awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₀ : Eq (α := X ⟶ cokernel (f ≫ g))
  (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
    (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
  (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₀ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
      (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₁ : Eq (α := X ⟶ cokernel (f ≫ g))
  (f ≫
    CategoryStruct.comp (Y := biprod.{v, u} Y Z) biprod.inl (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
  (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₀ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
      (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₁ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (f ≫
      CategoryStruct.comp (Y := biprod.{v, u} Y Z) biprod.inl
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₂ : Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g)) (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₀ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
      (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₁ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (f ≫
      CategoryStruct.comp (Y := biprod.{v, u} Y Z) biprod.inl
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₂ :
  Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₃ : Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g)) 0
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₀ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
      (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₁ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (f ≫
      CategoryStruct.comp (Y := biprod.{v, u} Y Z) biprod.inl
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₂ :
  Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₃ : Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g)) 0
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₄ : Eq (α := X ⟶ cokernel (f ≫ g)) ((f ≫ g) ≫ cokernel.π (f ≫ g)) 0
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₀ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (f ≫ biprod.inl)
      (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₁ :
  Eq (α := X ⟶ cokernel (f ≫ g))
    (f ≫
      CategoryStruct.comp (Y := biprod.{v, u} Y Z) biprod.inl
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₂ :
  Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
h₃ : Eq (α := X ⟶ cokernel (f ≫ g)) (f ≫ g ≫ cokernel.π (f ≫ g)) 0
h₄ : Eq (α := X ⟶ cokernel (f ≫ g)) ((f ≫ g) ≫ cokernel.π (f ≫ g)) 0
⊢ Eq (α := X ⟶ cokernel (f ≫ g))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl
      (CategoryStruct.comp (Y := biprod.{v, u} Y Z) (biprod.desc (f ≫ biprod.inl) (biprod.lift (-𝟙 Y) g))
        (biprod.desc (g ≫ cokernel.π (f ≫ g)) (cokernel.π (f ≫ g)))))
    (CategoryStruct.comp (Y := biprod.{v, u} X Y) biprod.inl 0)
have h₅ : Eq.{v + 1} (α := X ⟶ cokernel (f ≫ g)) 0 0
Tactic: rw [biprod.inl_desc_assoc, assoc, biprod.inl_desc, comp_zero,
      ← assoc, cokernel.condition]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : A ⟶ biprod.{v, u} X Y
hk : k ≫ φ f g = 0
⊢ { l : A ⟶ kernel (f ≫ g) // l ≫ ι f g = k }
have refine_1 : (k ≫ biprod.fst) ≫ f ≫ g = 0
Tactic: refine ⟨kernel.lift _ (k ≫ biprod.fst) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : A ⟶ biprod.{v, u} X Y
hk : k ≫ φ f g = 0
refine_1 : (k ≫ biprod.fst) ≫ f ≫ g = 0
⊢ { l : A ⟶ kernel (f ≫ g) // l ≫ ι f g = k }
have refine_2 : kernel.lift (f ≫ g) (k ≫ biprod.fst) sorry ≫ ι f g = k
Tactic: refine ⟨kernel.lift _ (k ≫ biprod.fst) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : A ⟶ biprod.{v, u} X Y
hk : k ≫ φ f g = 0
⊢ (k ≫ biprod.fst) ≫ f ≫ g = 0
have refine_1 : ∀ (k₁ : A ⟶ X) (k₂ : A ⟶ Y)
