l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by
  letI := Classical.decEq (Ideal A)
  letI := Classical.decEq (Associates (Ideal A))
  letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
  have hgcd : gcd I J = I ⊔ J := by
    rw [gcd_eq_normalize _ _, normalize_eq]
    · rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
      exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
    · rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
      simp
  have hlcm : lcm I J = I ⊓ J := by
    rw [lcm_eq_normalize _ _, normalize_eq]
    · rw [lcm_dvd_iff, dvd_iff_le, dvd_iff_le]
      

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 : Ideal A
this✝¹ : DecidableEq (Ideal A) := Classical.decEq (Ideal A)
this✝ : DecidableEq (Associates (Ideal A)) := Classical.decEq (Associates (Ideal A))
this : NormalizedGCDMonoid (Ideal A) := toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ⊓ J ≤ I ∧ I ⊓ J ≤ J

Next tactic:
simp
@[simp]
theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by
  letI := Classical.decEq (Ideal A)
  letI := Classical.decEq (Associates (Ideal A))
  letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
  have hgcd : gcd I J = I ⊔ J := by
    rw [gcd_eq_normalize _ _, normalize_eq]
    · rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
      exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
    · rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
      simp
  have hlcm : lcm I J = I ⊓ J := by
    rw [lcm_eq_normalize _ _, normalize_eq]
    · rw [lcm_dvd_iff, dvd_iff_le, dvd_iff_le]
      simp
    · 

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 : Ideal A
this✝¹ : DecidableEq (Ideal A) := Classical.decEq (Ideal A)
this✝ : DecidableEq (Associates (Ideal A)) := Classical.decEq (Associates (Ideal A))
this : NormalizedGCDMonoid (Ideal A) := toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ⊓ J ∣ lcm I J

Next tactic:
rw [dvd_iff_le, le_inf_iff, ← dvd_iff_le, ← dvd_iff_le]
@[simp]
theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by
  letI := Classical.decEq (Ideal A)
  letI := Classical.decEq (Associates (Ideal A))
  letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
  have hgcd : gcd I J = I ⊔ J := by
    rw [gcd_eq_normalize _ _, normalize_eq]
    · rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
      exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
    · rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
      simp
  have hlcm : lcm I J = I ⊓ J := by
    rw [lcm_eq_normalize _ _, normalize_eq]
    · rw [lcm_dvd_iff, dvd_iff_le, dvd_iff_le]
      simp
    · rw [dvd_iff_le, le_inf_iff, ← dvd_iff_le, ← dvd_iff_le]
      

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 : Ideal A
this✝¹ : DecidableEq (Ideal A) := Classical.decEq (Ideal A)
this✝ : DecidableEq (Associates (Ideal A)) := Classical.decEq (Associates (Ideal A))
this : NormalizedGCDMonoid (Ideal A) := toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
⊢ I ∣ lcm I J ∧ J ∣ lcm I J

Next tactic:
exact ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩
@[simp]
theorem sup_mul_inf (I J : Ideal A) : (I ⊔ J) * (I ⊓ J) = I * J := by
  letI := Classical.decEq (Ideal A)
  letI := Classical.decEq (Associates (Ideal A))
  letI := UniqueFactorizationMonoid.toNormalizedGCDMonoid (Ideal A)
  have hgcd : gcd I J = I ⊔ J := by
    rw [gcd_eq_normalize _ _, normalize_eq]
    · rw [dvd_iff_le, sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
      exact ⟨gcd_dvd_left _ _, gcd_dvd_right _ _⟩
    · rw [dvd_gcd_iff, dvd_iff_le, dvd_iff_le]
      simp
  have hlcm : lcm I J = I ⊓ J := by
    rw [lcm_eq_normalize _ _, normalize_eq]
    · rw [lcm_dvd_iff, dvd_iff_le, dvd_iff_le]
      simp
    · rw [dvd_iff_le, le_inf_iff, ← dvd_iff_le, ← dvd_iff_le]
      exact ⟨dvd_lcm_left _ _, dvd_lcm_right _ _⟩
  

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 : Ideal A
this✝¹ : DecidableEq (Ideal A) := Classical.decEq (Ideal A)
this✝ : DecidableEq (Associates (Ideal A)) := Classical.decEq (Associates (Ideal A))
this : NormalizedGCDMonoid (Ideal A) := toNormalizedGCDMonoid (Ideal A)
hgcd : gcd I J = I ⊔ J
hlcm : lcm I J = I ⊓ J
⊢ (I ⊔ J) * (I ⊓ J) = I * J

Next tactic:
rw [← hgcd, ← hlcm, associated_iff_eq.mp (gcd_mul_lcm _ _)]
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
x✝¹ x✝ : Ideal A
⊢ (fun x x_1 => x ⊔ x_1) x✝¹ x✝ ∣ x✝¹

Next tactic:
simpa only [dvd_iff_le] using le_sup_left
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
x✝¹ x✝ : Ideal A
⊢ (fun x x_1 => x ⊔ x_1) x✝¹ x✝ ∣ x✝

Next tactic:
simpa only [dvd_iff_le] using le_sup_right
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
    dvd_gcd := 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
⊢ ∀ {a b c : Ideal A}, a ∣ c → a ∣ b → a ∣ (fun x x_1 => x ⊔ x_1) c b

Next tactic:
simp only [dvd_iff_le]
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
    dvd_gcd := by
      simp only [dvd_iff_le]
      

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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
⊢ ∀ {a b c : Ideal A}, c ≤ a → b ≤ a → c ⊔ b ≤ a

