-- Module: Mathlib.RingTheory.DedekindDomain.Factorization
Goal:
R : Type u_1
inst : CommRing R
K : Type u_2
inst_1 : Field K
inst_2 : Algebra R K
inst_3 : IsFractionRing R K
inst_4 : IsDedekindDomain R
I : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K
hI : I ≠ 0
h :
  I =
    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      (spanSingleton R⁰
          ((algebraMap R K : (a : R) → K)
              (Classical.choose (p := fun (x : R) ↦
                ∃ (aI : Ideal R),
                  x ≠ 0 ∧
                    I =
                      HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                        (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                          FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                        ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                ⋯))⁻¹ :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
      ((↑(Classical.choose (p := fun (x : Ideal R) ↦
              Classical.choose (p := fun (x : R) ↦
                    ∃ (aI : Ideal R),
                      x ≠ 0 ∧
                        I =
                          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                    ⋯ ≠
                  0 ∧
                I =
                  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                    (spanSingleton R⁰
                        ((algebraMap R K : (a : R) → K)
                            (Classical.choose (p := fun (x : R) ↦
                              ∃ (aI : Ideal R),
                                x ≠ 0 ∧
                                  I =
                                    HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                      (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                      ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                              ⋯))⁻¹ :
                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                    ((↑x : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
              ⋯) :
          FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
        FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
h₁ :
  ∀ (x : HeightOneSpectrum R),
    HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑x.asIdeal : FractionalIdeal R⁰ K) (count K x I) =
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑x.asIdeal : FractionalIdeal R⁰ K)
        (HSub.hSub (α := ℤ)
          (↑(Associates.count (Associates.mk x.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Classical.choose (p := fun (x : Ideal R) ↦
                      Classical.choose (p := fun (x : R) ↦
                            ∃ (aI : Ideal R),
                              x ≠ 0 ∧
                                I =
                                  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                    ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                            ⋯ ≠
                          0 ∧
                        I =
                          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            (spanSingleton R⁰
                                ((algebraMap R K : (a : R) → K)
                                    (Classical.choose (p := fun (x : R) ↦
                                      ∃ (aI : Ideal R),
                                        x ≠ 0 ∧
                                          I =
                                            HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                              (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                                FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                              ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                                FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                                      ⋯))⁻¹ :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            ((↑x : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                      ⋯)))) :
            ℤ)
          (↑(Associates.count (Associates.mk x.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Ideal.span
                      {Classical.choose (p := fun (x : R) ↦
                          ∃ (aI : Ideal R),
                            x ≠ 0 ∧
                              I =
                                HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                  FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                  (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                  ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                          ⋯})))) :
            ℤ))
⊢ Eq.{u_2 + 1} (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
    (∏ᶠ (v : HeightOneSpectrum R),
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑v.asIdeal : FractionalIdeal R⁰ K) (count K v I))
    (∏ᶠ (v : HeightOneSpectrum R),
      HPow.hPow (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (↑v.asIdeal : FractionalIdeal R⁰ K)
        (HSub.hSub (α := ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Classical.choose (p := fun (x : Ideal R) ↦
                      Classical.choose (p := fun (x : R) ↦
                            ∃ (aI : Ideal R),
                              x ≠ 0 ∧
                                I =
                                  HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                    (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                    ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                      FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                            ⋯ ≠
                          0 ∧
                        I =
                          HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                            FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            (spanSingleton R⁰
                                ((algebraMap R K : (a : R) → K)
                                    (Classical.choose (p := fun (x : R) ↦
                                      ∃ (aI : Ideal R),
                                        x ≠ 0 ∧
                                          I =
                                            HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                              (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                                FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                              ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                                FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                                      ⋯))⁻¹ :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                            ((↑x : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                              FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                      ⋯)))) :
            ℤ)
          (↑(Associates.count (Associates.mk v.asIdeal)
                (Associates.factors
                  (Associates.mk
                    (Ideal.span
                      {Classical.choose (p := fun (x : R) ↦
                          ∃ (aI : Ideal R),
                            x ≠ 0 ∧
                              I =
                                HMul.hMul (α := FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) (β :=
                                  FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                  (spanSingleton R⁰ ((algebraMap R K : (a : R) → K) x)⁻¹ :
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K)
                                  ((↑aI : FractionalIdeal.{u_1, u_2} (R := R) R⁰ K) :
                                    FractionalIdeal.{u_1, u_2} (R := R) R⁰ K))
                          ⋯})))) :
            ℤ)))

Tactic:
apply finprod_congr

Premises used: finprod_congr