(PRIMES::MOD-OF-EXPT-OF-MOD
 (180 74 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (164 164 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (74 74 (:TYPE-PRESCRIPTION NATP))
 (73 5 (:DEFINITION EXPT))
 (69 7 (:REWRITE RTL::MOD-DOES-NOTHING))
 (62 4 (:LINEAR RTL::MOD-BND-2))
 (37 17 (:REWRITE DEFAULT-<-1))
 (17 17 (:REWRITE DEFAULT-<-2))
 (15 5 (:REWRITE DEFAULT-*-2))
 (15 5 (:REWRITE COMMUTATIVITY-OF-+))
 (10 10 (:REWRITE DEFAULT-+-2))
 (10 10 (:REWRITE DEFAULT-+-1))
 (9 9 (:REWRITE MOD-OF-EXPT-WHEN-EQUAL-OF-MOD-SUBST-CONSTANT))
 (6 5 (:DEFINITION FIX))
 (5 5 (:REWRITE ZIP-OPEN))
 (5 5 (:REWRITE DEFAULT-*-1))
 (4 4 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::MOD-OF-*-OF-EXPT-OF-MOD
 (188 78 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (173 173 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (78 78 (:TYPE-PRESCRIPTION NATP))
 (73 5 (:DEFINITION EXPT))
 (69 7 (:REWRITE RTL::MOD-DOES-NOTHING))
 (62 4 (:LINEAR RTL::MOD-BND-2))
 (37 17 (:REWRITE DEFAULT-<-1))
 (31 10 (:REWRITE DEFAULT-*-2))
 (17 17 (:REWRITE DEFAULT-<-2))
 (15 5 (:REWRITE COMMUTATIVITY-OF-+))
 (10 10 (:REWRITE DEFAULT-+-2))
 (10 10 (:REWRITE DEFAULT-+-1))
 (10 10 (:REWRITE DEFAULT-*-1))
 (6 5 (:DEFINITION FIX))
 (5 5 (:REWRITE ZIP-OPEN))
 (4 4 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::NATP-2^X
 (21 17 (:REWRITE DEFAULT-<-1))
 (17 17 (:REWRITE DEFAULT-<-2))
 (16 6 (:REWRITE DEFAULT-*-2))
 (12 12 (:REWRITE DEFAULT-+-2))
 (12 12 (:REWRITE DEFAULT-+-1))
 (6 6 (:REWRITE DEFAULT-*-1))
 (5 5 (:REWRITE ZIP-OPEN))
 )
(PRIMES::EXPONENTS-ADD-FOR-NONNEG-EXPONENTS
 (273 273 (:TYPE-PRESCRIPTION EVENP))
 (182 91 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (182 91 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (182 91 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (91 91 (:TYPE-PRESCRIPTION INTEGERP-OF-EXPT-TYPE))
 (91 91 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NON-POSITIVE-EXPONENT))
 (91 91 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (91 91 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP))
 (91 91 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-POSITIVE))
 (91 91 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONZERO))
 (91 91 (:TYPE-PRESCRIPTION <=-OF-0-AND-EXPT-TYPE))
 (91 91 (:TYPE-PRESCRIPTION <-OF-0-AND-EXPT-TYPE))
 (34 2 (:REWRITE DEFAULT-*-2))
 (34 2 (:REWRITE DEFAULT-*-1))
 (4 4 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (4 4 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 (1 1 (:REWRITE EXPONENTS-ADD))
 (1 1 (:REWRITE DEFAULT-+-2))
 (1 1 (:REWRITE DEFAULT-+-1))
 )
(PRIMES::EXPT-HALF-LINEAR
 (136 68 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (68 68 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (68 68 (:TYPE-PRESCRIPTION EVENP))
 (46 46 (:TYPE-PRESCRIPTION INTEGERP-OF-EXPT-TYPE))
 (22 6 (:REWRITE DEFAULT-+-2))
 (22 6 (:REWRITE DEFAULT-+-1))
 (20 20 (:REWRITE DEFAULT-<-2))
 (20 20 (:REWRITE DEFAULT-<-1))
 (18 6 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (14 6 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (9 3 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (9 3 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (7 7 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (7 7 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (7 3 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (7 3 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (6 6 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (4 4 (:REWRITE EXPONENTS-ADD-FOR-NONNEG-EXPONENTS))
 (4 4 (:REWRITE EXPONENTS-ADD))
 )
(PRIMES::EXPONENTS-MULTIPLY
 (249 249 (:TYPE-PRESCRIPTION EVENP))
 (166 83 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (166 83 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (166 83 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (143 83 (:TYPE-PRESCRIPTION INTEGERP-OF-EXPT-TYPE))
 (143 83 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP))
 (83 83 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NON-POSITIVE-EXPONENT))
 (83 83 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (83 83 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-POSITIVE))
 (83 83 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONZERO))
 (83 83 (:TYPE-PRESCRIPTION <=-OF-0-AND-EXPT-TYPE))
 (83 83 (:TYPE-PRESCRIPTION <-OF-0-AND-EXPT-TYPE))
 (16 16 (:REWRITE DEFAULT-<-2))
 (16 16 (:REWRITE DEFAULT-<-1))
 (12 4 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (12 4 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (6 2 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (6 2 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (6 2 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (6 2 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (4 4 (:LINEAR <=-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (4 4 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (3 3 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (3 3 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (2 2 (:REWRITE DEFAULT-*-2))
 (2 2 (:REWRITE DEFAULT-*-1))
 )
(PRIMES::POSP!=0)
(PRIMES::LRS-AUX-LEMMA-1
 (468 55 (:REWRITE RTL::MOD-DOES-NOTHING))
 (408 12 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (372 12 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (179 120 (:REWRITE DEFAULT-<-1))
 (140 140 (:TYPE-PRESCRIPTION NATP))
 (120 120 (:REWRITE DEFAULT-<-2))
 (96 48 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (96 12 (:REWRITE COMMUTATIVITY-OF-*))
 (92 92 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (86 86 (:REWRITE DEFAULT-+-2))
 (86 86 (:REWRITE DEFAULT-+-1))
 (61 57 (:REWRITE DEFAULT-*-2))
 (61 57 (:REWRITE DEFAULT-*-1))
 (60 12 (:REWRITE DISTRIBUTIVITY))
 (48 48 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (48 48 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (24 24 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (17 17 (:LINEAR RTL::MOD-BND-3))
 (12 12 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (12 12 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 )
(PRIMES::LRS-AUX-LEMMA-2
 (170 5 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (155 5 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (137 11 (:REWRITE RTL::MOD-DOES-NOTHING))
 (49 31 (:REWRITE DEFAULT-<-1))
 (42 42 (:TYPE-PRESCRIPTION NATP))
 (40 20 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (40 5 (:REWRITE COMMUTATIVITY-OF-*))
 (32 24 (:REWRITE DEFAULT-*-2))
 (32 24 (:REWRITE DEFAULT-*-1))
 (31 31 (:REWRITE DEFAULT-<-2))
 (25 5 (:REWRITE DISTRIBUTIVITY))
 (23 23 (:REWRITE DEFAULT-+-2))
 (23 23 (:REWRITE DEFAULT-+-1))
 (22 22 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (20 20 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (20 20 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (10 10 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (5 5 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (5 5 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (1 1 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::LRS-AUX-LEMMA-3
 (560 70 (:REWRITE RTL::MOD-DOES-NOTHING))
 (408 12 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (372 12 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (234 160 (:REWRITE DEFAULT-<-1))
 (160 160 (:TYPE-PRESCRIPTION NATP))
 (160 160 (:REWRITE DEFAULT-<-2))
 (112 112 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (96 48 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (96 12 (:REWRITE COMMUTATIVITY-OF-*))
 (60 12 (:REWRITE DISTRIBUTIVITY))
 (56 56 (:REWRITE DEFAULT-*-2))
 (56 56 (:REWRITE DEFAULT-*-1))
 (49 49 (:REWRITE DEFAULT-+-2))
 (49 49 (:REWRITE DEFAULT-+-1))
 (48 48 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (48 48 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (25 25 (:LINEAR RTL::MOD-BND-3))
 (24 24 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (12 12 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (12 12 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (4 4 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 )
(PRIMES::LRS-AUX-LEMMA-4-1
 (1794 30 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (1769 82 (:REWRITE PRIMES::LRS-AUX-LEMMA-3))
 (599 599 (:TYPE-PRESCRIPTION NATP))
 (578 17 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (573 531 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (527 17 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (448 314 (:REWRITE DEFAULT-<-1))
 (397 175 (:REWRITE DEFAULT-*-2))
 (359 359 (:REWRITE DEFAULT-+-2))
 (359 359 (:REWRITE DEFAULT-+-1))
 (314 314 (:REWRITE DEFAULT-<-2))
 (205 175 (:REWRITE DEFAULT-*-1))
 (136 68 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (136 17 (:REWRITE COMMUTATIVITY-OF-*))
 (85 17 (:REWRITE DISTRIBUTIVITY))
 (68 68 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (68 68 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (67 19 (:REWRITE PRIMES::LRS-AUX-LEMMA-1))
 (36 36 (:REWRITE DEFAULT-UNARY-MINUS))
 (34 34 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (17 17 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (17 17 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (17 11 (:REWRITE DEFAULT-UNARY-/))
 (10 10 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::LRS-AUX-LEMMA-4
 (1508 123 (:REWRITE RTL::MOD-DOES-NOTHING))
 (1167 35 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (1059 35 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (726 439 (:REWRITE DEFAULT-<-1))
 (677 519 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (439 439 (:REWRITE DEFAULT-<-2))
 (326 214 (:REWRITE DEFAULT-*-2))
 (272 136 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (272 34 (:REWRITE COMMUTATIVITY-OF-*))
 (242 214 (:REWRITE DEFAULT-*-1))
 (219 219 (:REWRITE DEFAULT-+-2))
 (219 219 (:REWRITE DEFAULT-+-1))
 (170 34 (:REWRITE DISTRIBUTIVITY))
 (136 136 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (136 136 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (117 31 (:REWRITE PRIMES::LRS-AUX-LEMMA-1))
 (87 17 (:REWRITE ZIP-OPEN))
 (74 74 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (36 4 (:REWRITE ASSOCIATIVITY-OF-*))
 (35 35 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (35 35 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (31 31 (:LINEAR RTL::MOD-BND-3))
 (25 25 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-AUX-LESS-THAN-M))
 (10 10 (:REWRITE DEFAULT-UNARY-/))
 (8 8 (:REWRITE DEFAULT-UNARY-MINUS))
 (3 1 (:REWRITE PRIMES::HAS-SQUARE-ROOT-OF-0?))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-LEMMA5
 (4088 11 (:REWRITE RTL::MOD-DOES-NOTHING))
 (2012 11 (:REWRITE CANCEL-MOD-+))
 (1939 11 (:REWRITE MOD-X-Y-=-X . 4))
 (1934 11 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (1757 11 (:REWRITE MOD-X-Y-=-X . 3))
 (1620 11 (:REWRITE MOD-ZERO . 4))
 (1368 90 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (1305 1305 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (1305 1305 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (1305 1305 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (1305 1305 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (1305 1305 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (1084 204 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (1039 11 (:REWRITE MOD-ZERO . 3))
 (895 895 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (797 152 (:REWRITE DEFAULT-TIMES-1))
 (740 64 (:REWRITE |(* y x)|))
 (641 11 (:REWRITE |(integerp (- x))|))
 (636 152 (:REWRITE DEFAULT-TIMES-2))
 (608 219 (:REWRITE DEFAULT-LESS-THAN-1))
 (608 11 (:REWRITE DEFAULT-MOD-RATIO))
 (608 11 (:REWRITE |(* (- x) y)|))
 (441 441 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (441 441 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (441 441 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (387 11 (:REWRITE |(equal (expt x n) 0)|))
 (374 219 (:REWRITE DEFAULT-LESS-THAN-2))
 (365 11 (:REWRITE MOD-X-Y-=-X+Y . 2))
 (365 11 (:REWRITE MOD-X-Y-=-X . 2))
 (365 11 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (365 11 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (336 46 (:LINEAR EXPT->=-1-TWO))
 (336 46 (:LINEAR EXPT->-1-TWO))
 (336 46 (:LINEAR EXPT-<-1-ONE))
 (306 46 (:LINEAR EXPT-X->=-X))
 (306 46 (:LINEAR EXPT->=-1-ONE))
 (306 46 (:LINEAR EXPT-<=-1-TWO))
 (306 46 (:LINEAR EXPT-<=-1-ONE))
 (300 60 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (300 60 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (300 60 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (300 60 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (296 46 (:LINEAR EXPT-X->-X))
 (296 46 (:LINEAR EXPT->-1-ONE))
 (296 46 (:LINEAR EXPT-<-1-TWO))
 (268 34 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (266 92 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (265 34 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (242 92 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (219 219 (:REWRITE THE-FLOOR-BELOW))
 (219 219 (:REWRITE THE-FLOOR-ABOVE))
 (219 219 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (219 219 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (219 219 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (219 20 (:REWRITE DEFAULT-MINUS))
 (210 11 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (210 11 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (204 204 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (204 204 (:REWRITE INTEGERP-<-CONSTANT))
 (204 204 (:REWRITE CONSTANT-<-INTEGERP))
 (204 204 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (204 204 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (204 204 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (204 204 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (204 204 (:REWRITE |(< c (- x))|))
 (204 204 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (204 204 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (204 204 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (204 204 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (204 204 (:REWRITE |(< (/ x) (/ y))|))
 (204 204 (:REWRITE |(< (- x) c)|))
 (204 204 (:REWRITE |(< (- x) (- y))|))
 (195 6 (:REWRITE PRIMES::LRS-AUX-LEMMA-1))
 (166 11 (:REWRITE DEFAULT-MOD-1))
 (164 11 (:REWRITE MOD-CANCEL-*-CONST))
 (126 11 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (126 11 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (126 11 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (119 1 (:LINEAR MOD-BOUNDS-3))
 (95 11 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (95 11 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (95 11 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (92 92 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (92 92 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (90 90 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (90 90 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (90 90 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (81 81 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (79 69 (:REWRITE DEFAULT-PLUS-1))
 (69 69 (:REWRITE DEFAULT-PLUS-2))
 (66 6 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (64 64 (:REWRITE |(< 0 (* x y))|))
 (60 60 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (60 60 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (60 60 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (60 60 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (60 60 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (49 49 (:REWRITE |(< 0 (/ x))|))
 (48 48 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (48 48 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (48 2 (:LINEAR MOD-BOUNDS-2))
 (46 46 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (46 46 (:LINEAR EXPT-LINEAR-UPPER-<))
 (46 46 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (46 46 (:LINEAR EXPT-LINEAR-LOWER-<))
 (45 45 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (45 45 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (45 45 (:REWRITE |(equal c (/ x))|))
 (45 45 (:REWRITE |(equal c (- x))|))
 (45 45 (:REWRITE |(equal (/ x) c)|))
 (45 45 (:REWRITE |(equal (/ x) (/ y))|))
 (45 45 (:REWRITE |(equal (- x) c)|))
 (45 45 (:REWRITE |(equal (- x) (- y))|))
 (34 34 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (34 34 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (34 34 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (34 34 (:REWRITE DEFAULT-DIVIDE))
 (30 30 (:REWRITE |(* a (/ a))|))
 (28 28 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (23 23 (:REWRITE REDUCE-INTEGERP-+))
 (23 23 (:REWRITE INTEGERP-MINUS-X))
 (23 23 (:META META-INTEGERP-CORRECT))
 (18 18 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (17 8 (:REWRITE DEFAULT-EXPT-2))
 (16 16 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (15 15 (:REWRITE |(<= (/ x) y) with (< x 0)|))
 (15 15 (:REWRITE |(<= (/ x) y) with (< 0 x)|))
 (15 15 (:REWRITE |(< x (/ y)) with (< y 0)|))
 (15 15 (:REWRITE |(< x (/ y)) with (< 0 y)|))
 (11 11 (:REWRITE DEFAULT-MOD-2))
 (11 11 (:REWRITE |(mod x (- y))| . 3))
 (11 11 (:REWRITE |(mod x (- y))| . 2))
 (11 11 (:REWRITE |(mod x (- y))| . 1))
 (11 11 (:REWRITE |(mod (- x) y)| . 3))
 (11 11 (:REWRITE |(mod (- x) y)| . 2))
 (11 11 (:REWRITE |(mod (- x) y)| . 1))
 (11 11 (:REWRITE |(- (* c x))|))
 (8 8 (:REWRITE DEFAULT-EXPT-1))
 (8 8 (:REWRITE |(expt 1/c n)|))
 (8 8 (:REWRITE |(expt (- x) n)|))
 (6 6 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (6 6 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (5 5 (:REWRITE |(< (+ c/d x) y)|))
 (5 5 (:REWRITE |(< (+ (- c) x) y)|))
 (5 5 (:REWRITE |(+ c (+ d x))|))
 (3 3 (:REWRITE |(< y (+ (- c) x))|))
 (3 3 (:REWRITE |(< x (+ c/d y))|))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 (1 1 (:REWRITE |(< (/ x) 0)|))
 (1 1 (:REWRITE |(< (* x y) 0)|))
 (1 1 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::INTEGERP-MOD
 (15 3 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (7 3 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (7 3 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (7 3 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (6 6 (:TYPE-PRESCRIPTION NOT-INTEGERP-4B))
