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 plog_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) = plog 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 \lnot (mod r (nth_prime (max_prime_factor n))) = O.
91 apply Hcut1.apply divides_to_mod_O.
92 apply lt_O_nth_prime_n.assumption.
93 apply plog_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 plog_aux_to_exp n n ? q r.
103 apply lt_O_nth_prime_n.assumption.
106 theorem plog_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) = plog 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 plog_aux_to_exp n n.
117 apply lt_O_nth_prime_n.
118 assumption.assumption.
119 apply plog_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 plog 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 plog (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 defactorize_aux_factorize_aux :
169 \forall p,n:nat.\forall acc:nat_fact.O < n \to
170 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
171 defactorize_aux (factorize_aux p n acc) O = n*(defactorize_aux acc p).
172 intro.elim p.simplify.
173 elim H1.elim H2.rewrite > H3.
174 rewrite > sym_times. apply times_n_SO.
175 apply False_ind.apply not_le_Sn_O (max_prime_factor n) H2.
177 (* generalizing the goal: I guess there exists a better way *)
178 cut \forall q,r.(pair nat nat q r) = (plog_aux n1 n1 (nth_prime n)) \to
179 defactorize_aux match (plog_aux n1 n1 (nth_prime n)) with
180 [(pair q r) \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
181 n1*defactorize_aux acc (S n).
182 apply Hcut (fst ? ? (plog_aux n1 n1 (nth_prime n)))
183 (snd ? ? (plog_aux n1 n1 (nth_prime n))).
184 apply sym_eq.apply eq_pair_fst_snd.
188 cut n1 = r * (nth_prime n) \sup q.
190 simplify.rewrite < assoc_times.
191 rewrite < Hcut.reflexivity.
192 cut O < r \lor O = r.
193 elim Hcut1.assumption.absurd n1 = O.
194 rewrite > Hcut.rewrite < H4.reflexivity.
195 simplify. intro.apply not_le_Sn_O O.
196 rewrite < H5 in \vdash (? ? %).assumption.
197 apply le_to_or_lt_eq.apply le_O_n.
198 cut (S O) < r \lor \lnot (S O) < r.
201 apply plog_to_lt_max_prime_factor1 n1 ? q r.
205 apply not_eq_O_S n.apply sym_eq.assumption.
208 assumption.assumption.
211 left.split.assumption.reflexivity.
212 intro.right.rewrite > Hcut2.
213 simplify.apply le_S_S.apply le_O_n.
214 cut r \lt (S O) \or r=(S O).
215 elim Hcut2.absurd O=r.
216 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
220 rewrite > Hcut3 in \vdash (? ? %).
221 assumption.rewrite > Hcut.
222 rewrite < H6.reflexivity.
224 apply le_to_or_lt_eq r (S O).
225 apply not_lt_to_le.assumption.
226 apply decidable_lt (S O) r.
228 apply plog_aux_to_exp n1 n1.
229 apply lt_O_nth_prime_n.assumption.
232 theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
234 apply nat_case n.reflexivity.
235 intro.apply nat_case m.reflexivity.
237 let p \def (max (S(S m1)) (\lambda p:nat.eqb (mod (S(S m1)) (nth_prime p)) O)) in
238 defactorize (match plog (S(S m1)) (nth_prime p) with
239 [ (pair q r) \Rightarrow
240 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
242 (* generalizing the goal; find a better way *)
243 cut \forall q,r.(pair nat nat q r) = (plog (S(S m1)) (nth_prime p)) \to
244 defactorize (match plog (S(S m1)) (nth_prime p) with
245 [ (pair q r) \Rightarrow
246 nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
247 apply Hcut (fst ? ? (plog (S(S m1)) (nth_prime p)))
248 (snd ? ? (plog (S(S m1)) (nth_prime p))).
249 apply sym_eq.apply eq_pair_fst_snd.
253 defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1)).
254 cut (S(S m1)) = (nth_prime p) \sup q *r.
256 rewrite > defactorize_aux_factorize_aux.
257 change with r*(nth_prime p) \sup (S (pred q)) = (S(S m1)).
258 cut (S (pred q)) = q.
262 apply plog_aux_to_exp (S(S m1)).
263 apply lt_O_nth_prime_n.
266 apply sym_eq. apply S_pred.
267 cut O < q \lor O = q.
268 elim Hcut2.assumption.
269 absurd nth_prime p \divides S (S m1).
270 apply divides_max_prime_factor_n (S (S m1)).
