pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right

English

In a commutative ring with order, for a CHSH tuple satisfying A_i^2 = 1 and B_j^2 = 1, the CHSH expression is bounded above by 2.

Русский

В упорядоченном коммутативном кольце для CHSH-кортежа с A_i^2 = 1 и B_j^2 = 1, CHSH-выражение не превосходит 2.

LaTeX

$$$A_0 B_0 + A_0 B_1 + A_1 B_0 - A_1 B_1 \\le 2.$$$

Lean4

theorem pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right {k : ℕ} (hx : Squarefree x) (hp : Prime p)
    (h : p ^ (k + 1) ∣ x * y) : p ^ k ∣ y := by
  by_cases hxp : p ∣ x
  · obtain ⟨x', rfl⟩ := hxp
    have hx' : ¬p ∣ x' := fun contra ↦ hp.not_unit <| hx p (mul_dvd_mul_left p contra)
    replace h : p ^ k ∣ x' * y := by
      rw [pow_succ', mul_assoc] at h
      exact (mul_dvd_mul_iff_left hp.ne_zero).mp h
    exact hp.pow_dvd_of_dvd_mul_left _ hx' h
  · exact (pow_dvd_pow _ k.le_succ).trans (hp.pow_dvd_of_dvd_mul_left _ hxp h)

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