awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
A : Type u_2
inst : CommRing R
inst_1 : CommRing A
inst_2 : IsDedekindDomain A
I : Ideal R
J : Ideal A
inst_3 : IsDedekindDomain R
f : RingEquiv.{u_1, u_2} (R ⧸ I) (A ⧸ J)
hI : I ≠ ⊥
hJ : J ≠ ⊥
L : Ideal R
hL : L ∈ normalizedFactors I
⊢ ∀ (l l' : { l : Ideal R // l ∣ I }),
    Dvd.dvd.{u_2} (α := Ideal A)
        (↑(((idealFactorsEquivOfQuotEquiv f).toEquiv (r := fun (x1 x2 : (↑{p : Ideal R | p ∣ I} : Type u_1)) ↦ x1 ≤ x2)
                  (s := fun (x1 x2 : (↑{p : Ideal A | p ∣ J} : Type u_2)) ↦ x1 ≤ x2) :
                (a : { l : Ideal R // l ∣ I }) → { l : Ideal A // l ∣ J })
              l) :
          Ideal A)
        (↑(((idealFactorsEquivOfQuotEquiv f).toEquiv (r := fun (x1 x2 : (↑{p : Ideal R | p ∣ I} : Type u_1)) ↦ x1 ≤ x2)
                  (s := fun (x1 x2 : (↑{p : Ideal A | p ∣ J} : Type u_2)) ↦ x1 ≤ x2) :
                (a : { l : Ideal R // l ∣ I }) → { l : Ideal A // l ∣ J })
              l') :
          Ideal A) ↔
      (↑l : Ideal R) ∣ (↑l' : Ideal R)

Tactic:
exact fun ⟨l, hl⟩ ⟨l', hl'⟩ => idealFactorsEquivOfQuotEquiv_is_dvd_iso f hl hl'
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : DimensionLEOne R
P Q : Ideal R
hP : P.IsPrime
hQ : Q.IsPrime
hP0 : P ≠ ⊥
⊢ P ≤ Q ↔ P = Q

Tactic:
⟨(hP.isMaximal hP0).eq_of_le hQ.ne_top, Eq.le⟩

Premises used: Ideal.IsPrime.isMaximal, Eq.le, Ideal.IsMaximal.eq_of_le, Iff.intro, Ideal.IsPrime.ne_top
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ n
zero : ∀ (h : a * b ∈ HPow.hPow (β := ℕ) I 0), a ∈ I ∨ b ∈ HPow.hPow (β := ℕ) I 0
succ : ∀ (n : ℕ) (h : a * b ∈ I ^ (n + 1)), a ∈ I ∨ b ∈ I ^ (n + 1)
⊢ a ∈ I ∨ b ∈ I ^ n

Tactic:
cases n
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
h : a * b ∈ HPow.hPow (β := ℕ) I 0
⊢ a ∈ I ∨ b ∈ HPow.hPow (β := ℕ) I 0

Tactic:
simp
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ (n + 1)
pos : ∀ (hI0 : I = ⊥), a ∈ I ∨ b ∈ I ^ (n + 1)
neg : ∀ (hI0 : ¬I = ⊥), a ∈ I ∨ b ∈ I ^ (n + 1)
⊢ a ∈ I ∨ b ∈ I ^ (n + 1)

Tactic:
by_cases hI0 : I = ⊥
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ (n + 1)
hI0 : I = ⊥
⊢ a ∈ I ∨ b ∈ I ^ (n + 1)

Tactic:
simpa [pow_succ, hI0] using h

Premises used: pow_succ
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ (n + 1)
hI0 : ¬I = ⊥
⊢ I.IsMaximal

Tactic:
have : I.IsMaximal := hI.isMaximal hI0
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ (n + 1)
hI0 : ¬I = ⊥
this : I.IsMaximal
⊢ a ∈ I ∨ b ∈ I ^ (n + 1)

