Definition world := (env var adr * env adr adr)%type.Instance Join_world: Join world := _.Instance Perm_world : Perm_alg world := _.Instance Sep_world : Perm_alg world := _.Instance Canc_world : Perm_alg world := _.Instance Disj_world : Perm_alg world := _.Fixpoint den_env {key}{A}{KE: EqDec key} (rho: table key A)  : env key A :=
  match rho with
  | (x,v)::xs => env_set x v (den_env xs)
  | nil => empty_env
  end.Definition den (s: state) : world := (den_env (fst s), den_env (snd s)).Definition world_op (P: pred (env var adr)) (Q: pred (env adr adr)) : pred world :=
    fun w => P (fst w) /\ Q (snd w).Notation "x '=#' v" := (eval_var x v) (at level 25, no associativity): pred.Definition world_op_sepcon:
     forall P P' Q Q',
       (world_op P P' * world_op Q Q' = world_op (P*Q) (P'*Q'))%pred.Definition subst (x: var) (v: option (pshare * adr)) (P: pred world) : pred world :=
   fun w => P (env_set_sh x v (fst w), snd w).(*
Definition equal (x y: var) : pred world :=
            fun w => env_get (fst w) x = env_get (fst 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.(*
Definition equal (x y: var) : pred world :=
            fun w => env_get (fst w) x = env_get (fst 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.(*
Definition equal (x y: var) : pred world :=
            fun w => env_get (fst w) x = env_get (fst 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.(*
Definition equal (x y: var) : pred world :=
            fun w => env_get (fst w) x = env_get (fst 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.(*
Definition equal (x y: var) : pred world :=
            fun w => env_get (fst w) x = env_get (fst 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.Definition subset (S1 S2: var -> Prop) :=
  forall x, S1 x -> S2 x.Definition semax (P: pred world) (c: command) (Q: pred world) : Prop :=
  forall F s,
    (P*F)%pred (den s) -> exists s', exec c s = Some s' /\ (Q*F)%pred (den s').

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