-- Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
theorem exact_of_pseudo_exact (S : ShortComplex C)
    (hS : ∀ b, S.g b = 0 → ∃ a, S.f a = b) : S.Exact :=
  (S.exact_iff_kernel_ι_comp_cokernel_π_zero).2 (by
      -- If we apply `g` to the pseudoelement induced by its kernel, we get 0 (of course!).
      have : S.g (kernel.ι S.g) = 0 := apply_eq_zero_of_comp_eq_zero _ _ (kernel.condition _)
      -- By pseudo-exactness, we get a preimage.
      obtain ⟨a', ha⟩ := hS _ this
      obtain ⟨a, ha'⟩ := Quotient.exists_rep a'
      rw [← ha'] at ha
      obtain ⟨Z, r, q, _, eq, comm⟩ := Quotient.exact ha
      -- Consider the pullback of `kernel.ι (cokernel.π f)` and `kernel.ι g`.
      -- The commutative diagram given by the pseudo-equality `f a = b` induces
      -- a cone over this pullback, so we get a factorization `z`.
      obtain ⟨z, _, hz₂⟩ := @pullback.lift' _ _ _ _ _ _ (kernel.ι (cokernel.π S.f))
        (kernel.ι S.g) _ (r ≫ a.hom ≫ Abelian.factorThruImage S.f) q (by
          simp only [Category.assoc, Abelian.image.fac]
          exact comm)
      -- Let's give a name to the second pullback morphism.
      let j : pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) ⟶ kernel S.g := pullback.snd _ _
      -- Since `q` is an epimorphism, in particular this means that `j` is an epimorphism.
      haveI pe : Epi j := epi_of_epi_fac hz₂
      -- But it is also a monomorphism, because `kernel.ι (cokernel.π f)` is: A kernel is
      -- always a monomorphism and the pullback of a monomorphism is a monomorphism.
      -- But mono + epi = iso, so `j` is an isomorphism.
      haveI : IsIso j := isIso_of_mono_of_epi _
      -- But then `kernel.ι g` can be expressed using all of the maps of the pullback square, and we
      -- are done.
      rw [(Iso.eq_inv_comp (asIso j)).2 pullback.condition.symm]
      simp only [Category.assoc, kernel.condition, HasZeroMorphisms.comp_zero])

Proof state:
case intro.intro.intro.intro.intro.intro.intro.mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
z : Z ⟶ pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g)
left✝ : z ≫ pullback.fst (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = r ≫ a.hom ≫ Abelian.factorThruImage S.f
hz₂ : z ≫ pullback.snd (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = q
j : pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) ⟶ kernel S.g :=
  pullback.snd (kernel.ι (cokernel.π S.f)) (kernel.ι S.g)
pe : Epi j
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Tactic:
haveI : IsIso j := isIso_of_mono_of_epi _

Result:
case intro.intro.intro.intro.intro.intro.intro.mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this✝ : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
z : Z ⟶ pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g)
left✝ : z ≫ pullback.fst (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = r ≫ a.hom ≫ Abelian.factorThruImage S.f
hz₂ : z ≫ pullback.snd (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = q
j : pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) ⟶ kernel S.g :=
  pullback.snd (kernel.ι (cokernel.π S.f)) (kernel.ι S.g)
pe : Epi j
this : IsIso j
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Tactic:
have : S.g (kernel.ι S.g) = 0 := apply_eq_zero_of_comp_eq_zero _ _ (kernel.condition _)

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Proof state:
case intro.intro.intro.intro.intro.intro.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Tactic:
obtain ⟨z, _, hz₂⟩ := @pullback.lift' _ _ _ _ _ _ (kernel.ι (cokernel.π S.f))
  (kernel.ι S.g) _ (r ≫ a.hom ≫ Abelian.factorThruImage S.f) q (by
    simp only [Category.assoc, Abelian.image.fac]
    exact comm)

Result:
case intro.intro.intro.intro.intro.intro.intro.mk.intro
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
z : Z ⟶ pullback (kernel.ι (cokernel.π S.f)) (kernel.ι S.g)
left✝ : z ≫ pullback.fst (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = r ≫ a.hom ≫ Abelian.factorThruImage S.f
hz₂ : z ≫ pullback.snd (kernel.ι (cokernel.π S.f)) (kernel.ι S.g) = q
⊢ kernel.ι S.g ≫ cokernel.π S.f = 0

Proof state:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
⊢ (r ≫ a.hom ≫ Abelian.factorThruImage S.f) ≫ kernel.ι (cokernel.π S.f) = q ≫ kernel.ι S.g

Tactic:
simp only [Category.assoc, Abelian.image.fac]

Result:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
hS : ∀ (b : Pseudoelement S.X₂), pseudoApply S.g b = 0 → ∃ a, pseudoApply S.f a = b
this : pseudoApply S.g (Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))) = 0
a' : Pseudoelement S.X₁
a : Over S.X₁
ha : pseudoApply S.f ⟦a⟧ = Quot.mk (PseudoEqual S.X₂) (Over.mk (kernel.ι S.g))
ha' : ⟦a⟧ = a'
Z : C
r : Z ⟶ ((fun g => app S.f g) a).left
q : Z ⟶ (Over.mk (kernel.ι S.g)).left
w✝ : Epi r
eq : Epi q
comm : r ≫ ((fun g => app S.f g) a).hom = q ≫ (Over.mk (kernel.ι S.g)).hom
⊢ r ≫ a.hom ≫ S.f = q ≫ kernel.ι S.g