l3lab/ntp-mathlib huggingface dataset:
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  by_cases hJ0 : J = 0
  · simpa only [hJ0, zero_mul] using hJ
  obtain ⟨I', rfl⟩ := hK
  rw [mul_comm]
  

Proof state:
case neg.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
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
⊢ K * J ∣ K * I'

Next tactic:
refine mul_dvd_mul_left K
    (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)
theorem Ideal.le_mul_of_no_prime_factors {I J K : Ideal R}
    (coprime : ∀ P, J ≤ P → K ≤ P → ¬IsPrime P) (hJ : I ≤ J) (hK : I ≤ K) : I ≤ J * K := by
  simp only [← Ideal.dvd_iff_le] at coprime hJ hK ⊢
  by_cases hJ0 : J = 0
  · simpa only [hJ0, zero_mul] using hJ
  obtain ⟨I', rfl⟩ := hK
  rw [mul_comm]
  refine mul_dvd_mul_left K
    (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)
  

Proof state:
case neg.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
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬IsPrime P
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
⊢ ∀ {d : Ideal R}, d ∣ J → d ∣ K → ¬Prime d

Next tactic:
exact fun hPJ hPK => mt Ideal.isPrime_of_prime (coprime _ hPJ hPK)
theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
    I ≤ P := 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
I P : Ideal R
hP : IsPrime P
n : ℕ
h : I ^ n ≤ P
⊢ I ≤ P

Next tactic:
by_cases hP0 : P = ⊥
theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
    I ≤ P := by
  by_cases hP0 : P = ⊥
  · 

Proof state:
case pos
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
I P : Ideal R
hP : IsPrime P
n : ℕ
h : I ^ n ≤ P
hP0 : P = ⊥
⊢ I ≤ P

Next tactic:
simp only [hP0, le_bot_iff] at h ⊢
theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
    I ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff] at h ⊢
    

Proof state:
case pos
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
I P : Ideal R
hP : IsPrime P
n : ℕ
hP0 : P = ⊥
h : I ^ n = ⊥
⊢ I = ⊥

Next tactic:
exact pow_eq_zero h
theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
    I ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff] at h ⊢
    exact pow_eq_zero h
  

Proof state:
case neg
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
I P : Ideal R
hP : IsPrime P
n : ℕ
h : I ^ n ≤ P
hP0 : ¬P = ⊥
⊢ I ≤ P

Next tactic:
rw [← Ideal.dvd_iff_le] at h ⊢
theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
    I ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff] at h ⊢
    exact pow_eq_zero h
  rw [← Ideal.dvd_iff_le] at h ⊢
  

Proof state:
case neg
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
I P : Ideal R
hP : IsPrime P
n : ℕ
h : P ∣ I ^ n
hP0 : ¬P = ⊥
⊢ P ∣ I

Next tactic:
exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h
theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := 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
f : ι → Ideal R
P : Ideal R
hP : IsPrime P
⊢ ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P

Next tactic:
by_cases hP0 : P = ⊥
theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
  by_cases hP0 : P = ⊥
  · 

Proof state:
case pos
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
f : ι → Ideal R
P : Ideal R
hP : IsPrime P
hP0 : P = ⊥
⊢ ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P

Next tactic:
simp only [hP0, le_bot_iff]
theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff]
    

Proof state:
case pos
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
f : ι → Ideal R
P : Ideal R
hP : IsPrime P
hP0 : P = ⊥
⊢ ∏ i in s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥

Next tactic:
rw [← Ideal.zero_eq_bot, Finset.prod_eq_zero_iff]
theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff]
    rw [← Ideal.zero_eq_bot, Finset.prod_eq_zero_iff]
  

Proof state:
case neg
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
f : ι → Ideal R
P : Ideal R
hP : IsPrime P
hP0 : ¬P = ⊥
⊢ ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P

Next tactic:
simp only [← Ideal.dvd_iff_le]
theorem Ideal.prod_le_prime {ι : Type*} {s : Finset ι} {f : ι → Ideal R} {P : Ideal R}
    [hP : P.IsPrime] : ∏ i in s, f i ≤ P ↔ ∃ i ∈ s, f i ≤ P := by
  by_cases hP0 : P = ⊥
  · simp only [hP0, le_bot_iff]
    rw [← Ideal.zero_eq_bot, Finset.prod_eq_zero_iff]
  simp only [← Ideal.dvd_iff_le]
  

Proof state:
case neg
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
f : ι → Ideal R
P : Ideal R
hP : IsPrime P
hP0 : ¬P = ⊥
⊢ P ∣ ∏ i in s, f i ↔ ∃ i ∈ s, P ∣ f i

Next tactic:
exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_finset_prod_iff _
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f 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
f : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (f i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j
⊢ (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i

Next tactic:
letI := Classical.decEq ι
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  

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
f : ι → Ideal R
e : ι → ℕ
prime : ∀ i ∈ s, Prime (f i)
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j
this : DecidableEq ι := Classical.decEq ι
⊢ (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i

