awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
hv' : v.maxPowDividing I = 1
⊢ False

Tactic:
hv hv'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
hv' : v.maxPowDividing I = 1
⊢ False

Tactic:
exact hv hv'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
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}
⊢ 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 [Ne, zpow_natCast, ← FractionalIdeal.coeIdeal_pow, FractionalIdeal.coeIdeal_eq_one]

Premises used: zpow_natCast, Ne, FractionalIdeal.coeIdeal_eq_one, FractionalIdeal.coeIdeal_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 |
      ¬x.asIdeal ^ Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I)) = 1}

Tactic:
hasFiniteMulSupport hI

Premises used: setOf, Ideal.hasFiniteMulSupport, Function.mulSupport, Function.HasFiniteMulSupport, IsDedekindDomain.HeightOneSpectrum.maxPowDividing
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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 |
      ¬x.asIdeal ^ Associates.count (Associates.mk x.asIdeal) (Associates.factors (Associates.mk I)) = 1}

Tactic:
exact hasFiniteMulSupport hI

Premises used: Ideal.hasFiniteMulSupport
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
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}
⊢ 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]

Premises used: inv_eq_one, Ne, zpow_neg
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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}

Tactic:
hasFiniteMulSupport_coe hI

Premises used: Ideal.hasFiniteMulSupport_coe, setOf, Function.mulSupport, Function.HasFiniteMulSupport
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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}

Tactic:
exact hasFiniteMulSupport_coe hI

Premises used: Ideal.hasFiniteMulSupport_coe
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I

Tactic:
have hf := hasFiniteMulSupport hI
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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.maxPowDividing I ≠ 0

Tactic:
have h_ne_zero : v.maxPowDividing I ≠ 0 := pow_ne_zero _ v.ne_bot
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
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
⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣
      ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I

Tactic:
rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]

Premises used: pow_add, finprod_cond_ne, pow_one, mul_finprod_cond_ne
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ Prime v.asIdeal

Tactic:
have hv_prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.isPrime
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
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
⊢ False

Tactic:
obtain ⟨w, hw, hvw'⟩ :=
    Prime.exists_mem_finset_dvd hv_prime ((mul_dvd_mul_iff_left h_ne_zero).mp h_contr)

Premises used: Prime.exists_mem_finset_dvd, mul_dvd_mul_iff_left, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ Prime w.asIdeal

Tactic:
have hw_prime : Prime w.asIdeal := Ideal.prime_of_isPrime w.ne_bot w.isPrime
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ v.asIdeal ∣ w.asIdeal

Tactic:
have hvw := Prime.dvd_of_dvd_pow hv_prime hvw'
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ v.asIdeal = w.asIdeal

Tactic:
rw [Prime.dvd_prime_iff_associated hv_prime hw_prime, associated_iff_eq] at hvw

Premises used: associated_iff_eq, Prime.dvd_prime_iff_associated
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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

Tactic:
(Finset.mem_erase.mp hw).1 (HeightOneSpectrum.ext hvw.symm)

Premises used: IsDedekindDomain.HeightOneSpectrum.ext, Finset.mem_erase, And.left, Iff.mp, Eq.symm
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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

Tactic:
exact (Finset.mem_erase.mp hw).1 (HeightOneSpectrum.ext hvw.symm)

Premises used: IsDedekindDomain.HeightOneSpectrum.ext, Finset.mem_erase, And.left, Iff.mp, Eq.symm
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
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
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I)) 0

Tactic:
rw [Associates.mk_ne_zero, finprod_def]

Premises used: Associates.mk_ne_zero, finprod_def
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
neg :
  (¬HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) →
    Ne.{u_1 + 1} (α := Ideal R) 1 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

Tactic:
split_ifs
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
pos :
  ∀ a ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h,
    a.maxPowDividing I ≠ 0
⊢ 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:
rw [Finset.prod_ne_zero_iff]

Premises used: Finset.prod_ne_zero_iff
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
v : HeightOneSpectrum R
pos :
  v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h →
    v.maxPowDividing I ≠ 0
⊢ v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h →
    v.maxPowDividing I ≠ 0

Tactic:
_
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
v : HeightOneSpectrum R
h_1 : v ∈ Finite.toFinset (s := mulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) h
⊢ v.maxPowDividing I ≠ 0

Tactic:
apply pow_ne_zero _ v.ne_bot

Premises used: pow_ne_zero, IsDedekindDomain.HeightOneSpectrum.ne_bot
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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.{u_1 + 1} (α := Ideal R) 1 0

Tactic:
one_ne_zero

Premises used: one_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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.{u_1 + 1} (α := Ideal R) 1 0

Tactic:
exact one_ne_zero

Premises used: one_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
neg neg₁ :
  (¬HasFiniteMulSupport (M := Ideal R) fun (v : HeightOneSpectrum R) ↦ v.maxPowDividing I) →
    Ne.{u_1 + 1} (α := Ideal R) 1 0
⊢ Ne.{u_1 + 1} (α := Associates (Ideal R)) (Associates.mk (∏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

Premises used: Finset.prod_ne_zero_iff, Associates.mk_ne_zero, finprod_def, one_ne_zero
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
v : HeightOneSpectrum R
I : Ideal R
hI : I ≠ 0
⊢ 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
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk v.asIdeal)

Tactic:
have hv : Irreducible (Associates.mk v.asIdeal) := v.associates_irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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)
⊢ v.maxPowDividing I ∣ ∏ᶠ (v : HeightOneSpectrum R), v.maxPowDividing I

Tactic:
have h_dvd := finprod_mem_dvd v (hasFiniteMulSupport hI)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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
⊢ ¬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
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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))
⊢ 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

Premises used: Associates.mk_dvd_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
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))))
⊢ ¬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

Premises used: Associates.mk_dvd_mk