(EDP)
(=_E)
(C_=_E
 (1 1 (:TYPE-PRESCRIPTION C_=_E))
 )
(+_E)
(*_E)
(-_E)
(|0_E|)
(|1_E|)
(SIZE1)
(Q1_E)
(R1_E)
(CLOSURE-LAWS)
(EQUIVALENCE-LAW)
(CONGRUENCE-LAWS)
(CLOSURE-OF-C_=_E)
(EQUIVALENCE-CLASS-LAWS
 (2 2 (:TYPE-PRESCRIPTION C_=_E))
 )
(COMMUTATIVITY-LAWS
 (3 3 (:REWRITE DEFAULT-+-2))
 (3 3 (:REWRITE DEFAULT-+-1))
 (3 3 (:REWRITE DEFAULT-*-2))
 (3 3 (:REWRITE DEFAULT-*-1))
 )
(ASSOCIATIVITY-LAWS)
(LEFT-DISTRIBUTIVITY-LAW
 (6 2 (:LINEAR X*Y>1-POSITIVE))
 (5 5 (:REWRITE DEFAULT-*-2))
 (5 5 (:REWRITE DEFAULT-*-1))
 (3 3 (:REWRITE DEFAULT-+-2))
 (3 3 (:REWRITE DEFAULT-+-1))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 )
(LEFT-UNICITY-LAWS)
(RIGHT-INVERSE-LAW)
(ZERO-DIVISOR-LAW
 (1 1 (:REWRITE DEFAULT-*-2))
 (1 1 (:REWRITE DEFAULT-*-1))
 )
(NATP-SIZE1
 (2 1 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(CONGRUENCE-FOR-SIZE1
 (4 2 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(CLOSURE-OF-Q1_E-&-R1_E)
(CONGRUENCE-FOR-Q1_E-&-R1_E)
(DIVISION-PROPERTY
 (6 2 (:REWRITE /R-WHEN-ABS-NUMERATOR=1))
 (6 2 (:LINEAR X*Y>1-POSITIVE))
 (2 2 (:REWRITE NUMERATOR-WHEN-INTEGERP))
 (2 2 (:REWRITE DEFAULT-NUMERATOR))
 (2 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE DEFAULT-<-1))
 (2 2 (:REWRITE DEFAULT-+-2))
 (2 2 (:REWRITE DEFAULT-+-1))
 (1 1 (:REWRITE FOLD-CONSTS-IN-*))
 (1 1 (:REWRITE DEFAULT-UNARY-/))
 (1 1 (:REWRITE DEFAULT-*-2))
 (1 1 (:REWRITE DEFAULT-*-1))
 )
(==_E)
(++_E)
(**_E)
(CLOSURE-LAWS-FOR-++_E-&-**_E)
(==_E-EQUIVALENCE-LAW
 (224 224 (:REWRITE CLOSURE-LAWS))
 )
(==_E-IMPLIES-IFF-EDP
 (87 37 (:REWRITE EQUIVALENCE-LAW))
 (50 50 (:REWRITE CLOSURE-LAWS))
 )
(==_E-IMPLIES-==_E-++_E-1
 (2590 16 (:REWRITE ZERO-DIVISOR-LAW))
 )
(==_E-IMPLIES-==_E-++_E-2
 (2488 16 (:REWRITE ZERO-DIVISOR-LAW))
 )
(==_E-IMPLIES-==_E-**_E-1
 (5180 32 (:REWRITE ZERO-DIVISOR-LAW))
 )
(==_E-IMPLIES-==_E-**_E-2
 (4976 32 (:REWRITE ZERO-DIVISOR-LAW))
 )
(C_==_E)
(==_E-C_==_E
 (596 596 (:REWRITE CLOSURE-LAWS))
 )
(==_E-IMPLIES-EQUAL-C_==_E
 (445 188 (:REWRITE EQUIVALENCE-LAW))
 (256 256 (:REWRITE CLOSURE-LAWS))
 (1 1 (:REWRITE CLOSURE-OF-C_=_E))
 )
(COMMUTATIVITY-LAWS-FOR-++_E-&-**_E)
(ASSOCIATIVITY-LAWS-FOR-++_E-&-**_E)
(LEFT-DISTRIBUTIVITY-LAW-FOR-++_E-&-**_E)
(LEFT-UNICITY-LAWS-FOR-++_E-&-**_E)
(RIGHT-INVERSE-LAW-FOR-++_E)
(ZERO-DIVISOR-LAW-FOR-**_E
 (208 82 (:REWRITE EQUIVALENCE-LAW))
 )
(NATP-SIZE1-==_E
 (80 31 (:REWRITE EQUIVALENCE-LAW))
 (1 1 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 )
(==_E-IMPLIES-EQUAL-SIZE1
 (134 56 (:REWRITE EQUIVALENCE-LAW))
 (78 78 (:REWRITE CLOSURE-LAWS))
 (68 58 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (58 58 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(QQ_E)
(RR_E)
(CLOSURE-OF-QQ_E-&-RR_E
 (236 91 (:REWRITE EQUIVALENCE-LAW))
 )
(==_E-IMPLIES-==_E-QQ_E-1)
