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/gcd".
17 include "nat/primes.ma".
18 include "nat/lt_arith.ma".
20 let rec gcd_aux p m n: nat \def
21 match divides_b n m with
26 |(S q) \Rightarrow gcd_aux q n (m \mod n)]].
28 definition gcd : nat \to nat \to nat \def
34 | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
38 | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
40 theorem divides_mod: \forall p,m,n:nat. O < n \to p \divides m \to p \divides n \to
41 p \divides (m \mod n).
42 intros.elim H1.elim H2.
43 (* apply (witness ? ? (n2 - n1*(m / n))). *)
45 rewrite > distr_times_minus.
46 rewrite < H3 in \vdash (? ? ? (? % ?)).
47 rewrite < assoc_times.
48 rewrite < H4 in \vdash (? ? ? (? ? (? % ?))).
49 apply sym_eq.apply plus_to_minus.
52 rewrite < (div_mod ? ? H).
57 theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
58 p \divides (m \mod n) \to p \divides n \to p \divides m.
59 intros.elim H1.elim H2.
60 apply (witness p m ((n1*(m / n))+n2)).
61 rewrite > distr_times_plus.
63 rewrite < assoc_times.
64 rewrite < H4.rewrite < sym_times.
65 apply div_mod.assumption.
68 theorem divides_gcd_aux_mn: \forall p,m,n. O < n \to n \le m \to n \le p \to
69 gcd_aux p m n \divides m \land gcd_aux p m n \divides n.
71 absurd (O < n).assumption.apply le_to_not_lt.assumption.
72 cut ((n1 \divides m) \lor (n1 \ndivides m)).
74 elim Hcut.rewrite > divides_to_divides_b_true.
76 split.assumption.apply (witness n1 n1 (S O)).apply times_n_SO.
77 assumption.assumption.
78 rewrite > not_divides_to_divides_b_false.
80 cut (gcd_aux n n1 (m \mod n1) \divides n1 \land
81 gcd_aux n n1 (m \mod n1) \divides mod m n1).
83 split.apply (divides_mod_to_divides ? ? n1).
84 assumption.assumption.assumption.assumption.
86 cut (O \lt m \mod n1 \lor O = mod m n1).
87 elim Hcut1.assumption.
88 apply False_ind.apply H4.apply mod_O_to_divides.
89 assumption.apply sym_eq.assumption.
90 apply le_to_or_lt_eq.apply le_O_n.
92 apply lt_mod_m_m.assumption.
94 apply (trans_le ? n1).
95 change with (m \mod n1 < n1).
96 apply lt_mod_m_m.assumption.assumption.
97 assumption.assumption.
98 apply (decidable_divides n1 m).assumption.
101 theorem divides_gcd_nm: \forall n,m.
102 gcd n m \divides m \land gcd n m \divides n.
104 (*CSC: simplify simplifies too much because of a redex in gcd *)
110 | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
114 | (S p) \Rightarrow gcd_aux (S p) n (S p) ] ] \divides m
120 | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
124 | (S p) \Rightarrow gcd_aux (S p) n (S p) ] ] \divides n).
125 apply (leb_elim n m).
127 simplify.intros.split.
128 apply (witness m m (S O)).apply times_n_SO.
129 apply (witness m O O).apply times_n_O.
131 (gcd_aux (S m1) m (S m1) \divides m
133 gcd_aux (S m1) m (S m1) \divides (S m1)).
134 apply divides_gcd_aux_mn.
135 unfold lt.apply le_S_S.apply le_O_n.
136 assumption.apply le_n.
139 simplify.intros.split.
140 apply (witness n O O).apply times_n_O.
141 apply (witness n n (S O)).apply times_n_SO.
143 (gcd_aux (S m1) n (S m1) \divides (S m1)
145 gcd_aux (S m1) n (S m1) \divides n).
146 cut (gcd_aux (S m1) n (S m1) \divides n
148 gcd_aux (S m1) n (S m1) \divides S m1).
149 elim Hcut.split.assumption.assumption.
150 apply divides_gcd_aux_mn.
151 unfold lt.apply le_S_S.apply le_O_n.
152 apply not_lt_to_le.unfold Not. unfold lt.intro.apply H.
153 rewrite > H1.apply (trans_le ? (S n)).
154 apply le_n_Sn.assumption.apply le_n.
157 theorem divides_gcd_n: \forall n,m. gcd n m \divides n.
159 exact (proj2 ? ? (divides_gcd_nm n m)).
162 theorem divides_gcd_m: \forall n,m. gcd n m \divides m.
