End DomEqLemmas.
Hint Resolve domeq_refl : core.
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(* some precision lemmas *)
(*************************)
Section Precision.
Variables (K : ordType) (V : Type) (U : union_map_class K V).
Implicit Types (x y : U).

(* Specifications: *)
Lemma prec_flip x1 x2 y1 y2 :
valid (x1 \+ y1) -> x1 \+ y1 = x2 \+ y2 ->
valid (y2 \+ x2) /\ y2 \+ x2 = y1 \+ x1.
Lemma prec_filtV x1 x2 y1 y2 :
valid (x1 \+ y1) -> x1 \+ y1 = x2 \+ y2 ->
reflect (x1 = um_filter (mem (dom x1)) x2) (valid (x1 \+ y2)).