awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
rw :
  v.asIdeal ^ count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₁ :
  v.asIdeal ^
      count (Associates.mk v.asIdeal)
        (↑(factors' I) : WithTop (Multiset { a : Associates (Ideal R) // Irreducible a })) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₂ :
  v.asIdeal ^
      Multiset.count (Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯)
        (factors' I) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₃ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map (Subtype.val (p := fun (a : Associates (Ideal R)) ↦ Irreducible a)) (factors' I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₄ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map Associates.mk (UniqueFactorizationMonoid.factors I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
rw₅ :
  v.asIdeal ^
      Multiset.count
        (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) ⋯) :
          Associates (Ideal R))
        (Multiset.map Associates.mk (normalizedFactors I)) =
    v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
⊢ v.maxPowDividing I = v.asIdeal ^ Multiset.count v.asIdeal (normalizedFactors I)
have rw₆ : v.asIdeal ^
    Multiset.count
      (↑(Subtype.mk (p := fun (a : Associates (Ideal R)) ↦ Irreducible a) (Associates.mk v.asIdeal) sorry) :
        Associates (Ideal R))
      (Multiset.map Associates.mk (normalizedFactors I)) =
  v.asIdeal ^
    Multiset.count (α := Associates (Ideal R)) (Associates.mk v.asIdeal)
      (Multiset.map Associates.mk (normalizedFactors I))
Tactic: rw [maxPowDividing, factors_mk _ hI, count_some (irreducible_mk.mpr v.irreducible),
    ← Multiset.count_map_eq_count' _ _ Subtype.val_injective, map_subtype_coe_factors',
    factors_eq_normalizedFactors, ← Multiset.count_map_eq_count' _ _ (mk_injective (M := Ideal R))]
-- 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
⊢ Finite.{u_1 + 1} { x : HeightOneSpectrum R // x.asIdeal ∣ I }
have haveI : ∀ (h_fin : Fintype { x : Ideal R // x ∣ I }), Finite.{u_1 + 1} { x : HeightOneSpectrum R // x.asIdeal ∣ I }
Tactic: haveI h_fin := fintypeSubtypeDvd 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
h_fin : Fintype { x : Ideal R // x ∣ I }
⊢ Finite.{u_1 + 1} { x : HeightOneSpectrum R // x.asIdeal ∣ I }
have v : Injective (β := { x : Ideal R // x ∣ I }) fun (v : { x : HeightOneSpectrum R // x.asIdeal ∣ I }) ↦
  Subtype.mk (p := fun (x : Ideal R) ↦ x ∣ I) (↑v : HeightOneSpectrum R).asIdeal sorry
Tactic: refine
    Finite.of_injective (fun v => (⟨(v : HeightOneSpectrum R).asIdeal, v.2⟩ : { x // x ∣ I })) ?_
-- 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.{u_1 + 1} (α := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R | ¬Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0}
    {v : HeightOneSpectrum R | v.asIdeal ∣ I}
have h : ∀ (v : HeightOneSpectrum R),
  Membership.mem (γ := Set (HeightOneSpectrum R))
      {v : HeightOneSpectrum R | ¬Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0}
      v ↔
    Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.asIdeal ∣ I} v
Tactic: ext v
-- 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 : HeightOneSpectrum R
⊢ Membership.mem (γ := Set (HeightOneSpectrum R))
      {v : HeightOneSpectrum R | ¬Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0}
      v ↔
    Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.asIdeal ∣ I} v
have h : Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R | ¬count (Associates.mk v.asIdeal) (factors (Associates.mk I)) = 0} v ↔
  Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.asIdeal ∣ I} v
Tactic: simp_rw [Int.natCast_eq_zero]
-- 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
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite,
    Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0
have h_supp : Eq.{u_1 + 1} (α := Set (HeightOneSpectrum R))
  {v : HeightOneSpectrum R | ¬Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0}
  {v : HeightOneSpectrum R | v.asIdeal ∣ I}
Tactic: have h_supp : {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count
      (Associates.mk I).factors : ℤ) = 0} = {v : HeightOneSpectrum R | v.asIdeal ∣ I} := by
    ext v
    simp_rw [Int.natCast_eq_zero]
    exact Associates.count_ne_zero_iff_dvd hI v.irreducible
-- 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
h_supp :
  Eq.{u_1 + 1} (α := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R | ¬Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0}
    {v : HeightOneSpectrum R | v.asIdeal ∣ I}
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite,
    Eq (α := ℤ) (↑(count (Associates.mk v.asIdeal) (factors (Associates.mk I))) : ℤ) 0
have rw : Set.Finite.{u_1} (α := HeightOneSpectrum R) {v : HeightOneSpectrum R | v.asIdeal ∣ I}
Tactic: rw [Filter.eventually_cofinite, h_supp]
-- 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 : HeightOneSpectrum R
hv : Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} v
h_zero : Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) 0