164 exact (proj1 ? ? (divides_gcd_nm n m)).
168 theorem divides_times_gcd_aux: \forall p,m,n,d,c.
169 O \lt c \to O < n \to n \le m \to n \le p \to
170 d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd_aux p m n.
175 | apply le_to_not_lt.
179 cut (n1 \divides m \lor n1 \ndivides m)
181 [ rewrite > divides_to_divides_b_true
187 | rewrite > not_divides_to_divides_b_false
191 | cut (O \lt m \mod n1 \lor O = m \mod n1)
194 | absurd (n1 \divides m)
195 [ apply mod_O_to_divides
203 | apply le_to_or_lt_eq.
209 | apply le_S_S_to_le.
210 apply (trans_le ? n1)
211 [ change with (m \mod n1 < n1).
217 | rewrite < times_mod
218 [ rewrite < (sym_times c m).
219 rewrite < (sym_times c n1).
221 [ rewrite > (S_pred c)
222 [ rewrite > (S_pred n1)
223 [ apply (lt_O_times_S_S)
239 | apply (decidable_divides n1 m).
245 (*a particular case of the previous theorem (setting c=1)*)
246 theorem divides_gcd_aux: \forall p,m,n,d. O < n \to n \le m \to n \le p \to
247 d \divides m \to d \divides n \to d \divides gcd_aux p m n.
249 rewrite > (times_n_SO (gcd_aux p m n)).
250 rewrite < (sym_times (S O)).
251 apply (divides_times_gcd_aux)
256 | rewrite > (sym_times (S O)).
257 rewrite < (times_n_SO m).
259 | rewrite > (sym_times (S O)).
260 rewrite < (times_n_SO n).
265 theorem divides_d_times_gcd: \forall m,n,d,c.
266 O \lt c \to d \divides (c*m) \to d \divides (c*n) \to d \divides c*gcd n m.
274 | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
278 | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]).
280 [ apply (nat_case1 n)
285 change with (d \divides c*gcd_aux (S m1) m (S m1)).
286 apply divides_times_gcd_aux
292 | apply (le_n (S m1))
298 | apply (nat_case1 m)
303 change with (d \divides c * gcd_aux (S m1) n (S m1)).
304 apply divides_times_gcd_aux
306 change with (O \lt c).
312 | apply (le_n (S m1)).
321 (*a particular case of the previous theorem (setting c=1)*)
322 theorem divides_d_gcd: \forall m,n,d.
323 d \divides m \to d \divides n \to d \divides gcd n m.
325 rewrite > (times_n_SO (gcd n m)).
326 rewrite < (sym_times (S O)).
327 apply (divides_d_times_gcd)
329 | rewrite > (sym_times (S O)).
330 rewrite < (times_n_SO m).
332 | rewrite > (sym_times (S O)).
333 rewrite < (times_n_SO n).
338 theorem eq_minus_gcd_aux: \forall p,m,n.O < n \to n \le m \to n \le p \to
339 \exists a,b. a*n - b*m = gcd_aux p m n \lor b*m - a*n = gcd_aux p m n.
344 |apply le_to_not_lt.assumption
347 [cut (n1 \divides m \lor n1 \ndivides m)
350 [rewrite > divides_to_divides_b_true
352 apply (ex_intro ? ? (S O)).
353 apply (ex_intro ? ? O).
362 |rewrite > not_divides_to_divides_b_false
364 (\exists a,b.a*n1 - b*m = gcd_aux n n1 (m \mod n1)
365 \lor b*m - a*n1 = gcd_aux n n1 (m \mod n1)).
367 (\exists a,b.a*(m \mod n1) - b*n1= gcd_aux n n1 (m \mod n1)
368 \lor b*n1 - a*(m \mod n1) = gcd_aux n n1 (m \mod n1))
369 [elim Hcut2.elim H5.elim H6
372 apply (ex_intro ? ? (a1+a*(m / n1))).
373 apply (ex_intro ? ? a).
376 rewrite < (sym_times n1).
377 rewrite > distr_times_plus.
378 rewrite > (sym_times n1).
379 rewrite > (sym_times n1).
380 rewrite > (div_mod m n1) in \vdash (? ? (? % ?) ?)
381 [rewrite > assoc_times.
383 rewrite > distr_times_plus.
384 rewrite < eq_minus_minus_minus_plus.
387 [rewrite < minus_n_n.reflexivity
394 apply (ex_intro ? ? (a1+a*(m / n1))).
395 apply (ex_intro ? ? a).
397 (* clear Hcut2.clear H5.clear H6.clear H. *)
399 rewrite > distr_times_plus.
