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 divides_div: \forall d,n. divides d n \to divides (n/d) n.
216 apply (witness ? ? d).
218 apply divides_to_div.
222 theorem div_div: \forall n,d:nat. O < n \to divides d n \to
225 apply (inj_times_l1 (n/d))
226 [apply (lt_times_n_to_lt d)
227 [apply (divides_to_lt_O ? ? H H1).
228 |rewrite > divides_to_div;assumption
230 |rewrite > divides_to_div
231 [rewrite > sym_times.
232 rewrite > divides_to_div
236 |apply (witness ? ? d).
238 apply divides_to_div.
244 theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
245 O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
249 rewrite > (sym_times c n2).
251 [ rewrite > (lt_O_to_div_times n2 c)
252 [ rewrite < assoc_times.
253 rewrite > (lt_O_to_div_times (a *n2) c)
259 | apply (divides_to_lt_O c b);
265 (* boolean divides *)
266 definition divides_b : nat \to nat \to bool \def
267 \lambda n,m :nat. (eqb (m \mod n) O).
269 theorem divides_b_to_Prop :
270 \forall n,m:nat. O < n \to
271 match divides_b n m with
272 [ true \Rightarrow n \divides m
273 | false \Rightarrow n \ndivides m].
274 intros.unfold divides_b.
276 intro.simplify.apply mod_O_to_divides.assumption.assumption.
277 intro.simplify.unfold Not.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
280 theorem divides_b_true_to_divides1:
281 \forall n,m:nat. O < n \to
282 (divides_b n m = true ) \to n \divides m.
286 [ true \Rightarrow n \divides m
287 | false \Rightarrow n \ndivides m].
288 rewrite < H1.apply divides_b_to_Prop.
292 theorem divides_b_true_to_divides:
293 \forall n,m:nat. divides_b n m = true \to n \divides m.
294 intros 2.apply (nat_case n)
296 [intro.apply divides_n_n
297 |simplify.intros.apply False_ind.
298 apply not_eq_true_false.apply sym_eq.
302 apply divides_b_true_to_divides1
303 [apply lt_O_S|assumption]
307 theorem divides_b_false_to_not_divides1:
308 \forall n,m:nat. O < n \to
309 (divides_b n m = false ) \to n \ndivides m.
313 [ true \Rightarrow n \divides m
314 | false \Rightarrow n \ndivides m].
315 rewrite < H1.apply divides_b_to_Prop.
319 theorem divides_b_false_to_not_divides:
320 \forall n,m:nat. divides_b n m = false \to n \ndivides m.
321 intros 2.apply (nat_case n)
323 [simplify.unfold Not.intros.
324 apply not_eq_true_false.assumption
325 |unfold Not.intros.elim H1.
326 apply (not_eq_O_S m1).apply sym_eq.
330 apply divides_b_false_to_not_divides1
331 [apply lt_O_S|assumption]
335 theorem decidable_divides: \forall n,m:nat.O < n \to
336 decidable (n \divides m).
337 intros.unfold decidable.
339 (match divides_b n m with
340 [ true \Rightarrow n \divides m
341 | false \Rightarrow n \ndivides m] \to n \divides m \lor n \ndivides m).
342 apply Hcut.apply divides_b_to_Prop.assumption.
343 elim (divides_b n m).left.apply H1.right.apply H1.
346 theorem divides_to_divides_b_true : \forall n,m:nat. O < n \to
347 n \divides m \to divides_b n m = true.
349 cut (match (divides_b n m) with
350 [ true \Rightarrow n \divides m
351 | false \Rightarrow n \ndivides m] \to ((divides_b n m) = true)).
352 apply Hcut.apply divides_b_to_Prop.assumption.
353 elim (divides_b n m).reflexivity.
354 absurd (n \divides m).assumption.assumption.
357 theorem divides_to_divides_b_true1 : \forall n,m:nat.
358 O < m \to n \divides m \to divides_b n m = true.
360 elim (le_to_or_lt_eq O n (le_O_n n))
361 [apply divides_to_divides_b_true
362 [assumption|assumption]
367 apply (not_le_Sn_O O).
