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/primes".
17 include "nat/div_and_mod.ma".
18 include "nat/minimization.ma".
19 include "nat/sigma_and_pi.ma".
20 include "nat/factorial.ma".
22 inductive divides (n,m:nat) : Prop \def
23 witness : \forall p:nat.m = times n p \to divides n m.
25 interpretation "divides" 'divides n m = (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m).
26 interpretation "not divides" 'ndivides n m =
27 (cic:/matita/logic/connectives/Not.con (cic:/matita/nat/primes/divides.ind#xpointer(1/1) n m)).
29 theorem reflexive_divides : reflexive nat divides.
32 exact (witness x x (S O) (times_n_SO x)).
35 theorem divides_to_div_mod_spec :
36 \forall n,m. O < n \to n \divides m \to div_mod_spec m n (m / n) O.
37 intros.elim H1.rewrite > H2.
38 constructor 1.assumption.
39 apply (lt_O_n_elim n H).intros.
41 rewrite > div_times.apply sym_times.
44 theorem div_mod_spec_to_divides :
45 \forall n,m,p. div_mod_spec m n p O \to n \divides m.
47 apply (witness n m p).
49 rewrite > (plus_n_O (p*n)).assumption.
52 theorem divides_to_mod_O:
53 \forall n,m. O < n \to n \divides m \to (m \mod n) = O.
54 intros.apply (div_mod_spec_to_eq2 m n (m / n) (m \mod n) (m / n) O).
55 apply div_mod_spec_div_mod.assumption.
56 apply divides_to_div_mod_spec.assumption.assumption.
59 theorem mod_O_to_divides:
60 \forall n,m. O< n \to (m \mod n) = O \to n \divides m.
62 apply (witness n m (m / n)).
63 rewrite > (plus_n_O (n * (m / n))).
66 (* Andrea: perche' hint non lo trova ?*)
71 theorem divides_n_O: \forall n:nat. n \divides O.
72 intro. apply (witness n O O).apply times_n_O.
75 theorem divides_n_n: \forall n:nat. n \divides n.
76 intro. apply (witness n n (S O)).apply times_n_SO.
79 theorem divides_SO_n: \forall n:nat. (S O) \divides n.
80 intro. apply (witness (S O) n n). simplify.apply plus_n_O.
83 theorem divides_plus: \forall n,p,q:nat.
84 n \divides p \to n \divides q \to n \divides p+q.
86 elim H.elim H1. apply (witness n (p+q) (n2+n1)).
87 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
90 theorem divides_minus: \forall n,p,q:nat.
91 divides n p \to divides n q \to divides n (p-q).
93 elim H.elim H1. apply (witness n (p-q) (n2-n1)).
94 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
97 theorem divides_times: \forall n,m,p,q:nat.
98 n \divides p \to m \divides q \to n*m \divides p*q.
100 elim H.elim H1. apply (witness (n*m) (p*q) (n2*n1)).
101 rewrite > H2.rewrite > H3.
102 apply (trans_eq nat ? (n*(m*(n2*n1)))).
103 apply (trans_eq nat ? (n*(n2*(m*n1)))).
106 apply (trans_eq nat ? ((n2*m)*n1)).
107 apply sym_eq. apply assoc_times.
108 rewrite > (sym_times n2 m).apply assoc_times.
109 apply sym_eq. apply assoc_times.
112 theorem transitive_divides: transitive ? divides.
115 elim H.elim H1. apply (witness x z (n2*n)).
116 rewrite > H3.rewrite > H2.
120 variant trans_divides: \forall n,m,p.
121 n \divides m \to m \divides p \to n \divides p \def transitive_divides.
123 theorem eq_mod_to_divides:\forall n,m,p. O< p \to
124 mod n p = mod m p \to divides p (n-m).
126 cut (n \le m \or \not n \le m).
130 apply (witness p O O).
132 apply eq_minus_n_m_O.
134 apply (witness p (n-m) ((div n p)-(div m p))).
135 rewrite > distr_times_minus.
137 rewrite > (sym_times p).
138 cut ((div n p)*p = n - (mod n p)).
140 rewrite > eq_minus_minus_minus_plus.
143 rewrite < div_mod.reflexivity.
150 apply (decidable_le n m).
