂ ≫ (F.obj j).g : s.pt.X₂ ⟶ (F.obj j).X₃)
    (s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃ : s.pt.X₂ ⟶ (F.obj j).X₃)
rw₃ : (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ h₃.lift (mapCone π₃ s) ≫ (c.π.app j).τ₃
rw₄ : (s.π.app j).τ₂ ≫ (F.obj j).g = s.pt.g ≫ (s.π.app j).τ₃
⊢ (h₂.lift (mapCone π₂ s) ≫ c.pt.g) ≫ (c.π.app j).τ₃ = (s.pt.g ≫ h₃.lift (mapCone π₃ s)) ≫ (c.π.app j).τ₃
have rw₅ : s.pt.g ≫ (s.π.app j).τ₃ = s.pt.g ≫ (s.π.app j).τ₃
Tactic: rw [assoc, assoc, ← eq₂₃, reassoc_of% eq₂, eq₃, eq₂₃']
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
⊢ { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ } ≫
      c.π.app j =
    s.π.app j
have h₄ : Hom.τ₁
    ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
        comm₂₃ := sorry } ≫
      c.π.app j) =
  (s.π.app j).τ₁
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
h₄ :
  Hom.τ₁
      ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j) =
    (s.π.app j).τ₁
⊢ { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ } ≫
      c.π.app j =
    s.π.app j
have h₅ : Hom.τ₂
    ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
        comm₂₃ := sorry } ≫
      c.π.app j) =
  (s.π.app j).τ₂
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
j : J
h₄ :
  Hom.τ₁
      ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j) =
    (s.π.app j).τ₁
h₅ :
  Hom.τ₂
      ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        c.π.app j) =
    (s.π.app j).τ₂
⊢ { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ } ≫
      c.π.app j =
    s.π.app j
have h₆ : Hom.τ₃
    ({ τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
        comm₂₃ := sorry } ≫
      c.π.app j) =
  (s.π.app j).τ₃
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
⊢ m =
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }
have h₄ : m.τ₁ =
  Hom.τ₁
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
      comm₂₃ := sorry }
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
h₄ :
  m.τ₁ =
    Hom.τ₁
      { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }
⊢ m =
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }
have h₅ : m.τ₂ =
  Hom.τ₂
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
      comm₂₃ := sorry }
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cone F
h₁ : IsLimit (F := F ⋙ π₁) (mapCone π₁ c)
h₂ : IsLimit (F := F ⋙ π₂) (mapCone π₂ c)
h₃ : IsLimit (F := F ⋙ π₃) (mapCone π₃ c)
s : Cone F
m : s.pt ⟶ c.pt
hm : ∀ (j : J), m ≫ c.π.app j = s.π.app j
h₄ :
  m.τ₁ =
    Hom.τ₁
      { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }
h₅ :
  m.τ₂ =
    Hom.τ₂
      { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }
⊢ m =
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }
have h₆ : m.τ₃ =
  Hom.τ₃
    { τ₁ := h₁.lift (mapCone π₁ s), τ₂ := h₂.lift (mapCone π₂ s), τ₃ := h₃.lift (mapCone π₃ s), comm₁₂ := sorry,
      comm₂₃ := sorry }
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
inst_3 : HasLimit (F ⋙ π₁)
inst_4 : HasLimit (F ⋙ π₂)
inst_5 : HasLimit (F ⋙ π₃)
x x_1 : J
f : x ⟶ x_1
⊢ Eq (α :=
    obj
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
        x ⟶
      F.obj x_1)
    (Functor.map
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
        f ≫
      { τ₁ := limit.π (F ⋙ π₁) x_1, τ₂ := limit.π (F ⋙ π₂) x_1, τ₃ := limit.π (F ⋙ π₃) x_1, comm₁₂ := ⋯, comm₂₃ := ⋯ })
