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