unfold cond_irr in |- *.
intros.
apply incl_syn_impl_le_h.
generalize (analysis_is_analysis h).
unfold is_analysis in |- *.
intro.
elim H0.
unfold analysis_cond in |- *.
intros.
apply H2.
split.
unfold inclusion in |- *.
intros.
apply SynInc.
trivial.
unfold stable_analysis in |- *.
unfold inclusion in |- *.
intros.
induction  H3
as
[M N H3| M1 M2 N1 N2 H3 HrecH3| M1 M2 N1 N2 H3 HrecH3| M N K L H3 HrecH3 H4];
trivial.
inversion HrecH3; trivial.
unfold no_pairs in H.
elim H.
intros.
generalize (H7 M1 M2 N1 N2).
tauto.
inversion HrecH3; trivial.
unfold no_pairs in H.
elim H.
intros.
generalize (H7 M1 M2 N1 N2).
tauto.
generalize (synthesis_idempotent h).
unfold equal in |- *.
intros.
elim H5.
intros.
unfold inclusion in H7.
generalize (H7 (inv K) (inv L) H4).
intro.
unfold cyph_dec in H.
elim H.
intros.
elim H10.
intros.
elim H12.
intros.
inversion HrecH3; trivial.
apply (H14 M N K L); trivial.
Qed.

Lemma cond_irr_impl_equiv_h_irr :
forall h : hedge, cond_irr h -> equiv_h h (irreducible h).