l3lab/ntp-mathlib huggingface dataset:
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · 

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : ¬multiplicity.Finite a b
⊢ multiplicity (span {a}) (span {b}) = multiplicity a b

Next tactic:
suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
      rw [h, this]
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : ¬multiplicity.Finite a b
this : ¬multiplicity.Finite (span {a}) (span {b})
⊢ multiplicity (span {a}) (span {b}) = multiplicity a b

Next tactic:
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : multiplicity a b = ⊤
this : multiplicity (span {a}) (span {b}) = ⊤
⊢ multiplicity (span {a}) (span {b}) = multiplicity a b

Next tactic:
rw [h, this]
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
      rw [h, this]
    

Proof state:
case neg
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : ¬multiplicity.Finite a b
⊢ ¬multiplicity.Finite (span {a}) (span {b})

Next tactic:
exact not_finite_iff_forall.mpr fun n => by
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
      exact not_finite_iff_forall.mp h n
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
      rw [h, this]
    exact not_finite_iff_forall.mpr fun n => by
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : ¬multiplicity.Finite a b
n : ℕ
⊢ span {a} ^ n ∣ span {b}

Next tactic:
rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
theorem multiplicity_eq_multiplicity_span [DecidableRel ((· ∣ ·) : R → R → Prop)]
    [DecidableRel ((· ∣ ·) : Ideal R → Ideal R → Prop)] {a b : R} :
    multiplicity (Ideal.span {a}) (Ideal.span ({b} : Set R)) = multiplicity a b := by
  by_cases h : Finite a b
  · rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
    refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm <;>
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
    exact pow_multiplicity_dvd h
    · exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
  · suffices ¬Finite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by
      rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
      rw [h, this]
    exact not_finite_iff_forall.mpr fun n => by
      rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]
      

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁷ : CommRing R
inst✝⁶ : CommRing A
inst✝⁵ : Field K
inst✝⁴ : IsDomain A
inst✝³ : IsDomain R
inst✝² : IsPrincipalIdealRing R
inst✝¹ : DecidableRel fun x x_1 => x ∣ x_1
inst✝ : DecidableRel fun x x_1 => x ∣ x_1
a b : R
h : ¬multiplicity.Finite a b
n : ℕ
⊢ a ^ n ∣ b

Next tactic:
exact not_finite_iff_forall.mp h n
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ ↑{d | d ∈ normalizedFactors r} ≃ ↑{I | I ∈ normalizedFactors (span {r})}

Next tactic:
refine Equiv.ofBijective ?_ ?_
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · 

Proof state:
case refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ ↑{d | d ∈ normalizedFactors r} → ↑{I | I ∈ normalizedFactors (span {r})}

Next tactic:
exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · 

Proof state:
case refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ Function.Bijective fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }

Next tactic:
refine ⟨?_, ?_⟩
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · 

Proof state:
case refine_2.refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ Function.Injective fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }

Next tactic:
rintro ⟨a, ha⟩ ⟨b, hb⟩ h
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      

Proof state:
case refine_2.refine_1.mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : a ∈ {d | d ∈ normalizedFactors r}
b : R
hb : b ∈ {d | d ∈ normalizedFactors r}
h :
  (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) })
      { val := a, property := ha } =
    (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) })
      { val := b, property := hb }
⊢ { val := a, property := ha } = { val := b, property := hb }

Next tactic:
rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      

Proof state:
case refine_2.refine_1.mk.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
a : R
ha : a ∈ {d | d ∈ normalizedFactors r}
b : R
hb : b ∈ {d | d ∈ normalizedFactors r}
h : Associated a b
⊢ { val := a, property := ha } = { val := b, property := hb }

Next tactic:
exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · 

Proof state:
case refine_2.refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
⊢ Function.Surjective fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }

Next tactic:
rintro ⟨i, hi⟩
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · rintro ⟨i, hi⟩
      

