#| Copyright (C) 1994 by David Russinoff. All Rights Reserved. You may copy and distribute verbatim copies of this Nqthm-1992 event script as you receive it, in any medium, including embedding it verbatim in derivative works, provided that you conspicuously and appropriately publish on each copy a valid copyright notice "Copyright (C) 1994 by David Russinoff. All Rights Reserved." NO WARRANTY David Russinoff PROVIDES ABSOLUTELY NO WARRANTY. THE EVENT SCRIPT IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, ANY IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE SCRIPT IS WITH YOU. SHOULD THE SCRIPT PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR CORRECTION. IN NO EVENT WILL David Russinoff BE LIABLE TO YOU FOR ANY DAMAGES, ANY LOST PROFITS, LOST MONIES, OR OTHER SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THIS SCRIPT (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY THIRD PARTIES), EVEN IF YOU HAVE ADVISED US OF THE POSSIBILITY OF SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY. |# ; Mon Sep 17 12:10:54 CDT 1990 ; This events file is an addition to the events files described in A ; Computational Logic Handbook. ; This file is a new version of the proof of Gauss's law of quadratic ; reciprocity. It was composed entirely by David Russinoff, who also ; composed the Wilson and Gauss events in basic.events. According to ; Russinoff, the version below is much better than the one in basic.events. ; This version also corresponds to a forthcoming paper of Russinoff in ; the Journal of Automated Reasoning. (NOTE-LIB "wilson" T) (DEFN SQUARES (N P) (IF (ZEROP N) (CONS 0 NIL) (CONS (REMAINDER (TIMES N N) P) (SQUARES (SUB1 N) P)))) (DEFN RESIDUE (A P) (MEMBER (REMAINDER A P) (SQUARES P P))) (PROVE-LEMMA ALL-SQUARES-1 NIL (IMPLIES (AND (NOT (ZEROP P)) (LEQ M N)) (MEMBER (REMAINDER (TIMES M M) P) (SQUARES N P)))) (PROVE-LEMMA ALL-SQUARES-2 NIL (EQUAL (REMAINDER (TIMES Y Y) P) (REMAINDER (TIMES (REMAINDER Y P) (REMAINDER Y P)) P)) ((USE (TIMES-MOD-1 (X Y) (N P)) (TIMES-MOD-3 (B (REMAINDER Y P)) (A Y) (N P))) (DISABLE TIMES-MOD-1 TIMES-MOD-3))) (PROVE-LEMMA ALL-SQUARES (REWRITE) (IMPLIES (AND (NOT (ZEROP P)) (NOT (MEMBER X (SQUARES P P)))) (NOT (EQUAL X (REMAINDER (TIMES Y Y) P)))) ((USE (ALL-SQUARES-1 (N P) (M (REMAINDER Y P))) (ALL-SQUARES-2)) (DISABLE TIMES-MOD-1 TIMES-MOD-3))) (PROVE-LEMMA EULER-1-1 NIL (IMPLIES (NOT (DIVIDES 2 P)) (EQUAL (TIMES 2 (QUOTIENT