unfold equal in |- *.
intros.
split.
apply inclusion_le_h_reduce_inclusion.
tauto.
apply inclusion_impl_le_h; tauto.
apply inclusion_le_h_reduce_inclusion; try tauto.
apply inclusion_impl_le_h; tauto.
Qed.

(** if (reduce g) is included in h and h <= (reduce g) then (reduce g) is included in (reduce h) *)
Lemma inclusion_le_h_impl_inclusion_red :
forall g h : hedge,
inclusion (reduce g) h ->
le_h h (reduce g) -> inclusion (reduce g) (reduce h).