Definition func_at (f: funspec): address -> pred rmap :=
  match f with
   | mk_funspec fsig cc A P Q _ _ => pureat (SomeP (SpecTT A) (packPQ P Q)) (FUN fsig cc)
  end.Definition NDmk_funspec (f: funsig) (cc: calling_convention)
  (A: Type) (Pre Post: A -> environ -> mpred): funspec :=
  mk_funspec f cc (rmaps.ConstType A) (fun _ => Pre) (fun _ => Post)
             (const_super_non_expansive _ _) (const_super_non_expansive _ _).Definition closed_wrt_vars {B} (S: ident -> Prop) (F: environ -> B) : Prop :=
  forall rho te',
     (forall i, S i \/ Map.get (te_of rho) i = Map.get te' i) ->
     F rho = F (mkEnviron (ge_of rho) (ve_of rho) te').Definition closed_wrt_lvars {B} (S: ident -> Prop) (F: environ -> B) : Prop :=
  forall rho ve',
     (forall i, S i \/ Map.get (ve_of rho) i = Map.get ve' i) ->
     F rho = F (mkEnviron (ge_of rho) ve' (te_of rho)).Definition not_a_param (params: list (ident * type)) (i : ident) : Prop :=
  ~ In i (map (@fst _ _) params).Definition is_a_local (vars: list (ident * type)) (i: ident) : Prop :=
  In  i (map (@fst _ _) vars) .Fixpoint sepcon_list {A}{JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{AG: ageable A} {AgeA: Age_alg A}
   (p: list (pred A)) : pred A :=
 match p with nil => emp | h::t => h * sepcon_list t end.Definition typed_true (t: type) (v: val)  : Prop := strict_bool_val v t
= Some true.Definition typed_false (t: type)(v: val) : Prop := strict_bool_val v t =
                                                   Some false.Definition subst {A} (x: ident) (v: val) (P: environ -> A) : environ -> A :=
  fun s => P (env_set s x v).

(* Specifications: *)
Lemma approx_sepcon: forall (P Q: mpred) n, approx n (P * Q) = approx n P * approx n Q.
Lemma approx_andp: forall (P Q: mpred) n, approx n (P && Q) = approx n P && approx n Q.
Lemma approx_jam {B: Type} {S': B -> Prop} (S: forall l, {S' l}+{~ S' l}) (P Q: B -> mpred) n (b : B) : approx n (jam S P Q b) = jam S (approx n oo P) (approx n oo Q) b.
Lemma approx_func_ptr: forall (A: Type) fsig0 cc (P Q: A -> environ -> mpred) (v: val) (n: nat), approx n (func_ptr (NDmk_funspec fsig0 cc A P Q) v) = approx n (func_ptr (NDmk_funspec fsig0 cc A (fun a rho => approx n (P a rho)) (fun a rho => approx n (Q a rho))) v).