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
P Q : C
f : P ⟶ Q
I : C := Abelian.image f
p : P ⟶ Abelian.image f := Abelian.factorThruImage f
i : kernel (Y := cokernel f) (cokernel.π f) ⟶ Q := kernel.ι (cokernel.π f)
R : C
g : I ⟶ R
hpg : p ≫ g = 0
u : kernel g ⟶ Q := kernel.ι g ≫ i
hu : NormalMono u
h : Q ⟶ @NormalMono.Z _ _ _ _ toHasZeroMorphisms u _ := NormalMono.g
t : P ⟶ kernel g
ht : t ≫ kernel.ι g = p
fh : f ≫ h = 0
l : cokernel f ⟶ @NormalMono.Z _ _ _ _ toHasZeroMorphisms u _
hl : cokernel.π f ≫ l = h
hih : i ≫ h = 0
s : kernel (Y := cokernel f) (cokernel.π f) ⟶ kernel g
hs : s ≫ u = i
hs' : (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i
this : Epi (X := kernel g) (kernel.ι g)
⊢ g = 0

Tactic:
zero_of_epi_comp _ (kernel.condition g)

Premises used: CategoryTheory.Limits.zero_of_epi_comp, CategoryTheory.Limits.kernel.condition
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.image f
p : P ⟶ Abelian.image f := Abelian.factorThruImage f
i : kernel (Y := cokernel f) (cokernel.π f) ⟶ Q := kernel.ι (cokernel.π f)
R : C
g : I ⟶ R
hpg : p ≫ g = 0
u : kernel g ⟶ Q := kernel.ι g ≫ i
hu : NormalMono u
h : Q ⟶ @NormalMono.Z _ _ _ _ toHasZeroMorphisms u _ := NormalMono.g
t : P ⟶ kernel g
ht : t ≫ kernel.ι g = p
fh : f ≫ h = 0
l : cokernel f ⟶ @NormalMono.Z _ _ _ _ toHasZeroMorphisms u _
hl : cokernel.π f ≫ l = h
hih : i ≫ h = 0
s : kernel (Y := cokernel f) (cokernel.π f) ⟶ kernel g
hs : s ≫ u = i
hs' : (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i
this : Epi (X := kernel g) (kernel.ι g)
⊢ g = 0

Tactic:
exact zero_of_epi_comp _ (kernel.condition g)

Premises used: CategoryTheory.Limits.zero_of_epi_comp, CategoryTheory.Limits.kernel.condition
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
inst_2 : Mono f
⊢ IsIso (Y := Abelian.image f) (Abelian.factorThruImage f)

Tactic:
isIso_of_mono_of_epi <| Abelian.factorThruImage f

Premises used: CategoryTheory.Abelian.factorThruImage, CategoryTheory.isIso_of_mono_of_epi
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
haveI : ∀ (hu : NormalEpi u), g = 0
⊢ g = 0

Tactic:
haveI hu := normalEpiOfEpi u
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
obtain : ∀ (t : cokernel g ⟶ Q) (ht : cokernel.π g ≫ t = i), g = 0
⊢ g = 0

Tactic:
obtain ⟨t, ht⟩ := cokernel.desc' g i hgi

Premises used: CategoryTheory.Limits.cokernel.desc'
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
⊢ h ≫ p ≫ cokernel.π g ≫ t = h ≫ u ≫ t

Tactic:
simp only [u, Category.assoc]

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
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
rw : (h ≫ u) ≫ t = 0 ≫ t
rw₁ rw₂ : Eq (α := @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ Q) (0 ≫ t) (0 ≫ t)
⊢ h ≫ u ≫ t = 0 ≫ t

Tactic:
rw [← Category.assoc, hu.w]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.NormalEpi.w
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
⊢ h ≫ f = 0

Tactic:
have hf : h ≫ f = 0 :=
      calc
        h ≫ f = h ≫ p ≫ i := (Abelian.coimage.fac f).symm ▸ rfl
        _ = h ≫ p ≫ cokernel.π g ≫ t := ht ▸ rfl
        _ = h ≫ u ≫ t := by simp only [u, Category.assoc]
        _ = 0 ≫ t := by rw [← Category.assoc, hu.w]
        _ = 0 := zero_comp
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
obtain : ∀ (l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f) (hl : l ≫ kernel.ι f = h), g = 0
⊢ g = 0

Tactic:
obtain ⟨l, hl⟩ := kernel.lift' f h hf

Premises used: CategoryTheory.Limits.kernel.lift'
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
rw : l ≫ kernel.ι f ≫ p = l ≫ 0
rw₁ rw₂ : Eq (α := @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ cokernel (X := kernel f) (kernel.ι f)) (l ≫ 0) (l ≫ 0)
⊢ (l ≫ kernel.ι f) ≫ p = l ≫ 0

Tactic:
rw [Category.assoc, cokernel.condition]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.cokernel.condition
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
⊢ h ≫ p = 0