Proof state:
case refine_2.refine_2.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi : i ∈ {I | I ∈ normalizedFactors (span {r})}
⊢ ∃ a,
    (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }) a =
      { val := i, property := hi }

Next tactic:
have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · rintro ⟨i, hi⟩
      have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
      

Proof state:
case refine_2.refine_2.mk
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi : i ∈ {I | I ∈ normalizedFactors (span {r})}
this : IsPrime i
⊢ ∃ a,
    (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }) a =
      { val := i, property := hi }

Next tactic:
have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · rintro ⟨i, hi⟩
      have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
      have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
      · 

Proof state:
case refine_2.refine_2.mk.refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi : i ∈ {I | I ∈ normalizedFactors (span {r})}
this✝ : IsPrime i
this : ∃ q ∈ normalizedFactors r, Associated (Submodule.IsPrincipal.generator i) q
⊢ ∃ a,
    (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }) a =
      { val := i, property := hi }

Next tactic:
obtain ⟨a, ha, ha'⟩ := this
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · rintro ⟨i, hi⟩
      have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
      have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
      · obtain ⟨a, ha, ha'⟩ := this
        

Proof state:
case refine_2.refine_2.mk.refine_2.intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi : i ∈ {I | I ∈ normalizedFactors (span {r})}
this : IsPrime i
a : R
ha : a ∈ normalizedFactors r
ha' : Associated (Submodule.IsPrincipal.generator i) a
⊢ ∃ a,
    (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) }) a =
      { val := i, property := hi }

Next tactic:
use ⟨a, ha⟩
/-- The bijection between the (normalized) prime factors of `r` and the (normalized) prime factors
    of `span {r}` -/
-- @[simps] -- Porting note: simpNF complains about the lemmas generated by simps
noncomputable def normalizedFactorsEquivSpanNormalizedFactors {r : R} (hr : r ≠ 0) :
    { d : R | d ∈ normalizedFactors r } ≃
      { I : Ideal R | I ∈ normalizedFactors (Ideal.span ({r} : Set R)) } := by
  refine Equiv.ofBijective ?_ ?_
  · exact fun d =>
      ⟨Ideal.span {↑d}, singleton_span_mem_normalizedFactors_of_mem_normalizedFactors d.prop⟩
  · refine ⟨?_, ?_⟩
    · rintro ⟨a, ha⟩ ⟨b, hb⟩ h
      rw [Subtype.mk_eq_mk, Ideal.span_singleton_eq_span_singleton, Subtype.coe_mk,
          Subtype.coe_mk] at h
      exact Subtype.mk_eq_mk.mpr (mem_normalizedFactors_eq_of_associated ha hb h)
    · rintro ⟨i, hi⟩
      have : i.IsPrime := isPrime_of_prime (prime_of_normalized_factor i hi)
      have := exists_mem_normalizedFactors_of_dvd hr
        (Submodule.IsPrincipal.prime_generator_of_isPrime i
        (prime_of_normalized_factor i hi).ne_zero).irreducible ?_
      · obtain ⟨a, ha, ha'⟩ := this
        use ⟨a, ha⟩
        

Proof state:
case h
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Field K
inst✝⁵ : IsDomain A
inst✝⁴ : IsDomain R
inst✝³ : IsPrincipalIdealRing R
inst✝² : DecidableEq R
inst✝¹ : DecidableEq (Ideal R)
inst✝ : NormalizationMonoid R
r : R
hr : r ≠ 0
i : Ideal R
hi : i ∈ {I | I ∈ normalizedFactors (span {r})}
this : IsPrime i
a : R
ha : a ∈ normalizedFactors r
ha' : Associated (Submodule.IsPrincipal.generator i) a
⊢ (fun d => { val := span {↑d}, property := (_ : span {↑d} ∈ normalizedFactors (span {r})) })
      { val := a, property := ha } =
    { val := i, property := hi }

Next tactic:
simp only [Subtype.coe_mk, Subtype.mk_eq_mk, ← span_singleton_eq_span_singleton.mpr ha',
            Ideal.span_singleton_generator]