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 theorem reflexive_divides : reflexive nat divides.
28 exact witness x x (S O) (times_n_SO x).
31 theorem divides_to_div_mod_spec :
32 \forall n,m. O < n \to divides n m \to div_mod_spec m n (div m n) O.
33 intros.elim H1.rewrite > H2.
34 constructor 1.assumption.
35 apply lt_O_n_elim n H.intros.
37 rewrite > div_times.apply sym_times.
40 theorem div_mod_spec_to_div :
41 \forall n,m,p. div_mod_spec m n p O \to divides n m.
45 rewrite > plus_n_O (p*n).assumption.
48 theorem divides_to_mod_O:
49 \forall n,m. O < n \to divides n m \to (mod m n) = O.
50 intros.apply div_mod_spec_to_eq2 m n (div m n) (mod m n) (div m n) O.
51 apply div_mod_spec_div_mod.assumption.
52 apply divides_to_div_mod_spec.assumption.assumption.
55 theorem mod_O_to_divides:
56 \forall n,m. O< n \to (mod m n) = O \to divides n m.
58 apply witness n m (div m n).
59 rewrite > plus_n_O (n*div m n).
62 (* perche' hint non lo trova ?*)
67 theorem divides_n_O: \forall n:nat. divides n O.
68 intro. apply witness n O O.apply times_n_O.
71 theorem divides_SO_n: \forall n:nat. divides (S O) n.
72 intro. apply witness (S O) n n. simplify.apply plus_n_O.
75 theorem divides_plus: \forall n,p,q:nat.
76 divides n p \to divides n q \to divides n (p+q).
78 elim H.elim H1. apply witness n (p+q) (n2+n1).
79 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_plus.
82 theorem divides_minus: \forall n,p,q:nat.
83 divides n p \to divides n q \to divides n (p-q).
85 elim H.elim H1. apply witness n (p-q) (n2-n1).
86 rewrite > H2.rewrite > H3.apply sym_eq.apply distr_times_minus.
89 theorem divides_times: \forall n,m,p,q:nat.
90 divides n p \to divides m q \to divides (n*m) (p*q).
92 elim H.elim H1. apply witness (n*m) (p*q) (n2*n1).
93 rewrite > H2.rewrite > H3.
94 apply trans_eq nat ? (n*(m*(n2*n1))).
95 apply trans_eq nat ? (n*(n2*(m*n1))).
98 apply trans_eq nat ? ((n2*m)*n1).
99 apply sym_eq. apply assoc_times.
100 rewrite > sym_times n2 m.apply assoc_times.
101 apply sym_eq. apply assoc_times.
104 theorem transitive_divides: \forall n,m,p.
105 divides n m \to divides m p \to divides n p.
107 elim H.elim H1. apply witness n p (n2*n1).
108 rewrite > H3.rewrite > H2.
113 theorem divides_to_le : \forall n,m. O < m \to divides n m \to n \le m.
114 intros. elim H1.rewrite > H2.cut O < n2.
115 apply lt_O_n_elim n2 Hcut.intro.rewrite < sym_times.
116 simplify.rewrite < sym_plus.
118 elim le_to_or_lt_eq O n2.
119 assumption.apply le_O_n.
120 absurd O<m.assumption.
121 rewrite > H2.rewrite < H3.rewrite < times_n_O.
125 theorem divides_to_lt_O : \forall n,m. O < m \to divides n m \to O < n.
127 elim le_to_or_lt_eq O n (le_O_n n).
129 rewrite < H3.absurd O < m.assumption.
130 rewrite > H2.rewrite < H3.
131 simplify.exact not_le_Sn_n O.
134 (* boolean divides *)
135 definition divides_b : nat \to nat \to bool \def
136 \lambda n,m :nat. (eqb (mod m n) O).
138 theorem divides_b_to_Prop :
139 \forall n,m:nat. O < n \to
140 match divides_b n m with
141 [ true \Rightarrow divides n m
142 | false \Rightarrow \lnot (divides n m)].
145 match eqb (mod m n) O with
146 [ true \Rightarrow divides n m
147 | false \Rightarrow \lnot (divides n m)].
149 intro.simplify.apply mod_O_to_divides.assumption.assumption.
150 intro.simplify.intro.apply H1.apply divides_to_mod_O.assumption.assumption.
153 theorem divides_b_true_to_divides :
154 \forall n,m:nat. O < n \to
155 (divides_b n m = true ) \to divides n m.
159 [ true \Rightarrow divides n m
160 | false \Rightarrow \lnot (divides n m)].
