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
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
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
⊢ (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_1₁ : ((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.snd =
  s.ι ≫ prod.snd
Tactic: apply Limits.prod.hom_ext
-- 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
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
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : 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
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
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
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
⊢ m = (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s
have refine_2 : Eq (α := s.pt ⟶ Limits.prod.{v, u} A A) (m ≫ prod.lift (𝟙 A) 0)
  ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ 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
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
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
⊢ Eq (α := s.pt ⟶ Limits.prod.{v, u} A A) (m ≫ prod.lift (𝟙 A) 0)
    ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0)
have h₁ : (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0 = s.ι
Tactic: convert! h
-- 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
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
⊢ (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0 = s.ι
have h₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A)
    ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0)
    prod.fst =
  s.ι ≫ prod.fst
Tactic: apply Limits.prod.hom_ext
-- 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
m : s.pt ⟶ (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp).pt
h : m ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι
this : Mono.{v, u} (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0)
h₁ :
  CategoryStruct.comp (Y := Limits.prod.{v, u} A A)
      ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0)
      prod.fst =
    s.ι ≫ prod.fst
⊢ (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0 = s.ι
have h₂ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A)
    ((fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s ≫ prod.lift (𝟙 A) 0)
    prod.snd =
  s.ι ≫ prod.snd
Tactic: apply Limits.prod.hom_ext
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
⊢ Epi.{v, u} (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (r A)
have hf : ∀ (Z : C) (g : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z), r A ≫ g = 0 → g = 0
Tactic: apply NormalMonoCategory.epi_of_zero_cancel
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
⊢ prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
have rw : (prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A)) ≫ z = 0
Tactic: rw [← Category.assoc, hz]
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
rw : (prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A)) ≫ z = 0
⊢ prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
have rw₁ : Eq.{v + 1} (α := A ⟶ Z) 0 0
Tactic: rw [← Category.assoc, hz]
-- 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
⊢ z = 0
have h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
Tactic: have h : prod.lift (𝟙 A) (0 : A ⟶ A) ≫ cokernel.π (diag A) ≫ z = 0 := by rw [← Category.assoc, hz]
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
⊢ z = 0
have hf : ∀ (t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z)
  (ht : Cofork.π (CokernelCofork.ofπ prod.snd hlp) ≫ t = cokernel.π (diag A) ≫ z), z = 0
Tactic: obtain ⟨t, ht⟩ := CokernelCofork.IsColimit.desc' hp2 _ h
-- 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : Cofork.π (CokernelCofork.ofπ prod.snd hlp) ≫ t = cokernel.π (diag A) ≫ z
⊢ z = 0
have ht₁ : prod.snd ≫ t = cokernel.π (diag A) ≫ z
Tactic: rw [CokernelCofork.π_ofπ] at ht
Cleared hypotheses: ht
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
⊢ t = 0
have rw : 𝟙 (CokernelCofork.ofπ prod.snd hlp).pt ≫ t = 0
Tactic: rw [← Category.id_comp t]
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
⊢ 𝟙 (CokernelCofork.ofπ prod.snd hlp).pt ≫ t = 0
have change : 𝟙 A ≫ t = 0
Tactic: change 𝟙 A ≫ t = 0
-- 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
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
⊢ 𝟙 A ≫ t = 0
have rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
⊢ 𝟙 A ≫ t = 0
have rw₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) (prod.snd ≫ t) = 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
rw₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) (prod.snd ≫ t) = 0
⊢ 𝟙 A ≫ t = 0
have rw₂ : prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A) ≫ z = 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
rw₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) (prod.snd ≫ t) = 0
rw₂ : prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A) ≫ z = 0
⊢ 𝟙 A ≫ t = 0
have rw₃ : (prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A)) ≫ z = 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
rw₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) (prod.snd ≫ t) = 0
rw₂ : prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A) ≫ z = 0
rw₃ : (prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A)) ≫ z = 0
⊢ 𝟙 A ≫ t = 0
have rw₄ : Eq (α := A ⟶ Z) (0 ≫ z) 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
rw : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) prod.snd ≫ t = 0
rw₁ : CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) (𝟙 A)) (prod.snd ≫ t) = 0
rw₂ : prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A) ≫ z = 0
rw₃ : (prod.lift (𝟙 A) (𝟙 A) ≫ cokernel.π (diag A)) ≫ z = 0
rw₄ : Eq (α := A ⟶ Z) (0 ≫ z) 0
⊢ 𝟙 A ≫ t = 0
have rw₅ : Eq.{v + 1} (α := A ⟶ Z) 0 0
Tactic: rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
⊢ z = 0
have htt : t = 0
Tactic: have htt : t = 0 := by
    rw [← Category.id_comp t]
    change 𝟙 A ≫ t = 0
    rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
htt : t = 0
⊢ z = 0
have hf : cokernel.π (diag A) ≫ z =
  CategoryStruct.comp (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (cokernel.π (diag A)) 0
Tactic: apply (cancel_epi (cokernel.π (diag 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
hlp : Eq.{v + 1} (α := A ⟶ A) (CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (prod.lift (𝟙 A) 0) prod.snd) 0
hp1 : IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) :=
  Fork.IsLimit.mk (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)
    (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) ⋯ ⋯
hp2 : IsColimit (CokernelCofork.ofπ prod.snd hlp) := epiIsCokernelOfKernel (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) hp1
Z : C
z : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z
hz : r A ≫ z = 0
h : prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0
t : (CokernelCofork.ofπ prod.snd hlp).pt ⟶ Z
ht : prod.snd ≫ t = cokernel.π (diag A) ≫ z
htt : t = 0
⊢ cokernel.π (diag A) ≫ z =
    CategoryStruct.comp (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (cokernel.π (diag A)) 0
have hf : Eq (α := Limits.prod.{v, u} A A ⟶ Z) (prod.snd ≫ t)
  (CategoryStruct.comp (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (cokernel.π (diag A)) 0)
Tactic: rw [← ht, htt, comp_zero, comp_zero]