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).

(** If (M,N) is in S(h) and M is a name then (M,N) is in h *)
Theorem nam_left_syn :
forall (h : hedge) (a : Nam) (N : Msg),
synthesis h (MNam a) N -> h (MNam a) N.