 (6 6 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (6 6 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (6 6 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (6 3 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (3 3 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (3 3 (:TYPE-PRESCRIPTION NATP))
 (3 3 (:TYPE-PRESCRIPTION MOD-ZERO . 4))
 (3 3 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (3 3 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (3 3 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (3 3 (:TYPE-PRESCRIPTION MOD-NONNEGATIVE))
 (3 3 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (3 3 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (2 2 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (2 2 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 )
(PRIMES::MOD-*A-B=
 (1330 4 (:LINEAR MOD-BOUNDS-3))
 (1140 228 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (1108 228 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (1088 1088 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (1088 1088 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (1088 1088 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (1060 228 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (1060 228 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (1024 4 (:LINEAR RTL::MOD-BND-2))
 (782 21 (:REWRITE |(* (* x y) z)|))
 (696 228 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (648 8 (:REWRITE CANCEL-MOD-+))
 (623 91 (:REWRITE DEFAULT-TIMES-2))
 (528 8 (:LINEAR MOD-BOUNDS-2))
 (528 8 (:LINEAR MOD-BOUNDS-1))
 (443 91 (:REWRITE DEFAULT-TIMES-1))
 (350 8 (:REWRITE |(integerp (- x))|))
 (336 31 (:META META-INTEGERP-CORRECT))
 (326 6 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (272 2 (:REWRITE |(* y (* x z))|))
 (263 8 (:REWRITE MOD-ZERO . 3))
 (251 8 (:REWRITE MOD-X-Y-=-X . 4))
 (251 8 (:REWRITE MOD-X-Y-=-X . 3))
 (228 228 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (228 228 (:TYPE-PRESCRIPTION NATP))
 (228 228 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (228 228 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (228 228 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (228 228 (:TYPE-PRESCRIPTION MOD-NONNEGATIVE))
 (228 228 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (228 228 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (193 65 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (193 65 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (193 65 (:REWRITE DEFAULT-LESS-THAN-1))
 (183 8 (:REWRITE RTL::MOD-DOES-NOTHING))
 (173 8 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (173 8 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (173 8 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (173 8 (:REWRITE MOD-X-Y-=-X+Y . 2))
 (170 8 (:REWRITE DEFAULT-MOD-RATIO))
 (170 8 (:REWRITE |(* (- x) y)|))
 (159 15 (:REWRITE INTEGERP-/))
 (132 8 (:REWRITE MOD-ZERO . 4))
 (72 2 (:REWRITE |(* y x)|))
 (65 65 (:REWRITE THE-FLOOR-BELOW))
 (65 65 (:REWRITE THE-FLOOR-ABOVE))
 (65 65 (:REWRITE SIMPLIFY-SUMS-<))
 (65 65 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (65 65 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (65 65 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (65 65 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (65 65 (:REWRITE INTEGERP-<-CONSTANT))
 (65 65 (:REWRITE DEFAULT-LESS-THAN-2))
 (65 65 (:REWRITE CONSTANT-<-INTEGERP))
 (65 65 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (65 65 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
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 (65 65 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
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 (65 65 (:REWRITE |(< (/ x) (/ y))|))
 (65 65 (:REWRITE |(< (- x) c)|))
 (65 65 (:REWRITE |(< (- x) (- y))|))
 (64 8 (:REWRITE DEFAULT-MINUS))
 (58 58 (:REWRITE NORMALIZE-FACTORS-GATHER-EXPONENTS))
 (50 50 (:TYPE-PRESCRIPTION NOT-INTEGERP-4E))
 (50 50 (:TYPE-PRESCRIPTION NOT-INTEGERP-3E))
 (50 50 (:TYPE-PRESCRIPTION NOT-INTEGERP-2E))
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 (40 8 (:REWRITE MOD-X-Y-=-X . 2))
 (40 8 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (40 8 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (40 8 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (40 8 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (40 8 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
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 (31 31 (:REWRITE REDUCE-INTEGERP-+))
 (31 31 (:REWRITE INTEGERP-MINUS-X))
 (29 29 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (29 29 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (29 29 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (29 29 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (29 29 (:REWRITE |(< 0 (/ x))|))
 (29 29 (:REWRITE |(< 0 (* x y))|))
 (29 29 (:REWRITE |(< (/ x) 0)|))
 (29 29 (:REWRITE |(< (* x y) 0)|))
 (27 27 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (27 27 (:REWRITE DEFAULT-DIVIDE))
 (21 21 (:REWRITE |(* c (* d x))|))
 (20 20 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (20 20 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (20 20 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (20 20 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (20 20 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (20 20 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (20 20 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (20 20 (:REWRITE |(equal c (/ x))|))
 (20 20 (:REWRITE |(equal c (- x))|))
 (20 20 (:REWRITE |(equal (/ x) c)|))
 (20 20 (:REWRITE |(equal (/ x) (/ y))|))
 (20 20 (:REWRITE |(equal (- x) c)|))
 (20 20 (:REWRITE |(equal (- x) (- y))|))
 (18 18 (:REWRITE REMOVE-WEAK-INEQUALITIES))
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 (8 8 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (8 8 (:REWRITE DEFAULT-MOD-2))
 (8 8 (:REWRITE DEFAULT-MOD-1))
 (8 8 (:REWRITE |(mod x (- y))| . 3))
 (8 8 (:REWRITE |(mod x (- y))| . 2))
 (8 8 (:REWRITE |(mod x (- y))| . 1))
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 (8 8 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (8 8 (:REWRITE |(- (* c x))|))
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 (5 5 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (4 4 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::MOD-*MOD-A*MOD-B=)
(PRIMES::MOD-TIMES-MOD)
(PRIMES::Y^2=1MODP-LEMMA1)
(PRIMES::Y^2=1MODP-1
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 (1482 42 (:REWRITE |(integerp (- x))|))
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 (1307 1307 (:REWRITE DEFAULT-EXPT-1))
 (1307 1307 (:REWRITE |(expt 1/c n)|))
 (1307 1307 (:REWRITE |(expt (- x) n)|))
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 (867 258 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
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 (838 838 (:REWRITE THE-FLOOR-ABOVE))
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 (838 838 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (838 838 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (838 838 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (838 838 (:REWRITE INTEGERP-<-CONSTANT))
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 (838 838 (:REWRITE |(< c (- x))|))
 (838 838 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
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 (675 129 (:LINEAR EXPT-<=-1-TWO))
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 (654 129 (:LINEAR EXPT->-1-ONE))
 (654 129 (:LINEAR EXPT-<-1-TWO))
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 (570 42 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
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 (560 560 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
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 (492 63 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (486 18 (:REWRITE |(equal (expt x n) 0)|))
 (473 473 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (473 473 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (467 304 (:REWRITE DEFAULT-PLUS-1))
 (462 6 (:LINEAR MOD-BOUNDS-3))
 (420 14 (:REWRITE BUBBLE-DOWN-*-MATCH-3))
 (416 234 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (390 42 (:REWRITE MOD-CANCEL-*-CONST))
 (378 378 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (378 378 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (378 378 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (354 42 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (354 42 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (354 42 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (252 12 (:REWRITE |(+ x (if a b c))|))
 (243 63 (:REWRITE DEFAULT-MOD-1))
 (234 234 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (234 234 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (219 219 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (216 12 (:REWRITE |(* x (expt x n))|))
 (187 78 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (182 182 (:TYPE-PRESCRIPTION NATP))
 (180 77 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (153 153 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (153 153 (:REWRITE DEFAULT-DIVIDE))
 (147 147 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (147 147 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (147 147 (:REWRITE |(< 0 (/ x))|))
 (147 147 (:REWRITE |(< 0 (* x y))|))
 (134 78 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
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 (129 129 (:LINEAR EXPT-LINEAR-UPPER-<))
 (129 129 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (129 129 (:LINEAR EXPT-LINEAR-LOWER-<))
 (120 12 (:LINEAR MOD-BOUNDS-2))
 (115 115 (:REWRITE |(* x (- y))|))
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 (111 111 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (111 111 (:REWRITE |(< (/ x) 0)|))
 (111 111 (:REWRITE |(< (* x y) 0)|))
 (98 97 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
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 (96 96 (:REWRITE |(equal (- x) c)|))
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 (90 90 (:META META-INTEGERP-CORRECT))
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 (78 78 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (78 42 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
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 (72 72 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
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 (42 42 (:REWRITE |(mod x (- y))| . 2))
 (42 42 (:REWRITE |(mod x (- y))| . 1))
 (42 42 (:REWRITE |(mod (- x) y)| . 3))
 (42 42 (:REWRITE |(mod (- x) y)| . 2))
 (42 42 (:REWRITE |(mod (- x) y)| . 1))
 (42 42 (:REWRITE |(- (* c x))|))
 (35 35 (:REWRITE DM::DIVIDES-SELF))
 (23 6 (:REWRITE |(equal (+ (- c) x) y)|))
 (21 21 (:REWRITE |(< (* x y) 0) rationalp (* x y)|))
 (14 14 (:REWRITE |(* c (* d x))|))
 (12 12 (:REWRITE |(* c (expt d n))|))
 (9 9 (:REWRITE |(equal (* x y) 0)|))
 (9 9 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (7 2 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (6 6 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (6 6 (:LINEAR RTL::MOD-BND-3))
 (5 5 (:REWRITE |(+ c (+ d x))|))
 (4 1 (:REWRITE |(+ x x)|))
 (2 2 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (1 1 (:TYPE-PRESCRIPTION ABS))
 )
(PRIMES::Y^2=1MODP-2
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 (6384 56 (:REWRITE CANCEL-MOD-+))
 (5284 60 (:REWRITE MOD-X-Y-=-X . 4))
 (5066 65 (:REWRITE MOD-X-Y-=-X-Y . 2))
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 (4024 504 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
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 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-POSITIVE-BASE))
 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-ODD-EXPONENT))
 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONNEGATIVE-BASE))
 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (2739 2739 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
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 (2309 2309 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
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 (2185 61 (:REWRITE |(* (- x) y)|))
 (2156 56 (:REWRITE |(integerp (- x))|))
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 (2050 102 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
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 (1602 452 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (1600 40 (:REWRITE |(* (* x y) z)|))
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 (1422 226 (:LINEAR EXPT->=-1-ONE))
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 (1080 40 (:REWRITE |(* y (* x z))|))
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 (880 56 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
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 (822 822 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
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 (822 822 (:REWRITE INTEGERP-<-CONSTANT))
 (822 822 (:REWRITE CONSTANT-<-INTEGERP))
 (822 822 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
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 (580 56 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (580 56 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
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 (452 452 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (452 452 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (428 229 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (401 401 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (389 389 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (389 389 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
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 (359 359 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (358 358 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (238 238 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (238 238 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (238 238 (:REWRITE |(< 0 (/ x))|))
 (238 238 (:REWRITE |(< 0 (* x y))|))
 (229 229 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
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 (226 226 (:LINEAR EXPT-LINEAR-UPPER-<))
 (226 226 (:LINEAR EXPT-LINEAR-LOWER-<=))
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 (199 199 (:TYPE-PRESCRIPTION NATP))
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 (185 185 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
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 (170 102 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (164 164 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (164 164 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (164 164 (:REWRITE |(< (/ x) 0)|))
 (164 164 (:REWRITE |(< (* x y) 0)|))
 (153 53 (:REWRITE DEFAULT-PLUS-2))
 (147 147 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (146 26 (:REWRITE NORMALIZE-ADDENDS))
 (134 133 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (133 133 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (132 132 (:REWRITE |(equal c (/ x))|))
 (132 132 (:REWRITE |(equal c (- x))|))
 (132 132 (:REWRITE |(equal (/ x) c)|))
 (132 132 (:REWRITE |(equal (/ x) (/ y))|))
 (132 132 (:REWRITE |(equal (- x) c)|))
 (132 132 (:REWRITE |(equal (- x) (- y))|))
 (122 122 (:REWRITE REDUCE-INTEGERP-+))
 (122 122 (:REWRITE INTEGERP-MINUS-X))
 (122 122 (:META META-INTEGERP-CORRECT))
 (105 12 (:REWRITE |(+ y x)|))
 (102 102 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (100 53 (:REWRITE DEFAULT-PLUS-1))
 (95 67 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (88 60 (:REWRITE DEFAULT-EXPT-1))
 (84 84 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (65 65 (:REWRITE DEFAULT-MOD-2))
 (65 5 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (60 60 (:REWRITE DEFAULT-EXPT-2))
 (60 60 (:REWRITE |(expt 1/c n)|))
 (60 60 (:REWRITE |(expt (- x) n)|))
 (56 56 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (56 56 (:REWRITE |(mod x (- y))| . 3))
 (56 56 (:REWRITE |(mod x (- y))| . 2))
 (56 56 (:REWRITE |(mod x (- y))| . 1))
 (56 56 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (56 56 (:REWRITE |(mod (- x) y)| . 3))
 (56 56 (:REWRITE |(mod (- x) y)| . 2))
 (56 56 (:REWRITE |(mod (- x) y)| . 1))
 (56 56 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (56 56 (:REWRITE |(- (* c x))|))
 (49 49 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (48 3 (:REWRITE |(+ y (+ x z))|))
 (32 15 (:REWRITE |(equal (+ (- c) x) y)|))
 (28 28 (:REWRITE |(< (* x y) 0) rationalp (* x y)|))
 (25 5 (:REWRITE |(+ 0 x)|))
 (24 4 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (19 19 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (16 16 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (16 16 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 (12 12 (:REWRITE |(equal (* x y) 0)|))
 (12 12 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (11 11 (:LINEAR RTL::MOD-BND-3))
 (10 5 (:REWRITE |(* -1 x)|))
 (7 2 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (6 6 (:REWRITE |(+ x (- x))|))
 (5 5 (:REWRITE |(+ c (+ d x))|))
 (4 4 (:REWRITE |(* 1 x)|))
 (4 1 (:REWRITE |(+ x x)|))
 (1 1 (:TYPE-PRESCRIPTION ABS))
 )
(PRIMES::Y^2=1MODP
 (80 16 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (48 16 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (28 16 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (24 24 (:TYPE-PRESCRIPTION NOT-INTEGERP-4B))
