/-- Specialization of `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` to Dedekind domains:
Let `I : Ideal A` be a nonzero ideal, where `A` is a Dedekind domain that is not a field.
Then `exists_primeSpectrum_prod_le_and_ne_bot_of_domain` states we can find a product of prime
ideals that is contained within `I`. This lemma extends that result by making the product minimal:
let `M` be a maximal ideal that contains `I`, then the product including `M` is contained within `I`
and the product excluding `M` is not contained within `I`. -/
theorem exists_multiset_prod_cons_le_and_prod_not_le [IsDedekindDomain A] (hNF : ¬IsField A)
    {I M : Ideal A} (hI0 : I ≠ ⊥) (hIM : I ≤ M) [hM : M.IsMaximal] :
    ∃ Z : Multiset (PrimeSpectrum A),
      (M ::ₘ Z.map PrimeSpectrum.asIdeal).prod ≤ I ∧
        ¬Multiset.prod (Z.map PrimeSpectrum.asIdeal) ≤ I := by
  -- Let `Z` be a minimal set of prime ideals such that their product is contained in `J`.
  obtain ⟨Z₀, hZ₀⟩ := PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
  obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
    wellFounded_lt.has_min
      {Z | (Z.map PrimeSpectrum.asIdeal).prod ≤ I ∧ (Z.map PrimeSpectrum.asIdeal).prod ≠ ⊥}
      ⟨Z₀, hZ₀.1, hZ₀.2⟩
  obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)
  -- Then in fact there is a `P ∈ Z` with `P ≤ M`.
  obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : Algebra A K
inst✝¹ : IsFractionRing A K
inst✝ : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : Ideal.IsMaximal M
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ :
  Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z₀) ≤ I ∧ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z₀) ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀
    x ∈
      {Z |
        Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
          Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≠ ⊥},
    ¬x < Z
hZI : Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I
hprodZ : Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map PrimeSpectrum.asIdeal Z
hPM : P.asIdeal ≤ M
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
classical
    have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
    obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).prod ≠ ⊥ := by
      rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
    -- By maximality of `P` and `M`, we have that `P ≤ M` implies `P = M`.
    have hPM' := (P.IsPrime.isMaximal hP0).eq_of_le hM.ne_top hPM
    subst hPM'
    -- By minimality of `Z`, erasing `P` from `Z` is exactly what we need.
    refine ⟨Z.erase P, ?_, ?_⟩
    · convert hZI
      rw [this, Multiset.cons_erase hPZ']
    · refine fun h => h_eraseZ (Z.erase P) ⟨h, ?_⟩ (Multiset.erase_lt.mpr hPZ)
      exact hZP0