271 simplify.apply le_S_S.apply le_S_S. apply le_O_n.
273 rewrite > Hcut3 in \vdash (? (? ? %)).
274 change with nth_prime p \divides r \to False.
276 apply plog_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r.
277 apply lt_SO_nth_prime_n.
278 simplify.apply le_S_S.apply le_O_n.apply le_n.
280 apply divides_to_mod_O.apply lt_O_nth_prime_n.assumption.
281 rewrite > times_n_SO in \vdash (? ? ? %).
283 rewrite > exp_n_O (nth_prime p).
284 rewrite > H1 in \vdash (? ? ? (? (? ? %) ?)).
286 apply le_to_or_lt_eq.apply le_O_n.assumption.
287 (* e adesso l'ultimo goal. TASSI: che ora non e' piu' l'ultimo :P *)
288 cut (S O) < r \lor \lnot (S O) < r.
291 apply plog_to_lt_max_prime_factor1 (S(S m1)) ? q r.
292 simplify.apply le_S_S. apply le_O_n.
294 assumption.assumption.
297 left.split.assumption.reflexivity.
298 intro.right.rewrite > Hcut3.
299 simplify.apply le_S_S.apply le_O_n.
300 cut r \lt (S O) \or r=(S O).
301 elim Hcut3.absurd O=r.
302 apply le_n_O_to_eq.apply le_S_S_to_le.exact H2.
305 rewrite > H3 in \vdash (? ? %).assumption.assumption.
306 apply le_to_or_lt_eq r (S O).
307 apply not_lt_to_le.assumption.
308 apply decidable_lt (S O) r.
310 cut O < r \lor O = r.
311 elim Hcut1.assumption.
313 apply not_eq_O_S (S m1).
314 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
315 apply le_to_or_lt_eq.apply le_O_n.
318 apply plog_aux_to_exp (S(S m1)).
319 apply lt_O_nth_prime_n.
326 [ (nf_last n) \Rightarrow O
327 | (nf_cons n g) \Rightarrow S (max_p g)].
329 let rec max_p_exponent f \def
331 [ (nf_last n) \Rightarrow n
332 | (nf_cons n g) \Rightarrow max_p_exponent g].
334 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat.
335 nth_prime ((max_p f)+i) \divides defactorize_aux f i.
337 elim f.simplify.apply witness ? ? ((nth_prime i) \sup n).
340 nth_prime (S(max_p n1)+i) \divides
341 (nth_prime i) \sup n *(defactorize_aux n1 (S i)).
345 rewrite > assoc_times.
347 apply witness ? ? (n2* (nth_prime i) \sup n).
351 theorem divides_exp_to_divides:
352 \forall p,n,m:nat. prime p \to
353 p \divides n \sup m \to p \divides n.
354 intros 3.elim m.simplify in H1.
355 apply transitive_divides p (S O).assumption.
357 cut p \divides n \lor p \divides n \sup n1.
358 elim Hcut.assumption.
359 apply H.assumption.assumption.
360 apply divides_times_to_divides.assumption.
364 theorem divides_exp_to_eq:
365 \forall p,q,m:nat. prime p \to prime q \to
366 p \divides q \sup m \to p = q.
370 apply divides_exp_to_divides p q m.
371 assumption.assumption.
372 simplify in H.elim H.assumption.
375 theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
376 i < j \to nth_prime i \ndivides defactorize_aux f j.
379 nth_prime i \divides (nth_prime j) \sup (S n) \to False.
380 intro.absurd (nth_prime i) = (nth_prime j).
381 apply divides_exp_to_eq ? ? (S n).
382 apply prime_nth_prime.apply prime_nth_prime.
384 change with (nth_prime i) = (nth_prime j) \to False.
386 apply not_le_Sn_n i.rewrite > Hcut in \vdash (? ? %).assumption.
387 apply injective_nth_prime ? ? H2.
389 nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False.
391 cut nth_prime i \divides (nth_prime j) \sup n
392 \lor nth_prime i \divides defactorize_aux n1 (S j).
394 absurd (nth_prime i) = (nth_prime j).
395 apply divides_exp_to_eq ? ? n.
396 apply prime_nth_prime.apply prime_nth_prime.
398 change with (nth_prime i) = (nth_prime j) \to False.
401 apply not_le_Sn_n i.rewrite > Hcut1 in \vdash (? ? %).assumption.
402 apply injective_nth_prime ? ? H4.
404 apply trans_lt ? j.assumption.simplify.apply le_n.
406 apply divides_times_to_divides.
