1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / Matita is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/factorization".
19 include "nat/nth_prime.ma".
21 (* the following factorization algorithm looks for the largest prime
23 definition max_prime_factor \def \lambda n:nat.
24 (max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)).
26 (* max_prime_factor is indeed a factor *)
27 theorem divides_max_prime_factor_n:
28 \forall n:nat. (S O) < n
29 \to nth_prime (max_prime_factor n) \divides n.
30 intros; apply divides_b_true_to_divides;
31 [ apply lt_O_nth_prime_n;
32 | apply (f_max_true (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n);
33 cut (\exists i. nth_prime i = smallest_factor n);
35 apply (ex_intro nat ? a);
37 [ apply (trans_le a (nth_prime a));
39 exact lt_nth_prime_n_nth_prime_Sn;
41 apply le_smallest_factor_n; ]
43 (*CSC: simplify here does something nasty! *)
44 change with (divides_b (smallest_factor n) n = true);
45 apply divides_to_divides_b_true;
46 [ apply (trans_lt ? (S O));
47 [ unfold lt; apply le_n;
48 | apply lt_SO_smallest_factor; assumption; ]
49 | letin x \def le.auto.
51 apply divides_smallest_factor_n;
52 apply (trans_lt ? (S O));
53 [ unfold lt; apply le_n;
54 | assumption; ] *) ] ]
55 | letin x \def prime. auto.
57 apply prime_to_nth_prime;
58 apply prime_smallest_factor_n;
62 theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
63 max_prime_factor n \le max_prime_factor m.
64 intros.unfold max_prime_factor.
67 apply le_max_n.apply divides_to_le.assumption.assumption.
68 change with (divides_b (nth_prime (max_prime_factor n)) m = true).
69 apply divides_to_divides_b_true.
70 cut (prime (nth_prime (max_prime_factor n))).
71 apply lt_O_nth_prime_n.apply prime_nth_prime.
72 cut (nth_prime (max_prime_factor n) \divides n).
76 [ apply (transitive_divides ? n);
77 [ apply divides_max_prime_factor_n.
81 | apply divides_b_true_to_divides;
82 [ apply lt_O_nth_prime_n.
83 | apply divides_to_divides_b_true;
84 [ apply lt_O_nth_prime_n.
85 | apply divides_max_prime_factor_n.
93 theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
94 p = max_prime_factor n \to
95 (pair nat nat q r) = p_ord n (nth_prime p) \to
96 (S O) < r \to max_prime_factor r < p.
99 cut (max_prime_factor r \lt max_prime_factor n \lor
100 max_prime_factor r = max_prime_factor n).
101 elim Hcut.assumption.
102 absurd (nth_prime (max_prime_factor n) \divides r).
104 apply divides_max_prime_factor_n.
105 assumption.unfold Not.
107 cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
108 [unfold Not in Hcut1.auto.
110 apply Hcut1.apply divides_to_mod_O;
111 [ apply lt_O_nth_prime_n.
116 cut(pair nat nat q r=p_ord_aux n n (nth_prime (max_prime_factor n)));
117 [2: rewrite < H1.assumption.|letin x \def le.auto width = 4]
118 (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
121 apply (p_ord_aux_to_not_mod_O n n ? q r);
122 [ apply lt_SO_nth_prime_n.
125 | rewrite < H1.assumption.
129 cut (n=r*(nth_prime p)\sup(q));
130 [letin www \def le.letin www1 \def divides.
133 apply (le_to_or_lt_eq (max_prime_factor r) (max_prime_factor n)).
134 apply divides_to_max_prime_factor.
135 assumption.assumption.
136 apply (witness r n ((nth_prime p) \sup q)).
140 apply (p_ord_aux_to_exp n n ? q r).
141 apply lt_O_nth_prime_n.assumption.
145 theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
146 max_prime_factor n \le p \to
147 (pair nat nat q r) = p_ord n (nth_prime p) \to
148 (S O) < r \to max_prime_factor r < p.
150 cut (max_prime_factor n < p \lor max_prime_factor n = p).