153 theorem antisymmetric_divides: antisymmetric nat divides.
154 unfold antisymmetric.intros.elim H. elim H1.
155 apply (nat_case1 n2).intro.
156 rewrite > H3.rewrite > H2.rewrite > H4.
157 rewrite < times_n_O.reflexivity.
159 apply (nat_case1 n).intro.
160 rewrite > H2.rewrite > H3.rewrite > H5.
161 rewrite < times_n_O.reflexivity.
163 apply antisymmetric_le.
164 rewrite > H2.rewrite > times_n_SO in \vdash (? % ?).
165 apply le_times_r.rewrite > H4.apply le_S_S.apply le_O_n.
166 rewrite > H3.rewrite > times_n_SO in \vdash (? % ?).
167 apply le_times_r.rewrite > H5.apply le_S_S.apply le_O_n.
171 theorem divides_to_le : \forall n,m. O < m \to n \divides m \to n \le m.
172 intros. elim H1.rewrite > H2.cut (O < n2).
173 apply (lt_O_n_elim n2 Hcut).intro.rewrite < sym_times.
174 simplify.rewrite < sym_plus.
176 elim (le_to_or_lt_eq O n2).
178 absurd (O<m).assumption.
179 rewrite > H2.rewrite < H3.rewrite < times_n_O.
180 apply (not_le_Sn_n O).
184 theorem divides_to_lt_O : \forall n,m. O < m \to n \divides m \to O < n.
186 elim (le_to_or_lt_eq O n (le_O_n n)).
188 rewrite < H3.absurd (O < m).assumption.
189 rewrite > H2.rewrite < H3.
190 simplify.exact (not_le_Sn_n O).
196 theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
198 elim (le_to_or_lt_eq O n (le_O_n n))
200 rewrite < (divides_to_mod_O ? ? H H1).
205 generalize in match H2.
214 theorem div_div: \forall n,d:nat. O < n \to divides d n \to
217 apply (inj_times_l1 (n/d))
218 [apply (lt_times_n_to_lt d)
219 [apply (divides_to_lt_O ? ? H H1).
220 |rewrite > divides_to_div;assumption
222 |rewrite > divides_to_div
223 [rewrite > sym_times.
224 rewrite > divides_to_div
228 |apply (witness ? ? d).
230 apply divides_to_div.
236 theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
237 O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
241 rewrite > (sym_times c n2).
243 [ rewrite > (lt_O_to_div_times n2 c)
244 [ rewrite < assoc_times.
245 rewrite > (lt_O_to_div_times (a *n2) c)
251 | apply (divides_to_lt_O c b);
257 (* boolean divides *)
258 definition divides_b : nat \to nat \to bool \def
259 \lambda n,m :nat. (eqb (m \mod n) O).
261 theorem divides_b_to_Prop :
262 \forall n,m:nat. O < n \to
263 match divides_b n m with
264 [ true \Rightarrow n \divides m
265 | false \Rightarrow n \ndivides m].
266 intros.unfold divides_b.
268 intro.simplify.apply mod_O_to_divides.assumption.assumption.
269 intro.simplify.unfold Not.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
272 theorem divides_b_true_to_divides1:
273 \forall n,m:nat. O < n \to
274 (divides_b n m = true ) \to n \divides m.
278 [ true \Rightarrow n \divides m
279 | false \Rightarrow n \ndivides m].
280 rewrite < H1.apply divides_b_to_Prop.
284 theorem divides_b_true_to_divides:
285 \forall n,m:nat. divides_b n m = true \to n \divides m.
286 intros 2.apply (nat_case n)
288 [intro.apply divides_n_n
289 |simplify.intros.apply False_ind.
290 apply not_eq_true_false.apply sym_eq.assumption
293 apply divides_b_true_to_divides1
294 [apply lt_O_S|assumption]
298 theorem divides_b_false_to_not_divides1:
299 \forall n,m:nat. O < n \to
300 (divides_b n m = false ) \to n \ndivides m.
304 [ true \Rightarrow n \divides m
305 | false \Rightarrow n \ndivides m].
306 rewrite < H1.apply divides_b_to_Prop.
310 theorem divides_b_false_to_not_divides:
311 \forall n,m:nat. divides_b n m = false \to n \ndivides m.
