Remark eval_builtin_1:
forall le id sg arg1 varg1 vres,
eval_expr ge sp e m le arg1 varg1 ->
builtin_implements id sg (varg1::nil) vres ->
eval_expr ge sp e m le (Ebuiltin (EF_builtin id sg) (arg1 ::: Enil)) vres.
Remark eval_builtin_2:
forall le id sg arg1 arg2 varg1 varg2 vres,
eval_expr ge sp e m le arg1 varg1 ->
eval_expr ge sp e m le arg2 varg2 ->
builtin_implements id sg (varg1::varg2::nil) vres ->
eval_expr ge sp e m le (Ebuiltin (EF_builtin id sg) (arg1 ::: arg2 ::: Enil)) vres.
Definition unary_constructor_sound (cstr: expr -> expr) (sem: val -> val) : Prop :=
forall le a x,
eval_expr ge sp e m le a x ->
exists v, eval_expr ge sp e m le (cstr a) v /\ Val.lessdef (sem x) v.
Definition binary_constructor_sound (cstr: expr -> expr -> expr) (sem: val -> val -> val) : Prop :=
forall le a x b y,
eval_expr ge sp e m le a x ->
eval_expr ge sp e m le b y ->
exists v, eval_expr ge sp e m le (cstr a b) v /\ Val.lessdef (sem x y) v.
Ltac EvalOp :=
eauto;
match goal with
| [ |- eval_exprlist _ _ _ _ _ Enil _ ] => constructor
| [ |- eval_exprlist _ _ _ _ _ (_:::_) _ ] => econstructor; EvalOp
| [ |- eval_expr _ _ _ _ _ (Eletvar _) _ ] => constructor; simpl; eauto
| [ |- eval_expr _ _ _ _ _ (Elet _ _) _ ] => econstructor; EvalOp
| [ |- eval_expr _ _ _ _ _ (lift _) _ ] => apply eval_lift; EvalOp
| [ |- eval_expr _ _ _ _ _ _ _ ] => eapply eval_Eop; [EvalOp | simpl; eauto]
| _ => idtac
end.

Theorem eval_longuofsingle:
forall le a x y,
eval_expr ge sp e m le a x ->
Val.longuofsingle x = Some y ->
exists v, eval_expr ge sp e m le (longuofsingle a) v /\ Val.lessdef y v.