-- Module: Mathlib.RingTheory.LaurentSeries
Goal:
K : Type u_2
inst : Field K
P : K[X]
hP : ¬P = 0
span_ne_zero : Ne (α := Ideal K[X]) (Ideal.span {P}) 0 ∧ Ne (α := Ideal K[X]) (Ideal.span {Polynomial.X}) 0
span_ne_zero' : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0 ∧ Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) 0
neg :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))
neg₁ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₂ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₃ : Ideal.IsPrime (asIdeal (idealX K))
neg₄ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₅ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₆ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₇ : Ideal.IsPrime (asIdeal (idealX K))
neg₈ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
neg₉ :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)}))
neg₁₀ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₁₁ : Ideal.IsPrime (asIdeal (idealX K))
neg₁₂ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
h :
  Multiset.count (α := Ideal K[X]) (Ideal.span {Polynomial.X})
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {P})) =
    Multiset.count Polynomial.X (UniqueFactorizationMonoid.normalizedFactors P)
h₁ :
  Multiset.count (α := Ideal K⟦X⟧) (asIdeal (idealX K))
      (UniqueFactorizationMonoid.normalizedFactors (Ideal.span {(↑P : K⟦X⟧)})) =
    Multiset.count X (UniqueFactorizationMonoid.normalizedFactors (↑P : K⟦X⟧))
neg₁₃ : Ne (α := Ideal K⟦X⟧) (Ideal.span {(↑P : K⟦X⟧)}) 0
neg₁₄ : Ideal.IsPrime (asIdeal (idealX K))
neg₁₅ : Ne (α := Ideal K⟦X⟧) (asIdeal (idealX K)) ⊥
⊢ Associates.count (α := Ideal K[X]) (Associates.mk (Ideal.span {Polynomial.X}))
      (Associates.factors (Associates.mk (Ideal.span {P}))) =
    Associates.count (α := Ideal K⟦X⟧) (Associates.mk (asIdeal (idealX K)))
      (Associates.factors (Associates.mk (Ideal.span {(↑P : K⟦X⟧)})))

Tactic:
classical
  rw [Ideal.count_associates_factors_eq span_ne_zero.1
    (Ideal.span_singleton_prime Polynomial.X_ne_zero |>.mpr prime_X) span_ne_zero.2,
    Ideal.count_associates_factors_eq]
  on_goal 1 => convert! (normalized_count_X_eq_of_coe hP).symm
  exacts [Ideal.count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,
    Ideal.count_span_normalizedFactors_eq_of_normUnit (by simp [hP]) normUnit_X X_prime,
    span_ne_zero'.1, (idealX K).isPrime, span_ne_zero'.2]

Premises used: Iff.mpr, PowerSeries.normalized_count_X_eq_of_coe, Ideal.count_span_normalizedFactors_eq_of_normUnit, PowerSeries.idealX, And.right, And.left, Ideal.count_associates_factors_eq, Polynomial.prime_X, Ideal.span_singleton_prime, IsDedekindDomain.HeightOneSpectrum.isPrime, Eq.symm