awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    v
hv_irred : Irreducible v.asIdeal
h_notMem : ¬(v.asIdeal ∣ J ∨ v.asIdeal ∣ Ideal.span {a})
⊢ ¬v.asIdeal ∣ J ∧ ¬v.asIdeal ∣ Ideal.span {a}

Tactic:
push Not at h_notMem

Premises used: Not
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    v
hv_irred : Irreducible v.asIdeal
h_notMem : ¬v.asIdeal ∣ J ∧ ¬v.asIdeal ∣ Ideal.span {a}
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0

Tactic:
rw [← Associates.count_ne_zero_iff_dvd ha_ne_zero hv_irred, not_not,
      ← Associates.count_ne_zero_iff_dvd hJ_ne_zero hv_irred, not_not] at h_notMem

Premises used: Classical.not_not, Associates.count_ne_zero_iff_dvd
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    v
hv_irred : Irreducible v.asIdeal
h_notMem :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0
⊢ ¬Eq (α := ℤ) 0 0

Tactic:
rw [mem_setOf_eq, h_notMem.1, h_notMem.2, sub_self] at hv

Premises used: sub_self, Set.mem_setOf_eq, And.right, And.left
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    v
hv_irred : Irreducible v.asIdeal
h_notMem :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0
hv₁ : ¬Eq (α := ℤ) 0 0
⊢ Irreducible v.asIdeal

Tactic:
rw [mem_setOf_eq, h_notMem.1, h_notMem.2, sub_self] at hv

Premises used: sub_self, Set.mem_setOf_eq, And.right, And.left
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv :
  Membership.mem (γ := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    v
hv_irred : Irreducible v.asIdeal
h_notMem :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0
hv₁ : ¬Eq (α := ℤ) 0 0
hv_irred₁ : Irreducible v.asIdeal
⊢ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0

Tactic:
rw [mem_setOf_eq, h_notMem.1, h_notMem.2, sub_self] at hv

Premises used: sub_self, Set.mem_setOf_eq, And.right, And.left
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv : ¬Eq (α := ℤ) 0 0
hv_irred : Irreducible v.asIdeal
h_notMem :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0
⊢ False

Tactic:
hv (Eq.refl 0)

Premises used: Eq.refl, OfNat.ofNat
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
v : HeightOneSpectrum R
hv : ¬Eq (α := ℤ) 0 0
hv_irred : Irreducible v.asIdeal
h_notMem :
  Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J)) = 0 ∧
    Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a}))) = 0
⊢ False

Tactic:
exact hv (Eq.refl 0)

Premises used: Eq.refl, OfNat.ofNat
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
⊢ Subset.{u_1} (α := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    ({v : HeightOneSpectrum R | v.asIdeal ∣ J} ∪ {v : HeightOneSpectrum R | v.asIdeal ∣ Ideal.span {a}})

Tactic:
have h_subset :
    {v : HeightOneSpectrum R | ¬((Associates.mk v.asIdeal).count (Associates.mk J).factors : ℤ) -
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) = 0} ⊆
    {v : HeightOneSpectrum R | v.asIdeal ∣ J} ∪
      {v : HeightOneSpectrum R | v.asIdeal ∣ Ideal.span {a}} := by
    intro v hv
    have hv_irred : Irreducible v.asIdeal := v.irreducible
    by_contra h_notMem
    rw [mem_union, mem_setOf_eq, mem_setOf_eq] at h_notMem
    push Not at h_notMem
    rw [← Associates.count_ne_zero_iff_dvd ha_ne_zero hv_irred, not_not,
      ← Associates.count_ne_zero_iff_dvd hJ_ne_zero hv_irred, not_not] at h_notMem
    rw [mem_setOf_eq, h_notMem.1, h_notMem.2, sub_self] at hv
    exact hv (Eq.refl 0)
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
h_subset :
  Subset.{u_1} (α := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    ({v : HeightOneSpectrum R | v.asIdeal ∣ J} ∪ {v : HeightOneSpectrum R | v.asIdeal ∣ Ideal.span {a}})
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite,
    HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
        (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
      0

Tactic:
Finite.subset (Finite.union (Ideal.finite_factors (ideal_factor_ne_zero hI haJ))
    (Ideal.finite_factors (constant_factor_ne_zero hI haJ))) h_subset

Premises used: Set.Finite.union, FractionalIdeal.ideal_factor_ne_zero, FractionalIdeal.constant_factor_ne_zero, Set.Mem, Filter.Eventually, setOf, Set.Finite, Set.Finite.subset, Ideal.finite_factors
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
a : R
J : Ideal R
haJ :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) a)⁻¹ : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑J : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
ha_ne_zero : Ne (α := Ideal R) (Ideal.span {a}) 0
hJ_ne_zero : J ≠ 0
h_subset :
  Subset.{u_1} (α := Set (HeightOneSpectrum R))
    {v : HeightOneSpectrum R |
      ¬HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
            (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
          0}
    ({v : HeightOneSpectrum R | v.asIdeal ∣ J} ∪ {v : HeightOneSpectrum R | v.asIdeal ∣ Ideal.span {a}})
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite,
    HSub.hSub (α := ℤ) (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk J))) : ℤ)
        (↑(Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk (Ideal.span {a})))) : ℤ) =
      0

Tactic:
exact Finite.subset (Finite.union (Ideal.finite_factors (ideal_factor_ne_zero hI haJ))
    (Ideal.finite_factors (constant_factor_ne_zero hI haJ))) h_subset

Premises used: Set.Finite.union, FractionalIdeal.ideal_factor_ne_zero, FractionalIdeal.constant_factor_ne_zero, Set.Finite.subset, Ideal.finite_factors
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
pos : ∀ (hI : I = 0), ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0
neg : ∀ (hI : ¬I = 0), ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0

