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 (mod n (nth_prime p)) O)).
26 (* max_prime_factor is indeed a factor *)
27 theorem divides_max_prime_factor_n: \forall n:nat. (S O) < n \to
28 nth_prime (max_prime_factor n) \divides n.
29 intros.apply divides_b_true_to_divides.
30 apply lt_O_nth_prime_n.
31 apply f_max_true (\lambda p:nat.eqb (mod n (nth_prime p)) O) n.
32 cut \exists i. nth_prime i = smallest_factor n.
34 apply ex_intro nat ? a.
36 apply trans_le a (nth_prime a).
38 exact lt_nth_prime_n_nth_prime_Sn.
39 rewrite > H1. apply le_smallest_factor_n.
41 change with divides_b (smallest_factor n) n = true.
42 apply divides_to_divides_b_true.
43 apply trans_lt ? (S O).simplify. apply le_n.
44 apply lt_SO_smallest_factor.assumption.
45 apply divides_smallest_factor_n.
46 apply trans_lt ? (S O). simplify. apply le_n. assumption.
47 apply prime_to_nth_prime.
48 apply prime_smallest_factor_n.assumption.
51 theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
52 max_prime_factor n \le max_prime_factor m.
54 (max n (\lambda p:nat.eqb (mod n (nth_prime p)) O)) \le
55 (max m (\lambda p:nat.eqb (mod m (nth_prime p)) O)).
58 apply le_max_n.apply divides_to_le.assumption.assumption.
59 change with divides_b (nth_prime (max_prime_factor n)) m = true.
60 apply divides_to_divides_b_true.
61 cut prime (nth_prime (max_prime_factor n)).
62 apply lt_O_nth_prime_n.apply prime_nth_prime.
63 cut nth_prime (max_prime_factor n) \divides n.
64 apply transitive_divides ? n.
65 apply divides_max_prime_factor_n.
66 assumption.assumption.
67 apply divides_b_true_to_divides.
68 apply lt_O_nth_prime_n.
69 apply divides_to_divides_b_true.
70 apply lt_O_nth_prime_n.
71 apply divides_max_prime_factor_n.
75 theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
76 p = max_prime_factor n \to
77 (pair nat nat q r) = p_ord n (nth_prime p) \to
78 (S O) < r \to max_prime_factor r < p.
81 cut max_prime_factor r \lt max_prime_factor n \lor
82 max_prime_factor r = max_prime_factor n.
84 absurd nth_prime (max_prime_factor n) \divides r.
86 apply divides_max_prime_factor_n.
88 change with nth_prime (max_prime_factor n) \divides r \to False.
90 cut mod r (nth_prime (max_prime_factor n)) \neq O.
91 apply Hcut1.apply divides_to_mod_O.
92 apply lt_O_nth_prime_n.assumption.
93 apply p_ord_aux_to_not_mod_O n n ? q r.
94 apply lt_SO_nth_prime_n.assumption.
96 rewrite < H1.assumption.
97 apply le_to_or_lt_eq (max_prime_factor r) (max_prime_factor n).
98 apply divides_to_max_prime_factor.
99 assumption.assumption.
100 apply witness r n ((nth_prime p) \sup q).
102 apply p_ord_aux_to_exp n n ? q r.
103 apply lt_O_nth_prime_n.assumption.
106 theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
107 max_prime_factor n \le p \to
108 (pair nat nat q r) = p_ord n (nth_prime p) \to
109 (S O) < r \to max_prime_factor r < p.
111 cut max_prime_factor n < p \lor max_prime_factor n = p.
112 elim Hcut.apply le_to_lt_to_lt ? (max_prime_factor n).
113 apply divides_to_max_prime_factor.assumption.assumption.
114 apply witness r n ((nth_prime p) \sup q).
116 apply p_ord_aux_to_exp n n.
117 apply lt_O_nth_prime_n.
118 assumption.assumption.
119 apply p_ord_to_lt_max_prime_factor n ? q.
120 assumption.apply sym_eq.assumption.assumption.assumption.
