awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
A : Type u_2
inst : CommRing A
inst_1 : IsDedekindDomain A
I J : Ideal A
rw : Codisjoint I J ↔ @gcd _ CommSemiring.toCommMonoidWithZero _ I J = 1
rw₁ : I ⊔ J = ⊤ ↔ @gcd _ CommSemiring.toCommMonoidWithZero _ I J = 1
rw₂ : I ⊔ J = ⊤ ↔ @gcd _ CommSemiring.toCommMonoidWithZero _ I J = ⊤
⊢ IsCoprime I J ↔ @gcd _ CommSemiring.toCommMonoidWithZero _ I J = 1
have rw₃ : I ⊔ J = ⊤ ↔ I ⊔ J = ⊤
Tactic: rw [isCoprime_iff_codisjoint, codisjoint_iff, one_eq_top, gcd_eq_sup]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inl : Eq (α := Multiset (Ideal K[X])) (factors (span {0})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors 0))
Tactic: rcases eq_or_ne p 0 with rfl | hp
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
inl : Eq (α := Multiset (Ideal K[X])) (factors (span {0})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors 0))
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inr : ∀ (hp : p ≠ 0),
  Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
Tactic: rcases eq_or_ne p 0 with rfl | hp
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
q : Ideal K[X]
hq : Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q
⊢ Prime q
have obtain : ∀ (r : K[X]) (hr : r ∈ factors p)
  (hq : Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) (span {r})),
  Prime (M := Ideal K[X]) (span {r})
Tactic: obtain ⟨r, hr, rfl⟩ := Multiset.mem_map.mp hq
Cleared hypotheses: q, hq
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have this : ∀ (q : Ideal K[X]),
  Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q → Prime q
Tactic: have : ∀ q ∈ (factors p).map (fun q ↦ span {q}), Prime q := fun q hq ↦ by
    obtain ⟨r, hr, rfl⟩ := Multiset.mem_map.mp hq
    exact prime_span_singleton_iff.mpr <| prime_of_factor r hr
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
this :
  ∀ (q : Ideal K[X]),
    Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q → Prime q
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inr : Eq (α := Multiset (Ideal K[X])) (factors (span {(factors p).prod}))
  (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
Tactic: rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
    factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
this :
  ∀ (q : Ideal K[X]),
    Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q → Prime q
inr :
  Eq (α := Multiset (Ideal K[X])) (factors (span {(factors p).prod}))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inr₁ : Eq (α := Multiset (Ideal K[X])) (factors (Multiset.prod (Multiset.map (fun (x : K[X]) ↦ span {x}) (factors p))))
  (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
Tactic: rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
    factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
this :
  ∀ (q : Ideal K[X]),
    Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q → Prime q
inr :
  Eq (α := Multiset (Ideal K[X])) (factors (span {(factors p).prod}))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
inr₁ :
  Eq (α := Multiset (Ideal K[X])) (factors (Multiset.prod (Multiset.map (fun (x : K[X]) ↦ span {x}) (factors p))))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inr₂ : Eq (α := Multiset (Ideal K[X]))
  (normalizedFactors (Multiset.prod (Multiset.map (fun (x : K[X]) ↦ span {x}) (factors p))))
  (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
Tactic: rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
    factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
K : Type u_3
inst : Field K
p : K[X]
hp : p ≠ 0
this :
  ∀ (q : Ideal K[X]),
    Membership.mem (γ := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p)) q → Prime q
inr :
  Eq (α := Multiset (Ideal K[X])) (factors (span {(factors p).prod}))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
inr₁ :
  Eq (α := Multiset (Ideal K[X])) (factors (Multiset.prod (Multiset.map (fun (x : K[X]) ↦ span {x}) (factors p))))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
inr₂ :
  Eq (α := Multiset (Ideal K[X]))
    (normalizedFactors (Multiset.prod (Multiset.map (fun (x : K[X]) ↦ span {x}) (factors p))))
    (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
⊢ Eq (α := Multiset (Ideal K[X])) (factors (span {p})) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
have inr₃ : Eq (α := Multiset (Ideal K[X])) (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
  (Multiset.map (fun (q : K[X]) ↦ span {q}) (factors p))
Tactic: rw [← span_singleton_eq_span_singleton.mpr (factors_prod hp), ← multiset_prod_span_singleton,
    factors_eq_normalizedFactors, normalizedFactors_prod_of_prime this]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : IsDedekindDomain A
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
I J : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K
⊢ (I ⊓ J) * (I ⊔ J) = I * J
have a : Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K)
  ((fun (a : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) ↦
      HMul.hMul (β := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) a
