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 new.
51 apply divides_smallest_factor_n;
52 apply (trans_lt ? (S O));
53 [ unfold lt; apply le_n;
54 | assumption; ] *) ] ]
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 new.
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 new]
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 apply (le_to_or_lt_eq (max_prime_factor r) (max_prime_factor n)).
130 apply divides_to_max_prime_factor.
131 assumption.assumption.
132 apply (witness r n ((nth_prime p) \sup q)).
134 apply (p_ord_aux_to_exp n n ? q r).
135 apply lt_O_nth_prime_n.assumption.
138 theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
139 max_prime_factor n \le p \to
140 (pair nat nat q r) = p_ord n (nth_prime p) \to
141 (S O) < r \to max_prime_factor r < p.
143 cut (max_prime_factor n < p \lor max_prime_factor n = p).
144 elim Hcut.apply (le_to_lt_to_lt ? (max_prime_factor n)).
145 apply divides_to_max_prime_factor.assumption.assumption.
146 apply (witness r n ((nth_prime p) \sup q)).
148 apply (p_ord_aux_to_exp n n).
149 apply lt_O_nth_prime_n.
150 assumption.assumption.
151 apply (p_ord_to_lt_max_prime_factor n ? q).
152 assumption.apply sym_eq.assumption.assumption.assumption.
153 apply (le_to_or_lt_eq ? p H1).
156 (* datatypes and functions *)
158 inductive nat_fact : Set \def
159 nf_last : nat \to nat_fact
160 | nf_cons : nat \to nat_fact \to nat_fact.
162 inductive nat_fact_all : Set \def
163 nfa_zero : nat_fact_all
164 | nfa_one : nat_fact_all
165 | nfa_proper : nat_fact \to nat_fact_all.
167 let rec factorize_aux p n acc \def
171 match p_ord n (nth_prime p1) with
172 [ (pair q r) \Rightarrow
173 factorize_aux p1 r (nf_cons q acc)]].
175 definition factorize : nat \to nat_fact_all \def \lambda n:nat.
177 [ O \Rightarrow nfa_zero
180 [ O \Rightarrow nfa_one
182 let p \def (max (S(S n2)) (\lambda p:nat.eqb ((S(S n2)) \mod (nth_prime p)) O)) in
183 match p_ord (S(S n2)) (nth_prime p) with
184 [ (pair q r) \Rightarrow
185 nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
187 let rec defactorize_aux f i \def
189 [ (nf_last n) \Rightarrow (nth_prime i) \sup (S n)
190 | (nf_cons n g) \Rightarrow
191 (nth_prime i) \sup n *(defactorize_aux g (S i))].
193 definition defactorize : nat_fact_all \to nat \def
194 \lambda f : nat_fact_all.
196 [ nfa_zero \Rightarrow O
197 | nfa_one \Rightarrow (S O)
198 | (nfa_proper g) \Rightarrow defactorize_aux g O].
200 theorem lt_O_defactorize_aux:
203 O < defactorize_aux f i.
207 rewrite > times_n_SO;
209 [ change with (O < nth_prime i);
210 apply lt_O_nth_prime_n;
212 change with (O < exp (nth_prime i) n);
214 apply lt_O_nth_prime_n;
215 | change with (O < defactorize_aux n1 (S i));
219 theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
220 S O < defactorize_aux f i.
221 intro.elim f.simplify.unfold lt.
222 rewrite > times_n_SO.
224 change with (S O < nth_prime i).
225 apply lt_SO_nth_prime_n.
226 change with (O < exp (nth_prime i) n).
228 apply lt_O_nth_prime_n.
230 rewrite > times_n_SO.
233 change with (O < exp (nth_prime i) n).
235 apply lt_O_nth_prime_n.
236 change with (S O < defactorize_aux n1 (S i)).
240 theorem defactorize_aux_factorize_aux :
241 \forall p,n:nat.\forall acc:nat_fact.O < n \to
242 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
243 defactorize_aux (factorize_aux p n acc) O = n*(defactorize_aux acc p).
244 intro.elim p.simplify.
245 elim H1.elim H2.rewrite > H3.
246 rewrite > sym_times. apply times_n_SO.
247 apply False_ind.apply (not_le_Sn_O (max_prime_factor n) H2).