 (24 24 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (24 24 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (24 24 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (16 16 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (16 16 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (16 16 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (16 16 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (16 16 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (16 16 (:TYPE-PRESCRIPTION MOD-NONNEGATIVE))
 (16 16 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (16 16 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (16 16 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (16 16 (:TYPE-PRESCRIPTION INTEGERP-MOD-1))
 (16 16 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (12 12 (:TYPE-PRESCRIPTION NATP))
 (6 6 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 )
(PRIMES::MODX^2-Y^2
 (7748 28 (:REWRITE RTL::MOD-DOES-NOTHING))
 (5859 60 (:REWRITE PRIMES::Y^2=1MODP-LEMMA1))
 (5338 97 (:REWRITE MOD-X-Y-=-X . 4))
 (5274 1138 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (5157 5157 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (5157 5157 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (5157 5157 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (5075 40 (:REWRITE CANCEL-MOD-+))
 (4253 1197 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (4253 1197 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (3905 100 (:REWRITE MOD-ZERO . 4))
 (3754 642 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (3160 1029 (:REWRITE DEFAULT-TIMES-2))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-ODD-EXPONENT))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (3146 3146 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (2872 8 (:REWRITE PRIMES::Y^2=1MODP-2))
 (2617 116 (:REWRITE DEFAULT-MOD-RATIO))
 (2599 1029 (:REWRITE DEFAULT-TIMES-1))
 (1837 7 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (1501 394 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (1501 394 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (1314 108 (:REWRITE MOD-X-Y-=-X+Y . 2))
 (1239 402 (:REWRITE DEFAULT-LESS-THAN-1))
 (1231 394 (:REWRITE SIMPLIFY-SUMS-<))
 (1197 1197 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (1189 1189 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (1189 1189 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (1138 1138 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (1138 1138 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (1112 220 (:REWRITE DEFAULT-PLUS-2))
 (1086 220 (:REWRITE DEFAULT-PLUS-1))
 (1052 157 (:REWRITE DEFAULT-MINUS))
 (1040 26 (:REWRITE |(* (* x y) z)|))
 (960 484 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (907 7 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (835 116 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (795 108 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (758 40 (:REWRITE MOD-X-Y-=-X . 2))
 (758 40 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (758 40 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (729 27 (:REWRITE |(equal (expt x n) 0)|))
 (702 26 (:REWRITE |(* y (* x z))|))
 (678 162 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (672 402 (:REWRITE DEFAULT-LESS-THAN-2))
 (646 402 (:REWRITE |(< (- x) c)|))
 (642 642 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (642 642 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (608 144 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (576 156 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (544 144 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (536 72 (:LINEAR EXPT->=-1-TWO))
 (536 72 (:LINEAR EXPT->-1-TWO))
 (536 72 (:LINEAR EXPT-<-1-ONE))
 (492 6 (:REWRITE |(+ x (if a b c))|))
 (488 72 (:LINEAR EXPT-X->=-X))
 (488 72 (:LINEAR EXPT->=-1-ONE))
 (488 72 (:LINEAR EXPT-<=-1-TWO))
 (488 72 (:LINEAR EXPT-<=-1-ONE))
 (484 484 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (484 484 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (479 40 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (479 40 (:REWRITE MOD-CANCEL-*-CONST))
 (476 476 (:TYPE-PRESCRIPTION NATP))
 (472 72 (:LINEAR EXPT-X->-X))
 (472 72 (:LINEAR EXPT->-1-ONE))
 (472 72 (:LINEAR EXPT-<-1-TWO))
 (459 36 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (459 36 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (415 3 (:REWRITE |(equal (mod a n) (mod b n))|))
 (402 402 (:REWRITE THE-FLOOR-BELOW))
 (402 402 (:REWRITE THE-FLOOR-ABOVE))
 (402 402 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (402 402 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (402 402 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (402 402 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (402 402 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (402 402 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (402 402 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (402 402 (:REWRITE |(< c (- x))|))
 (402 402 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (402 402 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (402 402 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (402 402 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (402 402 (:REWRITE |(< (/ x) (/ y))|))
 (402 402 (:REWRITE |(< (- x) (- y))|))
 (395 116 (:REWRITE DEFAULT-MOD-1))
 (394 394 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (394 394 (:REWRITE INTEGERP-<-CONSTANT))
 (394 394 (:REWRITE CONSTANT-<-INTEGERP))
 (308 4 (:LINEAR MOD-BOUNDS-3))
 (282 6 (:REWRITE |(+ (if a b c) x)|))
 (249 193 (:REWRITE |(equal (- x) c)|))
 (248 248 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (248 248 (:REWRITE DEFAULT-DIVIDE))
 (216 12 (:REWRITE |(* x (expt x n))|))
 (201 201 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (201 201 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (198 198 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (193 193 (:REWRITE |(equal c (/ x))|))
 (193 193 (:REWRITE |(equal c (- x))|))
 (193 193 (:REWRITE |(equal (/ x) c)|))
 (193 193 (:REWRITE |(equal (/ x) (/ y))|))
 (193 193 (:REWRITE |(equal (- x) (- y))|))
 (168 168 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (162 162 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (158 130 (:REWRITE INTEGERP-MINUS-X))
 (157 29 (:REWRITE |(equal (+ (- c) x) y)|))
 (144 144 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (144 144 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (144 144 (:REWRITE |(< 0 (/ x))|))
 (144 144 (:REWRITE |(< 0 (* x y))|))
 (144 144 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (144 144 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (130 130 (:REWRITE REDUCE-INTEGERP-+))
 (130 130 (:META META-INTEGERP-CORRECT))
 (129 129 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (118 118 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (118 118 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 (116 116 (:REWRITE DEFAULT-MOD-2))
 (110 110 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (106 106 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (94 94 (:REWRITE DEFAULT-EXPT-2))
 (94 94 (:REWRITE DEFAULT-EXPT-1))
 (94 94 (:REWRITE |(expt 1/c n)|))
 (94 94 (:REWRITE |(expt (- x) n)|))
 (80 8 (:LINEAR MOD-BOUNDS-2))
 (73 73 (:REWRITE |(- (* c x))|))
 (72 72 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (72 72 (:LINEAR EXPT-LINEAR-UPPER-<))
 (72 72 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (72 72 (:LINEAR EXPT-LINEAR-LOWER-<))
 (67 7 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (66 6 (:REWRITE |(- (if a b c))|))
 (50 2 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (40 40 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (40 40 (:REWRITE |(mod x (- y))| . 3))
 (40 40 (:REWRITE |(mod x (- y))| . 2))
 (40 40 (:REWRITE |(mod x (- y))| . 1))
 (40 40 (:REWRITE |(mod (- x) y)| . 3))
 (40 40 (:REWRITE |(mod (- x) y)| . 2))
 (36 36 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (36 36 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (18 18 (:REWRITE |(< (/ x) 0)|))
 (18 18 (:REWRITE |(< (* x y) 0)|))
 (16 8 (:REWRITE |(* -1 x)|))
 (12 12 (:REWRITE |(* c (expt d n))|))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (10 10 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (8 8 (:REWRITE |(equal (* x y) 0)|))
 (8 8 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (8 8 (:REWRITE |(* a (/ a))|))
 (4 4 (:LINEAR RTL::MOD-BND-3))
 (2 2 (:REWRITE INTEGERP-+-REDUCE-CONSTANT))
 (2 2 (:REWRITE |(+ c (+ d x))|))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-LEMMA1
 (272 8 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (264 33 (:REWRITE RTL::MOD-DOES-NOTHING))
 (248 8 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (132 132 (:TYPE-PRESCRIPTION NATP))
 (131 98 (:REWRITE DEFAULT-<-1))
 (100 100 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (99 98 (:REWRITE DEFAULT-<-2))
 (64 32 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (64 8 (:REWRITE COMMUTATIVITY-OF-*))
 (40 8 (:REWRITE DISTRIBUTIVITY))
 (39 39 (:REWRITE DEFAULT-*-2))
 (39 39 (:REWRITE DEFAULT-*-1))
 (35 35 (:REWRITE DEFAULT-+-2))
 (35 35 (:REWRITE DEFAULT-+-1))
 (32 32 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (32 32 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (16 16 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (9 9 (:LINEAR RTL::MOD-BND-3))
 (8 8 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (8 8 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-LEMMA2
 (4107 4107 (:TYPE-PRESCRIPTION EVENP))
 (4098 77 (:REWRITE RTL::MOD-DOES-NOTHING))
 (2738 1369 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (2738 1369 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (2738 1369 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (1539 19 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (1369 1369 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (748 22 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (705 157 (:REWRITE DEFAULT-<-1))
 (682 22 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (474 42 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (434 434 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (240 36 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (238 178 (:REWRITE DEFAULT-+-2))
 (238 178 (:REWRITE DEFAULT-+-1))
 (237 21 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (237 21 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (237 21 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (237 21 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (190 190 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (176 88 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (176 22 (:REWRITE COMMUTATIVITY-OF-*))
 (157 157 (:REWRITE DEFAULT-<-2))
 (150 112 (:REWRITE DEFAULT-*-2))
 (150 112 (:REWRITE DEFAULT-*-1))
 (110 22 (:REWRITE DISTRIBUTIVITY))
 (88 88 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (88 88 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (45 36 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (44 44 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (42 42 (:LINEAR <=-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (42 42 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (28 4 (:REWRITE INTEGERP-OF-EXPT-WHEN-NATP))
 (24 4 (:REWRITE PRIMES::NATP-2^X))
 (22 22 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (22 22 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (16 4 (:REWRITE NATP-OF-EXPT))
 (13 13 (:LINEAR RTL::MOD-BND-3))
 (12 12 (:REWRITE DEFAULT-UNARY-MINUS))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-LEMMA3
 (12428 183 (:REWRITE RTL::MOD-DOES-NOTHING))
 (8115 8115 (:TYPE-PRESCRIPTION EVENP))
 (5410 2705 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (5410 2705 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (5410 2705 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (2705 2705 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (2256 28 (:REWRITE PRIMES::MOD-OF-EXPT-OF-MOD))
 (2118 491 (:REWRITE DEFAULT-<-1))
 (1516 116 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (1516 116 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (1154 38 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (1022 38 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (977 977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (849 226 (:REWRITE DEFAULT-*-2))
 (758 58 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (758 58 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (758 58 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (758 58 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (502 26 (:REWRITE ZIP-OPEN))
 (497 497 (:TYPE-PRESCRIPTION NATP))
 (491 491 (:REWRITE DEFAULT-<-2))
 (465 317 (:REWRITE DEFAULT-+-2))
 (461 317 (:REWRITE DEFAULT-+-1))
 (384 77 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (313 313 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (285 41 (:REWRITE INTEGERP-OF-EXPT-WHEN-NATP))
 (264 226 (:REWRITE DEFAULT-*-1))
 (256 128 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (256 32 (:REWRITE COMMUTATIVITY-OF-*))
 (192 28 (:REWRITE PRIMES::NATP-2^X))
 (160 32 (:REWRITE DISTRIBUTIVITY))
 (155 75 (:REWRITE FOLD-CONSTS-IN-+))
 (128 128 (:TYPE-PRESCRIPTION MOD-EXPT-FAST))
 (128 128 (:TYPE-PRESCRIPTION <=-OF-0-AND-MOD-EXPT-FAST))
 (124 28 (:REWRITE NATP-OF-EXPT))
 (116 116 (:LINEAR <=-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (116 116 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (100 100 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (89 77 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (40 40 (:LINEAR RTL::MOD-BND-3))
 (38 38 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (38 38 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (19 19 (:REWRITE DEFAULT-UNARY-MINUS))
 (18 6 (:REWRITE PRIMES::HAS-SQUARE-ROOT-OF-0?))
 (13 13 (:REWRITE <-OF-0-AND-EXPT))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-LEMMA4
 (245 7 (:REWRITE RTL::MOD-DOES-NOTHING))
 (195 195 (:TYPE-PRESCRIPTION EVENP))
 (130 65 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (130 65 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (130 65 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (75 3 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (66 3 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (65 65 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (43 17 (:REWRITE DEFAULT-<-1))
 (28 28 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (24 3 (:REWRITE COMMUTATIVITY-OF-*))
 (24 2 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (24 2 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (21 21 (:TYPE-PRESCRIPTION NONNEGATIVE-PRODUCT))
 (17 17 (:REWRITE DEFAULT-<-2))
 (16 5 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (15 3 (:REWRITE DISTRIBUTIVITY))
 (15 3 (:DEFINITION MOD-EXPT-FAST))
 (12 12 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (12 12 (:REWRITE DEFAULT-*-2))
 (12 12 (:REWRITE DEFAULT-*-1))
 (12 1 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (12 1 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (12 1 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (12 1 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (8 8 (:REWRITE DEFAULT-+-2))
 (8 8 (:REWRITE DEFAULT-+-1))
 (5 5 (:REWRITE EXPT-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (3 3 (:TYPE-PRESCRIPTION NATP))
 (3 3 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (3 3 (:TYPE-PRESCRIPTION PRIMES::HAS-SQUARE-ROOT?))
 (3 3 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (3 3 (:REWRITE EXPT-OF-1-ARG1))
 (3 1 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-AUX-LESS-THAN-M))
 (2 2 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (2 2 (:LINEAR <=-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 (2 2 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
 )
(PRIMES::LEAST-REPEATED-SQUARE-EQUIV
 (1856 25 (:REWRITE RTL::MOD-DOES-NOTHING))
 (1272 1272 (:TYPE-PRESCRIPTION EVENP))
 (848 424 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (848 424 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-2))
 (848 424 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (424 424 (:TYPE-PRESCRIPTION EXPT-TYPE-SMALL-BASE-NEGATIVE-EXPONENT))
 (275 37 (:REWRITE DEFAULT-<-1))
 (242 20 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (242 20 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (171 171 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (150 6 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (132 6 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
 (121 10 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-4))
 (121 10 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-3))
 (121 10 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-2))
 (121 10 (:LINEAR EXPT-WHEN-NEGATIVE-EXPONENT-LINEAR-1))
 (48 6 (:REWRITE COMMUTATIVITY-OF-*))
 (37 37 (:REWRITE DEFAULT-<-2))
 (34 10 (:REWRITE EXPT-WHEN-NOT-INTEGERP-ARG1-CHEAP))
 (30 6 (:REWRITE DISTRIBUTIVITY))
 (30 6 (:DEFINITION MOD-EXPT-FAST))
 (24 24 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (24 24 (:REWRITE DEFAULT-*-2))
 (24 24 (:REWRITE DEFAULT-*-1))
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 (20 20 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
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 (19 19 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (14 14 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
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 (6 6 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (6 6 (:REWRITE EXPT-OF-1-ARG1))
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 )
(PRIMES::LEAST-REPEATED-SQUARE-TT^2^LRS=1
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 (6081 6081 (:TYPE-PRESCRIPTION EVENP))
 (5949 27 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-EQUIV))
 (4489 130 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (4200 2173 (:TYPE-PRESCRIPTION EXPT-TYPE-ODD-EXPONENT-NEGATIVE-BASE))
 (4200 2173 (:TYPE-PRESCRIPTION EXPT-TYPE-EVEN-EXPONENT-1))
 (2650 106 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (2332 106 (:DEFINITION PRIMES::HAS-SQUARE-ROOT?))