401 rewrite > (sym_times n1).
402 rewrite > (div_mod m n1) in \vdash (? ? (? ? %) ?)
403 [rewrite > distr_times_plus.
404 rewrite > assoc_times.
405 rewrite < eq_minus_minus_minus_plus.
408 [rewrite < minus_n_n.reflexivity
414 |apply (H n1 (m \mod n1))
415 [cut (O \lt m \mod n1 \lor O = m \mod n1)
418 |absurd (n1 \divides m)
419 [apply mod_O_to_divides
426 |apply le_to_or_lt_eq.
433 apply (trans_le ? n1)
434 [change with (m \mod n1 < n1).
445 |apply (decidable_divides n1 m).
448 |apply (lt_to_le_to_lt ? n1);assumption
453 theorem eq_minus_gcd:
454 \forall m,n.\exists a,b.a*n - b*m = (gcd n m) \lor b*m - a*n = (gcd n m).
457 apply (leb_elim n m).
460 apply (ex_intro ? ? O).
461 apply (ex_intro ? ? (S O)).
464 apply sym_eq.apply minus_n_O.
468 a*(S m1) - b*m = (gcd_aux (S m1) m (S m1))
469 \lor b*m - a*(S m1) = (gcd_aux (S m1) m (S m1))).
470 apply eq_minus_gcd_aux.
471 unfold lt. apply le_S_S.apply le_O_n.
472 assumption.apply le_n.
475 apply (ex_intro ? ? (S O)).
476 apply (ex_intro ? ? O).
479 apply sym_eq.apply minus_n_O.
483 a*n - b*(S m1) = (gcd_aux (S m1) n (S m1))
484 \lor b*(S m1) - a*n = (gcd_aux (S m1) n (S m1))).
487 a*(S m1) - b*n = (gcd_aux (S m1) n (S m1))
489 b*n - a*(S m1) = (gcd_aux (S m1) n (S m1))).
490 elim Hcut.elim H2.elim H3.
491 apply (ex_intro ? ? a1).
492 apply (ex_intro ? ? a).
494 apply (ex_intro ? ? a1).
495 apply (ex_intro ? ? a).
497 apply eq_minus_gcd_aux.
498 unfold lt. apply le_S_S.apply le_O_n.
499 apply lt_to_le.apply not_le_to_lt.assumption.
503 (* some properties of gcd *)
505 theorem gcd_O_n: \forall n:nat. gcd O n = n.
506 intro.simplify.reflexivity.
509 theorem gcd_O_to_eq_O:\forall m,n:nat. (gcd m n) = O \to
511 intros.cut (O \divides n \land O \divides m).
512 elim Hcut.elim H2.split.
513 assumption.elim H1.assumption.
515 apply divides_gcd_nm.
518 theorem lt_O_gcd:\forall m,n:nat. O < n \to O < gcd m n.
520 apply (nat_case1 (gcd m n)).
522 generalize in match (gcd_O_to_eq_O m n H1).
524 rewrite < H4 in \vdash (? ? %).assumption.
525 intros.unfold lt.apply le_S_S.apply le_O_n.
528 theorem gcd_n_n: \forall n.gcd n n = n.
531 |apply le_to_le_to_eq
537 [apply lt_O_gcd.apply lt_O_S
539 [apply divides_n_n|apply divides_n_n]
545 theorem gcd_SO_to_lt_O: \forall i,n. (S O) < n \to gcd i n = (S O) \to
548 elim (le_to_or_lt_eq ? ? (le_O_n i))
550 |absurd ((gcd i n) = (S O))
555 apply (lt_to_not_eq (S O) n H).
556 apply sym_eq.assumption
561 theorem gcd_SO_to_lt_n: \forall i,n. (S O) < n \to i \le n \to gcd i n = (S O) \to
564 elim (le_to_or_lt_eq ? ? H1)
566 |absurd ((gcd i n) = (S O))
571 apply (lt_to_not_eq (S O) n H).
572 apply sym_eq.assumption
577 theorem gcd_n_times_nm: \forall n,m. O < m \to gcd n (n*m) = n.
578 intro.apply (nat_case n)
583 [apply lt_O_S|apply divides_gcd_n]
585 [apply lt_O_gcd.rewrite > (times_n_O O).
586 apply lt_times[apply lt_O_S|assumption]
588 [apply (witness ? ? m1).reflexivity
596 theorem symmetric_gcd: symmetric nat gcd.
597 (*CSC: bug here: unfold symmetric does not work *)
599 (\forall n,m:nat. gcd n m = gcd m n).
