irreducible_def preLiftAlgHom {s : A → A → Prop} { f : A →ₐ[S] B }
  (h : ∀ ⦃x y⦄, s x y → f x = f y) : RingQuot s →ₐ[S] B :=
{ toFun := fun x ↦ Quot.lift f
            (by
              rintro _ _ r
              induction r with
              | of r => exact h r
              | add_left _ r' => simp only [map_add, r']
              | mul_left _ r' => simp only [map_mul, r']
              | mul_right _ r' => simp only [map_mul, r'])
            x.toQuot
  map_zero' := by simp only [← zero_quot, f.map_zero]
  map_add' := by
    

Proof state:
R : Type uR
inst✝⁶ : Semiring R
S : Type uS
inst✝⁵ : CommSemiring S
T : Type uT
A : Type uA
inst✝⁴ : Semiring A
inst✝³ : Algebra S A
r : R → R → Prop
inst✝² : Semiring T
B : Type u₄
inst✝¹ : Semiring B
inst✝ : Algebra S B
s : A → A → Prop
f : A →ₐ[S] B
h : ∀ ⦃x y : A⦄, s x y → f x = f y
⊢ ∀ (x y : RingQuot s),
    OneHom.toFun
        (↑{
            toOneHom :=
              { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) },
            map_mul' :=
              (_ :
                ∀ (x y : RingQuot s),
                  OneHom.toFun
                      { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                        map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                      (x * y) =
                    OneHom.toFun
                        { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                          map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                        x *
                      OneHom.toFun
                        { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                          map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                        y) })
        (x + y) =
      OneHom.toFun
          (↑{
              toOneHom :=
                { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                  map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) },
              map_mul' :=
                (_ :
                  ∀ (x y : RingQuot s),
                    OneHom.toFun
                        { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                          map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                        (x * y) =
                      OneHom.toFun
                          { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                            map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                          x *
                        OneHom.toFun
                          { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                            map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                          y) })
          x +
        OneHom.toFun
          (↑{
              toOneHom :=
                { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                  map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) },
              map_mul' :=
                (_ :
                  ∀ (x y : RingQuot s),
                    OneHom.toFun
                        { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                          map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                        (x * y) =
                      OneHom.toFun
                          { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                            map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                          x *
                        OneHom.toFun
                          { toFun := fun x => Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) x.toQuot,
                            map_one' := (_ : Quot.lift ⇑f (_ : ∀ (a b : A), Rel s a b → f a = f b) 1.toQuot = 1) }
                          y) })
          y

Next tactic:
rintro ⟨⟨x⟩⟩ ⟨⟨y⟩⟩