Theorems · Theorem · order theory

Filter.liminf_eq_iSup_iInf

∀ {α : Type u_1} {β : Type u_2} [inst : CompleteLattice α] {f : Filter β} {u : β → α},
  Filter.liminf u f = ⨆ s ∈ f, ⨅ a ∈ s, u a

In a complete lattice, the liminf of a function is the infimum over sets s in the filter of the supremum of the function over s

Defined in

Mathlib.Order.LiminfLimsup

Cited by

4 results in Mathlib

Foundations

Depth 17 from the axioms · uses propext, Quot.sound

Assumes

CompleteLattice