unfold irreducible in |- *.
intros.
apply inclusion_le_h_impl_inclusion_red.
generalize (analysis_is_analysis (reduce (analysis_def h))).
unfold is_analysis in |- *.
unfold analysis_cond in |- *.
tauto.
apply le_h_trans with (analysis_def h).
apply inclusion_impl_le_h.
fold (irreducible h) in |- *.
apply ana_irr_incl_ana.
fold (irreducible h) in |- *.
generalize ana_h_equiv_irr_h.
unfold equiv_h in |- *.
intros.
elim (H h); auto.
Qed.

(** I(h) is irreducible *)
Theorem irreducible_is_irreducible :
forall h : hedge, is_irreducible (irreducible h).