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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).

(* Specifications: *)
Lemma umpleq_undef x : [pcm x <= um_undef].
(* monotonicity lemmas *)
Lemma umpleq_filt2 x y p : [pcm x <= y] -> [pcm um_filter p x <= um_filter p y].
(* filter p x is lower bound for x *)
Lemma umpleq_filtI x p : [pcm um_filter p x <= x].
Hint Resolve umpleq_filtI : core.
(* in valid case, we can define the order by means of filter *)
Lemma umpleq_filtE a x :
valid x -> [pcm a <= x] <-> um_filter (mem (dom a)) x = a.