601 cut (O < (gcd n m) \lor O = (gcd n m)).
603 cut (O < (gcd m n) \lor O = (gcd m n)).
606 apply divides_to_le.assumption.
607 apply divides_d_gcd.apply divides_gcd_n.apply divides_gcd_m.
608 apply divides_to_le.assumption.
609 apply divides_d_gcd.apply divides_gcd_n.apply divides_gcd_m.
612 elim Hcut2.rewrite > H2.rewrite > H3.reflexivity.
613 apply gcd_O_to_eq_O.apply sym_eq.assumption.
614 apply le_to_or_lt_eq.apply le_O_n.
617 elim Hcut1.rewrite > H1.rewrite > H2.reflexivity.
618 apply gcd_O_to_eq_O.apply sym_eq.assumption.
619 apply le_to_or_lt_eq.apply le_O_n.
622 variant sym_gcd: \forall n,m:nat. gcd n m = gcd m n \def
625 theorem le_gcd_times: \forall m,n,p:nat. O< p \to gcd m n \le gcd m (n*p).
627 apply (nat_case n).apply le_n.
631 rewrite > (times_n_O O).
632 apply lt_times.unfold lt.apply le_S_S.apply le_O_n.assumption.
634 apply (transitive_divides ? (S m1)).
636 apply (witness ? ? p).reflexivity.
640 theorem gcd_times_SO_to_gcd_SO: \forall m,n,p:nat. O < n \to O < p \to
641 gcd m (n*p) = (S O) \to gcd m n = (S O).
643 apply antisymmetric_le.
645 apply le_gcd_times.assumption.
646 change with (O < gcd m n).
647 apply lt_O_gcd.assumption.
650 (* for the "converse" of the previous result see the end of this development *)
652 theorem eq_gcd_SO_to_not_divides: \forall n,m. (S O) < n \to
653 (gcd n m) = (S O) \to \lnot (divides n m).
656 generalize in match H1.
660 [elim (gcd_times_SO_to_gcd_SO n n n2 ? ? H4)
661 [cut (gcd n (n*n2) = n)
662 [apply (lt_to_not_eq (S O) n)
663 [assumption|rewrite < H4.assumption]
664 |apply gcd_n_times_nm.assumption
666 |apply (trans_lt ? (S O))[apply le_n|assumption]
669 |elim (le_to_or_lt_eq O n2 (le_O_n n2));
672 apply (le_to_not_lt n (S O))
675 [rewrite > H4.apply lt_O_S
677 [apply (witness ? ? n2).reflexivity
687 theorem gcd_SO_n: \forall n:nat. gcd (S O) n = (S O).
689 apply antisym_le.apply divides_to_le.unfold lt.apply le_n.
691 cut (O < gcd (S O) n \lor O = gcd (S O) n).
692 elim Hcut.assumption.
694 apply (not_eq_O_S O).
695 cut ((S O)=O \land n=O).
696 elim Hcut1.apply sym_eq.assumption.
697 apply gcd_O_to_eq_O.apply sym_eq.assumption.
698 apply le_to_or_lt_eq.apply le_O_n.
701 theorem divides_gcd_mod: \forall m,n:nat. O < n \to
702 divides (gcd m n) (gcd n (m \mod n)).
705 apply divides_mod.assumption.
711 theorem divides_mod_gcd: \forall m,n:nat. O < n \to
712 divides (gcd n (m \mod n)) (gcd m n) .
716 apply (divides_mod_to_divides ? ? n).
722 theorem gcd_mod: \forall m,n:nat. O < n \to
723 (gcd n (m \mod n)) = (gcd m n) .
725 apply antisymmetric_divides.
726 apply divides_mod_gcd.assumption.
727 apply divides_gcd_mod.assumption.
732 theorem prime_to_gcd_SO: \forall n,m:nat. prime n \to n \ndivides m \to
734 intros.unfold prime in H.
737 apply not_lt_to_le.unfold Not.unfold lt.
739 apply H1.rewrite < (H3 (gcd n m)).
741 apply divides_gcd_n.assumption.
742 cut (O < gcd n m \lor O = gcd n m).
743 elim Hcut.assumption.
745 apply (not_le_Sn_O (S O)).
747 elim Hcut1.rewrite < H5 in \vdash (? ? %).assumption.
748 apply gcd_O_to_eq_O.apply sym_eq.assumption.
749 apply le_to_or_lt_eq.apply le_O_n.
752 theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
753 n \divides p \lor n \divides q.
