case (eq_nat_dec n 0);intro.
right;trivial.
case (eq_nat_dec n 1).
intro;left;exists (nil (A:=nat*nat)).
split;[apply nil_is_wf | simpl;trivial].
generalize n0.
apply (lt_wf_rec n (fun n:nat => n<>0 -> n <> 1 -> {l : list (nat * nat) | is_wf l /\ n = refactor l}+{n=0}));intros.
elim (nat_factor_prime n1 H0 H1).
intro p;intro.
elim p0;intro m;intro.
elim p1;intro q;intro.
elim p2;intros.
elim H3;intros.
elim H5;intros.
elim H7;intros.
case (eq_nat_dec q 1);intro.
left;exists (cons (p,m) nil);simpl;rewrite e in H6.
split;trivial.
apply cons_is_wf;auto;try (apply nil_is_wf).
unfold rel_prime;simpl;rewrite e in H8;trivial.
assert (q<>0).
intro;rewrite H10 in H6;rewrite mult_comm in H6;simpl in H6;auto.
elim (H q H9 H10 n2).
intro.
elim a;intro l;intro.
elim p3;intros.
left;exists (cons (p,m) l);simpl;rewrite H12 in H6;split;trivial.
apply cons_is_wf;auto.
rewrite <- H12;unfold rel_prime;trivial.
intro;tauto.
Qed.

(* Specifications: *)
(** we now show that a factorisation is unique upto permutations *)
(** if (p,n) is in l then p^n | (refactor l) *)
Lemma factor_divides_refactor : forall (x:nat*nat)(l:list (nat*nat)),(In x l)->(divides (refactor l) (power (fst x) (snd x))).
induction l;simpl;try tauto.