unfold irreducible in |- *.
intro.
apply inclusion_le_h_reduce_inclusion.
replace (reduce (analysis_def h)) with (irreducible h).
apply ana_irr_incl_ana.
unfold irreducible in |- *.
trivial.
replace (reduce (analysis_def h)) with (irreducible h).
apply le_h_trans with (irreducible h).
generalize ana_h_equiv_irr_h.
unfold equiv_h in |- *.
intros.
elim (H h); tauto.
apply h_le_h_ana_h.
unfold irreducible in |- *.
trivial.
Qed.

(* I(h) is included in I(I(h)) *)
Lemma irr_incl_irr_irr :
forall h : hedge, inclusion (irreducible h) (irreducible (irreducible h)).