awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
⊢ 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 _ _)
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)
⊢ x = 0

Tactic:
apply (cancel_mono (prod.lift (𝟙 A) (0 : A ⟶ A))).1

Premises used: CategoryTheory.CategoryStruct.id, CategoryTheory.cancel_mono, Iff.mp, OfNat.ofNat, CategoryTheory.Limits.prod.lift
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)
hf₃ hf₄ : Eq.{v + 1} (α := Z ⟶ Limits.prod.{v, u} A A) 0 0
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (x ≫ prod.lift (𝟙 A) 0) (0 ≫ prod.lift (𝟙 A) 0)

Tactic:
rw [← hy, hyy, zero_comp, zero_comp]

Premises used: CategoryTheory.Limits.zero_comp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
⊢ 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 _ _
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
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
⊢ IsLimit (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp)

Tactic:
refine Fork.IsLimit.mk _ (fun s => Fork.ι s ≫ Limits.prod.fst) ?_ ?_

Premises used: CategoryTheory.Limits.Fork.IsLimit.mk, CategoryTheory.Limits.Fork.ι, CategoryTheory.CategoryStruct.comp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
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
⊢ (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:
apply Limits.prod.hom_ext

Premises used: CategoryTheory.Limits.prod.hom_ext
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)) ≫
      prod.fst =
    s.ι ≫ prod.fst

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)) ≫
      prod.snd =
    s.ι ≫ prod.snd

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
⊢ 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 _ _)
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)
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)
⊢ m = (fun (s : Fork.{v, u} (C := C) (X := Limits.prod.{v, u} A A) prod.snd 0) ↦ s.ι ≫ prod.fst) s

Tactic:
apply (cancel_mono (prod.lift (𝟙 A) (0 : A ⟶ A))).1

Premises used: CategoryTheory.CategoryStruct.id, CategoryTheory.cancel_mono, Iff.mp, OfNat.ofNat, CategoryTheory.Limits.prod.lift
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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 ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι

Tactic:
h
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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 :
  (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
⊢ 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:
convert! h
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
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
⊢ (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:
apply Limits.prod.hom_ext

Premises used: CategoryTheory.Limits.prod.hom_ext
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)
⊢ 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:
simp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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)
⊢ 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:
simp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
hf : ∀ (Z : C) (g : cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A) ⟶ Z), r A ≫ g = 0 → g = 0
⊢ Epi.{v, u} (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (r A)

Tactic:
apply NormalMonoCategory.epi_of_zero_cancel

Premises used: CategoryTheory.NormalMonoCategory.epi_of_zero_cancel
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
rw₁ rw₂ : Eq.{v + 1} (α := A ⟶ Z) 0 0
⊢ prod.lift (𝟙 A) 0 ≫ cokernel.π (diag A) ≫ z = 0

Tactic:
rw [← Category.assoc, hz]

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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

Tactic:
have h : prod.lift (𝟙 A) (0 : A ⟶ A) ≫ cokernel.π (diag A) ≫ z = 0 := by rw [← Category.assoc, hz]
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
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
⊢ z = 0

Tactic:
obtain ⟨t, ht⟩ := CokernelCofork.IsColimit.desc' hp2 _ h

Premises used: CategoryTheory.Limits.CokernelCofork.IsColimit.desc'
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
⊢ prod.snd ≫ t = cokernel.π (diag A) ≫ z

Tactic:
rw [CokernelCofork.π_ofπ] at ht

Premises used: CategoryTheory.Limits.CokernelCofork.π_ofπ
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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 : 𝟙 (CokernelCofork.ofπ prod.snd hlp).pt ≫ t = 0
⊢ t = 0

Tactic:
rw [← Category.id_comp t]

Premises used: CategoryTheory.Category.id_comp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
change : 𝟙 A ≫ t = 0
⊢ 𝟙 (CokernelCofork.ofπ prod.snd hlp).pt ≫ t = 0

Tactic:
change 𝟙 A ≫ t = 0

Premises used: CategoryTheory.CategoryStruct.id
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
rw₅ rw₆ : Eq.{v + 1} (α := A ⟶ Z) 0 0
⊢ 𝟙 A ≫ t = 0

Tactic:
rw [← Limits.prod.lift_snd (𝟙 A) (𝟙 A), Category.assoc, ht, ← Category.assoc,
      cokernel.condition, zero_comp]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.cokernel.condition, CategoryTheory.CategoryStruct.id, CategoryTheory.Limits.prod.lift_snd, CategoryTheory.Limits.zero_comp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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

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]
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
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
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
⊢ z = 0

Tactic:
apply (cancel_epi (cokernel.π (diag A))).1

Premises used: Iff.mp, CategoryTheory.Limits.cokernel.π, CategoryTheory.Limits.diag, CategoryTheory.cancel_epi