121 apply le_to_or_lt_eq ? p H1.
124 (* datatypes and functions *)
126 inductive nat_fact : Set \def
127 nf_last : nat \to nat_fact
128 | nf_cons : nat \to nat_fact \to nat_fact.
130 inductive nat_fact_all : Set \def
131 nfa_zero : nat_fact_all
132 | nfa_one : nat_fact_all
133 | nfa_proper : nat_fact \to nat_fact_all.
135 let rec factorize_aux p n acc \def
139 match p_ord n (nth_prime p1) with
140 [ (pair q r) \Rightarrow
141 factorize_aux p1 r (nf_cons q acc)]].
143 definition factorize : nat \to nat_fact_all \def \lambda n:nat.
145 [ O \Rightarrow nfa_zero
148 [ O \Rightarrow nfa_one
150 let p \def (max (S(S n2)) (\lambda p:nat.eqb (mod (S(S n2)) (nth_prime p)) O)) in
151 match p_ord (S(S n2)) (nth_prime p) with
152 [ (pair q r) \Rightarrow
153 nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
155 let rec defactorize_aux f i \def
157 [ (nf_last n) \Rightarrow (nth_prime i) \sup (S n)
158 | (nf_cons n g) \Rightarrow
159 (nth_prime i) \sup n *(defactorize_aux g (S i))].
161 definition defactorize : nat_fact_all \to nat \def
162 \lambda f : nat_fact_all.
164 [ nfa_zero \Rightarrow O
165 | nfa_one \Rightarrow (S O)
166 | (nfa_proper g) \Rightarrow defactorize_aux g O].
168 theorem lt_O_defactorize_aux: \forall f:nat_fact.\forall i:nat.
169 O < defactorize_aux f i.
170 intro.elim f.simplify.
171 rewrite > times_n_SO.
173 change with O < nth_prime i.
174 apply lt_O_nth_prime_n.
175 change with O < exp (nth_prime i) n.
177 apply lt_O_nth_prime_n.
179 rewrite > times_n_SO.
181 change with O < exp (nth_prime i) n.
183 apply lt_O_nth_prime_n.
184 change with O < defactorize_aux n1 (S i).
188 theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
189 S O < defactorize_aux f i.
190 intro.elim f.simplify.
191 rewrite > times_n_SO.
193 change with S O < nth_prime i.
194 apply lt_SO_nth_prime_n.
195 change with O < exp (nth_prime i) n.
197 apply lt_O_nth_prime_n.
199 rewrite > times_n_SO.
202 change with O < exp (nth_prime i) n.
204 apply lt_O_nth_prime_n.
205 change with S O < defactorize_aux n1 (S i).
209 theorem defactorize_aux_factorize_aux :
210 \forall p,n:nat.\forall acc:nat_fact.O < n \to
211 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
212 defactorize_aux (factorize_aux p n acc) O = n*(defactorize_aux acc p).
213 intro.elim p.simplify.
214 elim H1.elim H2.rewrite > H3.
215 rewrite > sym_times. apply times_n_SO.
216 apply False_ind.apply not_le_Sn_O (max_prime_factor n) H2.
218 (* generalizing the goal: I guess there exists a better way *)
219 cut \forall q,r.(pair nat nat q r) = (p_ord_aux n1 n1 (nth_prime n)) \to
220 defactorize_aux match (p_ord_aux n1 n1 (nth_prime n)) with
221 [(pair q r) \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
222 n1*defactorize_aux acc (S n).
223 apply Hcut (fst ? ? (p_ord_aux n1 n1 (nth_prime n)))
224 (snd ? ? (p_ord_aux n1 n1 (nth_prime n))).
225 apply sym_eq.apply eq_pair_fst_snd.
229 cut n1 = r * (nth_prime n) \sup q.
231 simplify.rewrite < assoc_times.
232 rewrite < Hcut.reflexivity.
233 cut O < r \lor O = r.
234 elim Hcut1.assumption.absurd n1 = O.