⊢ v.maxPowDividing I = 1
have rw : v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) = 1
Tactic: rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
        pow_zero _]
-- 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 : HeightOneSpectrum R
hv : Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} v
h_zero : Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) 0
rw : v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) = 1
⊢ v.maxPowDividing I = 1
have rw₁ : HPow.hPow (β := ℕ) v.asIdeal 0 = 1
Tactic: rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
        pow_zero _]
-- 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 : HeightOneSpectrum R
hv : Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} v
h_zero : Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) 0
rw : v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) = 1
rw₁ : HPow.hPow (β := ℕ) v.asIdeal 0 = 1
⊢ v.maxPowDividing I = 1
have rw₂ : Eq.{u_1 + 1} (α := Ideal R) 1 1
Tactic: rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
        pow_zero _]
-- 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 : HeightOneSpectrum R
hv : Membership.mem (γ := Set (HeightOneSpectrum R)) {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} v
h_zero : Eq (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) 0
⊢ False
have hv' : v.maxPowDividing I = 1
Tactic: have hv' : v.maxPowDividing I = 1 := by
      rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
        pow_zero _]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ Set.Finite.{u_1} (α := HeightOneSpectrum R)
    {x : HeightOneSpectrum R |
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := ℤ) (↑x.asIdeal : FractionalIdeal R⁰ K)
          (↑(Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) ≠
        1}
have simp_rw : Set.Finite.{u_1} (α := HeightOneSpectrum R)
  {x : HeightOneSpectrum R |
    ¬x.asIdeal ^ Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I)) = 1}
Tactic: simp_rw [Ne, zpow_natCast, ← FractionalIdeal.coeIdeal_pow, FractionalIdeal.coeIdeal_eq_one]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : Ideal R
hI : I ≠ 0
⊢ Set.Finite.{u_1} (α := HeightOneSpectrum R)
    {x : HeightOneSpectrum R |
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := ℤ) (↑x.asIdeal : FractionalIdeal R⁰ K)
          (-(↑(Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I))) : ℤ)) ≠
        1}
have simp_rw : Set.Finite.{u_1} (α := HeightOneSpectrum R)
  {x : HeightOneSpectrum R |
    ¬HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := ℤ) (↑x.asIdeal : FractionalIdeal R⁰ K)
          (↑(Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I))) : ℤ) =
        1}
Tactic: simp_rw [zpow_neg, Ne, inv_eq_one]
-- 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
⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
have hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
Tactic: have hf := 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
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
have h_ne_zero : v.maxPowDividing I ≠ 0
Tactic: have h_ne_zero : v.maxPowDividing I ≠ 0 := pow_ne_zero _ v.ne_bot
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I
have rw : ¬v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
Tactic: rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]
-- 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
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
h_contr :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
⊢ False
have hv_prime : Prime v.asIdeal
Tactic: have hv_prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Factorization (goal)
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
h_contr :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
hv_prime : Prime v.asIdeal
⊢ False
have obtain : ∀ (w : HeightOneSpectrum R)
  (hw :
    w ∈
      (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase v)
  (hvw' : v.asIdeal ∣ w.maxPowDividing I), False
Tactic: obtain ⟨w, hw, hvw'⟩ :=
    Prime.exists_mem_finset_dvd hv_prime ((mul_dvd_mul_iff_left h_ne_zero).mp h_contr)
-- 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
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
h_contr :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
hv_prime : Prime v.asIdeal
w : HeightOneSpectrum R
hw :
  w ∈ (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase v
hvw' : v.asIdeal ∣ w.maxPowDividing I
⊢ False
have hw_prime : Prime w.asIdeal
Tactic: have hw_prime : Prime w.asIdeal := Ideal.prime_of_isPrime w.ne_bot w.isPrime
-- 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
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
h_contr :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
hv_prime : Prime v.asIdeal
w : HeightOneSpectrum R
hw :
  w ∈ (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase v
hvw' : v.asIdeal ∣ w.maxPowDividing I
hw_prime : Prime w.asIdeal
⊢ False
have hvw : v.asIdeal ∣ w.asIdeal
Tactic: have hvw := Prime.dvd_of_dvd_pow hv_prime hvw'
-- 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