Tactic:
by_cases hI : I = 0
-- 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 : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I = 0
⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0

Tactic:
simp only [hI, count_zero, Filter.eventually_cofinite, not_true_eq_false, setOf_false,
      finite_empty]

Premises used: Set.finite_empty, Filter.eventually_cofinite, not_true_eq_false, FractionalIdeal.count_zero, Set.setOf_false
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hIJ : I ≤ J
hI : I ≠ 0
obtain : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : J * I ≠ 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
⊢ ∃ (a : R), I ⊔ Ideal.span {a} = J

Tactic:
obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ

Premises used: Iff.mpr, Ideal.dvd_iff_le, rfl
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hI : J * I ≠ 0
⊢ ¬J = 0 ∧ ¬I = 0

Tactic:
simp only [ne_eq, mul_eq_zero, not_or] at hI

Premises used: mul_eq_zero, ne_eq, not_or
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hI : ¬J = 0 ∧ ¬I = 0
⊢ ¬J = 0

Tactic:
obtain ⟨hJ, hI⟩ := hI
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hI : ¬J = 0 ∧ ¬I = 0
hJ : ¬J = 0
⊢ ¬I = 0

Tactic:
obtain ⟨hJ, hI⟩ := hI
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
this : ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
obtain : ∀ (a : R) (K : Ideal R) (e : J * K = Ideal.span {a}) (e' : I + K = ⊤), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
⊢ ∃ (a : R), J * I ⊔ Ideal.span {a} = J

Tactic:
obtain ⟨a, K, e, e'⟩ := this
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
a : R
K : Ideal R
e : J * K = Ideal.span {a}
e' : I + K = ⊤
rw : J * I ⊔ J * K = J
rw₁ : J * I + J * K = J
rw₂ : J * (I + K) = J
rw₃ : J * ⊤ = J
rw₄ rw₅ : J = J
⊢ J * I ⊔ Ideal.span {a} = J

Tactic:
rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]

Premises used: Ideal.add_eq_sup, mul_add, Ideal.mul_top
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
a : R
K : Ideal R
e : J * K = Ideal.span {a}
e' : I + K = ⊤
⊢ ∃ (a : R), J * I ⊔ Ideal.span {a} = J

Tactic:
exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
suffices : ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
⊢ ∃ (a : R), J * I ⊔ Ideal.span {a} = J

Tactic:
suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
    obtain ⟨a, K, e, e'⟩ := this
    exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
rw : (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1
| J * ∏ q ∈ s \ {p}, q.asIdeal

Tactic:
rw [← mul_one (J * _)]

Premises used: HMul.hMul, mul_one
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
conv_rhs : J * ∏ q ∈ s, q.asIdeal < (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1
⊢ J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal

Tactic:
conv_rhs => rw [← mul_one (J * _)]

Premises used: HMul.hMul, mul_one
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
rw : (J * ∏ x ∈ s \ {p}, x.asIdeal) * p.asIdeal < (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1
⊢ J * ∏ q ∈ s, q.asIdeal < (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1

Tactic:
rw [Finset.prod_eq_mul_prod_diff_singleton_of_mem hps, ← mul_assoc,
      mul_right_comm _ p.asIdeal]

Premises used: mul_assoc, IsDedekindDomain.HeightOneSpectrum.asIdeal, Finset.prod_eq_mul_prod_diff_singleton_of_mem, mul_right_comm
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
⊢ (J * ∏ x ∈ s \ {p}, x.asIdeal) * p.asIdeal < (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1

Tactic:
mul_lt_mul_of_pos_left ?_ ?_

Premises used: Set.Mem, mul_lt_mul_of_pos_left
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
refine_1 : p.asIdeal < 1
refine_2 : 0 < J * ∏ x ∈ s \ {p}, x.asIdeal
⊢ (J * ∏ x ∈ s \ {p}, x.asIdeal) * p.asIdeal < (J * ∏ q ∈ s \ {p}, q.asIdeal) * 1

Tactic:
refine mul_lt_mul_of_pos_left ?_ ?_

Premises used: mul_lt_mul_of_pos_left
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
refine_1 : p.asIdeal ≠ ⊤
⊢ p.asIdeal < 1

Tactic:
rw [Ideal.one_eq_top, lt_top_iff_ne_top]

Premises used: Ideal.one_eq_top, lt_top_iff_ne_top
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
⊢ p.asIdeal ≠ ⊤

Tactic:
p.2.ne_top

Premises used: IsDedekindDomain.HeightOneSpectrum.isPrime, Ideal.IsPrime.ne_top
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
⊢ p.asIdeal ≠ ⊤

Tactic:
exact p.2.ne_top

Premises used: IsDedekindDomain.HeightOneSpectrum.isPrime, Ideal.IsPrime.ne_top
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
refine_2 : J ≠ 0 ∧ ∀ a ∈ s \ {p}, a.asIdeal ≠ 0
⊢ 0 < J * ∏ x ∈ s \ {p}, x.asIdeal

Tactic:
rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]

Premises used: bot_lt_iff_ne_bot, Finset.prod_ne_zero_iff, Ideal.zero_eq_bot, mul_ne_zero_iff
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
⊢ J ≠ 0 ∧ ∀ a ∈ s \ {p}, a.asIdeal ≠ 0

Tactic:
⟨hJ, fun x _ ↦ x.3⟩

Premises used: And.intro, IsDedekindDomain.HeightOneSpectrum.ne_bot
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
p : HeightOneSpectrum R
hps : p ∈ s
⊢ J ≠ 0 ∧ ∀ a ∈ s \ {p}, a.asIdeal ≠ 0

Tactic:
exact ⟨hJ, fun x _ ↦ x.3⟩