161 rewrite < H1.apply divides_b_to_Prop.
165 theorem divides_b_false_to_not_divides :
166 \forall n,m:nat. O < n \to
167 (divides_b n m = false ) \to \lnot (divides n m).
171 [ true \Rightarrow divides n m
172 | false \Rightarrow \lnot (divides n m)].
173 rewrite < H1.apply divides_b_to_Prop.
177 theorem decidable_divides: \forall n,m:nat.O < n \to
178 decidable (divides n m).
179 intros.change with (divides n m) \lor \not (divides n m).
181 match divides_b n m with
182 [ true \Rightarrow divides n m
183 | false \Rightarrow \not (divides n m)] \to (divides n m) \lor \not (divides n m).
184 apply Hcut.apply divides_b_to_Prop.assumption.
185 elim (divides_b n m).left.apply H1.right.apply H1.
188 theorem divides_to_divides_b_true : \forall n,m:nat. O < n \to
189 divides n m \to divides_b n m = true.
191 cut match (divides_b n m) with
192 [ true \Rightarrow (divides n m)
193 | false \Rightarrow \not (divides n m)] \to ((divides_b n m) = true).
194 apply Hcut.apply divides_b_to_Prop.assumption.
195 elim divides_b n m.reflexivity.
196 absurd (divides n m).assumption.assumption.
199 theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
200 \not(divides n m) \to (divides_b n m) = false.
202 cut match (divides_b n m) with
203 [ true \Rightarrow (divides n m)
204 | false \Rightarrow \not (divides n m)] \to ((divides_b n m) = false).
205 apply Hcut.apply divides_b_to_Prop.assumption.
207 absurd (divides n m).assumption.assumption.
212 theorem divides_f_pi_f : \forall f:nat \to nat.\forall n,i:nat.
213 i < n \to divides (f i) (pi n f).
214 intros 3.elim n.apply False_ind.apply not_le_Sn_O i H.
216 apply le_n_Sm_elim (S i) n1 H1.
218 apply transitive_divides ? (pi n1 f).
219 apply H.simplify.apply le_S_S_to_le. assumption.
220 apply witness ? ? (f n1).apply sym_times.
223 apply witness ? ? (pi n1 f).reflexivity.
224 apply inj_S.assumption.
227 theorem mod_S_pi: \forall f:nat \to nat.\forall n,i:nat.
228 i < n \to (S O) < (f i) \to mod (S (pi n f)) (f i) = (S O).
229 intros.cut mod (pi n f) (f i) = O.
231 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
232 rewrite > Hcut.assumption.
233 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
234 apply divides_f_pi_f.assumption.
237 (* divides and fact *)
238 theorem divides_fact : \forall n,i:nat.
239 O < i \to i \le n \to divides i (fact n).
240 intros 3.elim n.absurd O<i.assumption.apply le_n_O_elim i H1.
242 change with divides i ((S n1)*(fact n1)).
243 apply le_n_Sm_elim i n1 H2.
245 apply transitive_divides ? (fact n1).
246 apply H1.apply le_S_S_to_le. assumption.
247 apply witness ? ? (S n1).apply sym_times.
250 apply witness ? ? (fact n1).reflexivity.
253 theorem mod_S_fact: \forall n,i:nat.
254 (S O) < i \to i \le n \to mod (S (fact n)) i = (S O).
255 intros.cut mod (fact n) i = O.
257 apply mod_S.apply trans_lt O (S O).apply le_n (S O).assumption.
258 rewrite > Hcut.assumption.
259 apply divides_to_mod_O.apply trans_lt O (S O).apply le_n (S O).assumption.
260 apply divides_fact.apply trans_lt O (S O).apply le_n (S O).assumption.
264 theorem not_divides_S_fact: \forall n,i:nat.
265 (S O) < i \to i \le n \to \not (divides i (S (fact n))).
267 apply divides_b_false_to_not_divides.
268 apply trans_lt O (S O).apply le_n (S O).assumption.
269 change with (eqb (mod (S (fact n)) i) O) = false.
270 rewrite > mod_S_fact.simplify.reflexivity.
271 assumption.assumption.
275 definition prime : nat \to Prop \def
276 \lambda n:nat. (S O) < n \land
277 (\forall m:nat. divides m n \to (S O) < m \to m = n).
279 theorem not_prime_O: \lnot (prime O).
280 simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
283 theorem not_prime_SO: \lnot (prime (S O)).
284 simplify.intro.elim H.apply not_le_Sn_n (S O) H1.