Tactic:
IsMaximal.mul_mem_pow I h

Premises used: Ideal.IsMaximal.mul_mem_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ (n + 1)
hI0 : ¬I = ⊥
this : I.IsMaximal
⊢ a ∈ I ∨ b ∈ I ^ (n + 1)

Tactic:
exact IsMaximal.mul_mem_pow I h

Premises used: Ideal.IsMaximal.mul_mem_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : a * b ∈ I ^ n
⊢ b * a ∈ I ^ n

Tactic:
rw [mul_comm] at h

Premises used: mul_comm
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : b * a ∈ I ^ n
rw : b ∈ I ∨ a ∈ I ^ n
⊢ a ∈ I ^ n ∨ b ∈ I

Tactic:
rw [or_comm]

Premises used: or_comm
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : b * a ∈ I ^ n
⊢ b ∈ I ∨ a ∈ I ^ n

Tactic:
IsPrime.mul_mem_pow _ h

Premises used: Ideal.IsPrime.mul_mem_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I : Ideal R
hI : I.IsPrime
a b : R
n : ℕ
h : b * a ∈ I ^ n
⊢ b ∈ I ∨ a ∈ I ^ n

Tactic:
exact IsPrime.mul_mem_pow _ h

Premises used: Ideal.IsPrime.mul_mem_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
p x : Ideal R
hp : p.IsPrime
n : ℕ
hle : x ≤ p ^ n
hlt : ¬x ≤ p ^ (n + 1)
⊢ Multiset.count p (normalizedFactors x) = n

Tactic:
count_normalizedFactors_eq' ((isPrime_iff_bot_or_prime.mp hp).imp_right Prime.irreducible)
    (normalize_eq _) (dvd_iff_le.mpr hle) (mt le_of_dvd hlt)

Premises used: Prime.irreducible, Iff.mpr, Ideal.dvd_iff_le, Ideal.isPrime_iff_bot_or_prime, Ideal.le_of_dvd, mt, Iff.mp, UniqueFactorizationMonoid.count_normalizedFactors_eq', Or.imp_right, normalize_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hI : I ≠ 0
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
⊢ Ne (α := Associates (Ideal R)) (Associates.mk I) 0

Tactic:
replace hI : Associates.mk I ≠ 0 := Associates.mk_ne_zero.mpr hI
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk J)

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

Premises used: Prime.irreducible, Associates.irreducible_mk, Ideal.prime_of_isPrime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
⊢ Irreducible (M := Associates (Ideal R)) (Associates.mk J)

Tactic:
have hJ' : Irreducible (Associates.mk J) := by
    simpa only [Associates.irreducible_mk] using (prime_of_isPrime hJ₀ hJ).irreducible
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
apply : I ≤ J ^ Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))
apply₁ : ¬I ≤ J ^ (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1)
⊢ Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) = Multiset.count J (normalizedFactors I)

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

Premises used: Ideal.count_normalizedFactors_eq, Eq.symm
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
rw :
  HPow.hPow (α := Associates (Ideal R)) (Associates.mk J)
      (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))) ∣
    Associates.mk I
⊢ I ≤ J ^ Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))

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

Premises used: Ideal.dvd_iff_le, Associates.mk_pow, Associates.mk_dvd_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
rw rw₁ :
  Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) ≤
    Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))
⊢ HPow.hPow (α := Associates (Ideal R)) (Associates.mk J)
      (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))) ≤
    Associates.mk I

Tactic:
rw [Associates.prime_pow_dvd_iff_le hI hJ']

Premises used: Associates.prime_pow_dvd_iff_le
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
rw :
  ¬HPow.hPow (α := Associates (Ideal R)) (Associates.mk J)
        (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1) ∣
      Associates.mk I
⊢ ¬I ≤ J ^ (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1)

