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 ins_ins k1 k2 v1 v2 (s : fmap) :
ins k1 v1 (ins k2 v2 s) = if k1 == k2 then ins k1 v1 s
else ins k2 v2 (ins k1 v1 s).