Definition tp : Type := @UM.base nat_ordType A nonz.Definition nm_undef := @UM.Undef nat_ordType A nonz.Definition defined f := @UM.valid nat_ordType A nonz f.Definition empty := @UM.empty nat_ordType A nonz.Definition upd k v f := @UM.upd nat_ordType A nonz k v f.Definition dom f := @UM.dom nat_ordType A nonz f.Definition dom_eq f1 f2 := @UM.dom_eq nat_ordType A nonz f1 f2.Definition free k f := @UM.free nat_ordType A nonz k f.Definition find k f := @UM.find nat_ordType A nonz k f.Definition union f1 f2 := @UM.union nat_ordType A nonz f1 f2.Definition nm_filter q f := @UM.um_filter nat_ordType A nonz q f.Definition empb f := @UM.empb nat_ordType A nonz f.Definition undefb f := @UM.undefb nat_ordType A nonz f.Definition pts k v := @UM.pts nat_ordType A nonz k v.Definition from (f : tp) : @UM.base nat_ordType A nonz := f.Definition to (b : @UM.base nat_ordType A nonz) : tp := b.

Lemma freeE k f : free k f = to (UM.free k (from f)).