l3lab/ntp-mathlib huggingface dataset:
/-- 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`.
  

Proof state:
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
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
obtain ⟨Z₀, hZ₀⟩ := PrimeSpectrum.exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
/-- 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
  

Proof state:
case 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.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
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⟩
/-- 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⟩
  

Proof state:
case 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) ≠ ⊥
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)
/-- 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`.
  

Proof state:
case 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) ≠ ⊥
w✝ : Ideal A
hPZ' : w✝ ∈ Multiset.map PrimeSpectrum.asIdeal Z
hPM : w✝ ≤ M
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'
/-- 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'
  classical
    

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:
have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
/-- 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'
  classical
    have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
    

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
this :
  Multiset.map PrimeSpectrum.asIdeal (Multiset.erase Z P) =
    Multiset.erase (Multiset.map PrimeSpectrum.asIdeal Z) P.asIdeal
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
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
/-- 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'
  classical
    have := Multiset.map_erase PrimeSpectrum.asIdeal PrimeSpectrum.ext P Z
    obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map PrimeSpectrum.asIdeal).prod ≠ ⊥ := by
      

Proof state:
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
this :
  Multiset.map PrimeSpectrum.asIdeal (Multiset.erase Z P) =
    Multiset.erase (Multiset.map PrimeSpectrum.asIdeal Z) P.asIdeal
⊢ P.asIdeal ≠ ⊥ ∧ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal (Multiset.erase Z P)) ≠ ⊥

Next tactic:
rwa [Ne.def, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
/-- 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'
  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`.
    

Proof state:
case intro.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
this :
  Multiset.map PrimeSpectrum.asIdeal (Multiset.erase Z P) =
    Multiset.erase (Multiset.map PrimeSpectrum.asIdeal Z) P.asIdeal
hP0 : P.asIdeal ≠ ⊥
hZP0 : Multiset.prod (Multiset.map PrimeSpectrum.asIdeal (Multiset.erase Z P)) ≠ ⊥
⊢ ∃ Z,
    Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ I ∧
      ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ I

Next tactic:
have hPM' := (P.IsPrime.isMaximal hP0).eq_of_le hM.ne_top hPM
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  

Proof state:
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 : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
⊢ ∃ x ∈ (↑I)⁻¹, x ∉ 1

Next tactic:
suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>
        rw [coeIdeal_ne_zero] <;> assumption
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    

Proof state:
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 : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
⊢ ∃ x ∈ (↑I)⁻¹, x ∉ 1

Next tactic:
obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    

Proof state:
case 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 : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
⊢ ∃ x ∈ (↑I)⁻¹, x ∉ 1

Next tactic:
have hM0 := (M.bot_lt_of_maximal hNF).ne'
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    

Proof state:
case 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 : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
⊢ ∃ x ∈ (↑I)⁻¹, x ∉ 1

Next tactic:
obtain ⟨x, hxM, hx1⟩ := this hM
theorem exists_not_mem_one_of_ne_bot [IsDedekindDomain A] (hNF : ¬IsField A) {I : Ideal A}
    (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) :
    ∃ x : K, x ∈ (I⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) := by
  -- WLOG, let `I` be maximal.
  suffices
    ∀ {M : Ideal A} (_hM : M.IsMaximal),
      ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by
    obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1
    have hM0 := (M.bot_lt_of_maximal hNF).ne'
    obtain ⟨x, hxM, hx1⟩ := this hM
    

Proof state:
case 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 : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
this : ∀ {M : Ideal A}, IsMaximal M → ∃ x ∈ (↑M)⁻¹, x ∉ 1
M : Ideal A
hM : IsMaximal M
hIM : I ≤ M
hM0 : M ≠ ⊥
x : K
hxM : x ∈ (↑M)⁻¹
hx1 : x ∉ 1
⊢ ∃ x ∈ (↑I)⁻¹, x ∉ 1

Next tactic:
refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