unfold is_irreducible in |- *.
unfold equal in |- *.
intros.
split.
apply cond_irr_impl_h_incl_irr; trivial.
apply cond_irr_impl_irr_incl_h; trivial.
Qed.

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