Inductive localdef : Type :=
 | temp: ident -> val -> localdef
 | lvar: ident -> type -> val -> localdef   (* local variable *)
 | gvars: globals -> localdef.Inductive localdef : Type :=
 | temp: ident -> val -> localdef
 | lvar: ident -> type -> val -> localdef   (* local variable *)
 | gvars: globals -> localdef.Inductive localdef : Type :=
 | temp: ident -> val -> localdef
 | lvar: ident -> type -> val -> localdef   (* local variable *)
 | gvars: globals -> localdef.Inductive localdef : Type :=
 | temp: ident -> val -> localdef
 | lvar: ident -> type -> val -> localdef   (* local variable *)
 | gvars: globals -> localdef.Definition lvar_denote (i: ident) (t: type) (v: val) rho :=
     match Map.get (ve_of rho) i with
         | Some (b, ty') => t=ty' /\ v = Vptr b Ptrofs.zero
         | None => False
         end.Definition gvars_denote (gv: globals) rho :=
   gv = (fun i => match Map.get (ge_of rho) i with Some b => Vptr b Ptrofs.zero | None => Vundef end).Fixpoint fold_right_andp rho (l: list (environ -> Prop)) : Prop :=
 match l with
 | nil => True
 | b::nil => b rho
 | b::r => b rho /\ fold_right_andp rho r
 end.Definition PROPx (P: list Prop): forall (Q: environ->mpred), environ->mpred :=
     andp (prop (fold_right and True P)).Notation " 'ENTAIL' d ',' P '|--' Q " :=
  (@derives (environ->mpred) _ (andp (local (tc_environ d)) P%assert) Q%assert) (at level 80, P at level 79, Q at level 79).Definition SEPx (R: list mpred) : environ->mpred :=
    fun _ => (fold_right_sepcon R).

Lemma LOCALx_nonexpansive: forall Q R rho, nonexpansive (fun S => R S rho) -> nonexpansive (fun S => LOCALx Q (R S) rho).