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
    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 _⟩
  · rw [add_mul]
    

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 * (M + span A {y}) + span A {x} * (M + span A {y}) ≤ M

Next tactic:
apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
/-- 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 _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    

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) ≠ ⊥
⊢ span A {x} * (M + span A {y}) ≤ M

Next tactic:
rw [mul_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]
  refine' ⟨le_trans (Submodule.mul_le_mul h_Wx_le h_Wy_le) _, mt Ideal.mul_eq_bot.mp _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    rw [mul_add]
    

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) ≠ ⊥
⊢ span A {x} * M + span A {x} * span A {y} ≤ M

Next tactic:
apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
/-- 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 _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    rw [mul_add]
    apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
    

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) ≠ ⊥
⊢ span A {x} * span A {y} ≤ M

Next tactic:
rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]
/-- 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 _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    rw [mul_add]
    apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
    rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]
  · 

Proof state:
case h.refine'_2
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) = ⊥)

Next tactic:
rintro (hx | 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))
  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 _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    rw [mul_add]
    apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
    rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]
  · rintro (hx | hy) <;> 

Proof state:
case h.refine'_2.inl
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) ≠ ⊥
hx : Multiset.prod (Multiset.map asIdeal Wx) = ⊥
⊢ False

Next tactic:
contradiction
/-- 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 _⟩
  · rw [add_mul]
    apply sup_le (show M * (M + span A {y}) ≤ M from Ideal.mul_le_right)
    rw [mul_add]
    apply sup_le (show span A {x} * M ≤ M from Ideal.mul_le_left)
    rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem]
  · rintro (hx | hy) <;> 

Proof state:
case h.refine'_2.inr
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) ≠ ⊥
hy : Multiset.prod (Multiset.map asIdeal Wy) = ⊥
⊢ False

Next tactic:
contradiction
instance : NoetherianSpace (PrimeSpectrum R) := by
  

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
⊢ NoetherianSpace (PrimeSpectrum R)

Next tactic:
apply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr
instance : NoetherianSpace (PrimeSpectrum R) := by
  apply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr
  

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
⊢ WellFounded fun s t => s < t

Next tactic:
have H := ‹IsNoetherianRing R›
instance : NoetherianSpace (PrimeSpectrum R) := by
  apply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr
  have H := ‹IsNoetherianRing R›
  

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 : IsNoetherianRing R
⊢ WellFounded fun s t => s < t

Next tactic:
rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H
instance : NoetherianSpace (PrimeSpectrum R) := by
  apply ((noetherianSpace_TFAE <| PrimeSpectrum R).out 0 1).mpr
  have H := ‹IsNoetherianRing R›
  rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H
  

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 : WellFounded fun x x_1 => x > x_1
⊢ WellFounded fun s t => s < t

Next tactic:
exact (closedsEmbedding R).dual.wellFounded H
theorem fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
  ⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
    

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
N : Submodule R M
⊢ (∃ S, Set.Finite S ∧ span R S = N) → FG N

Next tactic:
rintro ⟨t', h, rfl⟩
theorem fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
  ⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
    rintro ⟨t', h, rfl⟩
    

Proof state:
case intro.intro
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
t' : Set M
h : Set.Finite t'
⊢ FG (span R t')

Next tactic:
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
theorem fg_def {N : Submodule R M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ span R S = N :=
  ⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
    rintro ⟨t', h, rfl⟩
    rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
    

Proof state:
case intro.intro.intro
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
t : Finset M
h : Set.Finite ↑t
⊢ FG (span R ↑t)

Next tactic:
exact ⟨t, rfl⟩
theorem fg_iff_addSubmonoid_fg (P : Submodule ℕ M) : P.FG ↔ P.toAddSubmonoid.FG :=
  ⟨fun ⟨S, hS⟩ => ⟨S, by 

Proof state:
R : Type u_1
M : Type u_2
inst✝² : Semiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
P : Submodule ℕ M
x✝ : FG P
S : Finset M
hS : span ℕ ↑S = P
⊢ AddSubmonoid.closure ↑S = P.toAddSubmonoid

Next tactic:
simpa [← span_nat_eq_addSubmonoid_closure] using hS