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 (1377 593 (:REWRITE DEFAULT-<-1))
 (984 984 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-ZERO-BASE))
 (848 106 (:REWRITE COMMUTATIVITY-OF-*))
 (742 742 (:TYPE-PRESCRIPTION NONNEGATIVE-PRODUCT))
 (599 593 (:REWRITE DEFAULT-<-2))
 (530 106 (:REWRITE DISTRIBUTIVITY))
 (530 106 (:DEFINITION MOD-EXPT-FAST))
 (528 48 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR-NEGATIVE-EXPONENT))
 (528 48 (:LINEAR <-OF-EXPT-AND-EXPT-SAME-EXPONENTS-LINEAR))
 (424 424 (:TYPE-PRESCRIPTION DM::PRIMEP))
 (424 424 (:REWRITE DEFAULT-*-2))
 (424 424 (:REWRITE DEFAULT-*-1))
 (387 359 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (359 359 (:TYPE-PRESCRIPTION NATP))
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 (48 48 (:LINEAR <=-OF-EXPT-AND-EXPT-SAME-BASE-LINEAR))
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 (6 6 (:TYPE-PRESCRIPTION POSP))
 (1 1 (:REWRITE FOLD-CONSTS-IN-+))
 (1 1 (:REWRITE DEFAULT-UNARY-MINUS))
 )
(PRIMES::LEAST-REPEATED-SQUARE-AUX-IS-LEAST
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 (117442 1048 (:REWRITE CANCEL-MOD-+))
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 (92648 3572 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (87665 1053 (:REWRITE MOD-ZERO . 4))
 (81335 65631 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (81335 65631 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (80006 726 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (66639 8420 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (65631 65631 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (65631 65631 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (65631 65631 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (59970 1053 (:REWRITE MOD-ZERO . 3))
 (48838 48838 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (46268 8541 (:REWRITE DEFAULT-TIMES-1))
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 (38974 38974 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (38974 38974 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
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 (30119 3336 (:LINEAR EXPT->-1-ONE))
 (28438 6672 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
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 (27322 5582 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (27050 5582 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
 (27050 5582 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (25753 3336 (:LINEAR EXPT->=-1-TWO))
 (25753 3336 (:LINEAR EXPT->-1-TWO))
 (25753 3336 (:LINEAR EXPT-<-1-ONE))
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 (23434 3336 (:LINEAR EXPT-<=-1-ONE))
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 (13418 13418 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
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 (13395 13395 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
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 (13395 13395 (:REWRITE |(< (/ x) (/ y))|))
 (13395 13395 (:REWRITE |(< (- x) (- y))|))
 (13385 13379 (:REWRITE REMOVE-STRICT-INEQUALITIES))
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 (13379 13379 (:REWRITE CONSTANT-<-INTEGERP))
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 (5742 5742 (:REWRITE REMOVE-WEAK-INEQUALITIES))
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 (5582 5582 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
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 (5582 5582 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
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 (4596 1048 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
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 (1048 1048 (:REWRITE |(mod x (- y))| . 3))
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 (913 913 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
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 (319 5 (:REWRITE |(* (if a b c) x)|))
 (249 249 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
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 (215 215 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
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 (165 165 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (165 165 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 (156 156 (:REWRITE |(expt x 0)|))
 (103 103 (:REWRITE FOLD-CONSTS-IN-+))
 (90 3 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-AUX-LESS-THAN-M))
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 (1 1 (:REWRITE |(< x (/ y)) with (< y 0)|))
 )
(PRIMES::LEAST-REPEATED-SQUARE-IS-LEAST
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 (29532 142 (:REWRITE RTL::MOD-DOES-NOTHING))
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 (14874 143 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (13464 142 (:REWRITE CANCEL-MOD-+))
 (12820 382 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
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 (9212 143 (:REWRITE MOD-ZERO . 4))
 (7266 1151 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (7163 143 (:REWRITE MOD-ZERO . 3))
 (7096 7096 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (7096 7096 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (7096 7096 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (7096 7096 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (7096 7096 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (4898 4898 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
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 (4619 1794 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (4571 4571 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (4571 4571 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (4522 142 (:REWRITE |(integerp (- x))|))
 (4339 742 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
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 (4084 371 (:LINEAR EXPT-X->-X))
 (4084 371 (:LINEAR EXPT->-1-ONE))
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 (3300 371 (:LINEAR EXPT->-1-TWO))
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 (2997 371 (:LINEAR EXPT-<=-1-ONE))
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 (1794 1794 (:REWRITE REMOVE-STRICT-INEQUALITIES))
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 (1352 116 (:REWRITE |(equal (+ (- c) x) y)|))
 (1228 742 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
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 (933 142 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
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 (755 755 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (742 742 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (705 105 (:REWRITE NORMALIZE-ADDENDS))
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 (669 142 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (669 142 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (622 83 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (563 563 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (563 563 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
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 (563 563 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (517 517 (:REWRITE |(< 0 (* x y))|))
 (514 514 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (514 514 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (514 514 (:REWRITE |(< 0 (/ x))|))
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 (436 436 (:REWRITE |(expt 1/c n)|))
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 (435 435 (:REWRITE DEFAULT-DIVIDE))
 (427 270 (:REWRITE DEFAULT-PLUS-1))
 (408 7 (:LINEAR MOD-BOUNDS-3))
 (403 83 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (382 382 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (371 371 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (371 371 (:LINEAR EXPT-LINEAR-UPPER-<))
 (371 371 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (371 371 (:LINEAR EXPT-LINEAR-LOWER-<))
 (336 16 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (303 303 (:REWRITE REDUCE-INTEGERP-+))
 (303 303 (:REWRITE INTEGERP-MINUS-X))
 (303 303 (:META META-INTEGERP-CORRECT))
 (192 192 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (179 51 (:REWRITE |(+ 0 x)|))
 (166 14 (:LINEAR MOD-BOUNDS-2))
 (156 51 (:REWRITE |(+ c (+ d x))|))
 (144 144 (:REWRITE DEFAULT-MOD-2))
 (142 142 (:REWRITE |(mod x (- y))| . 3))
 (142 142 (:REWRITE |(mod x (- y))| . 2))
 (142 142 (:REWRITE |(mod x (- y))| . 1))
 (142 142 (:REWRITE |(mod (- x) y)| . 3))
 (142 142 (:REWRITE |(mod (- x) y)| . 2))
 (142 142 (:REWRITE |(mod (- x) y)| . 1))
 (142 142 (:REWRITE |(- (* c x))|))
 (135 6 (:REWRITE PRIMES::POSP!=0))
 (126 3 (:DEFINITION POSP))
 (79 79 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (78 6 (:REWRITE |(+ y (+ x z))|))
 (77 77 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (51 1 (:REWRITE |(* (if a b c) x)|))
 (42 42 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (31 31 (:TYPE-PRESCRIPTION NATP))
 (31 31 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (28 28 (:REWRITE |(+ x (- x))|))
 (27 27 (:REWRITE |(expt x 0)|))
 (24 12 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (20 20 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (18 18 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (18 18 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 (15 15 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (15 15 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (15 15 (:REWRITE |(< (/ x) 0)|))
 (15 15 (:REWRITE |(< (* x y) 0)|))
 (7 7 (:LINEAR RTL::MOD-BND-3))
 (3 3 (:TYPE-PRESCRIPTION POSP))
 (3 3 (:REWRITE |(< x (+ c/d y))|))
 (3 3 (:REWRITE |(< (+ c/d x) y)|))
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 (2 2 (:REWRITE |(* 1 x)|))
 )
(PRIMES::T-S-AUX-EULER-CRITERION
 (365253 77 (:LINEAR EXPT->=-1-ONE))
 (347008 8 (:REWRITE HELPER-0))
 (279460 1176 (:REWRITE RTL::MOD-DOES-NOTHING))
 (173715 2158 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (163744 48 (:REWRITE |(mod (+ x y) z) where (<= 0 z)| . 3))
 (160656 48 (:REWRITE |(mod (+ x y) z) where (<= 0 z)| . 2))
 (103824 216 (:LINEAR MOD-OF-2-UPPER-BOUND))
 (66232 1176 (:REWRITE CANCEL-MOD-+))
 (36384 1187 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (35569 35569 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (35569 35569 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (33800 4800 (:REWRITE INTEGERP-MINUS-X))
 (26954 1186 (:REWRITE MOD-ZERO . 3))
 (25496 1160 (:REWRITE |(mod x 2)| . 2))
 (20457 2337 (:REWRITE |(* (- x) y)|))
 (17592 1662 (:REWRITE DEFAULT-PLUS-2))
 (16280 8 (:REWRITE HELPER-1-AUX2))
 (16209 3431 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (16192 14660 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (16117 3431 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (16057 3431 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (14676 1195 (:REWRITE DEFAULT-MOD-RATIO))
 (14660 14660 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (14660 14660 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (14660 14660 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (13247 12341 (:REWRITE DEFAULT-TIMES-1))
 (13204 102 (:REWRITE |(+ y (+ x z))|))
 (12967 12341 (:REWRITE DEFAULT-TIMES-2))
 (11862 2342 (:REWRITE DEFAULT-MINUS))
 (10957 21 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (10456 2336 (:REWRITE |(- (* c x))|))
 (8720 2240 (:REWRITE DEFAULT-LESS-THAN-1))
 (8426 1186 (:REWRITE MOD-X-Y-=-X . 4))
 (7698 444 (:REWRITE |(+ c (+ d x))|))
 (7412 7412 (:REWRITE NORMALIZE-FACTORS-GATHER-EXPONENTS))
 (7184 3592 (:REWRITE INTEGERP-OF-*-OF-1/2-WHEN-EVENP-CHEAP))
 (6666 1186 (:REWRITE MOD-ZERO . 4))
 (6232 1176 (:REWRITE MOD-X-Y-=-X . 2))
 (6232 1176 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (6232 1176 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (6108 1187 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (6108 1187 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (5835 32 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (5832 1176 (:REWRITE MOD-CANCEL-*-CONST))
 (5816 1176 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (5816 1176 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (5018 210 (:LINEAR MOD-BOUNDS-3))
 (4648 220 (:REWRITE |(< x (+ c/d y))|))
 (3839 1759 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (3839 1759 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (3640 3640 (:REWRITE REDUCE-INTEGERP-+))
 (3640 3640 (:META META-INTEGERP-CORRECT))
 (3592 3592 (:TYPE-PRESCRIPTION EVENP))
 (3490 442 (:REWRITE |(+ 0 x)|))
 (3431 3431 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (3431 3431 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (3296 77 (:LINEAR EXPT-X->=-X))
 (3257 3257 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (3257 3257 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (3147 77 (:LINEAR EXPT-<=-1-TWO))
 (3096 3096 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-OF-2))
 (2777 19 (:REWRITE |(equal (+ (- c) x) y)|))
 (2304 2240 (:REWRITE DEFAULT-LESS-THAN-2))
 (2287 1759 (:REWRITE SIMPLIFY-SUMS-<))
 (2240 2240 (:REWRITE THE-FLOOR-BELOW))
 (2240 2240 (:REWRITE THE-FLOOR-ABOVE))
 (2158 2158 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (2158 2158 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (2155 21 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (2152 420 (:LINEAR MOD-BOUNDS-2))
 (1780 1780 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (1780 1780 (:REWRITE INTEGERP-<-CONSTANT))
 (1780 1780 (:REWRITE CONSTANT-<-INTEGERP))
 (1780 1780 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (1780 1780 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (1780 1780 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (1780 1780 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (1780 1780 (:REWRITE |(< c (- x))|))
 (1780 1780 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (1780 1780 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (1780 1780 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (1780 1780 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (1780 1780 (:REWRITE |(< (/ x) (/ y))|))
 (1780 1780 (:REWRITE |(< (- x) c)|))
 (1780 1780 (:REWRITE |(< (- x) (- y))|))
 (1745 1745 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (1745 1745 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (1745 1745 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (1745 1745 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (1745 1745 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (1737 1662 (:REWRITE DEFAULT-PLUS-1))
 (1475 1195 (:REWRITE DEFAULT-MOD-1))
 (1416 1176 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (1416 1176 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (1416 1176 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (1351 77 (:LINEAR EXPT->=-1-TWO))
 (1351 77 (:LINEAR EXPT-<=-1-ONE))
 (1302 77 (:LINEAR EXPT-X->-X))
 (1302 77 (:LINEAR EXPT->-1-ONE))
 (1302 77 (:LINEAR EXPT-<-1-TWO))
 (1195 1195 (:REWRITE DEFAULT-MOD-2))
 (1176 1176 (:REWRITE |(mod x (- y))| . 3))
 (1176 1176 (:REWRITE |(mod x (- y))| . 2))
 (1176 1176 (:REWRITE |(mod x (- y))| . 1))
 (1176 1176 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (1176 1176 (:REWRITE |(mod (- x) y)| . 3))
 (1176 1176 (:REWRITE |(mod (- x) y)| . 2))
 (1176 1176 (:REWRITE |(mod (- x) y)| . 1))