        (spanSingleton A⁰ ((algebraMap A K : (a : A) → K) ((↑I.den : A) * (↑I.den : A) * (↑J.den : A) * (↑J.den : A))) :
          FractionalIdeal.{u_2, u_3} (R := A) A⁰ K))
    ((I ⊓ J) * (I ⊔ J)))
  ((fun (a : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) ↦
      HMul.hMul (β := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) a
        (spanSingleton A⁰ ((algebraMap A K : (a : A) → K) ((↑I.den : A) * (↑I.den : A) * (↑J.den : A) * (↑J.den : A))) :
          FractionalIdeal.{u_2, u_3} (R := A) A⁰ K))
    (I * J))
Tactic: apply mul_left_injective₀ (b := spanSingleton A⁰ (algebraMap A K
    (I.den.1 * I.den.1 * J.den.1 * J.den.1))) (by simp [spanSingleton_eq_zero_iff])
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
A : Type u_2
K : Type u_3
inst : CommRing A
inst_1 : Field K
inst_2 : IsDedekindDomain A
inst_3 : Algebra A K
inst_4 : IsFractionRing A K
I J : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K
⊢ Eq.{u_3 + 1} (α := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K)
    ((fun (a : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) ↦
        HMul.hMul (β := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) a
          (spanSingleton A⁰
              ((algebraMap A K : (a : A) → K) ((↑I.den : A) * (↑I.den : A) * (↑J.den : A) * (↑J.den : A))) :
            FractionalIdeal.{u_2, u_3} (R := A) A⁰ K))
      ((I ⊓ J) * (I ⊔ J)))
    ((fun (a : FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) ↦
        HMul.hMul (β := FractionalIdeal.{u_2, u_3} (R := A) A⁰ K) a
          (spanSingleton A⁰
              ((algebraMap A K : (a : A) → K) ((↑I.den : A) * (↑I.den : A) * (↑J.den : A) * (↑J.den : A))) :
            FractionalIdeal.{u_2, u_3} (R := A) A⁰ K))
      (I * J))
have this : (Ideal.span {(↑J.den : A)} * I.num ⊔ Ideal.span {(↑I.den : A)} * J.num) *
    (Ideal.span {(↑J.den : A)} * I.num ⊓ Ideal.span {(↑I.den : A)} * J.num) =
  Ideal.span {(↑J.den : A)} * I.num * (Ideal.span {(↑I.den : A)} * J.num)
Tactic: have := Ideal.sup_mul_inf (Ideal.span {J.den.1} * I.num) (Ideal.span {I.den.1} * J.num)
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
have this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
Tactic: have := prod_inter_normalizedFactors_ne_zero I J
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
have a : I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
a : I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod
⊢ I ⊔ J = (normalizedFactors I ∩ normalizedFactors J).prod
have a₁ : (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
Tactic: apply le_antisymm
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ I ⊔ J ≤ (normalizedFactors I ∩ normalizedFactors J).prod
have a : (normalizedFactors I ∩ normalizedFactors J).prod ∣ I ∧ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
Tactic: rw [sup_le_iff, ← dvd_iff_le, ← dvd_iff_le]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ I ∧ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
have a : (normalizedFactors I ∩ normalizedFactors J).prod ∣ I
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
a : (normalizedFactors I ∩ normalizedFactors J).prod ∣ I
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ I ∧ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
have a₁ : (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ I
have a : normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors I
Tactic: rw [dvd_iff_normalizedFactors_le_normalizedFactors this (by assumption),
        normalizedFactors_prod_inter_eq_inter]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ∣ J
have a : normalizedFactors I ∩ normalizedFactors J ≤ normalizedFactors J
Tactic: rw [dvd_iff_normalizedFactors_le_normalizedFactors this (by assumption),
        normalizedFactors_prod_inter_eq_inter]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
have a : ∀ (a : Ideal T), count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)
Tactic: rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors ?H this,
      normalizedFactors_prod_inter_eq_inter, le_iff_count]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
a : ∀ (a : Ideal T), count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)
⊢ (normalizedFactors I ∩ normalizedFactors J).prod ≤ I ⊔ J
have rw : I ⊔ J ≠ 0
Tactic: rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors ?H this,
      normalizedFactors_prod_inter_eq_inter, le_iff_count]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : J ≠ ⊥
this : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0
a : Ideal T
⊢ count a (normalizedFactors (I ⊔ J)) ≤ count a (normalizedFactors I ∩ normalizedFactors J)
have a₁ : count a (normalizedFactors (I ⊔ J)) ≤ min (count a (normalizedFactors I)) (count a (normalizedFactors J))
Tactic: rw [Multiset.count_inter]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw₁ : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
rw₁ : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw₂ : Multiset.prod (replicate n ((normalize : (a : Ideal T) → Ideal T) J) ∩ normalizedFactors I) =
