intros A v N_dans_A_X.
unfold cut2, cut in |- *.
elim (cut_spec (cons A v)).
intros a temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inmon_a temp; elim temp; clear temp.
intros App_eg spec_or.
cut (inmonoid X a).
cut (Append a b = cons A v).
clear App_eg inmon_a.
generalize a; clear a; simple induction a.
auto.
intros x w Hyp App_eg_x_w inmon.
absurd (dans A X).
assumption.
replace A with x.
apply inmonoid_cons_inv2 with w.
assumption.
apply cons_cons_inv1 with (Append w b) v.
assumption.
assumption.
assumption.
Qed.

Lemma Deriveg_imp_Deriveg_cut :
forall x y : Word,
Deriveg X R x y ->
ex2 (fun w : Word => Append (cut1 x) w = y)
(fun w : Word => Deriveg X R (cut2 x) w).