/-- If `p` is a maximal ideal of `R`, and `S` is the integral closure of `R` in `L`,
then the dimension `[S/pS : R/p]` is equal to `[Frac(S) : Frac(R)]`. -/
theorem finrank_quotient_map [IsDomain R] [IsDomain S] [IsDedekindDomain R] [Algebra K L]
    [Algebra R L] [IsScalarTower R K L] [IsScalarTower R S L] [IsIntegralClosure S R L]
    [hp : p.IsMaximal] [IsNoetherian R S] :
    finrank (R ⧸ p) (S ⧸ map (algebraMap R S) p) = finrank K L := by
  -- Choose an arbitrary basis `b` for `[S/pS : R/p]`.
  -- We'll use the previous results to turn it into a basis on `[Frac(S) : Frac(R)]`.
  letI : Field (R ⧸ p) := Ideal.Quotient.field _
  let ι := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
  let b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Module.Free.chooseBasis _ _
  -- Namely, choose a representative `b' i : S` for each `b i : S / pS`.
  let b' : ι → S := fun i => (Ideal.Quotient.mk_surjective (b i)).choose
  have b_eq_b' : ⇑b = (Submodule.mkQ (map (algebraMap R S) p)).restrictScalars R ∘ b' :=
    funext fun i => (Ideal.Quotient.mk_surjective (b i)).choose_spec.symm
  -- We claim `b'` is a basis for `Frac(S)` over `Frac(R)` because it is linear independent
  -- and spans the whole of `Frac(S)`.
  let b'' : ι → L := algebraMap S L ∘ b'
  have b''_li : LinearIndependent K b'' := ?_
  have b''_sp : Submodule.span K (Set.range b'') = ⊤ := ?_
  -- Since the two bases have the same index set, the spaces have the same dimension.
  let c : Basis ι K L := Basis.mk b''_li b''_sp.ge
  rw [finrank_eq_card_basis b, finrank_eq_card_basis c]
  -- It remains to show that the basis is indeed linear independent and spans the whole space.
  · rw [Set.range_comp]
    refine FinrankQuotientMap.span_eq_top p hp.ne_top _ (top_le_iff.mp ?_)
    -- The nicest way to show `S ≤ span b' ⊔ pS` is by reducing both sides modulo pS.
    -- However, this would imply distinguishing between `pS` as `S`-ideal,
    -- and `pS` as `R`-submodule, since they have different (non-defeq) quotients.
    -- Instead we'll lift `x mod pS ∈ span b` to `y ∈ span b'` for some `y - x ∈ pS`.
    intro x _
    

Proof state:
case refine_2.refine_1
R : Type u
inst✝²⁶ : CommRing R
S : Type v
inst✝²⁵ : CommRing S
f : R →+* S
p : Ideal R
P : Ideal S
inst✝²⁴ : Algebra R S
K : Type u_1
inst✝²³ : Field K
inst✝²² : Algebra R K
hRK : IsFractionRing R K
L : Type u_2
inst✝²¹ : Field L
inst✝²⁰ : Algebra S L
inst✝¹⁹ : IsFractionRing S L
V : Type u_3
V' : Type u_4
V'' : Type u_5
inst✝¹⁸ : AddCommGroup V
inst✝¹⁷ : Module R V
inst✝¹⁶ : Module K V
inst✝¹⁵ : IsScalarTower R K V
inst✝¹⁴ : AddCommGroup V'
inst✝¹³ : Module R V'
inst✝¹² : Module S V'
inst✝¹¹ : IsScalarTower R S V'
inst✝¹⁰ : AddCommGroup V''
inst✝⁹ : Module R V''
inst✝⁸ : IsDomain R
inst✝⁷ : IsDomain S
inst✝⁶ : IsDedekindDomain R
inst✝⁵ : Algebra K L
inst✝⁴ : Algebra R L
inst✝³ : IsScalarTower R K L
inst✝² : IsScalarTower R S L
inst✝¹ : IsIntegralClosure S R L
hp : IsMaximal p
inst✝ : IsNoetherian R S
this : Field (R ⧸ p) := Quotient.field p
ι : Type v := Module.Free.ChooseBasisIndex (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b : Basis ι (R ⧸ p) (S ⧸ map (algebraMap R S) p) := Module.Free.chooseBasis (R ⧸ p) (S ⧸ map (algebraMap R S) p)
b' : ι → S := fun i => Exists.choose (_ : ∃ a, (Quotient.mk (map (algebraMap R S) p)) a = b i)
b_eq_b' : ⇑b = ⇑(↑R (Submodule.mkQ (map (algebraMap R S) p))) ∘ b'
b'' : ι → L := ⇑(algebraMap S L) ∘ b'
b''_li : LinearIndependent K b''
x : S
a✝ : x ∈ ⊤
⊢ x ∈ Submodule.span R (Set.range b') ⊔ Submodule.restrictScalars R (map (algebraMap R S) p)

Next tactic:
have mem_span_b : ((Submodule.mkQ (map (algebraMap R S) p)) x : S ⧸ map (algebraMap R S) p) ∈
        Submodule.span (R ⧸ p) (Set.range b) := b.mem_span _