-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
inst_1 : IsDedekindDomain R
I J : Ideal R
hIJ : I ≤ J
hI : I ≠ 0
obtain : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : J * I ≠ 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₁ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hI : ¬J = 0 ∧ ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₂ : ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0), ∃ (a : R), J * I ⊔ Ideal.span {a} = J
obtain₃ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
obtain₄ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
obtain₅ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.toFinset (s := {v : HeightOneSpectrum R | v.asIdeal ∣ I});
    ∀ (this : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \ {p}, q.asIdeal),
      ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
obtain₆ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      ∃ (a : R) (K : Ideal R), J * K = Ideal.span {a} ∧ I + K = ⊤
obtain₇ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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 = ⊤
obtain₈ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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), I + K = ⊤
obtain₉ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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 = ⊤), False
obtain₁₀ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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), False
obtain₁₁ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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), False
obtain₁₂ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (hp's : p' ∈ s), False
obtain₁₃ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (hp's : p' ∈ s) (H₁ : J * K ≤ J * p), False
obtain₁₄ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (hp's : p' ∈ s) (H₁ : ∑ p ∈ s, a p ∈ J * p), False
obtain₁₅ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (hp's : p' ∈ s) (H₁ : ∑ p ∈ s, a p ∈ J * p) (H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p), False
obtain₁₆ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (hp's : p' ∈ s) (H₁ : ∑ p ∈ s, a p ∈ J * p) (H₂ : ∑ q ∈ s \ {p'}, a q ∈ J * p), a p' ∈ J * ∏ q ∈ s, q.asIdeal
obtain₁₇ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))),
        a p' ∈ J * ∏ q ∈ s, q.asIdeal
obtain₁₈ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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)),
        a p' ∈ J * ∏ q ∈ s, q.asIdeal
obtain₁₉ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))
        (inst_2 : Nonempty.{max 1 (u_1 + 1)} ↥s), a p' ∈ J * ∏ q ∈ s, q.asIdeal
obtain₂₀ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))
        (inst_2 : Nonempty.{max 1 (u_1 + 1)} ↥s) (i : Subtype (Membership.mem s)),
        a p' ∈ J * (↑i : HeightOneSpectrum R).asIdeal
obtain₂₁ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))
        (inst_2 : Nonempty.{max 1 (u_1 + 1)} ↥s) (q : HeightOneSpectrum R) (hq : q ∈ s),
        a p' ∈ J * (↑(Subtype.mk (p := Membership.mem s) q hq) : HeightOneSpectrum R).asIdeal
pos :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))
        (inst_2 : Nonempty.{max 1 (u_1 + 1)} ↥s) (q : HeightOneSpectrum R) (hq : q ∈ s) (hqp : q = p'),
        a p' ∈ J * (↑(Subtype.mk (p := Membership.mem s) q hq) : HeightOneSpectrum R).asIdeal
neg neg₁ :
  ∀ (I : Ideal R) (hIJ : J * I ≤ J) (hJ : ¬J = 0) (hI : ¬I = 0),
    let s := ⋯.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),
      let p' := { asIdeal := p, isPrime := ⋯, ne_bot := ⋯ : HeightOneSpectrum R };
      ∀ (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))
        (inst_2 : Nonempty.{max 1 (u_1 + 1)} ↥s) (q : HeightOneSpectrum R) (hq : q ∈ s) (hqp : ¬q = p'),
        a p' ∈ J * (↑(Subtype.mk (p := Membership.mem s) q hq) : HeightOneSpectrum R).asIdeal
⊢ ∃ (a : R), I ⊔ Ideal.span {a} = J

Tactic:
classical
  obtain ⟨I, rfl⟩ := Ideal.dvd_iff_le.mpr hIJ
  simp only [ne_eq, mul_eq_zero, not_or] at hI
  obtain ⟨hJ, hI⟩ := hI
  suffices ∃ a, ∃ K, J * K = Ideal.span {a} ∧ I + K = ⊤ by
    obtain ⟨a, K, e, e'⟩ := this
    exact ⟨a, by rw [← e, ← Ideal.add_eq_sup, ← mul_add, e', Ideal.mul_top]⟩
  let s := (I.finite_factors hI).toFinset
  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⟩
  choose! a ha ha' using fun p hps ↦ SetLike.exists_of_lt (this p hps)
  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)
  refine ⟨_, _, hK.symm, ?_⟩
  by_contra H
  obtain ⟨p, hp, h⟩ := Ideal.exists_le_maximal _ H
  let p' : HeightOneSpectrum R := ⟨p, hp.isPrime, fun e ↦ hI (by simp_all)⟩
  have hp's : p' ∈ s := by simpa [p', s, Ideal.dvd_iff_le] using le_sup_left.trans h
  have H₁ : J * K ≤ J * p := Ideal.mul_mono_right (le_sup_right.trans h)
  replace H₁ := hK.trans_le H₁ (Ideal.mem_span_singleton_self _)
  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))
  apply ha' _ hp's
  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)
  simp only [pow_one] at this
  have inst : Nonempty {x // x ∈ s} := ⟨_, hp's⟩
  rw [← this, Finset.inf_eq_iInf, iInf_subtype', Ideal.mul_iInf, Ideal.mem_iInf]
  rintro ⟨q, hq⟩
  by_cases hqp : q = p'
  · subst hqp
    convert! sub_mem H₁ H₂
    rw [Finset.sum_eq_add_sum_diff_singleton_of_mem hp's, add_sub_cancel_right]
  · refine Ideal.mul_mono_right ?_ (ha p' hp's)
    exact Ideal.prod_le_inf.trans (Finset.inf_le (b := q) (by simpa [hq] using hqp))

Premises used: Iff.mpr, Ideal.dvd_iff_le, iInf_subtype', mul_eq_zero, ne_eq, not_or, Ideal.mem_iInf, pow_one, Finset.inf_eq_iInf, Ideal.mul_iInf, add_sub_cancel_right, rfl