 (1160 1160 (:REWRITE |(mod x 2)| . 1))
 (977 659 (:REWRITE NORMALIZE-ADDENDS))
 (784 16 (:REWRITE |(integerp (- x))|))
 (647 647 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (505 32 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (427 91 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (375 26 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (335 335 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (320 8 (:REWRITE |(< 1 (* x (/ y))) with (< 0 y)|))
 (256 8 (:REWRITE |(* (if a b c) x)|))
 (240 240 (:REWRITE |(< (+ c/d x) y)|))
 (240 48 (:REWRITE |(mod (+ x y) z) where (<= z 0)| . 3))
 (240 48 (:REWRITE |(mod (+ x y) z) where (<= z 0)| . 2))
 (210 210 (:LINEAR RTL::MOD-BND-3))
 (206 145 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (184 8 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (154 154 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (154 154 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (154 154 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (154 154 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (145 145 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (145 145 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (141 141 (:REWRITE |(< 0 (* x y))|))
 (138 13 (:REWRITE |(+ y x)|))
 (136 8 (:REWRITE MOD-ZERO . 1))
 (93 91 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (82 82 (:REWRITE |(* a (/ a) b)|))
 (77 77 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (77 77 (:LINEAR EXPT-LINEAR-UPPER-<))
 (77 77 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (77 77 (:LINEAR EXPT-LINEAR-LOWER-<))
 (77 77 (:LINEAR EXPT->-1-TWO))
 (77 77 (:LINEAR EXPT-<-1-ONE))
 (69 69 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (69 69 (:REWRITE DEFAULT-DIVIDE))
 (62 62 (:TYPE-PRESCRIPTION NATP))
 (58 13 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (51 9 (:REWRITE ACL2-NUMBERP-X))
 (49 49 (:REWRITE DEFAULT-EXPT-1))
 (49 49 (:REWRITE |(expt 1/c n)|))
 (49 49 (:REWRITE |(expt (- x) n)|))
 (48 48 (:REWRITE |(mod (+ x y) z) where (<= z 0)| . 1))
 (48 48 (:REWRITE |(mod (+ x y) z) where (<= 0 z)| . 1))
 (40 8 (:REWRITE MOD-ZERO . 2))
 (33 33 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (33 33 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (32 32 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (32 32 (:REWRITE |(equal c (/ x))|))
 (32 32 (:REWRITE |(equal c (- x))|))
 (32 32 (:REWRITE |(equal (/ x) c)|))
 (32 32 (:REWRITE |(equal (/ x) (/ y))|))
 (32 32 (:REWRITE |(equal (- x) c)|))
 (32 32 (:REWRITE |(equal (- x) (- y))|))
 (21 3 (:REWRITE RATIONALP-X))
 (20 20 (:DEFINITION FIX))
 (18 18 (:REWRITE |(* 1 x)|))
 (14 4 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (12 12 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (11 11 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (11 11 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (11 11 (:REWRITE |(< 0 (/ x))|))
 (10 10 (:REWRITE |(+ x (- x))|))
 (8 8 (:REWRITE |(< 1 (* x (/ y))) with (< y 0)|))
 (8 8 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (8 8 (:REWRITE |(< 0 (* x (/ y))) rationalp (* x (/ y))|))
 (8 2 (:REWRITE |(+ x x)|))
 (3 3 (:REWRITE REDUCE-RATIONALP-+))
 (3 3 (:REWRITE REDUCE-RATIONALP-*))
 (3 3 (:REWRITE RATIONALP-MINUS-X))
 (3 3 (:META META-RATIONALP-CORRECT))
 (2 2 (:TYPE-PRESCRIPTION ABS))
 (2 2 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (2 2 (:REWRITE FOLD-CONSTS-IN-+))
 (2 1 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (2 1 (:REWRITE |(* -1 x)|))
 )
(PRIMES::LEMMA-1
 (252 140 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (252 140 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (140 140 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (140 140 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (140 140 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (62 2 (:LINEAR EXPT-<=-1-ONE))
 (60 2 (:LINEAR EXPT->=-1-TWO))
 (60 2 (:LINEAR EXPT->-1-TWO))
 (60 2 (:LINEAR EXPT-<-1-ONE))
 (54 2 (:LINEAR EXPT-X->=-X))
 (54 2 (:LINEAR EXPT->=-1-ONE))
 (54 2 (:LINEAR EXPT-<=-1-TWO))
 (52 52 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (52 2 (:LINEAR EXPT-X->-X))
 (52 2 (:LINEAR EXPT->-1-ONE))
 (52 2 (:LINEAR EXPT-<-1-TWO))
 (42 6 (:REWRITE DEFAULT-TIMES-2))
 (37 37 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (37 37 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (37 37 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (34 34 (:TYPE-PRESCRIPTION NOT-INTEGERP-3E))
 (34 34 (:TYPE-PRESCRIPTION NOT-INTEGERP-2E))
 (34 34 (:TYPE-PRESCRIPTION NOT-INTEGERP-1E))
 (20 20 (:REWRITE THE-FLOOR-BELOW))
 (20 20 (:REWRITE THE-FLOOR-ABOVE))
 (20 20 (:REWRITE SIMPLIFY-SUMS-<))
 (20 20 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (20 20 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (20 20 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (20 20 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (20 20 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (20 20 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (20 20 (:REWRITE INTEGERP-<-CONSTANT))
 (20 20 (:REWRITE DEFAULT-LESS-THAN-2))
 (20 20 (:REWRITE DEFAULT-LESS-THAN-1))
 (20 20 (:REWRITE CONSTANT-<-INTEGERP))
 (20 20 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (20 20 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (20 20 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (20 20 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (20 20 (:REWRITE |(< c (- x))|))
 (20 20 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (20 20 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (20 20 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (20 20 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (20 20 (:REWRITE |(< (/ x) (/ y))|))
 (20 20 (:REWRITE |(< (- x) c)|))
 (20 20 (:REWRITE |(< (- x) (- y))|))
 (16 6 (:REWRITE DEFAULT-TIMES-1))
 (8 8 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (7 7 (:REWRITE REDUCE-INTEGERP-+))
 (7 7 (:REWRITE INTEGERP-MINUS-X))
 (7 7 (:META META-INTEGERP-CORRECT))
 (6 6 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (6 6 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (6 6 (:REWRITE |(< 0 (/ x))|))
 (6 6 (:REWRITE |(< 0 (* x y))|))
 (5 5 (:REWRITE DEFAULT-EXPT-1))
 (5 5 (:REWRITE |(expt 1/c n)|))
 (5 5 (:REWRITE |(expt (- x) n)|))
 (4 4 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (4 4 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (4 4 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (4 4 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (4 4 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE NORMALIZE-ADDENDS))
 (2 2 (:REWRITE DEFAULT-PLUS-2))
 (2 2 (:REWRITE DEFAULT-PLUS-1))
 (2 2 (:REWRITE |(expt c (* d n))|))
 (2 2 (:REWRITE |(< (/ x) 0)|))
 (2 2 (:REWRITE |(< (* x y) 0)|))
 (2 2 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (2 2 (:LINEAR EXPT-LINEAR-UPPER-<))
 (2 2 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (2 2 (:LINEAR EXPT-LINEAR-LOWER-<))
 (1 1 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 )
(PRIMES::T-S-AUX-Z-IS-NON-RESIDUE
 (37555 5 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-EQUIV))
 (34645 10 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE))
 (29842 35 (:REWRITE RTL::MOD-DOES-NOTHING))
 (17435 10 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (9313 24 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (6989 6989 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (6989 6989 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (6989 6989 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (6673 54 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (6451 222 (:LINEAR EXPT-X->=-X))
 (6270 222 (:LINEAR EXPT-<=-1-TWO))
 (6080 35 (:REWRITE CANCEL-MOD-+))
 (5840 141 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (5584 222 (:LINEAR EXPT->=-1-ONE))
 (5449 1544 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (5423 222 (:LINEAR EXPT-X->-X))
 (5172 53 (:REWRITE MOD-X-Y-=-X . 4))
 (5153 222 (:LINEAR EXPT->-1-ONE))
 (5153 222 (:LINEAR EXPT-<=-1-ONE))
 (5150 222 (:LINEAR EXPT->=-1-TWO))
 (5040 1235 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (4934 53 (:REWRITE MOD-ZERO . 4))
 (4833 222 (:LINEAR EXPT-<-1-TWO))
 (4605 1235 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (4422 222 (:LINEAR EXPT->-1-TWO))
 (4422 222 (:LINEAR EXPT-<-1-ONE))
 (4113 4113 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (4113 4113 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (3553 1255 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (3543 1255 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (3235 53 (:REWRITE MOD-ZERO . 3))
 (3215 848 (:REWRITE DEFAULT-TIMES-1))
 (3204 848 (:REWRITE DEFAULT-TIMES-2))
 (3185 1235 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (3145 47 (:REWRITE |(equal (+ (- c) x) y)|))
 (3141 117 (:REWRITE |(< x (+ c/d y))|))
 (3029 1321 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (2965 1397 (:REWRITE DEFAULT-LESS-THAN-1))
 (2946 1341 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (2638 1255 (:REWRITE SIMPLIFY-SUMS-<))
 (2494 62 (:REWRITE DEFAULT-MOD-RATIO))
 (2167 1397 (:REWRITE DEFAULT-LESS-THAN-2))
 (2124 24 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (1943 35 (:REWRITE |(integerp (- x))|))
 (1875 1265 (:REWRITE CONSTANT-<-INTEGERP))
 (1839 36 (:REWRITE |(* (- x) y)|))
 (1544 1544 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (1544 1544 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (1544 1544 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (1481 740 (:REWRITE DEFAULT-PLUS-2))
 (1397 1397 (:REWRITE THE-FLOOR-BELOW))
 (1397 1397 (:REWRITE THE-FLOOR-ABOVE))
 (1321 1321 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (1320 800 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (1265 1265 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (1265 1265 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (1265 1265 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (1265 1265 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (1265 1265 (:REWRITE |(< c (- x))|))
 (1265 1265 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (1265 1265 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (1265 1265 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (1265 1265 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (1265 1265 (:REWRITE |(< (/ x) (/ y))|))
 (1265 1265 (:REWRITE |(< (- x) c)|))
 (1265 1265 (:REWRITE |(< (- x) (- y))|))
 (1255 1255 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (1255 1255 (:REWRITE INTEGERP-<-CONSTANT))
 (1244 1235 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (1244 504 (:REWRITE NORMALIZE-FACTORS-GATHER-EXPONENTS))
 (1235 1235 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (1235 1235 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (1230 1230 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (1230 1230 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (1097 35 (:REWRITE MOD-X-Y-=-X . 2))
 (1097 35 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (1097 35 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (971 141 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (861 573 (:REWRITE NORMALIZE-ADDENDS))
 (841 21 (:REWRITE |(equal (expt x n) 0)|))
 (809 800 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (809 800 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (808 740 (:REWRITE DEFAULT-PLUS-1))
 (728 54 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (728 54 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (641 40 (:REWRITE DEFAULT-MINUS))
 (636 35 (:REWRITE MOD-CANCEL-*-CONST))
 (628 62 (:REWRITE DEFAULT-MOD-1))
 (582 582 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (562 562 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (511 511 (:TYPE-PRESCRIPTION NATP))
 (486 136 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (459 179 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (444 444 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (444 444 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (444 444 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (444 444 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (430 216 (:REWRITE DEFAULT-EXPT-1))
 (405 20 (:REWRITE BUBBLE-DOWN-*-MATCH-1))
 (395 10 (:REWRITE |(* (expt x n) (expt y n))|))
 (358 2 (:LINEAR MOD-BOUNDS-3))
 (356 356 (:REWRITE |(< 0 (* x y))|))
 (341 35 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (341 35 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (341 35 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (330 330 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (330 35 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (330 35 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (330 35 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (293 75 (:REWRITE |(+ c (+ d x))|))
 (290 290 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (290 290 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (290 290 (:REWRITE |(< 0 (/ x))|))
 (280 20 (:REWRITE |(* y (* x z))|))
 (277 24 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (258 8 (:REWRITE |(* (if a b c) x)|))
 (244 6 (:REWRITE |(+ y (+ x z))|))
 (222 222 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (222 222 (:LINEAR EXPT-LINEAR-UPPER-<))
 (222 222 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (222 222 (:LINEAR EXPT-LINEAR-LOWER-<))
 (204 204 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (191 179 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (189 189 (:REWRITE |(expt 1/c n)|))
 (189 189 (:REWRITE |(expt (- x) n)|))
 (180 180 (:TYPE-PRESCRIPTION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (165 7 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (156 156 (:REWRITE REDUCE-INTEGERP-+))
 (156 156 (:REWRITE INTEGERP-MINUS-X))
 (156 156 (:REWRITE |(equal c (/ x))|))
 (156 156 (:REWRITE |(equal c (- x))|))
 (156 156 (:REWRITE |(equal (/ x) c)|))
 (156 156 (:REWRITE |(equal (/ x) (/ y))|))
 (156 156 (:REWRITE |(equal (- x) c)|))
 (156 156 (:REWRITE |(equal (- x) (- y))|))
 (156 156 (:META META-INTEGERP-CORRECT))
 (148 73 (:REWRITE |(+ 0 x)|))
 (144 4 (:LINEAR MOD-BOUNDS-2))
 (143 38 (:DEFINITION FIX))
 (141 141 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (136 31 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (134 134 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (134 134 (:REWRITE DEFAULT-DIVIDE))
 (122 11 (:REWRITE |(+ y x)|))
 (110 15 (:REWRITE PRIMES::POSP-Q*2^S-N-IS-EVEN))
 (96 76 (:REWRITE |(* a (/ a) b)|))
 (83 83 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (62 62 (:REWRITE DEFAULT-MOD-2))
 (55 25 (:REWRITE NOT-EVENP-WHEN-ODDP))
 (50 10 (:REWRITE |(* x x)|))
 (35 35 (:REWRITE EVENP-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (35 35 (:REWRITE |(mod x (- y))| . 3))
 (35 35 (:REWRITE |(mod x (- y))| . 2))
 (35 35 (:REWRITE |(mod x (- y))| . 1))
 (35 35 (:REWRITE |(mod (- x) y)| . 3))
 (35 35 (:REWRITE |(mod (- x) y)| . 2))
 (35 35 (:REWRITE |(mod (- x) y)| . 1))
 (35 35 (:REWRITE |(- (* c x))|))
 (32 32 (:TYPE-PRESCRIPTION ABS))
 (30 30 (:TYPE-PRESCRIPTION ODDP))
 (30 10 (:REWRITE |(< (/ x) y) with (< 0 x)|))
 (26 26 (:REWRITE |(* 1 x)|))
 (25 25 (:REWRITE |(expt c (* d n))|))
 (24 24 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (24 24 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (24 24 (:REWRITE |(< (/ x) 0)|))
 (24 24 (:REWRITE |(< (* x y) 0)|))