  (hk : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ φ f g = 0),
  (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) ≫ f ≫ g = 0
Tactic: obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : A ⟶ X
k₂ : A ⟶ Y
hk : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ φ f g = 0
⊢ (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) ≫ f ≫ g = 0
have hk₁ : k₁ ≫ f = k₂ ∧ k₂ ≫ g = 0
Tactic: simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
        comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
        BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : A ⟶ X
k₂ : A ⟶ Y
hk : k₁ ≫ f = k₂ ∧ k₂ ≫ g = 0
⊢ (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) ≫ f ≫ g = 0
have refine_1 : ∀ (hk : (k₁ ≫ f) ≫ g = 0),
  (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) ((k₁ ≫ f) ≫ biprod.inr) ≫ biprod.fst) ≫ f ≫ g = 0
Tactic: obtain ⟨rfl, hk⟩ := hk
Cleared hypotheses: k₂, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : A ⟶ biprod.{v, u} X Y
hk : k ≫ φ f g = 0
⊢ kernel.lift (f ≫ g) (k ≫ biprod.fst) ⋯ ≫ ι f g = k
have refine_2 : ∀ (k₁ : A ⟶ X) (k₂ : A ⟶ Y)
  (hk : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ φ f g = 0),
  kernel.lift (f ≫ g) (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) sorry ≫
      ι f g =
    HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr)
Tactic: obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_to k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : A ⟶ X
k₂ : A ⟶ Y
hk : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ φ f g = 0
⊢ kernel.lift (f ≫ g) (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) ⋯ ≫
      ι f g =
    HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr)
have hk₁ : k₁ ≫ f = k₂ ∧ k₂ ≫ g = 0
Tactic: simp only [biprod.ext_to_iff, add_comp, assoc, inl_φ, BinaryBicone.inl_fst,
        comp_id, inr_φ_fst, comp_neg, zero_comp, BinaryBicone.inl_snd, comp_zero, φ_snd,
        BinaryBicone.inr_snd_assoc, zero_add, add_neg_eq_zero] at hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : A ⟶ X
k₂ : A ⟶ Y
hk_1 : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ φ f g = 0
hk : k₁ ≫ f = k₂ ∧ k₂ ≫ g = 0
⊢ kernel.lift (f ≫ g) (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr) ≫ biprod.fst) ⋯ ≫
      ι f g =
    HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) (k₂ ≫ biprod.inr)
have refine_2 : ∀ (hk : HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) ((k₁ ≫ f) ≫ biprod.inr) ≫ φ f g = 0)
  (hk_2 : (k₁ ≫ f) ≫ g = 0),
  kernel.lift (f ≫ g) (HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) ((k₁ ≫ f) ≫ biprod.inr) ≫ biprod.fst)
        sorry ≫
      ι f g =
    HAdd.hAdd (α := A ⟶ biprod.{v, u} X Y) (k₁ ≫ biprod.inl) ((k₁ ≫ f) ≫ biprod.inr)
Tactic: obtain ⟨rfl, hk⟩ := hk
Cleared hypotheses: k₂, hk_1, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : biprod.{v, u} Y Z ⟶ A
hk : φ f g ≫ k = 0
⊢ { l : cokernel (f ≫ g) ⟶ A // π f g ≫ l = k }
have refine_1 : (f ≫ g) ≫ biprod.inr ≫ k = 0
Tactic: refine ⟨cokernel.desc _ (biprod.inr ≫ k) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : biprod.{v, u} Y Z ⟶ A
hk : φ f g ≫ k = 0
refine_1 : (f ≫ g) ≫ biprod.inr ≫ k = 0
⊢ { l : cokernel (f ≫ g) ⟶ A // π f g ≫ l = k }
have refine_2 : π f g ≫ cokernel.desc (f ≫ g) (biprod.inr ≫ k) sorry = k
Tactic: refine ⟨cokernel.desc _ (biprod.inr ≫ k) ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : biprod.{v, u} Y Z ⟶ A
hk : φ f g ≫ k = 0
⊢ (f ≫ g) ≫ biprod.inr ≫ k = 0
have refine_1 : ∀ (k₁ : Y ⟶ A) (k₂ : Z ⟶ A)
  (hk : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0),
  (f ≫ g) ≫ biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
Tactic: obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_from k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : Y ⟶ A
