l3lab/ntp-mathlib huggingface dataset:
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  refine ⟨?_, fun hxy => ?_⟩
  · rintro rfl
    

Proof state:
case refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : Prime ⊤
⊢ False

Next tactic:
rw [← Ideal.one_eq_top] at h
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  refine ⟨?_, fun hxy => ?_⟩
  · rintro rfl
    rw [← Ideal.one_eq_top] at h
    

Proof state:
case refine_1
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
h : Prime 1
⊢ False

Next tactic:
exact h.not_unit isUnit_one
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  refine ⟨?_, fun hxy => ?_⟩
  · rintro rfl
    rw [← Ideal.one_eq_top] at h
    exact h.not_unit isUnit_one
  · 

Proof state:
case refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
h : Prime P
x✝ y✝ : A
hxy : x✝ * y✝ ∈ P
⊢ x✝ ∈ P ∨ y✝ ∈ P

Next tactic:
simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
theorem Ideal.isPrime_of_prime {P : Ideal A} (h : Prime P) : IsPrime P := by
  refine ⟨?_, fun hxy => ?_⟩
  · rintro rfl
    rw [← Ideal.one_eq_top] at h
    exact h.not_unit isUnit_one
  · simp only [← Ideal.dvd_span_singleton, ← Ideal.span_singleton_mul_span_singleton] at hxy ⊢
    

Proof state:
case refine_2
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
h : Prime P
x✝ y✝ : A
hxy : P ∣ span {x✝} * span {y✝}
⊢ P ∣ span {x✝} ∨ P ∣ span {y✝}

Next tactic:
exact h.dvd_or_dvd hxy
theorem Ideal.prime_of_isPrime {P : Ideal A} (hP : P ≠ ⊥) (h : IsPrime P) : Prime P := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hP : P ≠ ⊥
h : IsPrime P
⊢ Prime P

Next tactic:
refine ⟨hP, mt Ideal.isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩
theorem Ideal.prime_of_isPrime {P : Ideal A} (hP : P ≠ ⊥) (h : IsPrime P) : Prime P := by
  refine ⟨hP, mt Ideal.isUnit_iff.mp h.ne_top, fun I J hIJ => ?_⟩
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P : Ideal A
hP : P ≠ ⊥
h : IsPrime P
I J : Ideal A
hIJ : P ∣ I * J
⊢ P ∣ I ∨ P ∣ J

Next tactic:
simpa only [Ideal.dvd_iff_le] using h.mul_le.mp (Ideal.le_of_dvd hIJ)
theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
    I ^ e < I := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
e : ℕ
he : 2 ≤ e
⊢ I ^ e < I

Next tactic:
convert I.pow_right_strictAnti hI0 hI1 he
theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
    I ^ e < I := by
  convert I.pow_right_strictAnti hI0 hI1 he
  

Proof state:
case h.e'_4
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
e : ℕ
he : 2 ≤ e
⊢ I = (fun x => I ^ x) 1

Next tactic:
dsimp only
theorem Ideal.pow_lt_self (I : Ideal A) (hI0 : I ≠ ⊥) (hI1 : I ≠ ⊤) (e : ℕ) (he : 2 ≤ e) :
    I ^ e < I := by
  convert I.pow_right_strictAnti hI0 hI1 he
  dsimp only
  

Proof state:
case h.e'_4
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I : Ideal A
hI0 : I ≠ ⊥
hI1 : I ≠ ⊤
e : ℕ
he : 2 ≤ e
⊢ I = I ^ 1

Next tactic:
rw [pow_one]
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
⊢ I = P ^ i

Next tactic:
have := Classical.decEq (Ideal A)
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this : DecidableEq (Ideal A)
⊢ I = P ^ i

Next tactic:
refine le_antisymm hle ?_
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this : DecidableEq (Ideal A)
⊢ P ^ i ≤ I

Next tactic:
have P_prime' := Ideal.prime_of_isPrime hP P_prime
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this : DecidableEq (Ideal A)
P_prime' : Prime P
⊢ P ^ i ≤ I

Next tactic:
have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this : DecidableEq (Ideal A)
P_prime' : Prime P
h1 : I ≠ ⊥
⊢ P ^ i ≤ I

Next tactic:
have := pow_ne_zero i hP
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
⊢ P ^ i ≤ I

Next tactic:
have h3 := pow_ne_zero (i + 1) hP
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hlt : P ^ (i + 1) < I
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ P ^ i ≤ I

