(** * Semantic preservation for instruction selection. *)
(** Relationship between the local environments. *)
Definition env_lessdef (e1 e2: env) : Prop :=
forall id v1, e1!id = Some v1 -> exists v2, e2!id = Some v2 /\ Val.lessdef v1 v2.

Lemma set_var_lessdef:
forall e1 e2 id v1 v2,
env_lessdef e1 e2 -> Val.lessdef v1 v2 ->
env_lessdef (PTree.set id v1 e1) (PTree.set id v2 e2).