287 (* smallest factor *)
288 definition smallest_factor : nat \to nat \def
294 [ O \Rightarrow (S O)
295 | (S q) \Rightarrow min_aux q (S(S q)) (\lambda m.(eqb (mod (S(S q)) m) O))]].
298 theorem example1 : smallest_prime_factor (S(S(S O))) = (S(S(S O))).
299 normalize.reflexivity.
302 theorem example2: smallest_prime_factor (S(S(S(S O)))) = (S(S O)).
303 normalize.reflexivity.
306 theorem example3 : smallest_prime_factor (S(S(S(S(S(S(S O))))))) = (S(S(S(S(S(S(S O))))))).
307 simplify.reflexivity.
310 theorem lt_SO_smallest_factor:
311 \forall n:nat. (S O) < n \to (S O) < (smallest_factor n).
313 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
314 intro.apply nat_case m.intro. apply False_ind.apply not_le_Sn_n (S O) H.
317 S O < min_aux m1 (S(S m1)) (\lambda m.(eqb (mod (S(S m1)) m) O)).
318 apply lt_to_le_to_lt ? (S (S O)).
320 cut (S(S O)) = (S(S m1)) - m1.
323 apply sym_eq.apply plus_to_minus.apply le_S.apply le_n_Sn.
324 rewrite < sym_plus.simplify.reflexivity.
327 theorem lt_O_smallest_factor: \forall n:nat. O < n \to O < (smallest_factor n).
329 apply nat_case n.intro.apply False_ind.apply not_le_Sn_n O H.
330 intro.apply nat_case m.intro.
332 intros.apply trans_lt ? (S O).
333 simplify. apply le_n.
334 apply lt_SO_smallest_factor.simplify. apply le_S_S.
335 apply le_S_S.apply le_O_n.
338 theorem divides_smallest_factor_n :
339 \forall n:nat. O < n \to divides (smallest_factor n) n.
341 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O O H.
342 intro.apply nat_case m.intro. simplify.
343 apply witness ? ? (S O). simplify.reflexivity.
345 apply divides_b_true_to_divides.
346 apply lt_O_smallest_factor ? H.
348 eqb (mod (S(S m1)) (min_aux m1 (S(S m1))
349 (\lambda m.(eqb (mod (S(S m1)) m) O)))) O = true.
350 apply f_min_aux_true.
351 apply ex_intro nat ? (S(S m1)).
353 apply le_minus_m.apply le_n.
354 rewrite > mod_n_n.reflexivity.
355 apply trans_lt ? (S O).apply le_n (S O).simplify.
356 apply le_S_S.apply le_S_S.apply le_O_n.
359 theorem le_smallest_factor_n :
360 \forall n:nat. smallest_factor n \le n.
361 intro.apply nat_case n.simplify.reflexivity.
362 intro.apply nat_case m.simplify.reflexivity.
363 intro.apply divides_to_le.
364 simplify.apply le_S_S.apply le_O_n.
365 apply divides_smallest_factor_n.
366 simplify.apply le_S_S.apply le_O_n.
369 theorem lt_smallest_factor_to_not_divides: \forall n,i:nat.
370 (S O) < n \to (S O) < i \to i < (smallest_factor n) \to \lnot (divides i n).
372 apply nat_case n.intro.apply False_ind.apply not_le_Sn_O (S O) H.
373 intro.apply nat_case m.intro. apply False_ind.apply not_le_Sn_n (S O) H.
375 apply divides_b_false_to_not_divides.
376 apply trans_lt O (S O).apply le_n (S O).assumption.
377 change with (eqb (mod (S(S m1)) i) O) = false.
378 apply lt_min_aux_to_false
379 (\lambda i:nat.eqb (mod (S(S m1)) i) O) (S(S m1)) m1 i.
380 cut (S(S O)) = (S(S m1)-m1).
381 rewrite < Hcut.exact H1.
382 apply sym_eq. apply plus_to_minus.
383 apply le_S.apply le_n_Sn.
384 rewrite < sym_plus.simplify.reflexivity.
388 theorem prime_smallest_factor_n :
389 \forall n:nat. (S O) < n \to prime (smallest_factor n).
390 intro. change with (S(S O)) \le n \to (S O) < (smallest_factor n) \land
391 (\forall m:nat. divides m (smallest_factor n) \to (S O) < m \to m = (smallest_factor n)).
393 apply lt_SO_smallest_factor.assumption.
395 cut le m (smallest_factor n).
