Corollary get_fail_iff : forall m k, get m k = None <-> ~In k (map fst m).

Lemma get_item_trace_map : forall h k v i h' l (Hwf : wf_hists h l) (Htrace : get_item_trace h k v i h') (Hk : k <> 0) (Hrepk : repable_signed k) (Hrepv : repable_signed v), match get (make_map h') k with | Some v' => v' = v /\ wf_hists h' (Z.max (i + 1) l) /\ incl (set (make_map h) k v) (make_map h') | None => l <= i /\ wf_hists h' i /\ v = 0 /\ incl (make_map h) (make_map h') end.