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(* PCM-induced ordering *)
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(* Union maps and pcm ordering. *)
Section UmpleqLemmas.
Variables (K : ordType) (A : Type) (U : union_map_class K A).
Implicit Types (x y a b : U) (p : pred K).

(* monotonicity lemmas *)
Lemma umpleq_filt2 x y p : [pcm x <= y] -> [pcm um_filter p x <= um_filter p y].