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 fnd_rem k1 k2 (s : fmap) :
fnd k1 (rem k2 s) = if k1 == k2 then None else fnd k1 s.