English
For any p, a, b in a semiring, min(emultiplicity p a, emultiplicity p b) ≤ emultiplicity p (a + b).
Русский
Для любых p, a, b в полусферах: минимальная эмп выпуклности ≤ эмп выпуклности сложения.
LaTeX
$$$\min(\operatorname{emultiplicity}_p(a), \operatorname{emultiplicity}_p(b)) \le \operatorname{emultiplicity}_p(a + b).$$$
Lean4
theorem min_le_emultiplicity_add {p a b : α} : min (emultiplicity p a) (emultiplicity p b) ≤ emultiplicity p (a + b) :=
by
cases hm : min (emultiplicity p a) (emultiplicity p b)
· simp only [top_le_iff, min_eq_top, emultiplicity_eq_top] at hm ⊢
contrapose hm
simp only [not_and_or, not_not] at hm ⊢
exact hm.or_of_add
· apply le_emultiplicity_of_pow_dvd
simp [dvd_add, pow_dvd_of_le_emultiplicity, ← hm]
