awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (hypothesis)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDomain A
h : IsDedekindDomainInv A
this : CommGroupWithZero (FractionalIdeal (R := A) A⁰ (FractionRing A)) := h.commGroupWithZero
P : Ideal A
P_ne : P ≠ ⊥
hP : P.IsPrime
M : Ideal A
hM : P < M
P'_ne : Ne (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑P : FractionalIdeal A⁰ (FractionRing A)) 0
M'_ne : Ne (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑M : FractionalIdeal A⁰ (FractionRing A)) 0
le_one :
  HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑M : FractionalIdeal A⁰ (FractionRing A))⁻¹
      (↑P : FractionalIdeal A⁰ (FractionRing A)) ≤
    1
y : A
_hy : Membership.mem (γ := Ideal A) ⊤ y
hx :
  Membership.mem (α := FractionRing A)
    (HMul.hMul (α := FractionalIdeal (R := A) A⁰ (FractionRing A)) (↑M : FractionalIdeal A⁰ (FractionRing A))⁻¹
      (↑P : FractionalIdeal A⁰ (FractionRing A)))
    ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) y : FractionRing A)
z : A
hzM : z ∈ M
hzp : z ∉ P
zy_mem :
  Membership.mem (α := FractionRing A) (γ := FractionalIdeal (R := A) A⁰ (FractionRing A))
    (↑P : FractionalIdeal A⁰ (FractionRing A))
    ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (z * y) : FractionRing A)
zy : A
hzy : zy ∈ P
zy_eq :
  Eq (α := FractionRing A) ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) zy : FractionRing A)
    ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) (z * y) : FractionRing A)
⊢ Membership.mem (α := FractionRing A) (γ := FractionalIdeal (R := A) A⁰ (FractionRing A))
    (↑P : FractionalIdeal A⁰ (FractionRing A))
    ((algebraMap A (FractionRing A) : (a : A) → FractionRing A) y : FractionRing A)
have hzy₁ : z * y ∈ P
Tactic: rw [IsFractionRing.injective A (FractionRing A) zy_eq] at hzy
Cleared hypotheses: hzy, zy_eq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDomain A
I : Ideal A
hI : I ≠ ⊥
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K)⁻¹ 1
have rw : ∀ (y : K),
  Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y →
    Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (1 * y)
Tactic: rw [mem_inv_iff (coeIdeal_ne_zero.mpr hI)]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : IsDomain A
I : Ideal A
hI : I ≠ ⊥
y : K
hy : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) (↑I : FractionalIdeal A⁰ K) y
⊢ Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 (1 * y)
have rw : Membership.mem (γ := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) 1 y
Tactic: rw [one_mul]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have obtain : ∀ (Z₀ : Multiset (PrimeSpectrum A)) (hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥),
  ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
Tactic: obtain ⟨Z₀, hZ₀⟩ := exists_primeSpectrum_prod_le_and_ne_bot_of_domain hNF hI0
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have obtain : ∀ (Z : Multiset (PrimeSpectrum A))
  (h_eraseZ :
    ∀ (x : Multiset (PrimeSpectrum A)),
      Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
          {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
        ¬x < Z)
  (hZI : (Multiset.map asIdeal Z).prod ≤ I) (hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥),
  ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
Tactic: obtain ⟨Z, ⟨hZI, hprodZ⟩, h_eraseZ⟩ :=
    wellFounded_lt.has_min
      {Z | (Z.map asIdeal).prod ≤ I ∧ (Z.map asIdeal).prod ≠ ⊥}
      ⟨Z₀, hZ₀.1, hZ₀.2⟩
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have obtain : ∀ (w : Ideal A) (hPZ' : w ∈ Multiset.map asIdeal Z) (hPM : w ≤ M),
  ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
Tactic: obtain ⟨_, hPZ', hPM⟩ := hM.isPrime.multiset_prod_le.mp (hZI.trans hIM)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
w : Ideal A
hPZ' : w ∈ Multiset.map asIdeal Z
hPM : w ≤ M
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have obtain : ∀ (P : PrimeSpectrum A) (hPZ : P ∈ Z) (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M),
  ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
Tactic: obtain ⟨P, hPZ, rfl⟩ := Multiset.mem_map.mp hPZ'
Cleared hypotheses: w, hPZ', hPM
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (hypothesis)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
Tactic: have := Multiset.map_erase asIdeal (fun _ _ => PrimeSpectrum.ext) P Z