Next tactic:
exact fun h1 h2 => @sup_le (Ideal A) _ _ _ _ h1 h2
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
    dvd_gcd := by
      simp only [dvd_iff_le]
      exact fun h1 h2 => @sup_le (Ideal A) _ _ _ _ h1 h2
    lcm := (· ⊓ ·)
    lcm_zero_left := fun _ => by simp only [zero_eq_bot, bot_inf_eq]
    lcm_zero_right := fun _ => by simp only [zero_eq_bot, inf_bot_eq]
    gcd_mul_lcm := fun _ _ => 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
x✝¹ x✝ : Ideal A
⊢ Associated ((fun x x_1 => x ⊔ x_1) x✝¹ x✝ * (fun x x_1 => x ⊓ x_1) x✝¹ x✝) (x✝¹ * x✝)

Next tactic:
rw [associated_iff_eq, sup_mul_inf]
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
    dvd_gcd := by
      simp only [dvd_iff_le]
      exact fun h1 h2 => @sup_le (Ideal A) _ _ _ _ h1 h2
    lcm := (· ⊓ ·)
    lcm_zero_left := fun _ => 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
x✝ : Ideal A
⊢ (fun x x_1 => x ⊓ x_1) 0 x✝ = 0

Next tactic:
simp only [zero_eq_bot, bot_inf_eq]
/-- Ideals in a Dedekind domain have gcd and lcm operators that (trivially) are compatible with
the normalization operator. -/
instance : NormalizedGCDMonoid (Ideal A) :=
  { Ideal.normalizationMonoid with
    gcd := (· ⊔ ·)
    gcd_dvd_left := fun _ _ => by simpa only [dvd_iff_le] using le_sup_left
    gcd_dvd_right := fun _ _ => by simpa only [dvd_iff_le] using le_sup_right
    dvd_gcd := by
      simp only [dvd_iff_le]
      exact fun h1 h2 => @sup_le (Ideal A) _ _ _ _ h1 h2
    lcm := (· ⊓ ·)
    lcm_zero_left := fun _ => by simp only [zero_eq_bot, bot_inf_eq]
    lcm_zero_right := fun _ => 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
src✝ : NormalizationMonoid (Ideal A) := normalizationMonoid
x✝ : Ideal A
⊢ (fun x x_1 => x ⊓ x_1) x✝ 0 = 0

Next tactic:
simp only [zero_eq_bot, inf_bot_eq]
theorem inf_eq_mul_of_coprime {I J : Ideal A} (coprime : IsCoprime I J) : I ⊓ J = I * J := 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 : Ideal A
coprime : IsCoprime I J
⊢ I ⊓ J = I * J

Next tactic:
rw [← associated_iff_eq.mp (gcd_mul_lcm I J), lcm_eq_inf I J, gcd_eq_sup, coprime.sup_eq, top_mul]
theorem isCoprime_iff_gcd {I J : Ideal A} : IsCoprime I J ↔ gcd I J = 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 : Ideal A
⊢ IsCoprime I J ↔ gcd I J = 1

Next tactic:
rw [Ideal.isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := 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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)

Next tactic:
have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := 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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J

Next tactic:
apply normalizedFactors_prod_of_prime
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    

Proof state:
case h
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ ∀ p ∈ normalizedFactors I ∩ normalizedFactors J, Prime p

Next tactic:
intro p hp
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    

Proof state:
case h
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
p : Ideal T
hp : p ∈ normalizedFactors I ∩ normalizedFactors J
⊢ Prime p

Next tactic:
rw [mem_inter] at hp
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    

Proof state:
case h
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
p : Ideal T
hp : p ∈ normalizedFactors I ∧ p ∈ normalizedFactors J
⊢ Prime p

Next tactic:
exact prime_of_normalized_factor p hp.left
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  

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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)

Next tactic:
have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  

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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ I ⊔ J = Multiset.prod (normalizedFactors I ∩ normalizedFactors J)

Next tactic:
apply le_antisymm
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  apply le_antisymm
  · 

Proof state:
case a
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ I ⊔ J ≤ Multiset.prod (normalizedFactors I ∩ normalizedFactors J)

Next tactic:
rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  apply le_antisymm
  · rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
    

Proof state:
case a
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I ∧
    Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J

Next tactic:
constructor
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  apply le_antisymm
  · rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
    constructor
    · 

Proof state:
case a.left
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ I

Next tactic:
rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  apply le_antisymm
  · rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
    constructor
    · rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]
      

Proof state:
case a.left
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors I

Next tactic:
exact inf_le_left
theorem sup_eq_prod_inf_factors (hI : I ≠ ⊥) (hJ : J ≠ ⊥) :
    I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod := by
  have H : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod =
      normalizedFactors I ∩ normalizedFactors J := by
    apply normalizedFactors_prod_of_prime
    intro p hp
    rw [mem_inter] at hp
    exact prime_of_normalized_factor p hp.left
  have := Multiset.prod_ne_zero_of_prime (normalizedFactors I ∩ normalizedFactors J) fun _ h =>
      prime_of_normalized_factor _ (Multiset.mem_inter.1 h).1
  apply le_antisymm
  · rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
    constructor
    · rw [dvd_iff_normalizedFactors_le_normalizedFactors this hI, H]
      exact inf_le_left
    · 

Proof state:
case a.right
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
T : Type u_4
inst✝² : CommRing T
inst✝¹ : IsDomain T
inst✝ : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
H :
  normalizedFactors (Multiset.prod (normalizedFactors I ∩ normalizedFactors J)) =
    normalizedFactors I ∩ normalizedFactors J
this : Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ≠ 0
⊢ Multiset.prod (normalizedFactors I ∩ normalizedFactors J) ∣ J

Next tactic:
rw [dvd_iff_normalizedFactors_le_normalizedFactors this hJ, H]