/-- Hilbert basis theorem: a polynomial ring over a noetherian ring is a noetherian ring. -/
protected theorem Polynomial.isNoetherianRing [inst : IsNoetherianRing R] : IsNoetherianRing R[X] :=
  isNoetherianRing_iff.2
    ⟨fun I : Ideal R[X] =>
      let M :=
        WellFounded.min (isNoetherian_iff_wellFounded.1 (by infer_instance))
          (Set.range I.leadingCoeffNth) ⟨_, ⟨0, rfl⟩⟩
      have hm : M ∈ Set.range I.leadingCoeffNth := WellFounded.min_mem _ _ _
      let ⟨N, HN⟩ := hm
      let ⟨s, hs⟩ := I.is_fg_degreeLE N
      have hm2 : ∀ k, I.leadingCoeffNth k ≤ M := fun k =>
        Or.casesOn (le_or_lt k N) (fun h => HN ▸ I.leadingCoeffNth_mono h) fun h x hx =>
          Classical.by_contradiction fun hxm =>
            haveI : IsNoetherian R R := inst
            have : ¬M < I.leadingCoeffNth k := by
              refine' WellFounded.not_lt_min (wellFounded_submodule_gt R R) _ _ _; exact ⟨k, rfl⟩
            this ⟨HN ▸ I.leadingCoeffNth_mono (le_of_lt h), fun H => hxm (H hx)⟩
      have hs2 : ∀ {x}, x ∈ I.degreeLE N → x ∈ Ideal.span (↑s : Set R[X]) :=
        hs ▸ fun hx =>
          Submodule.span_induction hx (fun _ hx => Ideal.subset_span hx) (Ideal.zero_mem _)
            (fun _ _ => Ideal.add_mem _) fun c f hf => f.C_mul' c ▸ Ideal.mul_mem_left _ _ hf
      ⟨s, le_antisymm (Ideal.span_le.2 fun x hx =>
          have : x ∈ I.degreeLE N := hs ▸ Submodule.subset_span hx
          this.2) <| by
        have : Submodule.span R[X] ↑s = Ideal.span ↑s := rfl
        rw [this]
        intro p hp
        generalize hn : p.natDegree = k
        induction' k using Nat.strong_induction_on with k ih generalizing p
        rcases le_or_lt k N with h | h
        · subst k
          refine' hs2 ⟨Polynomial.mem_degreeLE.2
            (le_trans Polynomial.degree_le_natDegree <| WithBot.coe_le_coe.2 h), hp⟩
        · have hp0 : p ≠ 0 := by
            rintro rfl
            cases hn
            exact Nat.not_lt_zero _ h
          have : (0 : R) ≠ 1 := by
            intro h
            apply hp0
            ext i
            refine' (mul_one _).symm.trans _
            rw [← h, mul_zero]
            rfl
          haveI : Nontrivial R := ⟨⟨0, 1, this⟩⟩
          have : p.leadingCoeff ∈ I.leadingCoeffNth N := by
            rw [HN]
            exact hm2 k ((I.mem_leadingCoeffNth _ _).2
              ⟨_, hp, hn ▸ Polynomial.degree_le_natDegree, rfl⟩)
          rw [I.mem_leadingCoeffNth] at this
          

Proof state:
case h.inr
R : Type u
S : Type u_1
σ : Type v
M✝ : Type w
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : AddCommGroup M✝
inst✝ : Module R M✝
inst : IsNoetherianRing R
I : Ideal R[X]
M : Submodule R R :=
  WellFounded.min (_ : WellFounded fun x x_1 => x > x_1) (Set.range (Ideal.leadingCoeffNth I))
    (_ : ∃ x, x ∈ Set.range (Ideal.leadingCoeffNth I))
hm : M ∈ Set.range (Ideal.leadingCoeffNth I)
N : ℕ
HN : Ideal.leadingCoeffNth I N = M
s : Finset R[X]
hs : Submodule.span R ↑s = Ideal.degreeLE I ↑N
hm2 : ∀ (k : ℕ), Ideal.leadingCoeffNth I k ≤ M
hs2 : ∀ {x : R[X]}, x ∈ Ideal.degreeLE I ↑N → x ∈ Ideal.span ↑s
this✝² : Submodule.span R[X] ↑s = Ideal.span ↑s
k : ℕ
ih : ∀ m < k, ∀ ⦃p : R[X]⦄, p ∈ I → natDegree p = m → p ∈ Ideal.span ↑s
p : R[X]
hp : p ∈ I
hn : natDegree p = k
h : N < k
hp0 : p ≠ 0
this✝¹ : 0 ≠ 1
this✝ : Nontrivial R
this : ∃ p_1 ∈ I, degree p_1 ≤ ↑N ∧ leadingCoeff p_1 = leadingCoeff p
⊢ p ∈ Ideal.span ↑s

Next tactic:
rcases this with ⟨q, hq, hdq, hlqp⟩