odd_sq_dvd_geom_sum₂_sub — Mathlib

English

Let p be an odd prime and x,y ∈ R with p | x − y and p ∤ x. Then for all n, emultiplicity p (x^n − y^n) = emultiplicity p (x − y) + emultiplicity p n.

Русский

Пусть p — нечетное простое, и x,y в кольце, причём p делит x−y и p не делит x. Тогда для любого n выполняется LTE: v_p(x^n − y^n) = v_p(x − y) + v_p(n).

LaTeX

$$Given p prime and Odd p, with p | (x−y) and p ∤ x, then ∀ n, emultiplicity p (x^n − y^n) = emultiplicity p (x − y) + emultiplicity p n.$$

Lean4

theorem odd_sq_dvd_geom_sum₂_sub (hp : Odd p) :
    (p : R) ^ 2 ∣ (∑ i ∈ range p, (a + p * b) ^ i * a ^ (p - 1 - i)) - p * a ^ (p - 1) :=
  by
  have h1 : ∀ (i : ℕ), (p : R) ^ 2 ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * i + a ^ i) :=
    by
    intro i
    calc
      ↑p ^ 2 ∣ (↑p * b) ^ 2 := by simp only [mul_pow, dvd_mul_right]
      _ ∣ (a + ↑p * b) ^ i - (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) := by
        simp only [sq_dvd_add_pow_sub_sub (↑p * b) a i, ← sub_sub]
  simp_rw [← mem_span_singleton, ← Ideal.Quotient.eq] at *
  let s : R := (p : R) ^ 2
  calc
    (Ideal.Quotient.mk (span { s })) (∑ i ∈ range p, (a + (p : R) * b) ^ i * a ^ (p - 1 - i)) =
        ∑ i ∈ Finset.range p, mk (span { s }) ((a ^ (i - 1) * (↑p * b) * ↑i + a ^ i) * a ^ (p - 1 - i)) :=
      by simp_rw [s, RingHom.map_geom_sum₂, ← map_pow, h1, ← map_mul]
    _ =
        mk (span { s }) (∑ x ∈ Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span { s }) (∑ x ∈ Finset.range p, a ^ (x + (p - 1 - x))) :=
      by
      ring_nf
      simp_rw [← map_sum, sum_add_distrib, map_add]
    _ =
        mk (span { s }) (∑ x ∈ Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span { s }) (∑ _x ∈ Finset.range p, a ^ (p - 1)) :=
      by
      rw [add_right_inj]
      have : ∀ (x : ℕ), (hx : x ∈ range p) → a ^ (x + (p - 1 - x)) = a ^ (p - 1) :=
        by
        intro x hx
        rw [← Nat.add_sub_assoc _ x, Nat.add_sub_cancel_left]
        exact Nat.le_sub_one_of_lt (Finset.mem_range.mp hx)
      rw [Finset.sum_congr rfl this]
    _ =
        mk (span { s }) (∑ x ∈ Finset.range p, a ^ (x - 1) * (a ^ (p - 1 - x) * (↑p * (b * ↑x)))) +
          mk (span { s }) (↑p * a ^ (p - 1)) :=
      by simp only [Finset.sum_const, Finset.card_range, nsmul_eq_mul]
    _ = mk (span { s }) (↑p * b * ∑ x ∈ Finset.range p, a ^ (p - 2) * x) + mk (span { s }) (↑p * a ^ (p - 1)) :=
      by
      simp only [Finset.mul_sum, ← mul_assoc, ← pow_add]
      rw [Finset.sum_congr rfl]
      rintro (⟨⟩ | ⟨x⟩) hx
      · rw [Nat.cast_zero, mul_zero, mul_zero]
      · have : x.succ - 1 + (p - 1 - x.succ) = p - 2 :=
          by
          rw [← Nat.add_sub_assoc (Nat.le_sub_one_of_lt (Finset.mem_range.mp hx))]
          exact congr_arg Nat.pred (Nat.add_sub_cancel_left _ _)
        rw [this]
        ring1
    _ = mk (span { s }) (↑p * a ^ (p - 1)) :=
      by
      have : Finset.sum (range p) (fun (x : ℕ) ↦ (x : R)) = ((Finset.sum (range p) (fun (x : ℕ) ↦ (x : ℕ)))) := by
        simp only [Nat.cast_sum]
      simp only [add_eq_right, ← Finset.mul_sum, this]
      norm_cast
      simp only [Finset.sum_range_id]
      norm_cast
      simp only [Nat.cast_mul, map_mul, Nat.mul_div_assoc p (even_iff_two_dvd.mp (Nat.Odd.sub_odd hp odd_one))]
      ring_nf
      rw [mul_assoc, mul_assoc]
      refine mul_eq_zero_of_left ?_ _
      refine Ideal.Quotient.eq_zero_iff_mem.mpr ?_
      simp [s, mem_span_singleton]

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