A Mec hanical Pro of of the Chinese Remainder Theorem Da vid M. Russino� Adv anced Micro Devices, Inc. david.russinoff@a md. co m http://www.onr.c om /us er /ru ss /da vid
Informal Statemen t L et m ; : : : ; m b e p airwise r elatively Theorem N 2 1 k prime mo duli and let a ; : : : ; a . Ther e exists x N N 2 2 1 k such that x a (mo d m ) � 1 1 x a (mo d m ) � 2 2 . . . x a (mo d m ) : � k k If x 0 satis�es the same c ongruenc es, then 0 x x (mo d m m m ) : � � � � k 1 2 1
A CL2 F ormalization (defun g-c-d (x y) (declare (xargs :measure (nfix (+ x y)))) (if (zp x) y (if (zp y) x (if (<= x y) (g-c-d x (- y x)) (g-c-d (- x y) y))))) (defun rel-prime (x y) (= (g-c-d x y) 1)) (defun congruent (x y m) (= (rem x m) (rem y m))) (defun congruent-all (x a m) (if (endp m) t (and (congruent x (car a) (car m)) (congruent-al l x (cdr a) (cdr m))))) (defthm chinese-remain de r-t he or em (implies (and (natp-all a) (rel-prime-modu li m) (= (len a) (len m))) (and (natp (crt-witness a m)) (congruent-all (crt-witness a m) a m)))) 2
Informal Pro of If x; y ar e r elatively prime, then ther e Lemma 1 N 2 exists s such that sy 1 (mo d x ). Z 2 � If x; y ; z and x is r elatively prime to b oth Lemma 2 N 2 y and z , then x is r elatively prime to y z . Pro of of CR T: Let M = m m m . F or i = 1 ; : : : ; k , let � � � 1 2 k M = M =m and �nd s suc h that s M 1 (mo d m ). Let � i i i i i i x = a s M + a s M + + a s M : � � � k k k 1 1 1 2 2 2 Then x a s M a (mo d m ) : � � i i i i i 3
Example Supp ose w e ha v e 10000 50000 and N � � 6 (mo d 25) N � 13 (mo d 36) N � 28 (mo d 49) N � Then w e ma y solv e for as follo ws: N = 25 36 49 = 44100 M � � = 36 49 = 1764 M � 1 = 25 49 = 1225 M � 2 = 25 36 = 900 M � 3 1764 s 1 (mo d 25) 14 s 1 (mo d 25) 9 (mo d 25) s � , � , � 1 1 1 1225 s 1 (mo d 36) 1 (mo d 36) s � , � 2 2 900 s 1 (mo d 49) 18 s 1 (mo d 49) 30 (mo d 49) s � , � , � 3 3 3 = 6, = 13, = 28 a a a 1 2 3 = + + x a M s a M s a M s 1 1 1 2 2 2 3 3 3 = 6 1764 9 + 13 1225 1 + 28 900 30 � � � � � � = 867281 29281 (mo d 44100) � = 29281 N 5
Pro of of Lemma 1 If x; y ar e r elatively prime, then ther e Lemma 1 N 2 exists s such that sy 1 (mo d x ). Z 2 � This is a sp ecial case of the follo wing: F or al l x; y , ther e exist r ; s such that 2 N 2 Z r x sy g cd ( x; y ) . + = The pro of is b y induction on x + y : (1) If x = 0, then r = 0 and s = 1. (2) If y = 0, then r = 1 and s = 0. (3) If 0 < x y , then �nd r 0 and s 0 suc h that � 0 0 r x + s ( y x ) = g cd ( x; y x ) = g cd ( x; y ) � � and let r = r 0 s 0 and s = s 0 . Then � 0 0 0 0 0 r x + sy = ( r s ) x + s y = r x + s ( y x ) = g cd ( x; y ) : � � < y < x , r s (4) If 0 then �nd 0 and 0 suc h that r 0 ( x y ) + s 0 y = g cd ( x y ; y ) = g cd ( x; y ) � � and let r = r 0 and s = s 0 r 0 . � 6
F ormal Pro of (mutual-recurs io n (defun r (x y) (declare (xargs :measure (nfix (+ x y)))) (if (zp x) 0 (if (zp y) 1 (if (<= x y) (- (r x (- y x)) (s x (- y x))) (r (- x y) y))))) (defun s (x y) (declare (xargs :measure (nfix (+ x y)))) (if (zp x) 1 (if (zp y) 0 (if (<= x y) (s x (- y x)) (- (s (- x y) y) (r (- x y) y)))))) ) (defthm r-s-lemma (implies (and (natp x) (natp y)) (= (+ (* (r x y) x) (* (s x y) y)) (g-c-d x y)))) 7
