English
Let α be a preorder and β a semilattice with infimum. If f and g are antitone on a set s, then their pointwise infimum h(x) = min{f(x), g(x)} is antitone on s.
Русский
Пусть α — предобразное множество, β — полузамкнутая по операции ⊓. Если f, g антимонотонны на s, то h(x) = min{f(x), g(x)} антимоонтонна на s.
LaTeX
$$$\forall x,y\in s:\ x\le y\Rightarrow \min\{f(x),g(x)\} \ge \min\{f(y),g(y)\}$$$
Lean4
/-- Pointwise infimum of two antitone functions is an antitone function. -/
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
(hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s :=
(hf.dual.sup hg.dual).dual