373 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
374 \lnot(n \divides m) \to (divides_b n m) = false.
376 cut (match (divides_b n m) with
377 [ true \Rightarrow n \divides m
378 | false \Rightarrow n \ndivides m] \to ((divides_b n m) = false)).
379 apply Hcut.apply divides_b_to_Prop.assumption.
380 elim (divides_b n m).
381 absurd (n \divides m).assumption.assumption.
385 theorem divides_b_div_true:
386 \forall d,n. O < n \to
387 divides_b d n = true \to divides_b (n/d) n = true.
389 apply divides_to_divides_b_true1
392 apply divides_b_true_to_divides.
397 theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
399 elim (le_to_or_lt_eq ? ? (le_O_n m))
405 apply (lt_to_not_eq O n H).
407 apply eqb_true_to_eq.
413 theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,m,i:nat.
414 m \le i \to i \le n+m \to f i \divides pi n f m.
415 intros 5.elim n.simplify.
416 cut (i = m).rewrite < Hcut.apply divides_n_n.
417 apply antisymmetric_le.assumption.assumption.
419 cut (i < S n1+m \lor i = S n1 + m).
421 apply (transitive_divides ? (pi n1 f m)).
422 apply H1.apply le_S_S_to_le. assumption.
423 apply (witness ? ? (f (S n1+m))).apply sym_times.
425 apply (witness ? ? (pi n1 f m)).reflexivity.
426 apply le_to_or_lt_eq.assumption.
430 theorem mod_S_pi: \forall f:nat \to nat.\forall n,i:nat.
431 i < n \to (S O) < (f i) \to (S (pi n f)) \mod (f i) = (S O).
432 intros.cut (pi n f) \mod (f i) = O.
434 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
435 rewrite > Hcut.assumption.
436 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
437 apply divides_f_pi_f.assumption.
441 (* divides and fact *)
442 theorem divides_fact : \forall n,i:nat.
443 O < i \to i \le n \to i \divides n!.
444 intros 3.elim n.absurd (O<i).assumption.apply (le_n_O_elim i H1).
445 apply (not_le_Sn_O O).
446 change with (i \divides (S n1)*n1!).
447 apply (le_n_Sm_elim i n1 H2).
449 apply (transitive_divides ? n1!).
450 apply H1.apply le_S_S_to_le. assumption.
451 apply (witness ? ? (S n1)).apply sym_times.
454 apply (witness ? ? n1!).reflexivity.
457 theorem mod_S_fact: \forall n,i:nat.
458 (S O) < i \to i \le n \to (S n!) \mod i = (S O).
459 intros.cut (n! \mod i = O).
461 apply mod_S.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
462 rewrite > Hcut.assumption.
463 apply divides_to_mod_O.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
464 apply divides_fact.apply (trans_lt O (S O)).apply (le_n (S O)).assumption.
468 theorem not_divides_S_fact: \forall n,i:nat.
469 (S O) < i \to i \le n \to i \ndivides S n!.
471 apply divides_b_false_to_not_divides.
473 rewrite > mod_S_fact[simplify.reflexivity|assumption|assumption].
477 definition prime : nat \to Prop \def
478 \lambda n:nat. (S O) < n \land
479 (\forall m:nat. m \divides n \to (S O) < m \to m = n).
481 theorem not_prime_O: \lnot (prime O).
482 unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_O (S O) H1).
485 theorem not_prime_SO: \lnot (prime (S O)).
486 unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
489 theorem prime_to_lt_O: \forall p. prime p \to O < p.
490 intros.elim H.apply lt_to_le.assumption.
493 (* smallest factor *)
494 definition smallest_factor : nat \to nat \def
500 [ O \Rightarrow (S O)
501 | (S q) \Rightarrow min_aux q (S (S O)) (\lambda m.(eqb ((S(S q)) \mod m) O))]].
504 theorem example1 : smallest_factor (S(S(S O))) = (S(S(S O))).
505 normalize.reflexivity.
508 theorem example2: smallest_factor (S(S(S(S O)))) = (S(S O)).
