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 supp_ins k v (s : fmap) :
supp (ins k v s) =i predU (pred1 k) (mem (supp s)).