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 rem_ins k1 k2 v (s : fmap) :
rem k1 (ins k2 v s) =
if k1 == k2 then rem k1 s else ins k2 v (rem k1 s).