  J ^ min n (count J (normalizedFactors I))
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
rw₁ : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
rw₂ :
  Multiset.prod (replicate n ((normalize : (a : Ideal T) → Ideal T) J) ∩ normalizedFactors I) =
    J ^ min n (count J (normalizedFactors I))
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw₃ : (replicate n J ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
rw₁ : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
rw₂ :
  Multiset.prod (replicate n ((normalize : (a : Ideal T) → Ideal T) J) ∩ normalizedFactors I) =
    J ^ min n (count J (normalizedFactors I))
rw₃ : (replicate n J ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw₄ : (replicate (min n (count J (normalizedFactors I))) J).prod = J ^ min n (count J (normalizedFactors I))
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
rw : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min (count J (normalizedFactors I)) n
rw₁ : (normalizedFactors (J ^ n) ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
rw₂ :
  Multiset.prod (replicate n ((normalize : (a : Ideal T) → Ideal T) J) ∩ normalizedFactors I) =
    J ^ min n (count J (normalizedFactors I))
rw₃ : (replicate n J ∩ normalizedFactors I).prod = J ^ min n (count J (normalizedFactors I))
rw₄ : (replicate (min n (count J (normalizedFactors I))) J).prod = J ^ min n (count J (normalizedFactors I))
⊢ J ^ n ⊔ I = J ^ min (count J (normalizedFactors I)) n
have rw₅ : J ^ min n (count J (normalizedFactors I)) = J ^ min n (count J (normalizedFactors I))
Tactic: rw [sup_eq_prod_inf_factors (pow_ne_zero n hJ.ne_zero) hI, min_comm,
    normalizedFactors_of_irreducible_pow hJ, normalize_eq J, replicate_inter, prod_replicate]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
⊢ J ^ n ⊔ I = J ^ n
have pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
Tactic: by_cases hI : I = ⊥
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
⊢ J ^ n ⊔ I = J ^ n
have neg : ∀ (hI : ¬I = ⊥), J ^ n ⊔ I = J ^ n
Tactic: by_cases hI : I = ⊥
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ J ^ n ⊔ I = J ^ n
have neg : n ≤ count J (normalizedFactors I)
Tactic: rw [irreducible_pow_sup hI hJ, min_eq_right]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (hypothesis)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
hI : ¬I = ⊥
⊢ n ≤ count J (normalizedFactors I)
have hn₁ : LE.le (α := ℕ∞) (↑n : ℕ∞) (↑(count J (normalizedFactors I)) : ℕ∞)
Tactic: rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
Cleared hypotheses: hn, hI
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
⊢ J ^ n ⊔ I = J ^ n
have pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
Tactic: classical
  by_cases hI : I = ⊥
  · simp_all
  rw [irreducible_pow_sup hI hJ, min_eq_right]
  rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
  exact_mod_cast hn
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
⊢ J ^ n ⊔ I = J ^ n
have neg : ∀ (hI : ¬I = ⊥), J ^ n ⊔ I = J ^ n
Tactic: classical
  by_cases hI : I = ⊥
  · simp_all
  rw [irreducible_pow_sup hI hJ, min_eq_right]
  rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
  exact_mod_cast hn
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
neg neg₁ : ∀ (hI : ¬I = ⊥), J ^ n ⊔ I = J ^ n
⊢ J ^ n ⊔ I = J ^ n
have neg₂ : ∀ (hI : ¬I = ⊥), n ≤ count J (normalizedFactors I)
Tactic: classical
  by_cases hI : I = ⊥
  · simp_all
  rw [irreducible_pow_sup hI hJ, min_eq_right]
  rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
  exact_mod_cast hn
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hJ : Irreducible J
n : ℕ
hn : (↑n : ℕ∞) ≤ emultiplicity J I
pos : ∀ (hI : I = ⊥), J ^ n ⊔ I = J ^ n
neg neg₁ : ∀ (hI : ¬I = ⊥), J ^ n ⊔ I = J ^ n
neg₂ : ∀ (hI : ¬I = ⊥), n ≤ count J (normalizedFactors I)
⊢ J ^ n ⊔ I = J ^ n
have neg₃ : ∀ (hn : LE.le (α := ℕ∞) (↑n : ℕ∞) (↑(count J (normalizedFactors I)) : ℕ∞)) (hI : ¬I = ⊥),
  n ≤ count J (normalizedFactors I)
Tactic: classical
  by_cases hI : I = ⊥
  · simp_all
  rw [irreducible_pow_sup hI hJ, min_eq_right]
  rw [emultiplicity_eq_count_normalizedFactors hJ hI, normalize_eq J] at hn
  exact_mod_cast hn
Cleared hypotheses: hn
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ (↑n : ℕ∞)
⊢ J ^ n ⊔ I = J ^ multiplicity J I
have rw : J ^ count J (normalizedFactors I) = J ^ multiplicity J I
Tactic: rw [irreducible_pow_sup hI hJ, min_eq_left]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ (↑n : ℕ∞)
rw : J ^ count J (normalizedFactors I) = J ^ multiplicity J I
⊢ J ^ n ⊔ I = J ^ multiplicity J I
have rw₁ : count J (normalizedFactors I) ≤ n
Tactic: rw [irreducible_pow_sup hI hJ, min_eq_left]
-- Lean 4 proof step from Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas (goal)
Goal:
T : Type u_4
inst : CommRing T
inst_1 : IsDedekindDomain T
I J : Ideal T
hI : I ≠ ⊥
hJ : Irreducible J
n : ℕ
hn : emultiplicity J I ≤ (↑n : ℕ∞)
⊢ J ^ count J (normalizedFactors I) = J ^ multiplicity J I
have e_a : count J (normalizedFactors I) = multiplicity J I
Tactic: congr