isCoboundedUnder_le_of_isCobounded

English

If a monotone map g yields boundedness under atTop in one direction, then the same holds in the dual direction for the dual order.

Русский

Если монотонная отображение g даёт ограниченность при atTop в одном направлении, то аналогично в двойственном порядке.

LaTeX

$$Monotone.isCoboundedUnder_le_of_isCobounded$$

Lean4

theorem isCoboundedUnder_le_of_isCobounded {f : R → S} (f_incr : Monotone f) [NeBot F] (cobdd : IsCobounded (· ≤ ·) F) :
    F.IsCoboundedUnder (· ≤ ·) f :=
  by
  obtain ⟨l, hl⟩ := IsCobounded.frequently_ge cobdd
  exact IsCobounded.of_frequently_ge <| f_incr.frequently_ge_map_of_frequently_ge hl

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