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa : ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
rwa :
  ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa₁ : ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal ::ₘ (Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
rwa :
  ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₁ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal ::ₘ (Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa₂ : ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal * ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
rwa :
  ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₁ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal ::ₘ (Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₂ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal * ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa₃ : ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal = ⊥ ∨ ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥)) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
rwa :
  ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₁ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal ::ₘ (Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₂ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal * ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₃ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal = ⊥ ∨ ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥))
    (hPZ : P ∈ Z) (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa₄ : ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal = ⊥ ∧ ¬((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
rwa :
  ∀ (hprodZ : ¬(Multiset.map asIdeal Z).prod = ⊥) (P : PrimeSpectrum A) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₁ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal ::ₘ (Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₂ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal * ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₃ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬(P.asIdeal = ⊥ ∨ ((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥))
    (hPZ : P ∈ Z) (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
rwa₄ :
  ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal = ⊥ ∧ ¬((Multiset.map asIdeal Z).erase P.asIdeal).prod = ⊥) (hPZ : P ∈ Z)
    (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
    (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
    P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
⊢ P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
have rwa₅ : ∀ (P : PrimeSpectrum A) (hprodZ : ¬P.asIdeal = ⊥ ∧ ¬(Multiset.map asIdeal (Z.erase P)).prod = ⊥) (hPZ : P ∈ Z)
  (hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z) (hPM : P.asIdeal ≤ M)
  (this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal),
  P.asIdeal ≠ ⊥ ∧ (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
Cleared hypotheses: hprodZ, P, hPZ, hPZ', hPM, this
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (hypothesis)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have hP0 : P.asIdeal ≠ ⊥
Tactic: obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map asIdeal).prod ≠ ⊥ := by
      rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Basic (hypothesis)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
hNF : ¬IsField A
I M : Ideal A
hI0 : I ≠ ⊥
hIM : I ≤ M
hM : M.IsMaximal
Z₀ : Multiset (PrimeSpectrum A)
hZ₀ : (Multiset.map asIdeal Z₀).prod ≤ I ∧ (Multiset.map asIdeal Z₀).prod ≠ ⊥
Z : Multiset (PrimeSpectrum A)
h_eraseZ :
  ∀ (x : Multiset (PrimeSpectrum A)),
    Membership.mem (γ := Set (Multiset (PrimeSpectrum A)))
        {Z : Multiset (PrimeSpectrum A) | (Multiset.map asIdeal Z).prod ≤ I ∧ (Multiset.map asIdeal Z).prod ≠ ⊥} x →
      ¬x < Z
hZI : (Multiset.map asIdeal Z).prod ≤ I
hprodZ : (Multiset.map asIdeal Z).prod ≠ ⊥
P : PrimeSpectrum A
hPZ : P ∈ Z
hPZ' : P.asIdeal ∈ Multiset.map asIdeal Z
hPM : P.asIdeal ≤ M
this : Multiset.map asIdeal (Z.erase P) = (Multiset.map asIdeal Z).erase P.asIdeal
hP0 : P.asIdeal ≠ ⊥
⊢ ∃ (Z : Multiset (PrimeSpectrum A)), (M ::ₘ Multiset.map asIdeal Z).prod ≤ I ∧ ¬(Multiset.map asIdeal Z).prod ≤ I
have hZP0 : (Multiset.map asIdeal (Z.erase P)).prod ≠ ⊥
Tactic: obtain ⟨hP0, hZP0⟩ : P.asIdeal ≠ ⊥ ∧ ((Z.erase P).map asIdeal).prod ≠ ⊥ := by
      rwa [Ne, ← Multiset.cons_erase hPZ', Multiset.prod_cons, Ideal.mul_eq_bot, not_or, ←
        this] at hprodZ