(NAT-TREEP
 (48 23 (:REWRITE DEFAULT-+-2))
 (31 23 (:REWRITE DEFAULT-+-1))
 (30 4 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (30 2 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP-XXX))
 (24 2 (:DEFINITION TRUE-LIST-LISTP))
 (24 2 (:DEFINITION INTEGER-LISTP))
 (21 21 (:REWRITE DEFAULT-CDR))
 (20 4 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (19 1 (:DEFINITION MEMBER-EQUAL))
 (18 2 (:DEFINITION LENGTH))
 (16 4 (:REWRITE COMMUTATIVITY-OF-+))
 (16 4 (:DEFINITION INTEGER-ABS))
 (15 15 (:REWRITE DEFAULT-CAR))
 (14 2 (:DEFINITION RATIONAL-LISTP))
 (10 10 (:TYPE-PRESCRIPTION TRUE-LIST-LISTP))
 (10 10 (:TYPE-PRESCRIPTION RATIONAL-LISTP))
 (10 10 (:TYPE-PRESCRIPTION INTEGER-LISTP))
 (10 2 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (10 2 (:DEFINITION LEN))
 (6 6 (:REWRITE FOLD-CONSTS-IN-+))
 (6 5 (:REWRITE DEFAULT-<-2))
 (6 5 (:REWRITE DEFAULT-<-1))
 (6 2 (:DEFINITION APPLY$-BADGEP))
 (4 4 (:REWRITE DEFAULT-UNARY-MINUS))
 (4 2 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (3 3 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-3))
 (2 2 (:TYPE-PRESCRIPTION LEN))
 (2 2 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (2 2 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (2 2 (:REWRITE INTEGERP==>NUMERATOR-=-X))
 (2 2 (:REWRITE INTEGERP==>DENOMINATOR-=-1))
 (2 2 (:REWRITE DEFAULT-REALPART))
 (2 2 (:REWRITE DEFAULT-NUMERATOR))
 (2 2 (:REWRITE DEFAULT-IMAGPART))
 (2 2 (:REWRITE DEFAULT-DENOMINATOR))
 (2 2 (:REWRITE DEFAULT-COERCE-2))
 (2 2 (:REWRITE DEFAULT-COERCE-1))
 (2 2 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT-COROLLARY))
 (2 2 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-2))
 (1 1 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-1))
 )
(APPLY$-WARRANT-NAT-TREEP)
(APPLY$-WARRANT-NAT-TREEP-DEFINITION)
(APPLY$-WARRANT-NAT-TREEP-NECC)
(APPLY$-NAT-TREEP
 (2 2 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (2 2 (:REWRITE DEFAULT-CAR))
 (2 1 (:REWRITE APPLY$-PRIMP-BADGE))
 (2 1 (:REWRITE APPLY$-PRIMITIVE))
 )
(EXAMPLES-OF-NAT-TREEP
 (1 1 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 )
(COPY-NAT-TREE
 (1259 9 (:DEFINITION ACL2-COUNT))
 (823 16 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (810 3 (:DEFINITION INTEGER-LISTP))
 (789 9 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (783 3 (:DEFINITION APPLY$-BADGEP))
 (531 3 (:DEFINITION SUBSETP-EQUAL))
 (511 43 (:DEFINITION MEMBER-EQUAL))
 (306 21 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (302 302 (:REWRITE DEFAULT-CDR))
 (133 65 (:REWRITE DEFAULT-+-2))
 (91 91 (:REWRITE DEFAULT-CAR))
 (89 65 (:REWRITE DEFAULT-+-1))
 (63 63 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (63 7 (:DEFINITION LENGTH))
 (56 14 (:REWRITE COMMUTATIVITY-OF-+))
 (56 14 (:DEFINITION INTEGER-ABS))
 (56 10 (:DEFINITION LEN))
 (42 42 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (36 36 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (34 10 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (31 5 (:DEFINITION TRUE-LISTP))
 (30 2 (:REWRITE TRUE-LIST-LISTP-IMPLIES-TRUE-LISTP-XXX))
 (28 28 (:TYPE-PRESCRIPTION LEN))
 (24 12 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (24 2 (:DEFINITION TRUE-LIST-LISTP))
 (22 21 (:REWRITE DEFAULT-<-2))
 (22 21 (:REWRITE DEFAULT-<-1))
 (21 3 (:DEFINITION RATIONAL-LISTP))
 (20 20 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (18 9 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (18 3 (:DEFINITION ALL-NILS))
 (17 17 (:REWRITE FOLD-CONSTS-IN-+))
 (15 15 (:TYPE-PRESCRIPTION RATIONAL-LISTP))
 (15 15 (:TYPE-PRESCRIPTION INTEGER-LISTP))
 (15 15 (:TYPE-PRESCRIPTION ALL-NILS))
 (14 14 (:REWRITE DEFAULT-UNARY-MINUS))
 (12 12 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (12 3 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (10 10 (:TYPE-PRESCRIPTION TRUE-LIST-LISTP))
 (10 10 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (9 9 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-3))
 (7 7 (:REWRITE INTEGERP==>NUMERATOR-=-X))
 (7 7 (:REWRITE INTEGERP==>DENOMINATOR-=-1))
 (7 7 (:REWRITE DEFAULT-REALPART))
 (7 7 (:REWRITE DEFAULT-NUMERATOR))
 (7 7 (:REWRITE DEFAULT-IMAGPART))
 (7 7 (:REWRITE DEFAULT-DENOMINATOR))
 (7 7 (:REWRITE DEFAULT-COERCE-2))
 (7 7 (:REWRITE DEFAULT-COERCE-1))
 (6 6 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-2))
 (6 3 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (6 3 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (3 3 (:LINEAR MEMBER-EQUAL-ACL2-COUNT-SMALLER-1))
 (2 2 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT-COROLLARY))
 )
(APPLY$-WARRANT-COPY-NAT-TREE)
