awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : Q = P
hP₁ : Q.IsPrime
x_mem₁ : x ∈ Q
hxP2₁ : x ∉ HPow.hPow (β := ℕ) Q 2
⊢ Multiset.count Q (normalizedFactors (span {x})) = 1
have hxQ₁ : ∀ (Q_1 : Ideal R), Q_1.IsPrime → Q_1 ≠ Q → x ∉ Q_1
Tactic: subst hQ
Cleared hypotheses: P, hP, x_mem, hxP2, hxQ, hP0, hQ
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : Q = P
hP₁ : Q.IsPrime
x_mem₁ : x ∈ Q
hxP2₁ : x ∉ HPow.hPow (β := ℕ) Q 2
hxQ₁ : ∀ (Q_1 : Ideal R), Q_1.IsPrime → Q_1 ≠ Q → x ∉ Q_1
⊢ Multiset.count Q (normalizedFactors (span {x})) = 1
have hP0₁ : ¬Q = ⊥
Tactic: subst hQ
Cleared hypotheses: P, hP, x_mem, hxP2, hxQ, hP0, hQ
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
x : R
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hP : Q.IsPrime
x_mem : x ∈ Q
hxP2 : x ∉ HPow.hPow (β := ℕ) Q 2
hxQ : ∀ (Q_1 : Ideal R), Q_1.IsPrime → Q_1 ≠ Q → x ∉ Q_1
hP0 : ¬Q = ⊥
⊢ Multiset.count Q (normalizedFactors (span {x})) = 1
have pos : span {x} ≤ HPow.hPow (β := ℕ) Q 1
Tactic: apply Ideal.count_normalizedFactors_eq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
x : R
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hP : Q.IsPrime
x_mem : x ∈ Q
hxP2 : x ∉ HPow.hPow (β := ℕ) Q 2
hxQ : ∀ (Q_1 : Ideal R), Q_1.IsPrime → Q_1 ≠ Q → x ∉ Q_1
hP0 : ¬Q = ⊥
pos : span {x} ≤ HPow.hPow (β := ℕ) Q 1
⊢ Multiset.count Q (normalizedFactors (span {x})) = 1
have pos₁ : ¬span {x} ≤ Q ^ HAdd.hAdd (α := ℕ) 1 1
Tactic: apply Ideal.count_normalizedFactors_eq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : ¬Q = P
⊢ Multiset.count Q (normalizedFactors (span {x})) = 0
have pos : ∀ (hQp : Q.IsPrime), Multiset.count Q (normalizedFactors (span {x})) = 0
Tactic: by_cases hQp : IsPrime Q
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : ¬Q = P
pos : ∀ (hQp : Q.IsPrime), Multiset.count Q (normalizedFactors (span {x})) = 0
⊢ Multiset.count Q (normalizedFactors (span {x})) = 0
have neg : ∀ (hQp : ¬Q.IsPrime), Multiset.count Q (normalizedFactors (span {x})) = 0
Tactic: by_cases hQp : IsPrime Q
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : ¬Q = P
hQp : Q.IsPrime
⊢ Multiset.count Q (normalizedFactors (span {x})) = 0
have pos : span {x} ≤ HPow.hPow (β := ℕ) Q 0
Tactic: apply Ideal.count_normalizedFactors_eq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
hP0 : ¬P = ⊥
hspan0 : Ne (α := Ideal R) (span {x}) ⊥
Q : Ideal R
hQ : ¬Q = P
hQp : Q.IsPrime
pos : span {x} ≤ HPow.hPow (β := ℕ) Q 0
⊢ Multiset.count Q (normalizedFactors (span {x})) = 0
have pos₁ : ¬span {x} ≤ Q ^ HAdd.hAdd (α := ℕ) 0 1
Tactic: apply Ideal.count_normalizedFactors_eq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
⊢ P = span {x}
have pos : ∀ (hP0 : P = ⊥), P = span {x}
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
⊢ P = span {x}
have neg : ∀ (hP0 : ¬P = ⊥), P = span {x}
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
⊢ P = span {x}
have neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
⊢ P = span {x}
have neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
⊢ P = span {x}
have neg₄ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
  Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
⊢ P = span {x}
have neg₅ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
  (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
neg₅ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
⊢ P = span {x}
have neg₆ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
  Multiset.count Q (normalizedFactors (span {x})) = if Q = P then 1 else 0
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
neg₅ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
neg₆ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors (span {x})) = if Q = P then 1 else 0
⊢ P = span {x}
have pos₁ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : Q = P),
  Multiset.count Q (normalizedFactors (span {x})) = 1