 (20 10 (:REWRITE |(< (/ x) y) with (< x 0)|))
 (14 4 (:REWRITE BUBBLE-DOWN-+-MATCH-1))
 (10 10 (:TYPE-PRESCRIPTION PRIMES::LEAST-REPEATED-SQUARE))
 (10 10 (:REWRITE |(equal (* x y) 0)|))
 (10 10 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (9 9 (:REWRITE |(+ x (- x))|))
 (8 2 (:REWRITE |(+ x x)|))
 (5 5 (:REWRITE |(< y (+ (- c) x))|))
 (2 2 (:REWRITE FOLD-CONSTS-IN-+))
 (2 1 (:REWRITE |(* -1 x)|))
 (1 1 (:LINEAR RTL::MOD-BND-3))
 )
(PRIMES::T-S-AUX-N-IS-RESIDUE
 (80696 8 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-EQUIV))
 (73408 24 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE))
 (56136 8 (:REWRITE PRIMES::T-S-AUX-Z-IS-NON-RESIDUE))
 (52364 101 (:REWRITE RTL::MOD-DOES-NOTHING))
 (40624 24 (:DEFINITION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (17415 122 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (15412 101 (:REWRITE CANCEL-MOD-+))
 (12922 343 (:LINEAR EXPT-X->=-X))
 (10681 1288 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (10398 10398 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (10398 10398 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (10398 10398 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (10398 10398 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (8765 343 (:LINEAR EXPT-<=-1-TWO))
 (8480 343 (:LINEAR EXPT-X->-X))
 (8206 1782 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (8114 122 (:REWRITE MOD-ZERO . 3))
 (7747 1702 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (7743 343 (:LINEAR EXPT->=-1-ONE))
 (7624 343 (:LINEAR EXPT->-1-ONE))
 (7047 1618 (:REWRITE DEFAULT-TIMES-1))
 (6600 343 (:LINEAR EXPT-<-1-TWO))
 (6449 343 (:LINEAR EXPT->=-1-TWO))
 (6449 343 (:LINEAR EXPT-<=-1-ONE))
 (6434 1618 (:REWRITE DEFAULT-TIMES-2))
 (6425 269 (:REWRITE |(< x (+ c/d y))|))
 (5661 135 (:REWRITE DEFAULT-MOD-RATIO))
 (5590 1484 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (5510 1484 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (5383 1896 (:REWRITE DEFAULT-LESS-THAN-1))
 (5049 121 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (4967 101 (:REWRITE |(integerp (- x))|))
 (4835 1484 (:REWRITE SIMPLIFY-SUMS-<))
 (4664 101 (:REWRITE |(* (- x) y)|))
 (4650 4650 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (4650 4650 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (4547 59 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (3964 1524 (:REWRITE CONSTANT-<-INTEGERP))
 (3961 122 (:REWRITE MOD-X-Y-=-X . 4))
 (3105 621 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
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 (2844 122 (:REWRITE MOD-ZERO . 4))
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 (2747 1896 (:REWRITE DEFAULT-LESS-THAN-2))
 (2739 101 (:REWRITE MOD-X-Y-=-X . 2))
 (2739 101 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (2739 101 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (2625 621 (:TYPE-PRESCRIPTION MOD-POSITIVE . 2))
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 (2532 804 (:REWRITE NORMALIZE-FACTORS-GATHER-EXPONENTS))
 (1896 1896 (:REWRITE THE-FLOOR-BELOW))
 (1896 1896 (:REWRITE THE-FLOOR-ABOVE))
 (1727 122 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (1727 122 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (1702 1702 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (1650 121 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1625 104 (:REWRITE DEFAULT-MINUS))
 (1622 101 (:REWRITE MOD-CANCEL-*-CONST))
 (1524 1524 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
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 (1524 1524 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (1524 1524 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (1524 1524 (:REWRITE |(< c (- x))|))
 (1524 1524 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (1524 1524 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (1524 1524 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (1524 1524 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (1524 1524 (:REWRITE |(< (/ x) (/ y))|))
 (1524 1524 (:REWRITE |(< (- x) c)|))
 (1524 1524 (:REWRITE |(< (- x) (- y))|))
 (1484 1484 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (1484 1484 (:REWRITE INTEGERP-<-CONSTANT))
 (1433 933 (:REWRITE DEFAULT-PLUS-2))
 (1421 135 (:REWRITE DEFAULT-MOD-1))
 (1293 101 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (1293 101 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (1293 101 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (1288 1288 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
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 (1288 1288 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (1120 80 (:REWRITE |(* y (* x z))|))
 (987 205 (:REWRITE |(+ c (+ d x))|))
 (963 933 (:REWRITE DEFAULT-PLUS-1))
 (960 48 (:REWRITE BUBBLE-DOWN-*-MATCH-1))
 (936 24 (:REWRITE |(* (expt x n) (expt y n))|))
 (890 10 (:LINEAR MOD-BOUNDS-3))
 (870 54 (:REWRITE |(equal (+ (- c) x) y)|))
 (841 169 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (743 343 (:LINEAR EXPT->-1-TWO))
 (743 343 (:LINEAR EXPT-<-1-ONE))
 (694 694 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (686 686 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (686 686 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (686 686 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (686 686 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (672 8 (:REWRITE |(integerp (expt x n))|))
 (638 59 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (625 517 (:REWRITE NORMALIZE-ADDENDS))
 (621 621 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (621 621 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (621 621 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (621 621 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (621 621 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (514 514 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (456 8 (:REWRITE ODD-EXPT-THM))
 (438 219 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (432 432 (:TYPE-PRESCRIPTION PRIMES::LEAST-REPEATED-SQUARE-AUX))
 (430 101 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (430 101 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (430 101 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
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 (360 20 (:LINEAR MOD-BOUNDS-2))
 (347 347 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (347 347 (:REWRITE DEFAULT-DIVIDE))
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 (345 345 (:REWRITE |(expt (- x) n)|))
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 (343 343 (:LINEAR EXPT-LINEAR-UPPER-<))
 (343 343 (:LINEAR EXPT-LINEAR-LOWER-<=))
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 (320 205 (:REWRITE |(+ 0 x)|))
 (294 54 (:DEFINITION FIX))
 (293 293 (:REWRITE |(< 0 (* x y))|))
 (291 51 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
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 (271 271 (:REWRITE REDUCE-INTEGERP-+))
 (271 271 (:REWRITE INTEGERP-MINUS-X))
 (271 271 (:META META-INTEGERP-CORRECT))
 (219 219 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (219 219 (:TYPE-PRESCRIPTION NATP))
 (219 219 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (193 118 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (169 169 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (135 135 (:REWRITE DEFAULT-MOD-2))
 (125 59 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (121 121 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (121 121 (:REWRITE |(equal c (/ x))|))
 (121 121 (:REWRITE |(equal c (- x))|))
 (121 121 (:REWRITE |(equal (/ x) c)|))
 (121 121 (:REWRITE |(equal (/ x) (/ y))|))
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 (121 121 (:REWRITE |(equal (- x) (- y))|))
 (120 40 (:REWRITE |(< (/ x) y) with (< 0 x)|))
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 (115 115 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (115 115 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (115 115 (:REWRITE |(< 0 (/ x))|))
 (102 102 (:REWRITE |(expt c (* d n))|))
 (101 101 (:REWRITE |(mod x (- y))| . 3))
 (101 101 (:REWRITE |(mod x (- y))| . 2))
 (101 101 (:REWRITE |(mod x (- y))| . 1))
 (101 101 (:REWRITE |(mod (- x) y)| . 3))
 (101 101 (:REWRITE |(mod (- x) y)| . 2))
 (101 101 (:REWRITE |(mod (- x) y)| . 1))
 (101 101 (:REWRITE |(- (* c x))|))
 (88 88 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (84 12 (:REWRITE ACL2-NUMBERP-X))
 (80 80 (:TYPE-PRESCRIPTION ABS))
 (80 40 (:REWRITE |(< (/ x) y) with (< x 0)|))
 (69 3 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (57 6 (:REWRITE |(+ y x)|))
 (40 40 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (36 9 (:REWRITE RATIONALP-X))
 (34 34 (:REWRITE |(* 1 x)|))
 (29 29 (:REWRITE |(< (* x y) 0)|))
 (24 24 (:TYPE-PRESCRIPTION PRIMES::LEAST-REPEATED-SQUARE))
 (24 24 (:REWRITE |(< (+ c/d x) y)|))
 (22 3 (:REWRITE PRIMES::POSP-Q*2^S-N-IS-EVEN))
 (15 15 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (11 5 (:REWRITE NOT-EVENP-WHEN-ODDP))
 (10 10 (:LINEAR RTL::MOD-BND-3))
 (9 9 (:REWRITE REDUCE-RATIONALP-+))
 (9 9 (:REWRITE REDUCE-RATIONALP-*))
 (9 9 (:REWRITE RATIONALP-MINUS-X))
 (9 9 (:META META-RATIONALP-CORRECT))
 (8 8 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (8 8 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (7 7 (:REWRITE EVENP-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (6 6 (:TYPE-PRESCRIPTION ODDP))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (5 5 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (5 5 (:REWRITE |(< (/ x) 0)|))
 (3 3 (:REWRITE |(equal (* x y) 0)|))
 (3 3 (:REWRITE |(+ x (- x))|))
 (1 1 (:REWRITE |(< y (+ (- c) x))|))
 )
(PRIMES::HYPS-TRUE-T-S-AUX
 (66846 66 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (53216 162 (:REWRITE RTL::MOD-DOES-NOTHING))
 (39543 275 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (30936 6855 (:TYPE-PRESCRIPTION MOD-ZERO . 2))
 (30414 162 (:REWRITE CANCEL-MOD-+))
 (27096 6855 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (26409 26017 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (26017 26017 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (26017 26017 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (26017 26017 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (25682 219 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (24160 24160 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (24160 24160 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (20424 456 (:LINEAR EXPT->=-1-ONE))
 (19348 6855 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (17843 456 (:LINEAR EXPT-<=-1-TWO))
 (17757 456 (:LINEAR EXPT-X->=-X))
 (12779 2180 (:REWRITE DEFAULT-TIMES-2))
 (12182 215 (:REWRITE MOD-ZERO . 3))
 (11755 2180 (:REWRITE DEFAULT-TIMES-1))
 (10991 327 (:REWRITE |(< x (+ c/d y))|))
 (10190 215 (:REWRITE MOD-X-Y-=-X . 4))
 (10019 10019 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (10019 10019 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (10019 10019 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (9663 162 (:REWRITE |(integerp (- x))|))
 (9567 219 (:REWRITE DEFAULT-MOD-RATIO))
 (9560 2296 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (9530 2296 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (9221 166 (:REWRITE |(* (- x) y)|))
 (8725 215 (:REWRITE MOD-ZERO . 4))
 (8037 2663 (:REWRITE DEFAULT-LESS-THAN-1))
 (7872 2059 (:REWRITE DEFAULT-PLUS-2))
 (7669 2503 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (6855 6855 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (6855 6855 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (6855 6855 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (6599 6599 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (6599 6599 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (6520 3709 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (6412 456 (:LINEAR EXPT->=-1-TWO))
 (6280 456 (:LINEAR EXPT-<=-1-ONE))
 (6056 456 (:LINEAR EXPT-X->-X))
 (6056 456 (:LINEAR EXPT->-1-ONE))
 (6056 456 (:LINEAR EXPT-<-1-TWO))
 (5524 162 (:REWRITE MOD-X-Y-=-X . 2))
 (5524 162 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (5524 162 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (5205 2663 (:REWRITE DEFAULT-LESS-THAN-2))
 (4954 66 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (3709 3709 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (3709 3709 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
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 (3440 219 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (3295 123 (:REWRITE |(equal (+ (- c) x) y)|))
 (3219 214 (:REWRITE DEFAULT-MINUS))
 (3167 162 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (3167 162 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (3167 162 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (2956 2059 (:REWRITE DEFAULT-PLUS-1))
 (2663 2663 (:REWRITE THE-FLOOR-BELOW))
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 (2576 219 (:REWRITE DEFAULT-MOD-1))
 (2503 2503 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (2503 2503 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (2419 162 (:REWRITE MOD-CANCEL-*-CONST))
 (2418 696 (:REWRITE DEFAULT-EXPT-1))
 (2339 2339 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (2339 2339 (:REWRITE INTEGERP-<-CONSTANT))
 (2339 2339 (:REWRITE CONSTANT-<-INTEGERP))
 (2339 2339 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (2339 2339 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (2339 2339 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (2339 2339 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (2339 2339 (:REWRITE |(< c (- x))|))
 (2339 2339 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (2339 2339 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (2339 2339 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (2339 2339 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (2339 2339 (:REWRITE |(< (/ x) (/ y))|))
 (2339 2339 (:REWRITE |(< (- x) c)|))
 (2339 2339 (:REWRITE |(< (- x) (- y))|))
 (2039 275 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (1954 227 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (1918 52 (:REWRITE BUBBLE-DOWN-+-MATCH-3))
 (1732 456 (:LINEAR EXPT->-1-TWO))
 (1732 456 (:LINEAR EXPT-<-1-ONE))
 (1656 16 (:REWRITE |(equal (expt x n) 0)|))
 (1416 1416 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (1111 1111 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (1106 1106 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (1043 216 (:REWRITE |(+ 0 x)|))
 (912 912 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (912 912 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (912 912 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (912 912 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (805 66 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (724 216 (:REWRITE |(+ c (+ d x))|))
 (636 636 (:REWRITE |(expt 1/c n)|))
 (636 636 (:REWRITE |(expt (- x) n)|))
 (501 501 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (501 501 (:REWRITE DEFAULT-DIVIDE))
 (456 456 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (456 456 (:LINEAR EXPT-LINEAR-UPPER-<))
 (456 456 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (456 456 (:LINEAR EXPT-LINEAR-LOWER-<))
 (444 444 (:REWRITE |(< 0 (* x y))|))
 (355 355 (:REWRITE REDUCE-INTEGERP-+))
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 (355 355 (:META META-INTEGERP-CORRECT))