(APPLY$-WARRANT-COPY-NAT-TREE-DEFINITION)
(APPLY$-WARRANT-COPY-NAT-TREE-NECC)
(APPLY$-COPY-NAT-TREE
 (2 2 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (2 2 (:REWRITE DEFAULT-CAR))
 (2 1 (:REWRITE APPLY$-PRIMP-BADGE))
 (2 1 (:REWRITE APPLY$-PRIMITIVE))
 )
(FAILURE-0
 (338 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (290 12 (:REWRITE BETA-REDUCTION))
 (212 6 (:DEFINITION COLLECT$))
 (174 6 (:DEFINITION ALWAYS$))
 (100 95 (:REWRITE DEFAULT-CDR))
 (88 24 (:DEFINITION PAIRLIS$))
 (88 16 (:REWRITE DEFAULT-+-2))
 (66 61 (:REWRITE DEFAULT-CAR))
 (40 8 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (30 6 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (24 12 (:REWRITE APPLY$-PRIMITIVE))
 (18 6 (:DEFINITION APPLY$-BADGEP))
 (16 16 (:REWRITE DEFAULT-+-1))
 (13 13 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (13 13 (:REWRITE DEFAULT-<-2))
 (13 13 (:REWRITE DEFAULT-<-1))
 (12 12 (:TYPE-PRESCRIPTION LEN))
 (12 12 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (12 6 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (8 8 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (8 8 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (6 6 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (6 6 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (1 1 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (1 1 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 )
(FAILURE-1
 (338 12 (:REWRITE APPLY$-CONSP-ARITY-1))
 (290 12 (:REWRITE BETA-REDUCTION))
 (212 6 (:DEFINITION COLLECT$))
 (174 6 (:DEFINITION ALWAYS$))
 (100 95 (:REWRITE DEFAULT-CDR))
 (88 24 (:DEFINITION PAIRLIS$))
 (88 16 (:REWRITE DEFAULT-+-2))
 (66 61 (:REWRITE DEFAULT-CAR))
 (40 8 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (30 6 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (24 12 (:REWRITE APPLY$-PRIMITIVE))
 (18 6 (:DEFINITION APPLY$-BADGEP))
 (16 16 (:REWRITE DEFAULT-+-1))
 (13 13 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (13 13 (:REWRITE DEFAULT-<-2))
 (13 13 (:REWRITE DEFAULT-<-1))
 (12 12 (:TYPE-PRESCRIPTION LEN))
 (12 12 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (12 6 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (8 8 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (8 8 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (8 8 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (8 8 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (6 6 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (6 6 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 )
(INDUCTION-HINT)
(FAILURE-2
 (3625 124 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (2782 2740 (:REWRITE DEFAULT-CDR))
 (2066 112 (:DEFINITION LOOP$-AS))
 (1941 1899 (:REWRITE DEFAULT-CAR))
 (1473 59 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (1436 270 (:REWRITE DEFAULT-+-2))
 (1367 53 (:DEFINITION APPLY$-BADGEP))
 (885 5 (:DEFINITION SUBSETP-EQUAL))
 (673 110 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (673 110 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (645 129 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (567 112 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (497 249 (:REWRITE APPLY$-PRIMITIVE))
 (270 270 (:REWRITE DEFAULT-+-1))
 (250 250 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (248 248 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (248 248 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (248 248 (:REWRITE DEFAULT-<-2))
 (248 248 (:REWRITE DEFAULT-<-1))
 (220 220 (:TYPE-PRESCRIPTION LOOP$-AS))
 (213 213 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (130 130 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (130 130 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (129 129 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (129 129 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (116 53 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (86 86 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (35 5 (:DEFINITION LEN))
 (30 5 (:DEFINITION ALL-NILS))
 (25 25 (:TYPE-PRESCRIPTION ALL-NILS))
 (22 11 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (20 20 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (20 10 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (6 3 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (6 3 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(FAILURE-3
 (76801 649 (:DEFINITION APPLY$-BADGEP))
 (57706 642 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (54621 297 (:DEFINITION SUBSETP-EQUAL))
 (53938 4283 (:DEFINITION MEMBER-EQUAL))