hf : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
h_ne_zero : v.maxPowDividing I ≠ 0
h_contr :
  v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣
    v.maxPowDividing I *
      ∏
        i ∈
          (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase
            v,
        i.maxPowDividing I
hv_prime : Prime v.asIdeal
w : HeightOneSpectrum R
hw :
  w ∈ (Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) hf).erase v
hvw' : v.asIdeal ∣ w.maxPowDividing I
hw_prime : Prime w.asIdeal
hvw : v.asIdeal ∣ w.asIdeal
⊢ False
have hvw₁ : v.asIdeal = w.asIdeal
Tactic: rw [Prime.dvd_prime_iff_associated hv_prime hw_prime, associated_iff_eq] at hvw
Cleared hypotheses: hvw
-- 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
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
have rw : Ne.{u_1 + 1} (α := Ideal R)
  (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
    ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
      i.maxPowDividing I
  else 1)
  0
Tactic: rw [Associates.mk_ne_zero, finprod_def]
-- 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
⊢ Ne.{u_1 + 1} (α := Ideal R)
    (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
      ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        i.maxPowDividing I
    else 1)
    0
have pos : ∀ (h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I),
  Ne (α := Ideal R)
    (∏ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
      v.maxPowDividing I)
    0
Tactic: split_ifs
-- 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
pos :
  ∀ (h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I),
    Ne (α := Ideal R)
      (∏ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        v.maxPowDividing I)
      0
⊢ Ne.{u_1 + 1} (α := Ideal R)
    (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
      ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        i.maxPowDividing I
    else 1)
    0
have neg : (¬HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) →
  Ne.{u_1 + 1} (α := Ideal R) 1 0
Tactic: split_ifs
-- 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
h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I
⊢ Ne (α := Ideal R)
    (∏ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
      v.maxPowDividing I)
    0
have pos : ∀ a ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
  a.maxPowDividing I ≠ 0
Tactic: rw [Finset.prod_ne_zero_iff]
-- 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
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
have rw : Ne.{u_1 + 1} (α := Ideal R) (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I) 0
Tactic: classical
  rw [Associates.mk_ne_zero, finprod_def]
  split_ifs
  · rw [Finset.prod_ne_zero_iff]
    intro v _
    apply pow_ne_zero _ v.ne_bot
  · exact one_ne_zero
-- 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
rw : Ne.{u_1 + 1} (α := Ideal R) (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I) 0
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
have rw₁ : Ne.{u_1 + 1} (α := Ideal R)
  (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
    ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
      i.maxPowDividing I
  else 1)
  0
Tactic: classical
  rw [Associates.mk_ne_zero, finprod_def]
  split_ifs
  · rw [Finset.prod_ne_zero_iff]
    intro v _
    apply pow_ne_zero _ v.ne_bot
  · exact one_ne_zero
-- 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
rw : Ne.{u_1 + 1} (α := Ideal R) (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I) 0
rw₁ rw₂ rw₃ :
  Ne.{u_1 + 1} (α := Ideal R)
    (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
      ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        i.maxPowDividing I
    else 1)
    0
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
have pos : ∀ (h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I),
  Ne (α := Ideal R)
    (∏ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
      v.maxPowDividing I)
    0
Tactic: classical
  rw [Associates.mk_ne_zero, finprod_def]
  split_ifs
  · rw [Finset.prod_ne_zero_iff]
    intro v _
    apply pow_ne_zero _ v.ne_bot
  · exact one_ne_zero
-- 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
rw : Ne.{u_1 + 1} (α := Ideal R) (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I) 0
rw₁ rw₂ rw₃ :
  Ne.{u_1 + 1} (α := Ideal R)
    (if h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I then
      ∏ i ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        i.maxPowDividing I
    else 1)
    0
pos :
  ∀ (h : HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I),
    Ne (α := Ideal R)
      (∏ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
        v.maxPowDividing I)
      0
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0
have neg : (¬HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) →
  Ne.{u_1 + 1} (α := Ideal R) 1 0
Tactic: classical
  rw [Associates.mk_ne_zero, finprod_def]
  split_ifs
  · rw [Finset.prod_ne_zero_iff]
    intro v _
    apply pow_ne_zero _ v.ne_bot
  · exact one_ne_zero