(==_E-IMPLIES-==_E-QQ_E-2)
(==_E-IMPLIES-==_E-RR_E-1)
(==_E-IMPLIES-==_E-RR_E-2)
(DIVISION-PROPERTY-FOR-QQ_E-&-RR_E
 (28 2 (:REWRITE DEFAULT-<-1))
 (14 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (6 2 (:REWRITE DEFAULT-<-2))
 )
(EQUAL-C_==_E-IMPLIES-==_E)
(C_E-PROPERTY
 (46 46 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (46 46 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(C_E-PROPERTIES-SIZE1
 (69 3 (:REWRITE C_E-PROPERTY))
 (44 44 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (44 44 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(NULLITY-LAWS)
(|1_E-0_E|)
(C_E-PROPERTIES-SIZE1-1_E
 (21 21 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (9 3 (:REWRITE C_E-PROPERTY))
 )
(FIND-SMALLEST-N
 (108 108 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (76 48 (:REWRITE DEFAULT-<-1))
 (65 52 (:REWRITE DEFAULT-+-1))
 (61 52 (:REWRITE DEFAULT-+-2))
 (52 48 (:REWRITE DEFAULT-<-2))
 (16 1 (:REWRITE <-+-NEGATIVE-0-2))
 (14 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (13 5 (:TYPE-PRESCRIPTION |x < y  =>  0 < y-x|))
 (10 10 (:REWRITE DEFAULT-UNARY-MINUS))
 (9 9 (:REWRITE C_E-PROPERTY))
 (6 6 (:REWRITE FOLD-CONSTS-IN-+))
 )
(INTEGERP-FIND-SMALLEST-N)
(NATP-FIND-SMALLEST-N
 (133 77 (:REWRITE DEFAULT-<-1))
 (89 77 (:REWRITE DEFAULT-<-2))
 (86 86 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (54 18 (:REWRITE COMMUTATIVITY-OF-+))
 (36 36 (:REWRITE DEFAULT-+-2))
 (36 36 (:REWRITE DEFAULT-+-1))
 (14 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 )
(FIND-SMALLEST-N-SIZE1-X
 (481 481 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (267 154 (:REWRITE DEFAULT-<-1))
 (247 154 (:REWRITE DEFAULT-<-2))
 (128 128 (:REWRITE DEFAULT-+-2))
 (128 128 (:REWRITE DEFAULT-+-1))
 (90 30 (:REWRITE FOLD-CONSTS-IN-+))
 (90 24 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (10 10 (:REWRITE EQUAL-CONSTANT-+))
 )
(C_E-FIND-SMALLEST-N
 (1014 1014 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (973 973 (:REWRITE DEFAULT-+-2))
 (973 973 (:REWRITE DEFAULT-+-1))
 (693 231 (:REWRITE FOLD-CONSTS-IN-+))
 (519 372 (:REWRITE DEFAULT-<-2))
 (391 372 (:REWRITE DEFAULT-<-1))
 (77 77 (:REWRITE EQUAL-CONSTANT-+))
 )
(NOT-SIZE1-C_E-N-=-N
 (584 584 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (247 247 (:REWRITE DEFAULT-+-2))
 (247 247 (:REWRITE DEFAULT-+-1))
 (218 154 (:REWRITE DEFAULT-<-2))
 (154 154 (:REWRITE DEFAULT-<-1))
 (153 51 (:REWRITE FOLD-CONSTS-IN-+))
 (17 17 (:REWRITE EQUAL-CONSTANT-+))
 (4 2 (:REWRITE C_E-PROPERTIES-SIZE1))
 (3 2 (:REWRITE CLOSURE-LAWS-FOR-++_E-&-**_E))
 (3 1 (:REWRITE NATP-SIZE1-==_E))
 (2 1 (:REWRITE NATP-SIZE1))
 )
(NATP-FIND-SMALLEST-N-0)
(C_E-FIND-SMALLEST-N-0
 (70 5 (:DEFINITION FIND-SMALLEST-N))
 (18 18 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (15 15 (:REWRITE C_E-PROPERTY))
