Hint Resolve identity_None.
Definition val2loc (v: val) : option loc :=
match v with Some (S n) => Some (S n) | _ => None end.
Definition nil_val : val := Some O.

(* Specifications: *)
Lemma nil_not_loc: val2loc nil_val = None.
Definition full (v: val) := forall v2, joins v v2 -> identity v2.
Lemma val2loc_full v l : val2loc v = Some l -> full v.
intro H; destruct v as [[|n]|]; inv H.