    ({ τ₁ := limit.π (F ⋙ π₁) x, τ₂ := limit.π (F ⋙ π₂) x, τ₃ := limit.π (F ⋙ π₃) x, comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫
      F.map f)
have h₁ : Eq (α :=
  (obj
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) sorry))
        x).X₁ ⟶
    (F.obj x_1).X₁)
  (Hom.τ₁
    (Functor.map
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) sorry))
        f ≫
      { τ₁ := limit.π (F ⋙ π₁) x_1, τ₂ := limit.π (F ⋙ π₂) x_1, τ₃ := limit.π (F ⋙ π₃) x_1, comm₁₂ := sorry, comm₂₃ := sorry }))
  (Hom.τ₁
    ({ τ₁ := limit.π (F ⋙ π₁) x, τ₂ := limit.π (F ⋙ π₂) x, τ₃ := limit.π (F ⋙ π₃) x, comm₁₂ := sorry, comm₂₃ := sorry } ≫
      F.map f))
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (goal)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
inst_3 : HasLimit (F ⋙ π₁)
inst_4 : HasLimit (F ⋙ π₂)
inst_5 : HasLimit (F ⋙ π₃)
x x_1 : J
f : x ⟶ x_1
h₁ :
  Eq (α :=
    (obj
          (obj (const J)
            (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
              (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
          x).X₁ ⟶
      (F.obj x_1).X₁)
    (Hom.τ₁
      (Functor.map
          (obj (const J)
            (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
              (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
          f ≫
        { τ₁ := limit.π (F ⋙ π₁) x_1, τ₂ := limit.π (F ⋙ π₂) x_1, τ₃ := limit.π (F ⋙ π₃) x_1, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }))
    (Hom.τ₁
      ({ τ₁ := limit.π (F ⋙ π₁) x, τ₂ := limit.π (F ⋙ π₂) x, τ₃ := limit.π (F ⋙ π₃) x, comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫
        F.map f))
⊢ Eq (α :=
    obj
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
        x ⟶
      F.obj x_1)
    (Functor.map
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) ⋯))
        f ≫
      { τ₁ := limit.π (F ⋙ π₁) x_1, τ₂ := limit.π (F ⋙ π₂) x_1, τ₃ := limit.π (F ⋙ π₃) x_1, comm₁₂ := ⋯, comm₂₃ := ⋯ })
    ({ τ₁ := limit.π (F ⋙ π₁) x, τ₂ := limit.π (F ⋙ π₂) x, τ₃ := limit.π (F ⋙ π₃) x, comm₁₂ := ⋯, comm₂₃ := ⋯ } ≫
      F.map f)
have h₂ : Eq (α :=
  (obj
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) sorry))
        x).X₂ ⟶
    (F.obj x_1).X₂)
  (Hom.τ₂
    (Functor.map
        (obj (const J)
          (mk (X₁ := limit (F ⋙ π₁)) (X₂ := limit (F ⋙ π₂)) (X₃ := limit (F ⋙ π₃)) (limMap (F.whiskerLeft π₁Toπ₂))
            (limMap (F.whiskerLeft π₂Toπ₃)) sorry))
        f ≫
      { τ₁ := limit.π (F ⋙ π₁) x_1, τ₂ := limit.π (F ⋙ π₂) x_1, τ₃ := limit.π (F ⋙ π₃) x_1, comm₁₂ := sorry, comm₂₃ := sorry }))
  (Hom.τ₂
    ({ τ₁ := limit.π (F ⋙ π₁) x, τ₂ := limit.π (F ⋙ π₂) x, τ₃ := limit.π (F ⋙ π₃) x, comm₁₂ := sorry, comm₂₃ := sorry } ≫
      F.map f))
Tactic: ext
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (hypothesis)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cocone F
h₁ : IsColimit (F := F ⋙ π₁) (mapCocone π₁ c)
h₂ : IsColimit (F := F ⋙ π₂) (mapCocone π₂ c)
h₃ : IsColimit (F := F ⋙ π₃) (mapCocone π₃ c)
s : Cocone F
j : J
⊢ Eq (α := (F ⋙ π₁).obj j ⟶ s.pt.X₂)
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s) ≫ s.pt.f))
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (c.pt.f ≫ h₂.desc (mapCocone π₂ s)))
have eq₁ : ∀ (j : J),
  Eq (α := (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
        (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s)) :
      (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