 (35462 2195 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (13482 13398 (:REWRITE DEFAULT-CAR))
 (11341 217 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (11088 1008 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (9952 9952 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (6938 1457 (:REWRITE DEFAULT-+-2))
 (6407 6407 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (5159 453 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (4328 4328 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (2934 603 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (2707 2707 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (2535 134 (:DEFINITION LOOP$-AS))
 (2079 297 (:DEFINITION LEN))
 (2019 1010 (:REWRITE APPLY$-PRIMITIVE))
 (1964 649 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (1782 297 (:DEFINITION ALL-NILS))
 (1485 1485 (:TYPE-PRESCRIPTION ALL-NILS))
 (1457 1457 (:REWRITE DEFAULT-+-1))
 (1303 1222 (:REWRITE DEFAULT-<-1))
 (1226 1226 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (1222 1222 (:REWRITE DEFAULT-<-2))
 (1188 1188 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (1018 509 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (1009 1009 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (815 134 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (815 134 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (800 217 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 (690 134 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (603 603 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (603 603 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (381 381 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (381 381 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (297 297 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (268 268 (:TYPE-PRESCRIPTION LOOP$-AS))
 )
(FINALLY-SUCCESS!
 (4704 322 (:DEFINITION MEMBER-EQUAL))
 (4668 216 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (4639 227 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3978 3973 (:REWRITE DEFAULT-CDR))
 (3113 45 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (3032 48 (:DEFINITION APPLY$-BADGEP))
 (2320 132 (:DEFINITION LOOP$-AS))
 (2219 2214 (:REWRITE DEFAULT-CAR))
 (2160 12 (:DEFINITION SUBSETP-EQUAL))
 (1294 1294 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (768 128 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (768 128 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (670 136 (:REWRITE DEFAULT-+-2))
 (652 132 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (630 315 (:REWRITE APPLY$-PRIMITIVE))
 (428 428 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (332 332 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (315 315 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (290 58 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (256 256 (:TYPE-PRESCRIPTION LOOP$-AS))
 (250 250 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (250 250 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (136 136 (:REWRITE DEFAULT-+-1))
 (124 124 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (124 124 (:REWRITE DEFAULT-<-2))
 (124 124 (:REWRITE DEFAULT-<-1))
 (120 48 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (102 102 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (84 12 (:DEFINITION LEN))
 (72 36 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (72 12 (:DEFINITION ALL-NILS))
 (60 60 (:TYPE-PRESCRIPTION ALL-NILS))
 (58 58 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (58 58 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (48 48 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (24 24 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (24 12 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (15 3 (:REWRITE APPLY$-BADGEP-PROPERTIES . 2))
 (12 12 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (8 2 (:LINEAR APPLY$-BADGEP-PROPERTIES . 2))
 )
(MR-COPY-NAT-TREE)
(MR-COPY-NAT-TREE-COPIES
 (3650 212 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3188 139 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (2924 194 (:DEFINITION MEMBER-EQUAL))
 (2657 2657 (:REWRITE DEFAULT-CDR))
 (1816 104 (:DEFINITION LOOP$-AS))
 (1541 1541 (:REWRITE DEFAULT-CAR))
 (1235 15 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (1205 15 (:DEFINITION APPLY$-BADGEP))
 (885 885 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (885 5 (:DEFINITION SUBSETP-EQUAL))
 (600 100 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (600 100 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (512 104 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (424 212 (:REWRITE APPLY$-PRIMITIVE))
