l3lab/ntp-mathlib huggingface dataset:
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
z : A
hz : z ∉ M
⊢ M < M + span A {z}

Next tactic:
refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
z : A
hz : z ∉ M
m_eq : M = M + span A {z}
⊢ z ∈ M

Next tactic:
rw [m_eq]
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    

Proof state:
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
z : A
hz : z ∉ M
m_eq : M = M + span A {z}
⊢ z ∈ M + span A {z}

Next tactic:
exact mem_sup_right (mem_span_singleton_self z)
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  

Proof state:
case neg.intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
Wx : Multiset (PrimeSpectrum A)
h_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}
h_Wx_ne : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
  obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
  

Proof state:
case neg.intro.intro.intro.intro.intro.intro.intro.intro
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
Wx : Multiset (PrimeSpectrum A)
h_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}
h_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥
Wy : Multiset (PrimeSpectrum A)
h_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}
h_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
use Wx + Wy
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
  obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
  use Wx + Wy
  

Proof state:
case h
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
Wx : Multiset (PrimeSpectrum A)
h_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}
h_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥
Wy : Multiset (PrimeSpectrum A)
h_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}
h_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥
⊢ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≤ M ∧ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≠ ⊥

Next tactic:
rw [Multiset.map_add, Multiset.prod_add]
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
  obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  

Proof state:
case h
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
Wx : Multiset (PrimeSpectrum A)
h_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}
h_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥
Wy : Multiset (PrimeSpectrum A)
h_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}
h_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥
⊢ Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≤ M ∧
    Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥

Next tactic:
refine' ⟨le_trans (Submodule.mul_le_mul h_Wx_le h_Wy_le) _, mt Ideal.mul_eq_bot.mp _⟩
/-- In a noetherian integral domain which is not a field, every non-zero ideal contains a non-zero
  product of prime ideals; in a field, the whole ring is a non-zero ideal containing only 0 as
  product or prime ideals ([samuel, § 3.3, Lemma 3]) -/
theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
    (h_nzI : I ≠ ⊥) :
    ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by
  revert h_nzI
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
      Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
    (fun (M : Ideal A) hgt => _) I
  intro h_nzM
  have hA_nont : Nontrivial A
  apply IsDomain.toNontrivial
  by_cases h_topM : M = ⊤
  · rcases h_topM with rfl
    obtain ⟨p_id, h_nzp, h_pp⟩ : ∃ p : Ideal A, p ≠ ⊥ ∧ p.IsPrime := by
      apply Ring.not_isField_iff_exists_prime.mp h_fA
    use ({⟨p_id, h_pp⟩} : Multiset (PrimeSpectrum A)), le_top
    rwa [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases h_prM : M.IsPrime
  · use ({⟨M, h_prM⟩} : Multiset (PrimeSpectrum A))
    rw [Multiset.map_singleton, Multiset.prod_singleton]
    exact ⟨le_rfl, h_nzM⟩
  obtain ⟨x, hx, y, hy, h_xy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left h_topM
  have lt_add : ∀ z ∉ M, M < M + span A {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact mem_sup_right (mem_span_singleton_self z)
  obtain ⟨Wx, h_Wx_le, h_Wx_ne⟩ := hgt (M + span A {x}) (lt_add _ hx) (ne_bot_of_gt (lt_add _ hx))
  obtain ⟨Wy, h_Wy_le, h_Wx_ne⟩ := hgt (M + span A {y}) (lt_add _ hy) (ne_bot_of_gt (lt_add _ hy))
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  refine' ⟨le_trans (Submodule.mul_le_mul h_Wx_le h_Wy_le) _, mt Ideal.mul_eq_bot.mp _⟩
  · 

Proof state:
case h.refine'_1
R : Type u
inst✝⁴ : CommRing R
inst✝³ : IsNoetherianRing R
A : Type u
inst✝² : CommRing A
inst✝¹ : IsDomain A
inst✝ : IsNoetherianRing A
h_fA : ¬IsField A
I M : Ideal A
hgt :
  ∀ J > M,
    (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) J
h_nzM : M ≠ ⊥
hA_nont : Nontrivial A
h_topM : ¬M = ⊤
h_prM : ¬Ideal.IsPrime M
x : A
hx : x ∉ M
y : A
hy : y ∉ M
h_xy : x * y ∈ M
lt_add : ∀ z ∉ M, M < M + span A {z}
Wx : Multiset (PrimeSpectrum A)
h_Wx_le : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span A {x}
h_Wx_ne✝ : Multiset.prod (Multiset.map asIdeal Wx) ≠ ⊥
Wy : Multiset (PrimeSpectrum A)
h_Wy_le : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span A {y}
h_Wx_ne : Multiset.prod (Multiset.map asIdeal Wy) ≠ ⊥
⊢ (M + span A {x}) * (M + span A {y}) ≤ M

Next tactic:
rw [add_mul]