-- Mathlib/RingTheory/DedekindDomain/Ideal.lean
theorem count_associates_factors_eq [DecidableEq (Ideal R)] [DecidableEq <| Associates (Ideal R)]
    [∀ (p : Associates <| Ideal R), Decidable (Irreducible p)]
    {I J : Ideal R} (hI : I ≠ 0) (hJ : J.IsPrime) (hJ₀ : J ≠ ⊥) :
    (Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I)

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬Associates.mk J ^ ((Associates.mk J).count (Associates.mk I).factors + 1) ≤ Associates.mk I

Tactic:
rw [Associates.prime_pow_dvd_iff_le hI hJ']

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬(Associates.mk J).count (Associates.mk I).factors + 1 ≤ (Associates.mk J).count (Associates.mk I).factors

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬I ≤ J ^ ((Associates.mk J).count (Associates.mk I).factors + 1)

Tactic:
rw [← Ideal.dvd_iff_le, ← Associates.mk_dvd_mk, Associates.mk_pow]

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬Associates.mk J ^ ((Associates.mk J).count (Associates.mk I).factors + 1) ∣ Associates.mk I

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬Associates.mk J ^ ((Associates.mk J).count (Associates.mk I).factors + 1) ∣ Associates.mk I

Tactic:
simp only [Associates.dvd_eq_le]

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬Associates.mk J ^ ((Associates.mk J).count (Associates.mk I).factors + 1) ≤ Associates.mk I

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
⊢ Irreducible (Associates.mk J)

Tactic:
simpa only [Associates.irreducible_mk] using (Ideal.prime_of_isPrime hJ₀ hJ).irreducible

Result:
no goals

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hI : I ≠ 0
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
⊢ (Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I)

Tactic:
replace hI : Associates.mk I ≠ 0 := Associates.mk_ne_zero.mpr hI

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
⊢ (Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I)

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ (Associates.mk J).count (Associates.mk I).factors = Multiset.count J (normalizedFactors I)

Tactic:
apply (Ideal.count_normalizedFactors_eq (p := J) (x := I) _ _).symm

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ I ≤ J ^ (Associates.mk J).count (Associates.mk I).factors

R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬I ≤ J ^ ((Associates.mk J).count (Associates.mk I).factors + 1)

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬(Associates.mk J).count (Associates.mk I).factors + 1 ≤ (Associates.mk J).count (Associates.mk I).factors

Tactic:
omega

Result:
no goals

Proof state:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ I ≤ J ^ (Associates.mk J).count (Associates.mk I).factors

R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬I ≤ J ^ ((Associates.mk J).count (Associates.mk I).factors + 1)

Tactic:
all_goals
  rw [← Ideal.dvd_iff_le, ← Associates.mk_dvd_mk, Associates.mk_pow]
  simp only [Associates.dvd_eq_le]
  rw [Associates.prime_pow_dvd_iff_le hI hJ']

Result:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
inst✝² : DecidableEq (Ideal R)
inst✝¹ : DecidableEq (Associates (Ideal R))
inst✝ : (p : Associates (Ideal R)) → Decidable (Irreducible p)
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Associates.mk I ≠ 0
hJ' : Irreducible (Associates.mk J)
⊢ ¬(Associates.mk J).count (Associates.mk I).factors + 1 ≤ (Associates.mk J).count (Associates.mk I).factors