k₂ : Z ⟶ A
hk : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
⊢ (f ≫ g) ≫ biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
have hk₁ : f ≫ k₁ = 0 ∧ k₁ = g ≫ k₂
Tactic: simp only [comp_add, φ_snd_assoc, biprod.ext_from_iff, inl_φ_assoc,
        BinaryBicone.inl_fst_assoc, BinaryBicone.inl_snd_assoc, zero_comp, add_zero, comp_zero,
        inr_φ_fst_assoc, neg_comp, id_comp, BinaryBicone.inr_snd_assoc, neg_add_eq_zero] at hk
Cleared hypotheses: hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : Y ⟶ A
k₂ : Z ⟶ A
hk : f ≫ k₁ = 0 ∧ k₁ = g ≫ k₂
⊢ (f ≫ g) ≫ biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
have refine_1 : ∀ (hk : f ≫ g ≫ k₂ = 0),
  (f ≫ g) ≫ biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ g ≫ k₂) (biprod.snd ≫ k₂) = 0
Tactic: obtain ⟨hk, rfl⟩ := hk
Cleared hypotheses: k₁, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k : biprod.{v, u} Y Z ⟶ A
hk : φ f g ≫ k = 0
⊢ π f g ≫ cokernel.desc (f ≫ g) (biprod.inr ≫ k) ⋯ = k
have refine_2 : ∀ (k₁ : Y ⟶ A) (k₂ : Z ⟶ A)
  (hk : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0),
  π f g ≫
      cokernel.desc (f ≫ g) (biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂))
        sorry =
    HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂)
Tactic: obtain ⟨k₁, k₂, rfl⟩ := biprod.decomp_hom_from k
Cleared hypotheses: k, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : Y ⟶ A
k₂ : Z ⟶ A
hk : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
⊢ π f g ≫
      cokernel.desc (f ≫ g) (biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂))
        ⋯ =
    HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂)
have hk₁ : f ≫ k₁ = 0 ∧ k₁ = g ≫ k₂
Tactic: simp only [comp_add, φ_snd_assoc, biprod.ext_from_iff, inl_φ_assoc,
        BinaryBicone.inl_fst_assoc, BinaryBicone.inl_snd_assoc, zero_comp, add_zero, comp_zero,
        inr_φ_fst_assoc, neg_comp, id_comp, BinaryBicone.inr_snd_assoc, neg_add_eq_zero] at hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
A : C
k₁ : Y ⟶ A
k₂ : Z ⟶ A
hk_1 : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂) = 0
hk : f ≫ k₁ = 0 ∧ k₁ = g ≫ k₂
⊢ π f g ≫
      cokernel.desc (f ≫ g) (biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂))
        ⋯ =
    HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ k₁) (biprod.snd ≫ k₂)
have refine_2 : ∀ (hk : φ f g ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ g ≫ k₂) (biprod.snd ≫ k₂) = 0)
  (hk_2 : f ≫ g ≫ k₂ = 0),
  π f g ≫
      cokernel.desc (f ≫ g)
        (biprod.inr ≫ HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ g ≫ k₂) (biprod.snd ≫ k₂)) sorry =
    HAdd.hAdd (α := biprod.{v, u} Y Z ⟶ A) (biprod.fst ≫ g ≫ k₂) (biprod.snd ≫ k₂)
Tactic: obtain ⟨hk, rfl⟩ := hk
Cleared hypotheses: k₁, hk_1, hk
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ IsLimit (C := ShortComplex.{v, u} C)
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯)
have h₁ : IsLimit (F :=
  Functor.comp (D := ShortComplex.{v, u} C)
    (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd sorry) (Y :=
      ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd sorry)
      { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := sorry, comm₂₃ := sorry } 0)
    ShortComplex.π₁)
  (Functor.mapCone ShortComplex.π₁
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := sorry, comm₂₃ := sorry } sorry))
Tactic: apply ShortComplex.isLimitOfIsLimitπ
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₁ :
  IsLimit (F :=
    Functor.comp (D := ShortComplex.{v, u} C)
      (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd ⋯) (Y :=
        ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd ⋯)
        { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := ⋯, comm₂₃ := ⋯ } 0)