407 apply prime_nth_prime.assumption.
410 theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
411 defactorize_aux f i = defactorize_aux g i \to f = g.
414 generalize in match H.
417 apply inj_S. apply inj_exp_r (nth_prime i).
418 apply lt_SO_nth_prime_n.
420 absurd defactorize_aux (nf_last n) i =
421 defactorize_aux (nf_cons n1 n2) i.
422 rewrite > H2.reflexivity.
423 absurd nth_prime (S(max_p n2)+i) \divides defactorize_aux (nf_cons n1 n2) i.
424 apply divides_max_p_defactorize.
427 (nth_prime (S(max_p n2)+i) \divides (nth_prime i) \sup (S n)) \to False.
429 absurd nth_prime (S (max_p n2) + i) = nth_prime i.
430 apply divides_exp_to_eq ? ? (S n).
431 apply prime_nth_prime.apply prime_nth_prime.assumption.
432 change with nth_prime (S (max_p n2) + i) = nth_prime i \to False.
433 intro.apply not_le_Sn_n i.
434 cut S(max_p n2)+i= i.
435 rewrite < Hcut in \vdash (? ? %).
436 simplify.apply le_S_S.
438 apply injective_nth_prime ? ? H4.
439 generalize in match H1.
441 absurd defactorize_aux (nf_last n2) i =
442 defactorize_aux (nf_cons n n1) i.
443 apply sym_eq. assumption.
444 absurd nth_prime (S(max_p n1)+i) \divides defactorize_aux (nf_cons n n1) i.
445 apply divides_max_p_defactorize.
448 (nth_prime (S(max_p n1)+i) \divides (nth_prime i) \sup (S n2)) \to False.
450 absurd nth_prime (S (max_p n1) + i) = nth_prime i.
451 apply divides_exp_to_eq ? ? (S n2).
452 apply prime_nth_prime.apply prime_nth_prime.assumption.
453 change with nth_prime (S (max_p n1) + i) = nth_prime i \to False.
454 intro.apply not_le_Sn_n i.
455 cut S(max_p n1)+i= i.
456 rewrite < Hcut in \vdash (? ? %).
457 simplify.apply le_S_S.
459 apply injective_nth_prime ? ? H4.
461 generalize in match H3.
462 apply nat_elim2 (\lambda n,n2.
463 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
464 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
465 nf_cons n n1 = nf_cons n2 n3).
471 rewrite > plus_n_O (defactorize_aux n3 (S i)).assumption.
473 apply not_divides_defactorize_aux n1 i (S i).
474 simplify. apply le_n.
476 rewrite > plus_n_O (defactorize_aux n1 (S i)).
478 rewrite > assoc_times.
479 apply witness ? ? (((nth_prime i) \sup n5)*(defactorize_aux n3 (S i))).
483 apply not_divides_defactorize_aux n3 i (S i).
484 simplify. apply le_n.
486 rewrite > plus_n_O (defactorize_aux n3 (S i)).
488 rewrite > assoc_times.
489 apply witness ? ? (((nth_prime i) \sup n4)*(defactorize_aux n1 (S i))).
492 cut nf_cons n4 n1 = nf_cons m n3.
495 rewrite > Hcut1.rewrite > Hcut2.reflexivity.
497 match nf_cons n4 n1 with
498 [ (nf_last m) \Rightarrow n1
499 | (nf_cons m g) \Rightarrow g ] = n3.
500 rewrite > Hcut.simplify.reflexivity.
502 match nf_cons n4 n1 with
503 [ (nf_last m) \Rightarrow m
504 | (nf_cons m g) \Rightarrow m ] = m.
505 rewrite > Hcut.simplify.reflexivity.
506 apply H4.simplify in H5.
507 apply inj_times_r1 (nth_prime i).
508 apply lt_O_nth_prime_n.
509 rewrite < assoc_times.rewrite < assoc_times.assumption.
512 theorem injective_defactorize_aux: \forall i:nat.
513 injective nat_fact nat (\lambda f.defactorize_aux f i).
514 change with \forall i:nat.\forall f,g:nat_fact.
515 defactorize_aux f i = defactorize_aux g i \to f = g.
517 apply eq_defactorize_aux_to_eq f g i H.
521 theorem injective_defactorize:
522 injective nat_fact_all nat defactorize.
523 change with \forall f,g:nat_fact_all.
524 defactorize f = defactorize g \to f = g.
526 generalize in match H.elim g.
527 reflexivity.simplify in H1.