l3lab/ntp-mathlib huggingface dataset:
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  

Proof state:
case neg
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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  

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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  

Proof state:
case neg.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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  

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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) M

Next tactic:
use Wx + Wy
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ Multiset.prod (Multiset.map asIdeal (Wx + Wy)) ≤ M

Next tactic:
rw [Multiset.map_add, Multiset.prod_add]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) ≤ M

Next tactic:
apply le_trans (Submodule.mul_le_mul h_Wx h_Wy)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  apply le_trans (Submodule.mul_le_mul h_Wx h_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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ (M + span R {x}) * (M + span R {y}) ≤ M

Next tactic:
rw [add_mul]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  apply le_trans (Submodule.mul_le_mul h_Wx h_Wy)
  rw [add_mul]
  

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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ M * (M + span R {y}) + span R {x} * (M + span R {y}) ≤ M

Next tactic:
apply sup_le (show M * (M + span R {y}) ≤ M from Ideal.mul_le_right)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  apply le_trans (Submodule.mul_le_mul h_Wx h_Wy)
  rw [add_mul]
  apply sup_le (show M * (M + span R {y}) ≤ M from Ideal.mul_le_right)
  

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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ span R {x} * (M + span R {y}) ≤ M

Next tactic:
rw [mul_add]
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  apply le_trans (Submodule.mul_le_mul h_Wx h_Wy)
  rw [add_mul]
  apply sup_le (show M * (M + span R {y}) ≤ M from Ideal.mul_le_right)
  rw [mul_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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ span R {x} * M + span R {x} * span R {y} ≤ M

Next tactic:
apply sup_le (show span R {x} * M ≤ M from Ideal.mul_le_left)
/-- In a noetherian ring, every ideal contains a product of prime ideals
([samuel, § 3.3, Lemma 3])-/
theorem exists_primeSpectrum_prod_le (I : Ideal R) :
    ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I := by
  -- Porting note: Need to specify `P` explicitly
  refine' IsNoetherian.induction
    (P := fun I => ∃ Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) ≤ I)
    (fun (M : Ideal R) hgt => _) I
  by_cases h_prM : M.IsPrime
  · use {⟨M, h_prM⟩}
    rw [Multiset.map_singleton, Multiset.prod_singleton]
  by_cases htop : M = ⊤
  · rw [htop]
    exact ⟨0, le_top⟩
  have lt_add : ∀ z ∉ M, M < M + span R {z} := by
    intro z hz
    refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _
    rw [m_eq]
    exact Ideal.mem_sup_right (mem_span_singleton_self z)
  obtain ⟨x, hx, y, hy, hxy⟩ := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop
  obtain ⟨Wx, h_Wx⟩ := hgt (M + span R {x}) (lt_add _ hx)
  obtain ⟨Wy, h_Wy⟩ := hgt (M + span R {y}) (lt_add _ hy)
  use Wx + Wy
  rw [Multiset.map_add, Multiset.prod_add]
  apply le_trans (Submodule.mul_le_mul h_Wx h_Wy)
  rw [add_mul]
  apply sup_le (show M * (M + span R {y}) ≤ M from Ideal.mul_le_right)
  rw [mul_add]
  apply sup_le (show span R {x} * M ≤ M from Ideal.mul_le_left)
  

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
I M : Ideal R
hgt : ∀ J > M, (fun I => ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I) J
h_prM : ¬Ideal.IsPrime M
htop : ¬M = ⊤
lt_add : ∀ z ∉ M, M < M + span R {z}
x : R
hx : x ∉ M
y : R
hy : y ∉ M
hxy : x * y ∈ M
Wx : Multiset (PrimeSpectrum R)
h_Wx : Multiset.prod (Multiset.map asIdeal Wx) ≤ M + span R {x}
Wy : Multiset (PrimeSpectrum R)
h_Wy : Multiset.prod (Multiset.map asIdeal Wy) ≤ M + span R {y}
⊢ span R {x} * span R {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
  

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 : Ideal A
h_nzI : I ≠ ⊥
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
revert h_nzI
/-- 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
  

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 : Ideal A
⊢ I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
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
/-- 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
  

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
⊢ (fun I => I ≠ ⊥ → ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ I ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥) M

Next tactic:
intro h_nzM
/-- 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
  

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 ≠ ⊥
⊢ ∃ Z, Multiset.prod (Multiset.map asIdeal Z) ≤ M ∧ Multiset.prod (Multiset.map asIdeal Z) ≠ ⊥

Next tactic:
have hA_nont : Nontrivial A