235 rewrite > Hcut.rewrite < H4.reflexivity.
236 simplify. intro.apply not_le_Sn_O O.
237 rewrite < H5 in \vdash (? ? %).assumption.
238 apply le_to_or_lt_eq.apply le_O_n.
239 cut (S O) < r \lor (S O) \nlt r.
242 apply p_ord_to_lt_max_prime_factor1 n1 ? q r.
246 apply not_eq_O_S n.apply sym_eq.assumption.
249 assumption.assumption.
252 left.split.assumption.reflexivity.
253 intro.right.rewrite > Hcut2.
254 simplify.apply le_S_S.apply le_O_n.
255 cut r \lt (S O) \or r=(S O).
256 elim Hcut2.absurd O=r.
257 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
261 rewrite > Hcut3 in \vdash (? ? %).
262 assumption.rewrite > Hcut.
263 rewrite < H6.reflexivity.
265 apply le_to_or_lt_eq r (S O).
266 apply not_lt_to_le.assumption.
267 apply decidable_lt (S O) r.
269 apply p_ord_aux_to_exp n1 n1.
270 apply lt_O_nth_prime_n.assumption.
273 theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
275 apply nat_case n.reflexivity.
276 intro.apply nat_case m.reflexivity.
278 let p \def (max (S(S m1)) (\lambda p:nat.eqb (mod (S(S m1)) (nth_prime p)) O)) in
279 defactorize (match p_ord (S(S m1)) (nth_prime p) with
280 [ (pair q r) \Rightarrow
281 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
283 (* generalizing the goal; find a better way *)
284 cut \forall q,r.(pair nat nat q r) = (p_ord (S(S m1)) (nth_prime p)) \to
285 defactorize (match p_ord (S(S m1)) (nth_prime p) with
286 [ (pair q r) \Rightarrow
287 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
288 apply Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
289 (snd ? ? (p_ord (S(S m1)) (nth_prime p))).
290 apply sym_eq.apply eq_pair_fst_snd.
294 defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1)).
295 cut (S(S m1)) = (nth_prime p) \sup q *r.
297 rewrite > defactorize_aux_factorize_aux.
298 change with r*(nth_prime p) \sup (S (pred q)) = (S(S m1)).
299 cut (S (pred q)) = q.
303 apply p_ord_aux_to_exp (S(S m1)).
304 apply lt_O_nth_prime_n.
307 apply sym_eq. apply S_pred.
308 cut O < q \lor O = q.
309 elim Hcut2.assumption.
310 absurd nth_prime p \divides S (S m1).
311 apply divides_max_prime_factor_n (S (S m1)).
312 simplify.apply le_S_S.apply le_S_S. apply le_O_n.
314 rewrite > Hcut3 in \vdash (? (? ? %)).
315 change with nth_prime p \divides r \to False.
317 apply p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r.
318 apply lt_SO_nth_prime_n.
319 simplify.apply le_S_S.apply le_O_n.apply le_n.
321 apply divides_to_mod_O.apply lt_O_nth_prime_n.assumption.
322 rewrite > times_n_SO in \vdash (? ? ? %).
324 rewrite > exp_n_O (nth_prime p).
325 rewrite > H1 in \vdash (? ? ? (? (? ? %) ?)).
327 apply le_to_or_lt_eq.apply le_O_n.assumption.
328 (* e adesso l'ultimo goal. TASSI: che ora non e' piu' l'ultimo :P *)
329 cut (S O) < r \lor S O \nlt r.
332 apply p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r.
333 simplify.apply le_S_S. apply le_O_n.
335 assumption.assumption.
338 left.split.assumption.reflexivity.
339 intro.right.rewrite > Hcut3.
340 simplify.apply le_S_S.apply le_O_n.
341 cut r \lt (S O) \or r=(S O).
342 elim Hcut3.absurd O=r.
343 apply le_n_O_to_eq.apply le_S_S_to_le.exact H2.
346 rewrite > H3 in \vdash (? ? %).assumption.assumption.
