monovaryOn_iff_forall_smul_nonneg

English

For f,g, the monovary-on relation with set s is equivalent to a pointwise nonnegativity condition involving the smul of differences.

Русский

Для f,g моновариантность-на-множество s эквивалентна побочной неотрицательности в точке через произведение разностей под скаляром.

LaTeX

$$$\\text{MonovaryOn}(f,g,s) \\iff \\forall i\\in s,\\ \\forall j\\in s,\\ 0 \\le (f(j)-f(i)) \\; \\bullet \\; (g(j)-g(i))$$$

Lean4

theorem monovaryOn_iff_forall_smul_nonneg :
    MonovaryOn f g s ↔ ∀ ⦃i⦄, i ∈ s → ∀ ⦃j⦄, j ∈ s → 0 ≤ (f j - f i) • (g j - g i) :=
  by
  simp_rw [smul_nonneg_iff_pos_imp_nonneg, sub_pos, sub_nonneg, forall_and]
  exact (and_iff_right_of_imp MonovaryOn.symm).symm

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