312 intros 2.apply (nat_case n)
314 [simplify.unfold Not.intros.
315 apply not_eq_true_false.assumption
316 |unfold Not.intros.elim H1.
317 apply (not_eq_O_S m1).apply sym_eq.
321 apply divides_b_false_to_not_divides1
322 [apply lt_O_S|assumption]
326 theorem decidable_divides: \forall n,m:nat.O < n \to
327 decidable (n \divides m).
328 intros.unfold decidable.
330 (match divides_b n m with
331 [ true \Rightarrow n \divides m
332 | false \Rightarrow n \ndivides m] \to n \divides m \lor n \ndivides m).
333 apply Hcut.apply divides_b_to_Prop.assumption.
334 elim (divides_b n m).left.apply H1.right.apply H1.
337 theorem divides_to_divides_b_true : \forall n,m:nat. O < n \to
338 n \divides m \to divides_b n m = true.
340 cut (match (divides_b n m) with
341 [ true \Rightarrow n \divides m
342 | false \Rightarrow n \ndivides m] \to ((divides_b n m) = true)).
343 apply Hcut.apply divides_b_to_Prop.assumption.
344 elim (divides_b n m).reflexivity.
345 absurd (n \divides m).assumption.assumption.
348 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
349 \lnot(n \divides m) \to (divides_b n m) = false.
351 cut (match (divides_b n m) with
352 [ true \Rightarrow n \divides m
353 | false \Rightarrow n \ndivides m] \to ((divides_b n m) = false)).
354 apply Hcut.apply divides_b_to_Prop.assumption.
355 elim (divides_b n m).
356 absurd (n \divides m).assumption.assumption.
360 theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
362 elim (le_to_or_lt_eq ? ? (le_O_n m))
368 apply (lt_to_not_eq O n H).
370 apply eqb_true_to_eq.
376 theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,m,i:nat.
377 m \le i \to i \le n+m \to f i \divides pi n f m.
378 intros 5.elim n.simplify.
379 cut (i = m).rewrite < Hcut.apply divides_n_n.
380 apply antisymmetric_le.assumption.assumption.
382 cut (i < S n1+m \lor i = S n1 + m).
384 apply (transitive_divides ? (pi n1 f m)).
385 apply H1.apply le_S_S_to_le. assumption.
386 apply (witness ? ? (f (S n1+m))).apply sym_times.
388 apply (witness ? ? (pi n1 f m)).reflexivity.
389 apply le_to_or_lt_eq.assumption.
393 theorem mod_S_pi: \forall f:nat \to nat.\forall n,i:nat.
394 i < n \to (S O) < (f i) \to (S (pi n f)) \mod (f i) = (S O).
395 intros.cut (pi n f) \mod (f i) = O.
397 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
398 rewrite > Hcut.assumption.
399 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
400 apply divides_f_pi_f.assumption.
404 (* divides and fact *)
405 theorem divides_fact : \forall n,i:nat.
406 O < i \to i \le n \to i \divides n!.
407 intros 3.elim n.absurd (O<i).assumption.apply (le_n_O_elim i H1).
408 apply (not_le_Sn_O O).
409 change with (i \divides (S n1)*n1!).
410 apply (le_n_Sm_elim i n1 H2).
412 apply (transitive_divides ? n1!).
413 apply H1.apply le_S_S_to_le. assumption.
414 apply (witness ? ? (S n1)).apply sym_times.
417 apply (witness ? ? n1!).reflexivity.
420 theorem mod_S_fact: \forall n,i:nat.
421 (S O) < i \to i \le n \to (S n!) \mod i = (S O).
422 intros.cut (n! \mod i = O).
424 apply mod_S.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
425 rewrite > Hcut.assumption.
426 apply divides_to_mod_O.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
427 apply divides_fact.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
431 theorem not_divides_S_fact: \forall n,i:nat.
432 (S O) < i \to i \le n \to i \ndivides S n!.
434 apply divides_b_false_to_not_divides.
436 rewrite > mod_S_fact[simplify.reflexivity|assumption|assumption].
440 definition prime : nat \to Prop \def
441 \lambda n:nat. (S O) < n \land
442 (\forall m:nat. m \divides n \to (S O) < m \to m = n).
