l3lab/ntp-mathlib huggingface dataset:
/-- 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))
  refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
  · 

Proof state:
case refine_2.intro.intro.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✝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)
⊢ a ≠ b

Next tactic:
exact (ne_of_mem_of_not_mem hb ha).symm
/-- 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))
  refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
  · exact (ne_of_mem_of_not_mem hb ha).symm
  · 

Proof state:
case refine_2.intro.intro.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✝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)
⊢ f a = f b

Next tactic:
refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPa)).trans
      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPb)).symm
/-- 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))
  refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
  · exact (ne_of_mem_of_not_mem hb ha).symm
  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPa)).trans
      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPb)).symm
    · 

Proof state:
case refine_2.intro.intro.refine_2.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✝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)
⊢ f a ≠ ⊥

Next tactic:
exact (prime a (Finset.mem_insert_self a s)).ne_zero
/-- 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))
  refine coprime a (Finset.mem_insert_self a s) b (Finset.mem_insert_of_mem hb) ?_ ?_
  · exact (ne_of_mem_of_not_mem hb ha).symm
  · refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPa)).trans
      ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
      (Ideal.le_of_pow_le_prime hPb)).symm
    · exact (prime a (Finset.mem_insert_self a s)).ne_zero
    · 

Proof state:
case refine_2.intro.intro.refine_2.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✝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)
⊢ f b ≠ ⊥

Next tactic:
exact (prime b (Finset.mem_insert_of_mem hb)).ne_zero
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P i)
coprime : ∀ (i j : ι), i ≠ j → P i ≠ P j
prod_eq : ∏ i : ι, P i ^ e i = I
⊢ I = ⨅ i, P i ^ e i

Next tactic:
simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j]
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P i)
coprime : ∀ (i j : ι), i ≠ j → P i ≠ P j
prod_eq : ∏ i : ι, P i ^ e i = I
i j : ι
hij : i ≠ j
⊢ ∀ (P_1 : Ideal R), P i ^ e i ≤ P_1 → P j ^ e j ≤ P_1 → ¬IsPrime P_1

Next tactic:
intro P hPi hPj hPp
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      

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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
⊢ False

Next tactic:
haveI := Ideal.isPrime_of_prime (prime i)
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      haveI := Ideal.isPrime_of_prime (prime 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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
this : IsPrime (P✝ i)
⊢ False

Next tactic:
haveI := Ideal.isPrime_of_prime (prime j)
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      haveI := Ideal.isPrime_of_prime (prime i)
      haveI := Ideal.isPrime_of_prime (prime j)
      

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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
this✝ : IsPrime (P✝ i)
this : IsPrime (P✝ j)
⊢ False

Next tactic:
refine coprime i j hij ?_
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      haveI := Ideal.isPrime_of_prime (prime i)
      haveI := Ideal.isPrime_of_prime (prime j)
      refine coprime 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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
this✝ : IsPrime (P✝ i)
this : IsPrime (P✝ j)
⊢ P✝ i = P✝ j

Next tactic:
refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
        (Ideal.le_of_pow_le_prime hPi)).trans
        ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
          (Ideal.le_of_pow_le_prime hPj)).symm
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      haveI := Ideal.isPrime_of_prime (prime i)
      haveI := Ideal.isPrime_of_prime (prime j)
      refine coprime i j hij ?_
      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
        (Ideal.le_of_pow_le_prime hPi)).trans
        ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
          (Ideal.le_of_pow_le_prime hPj)).symm
      · 

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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
this✝ : IsPrime (P✝ i)
this : IsPrime (P✝ j)
⊢ P✝ i ≠ ⊥

