intros.
generalize (le_h_impl_incl_syn g h H).
intro.
apply incl_syn_impl_le_h.
generalize (analysis_is_analysis g).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
intros.
elim H1.
intros.
apply H3.
split.
apply inclusion_trans with (synthesis h).
trivial.
generalize h_le_h_ana_h.
unfold le_h in |- *.
trivial.
apply stable_ana_impl_stable_syn_ana.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
tauto.
Qed.

(** if g <= h then I(g) <= I(h) *)
Theorem le_h_irr_le_h :
forall g h : hedge, le_h g h -> le_h (irreducible g) (irreducible h).