444 theorem not_prime_O: \lnot (prime O).
445 unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_O (S O) H1).
448 theorem not_prime_SO: \lnot (prime (S O)).
449 unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
452 theorem prime_to_lt_O: \forall p. prime p \to O < p.
453 intros.elim H.apply lt_to_le.assumption.
456 (* smallest factor *)
457 definition smallest_factor : nat \to nat \def
463 [ O \Rightarrow (S O)
464 | (S q) \Rightarrow min_aux q (S (S O)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
467 theorem example1 : smallest_factor (S(S(S O))) = (S(S(S O))).
468 normalize.reflexivity.
471 theorem example2: smallest_factor (S(S(S(S O)))) = (S(S O)).
472 normalize.reflexivity.
475 theorem example3 : smallest_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
476 simplify.reflexivity.
479 theorem lt_SO_smallest_factor:
480 \forall n:nat. (S O) < n \to (S O) < (smallest_factor n).
482 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
483 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
486 (S O < min_aux m1 (S (S O)) (\lambda m.(eqb ((S(S m1)) \mod m) O))).
487 apply (lt_to_le_to_lt ? (S (S O))).
488 apply (le_n (S(S O))).
489 cut ((S(S O)) = (S(S m1)) - m1).
492 apply sym_eq.apply plus_to_minus.
493 rewrite < sym_plus.simplify.reflexivity.
496 theorem lt_O_smallest_factor: \forall n:nat. O < n \to O < (smallest_factor n).
498 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_n O H).
499 intro.apply (nat_case m).intro.
500 simplify.unfold lt.apply le_n.
501 intros.apply (trans_lt ? (S O)).
502 unfold lt.apply le_n.
503 apply lt_SO_smallest_factor.unfold lt. apply le_S_S.
504 apply le_S_S.apply le_O_n.
507 theorem divides_smallest_factor_n :
508 \forall n:nat. O < n \to smallest_factor n \divides n.
510 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O O H).
511 intro.apply (nat_case m).intro. simplify.
512 apply (witness ? ? (S O)). simplify.reflexivity.
514 apply divides_b_true_to_divides.
516 (eqb ((S(S m1)) \mod (min_aux m1 (S (S O))
517 (\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true).
518 apply f_min_aux_true.
519 apply (ex_intro nat ? (S(S m1))).
521 apply (le_S_S_to_le (S (S O)) (S (S m1)) ?).
522 apply (minus_le_O_to_le (S (S (S O))) (S (S (S m1))) ?).
524 rewrite < sym_plus. simplify. apply le_n.
525 apply (eq_to_eqb_true (mod (S (S m1)) (S (S m1))) O ?).
526 apply (mod_n_n (S (S m1)) ?).
530 theorem le_smallest_factor_n :
531 \forall n:nat. smallest_factor n \le n.
532 intro.apply (nat_case n).simplify.apply le_n.
533 intro.apply (nat_case m).simplify.apply le_n.
534 intro.apply divides_to_le.
535 unfold lt.apply le_S_S.apply le_O_n.
536 apply divides_smallest_factor_n.
537 unfold lt.apply le_S_S.apply le_O_n.
540 theorem lt_smallest_factor_to_not_divides: \forall n,i:nat.
541 (S O) < n \to (S O) < i \to i < (smallest_factor n) \to i \ndivides n.
543 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
544 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
546 apply divides_b_false_to_not_divides.
547 apply (lt_min_aux_to_false
548 (\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S (S O)) m1 i).
553 theorem prime_smallest_factor_n :
554 \forall n:nat. (S O) < n \to prime (smallest_factor n).
555 intro.change with ((S(S O)) \le n \to (S O) < (smallest_factor n) \land
556 (\forall m:nat. m \divides smallest_factor n \to (S O) < m \to m = (smallest_factor n))).
558 apply lt_SO_smallest_factor.assumption.
560 cut (le m (smallest_factor n)).
561 elim (le_to_or_lt_eq m (smallest_factor n) Hcut).
562 absurd (m \divides n).
563 apply (transitive_divides m (smallest_factor n)).
565 apply divides_smallest_factor_n.
566 apply (trans_lt ? (S O)). unfold lt. apply le_n. exact H.
