Definition val_inject_opt (j: meminj) (v1 v2: option val) :=
match v1, v2 with Some v1', Some v2' => val_inject j v1' v2'
| None, None => True
| _, _ => False
end.

Lemma val_inject_split: forall v1 v3 j12 j23 (V: val_inject (compose_meminj j12 j23) v1 v3),
exists v2, val_inject j12 v1 v2 /\ val_inject j23 v2 v3.