unfold equiv_h in |- *.
unfold inclusion in |- *.
intros.
elim H.
intros.
generalize (le_h_impl_incl_syn (reduce g) h H1).
unfold inclusion in |- *.
intros.
generalize (H3 M N H0).
generalize H0.
unfold reduce in |- *.
case M.
intros.
apply nam_left_syn; trivial.
case N.
intros.
apply nam_right_syn; trivial.
intros.
elim H4.
intros.
inversion H5.
trivial.
case H7.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
case N.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
elim H4.
intros.
case H13.
intro.
cut False.
tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
cut False; try tauto.
apply H14.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *; auto.
intros.
inversion H5; trivial.
case N.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
inversion H5; trivial.
intros.
elim H4.
intros.
case (op_eq_dec o0 o).
intro.
rewrite e.
rewrite e in H5.
inversion H5; trivial.
case H7.
intro.
tauto.
intro.
cut False; try tauto.
apply H12.
generalize
(le_h_trans h (reduce g) g H2
(inclusion_impl_le_h (reduce g) g (red_incl_h g))).
unfold le_h in |- *.
unfold inclusion in |- *.
auto.
intro.
inversion H5; trivial.
tauto.
Qed.

(** if g <= h then (reduce g) <= (reduce h) *)
Lemma red_stable_le_h :
forall g h : hedge, le_h g h -> le_h (reduce g) (reduce h).