Section CMCONSTR.
Variable sp: val.
Variable e: env.
Variable m: mem.
(*(Sstore chunk addr al b) effsstep tge (State f (store chunk a1 a2) k sp e m) E0 (State f Sskip k sp e m').*)
(** Correctness of instruction selection for operators *)
End CMCONSTR.
(** Recognition of calls to built-in functions *)
(** Compatibility of evaluation functions with the "less defined than" relation. *)
Ltac TrivialExists :=
match goal with
| [ |- exists v, Some ?x = Some v /\ _ ] => exists x; split; auto
| _ => idtac
end.
(*Lemma eval_unop_lessdef: forall op v1 v1' v, eval_unop op v1 = Some v -> Val.lessdef v1 v1' -> exists v', eval_unop op v1' = Some v' /\ Val.lessdef v v'.  (** Lenb: replace eval_unop_lessdef with eval_unop from Cminorgenproof; this requires the additio of Require Import Float above Compatibility of [eval_unop] with respect to [val_inject]. *)
(*Lemma eval_binop_lessdef: forall op v1 v1' v2 v2' v m m', eval_binop op v1 v2 m = Some v -> Val.lessdef v1 v1' -> Val.lessdef v2 v2' -> Mem.extends m m' -> exists v', eval_binop op v1' v2' m' = Some v' /\ Val.lessdef v v'. *)
(** Lenb: same for binops Compatibility of [eval_binop] with respect to [val_inject]. *)
(*Lenb: fom Cminorgenproof: *)
Remark val_inject_val_of_bool:
forall f b, val_inject f (Val.of_bool b) (Val.of_bool b).
(*Lenb: fom Cminorgenproof: *)
Remark val_inject_val_of_optbool:
forall f ob, val_inject f (Val.of_optbool ob) (Val.of_optbool ob).
(*Lenb: fom Cminorgenproof: *)
Ltac TrivialExistsCMINORGEN :=
match goal with
| [ |- exists y, Some ?x = Some y /\ val_inject _ _ _ ] =>
exists x; split; [auto | try(econstructor; eauto)]
| [ |- exists y, _ /\ val_inject _ (Vint ?x) _ ] =>
exists (Vint x); split; [eauto with evalexpr | constructor]
| [ |- exists y, _ /\ val_inject _ (Vfloat ?x) _ ] =>
exists (Vfloat x); split; [eauto with evalexpr | constructor]
| [ |- exists y, _ /\ val_inject _ (Vlong ?x) _ ] =>
exists (Vlong x); split; [eauto with evalexpr | constructor]
| _ => idtac
end.
(** * 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. *)
Definition env_inject j (e1 e2: env) : Prop :=
forall id v1, e1!id = Some v1 -> exists v2, e2!id = Some v2 /\ val_inject j v1 v2.
(* Lemma set_locals_lessdef: forall e1 e2, env_lessdef e1 e2 -> forall il, env_lessdef (set_locals il e1) (set_locals il e2). Lemma set_locals_inject: forall j e1 e2, env_inject j e1 e2 -> forall il, env_inject j (set_locals il e1) (set_locals il e2).  Lemma lessdef_inject: forall v j (Hv: forall b1, getBlocks (v::nil) b1 = true -> j b1 = Some(b1,0)) v' (LD: Val.lessdef v v'), val_inject j v v'. (* Lemma inject_lessdef: forall v j (Hv: forall b1, getBlocks (v::nil) b1 = true -> j b1 = Some(b1,0)) v' (INJ: val_inject j v v'), Val.lessdef v v'. *)
(** Semantic preservation for expressions. *)
(*NEW*)
Definition sp_preserved (j:meminj) (sp sp':val) :=
exists b i b', sp = Vptr b i /\ sp' = Vptr b' i /\
j b = Some(b',0).

Lemma sel_expr_inject:
forall sp e m a v,
Cminor.eval_expr ge sp e m a v ->
forall j e' le m',
env_inject j e e' -> Mem.inject j m m' ->
forall (PG: meminj_preserves_globals ge j)
(GD: genvs_domain_eq ge tge)
(NoRepet: list_norepet (map fst (prog_defs prog)))
sp' (SP: sp_preserved j sp sp'),
exists v', CminorSel.eval_expr tge sp' e' m' le (sel_expr hf a) v' /\
val_inject j v v'.