-- Module: Mathlib.RingTheory.Etale.StandardEtale
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
P : StandardEtalePair R
refine_1 :
  Eq (α :=
    HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
      (Ideal.span
        (singleton (α := Localization.Away P.g)
          ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
            Localization.Away P.g))))
    (((aeval
              ((algebraMap R[X]
                    (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                      (Ideal.span
                        (singleton (α := Localization.Away P.g)
                          ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                            Localization.Away P.g)))) :
                  (a : R[X]) →
                    Localization.Away P.g ⧸
                      Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
                X) :
            R[X] →ₐ[R]
              HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                (Ideal.span
                  (singleton (α := Localization.Away P.g)
                    ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                      Localization.Away P.g)))) :
          (a : R[X]) →
            Localization.Away P.g ⧸
              Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
        P.f :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
    (0 :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
refine_2 :
  IsUnit (M :=
    HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
      (Ideal.span
        (singleton (α := Localization.Away P.g)
          ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
            Localization.Away P.g))))
    (((aeval
              ((algebraMap R[X]
                    (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                      (Ideal.span
                        (singleton (α := Localization.Away P.g)
                          ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                            Localization.Away P.g)))) :
                  (a : R[X]) →
                    Localization.Away P.g ⧸
                      Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
                X) :
            R[X] →ₐ[R]
              HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                (Ideal.span
                  (singleton (α := Localization.Away P.g)
                    ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                      Localization.Away P.g)))) :
          (a : R[X]) →
            Localization.Away P.g ⧸
              Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
        P.g :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
refine_3 :
  ∀ (a : R[X]) (b : Membership.mem (γ := Submonoid R[X]) (Submonoid.powers P.g) a),
    IsUnit (M := P.Ring)
      ((aeval P.X : (a : R[X]) → P.Ring)
          (↑(Subtype.mk (p := fun (a : R[X]) ↦ Membership.mem (γ := Submonoid R[X]) (Submonoid.powers P.g) a) a b) :
            R[X]) :
        P.Ring)
refine_4 :
  ∀ (a : Localization.Away P.g),
    Membership.mem (γ := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g)))
        a →
      Eq (α := P.Ring)
        ((IsLocalization.liftAlgHom (R := R[X]) (M := Submonoid.powers P.g) (f := aeval P.X) ⋯ :
              (a : Localization.Away P.g) → P.Ring)
            a :
          P.Ring)
        (0 : P.Ring)
refine_5 :
  Eq (α :=
    HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))) →ₐ[R]
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
    (AlgHom.comp
        (P.lift
          ((algebraMap R[X]
                (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                  (Ideal.span
                    (singleton (α := Localization.Away P.g)
                      ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                        Localization.Away P.g)))) :
              (a : R[X]) →
                Localization.Away P.g ⧸
                  Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
            X)
          ⋯)
        (Ideal.Quotient.liftₐ
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))
          (IsLocalization.liftAlgHom (R := R[X]) (M := Submonoid.powers P.g) (f := aeval P.X) ⋯) ⋯) :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))) →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
    (AlgHom.id R
        (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g)))) :
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))) →ₐ[R]
        HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
          (Ideal.span
            (singleton (α := Localization.Away P.g)
              ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                Localization.Away P.g))))
refine_6 :
  Eq (α := P.Ring →ₐ[R] P.Ring)
    (AlgHom.comp (B :=
      HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g))))
      (Ideal.Quotient.liftₐ
        (Ideal.span
          (singleton (α := Localization.Away P.g)
            ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
              Localization.Away P.g)))
        (IsLocalization.liftAlgHom (R := R[X]) (M := Submonoid.powers P.g) (f := aeval P.X) ⋯) ⋯)
      (P.lift
        ((algebraMap R[X]
              (HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
                (Ideal.span
                  (singleton (α := Localization.Away P.g)
                    ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f :
                      Localization.Away P.g)))) :
            (a : R[X]) →
              Localization.Away P.g ⧸
                Ideal.span {(algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f})
          X)
        ⋯))
    (AlgHom.id R P.Ring)
⊢ P.Ring ≃ₐ[R]
    HasQuotient.Quotient (Localization.Away P.g) (B := Ideal (Localization.Away P.g))
      (Ideal.span
        (singleton (α := Localization.Away P.g)
          ((algebraMap R[X] (Localization.Away P.g) : (a : R[X]) → Localization.Away P.g) P.f : Localization.Away P.g)))

Tactic:
refine .ofAlgHom (P.lift (algebraMap R[X] _ .X) ⟨?_, ?_⟩)
    (Ideal.Quotient.liftₐ _ (IsLocalization.liftAlgHom (M := .powers <| P.g)
      (f := aeval P.X) <| Subtype.forall.mpr ?_) ?_)
      ?_ ?_