intros a w dans_a_X.
unfold cut1, cut in |- *.
elim (cut_spec w).
intros x temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inmon_X_x temp; elim temp; clear temp.
elim (cut_spec (cons a w)).
intros x' temp; elim temp; clear temp.
intros b' temp; elim temp; clear temp.
intros inmon_X_x' temp; elim temp; clear temp.
intros App_eg' spec' App_eg spec.
simpl in |- *.
cut (inmonoid X x').
cut (Append x' b' = cons a w).
clear inmon_X_x' App_eg'.
generalize x'; clear x'; simple induction x'.
intro App_nil_b'_eg_cons_a_w.
absurd (dans a X).
elim spec'.
intro b'_nil.
absurd (cons a w = nil).
discriminate.
rewrite <- b'_nil.
rewrite <- App_nil_b'_eg_cons_a_w.
trivial.
intro temp; elim temp; clear temp.
intros x0 N_dans_x0_X temp; elim temp; clear temp.
intros w1 eg_b'_cons.
replace a with x0.
assumption.
apply cons_cons_inv1 with w1 w.
rewrite <- eg_b'_cons.
assumption.
assumption.
intros x0 w0 Hyp App_eg_cons inmon_cons.
apply cons_cons.
apply cons_cons_inv1 with (Append w0 b') w.
assumption.
apply cut_unicite with w b' b.
split.
apply inmonoid_cons_inv with x0; assumption.
split.
apply cons_cons_inv2 with x0 a.
assumption.
assumption.
split; auto.
assumption.
assumption.
Qed.

Lemma cut1_Append :
forall u v : Word, inmonoid X u -> cut1 (Append u v) = Append u (cut1 v).