Next tactic:
exact (prime i).ne_zero
/-- **Chinese remainder theorem** for a Dedekind domain: if the ideal `I` factors as
`∏ i, P i ^ e i`, then `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`. -/
noncomputable def IsDedekindDomain.quotientEquivPiOfProdEq {ι : Type*} [Fintype ι] (I : Ideal R)
    (P : ι → Ideal R) (e : ι → ℕ) (prime : ∀ i, Prime (P i)) (coprime : ∀ i j, i ≠ j → P i ≠ P j)
    (prod_eq : ∏ i, P i ^ e i = I) : R ⧸ I ≃+* ∀ i, R ⧸ P i ^ e i :=
  (Ideal.quotEquivOfEq
    (by
      simp only [← prod_eq, Finset.inf_eq_iInf, Finset.mem_univ, ciInf_pos,
        ← IsDedekindDomain.inf_prime_pow_eq_prod _ _ _ (fun i _ => prime i) fun i _ j _ =>
        coprime i j])).trans <|
    Ideal.quotientInfRingEquivPiQuotient _ fun i j hij => Ideal.coprime_of_no_prime_ge (by
      intro P hPi hPj hPp
      haveI := Ideal.isPrime_of_prime (prime i)
      haveI := Ideal.isPrime_of_prime (prime j)
      refine coprime i j hij ?_
      refine ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
        (Ideal.le_of_pow_le_prime hPi)).trans
        ((Ring.DimensionLeOne.prime_le_prime_iff_eq ?_).mp
          (Ideal.le_of_pow_le_prime hPj)).symm
      · exact (prime i).ne_zero
      · 

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 : IsDomain R
inst✝¹ : IsDedekindDomain R
ι : Type u_4
inst✝ : Fintype ι
I : Ideal R
P✝ : ι → Ideal R
e : ι → ℕ
prime : ∀ (i : ι), Prime (P✝ i)
coprime : ∀ (i j : ι), i ≠ j → P✝ i ≠ P✝ j
prod_eq : ∏ i : ι, P✝ i ^ e i = I
i j : ι
hij : i ≠ j
P : Ideal R
hPi : P✝ i ^ e i ≤ P
hPj : P✝ j ^ e j ≤ P
hPp : IsPrime P
this✝ : IsPrime (P✝ i)
this : IsPrime (P✝ j)
⊢ P✝ j ≠ ⊥

Next tactic:
exact (prime j).ne_zero
/-- **Chinese remainder theorem** for a Dedekind domain: `R ⧸ I` factors as `Π i, R ⧸ (P i ^ e i)`,
where `P i` ranges over the prime factors of `I` and `e i` over the multiplicities. -/
noncomputable def IsDedekindDomain.quotientEquivPiFactors {I : Ideal R} (hI : I ≠ ⊥) :
    R ⧸ I ≃+* ∀ P : (factors I).toFinset, R ⧸ (P : Ideal R) ^ (Multiset.count ↑P (factors I)) :=
  IsDedekindDomain.quotientEquivPiOfProdEq _ _ _
    (fun P : (factors I).toFinset => prime_of_factor _ (Multiset.mem_toFinset.mp P.prop))
    (fun i j hij => Subtype.coe_injective.ne hij)
    (calc
      (∏ P : (factors I).toFinset, (P : Ideal R) ^ (factors I).count (P : Ideal R)) =
          ∏ P in (factors I).toFinset, P ^ (factors I).count P :=
        (factors I).toFinset.prod_coe_sort fun P => P ^ (factors I).count P
      _ = ((factors I).map fun P => P).prod := (Finset.prod_multiset_map_count (factors I) id).symm
      _ = (factors I).prod := 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 : Ideal R
hI : I ≠ ⊥
⊢ Multiset.prod (Multiset.map (fun P => P) (factors I)) = Multiset.prod (factors I)

Next tactic:
rw [Multiset.map_id']
/-- 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
  

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 : (i : { x // x ∈ s }) → R ⧸ P ↑i ^ e ↑i
⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), (Ideal.Quotient.mk (P i ^ e i)) y = x { val := i, property := hi }

Next tactic:
let f := IsDedekindDomain.quotientEquivPiOfFinsetProdEq _ P e prime coprime rfl
/-- 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
  

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 : (i : { x // x ∈ s }) → R ⧸ P ↑i ^ e ↑i
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)
⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), (Ideal.Quotient.mk (P i ^ e i)) y = x { val := i, property := hi }

Next tactic:
obtain ⟨y, rfl⟩ := f.surjective x
/-- 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
  

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
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)
y : R ⧸ ∏ i in s, P i ^ e i
⊢ ∃ y_1, ∀ (i : ι) (hi : i ∈ s), (Ideal.Quotient.mk (P i ^ e i)) y_1 = f y { val := i, property := hi }

Next tactic:
obtain ⟨z, rfl⟩ := Ideal.Quotient.mk_surjective y