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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.

(** The inversion is involutive. *)
Lemma inv_invol : forall M : Msg, inv (inv M) = M.