P 2)) (SUB1 P)))) (PROVE-LEMMA EULER-1-2 NIL (IMPLIES (NOT (DIVIDES 2 P)) (EQUAL (EXP (TIMES I I) (QUOTIENT P 2)) (EXP I (SUB1 P)))) ((USE (EXP-EXP (J 2) (K (QUOTIENT P 2))) (EULER-1-1)) (DISABLE EXP-EXP))) (PROVE-LEMMA EULER-1-3 NIL (IMPLIES (EQUAL (REMAINDER A P) (REMAINDER B P)) (EQUAL (REMAINDER (EXP A C) P) (REMAINDER (EXP B C) P))) ((USE (REMAINDER-EXP (I C) (N P)) (REMAINDER-EXP (A B) (I C) (N P))) (DISABLE REMAINDER-EXP))) (PROVE-LEMMA EULER-1-4 NIL (IMPLIES (AND (PRIME P) (NOT (DIVIDES 2 P)) (NOT (DIVIDES P I))) (EQUAL (REMAINDER (EXP (TIMES I I) (QUOTIENT P 2)) P) 1)) ((USE (EULER-1-2)) (DISABLE LESSP-REMAINDER-DIVISOR PRIME))) (PROVE-LEMMA EULER-1-5 NIL (IMPLIES (AND (PRIME P) (NOT (DIVIDES P A)) (EQUAL (REMAINDER A P) (REMAINDER (TIMES I I) P))) (NOT (DIVIDES P I))) ((USE (PRIME-KEY-REWRITE (A I) (B I))) (DISABLE PRIME-KEY-REWRITE PRIME))) (PROVE-LEMMA EULER-1-6 NIL (IMPLIES (AND (PRIME P) (NOT (DIVIDES 2 P)) (NOT (DIVIDES P A)) (EQUAL (REMAINDER A P) (REMAINDER (TIMES I I) P))) (EQUAL (REMAINDER (EXP A (QUOTIENT P 2)) P) 1)) ((USE (EULER-1-4) (EULER-1-5) (EULER-1-3 (B (TIMES I I)) (C (QUOTIENT P 2)))) (DISABLE PRIME LESSP-REMAINDER-DIVISOR B-I-LEMMA2 LESSP SUB1-NNUMBERP REMAINDER-0-CROCK REMAINDER))) (PROVE-LEMMA EULER-1-7 (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES 2 P)) (NOT (DIVIDES P A)) (MEMBER (REMAINDER A P) (SQUARES I P))) (EQUAL (REMAINDER (EXP A (QUOTIENT P 2)) P) 1)) ((USE (EULER-1-6)) (INDUCT (SQUARES I P)) (DISABLE PRIME REMAINDER LESSP-REMAINDER-DIVISOR))) (PROVE-LEMMA EULER-1 (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES 2 P)) (NOT (DIVIDES P A)) (RESIDUE A P)) (EQUAL (REMAINDER (EXP A (QUOTIENT P 2)) P) 1)) ((DISABLE PRIME))) (DEFN COMPLEMENT (J A P) (REMAINDER (TIMES (INVERSE J P) A) P)) (DISABLE INVERSE) (PROVE-LEMMA COMPLEMENT-WORKS (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P J))) (EQUAL (REMAINDER (TIMES J (COMPLEMENT J A P)) P) (REMAINDER A P))) ((USE (INVERSE-INVERTS) (TIMES-MOD-3 (A (TIMES J (INVERSE J P))) (B A) (N P))) (DISABLE INVERSE-INVERTS TIMES-MOD-3 PRIME))) (PROVE-LEMMA BOUNDED-COMPLEMENT (REWRITE) (IMPLIES (NOT (ZEROP P)) (LESSP (COMPLEMENT J A P) P))) (DISABLE COMPLEMENT) (PROVE-LEMMA NON-ZEROP-COMPLEMENT (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P J)) (NOT (DIVIDES P A))) (NOT (ZEROP (COMPLEMENT J A P)))) ((USE (COMPLEMENT-WORKS)) (DISABLE COMPLEMENT-WORKS PRIME))) (PROVE-LEMMA COMPLEMENT-IS-UNIQUE (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P A)) (EQUAL (REMAINDER (TIMES J X) P) (REMAINDER A P))) (EQUAL (COMPLEMENT J A P) (REMAINDER X P))) ((USE (COMPLEMENT-WORKS) (THM-55-SPECIALIZED-TO-PRIMES (M J) (Y (COMPLEMENT J A P)))) (DISABLE COMPLEMENT-WORKS THM-55-SPECIALIZED-TO-PRIMES PRIME))) (DISABLE SQUARES) (PROVE-LEMMA NO-SELF-COMPLEMENT (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P J)) (NOT (DIVIDES P A)) (NOT (RESIDUE A P))) (NOT (EQUAL J (COMPLEMENT J A P)))) ((USE (COMPLEMENT-WORKS) (ALL-SQUARES (X (REMAINDER A P)) (Y J))) (DISABLE COMPLEMENT-WORKS ALL-SQUARES PRIME1))) (PROVE-LEMMA COMPLEMENT-OF-COMPLEMENT (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P J)) (NOT (DIVIDES P A))) (EQUAL (COMPLEMENT (COMPLEMENT J A P) A P) (REMAINDER J P))) ((USE (COMPLEMENT-WORKS) (COMPLEMENT-IS-UNIQUE (J (COMPLEMENT J A P)) (X J))) (DISABLE COMPLEMENT-WORKS COMPLEMENT-IS-UNIQUE))) (DEFN COMPLEMENTS (I A P) (IF (ZEROP I) NIL (IF (MEMBER I (COMPLEMENTS (SUB1 I) A P)) (COMPLEMENTS (SUB1 I) A P) (CONS I (CONS (COMPLEMENT I A P) (COMPLEMENTS (SUB1 I) A P)))))) (PROVE-LEMMA ALL-NON-ZEROP-COMPLEMENTS (REWRITE) (IMPLIES (AND (PRIME P) (LESSP I P) (NOT (DIVIDES P A))) (ALL-NON-ZEROP (COMPLEMENTS I A P))) ((USE (NON-ZEROP-COMPLEMENT (J I))) (INDUCT (COMPLEMENTS I A P)))) (PROVE-LEMMA BOUNDED-COMPLEMENTS (REWRITE) (IMPLIES (LESSP I P) (ALL-LESSEQP (COMPLEMENTS I A P) (SUB1 P))) ((USE (BOUNDED-COMPLEMENT (J I))) (INDUCT (COMPLEMENTS I A P)))) (PROVE-LEMMA SUBSETP-POSITIVES-COMPLEMENTS (REWRITE) (SUBSETP (POSITIVES N) (COMPLEMENTS N A P))) (PROVE-LEMMA COMPLEMENTS-CLOSED-1 NIL (IMPLIES (AND (PRIME P) (NOT (ZEROP I)) (LESSP I P) (NOT (DIVIDES P A)) (MEMBER J (COMPLEMENTS I A P))) (MEMBER (COMPLEMENT J A P) (COMPLEMENTS I A P))) ((USE (COMPLEMENT-OF-COMPLEMENT (J I))) (INDUCT (COMPLEMENTS I A P)) (DISABLE COMPLEMENT-OF-COMPLEMENT))) (PROVE-LEMMA COMPLEMENTS-CLOSED-2 NIL (IMPLIES (AND (PRIME P) (NOT (ZEROP I)) (NOT (ZEROP J)) (LESSP I P) (LESSP J P) (NOT (DIVIDES P A)) (MEMBER (COMPLEMENT J A P) (COMPLEMENTS I A P))) (MEMBER J (COMPLEMENTS I A P))) ((USE (COMPLEMENT-OF-COMPLEMENT) (COMPLEMENTS-CLOSED-1 (J (COMPLEMENT J A P)))) (DISABLE COMPLEMENT-OF-COMPLEMENT COMPLEMENTS))) (PROVE-LEMMA ALL-DISTINCT-COMPLEMENTS-1 NIL (IMPLIES (AND (PRIME P) (LESSP I P) (NOT (DIVIDES P A)) (NOT (RESIDUE A P)) (ALL-DISTINCT (COMPLEMENTS (SUB1 I) A P))) (ALL-DISTINCT (COMPLEMENTS I A P))) ((USE (COMPLEMENTS-CLOSED-2 (J I) (I (SUB1 I))) (NO-SELF-COMPLEMENT (J I))) (DISABLE PRIME))) (PROVE-LEMMA ALL-DISTINCT-COMPLEMENTS (REWRITE) (IMPLIES (AND (PRIME P) (LESSP I P) (NOT (DIVIDES P A)) (NOT (RESIDUE A P))) (ALL-DISTINCT (COMPLEMENTS I A P))) ((USE (ALL-DISTINCT-COMPLEMENTS-1)) (INDUCT 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PRIME REMAINDER TIMES-LIST COMPLEMENTS QUOTIENT LENGTH))) (PROVE-LEMMA SUB1-LENGTH-DELETE (REWRITE) (IMPLIES (MEMBER X B) (EQUAL (LENGTH (DELETE X B)) (SUB1 (LENGTH B))))) (PROVE-LEMMA EQUAL-LENGTH-PERM NIL (IMPLIES (PERM A B) (EQUAL (LENGTH A) (LENGTH B))) ((INDUCT (PERM A B)))) (PROVE-LEMMA LENGTH-POSITIVES (REWRITE) (EQUAL (LENGTH (POSITIVES N)) (FIX N)) ((INDUCT (POSITIVES N)))) (PROVE-LEMMA EULER-2-1 NIL (IMPLIES (AND (PRIME P) (NOT (DIVIDES P A)) (NOT (RESIDUE A P))) (EQUAL (REMAINDER (EXP A (QUOTIENT (LENGTH (COMPLEMENTS (SUB1 P) A P)) 2)) P) (SUB1 P))) ((USE (TIMES-COMPLEMENTS (I (SUB1 P))) (COMPLEMENTS-FACT) (WILSON-THM)) (DISABLE TIMES-COMPLEMENTS COMPLEMENTS-FACT))) (PROVE-LEMMA EULER-2-2 (REWRITE) (IMPLIES (AND (PRIME P) (NOT (DIVIDES P A)) (NOT (RESIDUE A P))) (EQUAL (LENGTH (COMPLEMENTS (SUB1 P) A P)) (SUB1 P))) ((USE (EQUAL-LENGTH-PERM (A (POSITIVES (SUB1 P))) (B (COMPLEMENTS (SUB1 P) A P))) (PERM-POSITIVES-COMPLEMENTS)) (DISABLE EQUAL-LENGTH-PERM 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(PROVE-LEMMA EVENP-EVEN (REWRITE) (EQUAL (EVEN P) (EVENP P))) (PROVE-LEMMA EVEN-PLUS (REWRITE) (EQUAL (EVEN (PLUS A B)) (EQUAL (EVEN A) (EVEN B))) ((DISABLE EVEN))) (PROVE-LEMMA EVEN-DIFF (REWRITE) (EQUAL (EVEN (DIFFERENCE P X)) (OR (LESSP P X) (EQUAL (EVEN P) (EVEN X)))) ((DISABLE EVEN))) (PROVE-LEMMA EVEN-TIMES (REWRITE) (EQUAL (EVEN (TIMES A B)) (OR (EVEN A) (EVEN B))) ((DISABLE EVEN))) (PROVE-LEMMA EVEN-REM (REWRITE) (IMPLIES (NOT (EVEN P)) (EQUAL (EVEN (DIFFERENCE P (REMAINDER X P))) (NOT (EVEN (REMAINDER X P))))) ((DISABLE EVEN))) (PROVE-LEMMA EVEN-ADD1 (REWRITE) (EQUAL (EVEN (ADD1 X)) (NOT (EVEN X))) ((DISABLE EVEN))) (DISABLE EVENP-EVEN) (PROVE-LEMMA EVEN-PRIME-2 (REWRITE) (IMPLIES (AND (PRIME P) (NOT (EQUAL P 2))) (NOT (EVEN P)))) (PROVE-LEMMA EVEN-PRIME (REWRITE) (IMPLIES (AND (PRIME P) (NOT (EQUAL P 2))) (NOT (EQUAL (REMAINDER P 2) 0))) ((DISABLE PRIME1))) (PROVE-LEMMA EULER-CRITERION NIL (IMPLIES (AND (PRIME P) (NOT (EQUAL P 2)) (NOT (DIVIDES P A))) (EQUAL (REMAINDER (EXP A 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