567 apply lt_smallest_factor_to_not_divides.
568 exact H.assumption.assumption.assumption.
570 apply (trans_lt O (S O)).
572 apply lt_SO_smallest_factor.
577 theorem prime_to_smallest_factor: \forall n. prime n \to
578 smallest_factor n = n.
579 intro.apply (nat_case n).intro.apply False_ind.apply (not_prime_O H).
580 intro.apply (nat_case m).intro.apply False_ind.apply (not_prime_SO H).
583 ((S O) < (S(S m1)) \land
584 (\forall m:nat. m \divides S(S m1) \to (S O) < m \to m = (S(S m1))) \to
585 smallest_factor (S(S m1)) = (S(S m1))).
586 intro.elim H.apply H2.
587 apply divides_smallest_factor_n.
588 apply (trans_lt ? (S O)).unfold lt. apply le_n.assumption.
589 apply lt_SO_smallest_factor.
593 (* a number n > O is prime iff its smallest factor is n *)
594 definition primeb \def \lambda n:nat.
596 [ O \Rightarrow false
599 [ O \Rightarrow false
600 | (S q) \Rightarrow eqb (smallest_factor (S(S q))) (S(S q))]].
603 theorem example4 : primeb (S(S(S O))) = true.
604 normalize.reflexivity.
607 theorem example5 : primeb (S(S(S(S(S(S O)))))) = false.
608 normalize.reflexivity.
611 theorem example6 : primeb (S(S(S(S((S(S(S(S(S(S(S O)))))))))))) = true.
612 normalize.reflexivity.
615 theorem example7 : primeb (S(S(S(S(S(S((S(S(S(S((S(S(S(S(S(S(S O))))))))))))))))))) = true.
616 normalize.reflexivity.
619 theorem primeb_to_Prop: \forall n.
621 [ true \Rightarrow prime n
622 | false \Rightarrow \lnot (prime n)].
624 apply (nat_case n).simplify.unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_O (S O) H1).
625 intro.apply (nat_case m).simplify.unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
628 match eqb (smallest_factor (S(S m1))) (S(S m1)) with
629 [ true \Rightarrow prime (S(S m1))
630 | false \Rightarrow \lnot (prime (S(S m1)))].
631 apply (eqb_elim (smallest_factor (S(S m1))) (S(S m1))).
634 apply prime_smallest_factor_n.
635 unfold lt.apply le_S_S.apply le_S_S.apply le_O_n.
637 change with (prime (S(S m1)) \to False).
639 apply prime_to_smallest_factor.
643 theorem primeb_true_to_prime : \forall n:nat.
644 primeb n = true \to prime n.
647 [ true \Rightarrow prime n
648 | false \Rightarrow \lnot (prime n)].
650 apply primeb_to_Prop.
653 theorem primeb_false_to_not_prime : \forall n:nat.
654 primeb n = false \to \lnot (prime n).
657 [ true \Rightarrow prime n
658 | false \Rightarrow \lnot (prime n)].
660 apply primeb_to_Prop.
663 theorem decidable_prime : \forall n:nat.decidable (prime n).
664 intro.unfold decidable.
667 [ true \Rightarrow prime n
668 | false \Rightarrow \lnot (prime n)] \to (prime n) \lor \lnot (prime n)).
669 apply Hcut.apply primeb_to_Prop.
670 elim (primeb n).left.apply H.right.apply H.
673 theorem prime_to_primeb_true: \forall n:nat.
674 prime n \to primeb n = true.
676 cut (match (primeb n) with
677 [ true \Rightarrow prime n
678 | false \Rightarrow \lnot (prime n)] \to ((primeb n) = true)).
679 apply Hcut.apply primeb_to_Prop.
680 elim (primeb n).reflexivity.
681 absurd (prime n).assumption.assumption.
684 theorem not_prime_to_primeb_false: \forall n:nat.
685 \lnot(prime n) \to primeb n = false.
687 cut (match (primeb n) with
688 [ true \Rightarrow prime n
689 | false \Rightarrow \lnot (prime n)] \to ((primeb n) = false)).
690 apply Hcut.apply primeb_to_Prop.
692 absurd (prime n).assumption.assumption.