l3lab/ntp-mathlib huggingface dataset:
/-- Corollary of the Chinese remainder theorem: given elements `x i : R / P i ^ e i`,
we can choose a representative `y : R` such that `y ≡ x i (mod P i ^ e i)`.-/
theorem IsDedekindDomain.exists_representative_mod_finset {ι : Type*} {s : Finset ι}
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : ∀ i : s, R ⧸ P i ^ e i) :
    ∃ y, ∀ (i) (hi : i ∈ s), Ideal.Quotient.mk (P i ^ e i) y = x ⟨i, hi⟩ := by
  let f := IsDedekindDomain.quotientEquivPiOfFinsetProdEq _ P e prime coprime rfl
  obtain ⟨y, rfl⟩ := f.surjective x
  obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y
  

Proof state:
case intro.intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
f : R ⧸ ∏ i in s, P i ^ e i ≃+* ((i : { x // x ∈ s }) → R ⧸ P ↑i ^ e ↑i) :=
  quotientEquivPiOfFinsetProdEq (∏ i in s, P i ^ e i) P e prime coprime (_ : ∏ i in s, P i ^ e i = ∏ i in s, P i ^ e i)
z : R
⊢ ∃ y,
    ∀ (i : ι) (hi : i ∈ s),
      (Ideal.Quotient.mk (P i ^ e i)) y = f ((Ideal.Quotient.mk (∏ i in s, P i ^ e i)) z) { val := i, property := hi }

Next tactic:
exact ⟨z, fun i _hi => rfl⟩
/-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
    ∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : { x // x ∈ s } → R
⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x { val := i, property := hi } ∈ P i ^ e i

Next tactic:
obtain ⟨y, hy⟩ :=
    IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>
      Ideal.Quotient.mk _ (x i)
/-- Corollary of the Chinese remainder theorem: given elements `x i : R`,
we can choose a representative `y : R` such that `y - x i ∈ P i ^ e i`.-/
theorem IsDedekindDomain.exists_forall_sub_mem_ideal {ι : Type*} {s : Finset ι} (P : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (P i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → P i ≠ P j) (x : s → R) :
    ∃ y, ∀ (i) (hi : i ∈ s), y - x ⟨i, hi⟩ ∈ P i ^ e i := by
  obtain ⟨y, hy⟩ :=
    IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>
      Ideal.Quotient.mk _ (x i)
  

Proof state:
case intro
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : Field K
inst✝² : IsDomain A
inst✝¹ : IsDomain R
inst✝ : IsDedekindDomain R
ι : Type u_4
s : Finset ι
P : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (P i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → P i ≠ P j
x : { x // x ∈ s } → R
y : R
hy :
  ∀ (i : ι) (hi : i ∈ s),
    (Ideal.Quotient.mk (P i ^ e i)) y =
      (Ideal.Quotient.mk (P ↑{ val := i, property := hi } ^ e ↑{ val := i, property := hi }))
        (x { val := i, property := hi })
⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x { val := i, property := hi } ∈ P i ^ e i

Next tactic:
exact ⟨y, fun i hi => Ideal.Quotient.eq.mp (hy i hi)⟩
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : 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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
⊢ span {a} ∈ normalizedFactors (span {b})

Next tactic:
by_cases hb : b = 0
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · 

Proof state:
case pos
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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : b = 0
⊢ span {a} ∈ normalizedFactors (span {b})

Next tactic:
rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    

Proof state:
case pos
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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : b = 0
⊢ span {a} ∈ 0

Next tactic:
rw [hb, normalizedFactors_zero] at ha
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    

Proof state:
case pos
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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ 0
hb : b = 0
⊢ span {a} ∈ 0

Next tactic:
exact absurd ha (Multiset.not_mem_zero a)
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · 

Proof state:
case neg
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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ span {a} ∈ normalizedFactors (span {b})

Next tactic:
suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : 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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (span {a})
⊢ span {a} ∈ normalizedFactors (span {b})

Next tactic:
obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      

Proof state:
case 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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (span {a})
c : Ideal R
hc : c ∈ normalizedFactors (span {b})
hc' : Associated (span {a}) c
⊢ span {a} ∈ normalizedFactors (span {b})

Next tactic:
rwa [associated_iff_eq.mp hc']
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · 

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (span {a})
⊢ span {b} ≠ 0

Next tactic:
by_contra h
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · by_contra h
      

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
this : Prime (span {a})
h : span {b} = 0
⊢ False

Next tactic:
exact hb (span_singleton_eq_bot.mp h)
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · by_contra h
      exact hb (span_singleton_eq_bot.mp h)
    

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ Prime (span {a})

Next tactic:
rw [prime_iff_isPrime]
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · by_contra h
      exact hb (span_singleton_eq_bot.mp h)
    rw [prime_iff_isPrime]
    · 

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ IsPrime (span {a})

Next tactic:
exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
        (prime_of_normalized_factor a ha)
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · by_contra h
      exact hb (span_singleton_eq_bot.mp h)
    rw [prime_iff_isPrime]
    · exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
        (prime_of_normalized_factor a ha)
    · 

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
⊢ span {a} ≠ ⊥

Next tactic:
by_contra h
theorem singleton_span_mem_normalizedFactors_of_mem_normalizedFactors [NormalizationMonoid R]
    [DecidableEq R] [DecidableEq (Ideal R)] {a b : R} (ha : a ∈ normalizedFactors b) :
    Ideal.span ({a} : Set R) ∈ normalizedFactors (Ideal.span ({b} : Set R)) := by
  by_cases hb : b = 0
  · rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]
    rw [hb, normalizedFactors_zero] at ha
    exact absurd ha (Multiset.not_mem_zero a)
  · suffices Prime (Ideal.span ({a} : Set R)) by
      obtain ⟨c, hc, hc'⟩ := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible
          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))
      rwa [associated_iff_eq.mp hc']
    · by_contra h
      exact hb (span_singleton_eq_bot.mp h)
    rw [prime_iff_isPrime]
    · exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr
        (prime_of_normalized_factor a ha)
    · by_contra h
      

Proof state:
case neg.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 : NormalizationMonoid R
inst✝¹ : DecidableEq R
inst✝ : DecidableEq (Ideal R)
a b : R
ha : a ∈ normalizedFactors b
hb : ¬b = 0
h : span {a} = ⊥
⊢ False

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

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
⊢ multiplicity (span {a}) (span {b}) = multiplicity a b

Next tactic:
by_cases h : Finite a b
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
  · 

Proof state:
case pos
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:
rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
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)]
    

Proof state:
case pos
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}) = ↑(Part.get (multiplicity a b) (_ : (multiplicity a b).Dom))

Next tactic:
refine (multiplicity.unique
      (show Ideal.span {a} ^ (multiplicity a b).get h ∣ Ideal.span {b} from ?_) ?_).symm
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 <;>
      

Proof state:
case pos.refine_1
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
⊢ span {a} ^ Part.get (multiplicity a b) h ∣ 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 <;>
      

Proof state:
case pos.refine_2
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
⊢ ¬span {a} ^ (Part.get (multiplicity a b) h + 1) ∣ 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]
    

Proof state:
case pos.refine_1
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
⊢ a ^ Part.get (multiplicity a b) h ∣ b
case pos.refine_2
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
⊢ ¬a ^ (Part.get (multiplicity a b) h + 1) ∣ b

Next tactic:
exact pow_multiplicity_dvd h
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
    · 

Proof state:
case pos.refine_2
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
⊢ ¬a ^ (Part.get (multiplicity a b) h + 1) ∣ b

Next tactic:
exact multiplicity.is_greatest
        ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))