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

Premises used: Ideal.dvd_iff_le, Associates.mk_pow, Associates.mk_dvd_mk
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
rw :
  ¬Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1 ≤
      Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))
⊢ ¬HPow.hPow (α := Associates (Ideal R)) (Associates.mk J)
        (Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1) ≤
      Associates.mk I

Tactic:
rw [Associates.prime_pow_dvd_iff_le hI hJ']

Premises used: Associates.prime_pow_dvd_iff_le
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hJ : J.IsPrime
hJ₀ : J ≠ ⊥
hI : Ne (α := Associates (Ideal R)) (Associates.mk I) 0
hJ' : Irreducible (M := Associates (Ideal R)) (Associates.mk J)
⊢ ¬Associates.count (Associates.mk J) (Associates.factors (Associates.mk I)) + 1 ≤
      Associates.count (Associates.mk J) (Associates.factors (Associates.mk I))

Tactic:
lia
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
⊢ x ≠ 0

Tactic:
have hx0 : x ≠ 0 := Prime.ne_zero hx
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hp : Irreducible (M := Ideal R) (span {x})
hnorm : Eq (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) (span {x}) : Ideal R) (span {x} : Ideal R)
hle : HPow.hPow (α := Ideal R) (span {x}) n ∣ span {a₀}
hlt : ¬HPow.hPow (α := Ideal R) (span {x}) (n + 1) ∣ span {a₀}
hI : Ne (α := Ideal R) (span {a₀}) 0
hJ : IsPrime (α := R) (span {x})
hJ₀ : Ne (α := Ideal R) (span {x}) ⊥
⊢ Associates.count (α := Ideal R) (Associates.mk (span {x})) (Associates.factors (Associates.mk (span {a₀}))) = n

Tactic:
rw [count_associates_factors_eq, UniqueFactorizationMonoid.count_normalizedFactors_eq]

Premises used: UniqueFactorizationMonoid.count_normalizedFactors_eq, Ideal.count_associates_factors_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Irreducible (M := Ideal R) (span {x})

Tactic:
(prime_span_singleton_iff.mpr hx).irreducible

Premises used: Prime.irreducible, Iff.mpr, Ideal.prime_span_singleton_iff
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Irreducible (M := Ideal R) (span {x})

Tactic:
exact (prime_span_singleton_iff.mpr hx).irreducible

Premises used: Prime.irreducible, Iff.mpr, Ideal.prime_span_singleton_iff
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Eq (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) (span {x}) : Ideal R) (span {x} : Ideal R)

Tactic:
normalize_eq _

Premises used: normalize_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Eq (α := Ideal R) ((normalize : (a : Ideal R) → Ideal R) (span {x}) : Ideal R) (span {x} : Ideal R)

Tactic:
exact normalize_eq _

Premises used: normalize_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hle : Membership.mem (γ := Ideal R) (span {x ^ n}) (x ^ n * a)
⊢ HPow.hPow (α := Ideal R) (span {x}) n ∣ span {a₀}

Tactic:
simp only [span_singleton_pow, heq, dvd_span_singleton]

Premises used: Ideal.dvd_span_singleton, Ideal.span_singleton_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Membership.mem (γ := Ideal R) (span {x ^ n}) (x ^ n * a)

Tactic:
mul_mem_right _ _ (mem_span_singleton_self (x ^ n))

Premises used: HPow.hPow, Ideal.mul_mem_right, Ideal.mem_span_singleton_self
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ Membership.mem (γ := Ideal R) (span {x ^ n}) (x ^ n * a)

Tactic:
exact mul_mem_right _ _ (mem_span_singleton_self (x ^ n))

Premises used: HPow.hPow, Ideal.mul_mem_right, Ideal.mem_span_singleton_self
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hlt : ¬x ^ (n + 1) ∣ x ^ n * a
⊢ ¬HPow.hPow (α := Ideal R) (span {x}) (n + 1) ∣ span {a₀}

Tactic:
simp only [span_singleton_pow, heq, dvd_span_singleton, mem_span_singleton]