151 elim Hcut.apply (le_to_lt_to_lt ? (max_prime_factor n)).
152 apply divides_to_max_prime_factor.assumption.assumption.
153 apply (witness r n ((nth_prime p) \sup q)).
155 apply (p_ord_aux_to_exp n n).
156 apply lt_O_nth_prime_n.
157 assumption.assumption.
158 apply (p_ord_to_lt_max_prime_factor n ? q).
159 assumption.apply sym_eq.assumption.assumption.assumption.
160 apply (le_to_or_lt_eq ? p H1).
163 (* datatypes and functions *)
165 inductive nat_fact : Set \def
166 nf_last : nat \to nat_fact
167 | nf_cons : nat \to nat_fact \to nat_fact.
169 inductive nat_fact_all : Set \def
170 nfa_zero : nat_fact_all
171 | nfa_one : nat_fact_all
172 | nfa_proper : nat_fact \to nat_fact_all.
174 let rec factorize_aux p n acc \def
178 match p_ord n (nth_prime p1) with
179 [ (pair q r) \Rightarrow
180 factorize_aux p1 r (nf_cons q acc)]].
182 definition factorize : nat \to nat_fact_all \def \lambda n:nat.
184 [ O \Rightarrow nfa_zero
187 [ O \Rightarrow nfa_one
189 let p \def (max (S(S n2)) (\lambda p:nat.eqb ((S(S n2)) \mod (nth_prime p)) O)) in
190 match p_ord (S(S n2)) (nth_prime p) with
191 [ (pair q r) \Rightarrow
192 nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
194 let rec defactorize_aux f i \def
196 [ (nf_last n) \Rightarrow (nth_prime i) \sup (S n)
197 | (nf_cons n g) \Rightarrow
198 (nth_prime i) \sup n *(defactorize_aux g (S i))].
200 definition defactorize : nat_fact_all \to nat \def
201 \lambda f : nat_fact_all.
203 [ nfa_zero \Rightarrow O
204 | nfa_one \Rightarrow (S O)
205 | (nfa_proper g) \Rightarrow defactorize_aux g O].
207 theorem lt_O_defactorize_aux:
210 O < defactorize_aux f i.
214 rewrite > times_n_SO;
216 [ change with (O < nth_prime i);
217 apply lt_O_nth_prime_n;
219 change with (O < exp (nth_prime i) n);
221 apply lt_O_nth_prime_n;
222 | change with (O < defactorize_aux n1 (S i));
226 theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
227 S O < defactorize_aux f i.
228 intro.elim f.simplify.unfold lt.
229 rewrite > times_n_SO.
231 change with (S O < nth_prime i).
232 apply lt_SO_nth_prime_n.
233 change with (O < exp (nth_prime i) n).
235 apply lt_O_nth_prime_n.
237 rewrite > times_n_SO.
240 change with (O < exp (nth_prime i) n).
242 apply lt_O_nth_prime_n.
243 change with (S O < defactorize_aux n1 (S i)).
247 theorem defactorize_aux_factorize_aux :
248 \forall p,n:nat.\forall acc:nat_fact.O < n \to
249 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
250 defactorize_aux (factorize_aux p n acc) O = n*(defactorize_aux acc p).
251 intro.elim p.simplify.
252 elim H1.elim H2.rewrite > H3.
253 rewrite > sym_times. apply times_n_SO.
254 apply False_ind.apply (not_le_Sn_O (max_prime_factor n) H2).
256 (* generalizing the goal: I guess there exists a better way *)
257 cut (\forall q,r.(pair nat nat q r) = (p_ord_aux n1 n1 (nth_prime n)) \to
258 defactorize_aux match (p_ord_aux n1 n1 (nth_prime n)) with
259 [(pair q r) \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
260 n1*defactorize_aux acc (S n)).
261 apply (Hcut (fst ? ? (p_ord_aux n1 n1 (nth_prime n)))
262 (snd ? ? (p_ord_aux n1 n1 (nth_prime n)))).
263 apply sym_eq.apply eq_pair_fst_snd.