509 normalize.reflexivity.
512 theorem example3 : smallest_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
513 simplify.reflexivity.
516 theorem lt_SO_smallest_factor:
517 \forall n:nat. (S O) < n \to (S O) < (smallest_factor n).
519 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
520 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
523 (S O < min_aux m1 (S (S O)) (\lambda m.(eqb ((S(S m1)) \mod m) O))).
524 apply (lt_to_le_to_lt ? (S (S O))).
525 apply (le_n (S(S O))).
526 cut ((S(S O)) = (S(S m1)) - m1).
529 apply sym_eq.apply plus_to_minus.
530 rewrite < sym_plus.simplify.reflexivity.
533 theorem lt_O_smallest_factor: \forall n:nat. O < n \to O < (smallest_factor n).
535 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_n O H).
536 intro.apply (nat_case m).intro.
537 simplify.unfold lt.apply le_n.
538 intros.apply (trans_lt ? (S O)).
539 unfold lt.apply le_n.
540 apply lt_SO_smallest_factor.unfold lt. apply le_S_S.
541 apply le_S_S.apply le_O_n.
544 theorem divides_smallest_factor_n :
545 \forall n:nat. O < n \to smallest_factor n \divides n.
547 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O O H).
548 intro.apply (nat_case m).intro. simplify.
549 apply (witness ? ? (S O)). simplify.reflexivity.
551 apply divides_b_true_to_divides.
553 (eqb ((S(S m1)) \mod (min_aux m1 (S (S O))
554 (\lambda m.(eqb ((S(S m1)) \mod m) O)))) O = true).
555 apply f_min_aux_true.
556 apply (ex_intro nat ? (S(S m1))).
558 apply (le_S_S_to_le (S (S O)) (S (S m1)) ?).
559 apply (minus_le_O_to_le (S (S (S O))) (S (S (S m1))) ?).
561 rewrite < sym_plus. simplify. apply le_n.
562 apply (eq_to_eqb_true (mod (S (S m1)) (S (S m1))) O ?).
563 apply (mod_n_n (S (S m1)) ?).
567 theorem le_smallest_factor_n :
568 \forall n:nat. smallest_factor n \le n.
569 intro.apply (nat_case n).simplify.apply le_n.
570 intro.apply (nat_case m).simplify.apply le_n.
571 intro.apply divides_to_le.
572 unfold lt.apply le_S_S.apply le_O_n.
573 apply divides_smallest_factor_n.
574 unfold lt.apply le_S_S.apply le_O_n.
577 theorem lt_smallest_factor_to_not_divides: \forall n,i:nat.
578 (S O) < n \to (S O) < i \to i < (smallest_factor n) \to i \ndivides n.
580 apply (nat_case n).intro.apply False_ind.apply (not_le_Sn_O (S O) H).
581 intro.apply (nat_case m).intro. apply False_ind.apply (not_le_Sn_n (S O) H).
583 apply divides_b_false_to_not_divides.
584 apply (lt_min_aux_to_false
585 (\lambda i:nat.eqb ((S(S m1)) \mod i) O) (S (S O)) m1 i).
590 theorem prime_smallest_factor_n :
591 \forall n:nat. (S O) < n \to prime (smallest_factor n).
592 intro.change with ((S(S O)) \le n \to (S O) < (smallest_factor n) \land
593 (\forall m:nat. m \divides smallest_factor n \to (S O) < m \to m = (smallest_factor n))).
595 apply lt_SO_smallest_factor.assumption.
597 cut (le m (smallest_factor n)).
598 elim (le_to_or_lt_eq m (smallest_factor n) Hcut).
599 absurd (m \divides n).
600 apply (transitive_divides m (smallest_factor n)).
602 apply divides_smallest_factor_n.
603 apply (trans_lt ? (S O)). unfold lt. apply le_n. exact H.
604 apply lt_smallest_factor_to_not_divides.
605 exact H.assumption.assumption.assumption.
607 apply (trans_lt O (S O)).
609 apply lt_SO_smallest_factor.
