awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
erw :
  y ≫
      𝟙
        (Cone.pt.{0, v, 0, u} (F :=
          parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
            (cokernel.π (diag A)) 0)
          (KernelFork.ofι (diag A) ⋯)) =
    0
erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
⊢ y = 0
have erw₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (y ≫ prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
Tactic: erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
erw :
  y ≫
      𝟙
        (Cone.pt.{0, v, 0, u} (F :=
          parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
            (cokernel.π (diag A)) 0)
          (KernelFork.ofι (diag A) ⋯)) =
    0
erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (y ≫ prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
⊢ y = 0
have erw₃ : Eq (α :=
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) sorry))
  (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) prod.snd) 0
Tactic: erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
erw :
  y ≫
      𝟙
        (Cone.pt.{0, v, 0, u} (F :=
          parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
            (cokernel.π (diag A)) 0)
          (KernelFork.ofι (diag A) ⋯)) =
    0
erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (y ≫ prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₃ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) prod.snd) 0
⊢ y = 0
have erw₄ : Eq (α :=
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) sorry))
  (x ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
Tactic: erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
erw :
  y ≫
      𝟙
        (Cone.pt.{0, v, 0, u} (F :=
          parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
            (cokernel.π (diag A)) 0)
          (KernelFork.ofι (diag A) ⋯)) =
    0
erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (y ≫ prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₃ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) prod.snd) 0
erw₄ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (x ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
⊢ y = 0
have erw₅ : Eq (α :=
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) sorry))
  (x ≫ 0) 0
Tactic: erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
erw :
  y ≫
      𝟙
        (Cone.pt.{0, v, 0, u} (F :=
          parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
            (cokernel.π (diag A)) 0)
          (KernelFork.ofι (diag A) ⋯)) =
    0
erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (y ≫ prod.lift (𝟙 A) (𝟙 A)) prod.snd = 0
erw₃ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) prod.snd) 0
erw₄ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (x ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
erw₅ :
  Eq (α :=
    Z ⟶
      Cone.pt.{0, v, 0, u} (F :=
        parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
          (cokernel.π (diag A)) 0)
        (KernelFork.ofι (diag A) ⋯))
    (x ≫ 0) 0
⊢ y = 0
have erw₆ : Eq.{v + 1} (α :=
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) sorry))
  0 0
Tactic: erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
⊢ x = 0
have hyy : y = 0
Tactic: have hyy : y = 0 := by
    erw [← Category.comp_id y, ← Limits.prod.lift_snd (𝟙 A) (𝟙 A), ← Category.assoc, hy,
      Category.assoc, prod.lift_snd, HasZeroMorphisms.comp_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
⊢ x = 0
have this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
Tactic: haveI : Mono (prod.lift (𝟙 A) (0 : A ⟶ A)) := mono_of_mono_fac (prod.lift_fst _ _)
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
⊢ x = 0
have hf : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
Tactic: apply (cancel_mono (prod.lift (𝟙 A) (0 : A ⟶ A))).1
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
have hf : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
Tactic: rw [← hy, hyy, zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
hf : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
have hf₁ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) (0 ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
Tactic: rw [← hy, hyy, zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
hf : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
hf₁ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) (0 ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
have hf₂ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) 0 (0 ≫ prod.lift (𝟙 A) 0)
Tactic: rw [← hy, hyy, zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hl : IsLimit.{0, v, 0, u} (F :=
  parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
    (cokernel.π (diag A)) 0)
  (KernelFork.ofι (diag A) ⋯) :=
  monoIsKernelOfCokernel (colimit.cocone (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
    (colimit.isColimit (parallelPair (Y := Limits.prod.{v, u} A A) (diag A) 0))
Z : C
x : Z ⟶ A
hx : Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (x ≫ r A) 0
hxx :
  Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) ((x ≫ prod.lift (𝟙 A) 0) ≫ cokernel.π (diag A)) 0
y :
  Z ⟶
    Cone.pt.{0, v, 0, u} (F :=
      parallelPair (X := Limits.prod.{v, u} A A) (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A))
        (cokernel.π (diag A)) 0)
      (KernelFork.ofι (diag A) ⋯)
hy : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)
hyy : y = 0
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
hf : Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
hf₁ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) (0 ≫ diag A) (0 ≫ prod.lift (𝟙 A) 0)
hf₂ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) 0 (0 ≫ prod.lift (𝟙 A) 0)
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
have hf₃ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) 0 0
Tactic: rw [← hy, hyy, zero_comp, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
⊢ Epi.{v, u} (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (r A)
have hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
Tactic: have hlp : prod.lift (𝟙 A) (0 : A ⟶ A) ≫ Limits.prod.snd = 0 := prod.lift_snd _ _
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
⊢ IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
have refine_1 : ∀ (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0),
  (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫
      Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) =
    s.ι
Tactic: refine Fork.IsLimit.mk _ (fun s => Fork.ι s ≫ Limits.prod.fst) ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
refine_1 :
  ∀ (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0),
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫
        Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) =
      s.ι
⊢ IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
have refine_2 : ∀ (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0)
  (m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt),
  m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι →
    m = (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s
Tactic: refine Fork.IsLimit.mk _ (fun s => Fork.ι s ≫ Limits.prod.fst) ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Abelian.NonPreadditive (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0
⊢ (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫
      Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) =
    s.ι
have refine_1 : ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫
      Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)) ≫
    prod.fst =
  s.ι ≫ prod.fst
Tactic: apply Limits.prod.hom_ext