awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_ne_zero : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
Tactic: have h_ne_zero := Associates.finprod_ne_zero I
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
Tactic: have hv : Irreducible (Associates.mk v.asIdeal) := v.associates_irreducible
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_dvd : v.maxPowDividing I ∣ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
Tactic: have h_dvd := finprod_mem_dvd v (hasFiniteMulSupport hI)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd : v.maxPowDividing I ∣ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_not_dvd : ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
    ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
Tactic: have h_not_dvd := Ideal.finprod_not_dvd v I hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) ∣
    ∏ᶠ (v : HeightOneSpectrum R),
      v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
h_not_dvd :
  ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_dvd₁ : Dvd.dvd (α := Associates (Ideal R))
  (Associates.mk (v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
  (Associates.mk
    (∏ᶠ (v : HeightOneSpectrum R),
      v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
Tactic: rw [← Associates.mk_dvd_mk] at h_dvd h_not_dvd
Cleared hypotheses: h_dvd, h_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) ∣
    ∏ᶠ (v : HeightOneSpectrum R),
      v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
h_not_dvd :
  ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
h_dvd₁ :
  Dvd.dvd (α := Associates (Ideal R))
    (Associates.mk (v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_not_dvd₁ : ¬Dvd.dvd (α := Associates (Ideal R))
    (Associates.mk
      (v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1)))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
Tactic: rw [← Associates.mk_dvd_mk] at h_dvd h_not_dvd
Cleared hypotheses: h_dvd, h_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd :
  LE.le (α := Associates (Ideal R))
    (Associates.mk (v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
h_not_dvd :
  ¬LE.le (α := Associates (Ideal R))
      (Associates.mk
        (v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1)))
      (Associates.mk
        (∏ᶠ (v : HeightOneSpectrum R),
          v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_dvd₁ : Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) ≤
  Associates.count (Associates.mk v.asIdeal)
    (Associates.factors
      (Associates.mk
        (∏ᶠ (v : HeightOneSpectrum R),
          v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)))))
Tactic: rw [Associates.mk_pow, Associates.prime_pow_dvd_iff_le h_ne_zero hv] at h_dvd h_not_dvd
Cleared hypotheses: h_dvd, h_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd :
  LE.le (α := Associates (Ideal R))
    (Associates.mk (v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
h_not_dvd :
  ¬LE.le (α := Associates (Ideal R))
      (Associates.mk
        (v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1)))
      (Associates.mk
        (∏ᶠ (v : HeightOneSpectrum R),
          v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
h_dvd₁ :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) ≤
    Associates.count (Associates.mk v.asIdeal)
      (Associates.factors
        (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R),
            v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)))))
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_not_dvd₁ : ¬Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1 ≤
    Associates.count (Associates.mk v.asIdeal)
      (Associates.factors
        (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R),
            v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)))))
Tactic: rw [Associates.mk_pow, Associates.prime_pow_dvd_iff_le h_ne_zero hv] at h_dvd h_not_dvd
Cleared hypotheses: h_dvd, h_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
h_ne_zero :
  Ne.{u_1 + 1} (α := Associates (Ideal R))
    (Associates.mk
      (∏ᶠ (v : HeightOneSpectrum R),
        v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))
    0
hv : Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)
h_dvd :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) ≤
    Associates.count (Associates.mk v.asIdeal)
      (Associates.factors
        (Associates.mk
          (∏ᶠ (v : HeightOneSpectrum R),
            v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)))))
h_not_dvd :
  ¬Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1 ≤
      Associates.count (Associates.mk v.asIdeal)
        (Associates.factors
          (Associates.mk
            (∏ᶠ (v : HeightOneSpectrum R),
              v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)))))
⊢ Associates.count (Associates.mk v.asIdeal)
      (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))
have h_not_dvd₁ : Associates.count (Associates.mk v.asIdeal)
    (Associates.factors
      (Associates.mk
        (∏ᶠ (v : HeightOneSpectrum R),
          v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))))) <
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1
Tactic: rw [not_le] at h_not_dvd
Cleared hypotheses: h_not_dvd
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I = I
have rw : Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
Tactic: rw [← associated_iff_eq, ← Associates.mk_eq_mk_iff_associated]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
