awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
⊢ ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal

Tactic:
have : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal := by
    intro p hps
    conv_rhs => rw [← mul_one (J * _)]
    rw [Finset.prod_eq_mul_prod_diff_singleton_of_mem hps, ← mul_assoc,
      mul_right_comm _ p.asIdeal]
    refine mul_lt_mul_of_pos_left ?_ ?_
    · rw [Ideal.one_eq_top, lt_top_iff_ne_top]
      exact p.2.ne_top
    · rw [Ideal.zero_eq_bot, bot_lt_iff_ne_bot, ← Ideal.zero_eq_bot,
        mul_ne_zero_iff, Finset.prod_ne_zero_iff]
      exact ⟨hJ, fun x _ ↦ x.3⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
choose! :
  ∀ (a : (p : HeightOneSpectrum R) → R)
    (ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal)
    (ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal),
    ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
⊢ ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤

Tactic:
choose! a ha ha' using fun p hps ↦ SetLike.exists_of_lt (this p hps)

Premises used: SetLike.exists_of_lt
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
rw : ∑ p ∈ s, a p ∈ J
⊢ J ∣ Ideal.span {∑ p ∈ s, a p}

Tactic:
rw [Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem]

Premises used: Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
⊢ ∑ p ∈ s, a p ∈ J

Tactic:
sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)

Premises used: Ideal.mul_le_right, sum_mem
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
⊢ ∑ p ∈ s, a p ∈ J

Tactic:
exact sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)

Premises used: Ideal.mul_le_right, sum_mem
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
obtain :
  ∀ (K : Ideal R) (hK : Ideal.span {∑ p ∈ s, a p} = J * K), ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
⊢ ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤

Tactic:
obtain ⟨K, hK⟩ : J ∣ Ideal.span {∑ p ∈ s, a p} := by
    rw [Ideal.dvd_iff_le, Ideal.span_singleton_le_iff_mem]
    exact sum_mem fun p hp ↦ Ideal.mul_le_right (ha p hp)

Premises used: Ideal.span, Singleton.singleton, Finset.sum
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
⊢ ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤

Tactic:
⟨_, _, hK.symm, ?_⟩

Premises used: And.intro, Exists.intro, Eq.symm
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
refine : I + K = ⊤
⊢ ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤

Tactic:
refine ⟨_, _, hK.symm, ?_⟩
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
by_contra : ∀ (H : ¬I + K = ⊤), False
⊢ I + K = ⊤

Tactic:
by_contra H
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
obtain : ∀ (p : Ideal R) (hp : p.IsMaximal) (h : I + K ≤ p), False
⊢ False

Tactic:
obtain ⟨p, hp, h⟩ := Ideal.exists_le_maximal _ H

Premises used: Ideal.exists_le_maximal
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
e : p = ⊥
⊢ I = 0

Tactic:
simp_all
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
⊢ p' ∈ s

Tactic:
simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h

Premises used: Ideal.dvd_iff_le, le_sup_left, LE.le.trans
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
⊢ p' ∈ s

Tactic:
have hp's : p' ∈ s := by simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
⊢ J * K ≤ J * p

Tactic:
have H₁ : J * K ≤ J * p := Ideal.mul_mono_right (le_sup_right.trans h)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : J * K ≤ J * p
⊢ ∑ p ∈ s, a p ∈ J * p

Tactic:
replace H₁ := hK.trans_le H₁ (Ideal.mem_span_singleton_self _)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
⊢ ∑ q ∈ s \ {p'}, a q ∈ J * p

Tactic:
sum_mem fun q hq ↦ ?_

Premises used: sum_mem
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
refine : ∀ (q : HeightOneSpectrum R) (hq : q ∈ s \ {p'}), a q ∈ J * p
⊢ ∑ q ∈ s \ {p'}, a q ∈ J * p

Tactic:
refine sum_mem fun q hq ↦ ?_

Premises used: sum_mem
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
q : HeightOneSpectrum R
hq : q ∈ s \ {p'}
⊢ q ∈ s ∧ ¬q = p'

Tactic:
rw [Finset.mem_sdiff, Finset.mem_singleton] at hq

Premises used: Finset.mem_singleton, Finset.mem_sdiff
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
q : HeightOneSpectrum R
hq : q ∈ s ∧ ¬q = p'
⊢ a q ∈ J * p

Tactic:
Ideal.mul_mono_right ?_ (ha q hq.1)

