intros.
apply inclusion_trans with (reduce h).
generalize (cond_irr_impl_stable_reduce h H).
unfold equal in |- *; tauto.
apply red_equiv_impl_red_incl.
apply equiv_h_trans with h.
apply equiv_h_sym.
apply h_equiv_h_red.
apply cond_irr_impl_equiv_h_irr.
trivial.
Qed.

(** if h satisfies cond_irr then h is irreducible *)
Lemma cond_irr_impl_irreducible :
forall h : hedge, cond_irr h -> is_irreducible h.