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
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
rw : s ≫ kernel.ι g ≫ i = 𝟙 I ≫ i
⊢ (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i
have rw₁ : i = 𝟙 I ≫ i
Tactic: rw [Category.assoc, hs, Category.id_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
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
rw : s ≫ kernel.ι g ≫ i = 𝟙 I ≫ i
rw₁ : i = 𝟙 I ≫ i
⊢ (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i
have rw₂ : i = i
Tactic: rw [Category.assoc, hs, Category.id_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
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
⊢ g = 0
have hs' : (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i
Tactic: have hs' : (s ≫ kernel.ι g) ≫ i = 𝟙 I ≫ i := by rw [Category.assoc, hs, Category.id_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
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
⊢ g = 0
have this : Epi (X := kernel g) (kernel.ι g)
Tactic: haveI : Epi (kernel.ι g) := epi_of_epi_fac ((cancel_mono _).1 hs')
-- 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
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
⊢ g = 0
have haveI : ∀ (hu : NormalEpi u), g = 0
Tactic: haveI hu := normalEpiOfEpi u
-- 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
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
⊢ g = 0
have obtain : ∀ (t : cokernel g ⟶ Q) (ht : cokernel.π g ≫ t = i), g = 0
Tactic: obtain ⟨t, ht⟩ := cokernel.desc' g i hgi
-- 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
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 ≫ u ≫ t = 0 ≫ t
have rw : (h ≫ u) ≫ t = 0 ≫ t
Tactic: rw [← Category.assoc, hu.w]
-- 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
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
⊢ h ≫ u ≫ t = 0 ≫ t
have rw₁ : Eq (α := @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ Q) (0 ≫ t) (0 ≫ t)
Tactic: rw [← Category.assoc, hu.w]
-- 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
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
⊢ g = 0
have hf : 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
-- 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
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
⊢ g = 0
have obtain : ∀ (l : @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ kernel f) (hl : l ≫ kernel.ι f = h), g = 0
Tactic: obtain ⟨l, hl⟩ := kernel.lift' f h hf
-- 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
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
⊢ (l ≫ kernel.ι f) ≫ p = l ≫ 0
have rw : l ≫ kernel.ι f ≫ p = l ≫ 0
Tactic: rw [Category.assoc, cokernel.condition]
-- 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
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
⊢ (l ≫ kernel.ι f) ≫ p = l ≫ 0
have rw₁ : Eq (α := @NormalEpi.W _ _ _ _ toHasZeroMorphisms u _ ⟶ cokernel (X := kernel f) (kernel.ι f)) (l ≫ 0) (l ≫ 0)
Tactic: rw [Category.assoc, cokernel.condition]
-- 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
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
⊢ g = 0
have hhp : 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
-- 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
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
⊢ g = 0
have obtain : ∀ (s : cokernel g ⟶ cokernel (X := kernel f) (kernel.ι f)) (hs : u ≫ s = p), g = 0
Tactic: obtain ⟨s, hs⟩ := NormalEpi.desc' u p hhp
-- 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
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
have rw : (p ≫ cokernel.π g) ≫ s = p ≫ 𝟙 I
Tactic: rw [← Category.assoc, hs, Category.comp_id]
-- 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
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
⊢ p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I
have rw₁ : p = p ≫ 𝟙 I
Tactic: rw [← Category.assoc, hs, Category.comp_id]
-- 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
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
⊢ p ≫ cokernel.π g ≫ s = p ≫ 𝟙 I
have rw₂ : p = p
Tactic: rw [← Category.assoc, hs, Category.comp_id]
-- 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
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
⊢ g = 0
have hs' : 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]
-- 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
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
⊢ g = 0
have this : Mono (Y := cokernel g) (cokernel.π g)
Tactic: haveI : Mono (cokernel.π g) := mono_of_mono_fac ((cancel_epi _).1 hs')
-- 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))
⊢ Mono.{v, u} (Y := cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) (r A)
have 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
Tactic: apply NormalEpiCategory.mono_of_cancel_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
⊢ 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
have 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
Tactic: rw [Category.assoc, hx]
-- 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
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
⊢ 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
have rw₁ : Eq.{v + 1} (α := Z ⟶ cokernel.{v, u} (Y := Limits.prod.{v, u} A A) (diag A)) 0 0
Tactic: rw [Category.assoc, hx]
-- 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
⊢ x = 0
have 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
Tactic: have hxx : (x ≫ prod.lift (𝟙 A) (0 : A ⟶ A)) ≫ cokernel.π (diag A) = 0 := by
    rw [Category.assoc, hx]
-- 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
⊢ x = 0
have 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) sorry))
  (hy : y ≫ Fork.ι (KernelFork.ofι (diag A) sorry) = x ≫ prod.lift (𝟙 A) 0), x = 0
Tactic: obtain ⟨y, hy⟩ := KernelFork.IsLimit.lift' hl _ hxx
-- 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 : y ≫ Fork.ι (KernelFork.ofι (diag A) ⋯) = x ≫ prod.lift (𝟙 A) 0
⊢ x = 0
have hy₁ : 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
Cleared hypotheses: hy
-- 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)
⊢ y = 0
have 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) sorry)) =
  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
⊢ y = 0
have erw₁ : y ≫ CategoryStruct.comp (Y := Limits.prod.{v, u} A A) (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]