 (10 5 (:REWRITE DEFAULT-<-2))
 (5 5 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (5 5 (:REWRITE DEFAULT-<-1))
 )
(NOT-SIZE1-C_E-N-=-N-1
 (34 34 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (31 31 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (14 1 (:DEFINITION FIND-SMALLEST-N))
 (11 8 (:REWRITE C_E-PROPERTY))
 (5 3 (:REWRITE DEFAULT-<-2))
 (3 3 (:REWRITE DEFAULT-<-1))
 )
(SIZE1->=-FIND-SMALLEST-N-0
 (60 60 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (57 57 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (15 2 (:REWRITE DEFAULT-<-1))
 (14 1 (:DEFINITION FIND-SMALLEST-N))
 (7 1 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (4 2 (:REWRITE DEFAULT-<-2))
 (3 3 (:REWRITE C_E-PROPERTY))
 (1 1 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(==_E-IMPLIES-<=-SIZE1-**_E-C-E-FIND-SMALLEST-N
 (110 110 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (95 95 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (66 5 (:DEFINITION FIND-SMALLEST-N))
 (13 13 (:REWRITE C_E-PROPERTY))
 (10 5 (:REWRITE DEFAULT-<-2))
 (5 5 (:REWRITE DEFAULT-<-1))
 (3 3 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(==_E-IMPLIES-EQUAL-SIZE1-**_E-C-E-FIND-SMALLEST-N
 (269 269 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (44 16 (:REWRITE DEFAULT-<-2))
 (42 42 (:REWRITE C_E-PROPERTY))
 (17 16 (:REWRITE DEFAULT-<-1))
 (12 12 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (7 1 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 )
(==_E-IMPLIES-EQUAL-FIND-SMALLEST-N-0
 (130 10 (:DEFINITION FIND-SMALLEST-N))
 (92 92 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (62 62 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (51 31 (:REWRITE C_E-PROPERTY))
 (20 10 (:REWRITE DEFAULT-<-2))
 (10 10 (:REWRITE DEFAULT-<-1))
 (6 6 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(==_E-IMPLIES-EQUAL-SIZE1-**_E-C-E-FIND-SMALLEST-N-A
 (95 95 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (80 6 (:DEFINITION FIND-SMALLEST-N))
 (77 77 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (16 16 (:REWRITE C_E-PROPERTY))
 (12 6 (:REWRITE DEFAULT-<-2))
 (6 6 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(==_E-IMPLIES-EQUAL-SIZE1-**_E-C-E-FIND-SMALLEST-N-B
 (99 99 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (20 20 (:REWRITE C_E-PROPERTY))
 (16 8 (:REWRITE DEFAULT-<-2))
 (8 8 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(C_E-FIND-SMALLEST-N-0-C_=_E
 (6 6 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (6 6 (:TYPE-PRESCRIPTION NATP-SIZE1))
 )
(SIZE
 (5 5 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (5 5 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (1 1 (:TYPE-PRESCRIPTION NATP-FIND-SMALLEST-N))
 )
(Q_E
