CoInductive supp_spec x (s : fmap) : bool -> Type :=
| supp_spec_some v of fnd x s = Some v : supp_spec x s true
| supp_spec_none of fnd x s = None : supp_spec x s false.

Lemma suppE (s1 s2 : fmap) : supp s1 =i supp s2 <-> supp s1 = supp s2.