614 theorem prime_to_smallest_factor: \forall n. prime n \to
615 smallest_factor n = n.
616 intro.apply (nat_case n).intro.apply False_ind.apply (not_prime_O H).
617 intro.apply (nat_case m).intro.apply False_ind.apply (not_prime_SO H).
620 ((S O) < (S(S m1)) \land
621 (\forall m:nat. m \divides S(S m1) \to (S O) < m \to m = (S(S m1))) \to
622 smallest_factor (S(S m1)) = (S(S m1))).
623 intro.elim H.apply H2.
624 apply divides_smallest_factor_n.
625 apply (trans_lt ? (S O)).unfold lt. apply le_n.assumption.
626 apply lt_SO_smallest_factor.
630 (* a number n > O is prime iff its smallest factor is n *)
631 definition primeb \def \lambda n:nat.
633 [ O \Rightarrow false
636 [ O \Rightarrow false
637 | (S q) \Rightarrow eqb (smallest_factor (S(S q))) (S(S q))]].
640 theorem example4 : primeb (S(S(S O))) = true.
641 normalize.reflexivity.
644 theorem example5 : primeb (S(S(S(S(S(S O)))))) = false.
645 normalize.reflexivity.
648 theorem example6 : primeb (S(S(S(S((S(S(S(S(S(S(S O)))))))))))) = true.
649 normalize.reflexivity.
652 theorem example7 : primeb (S(S(S(S(S(S((S(S(S(S((S(S(S(S(S(S(S O))))))))))))))))))) = true.
653 normalize.reflexivity.
656 theorem primeb_to_Prop: \forall n.
658 [ true \Rightarrow prime n
659 | false \Rightarrow \lnot (prime n)].
661 apply (nat_case n).simplify.unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_O (S O) H1).
662 intro.apply (nat_case m).simplify.unfold Not.unfold prime.intro.elim H.apply (not_le_Sn_n (S O) H1).
665 match eqb (smallest_factor (S(S m1))) (S(S m1)) with
666 [ true \Rightarrow prime (S(S m1))
667 | false \Rightarrow \lnot (prime (S(S m1)))].
668 apply (eqb_elim (smallest_factor (S(S m1))) (S(S m1))).
671 apply prime_smallest_factor_n.
672 unfold lt.apply le_S_S.apply le_S_S.apply le_O_n.
674 change with (prime (S(S m1)) \to False).
676 apply prime_to_smallest_factor.
680 theorem primeb_true_to_prime : \forall n:nat.
681 primeb n = true \to prime n.
684 [ true \Rightarrow prime n
685 | false \Rightarrow \lnot (prime n)].
687 apply primeb_to_Prop.
690 theorem primeb_false_to_not_prime : \forall n:nat.
691 primeb n = false \to \lnot (prime n).
694 [ true \Rightarrow prime n
695 | false \Rightarrow \lnot (prime n)].
697 apply primeb_to_Prop.
700 theorem decidable_prime : \forall n:nat.decidable (prime n).
701 intro.unfold decidable.
704 [ true \Rightarrow prime n
705 | false \Rightarrow \lnot (prime n)] \to (prime n) \lor \lnot (prime n)).
706 apply Hcut.apply primeb_to_Prop.
707 elim (primeb n).left.apply H.right.apply H.
710 theorem prime_to_primeb_true: \forall n:nat.
711 prime n \to primeb n = true.
713 cut (match (primeb n) with
714 [ true \Rightarrow prime n
715 | false \Rightarrow \lnot (prime n)] \to ((primeb n) = true)).
716 apply Hcut.apply primeb_to_Prop.
717 elim (primeb n).reflexivity.
718 absurd (prime n).assumption.assumption.
721 theorem not_prime_to_primeb_false: \forall n:nat.
722 \lnot(prime n) \to primeb n = false.
724 cut (match (primeb n) with
725 [ true \Rightarrow prime n
726 | false \Rightarrow \lnot (prime n)] \to ((primeb n) = false)).
727 apply Hcut.apply primeb_to_Prop.
729 absurd (prime n).assumption.assumption.