347 apply le_to_or_lt_eq r (S O).
348 apply not_lt_to_le.assumption.
349 apply decidable_lt (S O) r.
351 cut O < r \lor O = r.
352 elim Hcut1.assumption.
354 apply not_eq_O_S (S m1).
355 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
356 apply le_to_or_lt_eq.apply le_O_n.
359 apply p_ord_aux_to_exp (S(S m1)).
360 apply lt_O_nth_prime_n.
367 [ (nf_last n) \Rightarrow O
368 | (nf_cons n g) \Rightarrow S (max_p g)].
370 let rec max_p_exponent f \def
372 [ (nf_last n) \Rightarrow n
373 | (nf_cons n g) \Rightarrow max_p_exponent g].
375 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat.
376 nth_prime ((max_p f)+i) \divides defactorize_aux f i.
378 elim f.simplify.apply witness ? ? ((nth_prime i) \sup n).
381 nth_prime (S(max_p n1)+i) \divides
382 (nth_prime i) \sup n *(defactorize_aux n1 (S i)).
386 rewrite > assoc_times.
388 apply witness ? ? (n2* (nth_prime i) \sup n).
392 theorem divides_exp_to_divides:
393 \forall p,n,m:nat. prime p \to
394 p \divides n \sup m \to p \divides n.
395 intros 3.elim m.simplify in H1.
396 apply transitive_divides p (S O).assumption.
398 cut p \divides n \lor p \divides n \sup n1.
399 elim Hcut.assumption.
400 apply H.assumption.assumption.
401 apply divides_times_to_divides.assumption.
405 theorem divides_exp_to_eq:
406 \forall p,q,m:nat. prime p \to prime q \to
407 p \divides q \sup m \to p = q.
411 apply divides_exp_to_divides p q m.
412 assumption.assumption.
413 simplify in H.elim H.assumption.
416 theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
417 i < j \to nth_prime i \ndivides defactorize_aux f j.
420 nth_prime i \divides (nth_prime j) \sup (S n) \to False.
421 intro.absurd (nth_prime i) = (nth_prime j).
422 apply divides_exp_to_eq ? ? (S n).
423 apply prime_nth_prime.apply prime_nth_prime.
425 change with (nth_prime i) = (nth_prime j) \to False.
427 apply not_le_Sn_n i.rewrite > Hcut in \vdash (? ? %).assumption.
428 apply injective_nth_prime ? ? H2.
430 nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False.
432 cut nth_prime i \divides (nth_prime j) \sup n
433 \lor nth_prime i \divides defactorize_aux n1 (S j).
435 absurd (nth_prime i) = (nth_prime j).
436 apply divides_exp_to_eq ? ? n.
437 apply prime_nth_prime.apply prime_nth_prime.
439 change with (nth_prime i) = (nth_prime j) \to False.
442 apply not_le_Sn_n i.rewrite > Hcut1 in \vdash (? ? %).assumption.
443 apply injective_nth_prime ? ? H4.
445 apply trans_lt ? j.assumption.simplify.apply le_n.
447 apply divides_times_to_divides.
448 apply prime_nth_prime.assumption.
451 lemma not_eq_nf_last_nf_cons: \forall g:nat_fact.\forall n,m,i:nat.
452 \lnot (defactorize_aux (nf_last n) i= defactorize_aux (nf_cons m g) i).
455 exp (nth_prime i) (S n) = defactorize_aux (nf_cons m g) i \to False.
459 rewrite < Hcut in \vdash (? ? %).
460 simplify.apply le_S_S.
462 apply injective_nth_prime.
463 (* uffa, perche' semplifica ? *)
464 change with nth_prime (S(max_p g)+i)= nth_prime i.
465 apply divides_exp_to_eq ? ? (S n).
466 apply prime_nth_prime.apply prime_nth_prime.
468 change with divides (nth_prime ((max_p (nf_cons m g))+i))
469 (defactorize_aux (nf_cons m g) i).
470 apply divides_max_p_defactorize.
