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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.
intros.
unfold union in |- *.
unfold inter in |- *.
unfold inclusion in |- *.
tauto.
Qed.
(** The synthesis S(h) of h : it contains all the messages that can be synthetised by the hedge. *)
Inductive synthesis (h : hedge) : hedge :=
| SynInc : forall M N : Msg, h M N -> synthesis h M N
| SynEnc :
forall M N K L : Msg,
synthesis h M N -> synthesis h K L -> synthesis h (MEnc M K) (MEnc N L)
| SynPair :
forall M1 N1 M2 N2 : Msg,
synthesis h M1 N1 ->
synthesis h M2 N2 -> synthesis h (MPair M1 M2) (MPair N1 N2)
| SynOp :
forall (o : op) (M N : Msg),
synthesis h M N -> synthesis h (Mop o M) (Mop o N).
intros.
inversion H.
trivial.
Qed.
intros.
inversion H.
trivial.
Qed.
unfold inclusion in |- *.
intros.
apply SynInc.
trivial.
Qed.
(** g <= h if S(g) is included in S(h) *)
Definition le_h (g h : hedge) := inclusion (synthesis g) (synthesis h).
(** g ~ h if g <= h and h <= g *)
Definition equiv_h (g h : hedge) := le_h g h /\ le_h h g.
unfold le_h in |- *.
intro.
apply inclusion_refl.
Qed.
unfold le_h in |- *.
intros.
apply inclusion_trans with (synthesis g); trivial.
Qed.
unfold equiv_h in |- *.
tauto.
Qed.
unfold equiv_h in |- *.
intro.
split; apply le_h_refl.
Qed.
unfold equiv_h in |- *.
intros.
elim H.
elim H0.
intros.
split; apply le_h_trans with g; trivial.
Qed.
unfold equiv_h in |- *.
tauto.
Qed.
unfold le_h in |- *.
intros.
apply inclusion_trans with (synthesis g); trivial.
apply h_incl_syn.
Qed.
unfold le_h in |- *.
unfold inclusion in |- *.
intros.
induction  H0
as
[M
N
H0|
M
N
K
L
H0_1
HrecH0_1
H0_0
HrecH0_0|
M1
N1
M2
N2
H0_1
HrecH0_1
H0_0
HrecH0_0|
o
M
N
H0
HrecH0].
apply H; trivial.
apply SynEnc; trivial.
apply SynPair; trivial.
apply SynOp; trivial.
Qed.
split.
apply le_h_impl_incl_syn.
apply incl_syn_impl_le_h.
Qed.
intros.
apply incl_syn_impl_le_h.
apply inclusion_trans with h; trivial.
apply h_incl_syn.
Qed.
unfold equal in |- *.
unfold equiv_h in |- *.
intros.
elim H.
intros.
split.
apply inclusion_impl_le_h; trivial.
apply inclusion_impl_le_h; trivial.
Qed.
intros.
apply incl_syn_impl_le_h.
apply union_intro; apply le_h_impl_incl_syn; trivial.
Qed.

(** if g1 <= h1 and g2 <= h2 then (union g1 g2) <= (union h1 h2) *)
Theorem le_h_le_h_impl_le_h_union :
forall g1 g2 h1 h2 : hedge,
le_h g1 h1 -> le_h g2 h2 -> le_h (union g1 g2) (union h1 h2).