Tactic: apply Associates.eq_of_eq_counts
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
Tactic: apply Associates.eq_of_eq_counts
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have h : ∀ (p : Associates (Ideal R)),
  Irreducible p →
    p.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
      p.count (Associates.factors (Associates.mk I))
Tactic: apply Associates.eq_of_eq_counts
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
v : Associates (Ideal R)
hv : Irreducible v
⊢ v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
have h : ∀ (J : Ideal R) (hJv : Associates.mk J = v),
  v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
Tactic: obtain ⟨J, hJv⟩ := Associates.exists_rep v
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (hypothesis)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
v : Associates (Ideal R)
hv : Irreducible v
J : Ideal R
hJv : Associates.mk J = v
⊢ v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
have hv₁ : Irreducible J
Tactic: rw [← hJv, Associates.irreducible_mk] at hv
Cleared hypotheses: hv, hJv
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
v : Associates (Ideal R)
J : Ideal R
hv : Irreducible J
hJv : Associates.mk J = v
⊢ v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
have h : Associates.count (Associates.mk J)
    (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
  Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))
Tactic: rw [← hJv]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
Tactic: classical
  apply Associates.eq_of_eq_counts
  · apply Associates.finprod_ne_zero I
  · apply Associates.mk_ne_zero.mpr hI
  intro v hv
  obtain ⟨J, hJv⟩ := Associates.exists_rep v
  rw [← hJv, Associates.irreducible_mk] at hv
  rw [← hJv]
  apply Ideal.finprod_count
    ⟨J, Ideal.isPrime_of_prime (irreducible_iff_prime.mp hv), Irreducible.ne_zero hv⟩ I hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
Tactic: classical
  apply Associates.eq_of_eq_counts
  · apply Associates.finprod_ne_zero I
  · apply Associates.mk_ne_zero.mpr hI
  intro v hv
  obtain ⟨J, hJv⟩ := Associates.exists_rep v
  rw [← hJv, Associates.irreducible_mk] at hv
  rw [← hJv]
  apply Ideal.finprod_count
    ⟨J, Ideal.isPrime_of_prime (irreducible_iff_prime.mp hv), Irreducible.ne_zero hv⟩ I hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have h : ∀ (p : Associates (Ideal R)),
  Irreducible p →
    p.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
      p.count (Associates.factors (Associates.mk I))
Tactic: classical
  apply Associates.eq_of_eq_counts
  · apply Associates.finprod_ne_zero I
  · apply Associates.mk_ne_zero.mpr hI
  intro v hv
  obtain ⟨J, hJv⟩ := Associates.exists_rep v
  rw [← hJv, Associates.irreducible_mk] at hv
  rw [← hJv]
  apply Ideal.finprod_count
    ⟨J, Ideal.isPrime_of_prime (irreducible_iff_prime.mp hv), Irreducible.ne_zero hv⟩ I hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
h :
  ∀ (p : Associates (Ideal R)),
    Irreducible p →
      p.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
        p.count (Associates.factors (Associates.mk I))
hb₁ : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
h₁ h₂ h₃ :
  ∀ (v : Associates (Ideal R)) (hv : Irreducible v),
    v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
      v.count (Associates.factors (Associates.mk I))
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have h₄ : ∀ (v : Associates (Ideal R)) (hv : Irreducible v) (J : Ideal R) (hJv : Associates.mk J = v),
  v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
Tactic: classical
  apply Associates.eq_of_eq_counts
  · apply Associates.finprod_ne_zero I
  · apply Associates.mk_ne_zero.mpr hI
  intro v hv
  obtain ⟨J, hJv⟩ := Associates.exists_rep v
  rw [← hJv, Associates.irreducible_mk] at hv
  rw [← hJv]
  apply Ideal.finprod_count
    ⟨J, Ideal.isPrime_of_prime (irreducible_iff_prime.mp hv), Irreducible.ne_zero hv⟩ I hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
ha : Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
hb : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
h :
  ∀ (p : Associates (Ideal R)),
    Irreducible p →
      p.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
        p.count (Associates.factors (Associates.mk I))
hb₁ : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
h₁ h₂ h₃ :
  ∀ (v : Associates (Ideal R)) (hv : Irreducible v),
    v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
      v.count (Associates.factors (Associates.mk I))
h₄ :
  ∀ (v : Associates (Ideal R)) (hv : Irreducible v) (J : Ideal R) (hJv : Associates.mk J = v),
    v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
      v.count (Associates.factors (Associates.mk I))
⊢ Eq (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) (Associates.mk I)
have h₅ : ∀ (v : Associates (Ideal R)) (J : Ideal R) (hv : Irreducible J) (hJv : Associates.mk J = v),
  v.count (Associates.factors (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I))) =
    v.count (Associates.factors (Associates.mk I))
Tactic: classical
  apply Associates.eq_of_eq_counts
  · apply Associates.finprod_ne_zero I
  · apply Associates.mk_ne_zero.mpr hI
  intro v hv
  obtain ⟨J, hJv⟩ := Associates.exists_rep v
  rw [← hJv, Associates.irreducible_mk] at hv
  rw [← hJv]
  apply Ideal.finprod_count
    ⟨J, Ideal.isPrime_of_prime (irreducible_iff_prime.mp hv), Irreducible.ne_zero hv⟩ I hI