Next tactic:
rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ P ^ i ≤ I

Next tactic:
rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
    normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
  rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
    normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ normalizedFactors I ≤ Multiset.replicate i (normalize P)
case hx
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ I ≠ 0
case hy
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ P ^ i ≠ 0

Next tactic:
all_goals assumption
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
  rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
    normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
  all_goals 

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ normalizedFactors I ≤ Multiset.replicate i (normalize P)

Next tactic:
assumption
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
  rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
    normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
  all_goals 

Proof state:
case hx
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ I ≠ 0

Next tactic:
assumption
theorem Ideal.eq_prime_pow_of_succ_lt_of_le {P I : Ideal A} [P_prime : P.IsPrime] (hP : P ≠ ⊥)
    {i : ℕ} (hlt : P ^ (i + 1) < I) (hle : I ≤ P ^ i) : I = P ^ i := by
  have := Classical.decEq (Ideal A)
  refine le_antisymm hle ?_
  have P_prime' := Ideal.prime_of_isPrime hP P_prime
  have h1 : I ≠ ⊥ := (lt_of_le_of_lt bot_le hlt).ne'
  have := pow_ne_zero i hP
  have h3 := pow_ne_zero (i + 1) hP
  rw [← Ideal.dvdNotUnit_iff_lt, dvdNotUnit_iff_normalizedFactors_lt_normalizedFactors h1 h3,
    normalizedFactors_pow, normalizedFactors_irreducible P_prime'.irreducible,
    Multiset.nsmul_singleton, Multiset.lt_replicate_succ] at hlt
  rw [← Ideal.dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
    normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton]
  all_goals 

Proof state:
case hy
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
P I : Ideal A
P_prime : IsPrime P
hP : P ≠ ⊥
i : ℕ
hle : I ≤ P ^ i
this✝ : DecidableEq (Ideal A)
hlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)
P_prime' : Prime P
h1 : I ≠ ⊥
this : P ^ i ≠ 0
h3 : P ^ (i + 1) ≠ 0
⊢ P ^ i ≠ 0

Next tactic:
assumption
theorem Associates.le_singleton_iff (x : A) (n : ℕ) (I : Ideal A) :
    Associates.mk I ^ n ≤ Associates.mk (Ideal.span {x}) ↔ x ∈ I ^ n := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : A
n : ℕ
I : Ideal A
⊢ Associates.mk I ^ n ≤ Associates.mk (span {x}) ↔ x ∈ I ^ n

Next tactic:
simp_rw [← Associates.dvd_eq_le, ← Associates.mk_pow, Associates.mk_dvd_mk,
    Ideal.dvd_span_singleton]
lemma FractionalIdeal.le_inv_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
    I ≤ J⁻¹ ↔ J ≤ I⁻¹ := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : FractionalIdeal A⁰ K
hI : I ≠ 0
hJ : J ≠ 0
⊢ I ≤ J⁻¹ ↔ J ≤ I⁻¹

Next tactic:
rw [inv_eq, inv_eq, le_div_iff_mul_le hI, le_div_iff_mul_le hJ, mul_comm]
lemma FractionalIdeal.inv_le_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
    I⁻¹ ≤ J ↔ J⁻¹ ≤ I := by
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
I J : FractionalIdeal A⁰ K
hI : I ≠ 0
hJ : J ≠ 0
⊢ I⁻¹ ≤ J ↔ J⁻¹ ≤ I

Next tactic:
simpa using le_inv_comm (A := A) (K := K) (inv_ne_zero hI) (inv_ne_zero hJ)
/-- Strengthening of `IsLocalization.exist_integer_multiples`:
Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
to find a collection of elements of `A` that is not completely contained in `J`. -/
theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
    (f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
    ∃ a : K,
      (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
        ∃ i ∈ s, a * f i ∉ (J : FractionalIdeal A⁰ K) := by
  -- Consider the fractional ideal `I` spanned by the `f`s.
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
⊢ ∃ a, (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧ ∃ i ∈ s, a * f i ∉ ↑J

Next tactic:
let I : FractionalIdeal A⁰ K := spanFinset A s f
/-- Strengthening of `IsLocalization.exist_integer_multiples`:
Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
to find a collection of elements of `A` that is not completely contained in `J`. -/
theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
    (f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
    ∃ a : K,
      (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
        ∃ i ∈ s, a * f i ∉ (J : FractionalIdeal A⁰ K) := by
  -- Consider the fractional ideal `I` spanned by the `f`s.
  let I : FractionalIdeal A⁰ K := spanFinset A s f
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
I : FractionalIdeal A⁰ K := spanFinset A s f
⊢ ∃ a, (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧ ∃ i ∈ s, a * f i ∉ ↑J