Next tactic:
revert prime coprime
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  

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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
⊢ (∀ i ∈ s, Prime (f i)) →
    (∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i

Next tactic:
refine s.induction ?_ ?_
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · 

Proof state:
case refine_1
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
⊢ (∀ i ∈ ∅, Prime (f i)) →
    (∀ i ∈ ∅, ∀ j ∈ ∅, i ≠ j → f i ≠ f j) → (Finset.inf ∅ fun i => f i ^ e i) = ∏ i in ∅, f i ^ e i

Next tactic:
simp
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
⊢ ∀ ⦃a : ι⦄ {s : Finset ι},
    a ∉ s →
      ((∀ i ∈ s, Prime (f i)) →
          (∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i) →
        (∀ i ∈ insert a s, Prime (f i)) →
          (∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j) →
            (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i

Next tactic:
intro a s ha ih prime coprime
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
ih :
  (∀ i ∈ s, Prime (f i)) →
    (∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j) → (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i

Next tactic:
specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
⊢ (Finset.inf (insert a s) fun i => f i ^ e i) = ∏ i in insert a s, f i ^ e i

Next tactic:
rw [Finset.inf_insert, Finset.prod_insert ha, ih]
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
⊢ f a ^ e a ⊓ ∏ i in s, f i ^ e i = f a ^ e a * ∏ x in s, f x ^ e x

Next tactic:
refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
⊢ ∀ (P : Ideal R), f a ^ e a ≤ P → ∏ x in s, f x ^ e x ≤ P → ¬IsPrime P

Next tactic:
intro P hPa hPs hPp
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
  intro P hPa hPs hPp
  

Proof state:
case refine_2
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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
P : Ideal R
hPa : f a ^ e a ≤ P
hPs : ∏ x in s, f x ^ e x ≤ P
hPp : IsPrime P
⊢ False

Next tactic:
obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
  intro P hPa hPs hPp
  obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
  

Proof state:
case refine_2.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
f : ι → Ideal R
e : ι → ℕ
this : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
P : Ideal R
hPa : f a ^ e a ≤ P
hPs : ∏ x in s, f x ^ e x ≤ P
hPp : IsPrime P
b : ι
hb : b ∈ s
hPb : f b ^ e b ≤ P
⊢ False

Next tactic:
haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
  intro P hPa hPs hPp
  obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
  haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
  

Proof state:
case refine_2.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
f : ι → Ideal R
e : ι → ℕ
this✝ : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
P : Ideal R
hPa : f a ^ e a ≤ P
hPs : ∏ x in s, f x ^ e x ≤ P
hPp : IsPrime P
b : ι
hb : b ∈ s
hPb : f b ^ e b ≤ P
this : IsPrime (f a)
⊢ False

Next tactic:
haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
/-- The intersection of distinct prime powers in a Dedekind domain is the product of these
prime powers. -/
theorem IsDedekindDomain.inf_prime_pow_eq_prod {ι : Type*} (s : Finset ι) (f : ι → Ideal R)
    (e : ι → ℕ) (prime : ∀ i ∈ s, Prime (f i))
    (coprime : ∀ᵉ (i ∈ s) (j ∈ s), i ≠ j → f i ≠ f j) :
    (s.inf fun i => f i ^ e i) = ∏ i in s, f i ^ e i := by
  letI := Classical.decEq ι
  revert prime coprime
  refine s.induction ?_ ?_
  · simp
  intro a s ha ih prime coprime
  specialize
    ih (fun i hi => prime i (Finset.mem_insert_of_mem hi)) fun i hi j hj =>
      coprime i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj)
  rw [Finset.inf_insert, Finset.prod_insert ha, ih]
  refine' le_antisymm (Ideal.le_mul_of_no_prime_factors _ inf_le_left inf_le_right) Ideal.mul_le_inf
  intro P hPa hPs hPp
  obtain ⟨b, hb, hPb⟩ := Ideal.prod_le_prime.mp hPs
  haveI := Ideal.isPrime_of_prime (prime a (Finset.mem_insert_self a s))
  haveI := Ideal.isPrime_of_prime (prime b (Finset.mem_insert_of_mem hb))
  

Proof state:
case refine_2.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
f : ι → Ideal R
e : ι → ℕ
this✝¹ : DecidableEq ι := Classical.decEq ι
a : ι
s : Finset ι
ha : a ∉ s
prime : ∀ i ∈ insert a s, Prime (f i)
coprime : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → f i ≠ f j
ih : (Finset.inf s fun i => f i ^ e i) = ∏ i in s, f i ^ e i
P : Ideal R
hPa : f a ^ e a ≤ P
hPs : ∏ x in s, f x ^ e x ≤ P
hPp : IsPrime P
b : ι
hb : b ∈ s
hPb : f b ^ e b ≤ P
this✝ : IsPrime (f a)
this : IsPrime (f b)
⊢ False

Next tactic:
refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_