      ShortComplex.π₁)
    (Functor.mapCone ShortComplex.π₁
      (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯))
⊢ IsLimit (C := ShortComplex.{v, u} C)
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯)
have h₂ : IsLimit (F :=
  Functor.comp (D := ShortComplex.{v, u} C)
    (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd sorry) (Y :=
      ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd sorry)
      { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := sorry, comm₂₃ := sorry } 0)
    ShortComplex.π₂)
  (Functor.mapCone ShortComplex.π₂
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := sorry, comm₂₃ := sorry } sorry))
Tactic: apply ShortComplex.isLimitOfIsLimitπ
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
h₁ :
  IsLimit (F :=
    Functor.comp (D := ShortComplex.{v, u} C)
      (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd ⋯) (Y :=
        ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd ⋯)
        { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := ⋯, comm₂₃ := ⋯ } 0)
      ShortComplex.π₁)
    (Functor.mapCone ShortComplex.π₁
      (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯))
h₂ :
  IsLimit (F :=
    Functor.comp (D := ShortComplex.{v, u} C)
      (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd ⋯) (Y :=
        ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd ⋯)
        { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := ⋯, comm₂₃ := ⋯ } 0)
      ShortComplex.π₂)
    (Functor.mapCone ShortComplex.π₂
      (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯))
⊢ IsLimit (C := ShortComplex.{v, u} C)
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯)
have h₃ : IsLimit (F :=
  Functor.comp (D := ShortComplex.{v, u} C)
    (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd sorry) (Y :=
      ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd sorry)
      { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := sorry, comm₂₃ := sorry } 0)
    ShortComplex.π₃)
  (Functor.mapCone ShortComplex.π₃
    (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := sorry, comm₂₃ := sorry } sorry))
Tactic: apply ShortComplex.isLimitOfIsLimitπ
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : Abelian.{v, u} C
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
⊢ IsLimit (F :=
    Functor.comp (D := ShortComplex.{v, u} C)
      (parallelPair (X := ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd ⋯) (Y :=
        ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd ⋯)
        { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := ⋯, comm₂₃ := ⋯ } 0)
      ShortComplex.π₁)
    (Functor.mapCone ShortComplex.π₁
      (KernelFork.ofι { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := ⋯, comm₂₃ := ⋯ } ⋯))
have h₁ : IsLimit (F :=
  parallelPair (X :=
    Functor.obj (C := ShortComplex.{v, u} C) ShortComplex.π₁
      (ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd sorry))
    (Y :=
    Functor.obj (C := ShortComplex.{v, u} C) ShortComplex.π₁
      (ShortComplex.mk (X₁ := Y) (X₂ := biprod.{v, u} Y Z) biprod.inl biprod.snd sorry))
    (Functor.map ShortComplex.π₁ { τ₁ := f, τ₂ := φ f g, τ₃ := g, comm₁₂ := sorry, comm₂₃ := sorry }) 0)
  (KernelFork.map
    (KernelFork.ofι
      { τ₁ := kernel.ι f, τ₂ := ι f g, τ₃ := kernel.ι g, comm₁₂ := sorry, comm₂₃ := sorry :
        ShortComplex.Hom
          { X₁ := kernel f, X₂ := kernel (f ≫ g), X₃ := kernel g, f := kernel.map f (f ≫ g) (𝟙 X) g sorry,
            g := kernel.map (f ≫ g) g f (𝟙 Z) sorry, zero := sorry : ShortComplex.{v, u} C }
          (ShortComplex.mk (X₁ := X) (X₂ := biprod.{v, u} X Y) biprod.inl biprod.snd sorry) }
      sorry)
    ShortComplex.π₁)
Tactic: apply (KernelFork.isLimitMapConeEquiv _ _).2