awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (M := Ideal R) (span {a})
neg : ∀ (h : Eq (α := Ideal R) (span {b}) 0), False
⊢ Ne (α := Ideal R) (span {b}) 0

Tactic:
by_contra h
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (M := Ideal R) (span {a})
h : Eq (α := Ideal R) (span {b}) 0
⊢ False

Tactic:
hb (span_singleton_eq_bot.mp h)

Premises used: Ideal.span_singleton_eq_bot, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (M := Ideal R) (span {a})
h : Eq (α := Ideal R) (span {b}) 0
⊢ False

Tactic:
exact hb (span_singleton_eq_bot.mp h)

Premises used: Ideal.span_singleton_eq_bot, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
neg : IsPrime (α := R) (span {a})
neg₁ : Ne (α := Ideal R) (span {a}) ⊥
⊢ Prime (M := Ideal R) (span {a})

Tactic:
rw [prime_iff_isPrime]

Premises used: Ideal.prime_iff_isPrime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ IsPrime (α := R) (span {a})

Tactic:
(span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
        (prime_of_normalized_factor a ha)

Premises used: Iff.mpr, UniqueFactorizationMonoid.prime_of_normalized_factor, Prime.ne_zero, Ideal.span_singleton_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ IsPrime (α := R) (span {a})

Tactic:
exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
        (prime_of_normalized_factor a ha)

Premises used: Iff.mpr, UniqueFactorizationMonoid.prime_of_normalized_factor, Prime.ne_zero, Ideal.span_singleton_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
neg : ∀ (h : Eq (α := Ideal R) (span {a}) ⊥), False
⊢ Ne (α := Ideal R) (span {a}) ⊥

Tactic:
by_contra h
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
h : Eq (α := Ideal R) (span {a}) ⊥
⊢ False

Tactic:
(prime_of_normalized_factor a ha).ne_zero (span_singleton_eq_bot.mp h)

Premises used: Ideal.span_singleton_eq_bot, UniqueFactorizationMonoid.prime_of_normalized_factor, Prime.ne_zero, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
inst_3 : NormalizationMonoid R
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
h : Eq (α := Ideal R) (span {a}) ⊥
⊢ False

Tactic:
exact (prime_of_normalized_factor a ha).ne_zero (span_singleton_eq_bot.mp h)

Premises used: Ideal.span_singleton_eq_bot, UniqueFactorizationMonoid.prime_of_normalized_factor, Prime.ne_zero, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
pos : ∀ (h : FiniteMultiplicity a b), emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
neg : ∀ (h : ¬FiniteMultiplicity a b), emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b

Tactic:
by_cases h : FiniteMultiplicity a b

Premises used: FiniteMultiplicity
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : emultiplicity (α := Ideal R) (span {a}) (span {b}) = (↑(multiplicity a b) : ℕ∞)
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b

Tactic:
rw [h.emultiplicity_eq_multiplicity]

Premises used: FiniteMultiplicity.emultiplicity_eq_multiplicity
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b) ∣ span {b}
pos₁ : ¬HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b + 1) ∣ span {b}
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = (↑(multiplicity a b) : ℕ∞)

Tactic:
apply emultiplicity_eq_of_dvd_of_not_dvd

Premises used: emultiplicity_eq_of_dvd_of_not_dvd
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : a ^ multiplicity a b ∣ b
⊢ HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b) ∣ span {b}

Tactic:
rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]

Premises used: Ideal.span_singleton_dvd_span_singleton_iff_dvd, Ideal.span_singleton_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : ¬a ^ (multiplicity a b + 1) ∣ b
⊢ ¬HPow.hPow (α := Ideal R) (span {a}) (multiplicity a b + 1) ∣ span {b}

Tactic:
rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]

Premises used: Ideal.span_singleton_dvd_span_singleton_iff_dvd, Ideal.span_singleton_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ a ^ multiplicity a b ∣ b

Tactic:
pow_multiplicity_dvd a b

Premises used: pow_multiplicity_dvd
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ a ^ multiplicity a b ∣ b

Tactic:
exact pow_multiplicity_dvd a b

Premises used: pow_multiplicity_dvd
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
pos : multiplicity a b < multiplicity a b + 1
⊢ ¬a ^ (multiplicity a b + 1) ∣ b

Tactic:
apply h.not_pow_dvd_of_multiplicity_lt

Premises used: FiniteMultiplicity.not_pow_dvd_of_multiplicity_lt
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : FiniteMultiplicity a b
⊢ multiplicity a b < multiplicity a b + 1

Tactic:
apply lt_add_one

Premises used: lt_add_one
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
this : ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})
rw : emultiplicity (α := Ideal R) (span {a}) (span {b}) = ⊤
rw₁ rw₂ : Eq (α := ℕ∞) ⊤ ⊤
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b

Tactic:
rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]

Premises used: Iff.mpr, emultiplicity_eq_top
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
neg : ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})
⊢ emultiplicity (α := Ideal R) (span {a}) (span {b}) = emultiplicity a b

Tactic:
suffices ¬FiniteMultiplicity (span ({a} : Set R)) (span ({b} : Set R)) by
      rw [emultiplicity_eq_top.2 h, emultiplicity_eq_top.2 this]
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
n : ℕ
rw : a ^ n ∣ b
⊢ HPow.hPow (α := Ideal R) (span {a}) n ∣ span {b}

Tactic:
rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]