Pro of of Lemma 2 If x; y ; z and x is r elatively prime to b oth Lemma 2 N 2 y and z , then x is r elatively prime to y z . This is a consequence of the follo wing basic prop erties of g cd and primes: (1) g cd ( x; y ) divides b oth x and y . (2) If d divides b oth x and y , then d divides g cd ( x; y ) . (3) If x > 1 , then some prime divides x . (4) If a prime p divides ab , then p divides either a or b . It w ould tak e some w ork to pro v e these in A CL2. F ortunately , there is a more direct route to CR T. 8
Alternate Approac h L et x; y ; y ; : : : ; y and p = y y . If Lemma 3 N 2 � � � 1 2 k 1 k x is r elatively prime to e ach y , then ther e exist c; d Z 2 i such that cx + dp = 1 . Pro of: Let p 0 = y y . Assume that � � � 1 k � 1 r x + sy = 1 k and, b y induction, that 0 0 0 c x + d p = 1 : Then 0 0 0 ( sd ) p = ( sy )( d p ) k 0 = (1 r x )(1 c x ) � � 0 0 = 1 ( r + c r c x ) x: � � Th us, if c = r + c 0 r c 0 x and d = sd 0 , then � cx + dp = 1 : 9
F ormal Pro of (defun c (x l) (if (endp l) 0 (- (+ (r x (car l)) (c x (cdr l))) (* (r x (car l)) (c x (cdr l)) x)))) (defun d (x l) (if (endp l) 1 (* (s x (car l)) (d x (cdr l))))) (defthm c-d-lemma (implies (and (natp x) (natp-all l) (rel-prime-all x l)) (= (+ (* (c x l) x) (* (d x l) (prod l))) 1))) 10
De�nition of crt-witness (defun one-mod (x l) (* (d x l) (prod l) (d x l) (prod l))) (defthm rem-one-mod-1 (implies (and (natp x) (> x 1) (natp-all l) (rel-prime-all x l)) (= (rem (one-mod x l) x) 1))) (defthm rem-one-mod-0 (implies (and (natp x) (> x 1) (rel-prime-modu li l) (rel-prime-all x l) (member y l)) (= (rem (one-mod x l) y) 0))) (defun crt1 (a m l) (if (endp a) 0 (+ (* (car a) (one-mod (car m) (remove (car m) l))) (crt1 (cdr a) (cdr m) l)))) (defun crt-witness (a m) (crt1 a m m)) 11
The Main Lemma W e pro v e the follo wing generalization of CR T: (defthm crt1-lemma (implies (and (natp-all a) (rel-prime-modu li l) (sublistp m l) (= (len a) (len m))) (congruent-all (crt1 a m l) a m))) The pro of is b y induction, as suggested b y the de�nition: (defun crt1 (a m l) (if (endp a) 0 (+ (* (car a) (one-mod (car m) (remove (car m) l))) (crt1 (cdr a) (cdr m) l)))) In the inductiv e case, the conclusion of the lemma expands as follo ws: (and (congruent (+ (* (car a) (one-mod (car m) (remove (car m) l))) (crt1 (cdr a) (cdr m) l)) (car a) (car m)) (congruent-al l (+ (* (car a) (one-mod (car m) (remove (car m) l))) (crt1 (cdr a) (cdr m) l)) (cdr a) (cdr m))). 12
The Final Result CR T is deriv ed as an instance of crt1-lemma : (defthm crt1-lemma (implies (and (natp-all a) (rel-prime-modu li l) (sublistp m l) (= (len a) (len m))) (congruent-all (crt1 a m l) a m))) (defthm chinese-remain de r-t he or em (implies (and (natp-all a) (rel-prime-modu li m) (= (len a) (len m))) (and (natp (crt-witness a m)) (congruent-all (crt a m) a m)))) 13
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