 (342 342 (:REWRITE EVENP-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (299 295 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (295 295 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (291 291 (:REWRITE |(equal c (/ x))|))
 (291 291 (:REWRITE |(equal c (- x))|))
 (291 291 (:REWRITE |(equal (/ x) c)|))
 (291 291 (:REWRITE |(equal (/ x) (/ y))|))
 (291 291 (:REWRITE |(equal (- x) c)|))
 (291 291 (:REWRITE |(equal (- x) (- y))|))
 (288 288 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (288 288 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (288 288 (:REWRITE |(< 0 (/ x))|))
 (275 275 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (254 70 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (219 219 (:REWRITE DEFAULT-MOD-2))
 (202 202 (:REWRITE |(expt c (* d n))|))
 (162 162 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (162 162 (:REWRITE |(mod x (- y))| . 3))
 (162 162 (:REWRITE |(mod x (- y))| . 2))
 (162 162 (:REWRITE |(mod x (- y))| . 1))
 (162 162 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (162 162 (:REWRITE |(mod (- x) y)| . 3))
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 (162 162 (:REWRITE |(mod (- x) y)| . 1))
 (162 162 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (162 162 (:REWRITE |(- (* c x))|))
 (117 117 (:REWRITE |(< (/ x) 0)|))
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 (94 94 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (78 78 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (74 74 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (74 74 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (60 60 (:REWRITE |(< (if a b c) x)|))
 (60 60 (:REWRITE |(* x (if a b c))|))
 (60 60 (:REWRITE |(* (if a b c) x)|))
 (44 6 (:REWRITE PRIMES::POSP-Q*2^S-N-IS-EVEN))
 (40 40 (:REWRITE |(+ x (if a b c))|))
 (20 20 (:REWRITE |(mod (if a b c) x)|))
 (20 20 (:REWRITE |(equal x (if a b c))|))
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 (20 20 (:REWRITE |(equal (* x y) 0)|))
 (20 20 (:REWRITE |(+ (if a b c) x)|))
 (16 4 (:REWRITE |(+ x x)|))
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 (4 4 (:TYPE-PRESCRIPTION ABS))
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(PRIMES::LEMMA1)
(PRIMES::LEMMA2
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 (2 2 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (2 2 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 )
(PRIMES::LEMMA3)
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(PRIMES::LEMMA2)
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 )
(PRIMES::LEMMA4)
(PRIMES::LEMMA5
 (6 6 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (6 6 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (6 6 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (6 6 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (6 6 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (2 2 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (2 2 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-RATIONALP-BASE))
 )
(PRIMES::LEMMA6
 (454 32 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (175 3 (:LINEAR EXPT->=-1-ONE))
 (115 3 (:LINEAR EXPT-<=-1-TWO))
 (112 3 (:LINEAR EXPT->-1-ONE))
 (112 3 (:LINEAR EXPT-<-1-TWO))
 (106 3 (:LINEAR EXPT-X->=-X))
 (106 3 (:LINEAR EXPT-X->-X))
 (60 2 (:REWRITE PRIMES::LEMMA2))
 (53 53 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (53 53 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (53 53 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (53 53 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (53 53 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (48 12 (:REWRITE |(+ c (+ d x))|))
 (32 32 (:REWRITE THE-FLOOR-BELOW))
 (32 32 (:REWRITE THE-FLOOR-ABOVE))
 (32 32 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (32 32 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (32 32 (:REWRITE DEFAULT-LESS-THAN-2))
 (32 32 (:REWRITE DEFAULT-LESS-THAN-1))
 (26 26 (:REWRITE DEFAULT-PLUS-2))
 (26 26 (:REWRITE DEFAULT-PLUS-1))
 (21 3 (:REWRITE DEFAULT-TIMES-2))
 (20 20 (:REWRITE SIMPLIFY-SUMS-<))
 (20 20 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (20 20 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (20 20 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (20 20 (:REWRITE INTEGERP-<-CONSTANT))
 (20 20 (:REWRITE CONSTANT-<-INTEGERP))
 (20 20 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (20 20 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (20 20 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (20 20 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (20 20 (:REWRITE |(< c (- x))|))
 (20 20 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (20 20 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (20 20 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (20 20 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (20 20 (:REWRITE |(< (/ x) (/ y))|))
 (20 20 (:REWRITE |(< (- x) c)|))
 (20 20 (:REWRITE |(< (- x) (- y))|))
 (19 19 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (19 19 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (19 19 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (12 12 (:REWRITE |(+ 0 x)|))
 (8 8 (:REWRITE |(< x (+ c/d y))|))
 (8 8 (:REWRITE |(< 0 (* x y))|))
 (6 6 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (6 6 (:REWRITE |(< (* x y) 0)|))
 (6 6 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (6 6 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (6 6 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (6 6 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (4 4 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (4 4 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (4 4 (:REWRITE |(< 0 (/ x))|))
 (4 4 (:REWRITE |(< (+ c/d x) y)|))
 (4 3 (:REWRITE DEFAULT-TIMES-1))
 (3 3 (:REWRITE DEFAULT-EXPT-2))
 (3 3 (:REWRITE DEFAULT-EXPT-1))
 (3 3 (:REWRITE |(expt 1/c n)|))
 (3 3 (:REWRITE |(expt (- x) n)|))
 (3 3 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (3 3 (:LINEAR EXPT-LINEAR-UPPER-<))
 (3 3 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (3 3 (:LINEAR EXPT-LINEAR-LOWER-<))
 (3 3 (:LINEAR EXPT->=-1-TWO))
 (3 3 (:LINEAR EXPT->-1-TWO))
 (3 3 (:LINEAR EXPT-<=-1-ONE))
 (3 3 (:LINEAR EXPT-<-1-ONE))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (2 2 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (2 2 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2 2 (:REWRITE NORMALIZE-ADDENDS))
 (2 2 (:REWRITE |(< (/ x) 0)|))
 (1 1 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (1 1 (:REWRITE REDUCE-INTEGERP-+))
 (1 1 (:REWRITE INTEGERP-MINUS-X))
 (1 1 (:META META-INTEGERP-CORRECT))
 )
(PRIMES::T-S-AUX-INVARIANT2-LEMMA1
 (16434 6912 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (7747 50 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-EQUIV))
 (6260 6260 (:TYPE-PRESCRIPTION NATP))
 (2488 692 (:REWRITE DEFAULT-<-1))
 (1538 563 (:REWRITE DEFAULT-*-2))
 (1343 10 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (1251 1112 (:REWRITE DEFAULT-+-2))
 (1114 1112 (:REWRITE DEFAULT-+-1))
 (1017 25 (:REWRITE PRIMES::MOD-OF-EXPT-OF-MOD))
 (699 692 (:REWRITE DEFAULT-<-2))
 (697 563 (:REWRITE DEFAULT-*-1))
 (626 223 (:REWRITE FOLD-CONSTS-IN-+))
 (384 12 (:REWRITE RTL::INTEGERP-MOD))
 (384 12 (:REWRITE PRIMES::INTEGERP-MOD))
 (205 27 (:REWRITE PRIMES::EXPONENTS-ADD-FOR-NONNEG-EXPONENTS))
 (139 28 (:REWRITE PRIMES::LEMMA6))
 (116 116 (:LINEAR RTL::MOD-BND-3))
 (101 101 (:REWRITE DEFAULT-UNARY-/))
 (72 6 (:REWRITE PRIMES::POSP!=0))
 (64 32 (:REWRITE DM::P-1-EVEN))
 (63 7 (:REWRITE ASSOCIATIVITY-OF-*))
 (42 7 (:REWRITE DM::NOT-RES-NO-ROOT))
 (36 18 (:REWRITE PRIMES::NATP-2^X))
 (28 7 (:REWRITE DM::MOD-DISTINCT))
 (21 7 (:REWRITE PRIMES::RESIDUE-MEANING))
 (19 13 (:REWRITE DEFAULT-UNARY-MINUS))
 (14 14 (:TYPE-PRESCRIPTION ZP))
 (14 14 (:TYPE-PRESCRIPTION DM::RESIDUE))
 (11 7 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (10 10 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (7 7 (:REWRITE ZP-OPEN))
 (6 6 (:TYPE-PRESCRIPTION POSP))
 (1 1 (:REWRITE DM::POWER-OF-1))
 )
(PRIMES::T-S-AUX-INVARIANT2-LEMMA2
 (739182 306 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (419644 5016 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (374542 1974 (:REWRITE RTL::MOD-DOES-NOTHING))
 (368164 228866 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (368164 228866 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (281685 2049 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (273186 2040 (:REWRITE MOD-X-Y-=-X . 4))
 (247752 31101 (:REWRITE DEFAULT-TIMES-2))
 (244169 2040 (:REWRITE MOD-ZERO . 3))
 (228866 228866 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (228866 228866 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (228866 228866 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (227910 2040 (:REWRITE MOD-ZERO . 4))
 (212126 306 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (165443 2056 (:REWRITE DEFAULT-MOD-RATIO))
 (164511 31101 (:REWRITE DEFAULT-TIMES-1))
 (120826 120826 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (120826 120826 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (120066 24379 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (95296 24381 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (78164 16486 (:REWRITE DEFAULT-LESS-THAN-1))
 (57974 2002 (:REWRITE MOD-X-Y-=-X . 2))
 (57974 2002 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (57974 2002 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (57475 463 (:LINEAR MOD-BOUNDS-3))
 (48176 48176 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (48176 48176 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (48176 48176 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (44835 5020 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (38540 2068 (:LINEAR EXPT-X->=-X))
 (38276 2068 (:LINEAR EXPT-<=-1-TWO))
 (37465 16486 (:REWRITE DEFAULT-LESS-THAN-2))
 (34847 2490 (:REWRITE DEFAULT-MINUS))
 (34200 2049 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (34200 2049 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (31475 1992 (:REWRITE MOD-CANCEL-*-CONST))
 (31268 4672 (:META META-INTEGERP-CORRECT))
 (30408 6498 (:REWRITE DEFAULT-PLUS-2))
 (25904 1992 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
 (25888 1992 (:REWRITE |(mod x (* y (/ z))) rewriting-goal-literal|))
 (25888 1992 (:REWRITE |(mod (* x (/ y)) z) rewriting-goal-literal|))
 (25093 24379 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (24878 4821 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (24381 24381 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (24379 24379 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (24130 24130 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (24130 24130 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (23967 2056 (:REWRITE DEFAULT-MOD-1))
 (18914 2068 (:LINEAR EXPT->=-1-TWO))
 (18504 524 (:REWRITE |(equal (expt x n) 0)|))
 (17408 2068 (:LINEAR EXPT-<=-1-ONE))
 (17346 2068 (:LINEAR EXPT-X->-X))
 (17328 2068 (:LINEAR EXPT->-1-ONE))
 (16818 2068 (:LINEAR EXPT-<-1-TWO))
 (16626 2068 (:LINEAR EXPT->-1-TWO))
 (16626 2068 (:LINEAR EXPT-<-1-ONE))
 (16486 16486 (:REWRITE THE-FLOOR-BELOW))
 (16486 16486 (:REWRITE THE-FLOOR-ABOVE))
 (16312 16310 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (16312 16310 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (16093 16069 (:REWRITE |(< (- x) (- y))|))
 (16077 16069 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (16069 16069 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (16069 16069 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (16069 16069 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (16069 16069 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (16069 16069 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (16069 16069 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (16069 16069 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (16069 16069 (:REWRITE |(< (/ x) (/ y))|))
 (16053 16019 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (16019 16019 (:REWRITE INTEGERP-<-CONSTANT))
 (16019 16019 (:REWRITE CONSTANT-<-INTEGERP))
 (15743 64 (:REWRITE |(equal (mod a n) (mod b n))|))
 (15506 926 (:LINEAR MOD-BOUNDS-2))
 (14733 6498 (:REWRITE DEFAULT-PLUS-1))
 (12135 519 (:REWRITE |(< x (+ c/d y))|))
 (11821 4136 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (11601 4136 (:LINEAR EXPT-IS-INCREASING-FOR-BASE->-1))
 (11417 15 (:REWRITE DM::MOD-DISTINCT))
 (9936 1992 (:REWRITE MOD-CANCEL-*-NOT-REWRITING-GOAL-LITERAL))
 (9936 1992 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
 (9936 1992 (:REWRITE |(mod (* x (/ y)) z) not rewriting-goal-literal|))
 (9844 6976 (:REWRITE REDUCE-ADDITIVE-CONSTANT-EQUAL))
 (8344 4680 (:REWRITE REDUCE-INTEGERP-+))
 (7881 7167 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (7837 7123 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (6992 6976 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-EQUAL))
 (6899 6899 (:TYPE-PRESCRIPTION NATP))
 (6721 6721 (:REWRITE DEFAULT-DIVIDE))
 (6709 6709 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (5745 5373 (:REWRITE |(< 0 (/ x))|))
 (5595 5595 (:REWRITE |(< 0 (* x y))|))
 (5569 5541 (:REWRITE |(equal (- x) (- y))|))
 (5541 5541 (:REWRITE |(equal c (/ x))|))
 (5541 5541 (:REWRITE |(equal c (- x))|))
 (5541 5541 (:REWRITE |(equal (/ x) c)|))
 (5541 5541 (:REWRITE |(equal (/ x) (/ y))|))
 (5541 5541 (:REWRITE |(equal (- x) c)|))
 (5471 5319 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-REMAINDER))
 (5471 5319 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-AX+BX-RATIONAL-COMMON))
 (5318 4680 (:REWRITE INTEGERP-MINUS-X))
 (5020 5020 (:REWRITE EQUAL-OF-PREDICATES-REWRITE))
 (4584 4136 (:LINEAR EXPT-IS-WEAKLY-INCREASING-FOR-BASE->-1))
 (4408 8 (:REWRITE INTEGERP-OF-+-OF---AND--))
 (4368 4136 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (3746 3746 (:REWRITE |(* c (* d x))|))
 (3410 3410 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (3156 306 (:REWRITE |(equal (mod (+ x y) z) x)|))
 (3008 3008 (:TYPE-PRESCRIPTION NOT-INTEGERP-4D))
 (3008 3008 (:TYPE-PRESCRIPTION NOT-INTEGERP-3D))
 (3008 3008 (:TYPE-PRESCRIPTION NOT-INTEGERP-2D))
 (3008 3008 (:TYPE-PRESCRIPTION NOT-INTEGERP-1D))
 (2693 2541 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-=-0))
 (2272 36 (:REWRITE PRIMES::REPEATED-SQUARE-EQUIV))
 (2216 2216 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (2068 2068 (:LINEAR EXPT-LINEAR-UPPER-<=))
 (2068 2068 (:LINEAR EXPT-LINEAR-UPPER-<))
 (2068 2068 (:LINEAR EXPT-LINEAR-LOWER-<=))
 (2068 2068 (:LINEAR EXPT-LINEAR-LOWER-<))
 (2056 2056 (:REWRITE DEFAULT-MOD-2))
 (1992 1992 (:REWRITE |(mod x (- y))| . 3))
 (1992 1992 (:REWRITE |(mod x (- y))| . 2))
 (1992 1992 (:REWRITE |(mod x (- y))| . 1))
 (1992 1992 (:REWRITE |(mod (- x) y)| . 3))
 (1992 1992 (:REWRITE |(mod (- x) y)| . 2))
 (1992 1992 (:REWRITE |(mod (- x) y)| . 1))
 (1446 1446 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (1302 16 (:REWRITE |(* (if a b c) x)|))
 (1124 1124 (:REWRITE |(equal (* x y) 0)|))
 (1124 1124 (:REWRITE |(< 0 (* x y)) rationalp (* x y)|))
 (840 472 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (758 585 (:REWRITE DEFAULT-EXPT-1))
 (576 576 (:REWRITE |(expt 1/c n)|))
 (576 576 (:REWRITE |(expt (- x) n)|))
 (423 423 (:LINEAR RTL::MOD-BND-3))
 (369 9 (:REWRITE PRIMES::POSP!=0))
 (250 250 (:REWRITE |(< (* x y) 0)|))
 (245 245 (:REWRITE EVENP-WHEN-NOT-ACL2-NUMBERP-CHEAP))
