liminf_piecewise — Mathlib

English

For s ⊆ β decidable, liminf of the piecewise combination equals the infimum of the two bliminfs corresponding to membership and non-membership in s.

Русский

Для разбиения по s верно: liminf от разбиения по s равен инфиминум между двумя частями по membership и non-membership.

LaTeX

$$$\\operatorname{liminf}(s.{\\text{piecewise}}\\,u\,v) = \\operatorname{bliminf} u f (\\cdot \\in s) \\; \\sqcap \\; \\operatorname{bliminf} v f (\\cdot \\notin s)$$$

Lean4

theorem liminf_piecewise {s : Set β} [DecidablePred (· ∈ s)] {v} :
    liminf (s.piecewise u v) f = bliminf u f (· ∈ s) ⊓ bliminf v f (· ∉ s) :=
  limsup_piecewise (α := αᵒᵈ)

Похожие утверждения