unfold stable_analysis in |- *.
unfold inclusion in |- *.
intros.
induction  H0
as
[M N H0| M1 M2 N1 N2 H0 HrecH0| M1 M2 N1 N2 H0 HrecH0| M N K L H0 HrecH0 H1];
trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaSplitL h M1 M2 N1 N2).
apply AnaInc; trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaSplitR h M1 M2 N1 N2).
apply AnaInc; trivial.
inversion HrecH0; trivial.
apply SynInc.
apply H.
apply (AnaDec h M N K L); trivial.
apply AnaInc; trivial.
generalize (synthesis_idempotent h).
unfold equal in |- *.
intros.
elim H5.
unfold inclusion in |- *.
intros.
apply H7; trivial.
Qed.

(** if g <= h then A(g) <= A(h) *)
Lemma le_h_ana_le_h :
forall g h : hedge, le_h g h -> le_h (analysis_def g) (analysis_def h).