Tactic:
have hhp : h ≫ p = 0 :=
      calc
        h ≫ p = (l ≫ kernel.ι f) ≫ p := hl ▸ rfl
        _ = l ≫ 0 := by rw [Category.assoc, cokernel.condition]
        _ = 0 := comp_zero
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
obtain : ∀ (s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)) (hs : u ≫ s = p), g = 0
⊢ g = 0

Tactic:
obtain ⟨s, hs⟩ := NormalEpi.desc' u p hhp

Premises used: CategoryTheory.NormalEpi.desc'
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)
hs : u ≫ s = p
rw : (p ≫ cokernel.π g) ≫ s = p ≫ 𝟙 I
rw₁ : p = p ≫ 𝟙 I
rw₂ rw₃ : p = p
⊢ p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I

Tactic:
rw [← Category.assoc, hs, Category.comp_id]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Category.comp_id
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)
hs : u ≫ s = p
⊢ p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I

Tactic:
have hs' : p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I := by rw [← Category.assoc, hs, Category.comp_id]
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)
hs : u ≫ s = p
hs' : p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I
⊢ Mono (Y := cokernel g) (cokernel.π g)

Tactic:
haveI : Mono (cokernel.π g) := mono_of_mono_fac ((cancel_epi _).1 hs')
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)
hs : u ≫ s = p
hs' : p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I
this : Mono (Y := cokernel g) (cokernel.π g)
⊢ g = 0

Tactic:
zero_of_comp_mono _ (cokernel.condition g)

Premises used: CategoryTheory.Limits.cokernel.condition, CategoryTheory.Limits.zero_of_comp_mono
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
I : C := Abelian.coimage f
i : Abelian.coimage f ⟶ Q := Abelian.factorThruCoimage f
p : P ⟶ cokernel (X := kernel f) (kernel.ι f) := cokernel.π (kernel.ι f)
R : C
g : R ⟶ I
hgi : g ≫ i = 0
u : P ⟶ cokernel g := p ≫ cokernel.π g
hu : NormalEpi u
h : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ P := NormalEpi.g
t : cokernel g ⟶ Q
ht : cokernel.π g ≫ t = i
hf : h ≫ f = 0
l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f
hl : l ≫ kernel.ι f = h
hhp : h ≫ p = 0
s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)
hs : u ≫ s = p
hs' : p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I
this : Mono (Y := cokernel g) (cokernel.π g)
⊢ g = 0

Tactic:
exact zero_of_comp_mono _ (cokernel.condition g)

Premises used: CategoryTheory.Limits.cokernel.condition, CategoryTheory.Limits.zero_of_comp_mono
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
P Q : C
f : P ⟶ Q
inst_2 : Epi f
⊢ IsIso (X := Abelian.coimage f) (Abelian.factorThruCoimage f)

Tactic:
isIso_of_mono_of_epi _

Premises used: CategoryTheory.isIso_of_mono_of_epi
-- Module: Mathlib.CategoryTheory.Abelian.NonPreadditive
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : NonPreadditiveAbelian.{v, u} C
A : C
⊢ Mono.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)

Tactic:
mono_of_mono_fac <| prod.lift_fst _ _

Premises used: CategoryTheory.Limits.prod.lift_fst, CategoryTheory.mono_of_mono_fac
-- 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))
hf : ∀ (Z : C) (g : Z ⟶ A), Eq (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (g ≫ r A) 0 → g = 0
⊢ Mono.{v, u} (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (r A)

Tactic:
apply NormalEpiCategory.mono_of_cancel_zero

Premises used: CategoryTheory.NormalEpiCategory.mono_of_cancel_zero
-- 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
rw :
  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
rw₁ rw₂ : Eq.{v + 1} (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) 0 0
⊢ 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

Tactic:
rw [Category.assoc, hx]

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
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
⊢ 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

Tactic:
have hxx : (x ≫ prod.lift (𝟙 A) (0 : A ⟶ A)) ≫ cokernel.π (diag A) = 0 := by
    rw [Category.assoc, hx]
-- 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
hf :
  ∀
    (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 : y ≫ Fork.ι (KernelFork.ofι (diag A) ⋯) = x ≫ prod.lift (𝟙 A) 0), x = 0
⊢ x = 0

Tactic:
obtain ⟨y, hy⟩ := KernelFork.IsLimit.lift' hl _ hxx

Premises used: CategoryTheory.Limits.KernelFork.IsLimit.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 : y ≫ Fork.ι (KernelFork.ofι (diag A) ⋯) = x ≫ prod.lift (𝟙 A) 0
⊢ Eq (α := Z ⟶ Limits.prod.{v, u} A A) (y ≫ diag A) (x ≫ prod.lift (𝟙 A) 0)

Tactic:
rw [KernelFork.ι_ofι] at hy

Premises used: CategoryTheory.Limits.KernelFork.ι_ofι
-- 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)
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
erw₆ erw₇ :
  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) ⋯))
    0 0
⊢ y = 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]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.CategoryStruct.id, CategoryTheory.Category.comp_id, CategoryTheory.Limits.prod.lift_snd, CategoryTheory.Limits.HasZeroMorphisms.comp_zero
-- 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)
⊢ 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]