267 cut (n1 = r * (nth_prime n) \sup q).
269 simplify.rewrite < assoc_times.
270 rewrite < Hcut.reflexivity.
271 cut (O < r \lor O = r).
272 elim Hcut1.assumption.absurd (n1 = O).
273 rewrite > Hcut.rewrite < H4.reflexivity.
274 unfold Not. intro.apply (not_le_Sn_O O).
275 rewrite < H5 in \vdash (? ? %).assumption.
276 apply le_to_or_lt_eq.apply le_O_n.
277 cut ((S O) < r \lor (S O) \nlt r).
280 apply (p_ord_to_lt_max_prime_factor1 n1 ? q r).
284 apply (not_eq_O_S n).apply sym_eq.assumption.
287 assumption.assumption.
290 left.split.assumption.reflexivity.
291 intro.right.rewrite > Hcut2.
292 simplify.unfold lt.apply le_S_S.apply le_O_n.
293 cut (r < (S O) ∨ r=(S O)).
294 elim Hcut2.absurd (O=r).
295 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
298 apply (not_le_Sn_O O).
299 rewrite > Hcut3 in ⊢ (? ? %).
300 assumption.rewrite > Hcut.
301 rewrite < H6.reflexivity.
303 apply (le_to_or_lt_eq r (S O)).
304 apply not_lt_to_le.assumption.
305 apply (decidable_lt (S O) r).
307 apply (p_ord_aux_to_exp n1 n1).
308 apply lt_O_nth_prime_n.assumption.
311 theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
313 apply (nat_case n).reflexivity.
314 intro.apply (nat_case m).reflexivity.
315 intro.(*CSC: simplify here does something really nasty *)
317 (let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
318 defactorize (match p_ord (S(S m1)) (nth_prime p) with
319 [ (pair q r) \Rightarrow
320 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
322 (* generalizing the goal; find a better way *)
323 cut (\forall q,r.(pair nat nat q r) = (p_ord (S(S m1)) (nth_prime p)) \to
324 defactorize (match p_ord (S(S m1)) (nth_prime p) with
325 [ (pair q r) \Rightarrow
326 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
327 apply (Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
328 (snd ? ? (p_ord (S(S m1)) (nth_prime p)))).
329 apply sym_eq.apply eq_pair_fst_snd.
333 cut ((S(S m1)) = (nth_prime p) \sup q *r).
335 rewrite > defactorize_aux_factorize_aux.
336 (*CSC: simplify here does something really nasty *)
337 change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
338 cut ((S (pred q)) = q).
342 apply (p_ord_aux_to_exp (S(S m1))).
343 apply lt_O_nth_prime_n.
346 apply sym_eq. apply S_pred.
347 cut (O < q \lor O = q).
348 elim Hcut2.assumption.
349 absurd (nth_prime p \divides S (S m1)).
350 apply (divides_max_prime_factor_n (S (S m1))).
351 unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
353 rewrite > Hcut3 in \vdash (? (? ? %)).
354 (*CSC: simplify here does something really nasty *)
355 change with (nth_prime p \divides r \to False).
357 apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
358 apply lt_SO_nth_prime_n.
359 unfold lt.apply le_S_S.apply le_O_n.apply le_n.
361 apply divides_to_mod_O.apply lt_O_nth_prime_n.assumption.
362 rewrite > times_n_SO in \vdash (? ? ? %).
364 rewrite > (exp_n_O (nth_prime p)).
365 rewrite > H1 in \vdash (? ? ? (? (? ? %) ?)).
367 apply le_to_or_lt_eq.apply le_O_n.assumption.
368 (* e adesso l'ultimo goal. TASSI: che ora non e' piu' l'ultimo :P *)
369 cut ((S O) < r \lor S O \nlt r).
372 apply (p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r).
373 unfold lt.apply le_S_S. apply le_O_n.
375 assumption.assumption.
378 left.split.assumption.reflexivity.
379 intro.right.rewrite > Hcut3.
380 simplify.unfold lt.apply le_S_S.apply le_O_n.