755 cut (n \divides p \lor n \ndivides p)
759 cut (\exists a,b. a*n - b*p = (S O) \lor b*p - a*n = (S O))
760 [elim Hcut1.elim H3.elim H4
762 rewrite > (times_n_SO q).rewrite < H5.
763 rewrite > distr_times_minus.
764 rewrite > (sym_times q (a1*p)).
765 rewrite > (assoc_times a1).
769 applyS (witness n (n*(q*a-a1*n2)) (q*a-a1*n2))
771 applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
773 rewrite < (sym_times n).rewrite < assoc_times.
774 rewrite > (sym_times q).rewrite > assoc_times.
775 rewrite < (assoc_times a1).rewrite < (sym_times n).
776 rewrite > (assoc_times n).
777 rewrite < distr_times_minus.
778 apply (witness ? ? (q*a-a1*n2)).reflexivity
781 rewrite > (times_n_SO q).rewrite < H5.
782 rewrite > distr_times_minus.
783 rewrite > (sym_times q (a1*p)).
784 rewrite > (assoc_times a1).
785 elim H1.rewrite > H6.
786 rewrite < sym_times.rewrite > assoc_times.
787 rewrite < (assoc_times q).
788 rewrite < (sym_times n).
789 rewrite < distr_times_minus.
790 apply (witness ? ? (n2*a1-q*a)).reflexivity
791 ](* end second case *)
792 |rewrite < (prime_to_gcd_SO n p)
793 [apply eq_minus_gcd|assumption|assumption
797 |apply (decidable_divides n p).
798 apply (trans_lt ? (S O))
799 [unfold lt.apply le_n
800 |unfold prime in H.elim H. assumption
805 theorem eq_gcd_times_SO: \forall m,n,p:nat. O < n \to O < p \to
806 gcd m n = (S O) \to gcd m p = (S O) \to gcd m (n*p) = (S O).
808 apply antisymmetric_le.
811 cut (divides (smallest_factor (gcd m (n*p))) n \lor
812 divides (smallest_factor (gcd m (n*p))) p).
814 apply (not_le_Sn_n (S O)).
815 change with ((S O) < (S O)).
816 rewrite < H2 in \vdash (? ? %).
817 apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
818 apply lt_SO_smallest_factor.assumption.
820 rewrite > H2.unfold lt.apply le_n.
821 apply divides_d_gcd.assumption.
822 apply (transitive_divides ? (gcd m (n*p))).
823 apply divides_smallest_factor_n.
824 apply (trans_lt ? (S O)). unfold lt. apply le_n. assumption.
826 apply (not_le_Sn_n (S O)).
827 change with ((S O) < (S O)).
828 rewrite < H3 in \vdash (? ? %).
829 apply (lt_to_le_to_lt ? (smallest_factor (gcd m (n*p)))).
830 apply lt_SO_smallest_factor.assumption.
832 rewrite > H3.unfold lt.apply le_n.
833 apply divides_d_gcd.assumption.
834 apply (transitive_divides ? (gcd m (n*p))).
835 apply divides_smallest_factor_n.
836 apply (trans_lt ? (S O)). unfold lt. apply le_n. assumption.
838 apply divides_times_to_divides.
839 apply prime_smallest_factor_n.
841 apply (transitive_divides ? (gcd m (n*p))).
842 apply divides_smallest_factor_n.
843 apply (trans_lt ? (S O)).unfold lt. apply le_n. assumption.
845 change with (O < gcd m (n*p)).
847 rewrite > (times_n_O O).
848 apply lt_times.assumption.assumption.
851 theorem gcd_SO_to_divides_times_to_divides: \forall m,n,p:nat. O < n \to
852 gcd n m = (S O) \to n \divides (m*p) \to n \divides p.
854 cut (n \divides p \lor n \ndivides p)
857 |cut (\exists a,b. a*n - b*m = (S O) \lor b*m - a*n = (S O))
858 [elim Hcut1.elim H4.elim H5
860 rewrite > (times_n_SO p).rewrite < H6.
861 rewrite > distr_times_minus.
862 rewrite > (sym_times p (a1*m)).
863 rewrite > (assoc_times a1).
865 applyS (witness n ? ? (refl_eq ? ?)) (* timeout=50 *).
867 rewrite > (times_n_SO p).rewrite < H6.
868 rewrite > distr_times_minus.
869 rewrite > (sym_times p (a1*m)).
870 rewrite > (assoc_times a1).
872 applyS (witness n ? ? (refl_eq ? ?)).
873 ](* end second case *)
874 |rewrite < H1.apply eq_minus_gcd.
877 |apply (decidable_divides n p).