 (245 245 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (225 225 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (212 212 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (205 47 (:REWRITE DEFAULT-+-2))
 (200 200 (:TYPE-PRESCRIPTION LOOP$-AS))
 (186 186 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (85 17 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (80 80 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (80 80 (:REWRITE DEFAULT-<-2))
 (80 80 (:REWRITE DEFAULT-<-1))
 (47 47 (:REWRITE DEFAULT-+-1))
 (40 15 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (35 5 (:DEFINITION LEN))
 (30 30 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (30 5 (:DEFINITION ALL-NILS))
 (25 25 (:TYPE-PRESCRIPTION ALL-NILS))
 (20 20 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (20 10 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (17 17 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (17 17 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (10 5 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (5 5 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(COPY-NAT-TREE-IS-MR-COPY-NAT-TREE
 (4990 372 (:DEFINITION MEMBER-EQUAL))
 (4768 4768 (:REWRITE DEFAULT-CDR))
 (4656 232 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (4142 242 (:REWRITE ALWAYS$-P-LST-IMPLIES-P-ELEMENT))
 (3952 32 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (3888 32 (:DEFINITION APPLY$-BADGEP))
 (2832 16 (:DEFINITION SUBSETP-EQUAL))
 (2229 2229 (:REWRITE DEFAULT-CAR))
 (2048 120 (:DEFINITION LOOP$-AS))
 (1324 1324 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (1216 237 (:REWRITE DEFAULT-+-2))
 (672 112 (:DEFINITION CDR-LOOP$-AS-TUPLE))
 (672 112 (:DEFINITION CAR-LOOP$-AS-TUPLE))
 (656 328 (:REWRITE APPLY$-PRIMITIVE))
 (584 120 (:DEFINITION EMPTY-LOOP$-AS-TUPLEP))
 (535 107 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (476 476 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (360 360 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (344 344 (:REWRITE CAR-CONS))
 (336 336 (:REWRITE CDR-CONS))
 (328 328 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (277 277 (:REWRITE APPLY$-WARRANT-NAT-TREEP-NECC))
 (277 277 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (237 237 (:REWRITE DEFAULT-+-1))
 (224 224 (:TYPE-PRESCRIPTION LOOP$-AS))
 (175 175 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (175 175 (:REWRITE DEFAULT-<-2))
 (175 175 (:REWRITE DEFAULT-<-1))
 (112 16 (:DEFINITION LEN))
 (107 107 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (107 107 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (96 96 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (96 48 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (96 32 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (96 16 (:DEFINITION ALL-NILS))
 (80 80 (:TYPE-PRESCRIPTION ALL-NILS))
 (64 64 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (32 16 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (16 16 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 (9 3 (:REWRITE FOLD-CONSTS-IN-+))
 )
(FLAGGED-MR-COPY-NAT-TREE)
(FLAGGED-EQUIVALENCE
 (494 2 (:REWRITE APPLY$-BADGEP-PROPERTIES . 1))
 (490 2 (:DEFINITION APPLY$-BADGEP))
 (354 2 (:DEFINITION SUBSETP-EQUAL))
 (328 28 (:DEFINITION MEMBER-EQUAL))
 (279 278 (:REWRITE DEFAULT-CDR))
 (235 50 (:REWRITE DEFAULT-+-2))
 (204 14 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-1))
 (113 112 (:REWRITE DEFAULT-CAR))
 (103 23 (:REWRITE RATIONALP-IMPLIES-ACL2-NUMBERP))
 (62 62 (:TYPE-PRESCRIPTION MEMBER-EQUAL))
 (50 50 (:REWRITE DEFAULT-+-1))
 (42 42 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-3))
 (30 15 (:REWRITE APPLY$-PRIMITIVE))
 (28 28 (:REWRITE MEMBER-EQUAL-NEWVAR-COMPONENTS-2))
 (23 23 (:REWRITE RATIONAL-LISTP-IMPLIES-RATIONALP))
 (23 23 (:REWRITE ACL2-NUMBER-LISTP-IMPLIES-ACL2-NUMBERP))
 (20 20 (:REWRITE INTEGER-LISTP-IMPLIES-INTEGERP))
 (20 20 (:REWRITE DEFAULT-<-2))
 (20 20 (:REWRITE DEFAULT-<-1))
 (18 18 (:REWRITE APPLY$-WARRANT-COPY-NAT-TREE-NECC))
 (16 8 (:REWRITE APPLY$-BADGEP-PROPERTIES . 3))
 (15 15 (:TYPE-PRESCRIPTION APPLY$-PRIMP))
 (14 2 (:DEFINITION TRUE-LISTP))
 (14 2 (:DEFINITION LEN))
 (12 12 (:TYPE-PRESCRIPTION APPLY$-BADGEP))
 (12 2 (:DEFINITION ALL-NILS))
 (10 10 (:TYPE-PRESCRIPTION ALL-NILS))
 (8 8 (:TYPE-PRESCRIPTION TRUE-LISTP))
 (8 8 (:TYPE-PRESCRIPTION SUBSETP-EQUAL))
 (8 2 (:DEFINITION WEAK-APPLY$-BADGE-P))
 (6 2 (:REWRITE FOLD-CONSTS-IN-+))
 (4 2 (:LINEAR APPLY$-BADGEP-PROPERTIES . 1))
 (2 2 (:REWRITE LEN-MEMBER-EQUAL-LOOP$-AS))
 )
(ELIMINATE-COPY-NAT-TREE)
