(** * 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.

(* Specifications: *)
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).
Lemma set_optvar_lessdef:
forall e1 e2 optid v1 v2,
env_lessdef e1 e2 -> Val.lessdef v1 v2 ->
env_lessdef (set_optvar optid v1 e1) (set_optvar optid v2 e2).
Lemma set_params_lessdef:
forall il vl1 vl2,
Val.lessdef_list vl1 vl2 ->
env_lessdef (set_params vl1 il) (set_params vl2 il).
Lemma set_locals_lessdef:
forall e1 e2, env_lessdef e1 e2 ->
forall il, env_lessdef (set_locals il e1) (set_locals il e2).