249 (* generalizing the goal: I guess there exists a better way *)
250 cut (\forall q,r.(pair nat nat q r) = (p_ord_aux n1 n1 (nth_prime n)) \to
251 defactorize_aux match (p_ord_aux n1 n1 (nth_prime n)) with
252 [(pair q r) \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
253 n1*defactorize_aux acc (S n)).
254 apply (Hcut (fst ? ? (p_ord_aux n1 n1 (nth_prime n)))
255 (snd ? ? (p_ord_aux n1 n1 (nth_prime n)))).
256 apply sym_eq.apply eq_pair_fst_snd.
260 cut (n1 = r * (nth_prime n) \sup q).
262 simplify.rewrite < assoc_times.
263 rewrite < Hcut.reflexivity.
264 cut (O < r \lor O = r).
265 elim Hcut1.assumption.absurd (n1 = O).
266 rewrite > Hcut.rewrite < H4.reflexivity.
267 unfold Not. intro.apply (not_le_Sn_O O).
268 rewrite < H5 in \vdash (? ? %).assumption.
269 apply le_to_or_lt_eq.apply le_O_n.
270 cut ((S O) < r \lor (S O) \nlt r).
273 apply (p_ord_to_lt_max_prime_factor1 n1 ? q r).
277 apply (not_eq_O_S n).apply sym_eq.assumption.
280 assumption.assumption.
283 left.split.assumption.reflexivity.
284 intro.right.rewrite > Hcut2.
285 simplify.unfold lt.apply le_S_S.apply le_O_n.
286 cut (r < (S O) ∨ r=(S O)).
287 elim Hcut2.absurd (O=r).
288 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
291 apply (not_le_Sn_O O).
292 rewrite > Hcut3 in ⊢ (? ? %).
293 assumption.rewrite > Hcut.
294 rewrite < H6.reflexivity.
296 apply (le_to_or_lt_eq r (S O)).
297 apply not_lt_to_le.assumption.
298 apply (decidable_lt (S O) r).
300 apply (p_ord_aux_to_exp n1 n1).
301 apply lt_O_nth_prime_n.assumption.
304 theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
306 apply (nat_case n).reflexivity.
307 intro.apply (nat_case m).reflexivity.
308 intro.(*CSC: simplify here does something really nasty *)
310 (let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
311 defactorize (match p_ord (S(S m1)) (nth_prime p) with
312 [ (pair q r) \Rightarrow
313 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
315 (* generalizing the goal; find a better way *)
316 cut (\forall q,r.(pair nat nat q r) = (p_ord (S(S m1)) (nth_prime p)) \to
317 defactorize (match p_ord (S(S m1)) (nth_prime p) with
318 [ (pair q r) \Rightarrow
319 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
320 apply (Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
321 (snd ? ? (p_ord (S(S m1)) (nth_prime p)))).
322 apply sym_eq.apply eq_pair_fst_snd.
326 cut ((S(S m1)) = (nth_prime p) \sup q *r).
328 rewrite > defactorize_aux_factorize_aux.
329 (*CSC: simplify here does something really nasty *)
330 change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
331 cut ((S (pred q)) = q).
335 apply (p_ord_aux_to_exp (S(S m1))).
336 apply lt_O_nth_prime_n.
339 apply sym_eq. apply S_pred.
340 cut (O < q \lor O = q).
341 elim Hcut2.assumption.
342 absurd (nth_prime p \divides S (S m1)).
343 apply (divides_max_prime_factor_n (S (S m1))).
344 unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
346 rewrite > Hcut3 in \vdash (? (? ? %)).
347 (*CSC: simplify here does something really nasty *)
348 change with (nth_prime p \divides r \to False).
350 apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
351 apply lt_SO_nth_prime_n.
352 unfold lt.apply le_S_S.apply le_O_n.apply le_n.
354 apply divides_to_mod_O.apply lt_O_nth_prime_n.assumption.
355 rewrite > times_n_SO in \vdash (? ? ? %).
357 rewrite > (exp_n_O (nth_prime p)).
358 rewrite > H1 in \vdash (? ? ? (? (? ? %) ?)).
360 apply le_to_or_lt_eq.apply le_O_n.assumption.
