unfold inclusion in |- *.
intro.
simple induction M.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H.
tauto.
intro M'.
intro.
intro K.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1.
tauto.
intros N' L.
intros.
case (hedge_mem_dec (synthesis h) M' N').
case (hedge_mem_dec (synthesis h) K L).
intros.
apply SynEnc.
apply H; trivial.
apply H0; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1.
split; tauto.
tauto.
intro.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intro M1.
intro.
intro M2.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros N1 N2.
intro.
case (hedge_mem_dec (synthesis h) M1 N1).
case (hedge_mem_dec (synthesis h) M2 N2).
intros.
apply SynPair.
apply H; trivial.
apply H0; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intro.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H1; tauto.
intros oM M'.
intro.
simple destruct N.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros oN N'.
intro.
case (op_eq_dec oM oN).
intro.
rewrite e in H0.
rewrite e.
case (hedge_mem_dec (synthesis h) M' N').
intro.
apply SynOp.
apply H; trivial.
intros.
apply SynInc.
unfold reduce in |- *; simpl in |- *.
inversion H0; tauto.
intros.
apply SynInc.
unfold reduce in |- *.
inversion H0; tauto.
Qed.

(** (reduce h) ~ h *)
Lemma h_equiv_h_red : forall h : hedge, equiv_h h (reduce h).
unfold equiv_h in |- *.