/-- Any injective polynomial map over an algebraic extension of a finite field is surjective. -/
theorem ax_grothendieck_of_locally_finite {ι K R : Type*} [Field K] [Finite K] [CommRing R]
    [Finite ι] [Algebra K R] (alg : Algebra.IsAlgebraic K R) (ps : ι → MvPolynomial ι R)
    (hinj : Injective fun v i => MvPolynomial.eval v (ps i)) :
    Surjective fun v i => MvPolynomial.eval v (ps i) := by
  classical
    intro v
    cases nonempty_fintype ι
    /- `s` is the set of all coefficients of the polynomial, as well as all of
      the coordinates of `v`, the point I am trying to find the preimage of. -/
    let s : Finset R :=
      (Finset.biUnion (univ : Finset ι) fun i => (ps i).support.image fun x => coeff x (ps i)) ∪
        (univ : Finset ι).image v
    have hv : ∀ i, v i ∈ Algebra.adjoin K (s : Set R) := fun j =>
      Algebra.subset_adjoin (mem_union_right _ (mem_image.2 ⟨j, mem_univ _, rfl⟩))
    have hs₁ : ∀ (i : ι) (k : ι →₀ ℕ),
        k ∈ (ps i).support → coeff k (ps i) ∈ Algebra.adjoin K (s : Set R) :=
      fun i k hk => Algebra.subset_adjoin
        (mem_union_left _ (mem_biUnion.2 ⟨i, mem_univ _, mem_image_of_mem _ hk⟩))
    letI := isNoetherian_adjoin_finset s fun x _ => alg.isIntegral x
    letI := Module.IsNoetherian.finite K (Algebra.adjoin K (s : Set R))
    letI : Finite (Algebra.adjoin K (s : Set R)) :=
      FiniteDimensional.finite_of_finite K (Algebra.adjoin K (s : Set R))
    -- The restriction of the polynomial map, `ps`, to the subalgebra generated by `s`
    let res : (ι → Algebra.adjoin K (s : Set R)) → ι → Algebra.adjoin K (s : Set R) := fun x i =>
      ⟨eval (fun j : ι => (x j : R)) (ps i), eval_mem (hs₁ _) fun i => (x i).2⟩
    have hres_inj : Injective res := by
      intro x y hxy
      ext i
      simp only [Subtype.ext_iff, funext_iff] at hxy
      exact congr_fun (hinj (funext hxy)) i
    have hres_surj : Surjective res := Finite.injective_iff_surjective.1 hres_inj
    cases' hres_surj fun i => ⟨v i, hv i⟩ with w hw
    

Proof state:
case intro.intro
ι : Type u_1
K : Type u_2
R : Type u_3
inst✝⁴ : Field K
inst✝³ : Finite K
inst✝² : CommRing R
inst✝¹ : Finite ι
inst✝ : Algebra K R
alg : Algebra.IsAlgebraic K R
ps : ι → MvPolynomial ι R
hinj : Injective fun v i => (MvPolynomial.eval v) (ps i)
v : ι → R
val✝ : Fintype ι
s : Finset R :=
  (Finset.biUnion univ fun i => image (fun x => coeff x (ps i)) (MvPolynomial.support (ps i))) ∪ image v univ
hv : ∀ (i : ι), v i ∈ Algebra.adjoin K ↑s
hs₁ : ∀ (i : ι), ∀ k ∈ MvPolynomial.support (ps i), coeff k (ps i) ∈ Algebra.adjoin K ↑s
this✝¹ : IsNoetherian K ↥(Algebra.adjoin K ↑s) :=
  isNoetherian_adjoin_finset s fun x x_1 => Algebra.IsAlgebraic.isIntegral alg x
this✝ : Module.Finite K ↥(Algebra.adjoin K ↑s) := Module.IsNoetherian.finite K ↥(Algebra.adjoin K ↑s)
this : Finite ↥(Algebra.adjoin K ↑s) := FiniteDimensional.finite_of_finite K ↥(Algebra.adjoin K ↑s)
res : (ι → ↥(Algebra.adjoin K ↑s)) → ι → ↥(Algebra.adjoin K ↑s) :=
  fun x i =>
    { val := (MvPolynomial.eval fun j => ↑(x j)) (ps i),
      property := (_ : (MvPolynomial.eval fun j => ↑(x j)) (ps i) ∈ Algebra.adjoin K ↑s) }
hres_inj : Injective res
hres_surj : Surjective res
w : ι → ↥(Algebra.adjoin K ↑s)
hw : res w = fun i => { val := v i, property := (_ : v i ∈ Algebra.adjoin K ↑s) }
⊢ ∃ a, (fun v i => (MvPolynomial.eval v) (ps i)) a = v

Next tactic:
use fun i => w i