381 cut (r \lt (S O) \or r=(S O)).
382 elim Hcut3.absurd (O=r).
383 apply le_n_O_to_eq.apply le_S_S_to_le.exact H2.
385 apply (not_le_Sn_O O).
386 rewrite > H3 in \vdash (? ? %).assumption.assumption.
387 apply (le_to_or_lt_eq r (S O)).
388 apply not_lt_to_le.assumption.
389 apply (decidable_lt (S O) r).
391 cut (O < r \lor O = r).
392 elim Hcut1.assumption.
394 apply (not_eq_O_S (S m1)).
395 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
396 apply le_to_or_lt_eq.apply le_O_n.
399 apply (p_ord_aux_to_exp (S(S m1))).
400 apply lt_O_nth_prime_n.
407 [ (nf_last n) \Rightarrow O
408 | (nf_cons n g) \Rightarrow S (max_p g)].
410 let rec max_p_exponent f \def
412 [ (nf_last n) \Rightarrow n
413 | (nf_cons n g) \Rightarrow max_p_exponent g].
415 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat.
416 nth_prime ((max_p f)+i) \divides defactorize_aux f i.
418 elim f.simplify.apply (witness ? ? ((nth_prime i) \sup n)).
421 (nth_prime (S(max_p n1)+i) \divides
422 (nth_prime i) \sup n *(defactorize_aux n1 (S i))).
426 rewrite > assoc_times.
428 apply (witness ? ? (n2* (nth_prime i) \sup n)).
432 theorem divides_exp_to_divides:
433 \forall p,n,m:nat. prime p \to
434 p \divides n \sup m \to p \divides n.
435 intros 3.elim m.simplify in H1.
436 apply (transitive_divides p (S O)).assumption.
438 cut (p \divides n \lor p \divides n \sup n1).
439 elim Hcut.assumption.
440 apply H.assumption.assumption.
441 apply divides_times_to_divides.assumption.
445 theorem divides_exp_to_eq:
446 \forall p,q,m:nat. prime p \to prime q \to
447 p \divides q \sup m \to p = q.
451 apply (divides_exp_to_divides p q m).
452 assumption.assumption.
453 unfold prime in H.elim H.assumption.
456 theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
457 i < j \to nth_prime i \ndivides defactorize_aux f j.
460 (nth_prime i \divides (nth_prime j) \sup (S n) \to False).
461 intro.absurd ((nth_prime i) = (nth_prime j)).
462 apply (divides_exp_to_eq ? ? (S n)).
463 apply prime_nth_prime.apply prime_nth_prime.
464 assumption.unfold Not.
466 apply (not_le_Sn_n i).rewrite > Hcut in \vdash (? ? %).assumption.
467 apply (injective_nth_prime ? ? H2).
470 cut (nth_prime i \divides (nth_prime j) \sup n
471 \lor nth_prime i \divides defactorize_aux n1 (S j)).
473 absurd ((nth_prime i) = (nth_prime j)).
474 apply (divides_exp_to_eq ? ? n).
475 apply prime_nth_prime.apply prime_nth_prime.
476 assumption.unfold Not.
479 apply (not_le_Sn_n i).rewrite > Hcut1 in \vdash (? ? %).assumption.
480 apply (injective_nth_prime ? ? H4).
482 apply (trans_lt ? j).assumption.unfold lt.apply le_n.
484 apply divides_times_to_divides.
485 apply prime_nth_prime.assumption.
488 lemma not_eq_nf_last_nf_cons: \forall g:nat_fact.\forall n,m,i:nat.
489 \lnot (defactorize_aux (nf_last n) i= defactorize_aux (nf_cons m g) i).
492 (exp (nth_prime i) (S n) = defactorize_aux (nf_cons m g) i \to False).
494 cut (S(max_p g)+i= i).
495 apply (not_le_Sn_n i).
496 rewrite < Hcut in \vdash (? ? %).
497 simplify.apply le_S_S.
499 apply injective_nth_prime.
500 apply (divides_exp_to_eq ? ? (S n)).
501 apply prime_nth_prime.apply prime_nth_prime.