Premises used: Ideal.span_singleton_dvd_span_singleton_iff_dvd, Ideal.span_singleton_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
n : ℕ
⊢ a ^ n ∣ b

Tactic:
FiniteMultiplicity.not_iff_forall.mp h n

Premises used: FiniteMultiplicity.not_iff_forall, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
n : ℕ
⊢ a ^ n ∣ b

Tactic:
exact FiniteMultiplicity.not_iff_forall.mp h n

Premises used: FiniteMultiplicity.not_iff_forall, Iff.mp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDomain R
inst_2 : IsPrincipalIdealRing R
a b : R
h : ¬FiniteMultiplicity a b
⊢ ¬FiniteMultiplicity (α := Ideal R) (span {a}) (span {b})

Tactic:
exact FiniteMultiplicity.not_iff_forall.mpr fun n => by
      rw [span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
      exact FiniteMultiplicity.not_iff_forall.mp h n

Premises used: Iff.mpr, FiniteMultiplicity.not_iff_forall
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
refine_1 :
  (↑{d : R | d ∈ normalizedFactors r} : Type u_1) →
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)
refine_2 :
  Function.Bijective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
⊢ (↑{d : R | d ∈ normalizedFactors r} : Type u_1) ≃
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)

Tactic:
refine Equiv.ofBijective ?_ ?_

Premises used: Equiv.ofBijective
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ (↑{d : R | d ∈ normalizedFactors r} : Type u_1) →
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1)

Tactic:
exact fun d =>
      ⟨span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ Function.Bijective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯

Tactic:
⟨?_, ?_⟩

Premises used: And.intro, Function.Bijective
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
refine_2 :
  Function.Injective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
refine_2₁ :
  Function.Surjective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯
⊢ Function.Bijective (β :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
    Subtype.mk (p := fun (x : Ideal R) ↦
      Membership.mem (γ := Set (Ideal R))
        {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
      (span {(↑d : R)}) ⋯

Tactic:
refine ⟨?_, ?_⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a
b : R
hb : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} b
h :
  Eq.{u_1 + 1} (α :=
    (↑{I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} : Type u_1))
    ((fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
        (Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
            (span {(↑d : R)}) ⋯ :
          { x : Ideal R //
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha))
    ((fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
        (Subtype.mk (p := fun (x : Ideal R) ↦
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
            (span {(↑d : R)}) ⋯ :
          { x : Ideal R //
            Membership.mem (γ := Set (Ideal R))
              {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
      (Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) b hb))
⊢ Associated a b

Tactic:
rw [Subtype.mk_eq_mk, span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h

Premises used: Subtype.coe_mk, Ideal.span_singleton_eq_span_singleton, Subtype.mk_eq_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a
b : R
hb : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} b
h : Associated a b
⊢ Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha =
    Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) b hb

Tactic:
Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)

Premises used: Iff.mpr, UniqueFactorizationMonoid.mem_normalizedFactors_eq_of_associated, Subtype.mk_eq_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} a
b : R
hb : Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} b
h : Associated a b
⊢ Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) a ha =
    Subtype.mk (p := fun (x : R) ↦ Membership.mem (γ := Set R) {d : R | d ∈ normalizedFactors r} x) b hb

Tactic:
exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)

Premises used: Iff.mpr, UniqueFactorizationMonoid.mem_normalizedFactors_eq_of_associated, Subtype.mk_eq_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
⊢ i.IsPrime

Tactic:
have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this : i.IsPrime
refine_2 :
  ∀ (this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q),
    ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
      (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
            (Subtype.mk (p := fun (x : Ideal R) ↦
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
                (span {(↑d : R)}) ⋯ :
              { x : Ideal R //
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
          a =
        Subtype.mk (p := fun (x : Ideal R) ↦
          Membership.mem (γ := Set (Ideal R))
            {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
          i hi
refine_2₁ : Submodule.IsPrincipal.generator i ∣ r
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi

Tactic:
have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
K : Type u_3
inst : CommRing R
inst_1 : CommRing A
inst_2 : Field K
inst_3 : IsDomain R
inst_4 : IsPrincipalIdealRing R
inst_5 : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi :
  Membership.mem (γ := Set (Ideal R))
    {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} i
this_1 : i.IsPrime
this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q
refine_2 :
  ∀ (a : R) (ha : a ∈ normalizedFactors r) (ha' : Associated (Submodule.IsPrincipal.generator i) a),
    ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
      (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
            (Subtype.mk (p := fun (x : Ideal R) ↦
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
                (span {(↑d : R)}) ⋯ :
              { x : Ideal R //
                Membership.mem (γ := Set (Ideal R))
                  {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
          a =
        Subtype.mk (p := fun (x : Ideal R) ↦
          Membership.mem (γ := Set (Ideal R))
            {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
          i hi
⊢ ∃ (a : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)),
    (fun (d : (↑{d : R | d ∈ normalizedFactors r} : Type u_1)) ↦
          (Subtype.mk (p := fun (x : Ideal R) ↦
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
              (span {(↑d : R)}) ⋯ :
            { x : Ideal R //
              Membership.mem (γ := Set (Ideal R))
                {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x }))
        a =
      Subtype.mk (p := fun (x : Ideal R) ↦
        Membership.mem (γ := Set (Ideal R))
          {I : Ideal R | Membership.mem (γ := Multiset (Ideal R)) (normalizedFactors (span {r})) I} x)
        i hi

Tactic:
obtain ⟨a, ha, ha'⟩ := this