case (prime_dec n);intro.
exists n;exists 1;exists 1;simpl.
split;try tauto.
split;try omega.
split;try ring.
split;try omega.
apply gcd_sym;apply gcd_one.
elim (not_prime_impl_prime_divides n n0);intro;try tauto.
elim a;intro p;intro.
elim p0;intros.
elim (nat_factor n p H1).
intro m;intro.
elim p1;intros.
exists p;exists m;exists (quo n (power p m) H3).
split;trivial.
split.
destruct m;try omega.
elim H4;simpl;rewrite mult_comm;simpl;rewrite plus_comm;simpl;trivial.
split.
apply (quo_is_quo n (power p m) H3).
generalize (gcd_is_gcd p (quo n (power p m) H3));intro.
assert ((gcd p (quo n (power p m) H3))=1).
case (prime_gcd (gcd p (quo n (power p m) H3)) p (quo n (power p m) H3));trivial.
apply gcd_sym;trivial.
intro.
rewrite H6 in H5;elim H5;intros.
elim H7;intros.
elim H4;rewrite plus_comm;simpl.
generalize (quo_is_quo n (power p m) H3);intro.
elim H10;intros.
rewrite H12 in H11.
exists x;rewrite H11;ring.
rewrite H6 in H5;trivial.
generalize (quo_is_quo n (power p m) H3);intro.
split;trivial.
rewrite H7;rewrite mult_comm;apply mult_lemma3.
intro.
apply H;rewrite H7.
rewrite H8;ring.
red;apply power_lt.
destruct p.
elim (not_prime_zero H1).
destruct p;try omega.
elim H1;tauto.
destruct m;try omega.
elim H4;simpl;rewrite mult_comm;simpl;rewrite plus_comm;simpl;trivial.
trivial.
Qed.
(** this compute p1^n1*p2^n2*...*pm^nm if l = (p1,n1)::(p2,n2)::...::(pm,nm) *)
Fixpoint refactor (l:(list (nat*nat))) {struct l} : nat :=
match l with
nil => 1
| (cons (p,n) tail) => (power p n)*(refactor tail)
end.
(** a list l is well-formed (is a factorisation) if whenever (p,n) is in l then p is prime and n>0 and p is relatively prime to (refactor (l\(p,n))) and l\(p,n) is well-formed *)
Inductive is_wf : (list (nat*nat))->Prop :=
nil_is_wf : (is_wf nil)
|cons_is_wf : forall (p n:nat)(tail:(list (nat*nat))),(is_prime p)->(n>0)->(is_wf tail)->(rel_prime p (refactor tail))->(is_wf (cons (p,n) tail)).

(** this gives an inefficient algorithm to compute a factorisation of n *)
Lemma factorisation : forall (n:nat),{l:(list (nat*nat)) | (is_wf l)/\n=(refactor l)}+{n=0}.
intro.