361 (* e adesso l'ultimo goal. TASSI: che ora non e' piu' l'ultimo :P *)
362 cut ((S O) < r \lor S O \nlt r).
365 apply (p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r).
366 unfold lt.apply le_S_S. apply le_O_n.
368 assumption.assumption.
371 left.split.assumption.reflexivity.
372 intro.right.rewrite > Hcut3.
373 simplify.unfold lt.apply le_S_S.apply le_O_n.
374 cut (r \lt (S O) \or r=(S O)).
375 elim Hcut3.absurd (O=r).
376 apply le_n_O_to_eq.apply le_S_S_to_le.exact H2.
378 apply (not_le_Sn_O O).
379 rewrite > H3 in \vdash (? ? %).assumption.assumption.
380 apply (le_to_or_lt_eq r (S O)).
381 apply not_lt_to_le.assumption.
382 apply (decidable_lt (S O) r).
384 cut (O < r \lor O = r).
385 elim Hcut1.assumption.
387 apply (not_eq_O_S (S m1)).
388 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
389 apply le_to_or_lt_eq.apply le_O_n.
392 apply (p_ord_aux_to_exp (S(S m1))).
393 apply lt_O_nth_prime_n.
400 [ (nf_last n) \Rightarrow O
401 | (nf_cons n g) \Rightarrow S (max_p g)].
403 let rec max_p_exponent f \def
405 [ (nf_last n) \Rightarrow n
406 | (nf_cons n g) \Rightarrow max_p_exponent g].
408 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat.
409 nth_prime ((max_p f)+i) \divides defactorize_aux f i.
411 elim f.simplify.apply (witness ? ? ((nth_prime i) \sup n)).
414 (nth_prime (S(max_p n1)+i) \divides
415 (nth_prime i) \sup n *(defactorize_aux n1 (S i))).
419 rewrite > assoc_times.
421 apply (witness ? ? (n2* (nth_prime i) \sup n)).
425 theorem divides_exp_to_divides:
426 \forall p,n,m:nat. prime p \to
427 p \divides n \sup m \to p \divides n.
428 intros 3.elim m.simplify in H1.
429 apply (transitive_divides p (S O)).assumption.
431 cut (p \divides n \lor p \divides n \sup n1).
432 elim Hcut.assumption.
433 apply H.assumption.assumption.
434 apply divides_times_to_divides.assumption.
438 theorem divides_exp_to_eq:
439 \forall p,q,m:nat. prime p \to prime q \to
440 p \divides q \sup m \to p = q.
444 apply (divides_exp_to_divides p q m).
445 assumption.assumption.
446 unfold prime in H.elim H.assumption.
449 theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
450 i < j \to nth_prime i \ndivides defactorize_aux f j.
453 (nth_prime i \divides (nth_prime j) \sup (S n) \to False).
454 intro.absurd ((nth_prime i) = (nth_prime j)).
455 apply (divides_exp_to_eq ? ? (S n)).
456 apply prime_nth_prime.apply prime_nth_prime.
457 assumption.unfold Not.
459 apply (not_le_Sn_n i).rewrite > Hcut in \vdash (? ? %).assumption.
460 apply (injective_nth_prime ? ? H2).
463 cut (nth_prime i \divides (nth_prime j) \sup n
464 \lor nth_prime i \divides defactorize_aux n1 (S j)).
466 absurd ((nth_prime i) = (nth_prime j)).
467 apply (divides_exp_to_eq ? ? n).
468 apply prime_nth_prime.apply prime_nth_prime.
469 assumption.unfold Not.
472 apply (not_le_Sn_n i).rewrite > Hcut1 in \vdash (? ? %).assumption.
473 apply (injective_nth_prime ? ? H4).
475 apply (trans_lt ? j).assumption.unfold lt.apply le_n.
477 apply divides_times_to_divides.
478 apply prime_nth_prime.assumption.
481 lemma not_eq_nf_last_nf_cons: \forall g:nat_fact.\forall n,m,i:nat.
482 \lnot (defactorize_aux (nf_last n) i= defactorize_aux (nf_cons m g) i).
485 (exp (nth_prime i) (S n) = defactorize_aux (nf_cons m g) i \to False).
487 cut (S(max_p g)+i= i).
488 apply (not_le_Sn_n i).