    (NatTrans.app (Cocone.ι (mapCocone π₁ s)) j :
      (F ⋙ π₁).obj j ⟶ obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))) j)
Tactic: have eq₁ := h₁.fac (π₁.mapCocone s)
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (hypothesis)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cocone F
h₁ : IsColimit (F := F ⋙ π₁) (mapCocone π₁ c)
h₂ : IsColimit (F := F ⋙ π₂) (mapCocone π₂ c)
h₃ : IsColimit (F := F ⋙ π₃) (mapCocone π₃ c)
s : Cocone F
j : J
eq₁ :
  ∀ (j : J),
    Eq (α := (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
      (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
          (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s)) :
        (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
      (NatTrans.app (Cocone.ι (mapCocone π₁ s)) j :
        (F ⋙ π₁).obj j ⟶ obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))) j)
⊢ Eq (α := (F ⋙ π₁).obj j ⟶ s.pt.X₂)
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s) ≫ s.pt.f))
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (c.pt.f ≫ h₂.desc (mapCocone π₂ s)))
have eq₂ : ∀ (j : J),
  Eq (α := (F ⋙ π₂).obj j ⟶ Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ c))) j)
        (NatTrans.app (Cocone.ι (mapCocone π₂ c)) j) (h₂.desc (mapCocone π₂ s)) :
      (F ⋙ π₂).obj j ⟶ Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))
    (NatTrans.app (Cocone.ι (mapCocone π₂ s)) j :
      (F ⋙ π₂).obj j ⟶ obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))) j)
Tactic: have eq₂ := h₂.fac (π₂.mapCocone s)
-- Lean 4 proof step from Mathlib.Algebra.Homology.ShortComplex.Limits (hypothesis)
Goal:
J : Type u_1
C : Type u_2
inst : Category.{v_1, u_1} J
inst_1 : Category.{v_2, u_2} C
inst_2 : HasZeroMorphisms.{v_2, u_2} C
F : J ⥤ ShortComplex.{v_2, u_2} C
c : Cocone F
h₁ : IsColimit (F := F ⋙ π₁) (mapCocone π₁ c)
h₂ : IsColimit (F := F ⋙ π₂) (mapCocone π₂ c)
h₃ : IsColimit (F := F ⋙ π₃) (mapCocone π₃ c)
s : Cocone F
j : J
eq₁ :
  ∀ (j : J),
    Eq (α := (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
      (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
          (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s)) :
        (F ⋙ π₁).obj j ⟶ Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))
      (NatTrans.app (Cocone.ι (mapCocone π₁ s)) j :
        (F ⋙ π₁).obj j ⟶ obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ s))) j)
eq₂ :
  ∀ (j : J),
    Eq (α := (F ⋙ π₂).obj j ⟶ Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))
      (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ c))) j)
          (NatTrans.app (Cocone.ι (mapCocone π₂ c)) j) (h₂.desc (mapCocone π₂ s)) :
        (F ⋙ π₂).obj j ⟶ Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))
      (NatTrans.app (Cocone.ι (mapCocone π₂ s)) j :
        (F ⋙ π₂).obj j ⟶ obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₂) (mapCocone π₂ s))) j)
⊢ Eq (α := (F ⋙ π₁).obj j ⟶ s.pt.X₂)
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (h₁.desc (mapCocone π₁ s) ≫ s.pt.f))
    (CategoryStruct.comp (Y := obj.{v_1, v_2, u_1, u_2} (obj (const J) (Cocone.pt (F := F ⋙ π₁) (mapCocone π₁ c))) j)
      (NatTrans.app (Cocone.ι (mapCocone π₁ c)) j) (c.pt.f ≫ h₂.desc (mapCocone π₂ s)))
have eq₁₂ : ∀ (j : J), (c.ι.app j).τ₁ ≫ (obj (obj (const J) c.pt) j).f = (F.obj j).f ≫ (c.ι.app j).τ₂
Tactic: have eq₁₂ := fun j => (c.ι.app j).comm₁₂