 (231 231 (:REWRITE |(< (/ x) 0)|))
 (184 184 (:REWRITE |(< (+ c/d x) y)|))
 (169 169 (:REWRITE |(< (+ (- c) x) y)|))
 (158 158 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (158 158 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (156 12 (:TYPE-PRESCRIPTION FLOOR-ZERO . 2))
 (129 129 (:REWRITE |(< y (+ (- c) x))|))
 (110 110 (:REWRITE FOLD-CONSTS-IN-+))
 (108 12 (:TYPE-PRESCRIPTION FLOOR-POSITIVE . 1))
 (108 12 (:TYPE-PRESCRIPTION FLOOR-NONPOSITIVE . 1))
 (108 12 (:TYPE-PRESCRIPTION FLOOR-NEGATIVE . 1))
 (108 4 (:REWRITE BUBBLE-DOWN-*-BUBBLE-DOWN))
 (88 88 (:REWRITE INTEGERP-+-REDUCE-CONSTANT))
 (64 8 (:REWRITE |(+ x (if a b c))|))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-ZERO . 4))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-ZERO . 3))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-ZERO . 1))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-POSITIVE . 3))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-POSITIVE . 2))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-NONPOSITIVE . 3))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-NONPOSITIVE . 2))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-NONNEGATIVE . 3))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-NEGATIVE . 3))
 (60 12 (:TYPE-PRESCRIPTION FLOOR-NEGATIVE . 2))
 (56 14 (:REWRITE |(+ x x)|))
 (45 45 (:TYPE-PRESCRIPTION POSP))
 (44 12 (:REWRITE ACL2-NUMBERP-X))
 (33 7 (:REWRITE |(+ (if a b c) x)|))
 (20 20 (:TYPE-PRESCRIPTION ABS))
 (20 20 (:REWRITE |(<= (/ x) y) with (< x 0)|))
 (20 20 (:REWRITE |(<= (/ x) y) with (< 0 x)|))
 (20 20 (:REWRITE |(< x (/ y)) with (< y 0)|))
 (16 4 (:REWRITE RATIONALP-X))
 (15 15 (:TYPE-PRESCRIPTION ZP))
 (12 12 (:REWRITE |(/ (/ x))|))
 (9 9 (:REWRITE |(< (if a b c) x)|))
 (8 8 (:REWRITE |(* (/ c) (expt d n))|))
 (4 4 (:REWRITE REDUCE-RATIONALP-+))
 (4 4 (:REWRITE REDUCE-RATIONALP-*))
 (4 4 (:REWRITE RATIONALP-MINUS-X))
 (4 4 (:REWRITE BUBBLE-DOWN-*-MATCH-3))
 (4 4 (:REWRITE |(<= x (/ y)) with (< y 0)|))
 (4 4 (:REWRITE |(< (/ x) y) with (< x 0)|))
 (4 4 (:REWRITE |(< (/ x) y) with (< 0 x)|))
 (4 4 (:META META-RATIONALP-CORRECT))
 (3 3 (:REWRITE |(equal x (if a b c))|))
 )
(PRIMES::T-S-AUX-INVARIANT2
 (1990 70 (:DEFINITION EXPT))
 (989 8 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (662 323 (:REWRITE DEFAULT-<-1))
 (550 245 (:REWRITE DEFAULT-*-2))
 (523 111 (:REWRITE COMMUTATIVITY-OF-+))
 (393 393 (:REWRITE DEFAULT-+-2))
 (393 393 (:REWRITE DEFAULT-+-1))
 (342 323 (:REWRITE DEFAULT-<-2))
 (307 68 (:REWRITE ZIP-OPEN))
 (273 245 (:REWRITE DEFAULT-*-1))
 (228 92 (:REWRITE FOLD-CONSTS-IN-+))
 (186 6 (:DEFINITION PRIMES::REPEATED-SQUARE))
 (96 8 (:REWRITE PRIMES::POSP!=0))
 (70 7 (:REWRITE ASSOCIATIVITY-OF-*))
 (67 12 (:REWRITE PRIMES::REPEATED-SQUARE-EQUIV))
 (62 2 (:REWRITE DM::MOD-DISTINCT))
 (60 30 (:REWRITE DM::P-1-EVEN))
 (57 52 (:DEFINITION FIX))
 (49 49 (:LINEAR RTL::MOD-BND-3))
 (40 4 (:REWRITE PRIMES::EXPONENTS-ADD-FOR-NONNEG-EXPONENTS))
 (37 37 (:REWRITE DEFAULT-UNARY-/))
 (32 7 (:REWRITE DM::NOT-RES-NO-ROOT))
 (19 19 (:TYPE-PRESCRIPTION POSP))
 (16 10 (:TYPE-PRESCRIPTION INTEGERP-OF-MOD-EXPT-FAST))
 (16 4 (:REWRITE PRIMES::LEMMA6))
 (15 5 (:REWRITE PRIMES::RESIDUE-MEANING))
 (12 6 (:REWRITE PRIMES::LEMMA4))
 (10 10 (:TYPE-PRESCRIPTION DM::RESIDUE))
 (10 10 (:REWRITE DEFAULT-UNARY-MINUS))
 (8 4 (:REWRITE UNICITY-OF-0))
 (6 6 (:REWRITE PRIMES::LEMMA2))
 (6 6 (:REWRITE PRIMES::LEAST-REPEATED-SQUARE-LESS-THAN-M))
 (6 6 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (2 2 (:TYPE-PRESCRIPTION ZP))
 )
(PRIMES::T-S-AUX-INVARIANT3-LEMMA1
 (344990 164 (:REWRITE MOD-X-Y-=-X-Y . 1))
 (196594 2370 (:REWRITE PREFER-POSITIVE-ADDENDS-EQUAL))
 (169707 776 (:REWRITE RTL::MOD-DOES-NOTHING))
 (168057 125977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (168057 125977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (131868 852 (:REWRITE MOD-X-Y-=-X-Y . 2))
 (128409 843 (:REWRITE MOD-X-Y-=-X . 4))
 (125977 125977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (125977 125977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (125977 125977 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (125242 13144 (:REWRITE DEFAULT-TIMES-2))
 (123318 843 (:REWRITE MOD-ZERO . 3))
 (121709 19487 (:TYPE-PRESCRIPTION NOT-INTEGERP-4A))
 (120307 164 (:REWRITE MOD-X-Y-=-X+Y . 1))
 (107331 843 (:REWRITE MOD-ZERO . 4))
 (94074 13144 (:REWRITE DEFAULT-TIMES-1))
 (86914 858 (:REWRITE DEFAULT-MOD-RATIO))
 (65055 15228 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (58484 58484 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (58484 58484 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (50511 7043 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (50461 15230 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (38480 7187 (:REWRITE DEFAULT-LESS-THAN-1))
 (27092 812 (:REWRITE MOD-X-Y-=-X . 2))
 (27092 812 (:REWRITE |(mod (+ x (mod a b)) y)|))
 (27092 812 (:REWRITE |(mod (+ x (- (mod a b))) y)|))
 (26446 174 (:LINEAR MOD-BOUNDS-3))
 (23077 2370 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (21870 729 (:REWRITE PRIMES::LEMMA2))
 (20367 2760 (:REWRITE DEFAULT-PLUS-2))
 (19487 19487 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (19487 19487 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (19487 19487 (:TYPE-PRESCRIPTION NOT-INTEGERP-1A))
 (17798 1045 (:REWRITE DEFAULT-MINUS))
 (17644 7187 (:REWRITE DEFAULT-LESS-THAN-2))
 (17515 963 (:LINEAR EXPT-<=-1-TWO))
 (17419 963 (:LINEAR EXPT-X->=-X))
 (15964 2284 (:REWRITE SIMPLIFY-SUMS-EQUAL))
 (15749 852 (:REWRITE MOD-X-Y-=-X-Y . 3))
 (15749 852 (:REWRITE MOD-X-Y-=-X+Y . 3))
 (15522 15228 (:TYPE-PRESCRIPTION INTEGERP-MOD-2))
 (15378 1910 (:META META-INTEGERP-CORRECT))
 (15230 15230 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (15228 15228 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (15002 15002 (:TYPE-PRESCRIPTION MOD-ZERO . 3))
 (15002 15002 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (14635 808 (:REWRITE MOD-CANCEL-*-CONST))
 (13284 33 (:REWRITE |(equal (mod a n) (mod b n))|))
 (12124 858 (:REWRITE DEFAULT-MOD-1))
 (12078 808 (:REWRITE MOD-CANCEL-*-REWRITING-GOAL-LITERAL))
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 (4210 808 (:REWRITE |(mod x (* y (/ z))) not rewriting-goal-literal|))
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 (322 51 (:REWRITE PRIMES::REPEATED-SQUARE-EQUIV))
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 (209 25 (:REWRITE PRIMES::RESIDUE-MEANING))
 (160 160 (:REWRITE DEFAULT-UNARY-/))
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 (91 7 (:REWRITE PRIMES::EXPONENTS-ADD-FOR-NONNEG-EXPONENTS))
 (84 7 (:REWRITE PRIMES::POSP!=0))
 (65 4 (:REWRITE RTL::INTEGERP-MOD))
 (61 61 (:TYPE-PRESCRIPTION POSP))
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 (10 10 (:TYPE-PRESCRIPTION ZP))
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(PRIMES::T-S-AUX-LOOP-INVARIANTS
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 (266 100 (:REWRITE FOLD-CONSTS-IN-+))
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 )
(PRIMES::T-S-AUX-BASE-CASE
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(PRIMES::T-S-AUX-EQUIV
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 (3 3 (:TYPE-PRESCRIPTION POSP))
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(PRIMES::TONELLI-SHANKS-SQRT-AUX-IS-SQRT-MODP
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 (90 9 (:REWRITE PRIMES::INTEGERP-MOD))
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 (52 52 (:REWRITE PRIMES::T-S-AUX-BASE-CASE))
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 (6 6 (:TYPE-PRESCRIPTION DM::RESIDUE))
 (4 2 (:REWRITE UNICITY-OF-1))
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 )
(PRIMES::TONELLI-SHANKS-SQRT-AUX-IS-CORRECT
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 (440 440 (:TYPE-PRESCRIPTION NOT-INTEGERP-1C))
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 (144 144 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (110 110 (:REWRITE PRIMES::T-S-AUX-BASE-CASE))
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 (8 4 (:TYPE-PRESCRIPTION PRIMES::NATP-TONELLI-SHANKS-SQRT-AUX))
 )
(PRIMES::LEMMA1)
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 (8 6 (:REWRITE DEFAULT-TIMES-1))
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 (3 3 (:TYPE-PRESCRIPTION NATP))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-POSITIVE-BASE))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-ODD-EXPONENT))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NONPOSITIVE-BASE-EVEN-EXPONENT))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NON-0-BASE))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-ODD-EXPONENT))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-NEGATIVE-BASE-EVEN-EXPONENT))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-B))
 (3 3 (:TYPE-PRESCRIPTION EXPT-TYPE-PRESCRIPTION-INTEGERP-BASE-A))
 (3 3 (:REWRITE REDUCE-INTEGERP-+))
 (3 3 (:REWRITE INTEGERP-MINUS-X))
 (3 3 (:REWRITE PRIMES::HYPS-TRUE-T-S-AUX))
 (3 3 (:META META-INTEGERP-CORRECT))
 (2 2 (:TYPE-PRESCRIPTION BUBBLE-DOWN))
 (2 2 (:REWRITE DEFAULT-EXPT-2))
 (2 2 (:REWRITE DEFAULT-EXPT-1))
 (2 2 (:REWRITE |(expt 1/c n)|))
 (2 2 (:REWRITE |(expt (- x) n)|))
 (1 1 (:REWRITE THE-FLOOR-BELOW))
 (1 1 (:REWRITE THE-FLOOR-ABOVE))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-REMAINDER))
 (1 1 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-AX+BX-<-0-RATIONAL-COMMON))
 (1 1 (:REWRITE SIMPLIFY-SUMS-<))
 (1 1 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-<))
 (1 1 (:REWRITE REMOVE-WEAK-INEQUALITIES))
 (1 1 (:REWRITE REMOVE-STRICT-INEQUALITIES))
 (1 1 (:REWRITE REDUCE-RATIONAL-MULTIPLICATIVE-CONSTANT-<))
 (1 1 (:REWRITE REDUCE-MULTIPLICATIVE-CONSTANT-<))
 (1 1 (:REWRITE REDUCE-ADDITIVE-CONSTANT-<))
 (1 1 (:REWRITE PREFER-POSITIVE-ADDENDS-<))
 (1 1 (:REWRITE INTEGERP-<-CONSTANT))
 (1 1 (:REWRITE DEFAULT-LESS-THAN-2))
 (1 1 (:REWRITE DEFAULT-LESS-THAN-1))
 (1 1 (:REWRITE CONSTANT-<-INTEGERP))
 (1 1 (:REWRITE |(< c (/ x)) positive c --- present in goal|))
 (1 1 (:REWRITE |(< c (/ x)) positive c --- obj t or nil|))
 (1 1 (:REWRITE |(< c (/ x)) negative c --- present in goal|))
 (1 1 (:REWRITE |(< c (/ x)) negative c --- obj t or nil|))
 (1 1 (:REWRITE |(< c (- x))|))
 (1 1 (:REWRITE |(< (/ x) c) positive c --- present in goal|))
 (1 1 (:REWRITE |(< (/ x) c) positive c --- obj t or nil|))
 (1 1 (:REWRITE |(< (/ x) c) negative c --- present in goal|))
 (1 1 (:REWRITE |(< (/ x) c) negative c --- obj t or nil|))
 (1 1 (:REWRITE |(< (/ x) 0)|))
 (1 1 (:REWRITE |(< (/ x) (/ y))|))
 (1 1 (:REWRITE |(< (- x) c)|))
 (1 1 (:REWRITE |(< (- x) (- y))|))
 (1 1 (:REWRITE |(< (* x y) 0)|))
 )
(PRIMES::TONELLI-SHANKS-IS-SQRT-MODP
 (952 607 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (846 7 (:REWRITE PRIMES::NO-SQUARE-ROOT-FORALL))
 (687 687 (:TYPE-PRESCRIPTION NATP))
 (464 13 (:DEFINITION EXPT))
 (208 5 (:LINEAR RTL::MOD-BND-1))
 (172 78 (:REWRITE DEFAULT-<-1))
 (172 2 (:REWRITE PRIMES::MOD-OF-EXPT-OF-MOD))
 (157 5 (:LINEAR RTL::MOD-BND-2))
 (151 52 (:REWRITE PRIMES::LEMMA1))
 (145 76 (:REWRITE DEFAULT-*-2))
 (126 21 (:REWRITE COMMUTATIVITY-OF-+))
 (104 13 (:REWRITE ZIP-OPEN))
 (99 76 (:REWRITE DEFAULT-*-1))
 (85 85 (:REWRITE DEFAULT-+-2))
 (85 85 (:REWRITE DEFAULT-+-1))
 (80 78 (:REWRITE DEFAULT-<-2))
 (63 21 (:REWRITE FOLD-CONSTS-IN-+))
 (44 2 (:REWRITE DM::NOT-RES-NO-ROOT))
 (38 20 (:REWRITE DEFAULT-UNARY-MINUS))
 (34 2 (:REWRITE PRIMES::RESIDUE-MEANING))
 (30 15 (:REWRITE DM::P-1-EVEN))
 (30 2 (:REWRITE ASSOCIATIVITY-OF-*))
 (19 13 (:DEFINITION FIX))
 (12 12 (:REWRITE PRIMES::T-S-AUX-BASE-CASE))
 (10 7 (:REWRITE PRIMES::HAS-SQUARE-ROOT?-SUFF))
 (8 8 (:REWRITE DEFAULT-UNARY-/))
 (5 5 (:LINEAR RTL::MOD-BND-3))
 (4 4 (:TYPE-PRESCRIPTION DM::RESIDUE))
 (2 2 (:REWRITE PRIMES::HAS-SQUARE-ROOT-OF-0?))
 )
(PRIMES::TONELLI-SHANKS-IS-CORRECT
 (774217 3317 (:REWRITE RTL::MOD-DOES-NOTHING))
 (170999 6097 (:LINEAR EXPT->-1-TWO))
 (170999 6097 (:LINEAR EXPT-<-1-ONE))
 (143128 40792 (:TYPE-PRESCRIPTION MOD-ZERO . 1))
 (114336 6097 (:LINEAR EXPT-<-1-TWO))
 (112514 3444 (:REWRITE MOD-X-Y-=-X . 4))
 (110926 3356 (:REWRITE MOD-ZERO . 2))
 (97979 3356 (:REWRITE MOD-ZERO . 3))
 (67176 67176 (:TYPE-PRESCRIPTION NOT-INTEGERP-3B))
 (67176 67176 (:TYPE-PRESCRIPTION NOT-INTEGERP-2B))
 (67176 67176 (:TYPE-PRESCRIPTION NOT-INTEGERP-1B))
 (57963 21002 (:REWRITE DEFAULT-<-1))
 (44651 44595 (:TYPE-PRESCRIPTION NOT-INTEGERP-3A))
 (44651 44595 (:TYPE-PRESCRIPTION NOT-INTEGERP-2A))
 (41789 41026 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 2))
 (41789 41026 (:TYPE-PRESCRIPTION MOD-NEGATIVE . 1))
 (41555 40792 (:TYPE-PRESCRIPTION MOD-NONPOSITIVE))
 (41554 40791 (:TYPE-PRESCRIPTION MOD-POSITIVE . 1))
 (41332 8979 (:REWRITE DEFAULT-*-2))
 (38938 9326 (:REWRITE |(expt 2^i n)|))
 (38815 21002 (:REWRITE DEFAULT-<-2))
 (30839 16745 (:TYPE-PRESCRIPTION RTL::NATP-MOD))
 (22883 6712 (:REWRITE MOD-COMPLETION))
 (21938 7706 (:META META-INTEGERP-CORRECT))
 (21498 21486 (:REWRITE REMOVE-WEAK-INEQUALITIES-TWO))
 (21090 21090 (:REWRITE |(< (- x) (- y))|))
 (20877 3520 (:REWRITE MOD-CANCEL-*))
 (20606 16745 (:TYPE-PRESCRIPTION RTL::NATP-MOD-2))
 (20606 16745 (:TYPE-PRESCRIPTION RTL::INTEGERP-MOD))
 (19247 12194 (:LINEAR EXPT-IS-DECREASING-FOR-POS-BASE-<-1))
 (17245 3520 (:REWRITE MOD-MINUS-2))
 (15093 4989 (:REWRITE REDUCE-INTEGERP-+))
 (14137 13374 (:TYPE-PRESCRIPTION RTL::RATIONALP-MOD))
 (13858 12194 (:LINEAR EXPT-IS-WEAKLY-DECREASING-FOR-POS-BASE-<-1))
 (12605 8937 (:REWRITE DEFAULT-EXPT-1))
 (11791 6892 (:TYPE-PRESCRIPTION RATIONALP-EXPT-TYPE-PRESCRIPTION))
 (10737 2549 (:REWRITE DEFAULT-+-2))
 (9662 295 (:REWRITE |(< d (+ c x))|))
 (9272 9272 (:REWRITE |(expt x (- n))|))
 (9272 9272 (:REWRITE |(expt 1/c n)|))
 (9164 3356 (:REWRITE MOD-X-Y-=-X . 2))
 (8979 8979 (:REWRITE DEFAULT-*-1))
 (8518 1336 (:LINEAR MOD-BOUNDS-2))
 (8223 8223 (:REWRITE NORMALIZE-TERMS-SUCH-AS-1/AX+BX))
 (8223 8223 (:REWRITE DEFAULT-UNARY-/))
 (6911 2062 (:REWRITE SIMPLIFY-PRODUCTS-GATHER-EXPONENTS-EQUAL))
 (5362 301 (:REWRITE MOD-POSITIVE . 3))
 (4747 4747 (:REWRITE |(integerp (* c x))|))
 (3768 3768 (:REWRITE SIMPLIFY-MOD-+-MOD-WEAK))
 (3768 3768 (:REWRITE SIMPLIFY-MOD-+-MINUS-MOD))
 (2783 2549 (:REWRITE DEFAULT-+-1))
 (2770 2770 (:REWRITE |(equal (- x) (- y))|))
 (2586 883 (:REWRITE DEFAULT-UNARY-MINUS))
 (2549 2549 (:REWRITE NORMALIZE-TERMS-SUCH-AS-A/A+B-+-B/A+B))
 (2505 2505 (:REWRITE |(equal (- x) 0)|))
 (2179 2179 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-0-<-A+AB))
 (2179 2179 (:REWRITE |(< 0 (- x))|))
 (1780 1780 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-=-0))
 (1767 172 (:REWRITE MOD-+-CANCEL-0-WEAK))
 (1631 316 (:REWRITE |(< (- x) 0)|))
 (656 656 (:LINEAR RTL::MOD-BND-3))
 (535 181 (:REWRITE MOD-POSITIVE . 2))
 (321 321 (:REWRITE |(+ c (+ d x))|))
 (235 235 (:REWRITE SIMPLIFY-TERMS-SUCH-AS-A+AB-<-0))
 (89 89 (:REWRITE FOLD-CONSTS-IN-+))
 (72 8 (:REWRITE |(< x (if a b c))|))
 (16 10 (:REWRITE RTL::RATIONALP-MOD))
 (16 10 (:REWRITE RATIONALP-MOD))
 (13 1 (:REWRITE RTL::INTEGERP-MOD))
 (13 1 (:REWRITE PRIMES::INTEGERP-MOD))
 (13 1 (:REWRITE INTEGERP-MOD))
 (10 4 (:REWRITE MOD-NONNEGATIVE))
 (10 4 (:REWRITE MOD-NEGATIVE . 1))
 (8 8 (:REWRITE |(equal (+ d x) (+ c y))|))
 (8 8 (:REWRITE |(equal (+ c x) (+ d y))|))
 (5 5 (:REWRITE |(equal (+ c x y) d)|))
 (4 4 (:REWRITE MOD-NEGATIVE . 3))
 (4 4 (:REWRITE MOD-NEGATIVE . 2))
 (2 2 (:REWRITE |(< (+ c x) d)|))
 )