Premises used: Ideal.mem_span_singleton, Ideal.dvd_span_singleton, Ideal.span_singleton_pow
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hlt : ¬x ∣ a
⊢ ¬x ^ (n + 1) ∣ x ^ n * a

Tactic:
rw [pow_add, pow_one, mul_dvd_mul_iff_left (pow_ne_zero n hx0)]

Premises used: pow_add, pow_ne_zero, mul_dvd_mul_iff_left, pow_one
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ ¬x ∣ a

Tactic:
ha
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ ¬x ∣ a

Tactic:
exact ha
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hI : ¬a₀ = 0
⊢ Ne (α := Ideal R) (span {a₀}) 0

Tactic:
simp only [Submodule.zero_eq_bot, ne_eq, span_singleton_eq_bot]

Premises used: Ideal.span_singleton_eq_bot, ne_eq, Submodule.zero_eq_bot
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ ¬a₀ = 0

Tactic:
aesop
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ IsPrime (α := R) (span {x})

Tactic:
(span_singleton_prime hx0).mpr hx

Premises used: Iff.mpr, Ideal.span_singleton_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ IsPrime (α := R) (span {x})

Tactic:
exact (span_singleton_prime hx0).mpr hx

Premises used: Iff.mpr, Ideal.span_singleton_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
hJ₀ : ¬x = 0
⊢ Ne (α := Ideal R) (span {x}) ⊥

Tactic:
simp only [ne_eq, span_singleton_eq_bot]

Premises used: Ideal.span_singleton_eq_bot, ne_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ ¬x = 0

Tactic:
hx0
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a a₀ x : R
n : ℕ
hx : Prime x
ha : ¬x ∣ a
heq : a₀ = x ^ n * a
hx0 : x ≠ 0
⊢ ¬x = 0

Tactic:
exact hx0
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
hle : x ^ n ∣ a
hlt : ¬x ^ (n + 1) ∣ a
obtain :
  ∀ (q : R) (hq : a = x ^ n * q),
    Associates.count (α := Ideal R) (Associates.mk (span {x})) (Associates.factors (Associates.mk (span {a}))) = n
⊢ Associates.count (α := Ideal R) (Associates.mk (span {x})) (Associates.factors (Associates.mk (span {a}))) = n

Tactic:
obtain ⟨q, hq⟩ := hle
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
hlt : ¬x ^ (n + 1) ∣ a
q : R
hq : a = x ^ n * q
apply : ¬x ∣ q
⊢ Associates.count (α := Ideal R) (Associates.mk (span {x})) (Associates.factors (Associates.mk (span {a}))) = n

Tactic:
apply count_associates_eq hx _ hq

Premises used: Ideal.count_associates_eq
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
hlt : ¬x ^ (n + 1) ∣ a
q : R
hq : a = x ^ n * q
contrapose : ∀ (hdvd : x ∣ q), x ^ (n + 1) ∣ a
⊢ ¬x ∣ q

Tactic:
contrapose hlt with hdvd
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
q : R
hq : a = x ^ n * q
hdvd : x ∣ q
obtain : ∀ (q' : R) (hq' : q = x * q'), x ^ (n + 1) ∣ a
⊢ x ^ (n + 1) ∣ a

Tactic:
obtain ⟨q', hq'⟩ := hdvd
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
q : R
hq : a = x ^ n * q
q' : R
hq' : q = x * q'
h : a = x ^ (n + 1) * q'
⊢ x ^ (n + 1) ∣ a

Tactic:
use q'
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
q : R
hq : a = x ^ n * q
q' : R
hq' : q = x * q'
h : x ^ n * (x * q') = x ^ (n + 1) * q'
⊢ a = x ^ (n + 1) * q'

Tactic:
rw [hq, hq']
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
a x : R
hx : Prime x
n : ℕ
q : R
hq : a = x ^ n * q
q' : R
hq' : q = x * q'
⊢ x ^ n * (x * q') = x ^ (n + 1) * q'

