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.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.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 fmap_ind'' (P : fmap -> Prop) : P nil -> (forall k v s, (forall x, x \in supp s -> ord x k) -> P s -> P (ins k v s)) -> forall s, P s.