489 rewrite < Hcut in \vdash (? ? %).
490 simplify.apply le_S_S.
492 apply injective_nth_prime.
493 apply (divides_exp_to_eq ? ? (S n)).
494 apply prime_nth_prime.apply prime_nth_prime.
496 change with (divides (nth_prime ((max_p (nf_cons m g))+i))
497 (defactorize_aux (nf_cons m g) i)).
498 apply divides_max_p_defactorize.
501 lemma not_eq_nf_cons_O_nf_cons: \forall f,g:nat_fact.\forall n,i:nat.
502 \lnot (defactorize_aux (nf_cons O f) i= defactorize_aux (nf_cons (S n) g) i).
504 simplify.unfold Not.rewrite < plus_n_O.
506 apply (not_divides_defactorize_aux f i (S i) ?).
507 unfold lt.apply le_n.
509 rewrite > assoc_times.
510 apply (witness ? ? ((exp (nth_prime i) n)*(defactorize_aux g (S i)))).
514 theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
515 defactorize_aux f i = defactorize_aux g i \to f = g.
518 generalize in match H.
521 apply inj_S. apply (inj_exp_r (nth_prime i)).
522 apply lt_SO_nth_prime_n.
525 apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
526 generalize in match H1.
529 apply (not_eq_nf_last_nf_cons n1 n2 n i).
530 apply sym_eq. assumption.
532 generalize in match H3.
533 apply (nat_elim2 (\lambda n,n2.
534 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
535 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
536 nf_cons n n1 = nf_cons n2 n3)).
542 rewrite > (plus_n_O (defactorize_aux n3 (S i))).assumption.
544 apply (not_eq_nf_cons_O_nf_cons n1 n3 n5 i).assumption.
547 apply (not_eq_nf_cons_O_nf_cons n3 n1 n4 i).
548 apply sym_eq.assumption.
550 cut (nf_cons n4 n1 = nf_cons m n3).
553 rewrite > Hcut1.rewrite > Hcut2.reflexivity.
555 (match nf_cons n4 n1 with
556 [ (nf_last m) \Rightarrow n1
557 | (nf_cons m g) \Rightarrow g ] = n3).
558 rewrite > Hcut.simplify.reflexivity.
560 (match nf_cons n4 n1 with
561 [ (nf_last m) \Rightarrow m
562 | (nf_cons m g) \Rightarrow m ] = m).
563 rewrite > Hcut.simplify.reflexivity.
564 apply H4.simplify in H5.
565 apply (inj_times_r1 (nth_prime i)).
566 apply lt_O_nth_prime_n.
567 rewrite < assoc_times.rewrite < assoc_times.assumption.
570 theorem injective_defactorize_aux: \forall i:nat.
571 injective nat_fact nat (\lambda f.defactorize_aux f i).
574 apply (eq_defactorize_aux_to_eq x y i H).
577 theorem injective_defactorize:
578 injective nat_fact_all nat defactorize.
580 change with (\forall f,g.defactorize f = defactorize g \to f=g).
582 generalize in match H.elim g.
588 apply (not_eq_O_S O H1).
592 apply (not_le_Sn_n O).
593 rewrite > H1 in \vdash (? ? %).
594 change with (O < defactorize_aux n O).
595 apply lt_O_defactorize_aux.
596 generalize in match H.
601 apply (not_eq_O_S O).apply sym_eq. assumption.
607 apply (not_le_Sn_n (S O)).
608 rewrite > H1 in \vdash (? ? %).
609 change with ((S O) < defactorize_aux n O).
610 apply lt_SO_defactorize_aux.
611 generalize in match H.elim g.
615 apply (not_le_Sn_n O).
616 rewrite < H1 in \vdash (? ? %).
617 change with (O < defactorize_aux n O).
618 apply lt_O_defactorize_aux.
622 apply (not_le_Sn_n (S O)).
623 rewrite < H1 in \vdash (? ? %).
624 change with ((S O) < defactorize_aux n O).
625 apply lt_SO_defactorize_aux.
626 (* proper - proper *)
628 apply (injective_defactorize_aux O).
632 theorem factorize_defactorize:
633 \forall f,g: nat_fact_all. factorize (defactorize f) = f.
635 apply injective_defactorize.
636 apply defactorize_factorize.