Tactic:
ring
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), J ≤ P → K ≤ P → ¬P.IsPrime
hJ : I ≤ J
hK : I ≤ K
simp : ∀ (coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime) (hJ : J ∣ I) (hK : K ∣ I), J * K ∣ I
⊢ I ≤ J * K

Tactic:
simp only [← dvd_iff_le] at coprime hJ hK ⊢

Premises used: Ideal.dvd_iff_le
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ : J ∣ I
hK : K ∣ I
pos : ∀ (hJ0 : J = 0), J * K ∣ I
neg : ∀ (hJ0 : ¬J = 0), J * K ∣ I
⊢ J * K ∣ I

Tactic:
by_cases hJ0 : J = 0
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ : J ∣ I
hK : K ∣ I
hJ0 : J = 0
⊢ J * K ∣ I

Tactic:
simpa only [hJ0, zero_mul] using hJ

Premises used: MulZeroClass.zero_mul
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ : J ∣ I
hK : K ∣ I
hJ0 : ¬J = 0
neg : ∀ (I' : Ideal R) (hJ : J ∣ K * I'), J * K ∣ K * I'
⊢ J * K ∣ I

Tactic:
obtain ⟨I', rfl⟩ := hK

Premises used: rfl
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
neg : K * J ∣ K * I'
⊢ J * K ∣ K * I'

Tactic:
rw [mul_comm]

Premises used: mul_comm
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
⊢ K * J ∣ K * I'

Tactic:
mul_dvd_mul_left K
    (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)

Premises used: Set.Mem, mul_dvd_mul_left, UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
neg : ∀ {d : Ideal R}, d ∣ J → d ∣ K → ¬Prime d
⊢ K * J ∣ K * I'

Tactic:
refine mul_dvd_mul_left K
    (UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors (b := K) hJ0 ?_ hJ)

Premises used: mul_dvd_mul_left, UniqueFactorizationMonoid.dvd_of_dvd_mul_right_of_no_prime_factors
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
d : Ideal R
⊢ d ∣ J → d ∣ K → ¬Prime d

Tactic:
fun hPJ hPK => mt isPrime_of_prime (coprime _ hPJ hPK)

Premises used: mt, Ideal.isPrime_of_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J K : Ideal R
coprime : ∀ (P : Ideal R), P ∣ J → P ∣ K → ¬P.IsPrime
hJ0 : ¬J = 0
I' : Ideal R
hJ : J ∣ K * I'
⊢ ∀ {d : Ideal R}, d ∣ J → d ∣ K → ¬Prime d

Tactic:
exact fun hPJ hPK => mt isPrime_of_prime (coprime _ hPJ hPK)

Premises used: mt, Ideal.isPrime_of_prime
-- Module: Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas
Goal:
R : Type u_1
inst : CommRing R
ι : Type u_4
inst_1 : IsDedekindDomain R
s : Finset ι
e : ι → ℕ
f : ι → HeightOneSpectrum R
coprime : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → f i ≠ f j
rw : Eq.{u_1 + 1} (α := Ideal R) (s.inf fun (i : ι) ↦ (f i).asIdeal ^ e i) (⨅ i ∈ s, (f i).asIdeal ^ e i)
rw₁ :
  (↑s : Set ι).Pairwise
    (Function.onFun.{u_4 + 1, u_1 + 1, 1} (β := Ideal R) IsCoprime fun (i : ι) ↦ (f i).asIdeal ^ e i)
⊢ Eq.{u_1 + 1} (α := Ideal R) (s.inf fun (i : ι) ↦ (f i).asIdeal ^ e i) (∏ i ∈ s, (f i).asIdeal ^ e i)

Tactic:
rw [prod_eq_iInf_of_pairwise_isCoprime]

Premises used: Ideal.prod_eq_iInf_of_pairwise_isCoprime