396 elim le_to_or_lt_eq m (smallest_factor n) Hcut.
398 apply transitive_divides m (smallest_factor n).
400 apply divides_smallest_factor_n.
401 apply trans_lt ? (S O). simplify. apply le_n. exact H.
402 apply lt_smallest_factor_to_not_divides.
403 exact H.assumption.assumption.assumption.
405 apply trans_lt O (S O).
407 apply lt_SO_smallest_factor.
412 theorem prime_to_smallest_factor: \forall n. prime n \to
413 smallest_factor n = n.
414 intro.apply nat_case n.intro.apply False_ind.apply not_prime_O H.
415 intro.apply nat_case m.intro.apply False_ind.apply not_prime_SO H.
418 (S O) < (S(S m1)) \land
419 (\forall m:nat. divides m (S(S m1)) \to (S O) < m \to m = (S(S m1))) \to
420 smallest_factor (S(S m1)) = (S(S m1)).
421 intro.elim H.apply H2.
422 apply divides_smallest_factor_n.
423 apply trans_lt ? (S O).simplify. apply le_n.assumption.
424 apply lt_SO_smallest_factor.
428 (* a number n > O is prime iff its smallest factor is n *)
429 definition primeb \def \lambda n:nat.
431 [ O \Rightarrow false
434 [ O \Rightarrow false
435 | (S q) \Rightarrow eqb (smallest_factor (S(S q))) (S(S q))]].
438 theorem example4 : primeb (S(S(S O))) = true.
439 normalize.reflexivity.
442 theorem example5 : primeb (S(S(S(S(S(S O)))))) = false.
443 normalize.reflexivity.
446 theorem example6 : primeb (S(S(S(S((S(S(S(S(S(S(S O)))))))))))) = true.
447 normalize.reflexivity.
450 theorem example7 : primeb (S(S(S(S(S(S((S(S(S(S((S(S(S(S(S(S(S O))))))))))))))))))) = true.
451 normalize.reflexivity.
454 theorem primeb_to_Prop: \forall n.
456 [ true \Rightarrow prime n
457 | false \Rightarrow \not (prime n)].
459 apply nat_case n.simplify.intro.elim H.apply not_le_Sn_O (S O) H1.
460 intro.apply nat_case m.simplify.intro.elim H.apply not_le_Sn_n (S O) H1.
463 match eqb (smallest_factor (S(S m1))) (S(S m1)) with
464 [ true \Rightarrow prime (S(S m1))
465 | false \Rightarrow \not (prime (S(S m1)))].
466 apply eqb_elim (smallest_factor (S(S m1))) (S(S m1)).
467 intro.change with prime (S(S m1)).
469 apply prime_smallest_factor_n.
470 simplify.apply le_S_S.apply le_S_S.apply le_O_n.
471 intro.change with \not (prime (S(S m1))).
472 change with prime (S(S m1)) \to False.
474 apply prime_to_smallest_factor.
478 theorem primeb_true_to_prime : \forall n:nat.
479 primeb n = true \to prime n.
482 [ true \Rightarrow prime n
483 | false \Rightarrow \not (prime n)].
485 apply primeb_to_Prop.
488 theorem primeb_false_to_not_prime : \forall n:nat.
489 primeb n = false \to \not (prime n).
492 [ true \Rightarrow prime n
493 | false \Rightarrow \not (prime n)].
495 apply primeb_to_Prop.
498 theorem decidable_prime : \forall n:nat.decidable (prime n).
499 intro.change with (prime n) \lor \not (prime n).
502 [ true \Rightarrow prime n
503 | false \Rightarrow \not (prime n)] \to (prime n) \lor \not (prime n).
504 apply Hcut.apply primeb_to_Prop.
505 elim (primeb n).left.apply H.right.apply H.
508 theorem prime_to_primeb_true: \forall n:nat.
509 prime n \to primeb n = true.
511 cut match (primeb n) with
512 [ true \Rightarrow prime n
513 | false \Rightarrow \not (prime n)] \to ((primeb n) = true).
514 apply Hcut.apply primeb_to_Prop.
515 elim primeb n.reflexivity.
516 absurd (prime n).assumption.assumption.
519 theorem not_prime_to_primeb_false: \forall n:nat.
520 \not(prime n) \to primeb n = false.
522 cut match (primeb n) with
523 [ true \Rightarrow prime n
524 | false \Rightarrow \not (prime n)] \to ((primeb n) = false).
525 apply Hcut.apply primeb_to_Prop.
527 absurd (prime n).assumption.assumption.