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(* 02110-1301 USA                                                     *)
(** In this file, we prove properties about hedges. These properties are in section 6.1 of the paper "On Bisimulations for the Spi-Calculus" by J. Borgström and U. Nestmann. However, we consider here an extended message language. *)
(** Author: Sébastien Briais (sebastien.briais@epfl.ch) *)
Unset Standard Proposition Elimination Names.
(** We assume to have a (countably infinite) set of names. *)
Hypothesis Nam : Set.
(** We define three operators for building messages. *)
Inductive op : Set :=
| Pub : op
| Priv : op
| Hash : op.
simple destruct o1; simple destruct o2; try auto;
try (right; red in |- *; intro; inversion H).
Qed.
(** The inversion of an operator. *)
Definition inv_op (o : op) : op :=
match o with
| Pub => Priv
| Priv => Pub
| _ => o
end.
(** A message is either a name, or an encrypted message, or a pair of messages, or an operator applied to a message. *)
Inductive Msg : Set :=
| MNam : Nam -> Msg
| MEnc : Msg -> Msg -> Msg
| MPair : Msg -> Msg -> Msg
| Mop : op -> Msg -> Msg.
(** Inversion of a message. This is to manage public/private keys. *)
Definition inv (M : Msg) : Msg :=
match M with
| Mop op M => Mop (inv_op op) M
| _ => M
end.
simple induction M; simpl in |- *; trivial.
simple destruct o; simpl in |- *; trivial.
Qed.
(** A hedge is a set of pairs of message. *)
Definition hedge := Msg -> Msg -> Prop.
intros.
apply (classic (h M N)).
Qed.
(** g is included in h iff whenever (M,N) is in g then (M,N) is in h *)
Definition inclusion (g h : hedge) := forall M N : Msg, g M N -> h M N.
(** g = h iff g is included in h and h is included in g *)
Definition equal (g h : hedge) := inclusion g h /\ inclusion h g.
unfold inclusion in |- *.
trivial.
Qed.
unfold inclusion in |- *.
intros.
apply H0.
apply H; trivial.
Qed.
unfold equal in |- *.
tauto.
Qed.
unfold equal in |- *.
intro.
split; apply inclusion_refl.
Qed.
unfold equal in |- *.
intros.
elim H.
tauto.
Qed.
unfold equal in |- *.
intros.
elim H.
elim H0.
intros.
split; apply inclusion_trans with g; trivial.
Qed.
(** union and intersection of hedges *)
Definition union (g h : hedge) (M N : Msg) := g M N \/ h M N.
Definition inter (g h : hedge) (M N : Msg) := g M N /\ h M N.
unfold union in |- *.
unfold equal in |- *.
intros.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold equal in |- *.
intros.
unfold inclusion in |- *.
tauto.
Qed.
unfold union in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold union in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
unfold inclusion in |- *.
unfold union in |- *.
intros.
case H1.
apply H.
apply H0.
Qed.

(** the intersection of g and h is included into the union of g and h *)
Lemma inter_inclusion_union :
forall g h : hedge, inclusion (inter g h) (union g h).