503 change with (divides (nth_prime ((max_p (nf_cons m g))+i))
504 (defactorize_aux (nf_cons m g) i)).
505 apply divides_max_p_defactorize.
508 lemma not_eq_nf_cons_O_nf_cons: \forall f,g:nat_fact.\forall n,i:nat.
509 \lnot (defactorize_aux (nf_cons O f) i= defactorize_aux (nf_cons (S n) g) i).
511 simplify.unfold Not.rewrite < plus_n_O.
513 apply (not_divides_defactorize_aux f i (S i) ?).
514 unfold lt.apply le_n.
516 rewrite > assoc_times.
517 apply (witness ? ? ((exp (nth_prime i) n)*(defactorize_aux g (S i)))).
521 theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
522 defactorize_aux f i = defactorize_aux g i \to f = g.
525 generalize in match H.
528 apply inj_S. apply (inj_exp_r (nth_prime i)).
529 apply lt_SO_nth_prime_n.
532 apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
533 generalize in match H1.
536 apply (not_eq_nf_last_nf_cons n1 n2 n i).
537 apply sym_eq. assumption.
539 generalize in match H3.
540 apply (nat_elim2 (\lambda n,n2.
541 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
542 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
543 nf_cons n n1 = nf_cons n2 n3)).
549 rewrite > (plus_n_O (defactorize_aux n3 (S i))).assumption.
551 apply (not_eq_nf_cons_O_nf_cons n1 n3 n5 i).assumption.
554 apply (not_eq_nf_cons_O_nf_cons n3 n1 n4 i).
555 apply sym_eq.assumption.
557 cut (nf_cons n4 n1 = nf_cons m n3).
560 rewrite > Hcut1.rewrite > Hcut2.reflexivity.
562 (match nf_cons n4 n1 with
563 [ (nf_last m) \Rightarrow n1
564 | (nf_cons m g) \Rightarrow g ] = n3).
565 rewrite > Hcut.simplify.reflexivity.
567 (match nf_cons n4 n1 with
568 [ (nf_last m) \Rightarrow m
569 | (nf_cons m g) \Rightarrow m ] = m).
570 rewrite > Hcut.simplify.reflexivity.
571 apply H4.simplify in H5.
572 apply (inj_times_r1 (nth_prime i)).
573 apply lt_O_nth_prime_n.
574 rewrite < assoc_times.rewrite < assoc_times.assumption.
577 theorem injective_defactorize_aux: \forall i:nat.
578 injective nat_fact nat (\lambda f.defactorize_aux f i).
581 apply (eq_defactorize_aux_to_eq x y i H).
584 theorem injective_defactorize:
585 injective nat_fact_all nat defactorize.
587 change with (\forall f,g.defactorize f = defactorize g \to f=g).
589 generalize in match H.elim g.
595 apply (not_eq_O_S O H1).
599 apply (not_le_Sn_n O).
600 rewrite > H1 in \vdash (? ? %).
601 change with (O < defactorize_aux n O).
602 apply lt_O_defactorize_aux.
603 generalize in match H.
608 apply (not_eq_O_S O).apply sym_eq. assumption.
614 apply (not_le_Sn_n (S O)).
615 rewrite > H1 in \vdash (? ? %).
616 change with ((S O) < defactorize_aux n O).
617 apply lt_SO_defactorize_aux.
618 generalize in match H.elim g.
622 apply (not_le_Sn_n O).
623 rewrite < H1 in \vdash (? ? %).
624 change with (O < defactorize_aux n O).
625 apply lt_O_defactorize_aux.
629 apply (not_le_Sn_n (S O)).
630 rewrite < H1 in \vdash (? ? %).
631 change with ((S O) < defactorize_aux n O).
632 apply lt_SO_defactorize_aux.
633 (* proper - proper *)
635 apply (injective_defactorize_aux O).
639 theorem factorize_defactorize:
640 \forall f,g: nat_fact_all. factorize (defactorize f) = f.
642 apply injective_defactorize.
643 apply defactorize_factorize.