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
neg₅ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
neg₆ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors (span {x})) = if Q = P then 1 else 0
pos₁ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : Q = P),
    Multiset.count Q (normalizedFactors (span {x})) = 1
⊢ P = span {x}
have neg₇ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P),
  Multiset.count Q (normalizedFactors (span {x})) = 0
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
neg₅ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
neg₆ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors (span {x})) = if Q = P then 1 else 0
pos₁ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : Q = P),
    Multiset.count Q (normalizedFactors (span {x})) = 1
neg₇ neg₈ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P),
    Multiset.count Q (normalizedFactors (span {x})) = 0
⊢ P = span {x}
have pos₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P) (hQp : Q.IsPrime),
  Multiset.count Q (normalizedFactors (span {x})) = 0
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (goal)
Goal:
R : Type u_1
inst : CommRing R
P : Ideal R
hP : P.IsPrime
inst_1 : IsDedekindDomain R
x : R
x_mem : x ∈ P
hxP2 : x ∉ HPow.hPow (β := ℕ) P 2
hxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q
pos : ∀ (hP0 : P = ⊥), P = span {x}
neg neg₁ : ∀ (hP0 : ¬P = ⊥), P = span {x}
neg₂ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), P = span {x}
neg₃ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥), normalizedFactors P = normalizedFactors (span {x})
neg₄ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors P) = Multiset.count Q (normalizedFactors (span {x}))
neg₅ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    (if Q = P then 1 else 0) = Multiset.count Q (normalizedFactors (span {x}))
neg₆ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R),
    Multiset.count Q (normalizedFactors (span {x})) = if Q = P then 1 else 0
pos₁ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : Q = P),
    Multiset.count Q (normalizedFactors (span {x})) = 1
neg₇ neg₈ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P),
    Multiset.count Q (normalizedFactors (span {x})) = 0
pos₂ :
  ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P) (hQp : Q.IsPrime),
    Multiset.count Q (normalizedFactors (span {x})) = 0
⊢ P = span {x}
have neg₉ : ∀ (hP0 : ¬P = ⊥) (hspan0 : Ne (α := Ideal R) (span {x}) ⊥) (Q : Ideal R) (hQ : ¬Q = P) (hQp : ¬Q.IsPrime),
  Multiset.count Q (normalizedFactors (span {x})) = 0
Tactic: classical
  by_cases hP0 : P = ⊥
  · subst hP0
    rwa [eq_comm, span_singleton_eq_bot, ← mem_bot]
  have hspan0 : span {x} ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp (hxP2 <| · ▸ zero_mem _)
  rw [← associated_iff_eq, associated_iff_normalizedFactors_eq_normalizedFactors hP0 hspan0]
  refine Multiset.ext' fun Q ↦ ?_
  rw [normalizedFactors_irreducible (Ideal.prime_of_isPrime hP0 hP).irreducible, normalize_eq,
    Multiset.count_singleton]
  symm
  split_ifs with hQ
  · subst hQ
    apply Ideal.count_normalizedFactors_eq <;> simpa
  by_cases hQp : IsPrime Q
  · apply Ideal.count_normalizedFactors_eq <;> simp [hxQ _ hQp hQ]
  exact Multiset.count_eq_zero.mpr fun hQi ↦
    hQp <| isPrime_of_prime <| irreducible_iff_prime.mp <| irreducible_of_normalized_factor _ hQi
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.PID (hypothesis)
Goal:
R : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
S : Submonoid R
inst_3 : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ (↑I⁻¹ : FractionalIdeal S A)
h : Submodule.comap (Algebra.linearMap R A) ((↑(↑I : FractionalIdeal S A) : Submodule R A) * (R ∙ v)) = ⊤
⊢ (↑(↑I : FractionalIdeal S A) : Submodule R A).IsPrincipal
have hinv : (↑I : FractionalIdeal S A) * (↑I⁻¹ : FractionalIdeal S A) = 1
Tactic: have hinv := I.mul_inv