473 lemma not_eq_nf_cons_O_nf_cons: \forall f,g:nat_fact.\forall n,i:nat.
474 \lnot (defactorize_aux (nf_cons O f) i= defactorize_aux (nf_cons (S n) g) i).
476 simplify.rewrite < plus_n_O.
478 apply not_divides_defactorize_aux f i (S i) ?.
481 rewrite > assoc_times.
482 apply witness ? ? ((exp (nth_prime i) n)*(defactorize_aux g (S i))).
486 theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
487 defactorize_aux f i = defactorize_aux g i \to f = g.
490 generalize in match H.
493 apply inj_S. apply inj_exp_r (nth_prime i).
494 apply lt_SO_nth_prime_n.
497 apply not_eq_nf_last_nf_cons n2 n n1 i H2.
498 generalize in match H1.
501 apply not_eq_nf_last_nf_cons n1 n2 n i.
502 apply sym_eq. assumption.
504 generalize in match H3.
505 apply nat_elim2 (\lambda n,n2.
506 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
507 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
508 nf_cons n n1 = nf_cons n2 n3).
514 rewrite > plus_n_O (defactorize_aux n3 (S i)).assumption.
516 apply not_eq_nf_cons_O_nf_cons n1 n3 n5 i.assumption.
519 apply not_eq_nf_cons_O_nf_cons n3 n1 n4 i.
520 apply sym_eq.assumption.
522 cut nf_cons n4 n1 = nf_cons m n3.
525 rewrite > Hcut1.rewrite > Hcut2.reflexivity.
527 match nf_cons n4 n1 with
528 [ (nf_last m) \Rightarrow n1
529 | (nf_cons m g) \Rightarrow g ] = n3.
530 rewrite > Hcut.simplify.reflexivity.
532 match nf_cons n4 n1 with
533 [ (nf_last m) \Rightarrow m
534 | (nf_cons m g) \Rightarrow m ] = m.
535 rewrite > Hcut.simplify.reflexivity.
536 apply H4.simplify in H5.
537 apply inj_times_r1 (nth_prime i).
538 apply lt_O_nth_prime_n.
539 rewrite < assoc_times.rewrite < assoc_times.assumption.
542 theorem injective_defactorize_aux: \forall i:nat.
543 injective nat_fact nat (\lambda f.defactorize_aux f i).
544 change with \forall i:nat.\forall f,g:nat_fact.
545 defactorize_aux f i = defactorize_aux g i \to f = g.
547 apply eq_defactorize_aux_to_eq f g i H.
550 theorem injective_defactorize:
551 injective nat_fact_all nat defactorize.
552 change with \forall f,g:nat_fact_all.
553 defactorize f = defactorize g \to f = g.
555 generalize in match H.elim g.
561 apply not_eq_O_S O H1.
566 rewrite > H1 in \vdash (? ? %).
567 change with O < defactorize_aux n O.
568 apply lt_O_defactorize_aux.
569 generalize in match H.
574 apply not_eq_O_S O.apply sym_eq. assumption.
580 apply not_le_Sn_n (S O).
581 rewrite > H1 in \vdash (? ? %).
582 change with (S O) < defactorize_aux n O.
583 apply lt_SO_defactorize_aux.
584 generalize in match H.elim g.
589 rewrite < H1 in \vdash (? ? %).
590 change with O < defactorize_aux n O.
591 apply lt_O_defactorize_aux.
595 apply not_le_Sn_n (S O).
596 rewrite < H1 in \vdash (? ? %).
597 change with (S O) < defactorize_aux n O.
598 apply lt_SO_defactorize_aux.
599 (* proper - proper *)
601 apply injective_defactorize_aux O.
605 theorem factorize_defactorize:
606 \forall f,g: nat_fact_all. factorize (defactorize f) = f.
608 apply injective_defactorize.
609 (* uffa: perche' semplifica ??? *)
610 change with defactorize(factorize (defactorize f)) = (defactorize f).
611 apply defactorize_factorize.