 (2 2 (:TYPE-PRESCRIPTION NATP-FIND-SMALLEST-N))
 )
(R_E
 (1 1 (:TYPE-PRESCRIPTION NATP-FIND-SMALLEST-N))
 )
(NATP-SIZE-==_E
 (56 4 (:DEFINITION FIND-SMALLEST-N))
 (20 20 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (12 12 (:REWRITE C_E-PROPERTY))
 (9 5 (:REWRITE DEFAULT-<-2))
 (8 8 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (8 5 (:REWRITE DEFAULT-<-1))
 (4 4 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (3 3 (:TYPE-PRESCRIPTION NATP-FIND-SMALLEST-N))
 )
(CONGRUENCE-FOR-SIZE-==_E
 (28 2 (:DEFINITION FIND-SMALLEST-N))
 (10 10 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (6 6 (:REWRITE C_E-PROPERTY))
 (6 2 (:REWRITE C_E-FIND-SMALLEST-N-0-C_=_E))
 (4 4 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (4 2 (:REWRITE DEFAULT-<-2))
 (2 2 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (2 2 (:REWRITE DEFAULT-<-1))
 )
(CLOSURE-OF-Q_E-&-R_E-==_E
 (42 3 (:DEFINITION FIND-SMALLEST-N))
 (15 15 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (9 9 (:REWRITE C_E-PROPERTY))
 (6 6 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (6 3 (:REWRITE DEFAULT-<-2))
 (3 3 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (3 3 (:REWRITE DEFAULT-<-1))
 )
(CONGRUENCE-FOR-Q_E-&-R_E-==_E
 (168 12 (:DEFINITION FIND-SMALLEST-N))
 (60 60 (:TYPE-PRESCRIPTION NATP-SIZE1-==_E))
 (36 36 (:REWRITE C_E-PROPERTY))
 (24 24 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (24 12 (:REWRITE DEFAULT-<-2))
 (12 12 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (12 12 (:REWRITE DEFAULT-<-1))
 )
(DIVISION-PROPERTY-Q_E-&-R_E-==_E
 (220 220 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (186 138 (:REWRITE C_E-PROPERTY))
 (128 51 (:REWRITE DEFAULT-<-2))
 (88 51 (:REWRITE DEFAULT-<-1))
 (46 46 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 (27 5 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 )
(SIZE-**_E-PROPERTY-==_E
 (337 337 (:TYPE-PRESCRIPTION NATP-SIZE1))
 (98 29 (:REWRITE DEFAULT-<-2))
 (94 29 (:REWRITE DEFAULT-<-1))
 (78 74 (:REWRITE C_E-PROPERTY))
 (56 8 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (24 24 (:REWRITE NOT-SIZE1-C_E-N-=-N))
 )
(NATP-SIZE
 (52 19 (:REWRITE EQUIVALENCE-LAW))
 (1 1 (:REWRITE DEFAULT-<-2))
 (1 1 (:REWRITE DEFAULT-<-1))
 )
(CONGRUENCE-FOR-SIZE
 (10 10 (:REWRITE CLOSURE-LAWS))
 )
(CLOSURE-OF-Q_E-&-R_E
 (66 25 (:REWRITE EQUIVALENCE-LAW))
 )
(CONGRUENCE-FOR-Q_E-&-R_E
 (16703 16703 (:REWRITE CLOSURE-LAWS))
 )
(DIVISION-PROPERTY-Q_E-&-R_E
 (426 136 (:REWRITE EQUIVALENCE-LAW))
 (2 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (2 1 (:REWRITE DEFAULT-<-2))
 (2 1 (:REWRITE DEFAULT-<-1))
 )
(SIZE-*_E-PROPERTY
 (132 50 (:REWRITE EQUIVALENCE-LAW))
 (2 2 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (2 1 (:REWRITE DEFAULT-<-2))
 (2 1 (:REWRITE DEFAULT-<-1))
 )
