English
The integer valuation of a polynomial P agrees with the integer valuation of its image under the natural embedding into Laurent series.
Русский
Целочисленная valoración полинома P совпадает с valoración его изображения под естественным вложением в Лорантовые разложения.
LaTeX
$$$ (Polynomial.idealX K).intValuation P = (idealX K).intValuation (P : K\langle X\rangle).$$$
Lean4
theorem intValuation_eq_of_coe (P : K[X]) : (Polynomial.idealX K).intValuation P = (idealX K).intValuation (P : K⟦X⟧) :=
by
by_cases hP : P = 0
· rw [hP, Valuation.map_zero, Polynomial.coe_zero, Valuation.map_zero]
rw [intValuation_if_neg _ hP, intValuation_if_neg _ <| (by simp [hP])]
simp only [idealX_span, ofAdd_neg, inv_inj, WithZero.coe_inj, EmbeddingLike.apply_eq_iff_eq, Nat.cast_inj]
have span_ne_zero : (Ideal.span { P } : Ideal K[X]) ≠ 0 ∧ (Ideal.span { Polynomial.X } : Ideal K[X]) ≠ 0 := by
simp only [Ideal.zero_eq_bot, ne_eq, Ideal.span_singleton_eq_bot, hP, Polynomial.X_ne_zero, not_false_iff,
and_self_iff]
have span_ne_zero' : (Ideal.span {↑P} : Ideal K⟦X⟧) ≠ 0 ∧ ((idealX K).asIdeal : Ideal K⟦X⟧) ≠ 0 := by
simp only [Ideal.zero_eq_bot, ne_eq, Ideal.span_singleton_eq_bot, coe_eq_zero_iff, hP, not_false_eq_true, true_and,
(idealX K).3]
classical
rw [count_associates_factors_eq (span_ne_zero).1 (Ideal.span_singleton_prime Polynomial.X_ne_zero |>.mpr prime_X)
(span_ne_zero).2,
count_associates_factors_eq]
on_goal 1 => convert (normalized_count_X_eq_of_coe hP).symm
exacts [count_span_normalizedFactors_eq_of_normUnit hP Polynomial.normUnit_X prime_X,
count_span_normalizedFactors_eq_of_normUnit (by simp [hP]) normUnit_X X_prime, span_ne_zero'.1, (idealX K).isPrime,
span_ne_zero'.2]
