split.
unfold analysis_cond in |- *.
split.
unfold inclusion in |- *.
intros.
unfold analysis_def in |- *.
exists 0.
simpl in |- *.
trivial.
unfold stable_analysis in |- *.
unfold inclusion in |- *.
unfold analysis_def in |- *.
intros.
induction  H
as [M N H| M1 M2 N1 N2 H HrecH| M1 M2 N1 N2 H HrecH| M N K L H HrecH H0].
trivial.
elim HrecH.
intro n.
intro.
exists (S n).
simpl in |- *.
apply (AnaSplitL (analysis_seq h n) M1 M2 N1 N2).
apply AnaInc.
trivial.
elim HrecH.
intro n.
intro.
exists (S n).
apply (AnaSplitR (analysis_seq h n) M1 M2 N1 N2).
apply AnaInc; trivial.
elim HrecH.
intro n1.
intro.
generalize (synthesis_analysis_seq h (inv K) (inv L) H0).
intro.
elim H2.
intro n2.
intro.
exists (S (max n1 n2)).
simpl in |- *.
apply (AnaDec (analysis_seq h (max n1 n2)) M N K L).
apply AnaInc.
generalize (analysis_seq_increase n1 (max n1 n2) h (le_max_l n1 n2)).
unfold inclusion in |- *; auto.
generalize
(inclusion_impl_le_h (analysis_seq h n2) (analysis_seq h (max n1 n2))
(analysis_seq_increase n2 (max n1 n2) h (le_max_r n1 n2))).
unfold le_h in |- *; auto.
intros.
unfold inclusion in |- *.
unfold analysis_def in |- *.
intros.
elim H0.
intro n.
intro.
generalize (analysis_seq_incl_analysis_cond h h' H n).
unfold inclusion in |- *; auto.
Qed.
(** the irreducibles I(h) of h is (reduce (A(h))) *)
Definition irreducible (h : hedge) := reduce (analysis_def h).

(** here follows three lemma that make clear the definition of the irreducibles of h *)
Lemma irreducible_enc :
forall (h : hedge) (M N K L : Msg),
irreducible h (MEnc M K) (MEnc N L) <->
analysis_def h (MEnc M K) (MEnc N L) /\
(~ synthesis (analysis_def h) M N \/ ~ synthesis (analysis_def h) K L).