Instance ag_world: ageable world := mkAgeable _ _ _ af_world.Instance Age_world : Age_alg world.Definition den (lev: nat) (s: state) : world := (lev, (table_get (fst s), table_get (snd s))).Program Definition defined (y: var) : pred world :=
   fun w => exists v, fst (snd w) y = Some v.Definition fun_set (f: nat -> option nat) (x: nat) (y: option nat) :=
   fun i => if eq_dec i x then y else f i.Program Definition subst (x y: var) (P: pred world) : pred world :=
   fun w => P (fst w, (fun_set (fst (snd w)) x (fst (snd w) y), snd (snd w))).Program Definition mapsto (x y: var) : pred world :=
 fun w =>
    exists ax, fst (snd w) x = Some ax /\
    exists ay, fst (snd w) y = Some ay /\
    forall a, if eq_dec a ax then snd (snd w) a = Some ay else snd (snd w) a = None.Program Definition equal (x y: var) : pred world :=
            fun w => fst (snd w) x = fst (snd w) y.Inductive modvars : command -> var -> Prop :=
| mod_assign: forall x y, modvars (Assign x y) x
| mod_load: forall x y, modvars (Load x y) x
| mod_seq1: forall x c1 c2, modvars c1 x -> modvars (Seq c1 c2) x
| mod_seq2: forall x c1 c2, modvars c2 x -> modvars (Seq c1 c2) x.Inductive modvars : command -> var -> Prop :=
| mod_assign: forall x y, modvars (Assign x y) x
| mod_load: forall x y, modvars (Load x y) x
| mod_seq1: forall x c1 c2, modvars c1 x -> modvars (Seq c1 c2) x
| mod_seq2: forall x c1 c2, modvars c2 x -> modvars (Seq c1 c2) x.Inductive modvars : command -> var -> Prop :=
| mod_assign: forall x y, modvars (Assign x y) x
| mod_load: forall x y, modvars (Load x y) x
| mod_seq1: forall x c1 c2, modvars c1 x -> modvars (Seq c1 c2) x
| mod_seq2: forall x c1 c2, modvars c2 x -> modvars (Seq c1 c2) x.Inductive modvars : command -> var -> Prop :=
| mod_assign: forall x y, modvars (Assign x y) x
| mod_load: forall x y, modvars (Load x y) x
| mod_seq1: forall x c1 c2, modvars c1 x -> modvars (Seq c1 c2) x
| mod_seq2: forall x c1 c2, modvars c2 x -> modvars (Seq c1 c2) x.Inductive modvars : command -> var -> Prop :=
| mod_assign: forall x y, modvars (Assign x y) x
| mod_load: forall x y, modvars (Load x y) x
| mod_seq1: forall x c1 c2, modvars c1 x -> modvars (Seq c1 c2) x
| mod_seq2: forall x c1 c2, modvars c2 x -> modvars (Seq c1 c2) x.Definition nonfreevars (P: pred world) (x: var) : Prop :=
  forall lev stk hp v, P (lev, (stk,hp)) -> P (lev, (fun_set stk x v, hp)).Definition subset (S1 S2: var -> Prop) :=
  forall x, S1 x -> S2 x.Inductive safeN: nat -> (list command * state) -> Prop :=
| safe0: forall cs, safeN 0 cs
| safe_step: forall n cs1 cs2, step' cs1 cs2 -> safeN n cs2 -> safeN (S n) cs1
| safe_halt: forall n s, safeN (S n) (nil, s).Inductive safeN: nat -> (list command * state) -> Prop :=
| safe0: forall cs, safeN 0 cs
| safe_step: forall n cs1 cs2, step' cs1 cs2 -> safeN n cs2 -> safeN (S n) cs1
| safe_halt: forall n s, safeN (S n) (nil, s).Inductive safeN: nat -> (list command * state) -> Prop :=
| safe0: forall cs, safeN 0 cs
| safe_step: forall n cs1 cs2, step' cs1 cs2 -> safeN n cs2 -> safeN (S n) cs1
| safe_halt: forall n s, safeN (S n) (nil, s).Inductive safeN: nat -> (list command * state) -> Prop :=
| safe0: forall cs, safeN 0 cs
| safe_step: forall n cs1 cs2, step' cs1 cs2 -> safeN n cs2 -> safeN (S n) cs1
| safe_halt: forall n s, safeN (S n) (nil, s).Definition guards (P: pred world) (k: list command) : Prop :=
  forall lev s, P (den lev s) -> safeN lev (k,s).Definition semax (P: pred world) (c: command) (Q: pred world) : Prop :=
  forall F, subset (modvars c) (nonfreevars F) ->
  forall k, guards (Q*F) k -> guards (P*F) (c :: k).

Lemma frame: forall F P c Q, subset (modvars c) (nonfreevars F) -> semax P c Q -> semax (P * F) c (Q * F).