Premises used: Set.Mem, And.left, Ideal.mul_mono_right
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
q : HeightOneSpectrum R
hq : q ∈ s ∧ ¬q = p'
refine : ∏ q ∈ s \ {q}, q.asIdeal ≤ p
⊢ a q ∈ J * p

Tactic:
refine Ideal.mul_mono_right ?_ (ha q hq.1)

Premises used: And.left, Ideal.mul_mono_right
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
q : HeightOneSpectrum R
hq : q ∈ s ∧ ¬q = p'
⊢ p' ∈ s \ {q}

Tactic:
simpa [hp's] using Ne.symm hq.2

Premises used: Ne.symm, And.right
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
q : HeightOneSpectrum R
hq : q ∈ s ∧ ¬q = p'
⊢ ∏ q ∈ s \ {q}, q.asIdeal ≤ p

Tactic:
exact Ideal.prod_le_inf.trans (Finset.inf_le (b := p') (by simpa [hp's] using Ne.symm hq.2))

Premises used: Ideal.prod_le_inf, LE.le.trans, Finset.inf_le
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
⊢ ∑ q ∈ s \ {p'}, a q ∈ J * p

Tactic:
have H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p := by
    refine sum_mem fun q hq ↦ ?_
    rw [Finset.mem_sdiff, Finset.mem_singleton] at hq
    refine Ideal.mul_mono_right ?_ (ha q hq.1)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := p') (by simpa [hp's] using Ne.symm hq.2))
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p
apply : a p' ∈ J * ∏ q ∈ s, q.asIdeal
⊢ False

Tactic:
apply ha' _ hp's
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p
⊢ Eq (α := Ideal R)
    (s.inf fun (i : HeightOneSpectrum R) ↦
      HPow.hPow (α := Ideal R) (β := ℕ) ((fun (i : HeightOneSpectrum R) ↦ i.asIdeal) i)
        ((fun (x : HeightOneSpectrum R) ↦ (1 : ℕ)) i))
    (∏ i ∈ s,
      HPow.hPow (α := Ideal R) (β := ℕ) ((fun (i : HeightOneSpectrum R) ↦ i.asIdeal) i)
        ((fun (x : HeightOneSpectrum R) ↦ (1 : ℕ)) i))

Tactic:
have := IsDedekindDomain.inf_pow_eq_prod_of_prime s (fun i ↦ i.asIdeal) (fun _ ↦ 1)
    (fun i _ ↦ i.prime) (fun i _ j _ e ↦ mt HeightOneSpectrum.ext e)
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this_1 : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p
this :
  Eq (α := Ideal R)
    (s.inf fun (i : HeightOneSpectrum R) ↦
      HPow.hPow (α := Ideal R) (β := ℕ) ((fun (i : HeightOneSpectrum R) ↦ i.asIdeal) i)
        ((fun (x : HeightOneSpectrum R) ↦ (1 : ℕ)) i))
    (∏ i ∈ s,
      HPow.hPow (α := Ideal R) (β := ℕ) ((fun (i : HeightOneSpectrum R) ↦ i.asIdeal) i)
        ((fun (x : HeightOneSpectrum R) ↦ (1 : ℕ)) i))
⊢ Eq (α := Ideal R) (s.inf fun (i : HeightOneSpectrum R) ↦ i.asIdeal) (∏ i ∈ s, i.asIdeal)

Tactic:
simp only [pow_one] at this

Premises used: pow_one
-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
J I : Ideal R
hIJ : J * I ≤ J
hJ : ¬J = 0
hI : ¬I = 0
s : Finset (HeightOneSpectrum R) := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I})
this_1 : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal
a : (p : HeightOneSpectrum R) → R
ha : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∈ J * ∏ q ∈ s \ {p}, q.asIdeal
ha' : ∀ (p : HeightOneSpectrum R) (hps : p ∈ s), a p ∉ J * ∏ q ∈ s, q.asIdeal
K : Ideal R
hK : Ideal.span {∑ p ∈ s, a p} = J * K
H : ¬I + K = ⊤
p : Ideal R
hp : p.IsMaximal
h : I + K ≤ p
p' : HeightOneSpectrum R := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ }
hp's : p' ∈ s
H₁ : ∑ p ∈ s, a p ∈ J * p
H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p
this : Eq (α := Ideal R) (s.inf fun (i : HeightOneSpectrum R) ↦ i.asIdeal) (∏ i ∈ s, i.asIdeal)
⊢ Nonempty.{max 1 (u_1 + 1)} ↥s

Tactic:
have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