Next tactic:
have hI0 : I ≠ 0 := spanFinset_ne_zero.mpr ⟨j, hjs, hjf⟩
/-- Strengthening of `IsLocalization.exist_integer_multiples`:
Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
to find a collection of elements of `A` that is not completely contained in `J`. -/
theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
    (f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
    ∃ a : K,
      (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
        ∃ i ∈ s, a * f i ∉ (J : FractionalIdeal A⁰ K) := by
  -- Consider the fractional ideal `I` spanned by the `f`s.
  let I : FractionalIdeal A⁰ K := spanFinset A s f
  have hI0 : I ≠ 0 := spanFinset_ne_zero.mpr ⟨j, hjs, hjf⟩
  -- We claim the multiplier `a` we're looking for is in `I⁻¹ \ (J / I)`.
  

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
I : FractionalIdeal A⁰ K := spanFinset A s f
hI0 : I ≠ 0
⊢ ∃ a, (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧ ∃ i ∈ s, a * f i ∉ ↑J

Next tactic:
suffices ↑J / I < I⁻¹ by
    obtain ⟨_, a, hI, hpI⟩ := SetLike.lt_iff_le_and_exists.mp this
    rw [mem_inv_iff hI0] at hI
    refine' ⟨a, fun i hi => _, _⟩
    -- By definition, `a ∈ I⁻¹` multiplies elements of `I` into elements of `1`,
    -- in other words, `a * f i` is an integer.
    · exact (mem_one_iff _).mp (hI (f i) (Submodule.subset_span (Set.mem_image_of_mem f hi)))
    · contrapose! hpI
      -- And if all `a`-multiples of `I` are an element of `J`,
      -- then `a` is actually an element of `J / I`, contradiction.
      refine' (mem_div_iff_of_nonzero hI0).mpr fun y hy => Submodule.span_induction hy _ _ _ _
      · rintro _ ⟨i, hi, rfl⟩; exact hpI i hi
      · rw [mul_zero]; exact Submodule.zero_mem _
      · intro x y hx hy; rw [mul_add]; exact Submodule.add_mem _ hx hy
      · intro b x hx; rw [mul_smul_comm]; exact Submodule.smul_mem _ b hx
/-- Strengthening of `IsLocalization.exist_integer_multiples`:
Let `J ≠ ⊤` be an ideal in a Dedekind domain `A`, and `f ≠ 0` a finite collection
of elements of `K = Frac(A)`, then we can multiply the elements of `f` by some `a : K`
to find a collection of elements of `A` that is not completely contained in `J`. -/
theorem Ideal.exist_integer_multiples_not_mem {J : Ideal A} (hJ : J ≠ ⊤) {ι : Type*} (s : Finset ι)
    (f : ι → K) {j} (hjs : j ∈ s) (hjf : f j ≠ 0) :
    ∃ a : K,
      (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧
        ∃ i ∈ s, a * f i ∉ (J : FractionalIdeal A⁰ K) := by
  -- Consider the fractional ideal `I` spanned by the `f`s.
  let I : FractionalIdeal A⁰ K := spanFinset A s f
  have hI0 : I ≠ 0 := spanFinset_ne_zero.mpr ⟨j, hjs, hjf⟩
  -- We claim the multiplier `a` we're looking for is in `I⁻¹ \ (J / I)`.
  suffices ↑J / I < I⁻¹ by
    

Proof state:
R : Type u_1
A : Type u_2
K : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : IsDomain A
inst✝² : IsDedekindDomain A
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
J : Ideal A
hJ : J ≠ ⊤
ι : Type u_4
s : Finset ι
f : ι → K
j : ι
hjs : j ∈ s
hjf : f j ≠ 0
I : FractionalIdeal A⁰ K := spanFinset A s f
hI0 : I ≠ 0
this : ↑J / I < I⁻¹
⊢ ∃ a, (∀ i ∈ s, IsLocalization.IsInteger A (a * f i)) ∧ ∃ i ∈ s, a * f i ∉ ↑J

Next tactic:
obtain ⟨_, a, hI, hpI⟩ := SetLike.lt_iff_le_and_exists.mp this